Universit´ePierre et Marie Curie Sorbonne Universit´es
Ecole´ doctorale de Physique en ˆIle-de-France
Institut d’Astrophysique de Paris (IAP) et Institut de Physique Th´eorique(IPhT) du Commissariat `al’Energie´ Atomique et aux Energies´ Alternatives (CEA) de Saclay
Precision cosmology with the large-scale structure of the universe
par H´el`eneDupuy Th`esede doctorat en cosmologie
dirig´eepar Francis Bernardeau (chercheur CEA, directeur de l’IAP)
pr´esent´eeet soutenue publiquement le 11/09/2015
devant un jury compos´ede (par ordre alphab´etique):
Dr. Francis Bernardeau IAP, CEA Examinateur Dr. Diego Blas CERN (TH-Division) Examinateur Pr. Michael Joyce LPNHE Examinateur Pr. Eiichiro Komatsu Max-Planck-Institut f¨ur Rapporteur Astrophysik Pr. Julien Lesgourgues RWTH Aachen University Rapporteur Dr. Jean-Philippe Uzan IAP, CNRS, IHP Invit´e Contents
Introduction4
1 An illustration of what precision cosmology means9 1.1 Is the cosmological principle stringent enough for precision cosmology?9 1.2 Questioning the cosmological principle analytically...... 12 1.2.1 Redshift, luminosity distance and Hubble diagram...... 12 1.2.2 Misinterpretation of Hubble diagrams...... 13 1.2.3 An idea to test analytically the impact of inhomogeneities.. 15 1.3 Generalities about propagation of light in a Swiss-cheese model... 16 1.3.1 The homogeneous background...... 16 1.3.2 Geometry inside the holes...... 19 1.3.3 Junction of the two metrics...... 20 1.3.4 Light propagation...... 23 1.4 Impact of inhomogeneities on cosmology...... 27 1.5 Article “Interpretation of the Hubble diagram in a nonhomogeneous universe”...... 29 1.6 Article “Can all observations be accurately interpreted with a unique geometry?”...... 54 1.7 Continuation of this work...... 60
2 A glimpse of neutrino cosmology 61 2.1 Basic knowledge about neutrinos...... 61 2.2 Why are cosmologists interested in neutrinos?...... 62 2.2.1 An equilibrium that does not last long...... 63 2.2.2 The effective number of neutrino species, a parameter relevant for precision cosmology...... 64
1 Contents
2.2.3 Neutrino masses: small but decisive...... 70 2.3 How cosmic neutrinos are studied...... 74 2.3.1 Observation...... 74 2.3.2 Modeling...... 75
3 Refining perturbation theory to refine the description of the growth of structure 83 3.1 A brief presentation of cosmological perturbation theory...... 84 3.2 The Vlasov-Poisson system, cornerstone of standard perturbation theory...... 85 3.3 The single-flow approximation and its consequences...... 87 3.4 Perturbation theory at NLO and NNLO...... 90 3.5 Eikonal approximation and invariance properties...... 92 3.5.1 Eikonal approximation...... 92 3.5.2 Invariance properties...... 95 3.6 Perturbation theory applied to the study of massive neutrinos.... 97
4 A new way of dealing with the neutrino component in cosmology 99 4.1 Principle of the method...... 99 4.2 Derivation of the equations of motion...... 102 4.2.1 Conservation of the number of particles...... 102 4.2.2 Conservation of the energy-momentum tensor...... 104 4.2.3 An alternative: derivation from the Boltzmann and geodesic equations...... 105 4.3 Comparison with standard results...... 107 4.3.1 Linearized equations of motion...... 107 4.3.2 Linearized multipole energy distribution...... 109 4.3.3 Initial conditions...... 110 4.3.4 Numerical tests and discussion on the appropriateness of the multi-fluid approach...... 113 4.4 Article “Describing massive neutrinos in cosmology as a collection of independent flows”...... 115
5 Towards a relativistic generalization of nonlinear perturbation the- ory 141 5.1 Generic form of the relativistic equations of motion...... 141
2 Contents
5.2 Useful properties in the subhorizon limit...... 142 5.3 Consistency relations...... 147 5.4 Article “Cosmological Perturbation Theory for streams of relativistic particles”...... 147
6 A concrete application of the multi-fluid description of neutrinos 168 6.1 Generalized definitions of displacement fields in the eikonal approxi- mation...... 168 6.2 Impact on the nonlinear growth of structure...... 170 6.2.1 Qualitative description...... 170 6.2.2 Quantitative results in the case of streams of massive neutrinos172 6.3 Article “On the importance of nonlinear couplings in large-scale neu- trino streams”...... 174
Conclusions and perspectives 188
Acknowledgments 193
Remerciements 195
R´esum´ede la th`eseen fran¸cais 218
3 Introduction
The title of my PhD thesis, “Precision cosmology with the large-scale structure of the universe”, encompasses a wide range of searches, all equally exciting. Cosmology is an ancient questioning: understanding what the universe is made of, how it formed, how it evolved since then and what its future is. Yet it is entering a new era, whence the denomination precision cosmology. The investigation method is still the same, i.e. switching back and forth between observations of the sky and formulation of theories, but lately the level of description has amazingly evolved. The twentieth century has been a golden period for cosmology. On the theo- retical side, about a hundred years ago, Albert Einstein’s general relativity ([64]) brought the mathematical tools allowing to represent appropriately time and space in a gravitation theory. Soon after appeared the first models based on this theory to describe the structure of the universe and its time evolution. The expansion of the universe has been discovered at the same epoch, jointly thanks to the ob- servations of the astrophysicist Edwin Hubble (carried out and discussed between 1925 and 1929, [94]) and to the relativistic calculations of several theorists, such as Georges Lemaˆıtre(in 1927, [103]), Alexandre Friedmann (in 1922, [79]) and Willem de Sitter (in 1917, [53]). A cosmological model naturally emerged from the study of the physics at play in an expanding universe. It is the Big Bang theory, accord- ing to which the universe was born 13.8 billion of years ago (from a process called Big Bang and involving unimaginably high energies) and then progressively cooled down and dilated. In this context, cooling down means dropping from a temperature1 T 1019 ∼ GeV to T 2.73 K. Such different temperatures lead perforce to extremely di- ≈ 1In cosmology, it is common to use the electronvolt as unit of temperature, or more precisely the electronvolt divided by the Boltzmann constant since 1 eV / kB 11604.5 K, kB being the Boltz- 23 2 2 1 ≈ mann constant (kB = 1.3806488 10− m .kg.s− .K− ). In practice, one omits the Boltzmann × constant so that T (K) 11604.5 T (eV). ≈
4 versified physical processes. The chain of cosmological eras, each characterized by specific physical processes, is called thermal history of the universe. All stages of the thermal history are not equally understood. In particular, understanding mech- anisms involving energies that are not accessible in laboratories is very challenging. Some epochs are thus described very speculatively. Conversely, the Big Bang nu- cleosynthesis and the recombination era are particularly well depicted. The former is the process during which nuclei heavier than the lightest isotope of hydrogen are created thanks to the trapping of protons and neutrons that previously freely evolved in the cosmic plasma (T 1 MeV). The latter refers to the epoch at which ∼ electrons and protons first became bound to form electrically neutral hydrogen atoms (T 3000 K). ∼ One specificity of the Big Bang model is that it predicts the existence of a radiation which started to propagate freely about 380 000 years after the Big Bang. This “cosmic microwave background”, detected incidentally in 1964 by the two physicists Arno Penzias and Robert Wilson ([128]), is nothing but the photons that emerged when the temperature of the cosmic plasma became lower than the temperature of ionization of hydrogen. Its detection was such a strong argument in favor of the Big Bang model that this discovery has been awarded the Nobel Prize in 1978. Nowadays, observing and studying carefully the cosmic microwave background is still one of the driving force of cosmology. So far, the Big Bang model has not been ruled out by observations. Rather, each new set of cosmological data provided by exploration of space strengthens the so-called “standard model of cosmology”. This model relies on the Big Bang theory and assumes the existence of a cosmological constant, which mimics the acceleration of the expansion of the universe, and of cold dark matter, a form of matter proposed in response to some unexpected observations. More precisely, two observational projects realized in 1998 by the scientific teams headed by the cosmologists Saul Perlmutter, Brian P. Schmidt and Adam Riess showed independently that the expansion of the universe is accelerating ([129, 148]). This breakthrough has been honored with the Nobel Prize in 2011. Gravity being an attractive unstoppable force, astrophysicists originally thought that expansion could only decelerate. It has therefore been necessary to imagine a form of re- pulsive energy, capable of accelerating the expansion of the universe. This cosmic component of unknown nature has been called dark energy. In the standard model, it is characterized by a parameter denominated cosmological constant and repre-
5 senting about 70% of the energy content of the universe. Note nevertheless that several alternatives to the cosmological constant (and thus alternatives to the stan- dard model) can be imagined to describe the same effect. At this time, it is still very difficult to determine which representation of dark energy is the most credible. Hopefully, the Euclid space mission ([157]), proposed in 2005 to the European Space Agency and whose launch is planned for 2020, will enrich (or even conclude) the debate. Besides, many observations suggest the presence of an unfamiliar form of matter, invisible and with a non-zero mass, called dark matter. Dark matter would explain for instance why galaxies and clusters of galaxies seem much more massive when one studies their gravitational properties than when one infers their mass from the light they emit or why the cosmic microwave background displays spatial fluctuations too faint to initiate the formation of structures by gravitational instability (see more details about gravitational instability in section 1.1). In the standard model of cosmology, this dark matter is assumed to be cold, i.e. made of particles whose velocity is already much smaller than the speed of light in vacuum when it decouples from other species. Unveiling the nature and the properties of those dark elements is a major challenge, at the core of modern cosmology. It is the main raison d’ˆetre of the Euclid mission. Another very puzzling piece of modern cosmology is cosmic inflation. It is an idea that arose in the late seventies-early eighties to complement the Big Bang model (see the foundational works of the physicists Alexei Starobinsky ([172]), Alan Guth ([84]) and Andrei Linde ([110])). According to this theory, the primordial universe underwent an extremely fast and violent phase of expansion. Adding this step to the thermal history of the universe is useful in many ways to get a coherent whole. However, the paradigm of inflation is not yet perfectly controlled theoretically. The agreement between the standard model of cosmology and observational surveys is remarkably good. So, at the present time, most cosmologists focus their effort on the refinement of this model. The technical means available today allow to probe a huge quantity of astrophysical objects, of various nature, spread over very large areas of the sky. Specific developments, involving to a large extent numerical simulations and statistical physics, are necessary to deal with such an amount of data. But an exquisite exploration of the universe is useless if there is no comparable progress in theoretical cosmology. The present thesis is a tiny illustration of the theoretical efforts currently re-
6 alized in order to anticipate a project such as the Euclid mission. As already mentioned, it has been designed primarily to examine the dark content of the uni- verse. It will moreover supply a catalog of galaxies of unprecedented richness and refinement, showing how galaxies and clusters of galaxies arranged with time to form tremendous filaments, commonly referred to as “the large-scale structure of the universe”. The purpose of this thesis is twofold. A minor part of the time has been dedi- cated to a study designed to question a leading principle of cosmology, namely the 2 cosmological principle. According to it, on very large scales (& 100 Mpc), the dis- tribution of matter in the universe is homogeneous and isotropic. Many predictions of modern cosmology ensue from it. In collaboration with Dr. Jean-Philippe Uzan and his PhD student Pierre Fleury, I have participated in a project consisting in simulating observations of supernovae in an inhomogeneous model of the universe in order to infer the impact of inhomogeneities on cosmology. We highlighted in particular the fact that using the same kind of geometrical models to interpret ob- servations corresponding to very different spatial scales could become inappropriate in the precision cosmology era (see chapter1). The second topic I am specialized in is the enhancement of cosmological perturbation theory methods (see chapter3 for a quick presentation of the current challenges of cosmological perturbation theory), especially within the nonlinear and/or relativistic regime(s). It is the main task developed in this thesis (see chapters4,5 and6). In particular, I have proposed with my PhD advisor Dr. Francis Bernardeau a new analytic method, which is a multi-fluid approach, to study efficiently the nonlinear time evolution of non-cold species such as massive neutrinos (see chapter2 for a modest overview of neutrino cosmology). This research project is at the interface between two topical subjects: neutrino cosmology and cosmological perturbation theory beyond the linear regime. The manuscript is organized as follows. In chapter1, I take the example of an analytic questioning of the cosmological principle, in which I participated, to illustrate what is at stake in precision cosmology. Chapter2 introduces basics of neutrino cosmology, with an emphasis on the key role it plays in the study of the formation of the large-scale structure of the universe. In chapter3, I present standard results of cosmological perturbation theory on which I based most of my PhD work. The three next chapters expose the developments that I realized with my advisor with the aim of incorporating properly massive neutrinos, or any non-
2pc stands for parsec, defined so that 1 pc 3.1 1016m. ≈ ×
7 cold species, in analytic models of the nonlinear growth of structure. Chapter4 presents the method and questions its accuracy. Chapter5 shows that the results presented in chapter3, valid for cold dark matter, can then partly be extended to massive neutrinos. Finally, in chapter6, I explain how our method can be helpful to determine the scales at which nonlinear effects involving neutrinos are substantial and should be accounted for in models of structure formation.
8 Chapter 1
An illustration of what precision cosmology means
1.1 Is the cosmological principle stringent enough for precision cosmology?
A specificity of cosmology is that the system of interest is studied from inside. Furthermore, it is studied from a given position (the earth and surroundings) at a given time (epoch at which human beings are present on earth). Hence, since light propagates with a finite velocity and the universe has a finite age, only a portion of the universe can be reached observationally. This area is called observable universe and its boundary is called cosmological horizon. Consequently, cosmological models intended to describe the entire universe necessarily involve untestable assumptions (see e.g. [67] for discussions on this subject). In particular, as already mentioned, the standard model of cosmology relies on the cosmological principle, according to which the spatial distribution of matter is homogeneous and isotropic on large scales. This is encoded mathematically in the metric chosen to characterize the geometry of the universe, namely the Friedmann-Lemaˆıtremetric (see section 1.3.1). Of course, the universe is not perfectly smooth. Otherwise, growth of structures from gravitational instability wouldn’t have been achievable. It is indeed commonly assumed that density contrasts existed in the primordial universe and initiated the formation of the large-scale structure. More precisely, this scenario is encapsulated in the inflationary paradigm. During this stage, in theory, quantum fluctuations of a cosmic scalar field became macroscopic, generating fluctuations of the metric
9 1.1. Is the cosmological principle stringent enough for precision cosmology?
([7, 85, 89, 173, 30, 122]). Those metric fluctuations then resulted into density fluc- tuations, as predicted by general relativity, and thus into gravitational instability. The study of the time evolution of such fluctuations is generally performed thanks to cosmological perturbation theory (see chapter3). In practice, this means adding perturbations, assumed to be small compared with the background quantities, in the Friedmann-Lemaˆıtremetric. In its minimal version (in particular, in the absence of neutrinos), the standard model of cosmology allows to summarize the properties of the universe in six pa- rameters, called cosmological parameters. Defining them at this stage would be premature since none of the cosmological effects behind them has been introduced yet. Nevertheless, for the record, here is the list:
the primordial spectrum amplitude A , • s the primordial tilt n , • s the baryon density ω , • B the total non-relativistic matter density ω , • m the cosmological constant density fraction Ω , • Λ the optical depth at reionization τ . • reion Their values are not predicted by the model, whence the importance of constraining them observationally. Several kinds of sources provide this opportunity. The lead- ing ones are Type Ia supernovae (SNIa), baryon acoustic oscillations (BAO) and the cosmic microwave background (CMB). A supernova is an astrophysical object resulting from the highly energetic explosion of a star. SNIa are particular super- novae whose spectra contain silicon but no hydrogen. They are useful in particular for the tracking of the expansion history of the universe. More precisely, they give information about the equation of state of dark energy (see e.g. [38, 146]). BAO are periodic fluctuations experimented by the baryonic1 components of the uni- verse and the knowledge of their properties brings also much to cosmology (see e.g. [5]). CMB observations provide an estimation of cosmological parameters with an exquisite precision, especially when combined with other astrophysical data ([137]).
1Baryonic means made of particles called baryons. In particular, all the matter made of atoms is baryonic.
10 1.1. Is the cosmological principle stringent enough for precision cosmology?
Since its first detection, the CMB has been inspected in great detail by several satellites looking for anisotropies in it. The motivation of such an investigation is the evidence of a dipole anisotropy2, presented in 1977 in [168] (which brought another Nobel Prize to cosmologists, this time George Fitzgerald Smoot and John C. Mather, in 2006). Higher order anisotropies3 have then been measured by the satellites COBE (launched in 1989, [167]), WMAP (launched in 2001, [10, 90]) and Planck (launched in 2009, [134]). Those space projects are complementary to ground-based experiments ([145, 175, 165]) and to balloon-borne instruments such as BOOMERanG4 (launched in late 1998, [52]) and Archeops (launched in 2002, [11]). The combination “standard model of cosmology + perturbations” seems to be a satisfactory description. As proof, it is in agreement with most existing data. Such an accuracy is a bit surprising given the simplicity of the model and the thorough- ness with which the observable universe is explored. For instance, it is assumed that the matter is continuously distributed. This involves in particular a smooth- ing scale, not explicitly given in the model ([68]). Besides, some observables used in cosmology are “point sources” and thus probe scales at which the cosmological principle does not hold (and furthermore scales that are not accessible by numerical simulations, see section 1.2.2 for more details). This is particularly true for SNIa, which emit very narrow beams probing scales smaller than5 1 AU. For this reason, it has been argued in several references that the use of the cosmological principle might lead to misinterpretations (see for instance [33]). This is precisely the kind of questioning raised by the precision cosmology era.
2The earth is in motion in the universe. Consequently, when cosmologists measure the tem- perature of the CMB, one of the celestial hemispheres appears hotter than the other one. This phenomenon is called dipole anisotropy. 3Higher order anisotropies are reflective of plasma oscillations that develop when the last scat- tering surface is reached. They superimpose over dipole anisotropies. 4The BOOMERanG experiment has brought a strong support to the inflationary paradigm by showing that the geometry of the universe is Euclidean. 5AU stands for astronomical unit, defined so that 1 AU 1.5 1011 m. ≈ ×
11 1.2. Questioning the cosmological principle analytically
1.2 Questioning the cosmological principle analytically
1.2.1 Redshift, luminosity distance and Hubble diagram
Redshift
The universe being in expansion, one can observe that galaxies are moving away from the earth. This phenomenon is called recession of galaxies and is not due to a genuine proper motion of galaxies. Galaxies (and other astrophysical objects) are in fact taken away by the dilatation of the universe itself, even if some of them have a proper motion oriented towards us. Because of this relative displacement, the observed wavelengths λobs are shifted to longer wavelengths (compared to the emitted ones λem). It is known as cosmological redshift, denoted z and defined so that (see section 1.3.4 for a more general interpretation)
λ λ z = obs − em . (1.1) λem More precisely, in his contribution to the discovery of the expansion of the universe, 6 Edwin Hubble brought to light a law stating that the recession velocity , vrec, of galaxies is proportional to their distance7 d. Actually, this relation had been previously found by Georges Lemaˆıtrebut remained almost unnoticed since stated in French. It reads
vrec = H0d, (1.2) it is called Hubble’s law and H0 is Hubble’s constant. H0 is usually not considered as part of the six cosmological parameters of the standard model of cosmology but it can be readily computed once those parameters are known8. Its value is currently estimated to be H = (67.8 0.9) km.s 1.Mpc 1([137]). To get information about 0 ± − − the expansion of the universe, it is thus useful to make measurements of velocities (or equivalently redshifts) on the one hand and of distances on the other hand.
6The recession velocity is the observed velocity to which the peculiar velocity has been sub- tracted. 7The distance at play in Hubble’s law is the distance between the source and the observer at the observation time. Because of expansion and since the speed of light is finite, this distance is larger than what it was at the emission time. 8 Note nevertheless that, H0 being in one-to-one correspondence with the standard cosmological parameter ΩΛ, H0 is sometimes considered as a cosmological parameter instead of ΩΛ. In practice, the nuance is inconsequential (provided that dark energy is assumed to be a cosmological constant).
12 1.2. Questioning the cosmological principle analytically
Luminosity distance
Distances are not directly measurable. What observers can measure when they probe the sky is e.g. the angular diameter between two points or the amount of en- ergy collected by their telescopes. Cosmologists have introduced several definitions of distances from these observables. For instance, the “luminosity distance” of a source, DL, is easy to define with the help of the intrinsic luminosity of the source
Lsource (energy emitted per unit of time) and of the observed flux Fobs (energy received per unit of time and surface). It is given by
2 Lsource = 4πDLFobs. (1.3)
SNIa are extremely bright. Furthermore, after their apparition in the sky, the time evolution of their brightness is very well known, which allows one to calibrate their intrinsic luminosity. Those particularities make them precious suppliers of luminosity distances. Such sources are often called standard candles.
Hubble diagram
Building diagrams luminosity distance versus redshift is an efficient way of probing the history of the expansion of the universe since redshifts and distances are related via cosmological parameters. Such diagrams, naturally called Hubble diagrams, are largely used in modern cosmology (see e.g. [2]). An example is given in figure 1.1.
1.2.2 Misinterpretation of Hubble diagrams
The main message of [33] is that, to meet the requirements of precision cosmology, it is crucial to model accurately the propagation of the ultra-narrow beams produced by SNIa. Otherwise, using those observables as widely as it is done in cosmology would be inappropriate. The fact that there seems to be a paradox between the real universe and its smooth representation was already discussed in the sixties ([199, 50, 19, 83, 98, 144]). More recently, consequences regarding the interpretation of Hubble diagrams have been studied. In particular, according to general relativity, the presence of massive objects (such as clusters of galaxies) affects the geometry of the universe. Since the way light propagates depends on the geometry, this results into a modification of the observed luminosity of sources. This phenomenon is called gravitational
13 1.2. Questioning the cosmological principle analytically
Figure 1.1: One example of a Hubble diagram. It has been obtained from the observation of 42 high-redshift SNIa of the Supernova Cosmology Project and 18 low-redshift SNIa of the Cal`an/Tololo Supernova Survey. mB is the apparent magnitude in the spectral band B, the magnitude m being related to Lsource via m = 2.5 log L + 4.76. Authors: Perlmutter et al.,[129]. − source lensing and it has been pointed out that it probably induces a scattering in Hubble diagrams ([99, 80, 191, 190, 192]). Because of the thinness of the beams emitted by SNIa, such beams spend most of their propagation time in underdense regions (and rarely, but sometimes, encounter clusters of matter). They are a priori too narrow for the effects of such fluctuations of density to be negligible. Yet, Hubble diagrams are interpreted assuming that the geometry of the universe is on average homogeneous and isotropic. Perturbative approaches, i.e. descriptions in which perturbations are added in the metric, have been proposed ([92, 39, 54]). However, in these works, there is still a smoothing scale at play. It is of the order of 1 arcminute9 whereas lensing dispersion arises to a large extent from subarcminute scales (see e.g. [48]). Besides, performing an average is problematic since SNIa probe the sky in directions where the density of matter encountered is smaller than the cosmological average (contrary
9 The arcminute is a unit of angular measurement defined so that 1◦ = 60 arcminutes.
14 1.2. Questioning the cosmological principle analytically to the beams of BAO and CMB, which are large enough for the average density encountered to be representative of the overall cosmological density, [195]). Hubble diagrams are thus prone to an observational selection effect. All these problems are discussed in detail in [33]. Numerical simulations are an alternative to analytic modeling which can take inhomogeneities into account. However, it has been argued in [33] that, because of the limitation in resolution, only the distribution of matter encountered by beams larger than a few tens of kpc can be accurately described by N-body simulations. It is much wider than the characteristic diameter of a SNIa light beam.
1.2.3 An idea to test analytically the impact of inhomogeneities
Since a theoretical study seems necessary to investigate the way in which inhomo- geneities affect the aspect of Hubble diagrams, we decided to elaborate analytically a toy model, representing a very inhomogeneous universe, and to study propaga- tion of light in it. More precisely, we chose to consider a geometry of the type “Swiss cheese”. Swiss-cheese configurations are obtained from a homogeneous and isotropic basis (i.e. a Friedmann-Lemaˆıtremetric) by removing spheres of matter at some places and replacing them by point masses, equal to the mass removed (see figure 1.2 for an illustration and [65, 158] for the cornerstones of Swiss-cheese representations).
Figure 1.2: A Swiss-cheese configuration. The background is homogeneous. Spheres are empty except at their centers, where a point mass is present. The point mass is equal to the density of the background multiplied by the volume of the blank area surrounding it.
Our model is not intended to be realistic but rather to test the impact of inho- mogeneities on light propagation. From a theoretical point of view, it is as justified as the standard homogeneous and isotropic description because it is an exact solu- tion of general relativity which preserves the global dynamics (see section 1.3 for more details). As a first step, we studied analytically the effect of light crossing a single hole
15 1.3. Generalities about propagation of light in a Swiss-cheese model and implemented it in a Mathematica program. It was then easy to predict the effect of several holes. We proposed to do it with the help of Wronski matrices (see our paper, presented in section 1.5). Eventually, we simulated Hubble diagrams by imagining SNIa beams evolving in our Swiss-cheese universe. To do this, we took inspiration from the SNLS 3 data set ([38]). The cosmological parameters of this mock universe being under control, we have been able to test the impact of inhomogeneities on cosmology (see section 1.4). Note that using Swiss-cheese models to study the effect of inhomogeneities is not new at all. The specificity of our study is that it is neither a “modern” Lemaˆıtre- Tolman-Bondi (LTB) solution, which is an extension of Swiss-cheese solutions in which point masses are replaced by continuous gradients of density (here is only a sample of the existing LTB studies: [31, 116, 24, 185, 32, 35, 183, 176, 74, 200, 72, 73, 143, 151, 150, 121, 40, 149, 70]) nor a real step backwards into the original works on Swiss-cheese models ([98] and [62]) because our investigation is totally embedded in the framework of modern cosmology. Actually, we were interested in testing a discontinuous distribution of matter, which is not the case for LTB or perturbative approaches, in order to avoid the compensation effects due to a finite smoothing scale. Consequently, we did not expect to obtain results similar to those of the literature.
1.3 Generalities about propagation of light in a Swiss- cheese model
Rather than starting with general considerations about general relativity, I will merely introduce progressively the concepts that help to understand my PhD work. Comprehensive presentations of general relativity suited to the study of cosmology can be found in a broad collection of references (see in particular the indispensable [193] and [120]).
1.3.1 The homogeneous background
The standard model of cosmology assumes that gravitation is well described by gen- eral relativity. In general relativity, time and space are not independent quantities, whence the use of the concept of spacetime. In standard cosmology, spacetimes encompass four dimensions and are mathematically characterized by a quantity
16 1.3. Generalities about propagation of light in a Swiss-cheese model called metric. The Friedmann-Lemaˆıtremetric emerges naturally when one wants to write a metric in agreement with the cosmological principle and simple enough for analytic calculations to be manageable. Its expression is (see e.g. [194] for a demonstration)
ds2 = dT 2 + a2(T )γ xk dxidxj. (1.4) − ij Geometrically, ds2 is the square of the distance separating two points of the space- time infinitely close to each other, or equivalently the self scalar product of the infinitesimal vector separating them. In this expression, units are chosen so that the speed of light in vacuum, c, is equal to unity and Latin indices run from 1 to 3. Besides, the Einstein convention is used, which means that a summation over repeated indices is implicitly assumed. In other words, one postulates
3 3 γ xk dxidxj γ xk dxidxj. (1.5) ij ≡ ij Xi=1 Xj=1 The Einstein convention will be used in all this manuscript. In relativity, time depends on the frame in which it is measured. The proper time of an observer is the time corresponding to his rest frame. The cosmic time T that appears in (1.4) is the proper time of comoving observers, who are by definition observers moving along with the Hubble flow10 (without peculiar velocity). a(T ) is an arbitrary function of the cosmic time. In cosmology, it is called scale factor and it encodes the expansion of the universe, i.e. distances are expected to grow like a(T ) in a Friedmann- Lemaˆıtremetric11. This is the reason why the derivation of the metric (1.4) by A. Friedmann and G. Lemaˆıtrein 1922 and 1927 is regarded as part of the discovery of the expansion of the universe. So expansion appears spontaneously when general relativity is applied to cosmology. However, when he published the first cosmological solution of the Einstein equations12 in 1917, A. Einstein introduced artificially a quantity called cosmological constant to force the universe to be static. It is the observation of the recession of galaxies by E. Hubble that made him eventually
10The Hubble flow is the recession motion due to expansion. 11It is important to have in mind that it is true for “cosmological” distances only, that is to say for the very high distances separating objects that do not interact via electromagnetic forces or form gravitationally bound systems. For example, it is not true within the solar system or between the atoms of macroscopic objects. 12The Einstein equations are fundamental equations of general relativity describing gravitation as a result of a curvature of spacetime by matter and energy.
17 1.3. Generalities about propagation of light in a Swiss-cheese model accept the idea of an expanding universe. Nowadays, the cosmological constant has been reintroduced in the standard model of cosmology to mimic dark energy. The k quantities x of (1.4) are space coordinates called comoving coordinates. Lastly, γij are equal-time hypersurfaces, that is to say the geometrical figures obtained in the four-dimensional spacetime when imposing a given value to T . Assuming that in the universe space has a constant curvature K, one can show that there are only three possible expressions for γij (see e.g. [131]):
1 γ dxidxj = dχ2 + sin2(√Kχ)dΩ2, ij K γ dxidxj = dχ2 + χ2dΩ2, ij (1.6) 1 γ dxidxj = dχ2 sinh2(√ Kχ)dΩ2, ij − K − where χ is a radial coordinate, dΩ2 = dθ2 +sin2 θdϕ2 is the infinitesimal solid angle, ϕ varies from 0 to 2π, θ from π to π, K χ from 0 to π when K > 0 and from − | | 0 to + otherwise. The first possibility, for which K > 0, is characteristic of a ∞ p space with a spherical geometry. The second one, for which K = 0, corresponds to an Euclidean geometry. The last one, for which K < 0, describes a hyperbolic geometry. The Friedmann-Lemaˆıtremetric is often written using another definition of time, called conformal time. It is denoted η and it satisfies
dT dη = . (1.7) a(T ) In this framework, the metric reads
ds2 = a2(η) dη2 + γ dxidxj . (1.8) − ij The equations describing the time evolution of the scale factor are obtained from the Einstein equations. When they are written in the Friedmann-Lemaˆıtremetric, one gets (see [194] or any cosmology course)
κ K Λ H2 = ρ + (1.9) 3 − a2 3 and 1 d2a κ Λ = (ρ + 3P ) + . (1.10) a dT 2 − 6 3
18 1.3. Generalities about propagation of light in a Swiss-cheese model
1 da H is the Hubble parameter, defined as H . It gives the velocity at which ≡ a dT the universe expands. Λ is a cosmological constant. One can note that Λ could be intentionally fine-tuned to impose a static universe (like A. Einstein initially did) but in general there is no reason for H to be zero. κ 8πG, G being the ≡ gravitational constant, and ρ is the energy density of a cosmic fluid with a pressure P . Those equations are called Friedmann equations. The time evolution of the scale factor being given by second order equations, in addition to H, another observable is necessary to probe the expansion of the universe. It is generally incarnated by the deceleration parameter, denoted q and defined as
aa¨ q = , (1.11) − a˙ 2 where a dot stands for a time derivative. Equivalently, with the conformal time, the Friedmann equations read
κ Λ 2 = ρa2 K + a2, (1.12) H 3 − 3
d κ Λ H = a2 (ρ + 3P ) + a2, (1.13) dη − 6 3 1 da where . In our study, we consider pressureless fluids only and we use those H ≡ a dη results in the homogeneous background of the Swiss-cheese spacetime.
1.3.2 Geometry inside the holes
In astrophysics, it is common to study the gravitational field caused by objects with a spherical symmetry. There exists in general relativity an equivalent of Gauss’s law for Newtonian gravity, according to which the gravitational field outside a spherical object depends on the mass of this object only (and thus not on its structure). When a cosmological constant is taken into account, it is encoded in a metric known as the Kottler metric. It has been derived independently by Kottler in 1918 ([102]) and Weyl in 1919 ([197]). It is given by
2 2 1 2 2 2 ds = A(r)dt + A− (r)dr + r dΩ , (1.14) −
19 1.3. Generalities about propagation of light in a Swiss-cheese model where
2GM Λr2 A(r) = 1 . (1.15) − r − 3 M is the mass of the astrophysical object considered, t is a time coordinate, r a radial coordinate, Λ a cosmological constant and dΩ2 the infinitesimal solid angle. For Λ = 0, this metric is known as the Schwarzschild metric, which is a solution of the Einstein equations derived in 1915 by the astrophysicist Karl Schwarzschild. The Schwarzschild metric is the first solution of general relativity that includes a mass, allowing to study the gravitational effect of e.g. stars, planets or certain types of black holes.
1.3.3 Junction of the two metrics
In order to be a spacetime well-defined in the sense of general relativity, the space- time resulting from the junction of the Friedmann-Lemaˆıtreand Kottler metrics must satisfy the Israel junction conditions ([96]). They impose continuity of two quantities, namely the induced metric and the extrinsic curvature, on junction hy- persurfaces. In our study, junction hypersurfaces Σ are given by
Σ χ = χ , (1.16) ≡ { h} where χh is a constant, in the Friedmann-Lemaˆıtremetric and by
Σ r = r (t) (1.17) ≡ { h } in the Kottler metric.
Continuity of the induced metric
If the coordinates of the Friedmann-Lemaˆıtremetric are Xα = T, χ, θ, φ , one can { } define “intrinsic coordinates” on Σ as13 σa = T, θ, φ . The parametric equation of { } the hypersurface (1.16) is therefore X¯ α(σa) = T, χ , θ, φ . What is called induced { h } metric is the quantity α β hab = gαβ ja jb . (1.18) 13Latin indices run from 1 to 3 whereas Greek indices run from 0 to 3, the 0 index corresponding to the time coordinate.
20 1.3. Generalities about propagation of light in a Swiss-cheese model
gαβ is the metric tensor of the spacetime in which the hypersurface is immersed. It means that, in this spacetime, the scalar product is given by
α β ~u.~v = gαβu v . (1.19)
2 α β It imposes in particular ds = gαβdx dx . So the coordinates of the Friedmann- Lemaˆıtremetric tensor are given in (1.4) and those of the Kottler metric tensor are α given in (1.14). Besides, one defines ja as
∂X¯ α jα = . (1.20) a ∂σa Similarly, in the Kottler regions, one introduces Xα = t, r, θ, φ and one defines { } intrinsic coordinates as σa = t, θ, φ , whence X¯ α(σa) = t, r (t), θ, φ . { } { h } Using those definitions, the relations imposed by the continuity of the induced metric are eventually
rh(t) = a(T )χh,
2 (1.21) dT 1 drh = A (r ) . dt h − A (r ) dt s h Continuity of the extrinsic curvature
By definition, if the unit vector normal to the hypersurface Σ is denoted nµ and if one calls Kab the extrinsic curvature of this hypersurface, one has
α β Kab = nα;β ja jb . (1.22)
I use here the notation “ ; ” to indicate a covariant derivative. Its definition can be found in any course in general relativity: for any vector T ν,
T ν = T ν = ∂ T ν + Γν T α, (1.23) ;µ ∇µ µ µα ∂ where the notation ∂ stands for and where Γν are the Christoffel symbols, µ ∂xµ µα related to the metric tensor via
1 Γν = gνσ (∂ g + ∂ g ∂ g ) , (1.24) µα 2 µ σα α µσ − σ µα
21 1.3. Generalities about propagation of light in a Swiss-cheese model
ασ α α α with g gσβ = δ β, δ β being the Kronecker symbol, i.e. δ β = 1 if α = β and 0 otherwise. Let us mention here a useful property of the metric tensor allowing to raise and lower indices of vectors conveniently: the metric tensor is defined so that
β α αβ uα = gαβu and u = g uβ. (1.25)
Generally, the unit vector normal to a hypersurface defined by q = ... is given by { }
∂µq nµ = . (1.26) αβ g ∂αq ∂βq
When imposing continuity to Kab, thep whole calculation gives finally on Σ
rh A (rh) = a(T )χh, 1 dr 2 A (r ) h h s − A (rh) dt (1.27) 2 2 drh d rh 2 3 ∂rA (r = rh) 2 A (rh) A (rh) ∂rA (r = rh) = 0. dt − dt2 − Consequences on the properties of the holes
Equations (1.21) and (1.27) govern the dynamics of the hole boundary by connecting the time and space coordinates of the two metrics. When combined with (1.10), they impose equality between the cosmological constants of the two metrics and the (almost intuitive) relation
4π M = ρa3f 3 (χ ), (1.28) 3 K h where
sin √Kχ sinh √ Kχ fK (χ) = , χ or − (1.29) √K √ K − respectively for a positive, zero or negative spatial curvature. Note that, in the
Friedmann-Lemaˆıtremetric, the hole boundaries χh are defined with comoving co- ordinates, i.e. with coordinates not affected by expansion. Hence, holes are not expected to overlap at any stage. Such constraints ensure that the holes inserted in the homogeneous background do not modify its global dynamics. The resulting spacetime is indeed an exact
22 1.3. Generalities about propagation of light in a Swiss-cheese model solution of the Einstein equations. In this context, it is possible to insert as many holes as one wants provided that holes do not overlap initially.
1.3.4 Light propagation
Geodesics
The equivalence principle is a pillar of gravitation theory. In classical mechanics, it predicts that the acceleration of massive objects in free fall (i.e. falling under the sole influence of gravity) does not depend on their masses. In other words, gravitation appears as a property of space rather than of objects themselves. In general relativity, it is reflected by the use of the concept of geodesics. The successive positions occupied by free-falling objects in spacetime are interpreted as a consequence of a bending of this spacetime. Those trajectories, or “world lines”, are called geodesics. In metrics with a signature14 ( , +, +, +), geodesics are − curves that extremize the “distances” ds2 between two points. More precisely, time- like15 geodesics maximize the distances between time-like world lines connecting two points; space-like16 geodesics minimize the distances between space-like world lines connecting two points; null17 geodesics are defined so that the distances between null world lines connecting two points are zero. To study the propagation of light rays in a spacetime, one thus has to write that photons are particles following null geodesics. The mathematical implementation suited to our study is given in our paper (see section 1.5). As an illustration, for a null geodesic of a Friedmann-Lemaˆıtreuniverse, (1.4) gives
dT 2 = γ xk dxidxj. (1.30) a2(T ) ij 14With a signature ( , +, +, +) means that there exists a basis in which the scalar product − between two 4-dimensional vectors with coordinates (u0, u1, u2, u3) and (v0, v1, v2, v3) is u0v0 + − u1v1 + u2v2 + u3v3. It is the case for the metrics (1.4) and (1.14). 15A time-like vector is a vector whose self scalar product is negative and a time-like world line is a curve whose tangent vectors are in any point time-like vectors. 16A space-like vector is a vector whose self scalar product is positive and a space-like world line is a curve whose tangent vectors are in any point space-like vectors. 17A null vector is a non-zero vector whose self scalar product is zero and a null world line is a curve whose tangent vectors are in any point null vectors.
23 1.3. Generalities about propagation of light in a Swiss-cheese model
In the particular case of a radial trajectory18, for which dΩ2 = 0, it leads to
dT = dχ. (1.31) a(T )
Let us imagine two photons emitted at times T1 and T1 +δT1 and observed at times
T0 and T0 + δT0. Since χ is a comoving distance, it is a constant, whence
T1 dT T1+δT1 dT χ = = . (1.32) a(T ) a(T ) ZT0 ZT0+δT0 Subtracting integration over the time interval [T0 + δT0,T1] on each side, one gets
T0+δT0 dT T1+δT1 dT = . (1.33) a(T ) a(T ) ZT0 ZT1
If the lags are assumed to be very small compared to T0 and T1, the scale factor is expected to be almost constant in (1.33). In this context, one can write
δT δT 0 1 . (1.34) a(T0) ≈ a(T1)
Furthermore, one can decide that δT1 is the period of the electromagnetic wave at emission. In that case, δT0 is the period of the electromagnetic wave measured by the observer. Denoting λ1 and λ0 the associated wavelengths, one has (see (1.1))
a(T ) λ 0 = 0 = 1 + z. (1.35) a(T1) λ1 The cosmological redshift can thus be interpreted as a witness of the time evolution of the expansion rate of the universe.
Light beams
Despite their narrowness, beams emitted by SNIa have a finite section. They are collections of light rays. A study of the relative displacements between the rays is therefore required to infer the deformation of the beams, and consequently the change in luminosity distance, induced by the presence of structures in the space- time. It is encoded in the geodesic deviation equation, which governs the time evolution of the “separation vectors” that connect the geodesics of a beam. It is
18One can show that such trajectories are indeed geodesics, that is to say that they obey equations of general relativity called geodesic equations. They can be found in any general relativity book.
24 1.3. Generalities about propagation of light in a Swiss-cheese model
µ µ obtained by considering two geodesics γ0 and γ1 described by x (λ), x being the 19 coordinates and λ an affine parameter . Then, between γ0 and γ1, one defines a collection of geodesics, each of them being associated with a label s [0, 1] (γ ’s ∈ 0 label is 0 and γ1’s label is 1). To indicate both the geodesic considered and the value of the affine parameter, one uses the notation xµ(λ, s). Besides, maintain- ing λ constant and varying s, one gets another collection of curves (which are not µ µ geodesics in general). The separation vectors ξ = ∂sx are tangent to this col- lection of curves and establish a connection between the geodesics. Besides, the µ µ vectors k = ∂λx are tangent to the null geodesics (see figure 1.3 and more details in the paper presented in section 1.5).
Figure 1.3: A geodesic bundle.
Let us define the acceleration of the separation vector:
D2ξµ = kβ ξµ kν. (1.36) Dλ2 ∇ν ∇β In a flat spacetime, it is zero because in that case geodesics are straight lines so the variation of ξµ with λ is necessarily linear. Hence, a non zero acceleration reveals a curvature. In general relativity, it means that one expects this acceleration to be proportional to the Riemann tensor20 Rµ . Noticing that kβ ξα = ξβ kα and γβν ∇β ∇β ν µ µ using the geodesic equation k νk = 0, one can see that the quantity ξ kµ is con- ∇d (ξµk ) served along the geodesic γ µ = 0 . Consequently, one can parametrize 0 dλ 19An affine parameter λ is a parameter defined so that the geodesic equations governing the d2xα dxµ dxν behavior of geodesics described by xα(λ) take the form + Γα = 0. dλ2 µν dλ dλ 20 µ µ µ µ σ µ σ By definition, R = ∂αΓ ∂β Γ + Γ Γ Γ Γ . ναβ νβ − να σα νβ − σβ να
25 1.3. Generalities about propagation of light in a Swiss-cheese model
µ µ µ the curves so that, on γ0, ξ and k are always orthogonal (ξ kµ = 0). The same properties allow to compute the acceleration of γ1 with respect to γ0 (see e.g. [131]):
D2ξµ = ξβRµ kγkν . (1.37) Dλ2 − γβν It is the geodesic deviation equation. Once projected on a judicious basis, it leads to the so-called Sachs equation and finally the separation vectors can be related to observables such as the luminosity distance (see our paper, section 1.5).
Strategy adopted to determine the light path
In practice, what we did to determine the light path between a source and an observer is to start from the observer (assumed to stand in the Friedmann-Lemaˆıtre region) and to reconstruct, step by step, the trajectory of photons back to the source. The first step is the identification of the coordinates corresponding to a photon leaving the last hole encountered before reaching the observer (resolution of the geodesic equation in the Friedmann-Lemaˆıtremetric). Next, the resolution of the Sachs equation gives the behavior of the separation vector. It is then necessary to switch to the Kottler metric to infer the deviation caused by the point mass inside the hole. Identification of the geodesics relies on two conservation laws which reflect that the Kottler metric is static21 and with a spherical symmetry. It gives the coordinates associated with the event corresponding to photons entering the last hole. Then the Sachs equation is integrated again and one goes back to the Friedmann-Lemaˆıtremetric. One can iterate this procedure for an arbitrary number of holes, until one assumes that the source has been reached. Throughout the whole calculation, the evolution of the wavenumber is also studied, which gives the evolution of the cosmological redshift of the source (by comparing the wavenumber at emission and reception times). Thanks to the resolution of the Sachs equation, the luminosity distance is also part of the traceable observables. Consequently, Hubble diagrams can be computed once the observation conditions, the distribution of the holes, the value of the point masses and the redshifts of the sources are specified (full details are given in section 1.5). Besides, they can be easily compared to what they would be in a configuration similar but with no hole.
21The fact that it is static is a consequence of Birkhoff’s theorem, see e.g. [69].
26 1.4. Impact of inhomogeneities on cosmology
1.4 Impact of inhomogeneities on cosmology
Cosmological parameters are omnipresent in the equations of our study so the as- pect of Hubble diagrams strongly depends on them. We therefore introduced the quantities
Λ 8πGρ0 K ΩΛ = 2 , Ωm = 2 , ΩK = 2 2 , (1.38) 3H0 3H0 −a0H0 where an index 0 indicates that the quantity is evaluated at present time. ΩΛ is one of the six cosmological parameters of the standard model of cosmology. It charac- terizes the amount of dark energy in the universe. According to [135], results from the Planck satellite lead to the 1σ-constraint22 Ω = 0.686 0.020 (the meaning of Λ ± “1σ-constraint” is given below.). Ωm characterizes the amount of non-relativistic 2 matter (and is related to the standard cosmological parameter ωm via ωm = Ωmh , 1 with h defined so that H0 = 100 h km.s− ). It is the sum of the amounts of bary- onic matter and cold dark matter. Still from [135] and with the same precision, one estimates Ω = 0.315 0.017. Ω indicates what the spatial curvature of the uni- m ± K verse is. It is given by Ω = 1 Ω Ω . Observational constraints are therefore K − Λ − m compatible with ΩK = 0, which means that the universe can reasonably be assumed to have a flat geometry. On the one hand, we used our Hubble diagrams as mock data and tried to inter- pret them assuming a homogeneous and isotropic geometry. More precisely, we used the Chi-Square Goodness of Fit Test to estimate the values of the parameters (1.38) that best fit the resulting diagrams when the spacetime is entirely characterized by the Friedmann-Lemaˆıtremetric. Generally, this test summarizes the discrepancy between observed values O (here the mock data) and the values E expected under the model in question (here the homogeneous model). The generic formula is
(O E)2 χ2 = − , (1.39) ∆2 dataX where ∆ is an observational error bar23. The values of χ2 give the best fit, i.e. the values of E that best match observations, and confidence contours, i.e. areas in parameter space in which the parameters are expected to lie with a probability
22This estimation is yet highly model dependent. 23In our study, we estimated ∆ by imitating a real catalog of SNIa observations.
27 1.4. Impact of inhomogeneities on cosmology exceeding a given value24 (see e.g. [141] for more details). We found that the es- timated cosmological parameters can be very different from the actual parameters characterizing the Swiss-cheese spacetime in which we simulated light propagation. In other words, in some cases, the homogeneity assumption largely affects the esti- mation of the cosmological parameters (see the detailed study in section 1.5). Another test we performed is the estimation of the cosmological parameters of the Swiss-cheese model that best reproduce real observations. Using the same χ2 method, we found that replacing a Friedmann-Lemaˆıtre model by a Swiss-cheese one can in some cases induce a significant (regarding the accuracy intended by pre- cision cosmology, i.e. a few percent) difference in the prediction of the parameters (see section 1.5). It required in particular to estimate analytically a luminosity distance-redshift relation of a Swiss-cheese universe. Evaluating such effects is an important issue since the 2013 Planck results ([135]) highlighted a tension between the estimation of H0 and Ωm from CMB observations and their estimation from other observables, such as SNIa. In the paper presented in section 1.6, we demon- strate that Swiss-cheese descriptions make possible the reconciliation between the values of Ωm inferred from Planck and from SNIa. This is illustrated in figure 1.4.
Besides, the 2015 Planck results ([137]) show that the estimation of H0 from Planck is in fact consistent with its estimation from the recent “Joint Light-curve Analysis” sample of SNIa (constructed from the SNLS and SDSS supernovae data, together with several samples of low redshift supernovae, [20]).
24For a Gaussian probability distribution of the parameters, we call “1σ-contours” the contours associated with a 68.3% probability, “2σ-contours” the contours associated with a 95.4% probability and “3σ-contours” the contours associated with a 99.73% probability.
28 1.5. Article “Interpretation of the Hubble diagram in a nonhomogeneous universe”
Figure 1.4: Comparison of the constraints obtained by Planck on (Ωm, h), [135], and from the analysis of the Hubble diagram constructed from the SNLS 3 catalog, [86]. f is the “smoothness parameter”, which gives the ratio between the volume which is not in form of holes and the total volume. Hence, f = 1 corresponds to the Friedmann-Lemaˆıtremetric.
1.5 Article “Interpretation of the Hubble diagram in a nonhomogeneous universe”
29 PHYSICAL REVIEW D 87, 123526 (2013) Interpretation of the Hubble diagram in a nonhomogeneous universe
Pierre Fleury,1,2,* He´le`ne Dupuy,1,2,3,† and Jean-Philippe Uzan1,2,‡ 1Institut d’Astrophysique de Paris, UMR-7095 du CNRS, Universite´ Pierre et Marie Curie, 98 bis bd Arago, 75014 Paris, France 2Sorbonne Universite´s, Institut Lagrange de Paris, 98 bis bd Arago, 75014 Paris, France 3Institut de Physique The´orique, CEA, IPhT, URA 2306 CNRS, F-91191 Gif-sur-Yvette, France (Received 15 March 2013; published 24 June 2013) In the standard cosmological framework, the Hubble diagram is interpreted by assuming that the light emitted by standard candles propagates in a spatially homogeneous and isotropic spacetime. However, the light from ‘‘point sources’’—such as supernovae—probes the Universe on scales where the homogeneity principle is no longer valid. Inhomogeneities are expected to induce a bias and a dispersion of the Hubble diagram. This is investigated by considering a Swiss-cheese cosmological model, which (1) is an exact solution of the Einstein field equations, (2) is strongly inhomogeneous on small scales, but (3) has the same expansion history as a strictly homogeneous and isotropic universe. By simulating Hubble diagrams in such models, we quantify the influence of inhomogeneities on the measurement of the cosmological parameters. Though significant in general, the effects reduce drastically for a universe dominated by the cosmological constant.
DOI: 10.1103/PhysRevD.87.123526 PACS numbers: 98.80. k, 04.20. q, 42.15. i
can however imprint some observations, in particular I. INTRODUCTION regarding the propagation of light with narrow beams, as The standard physical model of cosmology relies on a discussed in detail in Ref. [8]. It was argued that such solution of general relativity describing a spatially homo- beams, as e.g. for supernova observations, probe the space- geneous and isotropic spacetime, known as the Friedmann- time structure on scales much smaller than those accessible Lemaıˆtre (FL) solution (see e.g. Ref. [1]). It is assumed to in numerical simulations. The importance of quantifying describe the geometry of our Universe smoothed on large the effects of inhomogeneities on light propagation was scales. Besides, the use of the perturbation theory allows first pointed out by Zel’dovich [9]. Arguing that photons one to understand the properties of the large scale struc- should mostly propagate in vacuum, he designed an ture, as well as its growth from initial conditions set by ‘‘empty beam’’ approximation, generalized later by Dyer inflation and constrained by the observation of the cosmic and Roeder as the ‘‘partially filled beam’’ approach [10]. microwave background. More generally, the early work of Ref. [9] stimulated many While this simple solution of the Einstein field equa- studies on this issue. [11–25]. tions, together with the perturbation theory, provides a The propagation of light in an inhomogeneous universe description of the Universe in agreement with all existing gives rise to both distortion and magnification induced by data, it raises many questions on the reason why it actually gravitational lensing. While most images are demagnified, gives such a good description. In particular, it involves a because most lines of sight probe underdense regions, smoothing scale which is not included in the model itself some are amplified because of strong lensing. Lensing [2]. This opened a lively debate on the fitting problem [3] can thus discriminate between a diffuse, smooth compo- (i.e. what is the best-fit FL model to the lumpy Universe?) nent, and the one of a gas of macroscopic, massive objects and on backreaction (i.e. the fact that local inhomogene- (this property has been used to probe the nature of dark ities may affect the cosmological dynamics). The ampli- matter [26–28]). Therefore, it is expected that lensing shall tude of backreaction is still actively debated [4–6], see induce a dispersion of the luminosities of the sources, and Ref. [7] for a critical review. thus an extra scatter in the Hubble diagram [29]. Indeed, Regardless of backreaction, the cosmological model such an effect does also appear at the perturbation level— assumes that the distribution of matter is continuous (i.e. i.e. with light propagating in a perturbed FL spacetime— it assumes that the fluid approximation holds on the scales and it was investigated in Refs. [30–35]. The dispersion of interest) both at the background and perturbation levels. due to the large-scale structure becomes comparable to the Indeed numerical simulations fill part of this gap by deal- intrinsic dispersion for redshifts z>1 [36] but this disper- ing with N-body gravitational systems in an expanding sion can actually be corrected [37–42]. Nevertheless, a space. The fact that matter is not continuously distributed considerable fraction of the lensing dispersion arises from sub-arc minute scales, which are not probed by shear *fl[email protected] maps smoothed on arc minute scales [43]. The typical †[email protected] angular size of the light beam associated with a supernova ‡[email protected] (SN) is typically of order 10 7 arc sec (e.g. for a source of
1550-7998=2013=87(12)=123526(24) 123526-1 Ó 2013 American Physical Society FLEURY, DUPUY, AND UZAN PHYSICAL REVIEW D 87, 123526 (2013) physical size 1AUat redshift z 1), while the typical exact solution of the Einstein field equations with strong observational aperture is of order 1 arc sec. This is smaller density fluctuations, but which keeps a well-defined FL than the mean distance between any massive objects. averaged behavior. Such conditions are satisfied by the One can estimate [27] that a gas composed of particles Swiss-cheese model [51]: one starts with a spatially homo- of mass M can be considered diffuse on the scale of geneous and isotropic FL geometry, and then cuts out the beam of an observed source of size s if M< spherical vacuoles in which individual masses are em- 23 2 3 2 10 M h ð s=1AUÞ . In the extreme case for which bedded. Thus, the masses are contained in vacua within a matter is composed only of macroscopic pointlike objects, spatially homogeneous fluid-filled cosmos (see bottom then most high-redshift SNeIa would appear fainter than in panel of Fig. 2). By construction, this exact solution is a universe with the same density distributed smoothly, with free from any backreaction: its cosmic dynamics is identi- some very rare events of magnified SNeIa [27,44,45]. This cal to the one of the underlying FL spacetime. makes explicit the connection between the Hubble diagram From the kinematical point of view, Swiss-cheese and the fluid approximation which underpins its standard models allow us to go further than perturbation theory, interpretation. because not only the density of matter exhibits finite fluc- The fluid approximation was first tackled in a very tuations, but also the metric itself. Hence, light propagation innovative work of Lindquist and Wheeler [46], using is expected to be very different in a Swiss-cheese universe a Schwarzschild cell method modeling an expanding compared to its underlying FL model. Moreover, the in- universe with spherical spatial sections. For simplicity, homogeneities of a Swiss cheese are introduced in a way they used a regular lattice which restricts the possibilities that addresses the so-called ‘‘Ricci-Weyl problem.’’ to the most homogeneous topologies of the 3-sphere [47]. Indeed, the standard FL geometry is characterized by a It has recently been revisited in Refs. [48] and in Refs. [49] vanishing Weyl tensor and a nonzero Ricci tensor, while in for Euclidean spatial sections. They both constructed reality light mostly travels in vacuum, where conversely the associated Hubble diagrams, but their spacetimes are the Ricci tensor vanishes—apart from the contribution of only approximate solutions of the Einstein field equations. , which does not focus light—and the Weyl tensor is An attempt to describe filaments and voids was also pro- nonzero (see Fig. 1). A Swiss-cheese model is closer to posed in Refs. [50]. the latter situation, because the Ricci tensor is zero inside These approaches are conceptually different from the the holes (see Fig. 2). It is therefore hoped to capture the solution we adopt in the present article. We consider an relevant optical properties of the Universe.
FL model FL model SN Ia SN Ia
Ricci = 0 Ricci = 0 Weyl = 0 Weyl = 0 Ricci = 0 Ricci = 0 Weyl = 0 Weyl = 0
Swiss−cheese model SN Ia SN Ia
Real Universe
FIG. 1 (color online). The standard interpretation of SNe data assumes that light propagates in purely homogeneous and iso- FIG. 2 (color online). Swiss-cheese models (bottom) allow us tropic space (top). However, thin light beams are expected to to model inhomogeneities beyond the continuous limit, while probe the inhomogeneous nature of the actual Universe (bottom) keeping the same dynamics and average properties as the FL down to a scale where the continuous limit is no longer valid. model (top).
123526-2 INTERPRETATION OF THE HUBBLE DIAGRAM IN A ... PHYSICAL REVIEW D 87, 123526 (2013) In fact, neither a Friedmann-Lemaıˆtre model nor Swiss-cheese models and solve the associated equations. a Swiss-cheese model can be considered a realistic The results allow us to investigate the effect of one hole description of the Universe. They share the property of (Sec. V) and of many holes (Sec. VI) on cosmological being exact solutions of the Einstein equations which satisfy observables, namely the redshift and the luminosity dis- the Copernican principle, either strictly or statistically. tance. Finally, the consequences on the determination of Swiss-cheese models can be characterized by an extra- the cosmological parameters are presented in Sec. VII. cosmological parameter describing the smoothness of their distribution of matter. Thus, a FL spacetime is nothing but a II. DESCRIPTION OF THE SWISS-CHEESE perfectly smooth Swiss cheese. It is legitimate to investigate COSMOLOGICAL MODEL to which extent observations can constrain the smoothness The construction of Swiss-cheese models is based on the cosmological parameter, and therefore to quantify how close Einstein-Straus method [51] for embedding a point-mass to a FL model the actual Universe is. within a homogeneous spacetime (the ‘‘cheese’’). It con- The propagation of light in a Swiss-cheese universe was sists in cutting off a spherical domain of the cheese and first investigated by Kantowski [52], and later by Dyer and concentrating the matter it contained at the center of the Roeder [53]. Both concluded that the effect, on the Hubble hole. This section presents the spacetime geometries inside diagram, of introducing ‘‘clumps’’ of matter was to lower and outside a hole (Sec. II A), and how they are glued the apparent deceleration parameter. The issue was revived together (Sec. II B). within the backreaction and averaging debates, and the Swiss-cheese models have been extended to allow for A. Spacetime patches more generic distributions of matter inside the holes— instead of just concentrating it at the center—where space- 1. The ‘‘cheese’’—Friedmann-Lemaıˆtre geometry time geometry is described by the Lemaıˆtre-Tolman-Bondi Outside the hole, the geometry is described by the (LTB) solution. The optical properties of such models have standard FL metric been extensively studied (see Refs. [54–62]) to finally 2 2 2 2 2 2 conclude that the average luminosity-redshift relation re- d s ¼ dT þ a ðTÞ½d þ fKð Þd ; (2.1) mains unchanged with respect to the purely homogeneous where a is the scale factor and T is the cosmic time. The case, contrary to the early results of Refs. [52,53]. function fKð Þ depends on the sign of K and thus of the In general, the relevance of ‘‘LTB holes’’ in Swiss- spatial geometry (spherical, Euclidean or hyperbolic), cheese models is justified by the fact that they allow one pffiffiffiffi pffiffiffiffiffiffiffiffi to reproduce the actual large-scale structure of the sin ffiffiffiffiK sinh ffiffiffiffiffiffiffiffi K fKð Þ¼ p ; or p (2.2) Universe (with voids and walls). However, though inho- K K mogeneous, the distribution of matter in this class of respectively for K>0, K ¼ 0 or K<0. The Einstein field models remains continuous at all scales. On the contrary, equations imply that the scale factor aðTÞ satisfies the the old-fashioned approach with ‘‘clumps’’ of matter inside Friedmann equation the holes breaks the continuous limit. Hence, it seems more 8 G K 1 da relevant for describing the small-scale structure probed by H2 ¼ þ ; with H ; (2.3) thin light beams. 3 a2 3 a dT In this article, we revisit and update the studies of 3 and where ¼ 0ða0=aÞ is the energy density of a Refs. [52,53] within the paradigm of modern cosmology. pressureless fluid. A subscript 0 indicates that the quantity For that purpose, we first provide a comprehensive study of is evaluated today. It is convenient to introduce the light propagation in the same class of Swiss-cheese mod- cosmological parameters els, including the cosmological constant. By generating 8 G 0 K mock Hubble diagrams, we then show that the inhomoge- m ¼ 2 ; K ¼ 2 2 ; ¼ 2 ; (2.4) neities induce a significant bias in the apparent luminosity- 3H0 a0H0 3H0 redshift relation, which affects the determination of the in terms of which the Friedmann equation takes the form cosmological parameters. As we shall see, the effect in- H 2 a 3 a 2 ¼ 0 þ 0 þ : (2.5) creases with the fraction of clustered matter but decreases H m a K a with . For a universe apparently dominated by dark 0 energy, the difference turns out to be small. The article is organized as follows. Section II describes 2. The ‘‘hole’’—Kottler geometry the construction and mathematical properties of the Swiss- Inside the hole, the geometry is described by the exten- cheese model. In Sec. III, we summarize the laws of light sion of the Schwarzschild metric to the case of a nonzero propagation, and introduce a new tool to deal with a patch- cosmological constant, known as the Kottler solution work of spacetimes, based on matrix multiplications. In [63,64] (see e.g. Ref. [65] for a review). In spherical Sec. IV, we apply the laws introduced in Sec. III to coordinates ðr; ; ’Þ, it reads
123526-3 FLEURY, DUPUY, AND UZAN PHYSICAL REVIEW D 87, 123526 (2013) T t d s2 ¼ AðrÞdt2 þ A 1ðrÞdr2 þ r2d 2; (2.6) Kottler Kottler
2 rS r Σ Σ with AðrÞ 1 ; (2.7) FL FL t r 3 T n χ M r h (t ) θ , ϕ M h θ , ϕ and where rS 2GM is the Schwarzschild radius associ- ated with the mass M at the center of the hole. It is easy to r check that the above metric describes a static spacetime. n χ The corresponding Killing vector ¼ 0 has norm g ¼ AðrÞ and is therefore timelike as long as A>0. Hence, there are two cases: (1) If 9ðGMÞ2 > 1, then AðrÞ < 0 for all r>0, so that is spacelike. In this case, the Kottler spacetime FIG. 3 (color online). The junction hypersurface as seen from the FL point of view with equation ¼ h (left); and from the contains no static region but it is spatially Kottler point of view with equation r ¼ r ðtÞ (right). homogeneous. h 2 (2) If 9ðGMÞ < 1, then AðrÞ > 0 for r between rb and rc >rb which are the two positive roots of the In the FL region, the normal vector to the hypersurface is ðFLÞ polynomial rAðrÞ, and correspond respectively to given by n ¼ =a. The 3-metric and the extrinsic the black hole and cosmological horizons. We have curvature induced by the FL geometry are respectively 2 c 2 2 2 2 2 ffiffiffiffi d s ¼ dT þ a ðTÞfKð hÞd ; (2.11) rc ¼ p cos þ ; (2.8) 3 3 ðFLÞ a b 0 2 Kab dx dx ¼ aðTÞfKð hÞfKð hÞd ; (2.12) 2 c ffiffiffiffi a rb ¼ p cos ; (2.9) where ðx Þ¼ðT; ; ’Þ are natural intrinsic coordinates for 3 3 . We stress carefully that, in the following and as long as pffiffiffiffi with cos c ¼ 3GM , so that there is no ambiguity, a dot can denote a time derivative with respect to T or t, so that a_ ¼ da=dT and r_ h ¼ drh=dt, 3 1ffiffiffiffi 3ffiffiffiffi while a prime can denote a derivative with respect to or r, rS
123526-4 INTERPRETATION OF THE HUBBLE DIAGRAM IN A ... PHYSICAL REVIEW D 87, 123526 (2013) It is straightforward to show that Eq. (2.18), together with v is the affine parameter along them. The relative behavior the Friedmann equation (2.3), imply that the Kottler and FL of two neighboring geodesics x ð ; Þ and x ð ; þ d Þ is regions have the same cosmological constant, and described by their separation vector dx =d . Hence, this vector encodes the whole information on the size and 4 3 3 M ¼ a fKð hÞ: (2.19) shape of the bundle. 3 Having chosen v ¼ 0 at the observation event—which is a vertex point of the bundle—ensures that the separation C. Summary vector field is everywhere orthogonal to the geodesics, Given a FL spacetime with pressureless matter and a k ¼ 0. In such conditions, the evolution of with v cosmological constant, aðTÞ is completely determined is governed by the geodesic deviation equation from the Friedmann equation. A spherical hole of k k r r ¼ R k k ; (3.4) comoving radius h, which contains a constant mass 3 3 M ¼ 4 a fKð hÞ=3 at its center, and whose geometry is where R is the Riemann tensor. described by the Kottler metric, can then be inserted any- where. The resulting spacetime geometry is an exact 2. Sachs equation solution of the Einstein field equations. By construction, the clump inside the hole does not Consider an observer with four-velocity u . In view of relating to observable quantities, we introduce the backreact on the surrounding FL region. It follows that many such holes can be inserted, as long as they do not Sachs basis ðsA ÞA2f1;2g, defined as an orthonormal basis overlap. Note that if two holes do not overlap initially, then of the plane orthogonal to both u and k , they will never do so, despite the expansion of the universe, s s ¼ ;s u ¼ s k ¼ 0; (3.5) because their boundaries are comoving. A B AB A A and parallel-transported along the geodesic bundle, III. PROPAGATION OF LIGHT k r sA ¼ 0: (3.6) A. Light rays The plane spanned by ðs1;s2Þ can be considered a screen on The past light cone of a given observer is a constant which the observer projects the light beam. The two-vector phase hypersurface w ¼ const. Its normal vector k of components A ¼ sA then represents the relative @ w (the wave four-vector) is a null vector satisfying the position, on the screen, of the light spots corresponding geodesic equation, and whose integral curves (light rays) to two neighboring rays separated by . are irrotational: The evolution of A, with light propagation, is deter- mined by projecting the geodesic deviation equation (3.4) k k ¼ 0;kr k ¼ 0; r½ k ¼ 0: (3.1) on the Sachs basis. The result is known as the Sachs For an emitter and an observer with respective four- equation [1,68,69], and reads velocities uem and uobs, we define the redshift by 2 d A B ¼ RAB ; (3.7) uemk ðvemÞ dv2 1 þ z ¼ ; (3.2) uobsk ð0Þ R where AB ¼ R k k sA sB is the screen-projected where v is an affine parameter along the geodesic, so that Riemann tensor, called optical tidal matrix. It is conven- k ¼ dx =dv, and v ¼ 0 at the observation event. The iently decomposed into a Ricci term and a Weyl term as wave four-vector can always be decomposed into temporal ! ! 00 0 Re 0 Im 0 and spatial components, ðRABÞ¼ þ (3.8) 0 00 Im 0 Re 0 k ¼ð1 þ zÞðu d Þ;du ¼ 0;dd ¼ 1; (3.3) with where d denotes the spatial direction of observation. In 1 1 00 R k k ; 0 C k k ; (3.9) Eq. (3.3), we have chosen an affine parameter adapted to 2 2 the observer, in the sense that 2 ¼ u k ð0Þ¼1. This 0 obs and where s1 is2 . convention is used in all the remainder of the article. 3. Notions of distance B. Light beams Since the light beam converges at the observation event, 1. Geodesic deviation equation we have Aðv ¼ 0Þ¼0. The linearity of the Sachs A light beam is a collection of light rays, that is, a bundle equation then implies the existence of a 2 2 matrix DA of null geodesics fx ðv; Þg, where labels the curves and B, called Jacobi matrix, such that
123526-5 FLEURY, DUPUY, AND UZAN PHYSICAL REVIEW D 87, 123526 (2013) d B d A DA ðvÞ¼Cðv; v Þ þ Dðv; v Þ ; ðvÞ¼ BðvÞ : (3.10) init v¼vinit init dv v¼0 dv v¼vinit From Eq. (3.7), we immediately deduce that this matrix (3.17) satisfies the Jacobi matrix equation as for any linear second order differential equation, solved from v to v. In the following, Cðv; v Þ is referred to as d2 init init DA ¼ RA DC ; (3.11) the scale matrix. It is easy to check that both the scale and 2 B C B dv Jacobi matrices satisfy the Jacobi matrix equation (3.11) with initial conditions but with different initial conditions: D dDA D 0 d 1 D A ð0Þ¼0; B ð0Þ¼ A: (3.12) ðvinit; vinitÞ¼ ; ðvinit; vinitÞ¼ ; (3.18) B dv B dv We shall also use the short-hand notation ¼ð AÞ and whereas D ¼ðDA Þ so that Eq. (3.11) reads d2D=dv2 ¼ R D, dC B C ðv ; v Þ¼1; ðv ; v Þ¼0: (3.19) with Dð0Þ¼0 and D_ ð0Þ¼1. init init dv init init Since the Jacobi matrix relates the shape of a light beam The most useful object for our problem turns out to be to its ‘‘initial’’ aperture, it is naturally related to the various the 4 4 Wronski matrix constructed from C and D, notions of distance used in astronomy and cosmology. The ! angular distance D is defined by comparing the emission Cðv; v Þ Dðv; v Þ A W init init 2 ðv; vinitÞ C D ; (3.20) cross-sectional area d Ssource of a source to the solid angle d ðv; v Þ d ðv; v Þ 2 dv init dv init d obs under which it is observed, in terms of which the general solution (3.17) reads d 2S ¼ D2 d 2 : (3.13) source A obs d ðvÞ¼W ðv; vinitÞ d ðvinitÞ: (3.21) It is related to the Jacobi matrix by dv dv qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D It is clear, from Eq. (3.21), that W satisfies the relation DA ¼ j det ðvsourceÞj; (3.14) W W W ðv1; v3Þ¼ ðv1; v2Þ ðv2; v3Þ: (3.22) where vsource is the affine parameter at emission. The luminosity distance DL is defined from the ratio Hence, the general solution of the Sachs equation in a between the observed flux Fobs and the intrinsic luminosity Swiss-cheese universe can be obtained by multiplying Lsource of the source, so that Wronski matrices, according to ¼ 2 W W ð1Þ W ð1Þ ð1Þ Lsource 4 DLFobs: (3.15) ðvsource;0Þ¼ FLðvsource; vin Þ Kðvin ; voutÞ W ð1Þ ð2Þ W ðNÞ It is related to the angular distance by the following FLðvout; vin Þ ... FLðvout ;0Þ distance duality law: (3.23) D ¼ð1 þ zÞ2D : (3.16) W W L A where FL and K are the Wronski matrices computed ðiÞ Hence, the theoretical determination of the luminosity respectively in the FL region and in the Kottler holes; vin ðiÞ distance relies on the computation of the Jacobi matrix. and vout are the values of the affine parameter respectively at the entrance and the exit of the ith hole. C. Solving the Sachs equation piecewise IV. INTEGRATION OF THE GEODESIC Since we work in a Swiss-cheese universe, we have to AND SACHS EQUATIONS compute the Jacobi matrix for a patchwork of spacetimes. It is tempting, in this context, to calculate the Jacobi matrix Consider an observer lying within a FL region, who for each patch independently, and then try to reconnect receives a photon after the latter has crossed a hole. In them. In fact, such an operation is unnatural, because the this section, we determine the light path from entrance to very definition of D imposes that the initial condition is a observation by solving the geodesic equation, and we vertex point of the light beam. Thus, juxtaposing two calculate the Wronski matrix for the Sachs equation. Jacobi matrices is only possible at a vertex point, which The main geometrical quantities are summarized in is of course too restrictive for us. Fig. 4. d is the direction of observation as defined in We can solve this problem by extending the Jacobi Eq. (3.3). The spatial sections of the FL region can be matrix formalism into a richer structure. This requires us described either by comoving spherical coordinates to consider the general solution of Eq. (3.7), for arbitrary ð ; ; ’Þ or, when the spatial sections are Euclidean, by initial conditions. Thus, we have comoving Cartesian coordinates ðX; Y; ZÞ.
123526-6 INTERPRETATION OF THE HUBBLE DIAGRAM IN A ... PHYSICAL REVIEW D 87, 123526 (2013)
µ FL k in out µ in new initial kout conditions θout
Z light propagation ϕ out µ our calculations d ρh
0 Y Kottler X
FIG. 4 (color online). A light ray propagates alternatively in FL and Kottler regions. The main geometrical quantities defined and used in Sec. IV are depicted in this simplified view of a single hole.
2 The hole is characterized by its comoving spatial a0 i k ¼ k0 : (4.3) position Xh, in terms of the FL coordinates, and its mass a M, or equivalently its comoving radius h. Note that, We stress that, in Eq. (4.3), ¼ 0 refers to components on contrary to Sec. II A, it is no longer denoted h, in order to avoid confusion with the radial comoving coordinate of @ , not on @T ¼ @ =a. the center of the hole. A photon enters into the hole with wave vector kin, exits 2. Intersection with the hole from it with wave vector k , and reaches the observer with out Once the geodesic equation has been solved and the wave vector k . We respectively denote E , E and E the 0 in out 0 position of the hole has been chosen, we can calculate associated events. The coordinates of the first two can be the intersection E between the light ray and the hole ðT ;Xi Þ out expressed either with respect to FL, e.g. as in in in boundary. In the particular case of a spatially Euclidean Cartesian coordinates, or with respect to the hole, e.g. as i E FL solution (K ¼ 0), the Cartesian coordinates Xout of out ðtin;rin; in;’inÞ in the Kottler spherical coordinate system. E satisfy the simple system of equations Our calculations go backward in time. Starting from 0, E W E 8 we first determine out, FLðvout; vobsÞ, and second in, < i i j j 2 W ijðXout XhÞðXout XhÞ¼ h Kðvin; voutÞ. The same operations can then be repeated (4.4) E : i i i starting from in and so on. Xout ¼ X0 þð 0 outÞd ;
A. Friedmann-Lemaıˆtre region (from E0 to Eout) i i where Xh and X0 are the respective Cartesian coordinates The geometry of the Friedmann-Lemaıˆtre region is of the hole and the observer, while di is the spatial direction given by the metric (2.1) which can be rewritten in terms of observation. Although conceptually similar, the deter- E of the conformal time , defined by d ¼ dT=aðTÞ,as mination of out for a FL solution with arbitrary spatial curvature is technically harder. d s2 ¼ a2ð Þ½ d 2 þ d 2 þ f2 ð Þd 2 : (4.1) K In general, we deduce from Eq. (4.3) that the wave vector at E is k ¼ða =a Þ2k , where a að Þ. 1. Geodesic equation out out 0 out 0 out out If one chooses the center ¼ 0 of the FL spherical 3. Wronski matrix coordinate system on the worldline of the (comoving) observer, then the geodesic equation is easily solved as In the FL region, the Sachs basis ðs1;s2Þ is defined with respect to the fundamental observers, comoving with four- ð Þ¼ 0 ; ¼ 0;’¼ ’0; (4.2) velocity u ¼ @T. The explicit form of this basis does not which corresponds to a purely radial trajectory. Note need to be specified here. however that for a generic origin, this is no longer true. The Sachs equation can be solved analytically by means The associated wave vector remains collinear to the of a conformal transformation to the static metric observed one, k . It is only subject to a redshift induced 0 2 2 2 2 2 by the cosmic expansion, so that d s~ ¼ a0½ d þ fKð Þd g~ dx dx : (4.5)
123526-7 FLEURY, DUPUY, AND UZAN PHYSICAL REVIEW D 87, 123526 (2013)
Because the geometries associated with g and g~ are to convert its components from the FL coordinate system to conformal, any null geodesic for g affinely parametrized the Kottler one. The result is by v is also a null geodesic for g~ affinely parametrized qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 t aout by v~, with a dv~ ¼ a0dv.Asdv ¼ða =a0Þd , it follows k ¼ k þ 1 Aðr Þk (4.11) out Aðr Þ out out out that v~ ¼ a0 . out For the static geometry, the optical tidal matrix reads qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R~ 2 1 ¼ ðK=a0Þ , so that the Sachs equation is simply r kout ¼ aout 1 AðroutÞkout þ kout (4.12) d2 ~ ¼ K ~: (4.6) 2 d kout ¼ kout (4.13) We then easily obtain the Jacobi and scale matrices: ’ ’ kout ¼ kout: (4.14) D~ 1 C~ 0 1 ¼ a0fKð initÞ ; ¼ fKð initÞ : (4.7) To go back to the original FL spacetime, we use that 2. Shifting to the equatorial plane dv ¼ a2d and the fact that the screen projections of Since the Kottler spacetime is spherically symmetric, it the separation vectors for both geometries are related by is easier to integrate the geodesic equation in the equatorial ~ a ¼ a0 . The final result is plane ¼ =2. In general, however, we must perform E a rotations to bring both out and kout into this plane. D ¼ a f ð Þ1; (4.8) E FL init a K init Starting from arbitrary initial conditions ð out;koutÞ,we 0 can shift to the equatorial plane in two steps. In the R a following, ið#Þ denotes the rotation of angle # about C ¼ ½f0 ð Þ H f ð Þ 1; (4.9) i FL a K init init K init the x -axis. The operations are depicted in Fig. 5. init E E (i) First, bring out to the point out;eq on the equatorial where H a0ð Þ=að Þ is the conformal Hubble function. plane by the action of two successive rotations, W R R This completely determines FL. zð ’outÞ followed by yð =2 outÞ. The wave Note that we can recover the standard expression of 0 vector after the two rotations is denoted kout. the angular distance by taking the initial condition at the (ii) Then, bring k0 to the equatorial plane with observer. The relation (3.14) then implies out Rxð c Þ, where c is the angle between the projec- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 0 D 0 tion of kout on the yz-plane and the y-axis. Note that DA ¼ det FL ¼ fKð sourceÞ; (4.10) E ð1 þ zÞ such a rotation leaves out;eq unchanged. It follows that, after the three rotations where z ¼ a0=a 1 is the redshift of a photon that only travels through a FL region. R ¼ R ð c Þ R R ð ’ Þ; (4.15) x y 2 out z out B. Kottler region (from Eout to Ein) E E out and kout are changed into out;eq and kout;eq which lie in 1. Initial condition at Eout the equatorial plane. In the following, we omit subscripts E ‘‘eq,’’ keeping in mind that we will have to apply R 1 to In the previous section, we have determined out and kout in terms of the FL coordinate system. However, in order to recover the original system of axes. proceed inside the hole, we need to express them in terms of the Kottler coordinate system ðt; r; ; ’Þ. 3. Null geodesics in Kottler geometry E A preliminary task consists in expressing out and kout in In the Kottler region, the existence of two Killing vec- terms of FL spherical coordinates, with origin at the center tors associated to statisticity and spherical symmetry im- of the hole. This operation is straightforward. The event plies the existence of two conserved quantities, the energy E out is then easily converted, since (a) we are free to set E and the angular momentum L of the photon. It follows 1 tout ¼ 0, (b) Eq. (2.15) implies rout ¼ að outÞ h, and that a null geodesic is a solution of (c) the angular coordinates out, ’out remain unchanged dt dr 2 L 2 d’ if the Kottler axes are chosen parallel to the FL ones. AðrÞ ¼ E; þ AðrÞ¼E2;r2 ¼ L: The first junction condition ensures that light is not dv dv r dv deflected when it crosses the boundary of the hole. (4.16) Indeed, the continuity of the metric implies that the connection does not diverge on . Integrating the geodesic 1 þ See e.g. Refs. [70,71] for early works on the propagation of equation dk ¼ k k dv between vout and vout then light rays in spacetimes with a nonvanishing cosmological E shows that k is continuous at out. Therefore, we just need constant.
123526-8 INTERPRETATION OF THE HUBBLE DIAGRAM IN A ... PHYSICAL REVIEW D 87, 123526 (2013) z
(a) µ kout
θout z z out
µ k’ y out ψ ϕ out,eq out (b) y y x out,eq x µ x kout,eq
FIG. 5 (color online). An arbitrary initial condition is rotated so that the geodesic lies in the equatorial plane ¼ =2. Left: Eout is brought (a) to ’ ¼ 0 by the rotation Rzð ’outÞ, and (b) to ¼ =2 by the rotation Ryð =2 outÞ. The resulting event and wave E 0 0 R c vector are denoted out;eq and kout. Middle: kout is brought to the equatorial plane by the rotation xð Þ. Right: Final situation. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Introducing the dimensionless variable u r =r and the sffiffiffiffiffiffiffiffiffiffiffiffiffiffi S " 4GM b2 impact parameter b ¼ L=E, Eqs. (4.16) imply 2 ’1 ¼ 2"1 1 þ 2 ¼ 1 þ (4.22) "1 b 3 du 2 u4 r2 ¼ PðuÞA2ðuÞ; (4.17) at lowest order in " and " . However, we cannot use this S dt "2 1 2 1 expression here—although it gives its typical order of magnitude—because ’1 represents the angle between du 2 ¼ PðuÞ; (4.18) the asymptotic incoming and outgoing directions of a ray, d’ whereas we must take into account the finite extension of the hole (see Fig. 6). r2 du 2 u4 In general, the deflection angle ’ ¼ ’ ’ is S ¼ ð Þ out in 2 2 P u ; (4.19) Z Z E dv "1 um du um du ’ ¼ pffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffi 2 (4.23) with uin PðuÞ uout PðuÞ 2 2 2 where PðuÞ is the polynomial defined in Eq. (4.20), and AðuÞ¼1 u "2u ;PðuÞ "1 u AðuÞ; u is the value of u at minimal approach. The integral (4.20) m involved in Eq. (4.23) can be rewritten as 2 2 3 and where "1 rS=b and "2 r =3. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S Z 0 um du 2 4 u2 u u2 u15 Our purpose is now to compute the coordinates pffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F arcsin ; ; 0 ðtin;rin;’inÞ and the components kin of the wave vector at u Pðu Þ u3 u2 u2 u1 u2 u3 E E the entrance event in, given those at out. The situation is summarized in Fig. 6. (4.24) The radius rin (or alternatively uin) and time tin at entrance are determined by comparing the radial dynamics y of the photon, governed by Eq. (4.17), to the one of the hole boundary. The latter is obtained from Eqs. (2.14) and (2.18). rout By introducing uh ¼ rS=rh, it reads qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi duh 2 µ rS ¼ uhAðuhÞ 1 AðuhÞ: (4.21) ϕ kout dt in rin out 2 x Equations (4.17) and (4.21) are then integrated as tphotonðuÞ and tholeðuhÞ. The entrance radius then results from solving numerically the equation tphotonðuinÞ¼tholeðuinÞ, which also provides t . in in µ The usual textbook calculation of the deflection angle kin ’1 of a light ray in Kottler geometry yields
2The integration can be performed either numerically, or FIG. 6 (color online). Null geodesic in the Kottler region. analytically in the case of Eq. (4.17) and perturbatively for Depicted with the Kottler coordinate system, the hole grows so Eq. (4.21). that the ray enters with rin and exits with rout >rin.
123526-9 FLEURY, DUPUY, AND UZAN PHYSICAL REVIEW D 87, 123526 (2013) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where u
where the normalization function is 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4. Final conditions at Ein r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2AðrÞ @ A: The last step consists in coming back to the original FL N 1 1 AðrÞ 1 2 (4.34) coordinate system. That means (a) using R 1 to recover the bAðrÞ r initial system of axes, and (b) converting the components Using the Sachs basis defined by Eqs. (4.30), (4.31), (4.32), of E and k in terms of the FL coordinate system. We in in and (4.33), we can finally compute the optical tidal matrix, have already described such operations in Secs. IV B 2 and and get IV B 1 respectively, except for the time coordinate ! (since we set t ¼ 0). RðrÞ 0 out R ¼ ; (4.35) The easiest way to compute the cosmic time Tin at 0 RðrÞ entrance is to use the relation rin ¼ aðTinÞfKð hÞ.Ina spatially Euclidean FL spacetime (K ¼ 0), we get where the function RðrÞ is 2 3 2 5 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 L rS 3=2 RðrÞ : (4.36) 2 4 rin 5 2 r2 r Tin ¼ pffiffiffiffiffiffiffiffi argsinh : (4.29) S 3H 1 a0 h 0 As expected from the general decomposition (3.8), R is With this last result, we have completely determined the trace free because only Weyl focusing is at work. Let us entrance event E . finally emphasize that does not appear in the expression in (4.36)ofRðrÞ, which is not surprising since a pure cosmological constant does not deflect light. 5. Sachs basis and optical tidal matrix Once the geodesic equation is completely solved, we are 6. Wronski matrix ready to integrate the Sachs equation in the Kottler region, W The Sachs equations can now be integrated in order to that is, to determine the Wronski matrix K. Such a task C D ð Þ determine the scale matrix K and the Jacobi matrix K requires us to first define the Sachs basis s1;s2 with W W that compose the Wronski matrix K. respect to which K will be calculated. R The four-velocity u is chosen to be the one of a radially First, since is diagonal, the Sachs equations (3.7) only free-falling observer, consist of the following two decoupled ordinary differen- tial equations: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 u @t þ 1 AðrÞ@r: (4.30) d 1 R AðrÞ ¼ ½rðvÞ 1ðvÞ; (4.37) dv2 This choice ensures the continuity of u through the hole 2 frontier, where u ¼ @ . The wave four-vector k is imposed d 2 R T ¼þ ½rðvÞ 2ðvÞ: (4.38) by the null geodesic equations, and reads dv2
123526-10 INTERPRETATION OF THE HUBBLE DIAGRAM IN A ... PHYSICAL REVIEW D 87, 123526 (2013) Clearly, the decoupling implies that the off-diagonal terms OneHole which takes, as input, the observation conditions C D E of K and K vanish, and the properties of the hole; and returns in, kin and W ðv ; v Þ¼W ðv ; v Þ W ðv ; v Þ.For C K ¼ CK ¼ DK ¼ DK ¼ source obs K in out FL out obs 12 21 12 21 0: (4.39) simplicity, this program has been written assuming that The calculation of the diagonal coefficients requires the FL region has Euclidean spatial sections (K ¼ 0). us to integrate Eqs. (4.37) and (4.38). This cannot be Iterating OneHole allows us to propagate a light signal performed analytically because there is no exact expres- back to an arbitrary emission event. Eventually, the redshift sion for r as a function of v along the null geodesic. Indeed, z is obtained by comparing the wave vector at emission we can write v as a function of r from Eq. (4.19) but this and reception; and the luminosity distance is extracted D relation is not invertible by hand. from the block ðvsource; vobsÞ of the Wronski matrix W Nevertheless, we are able to perform the integration ðvsource; vobsÞ, according to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi perturbatively in the regime where "2="1 "1 1, 2 D the relevance of which shall be justified by the orders of DL ¼ð1 þ zÞ det ðvsource; vobsÞ: (4.45) magnitude discussed in the next section. Solving Eq. (4.19) Note finally that, when iterating OneHole, we must also at leading order in "1, "2 leads to rotate the Sachs basis ðs1;s2Þ, to take into account that the " " uðvÞ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ O "2; 2 (4.40) plane of motion differs for two successive holes. 2 2 1 " 1 þðv vmÞ = v 1 V. EFFECT OF ONE HOLE with v b=E, and where vm denotes the value of the affine parameter v at the point of minimal approach. Our method is first applied to a Swiss cheese with a Equation (4.37) then becomes, at leading order in "1, "2, single hole. The purpose is to study the effects on the and using the dimensionless variable w ðv vmÞ= v, redshift and luminosity distance—for the light emitted by d2 3" 1 5=2 a standard candle—due to the presence of the hole. 1 ¼ 1 : (4.41) dw2 2 1 þ w2 1 A. Numerical values and ‘‘opacity’’ assumption The perturbative resolution of Eq. (4.41) from vinit to v finally leads to The mass M of the clump inside the hole depends on what object it is supposed to model. The choice must be 3" CK 1 0 driven by the typical scales probed by the light beams 11 ¼ 1 ½ B ðwinitÞðw winitÞþBðwÞ BðwinitÞ 2 involved in supernova observations. As discussed in the " introduction the typical width of such beams is AU; for þ O 2 2 "1; ; (4.42) comparison the typical interstellar distance within a galaxy "1 is pc. Hence, SN beams are sensitive to the very fine and structure of the Universe, including the internal content of 3" galaxies. This suggests that the clump inside the hole DK ¼ðv v Þþ 1 vfw ½BðwÞ Bðw Þ 11 init 2 init init should represent a star, so that the natural choice should be M M . Unfortunately, we cannot afford to deal with B0ðw Þðw w Þ ½CðwÞ Cðw Þ init init init such a fine description, for numerical reasons. " 0 O 2 2 Instead, the clump is chosen to stand for a gravitation- C ðwinitÞðw winitÞ g þ "1; ; (4.43) 11 "1 ally bound system, such as a galaxy (M 10 M ), or a cluster of galaxies (M 1015M ). By virtue of Eq. (2.19), where the functions B and C are given by the corresponding hole radii are respectively rh 1 Mpc 2 and r 20 Mpc. It is important to note that this choice 1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w h BðwÞ p and CðwÞ p : (4.44) keeps entirely relevant as far as the light beam does not 3 1 þ w2 3 1 þ w2 enter the clump (so that its internal structure does not CK DK The expressions of 22 and 22 are respectively obtained matter), that is, as long as from Eqs. (4.42) and (4.43) by turning "1 into "1. b>bmin rphys; (5.1) Note that in the limit "1, "2="1 ! 0, i.e. b !1and ¼ C ¼ 1 D ¼ð Þ1 0, we find and v vinit , which are the where r is the physical size of the clump. For a galaxy expected expressions in Minkowski spacetime. phys rphys 10 kpc, and for a cluster rphys 1 Mpc. We choose to work under the assumption of Eq. (5.1); in other words C. Practical implementation we proceed as if the clumps were opaque spheres. This section has described the complete resolution of the In the case of galactic clumps this ‘‘opacity’’ assumption equations for light propagation in a Swiss-cheese universe. can be justified by the three following arguments (in the All the results are included in a Mathematica program case of clusters, however, it is highly questionable).
123526-11 FLEURY, DUPUY, AND UZAN PHYSICAL REVIEW D 87, 123526 (2013) The effect of the cosmological constant will be studied in detail in Sec. VII. The value of the Hubble parameter is FL Kottler fixed to H0 ¼ h 100 km=s=Mpc, with h ¼ 0:72.
B. Setup rphys <
(FL) zsource TABLE I. Typical orders of magnitude for galaxylike (FL) 11 15 (M 10 M ) and clusterlike (M 10 M )Swiss-cheeseholes. DL
Type rS (pc) rphys (kpc) rh (Mpc) "1 "2 Galaxy 10 2 10 1 10 8–10 6 10 23 Cluster 100 1000 20 10 6–10 4 10 15 FIG. 8 (color online). Setup for evaluating the effect of one hole on the redshift and luminosity distance.
123526-12 INTERPRETATION OF THE HUBBLE DIAGRAM IN A ... PHYSICAL REVIEW D 87, 123526 (2013)
8 8 7 7 10 10 6 6 z 0.1 z FL h source FL z 0.1 z z 4 z 4 h source zh 0.5 z source z z z 0.5 z z z 0.9 z z h source h source 2 zh 0.9 z source 2
0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 min max min
14 14 12 12
7 10 7
10 10 10 8 8 6 15 z M 10 M 15 6 z
z M 10 M z 15 z 4 M 2 10 M z 4 M 2 1015 M 2 M 3 1015 M 2 M 3 1015 M 0 0 0.0 0.2 0.4 0.6 0.8 1.0
0.00 0.05 0.10 0.15 min max min
FIG. 10 (color online). Same as Fig. 9, but plotted in FIG. 9 (color online). Relative correction to the redshift z, due terms of the centered and normalized observation direction to the hole in the line of sight, as a function of the direction of ð min Þ=ð max min Þ. observation , for a source at zsource ¼ 0:05. Top panel: The 15 mass of the hole is M ¼ 10 M , and three positions between ðFLÞ Ref. [1]). As the boundary of the hole grows with time the source and the observer are tested, zh =zsource ¼ 0:1 (blue, dot-dashed), 0.5 (purple, dashed), and 0.9 (red, solid). (see Fig. 6), the light signal undergoes a stronger gravita- ðFLÞ tional potential at entrance than at exit. That induces a Bottom panel: The hole is at zh ¼ 0:5zsource and three values 15 for the mass are tested, M=10 M ¼ 3 (blue, dot-dashed), gravitational redshift zgrav which adds to the cosmological 2 (purple, dashed), and 1 (red, solid). one, and reads t ð Þ zsource ¼ 0:05 and various hole positions and masses. kin A rout 15 1 þ zgrav ¼ t ¼ : (5.4) We have chosen M 10 M because the effect is more kout AðrinÞ significant and displays fewer numerical artifacts than for 11 z M 10 M . The order of magnitude of grav can be evaluated as We only consider directions of observation such that the follows. Let r ¼ rout rin be the increase of the radius of the hole between entrance and exit. The expansion dynam- light beam crosses the hole. Thus, min < < max pffiffiffiffiffi where min and max depend on the physical cutoff ics implies r "1 t, where t ¼ tout tin rin, rout rphys, the radius rh of the hole, and its distance to the is the time spent by the photon inside the hole. Using ðFLÞ Eq. (5.4), we conclude that observer zh . Those dependences can be eliminated by plotting z as a function of ð Þ=ð Þ min max min 3=2 instead of , as displayed in Fig. 10. zgrav "1 : (5.5) As expected, z tends to zero when approaches max For M ¼ 1015M (clusterlike hole), the numerical values (light ray tangent to the hole boundary). We notice that z given in Table I yield z 10 6. This order of does not significantly depend on the distance between the grav;max observer and the hole. However, the effect clearly grows magnitude is compatible with the full numerical integra- with the mass of the hole. tion displayed in Figs. 9 and 10. Such an analytical estimate enables us to understand why z increases with M, that is, with the size of the 2. Analytical estimation of the effect hole. Indeed, the bigger the hole, the longer the photon The correction in redshift due to hole can be understood travel time so that the hole has more time to grow, and as an integrated Sachs-Wolfe effect (see e.g. Chapter 7 of finally AðroutÞ AðrinÞ is larger.
123526-13 FLEURY, DUPUY, AND UZAN PHYSICAL REVIEW D 87, 123526 (2013) D. Corrections to the luminosity distance 10 The effect of the hole on the luminosity distance can be 5 8 characterized in a similar way by 10 zh 0.1 z source z 0.5 z 6 h source
ðFLÞ FL D D zh 0.9 z source L L L FL
DL : (5.6) D ðFLÞ L 4
DL D L D The associated results, in the same conditions as in the 2 previous paragraph, are displayed in Figs. 11 and 12. 0 We notice that DL is maximum if the hole lies halfway 0.0 0.2 0.4 0.6 0.8 1.0 between the source and the observer, which is indeed min max min expected since the lensing effects scale as 14 D ðobserver; lensÞ D ðlens; sourceÞ A A ; (5.7) 12 D ðobserver; sourceÞ 5
A 10 10 which typically peaks for z z =2. The maximal 8 lens source FL 15 4 L amplitude of the correction is of order 10 , for masses FL M 10 M D 6 L 15 15 15 ranging from 10 M to 3 10 M . Just as for the red- D L M 2 10 M 4 shift, the effect increases with the size of the hole. D M 3 1015 M 2 10 0 0.0 0.2 0.4 0.6 0.8 1.0
5 min max min 8 z h 0.1 z source 10 z h 0.5 z source 6 FIG. 12 (color online). Same as Fig. 11, but plotted in z h 0.9 z source FL terms of the centered and normalized observation direction L FL
D ð Þ=ð Þ. L 4 min max min D L D 2 Note that DL can be related to the relative magnifica- tion , frequently used in the weak-lensing formalism, and 0 defined by 0.0 0.1 0.2 0.3 0.4 0.5 0.6 rad DðFLÞ 2 1 þ z 4 DðFLÞ 2 A ¼ L : (5.8) ðFLÞ DA 1 þ z DL 14 Hence, if the correction on z is negligible compared to the 12 5 one of DA, then the relation between DL and is 10 10 1 D pffiffiffiffi 1: (5.9) 8 L FL
L 15 FL M 10 M D 6 L
D 15 L M 2 10 M E. Summary
D 4 M 3 1015 M The presence of a single hole between the source and the 2 observer induces both a correction in redshift and luminos- 0 15 0.00 0.05 0.10 0.15 ity distance. For a hole with mass M 10 M , the rela- 7 6 rad tive amplitudes of those corrections are z 10 –10 11 and DL 100 z. The same study for M 10 M leads FIG. 11 (color online). Relative correction to the luminosity to similar results with z 10 10–10 9. Therefore, the distance DL, due to the hole in the line of sight, as a function of effects of a single hole seem negligible. the direction of observation , for a source at zsource ¼ 0:05.Top 15 panel: The mass of the hole is M ¼ 10 M , and three positions ðFLÞ VI. EFFECT OF SEVERAL HOLES between the source and the observer are tested, zh =zsource ¼ 0:1 (blue, dot-dashed), 0.5 (purple, dashed), and 0.9 (red, solid). We now investigate a Swiss-cheese model containing ðFLÞ many holes arranged on a regular lattice. Again, in this Bottom panel: The hole is at zh ¼ 0:5zsource and three values 15 for the mass are tested, M=10 M ¼ 3 (blue, dot-dashed), 2 entire section, the cosmological parameters characterizing (purple, dashed), and 1 (red, solid). the FL region are the EdS ones.
123526-14 INTERPRETATION OF THE HUBBLE DIAGRAM IN A ... PHYSICAL REVIEW D 87, 123526 (2013) A. Description of the arrangement of holes the influence of (a) the distance between the source and the 1. Smoothness parameter observer, (b) the smoothness parameter f, and (c) the mass M of the holes. The smoothness of the distribution of matter within a Swiss cheese can be quantified by a parameter f con- structed as follows. Choose a region of space with— 1. Setup comoving or physical—volume V, where V1=3 is large After having chosen the parameters ðf; MÞ of the model, compared to the typical distance between two holes. we arbitrarily choose the spatial position of the observer in Thus, this volume contains many holes, the total volume the FL region, and fix its direction of observation. The of which is Vholes, while the region left with homogeneous method is then identical to the one of Sec. V. The light matter occupies a volume VFL ¼ V Vholes. We define the beam is propagated from the observer until the redshift smoothness parameter by reaches the one of the source, z. The ending point defines the emission event E . We emphasize that only emis- V source f lim FL : (6.1) sion events occurring in the FL region are considered in V!1 V this article. In particular, f ¼ 1 corresponds to a Swiss cheese with no hole—that is, perfectly smooth—while f ¼ 0 corresponds 2. Influence of the smoothness parameter to the case where matter is under the form of clumps. Of In this paragraph, the mass of every hole is fixed to course, f also characterizes the ratio between the energy M ¼ 1011M (galactic holes). The relative corrections to density of the continuous matter and the mean energy the redshift z and luminosity distance D , as functions density. L of the redshift z of the source, have been computed and are displayed in Fig. 14 for different values of the smoothness 2. Lattice parameter f. We want to construct a Swiss cheese for which the smoothness parameter is as small as possible. If all holes are identical, this close-packing problem can be solved by 1.0 f 0.26 using, for instance, a hexagonal lattice. The corresponding 0.8 arrangement is pictured in Fig. 13. The minimal value of 5 f 0.6 10 f 0.9 the smoothness parameter is in this case 0.6 ffiffiffi fmin ¼ 1 p 0:26: (6.2) z 0.4 3 2 z z In order to reach a smoothness parameter smaller than 0.2 fmin , one would have to insert a second family of smaller holes. By iterating the process, one can in principle make f 0.0 0.0 0.5 1.0 1.5 2.0 as close as one wants to zero. z
B. Observations in a unique line of sight 15 We now focus on the corrections to the redshift and f 0.26 luminosity distance of a source whose light travels through f 0.6 the Swiss-cheese universe described previously. We study 10 f 0.9 FL L FL D L D L 5 D
0 0.0 0.5 1.0 1.5 2.0 z
FIG. 14 (color online). Relative corrections to the redshift z (top panel) and luminosity distance DL (bottom panel) as func- tions of z, for an arbitrary light beam traveling through a Swiss-cheese universe. All holes are identical; their mass is 11 FIG. 13 (color online). Hexagonal lattice of identical holes. On M ¼ 10 M . Three different smoothness parameters are tested: the left, the arrangement is close-packed, so that the smoothness f ¼ 0:26 (blue, dot-dashed), 0.6 (purple, dashed), and 0.9 parameter is f ¼ fmin 0:26. On the right, f ¼ 0:7. (red, solid).
123526-15 FLEURY, DUPUY, AND UZAN PHYSICAL REVIEW D 87, 123526 (2013)
1.0 Refs. [57,61], such a restrictive analysis can lead to over- estimate the mean corrections induced by inhomogeneities. 0.8 5 Besides, as stressed by Ref. [8], the dispersion of the data is
10 crucial for interpreting SN observations. Hence, the 0.6 conclusions of the previous subsection need to be com- M 10 11 M pleted by a statistical study, with randomized directions of z 0.4 z 11 observation. z M 2 10 M 0.2 11 Since the effect on the redshift is observationally negli- M 3 10 M gible, we focus on the luminosity distance. After having set 0.0 the parameters ðf; MÞ of the model, we fix the position of 0.0 0.5 1.0 1.5 2.0 the observer in the FL region. Then, for a given redshift z, z we consider a statistical sample of Nobs randomly distrib- uted directions of observation d~ 2 S2, and compute M 10 11 M D ðz; d~Þ for each one. 15 L M 2 10 11 M Figure 16 shows the probability distribution of DL for sources at redshifts z ¼ 0:1 (top panel) and z ¼ 1 (bottom M 3 10 11 M
FL 10 panel). We compare two Swiss-cheese models with the L FL D same smoothness parameter f ¼ fmin but with different L 11 D L values for the masses of their holes (M ¼ 10 M and D 5 15 10 M ). The histograms of Fig. 16 are generated from statistical samples which contain Nobs ¼ 200 directions of 0 observation each. 0.0 0.5 1.0 1.5 2.0 z
FIG. 15 (color online). Same as Fig. 14 but with f ¼ 0:26 11 0.30 and three different values for the masses: M=10 M ¼ 1 z 0.1 (blue, dot-dashed), 2 (purple, dashed), and 3 (red, solid). 0.25
0.20 While the corrections to the redshift remain small— typically z < 10 5—the cumulative effect of lensing on 0.15 Probability the luminosity distance is significant. For instance, a source 0.10 at z 1:5 would appear 10% farther in a Swiss cheese with 0.05 f ¼ 0:26, than in a strictly homogeneous universe. Both z 0.00 and DL increase with z and decrease with f, as intuitively 0.08 0.10 0.12 0.14 0.16 0.18 0.20 expected. Thus, the more holes, the stronger the effect. As D D FL D FL examples, the light beam crosses 300 holes for (f ¼ 0:26, L L L z ¼ 0:1)or(f ¼ 0:9, z ¼ 1), but it crosses 2000 holes for (f ¼ f , z ¼ 1). 0.25 min z 1 3. Influence of the mass of the holes 0.20 We now set the smoothness parameter to its minimal 0.15 value fmin 0:26, and repeat the previous analysis for various hole masses. The results are displayed in Fig. 15. Probability 0.10 We conclude that neither z nor D depends signifi- L 0.05 cantly on M, that is, on the size of the holes. Thus, what actually matters is not the number of holes intersected by 0.00 the beam, but rather the total time spent inside holes. 4 5 6 7 8 9 10 11 FL FL DL D L DL C. Statistical study for random directions of observation FIG. 16 (color online). Probability distribution of the relative correction to the luminosity distance for random directions of The previous study was restricted to a single line of sight, observation, at z ¼ 0:1 (top panel) and z ¼ 1 (bottom panel). but since a Swiss-cheese universe is not strictly homoge- The smoothness parameter is f ¼ fmin and two different values 11 neous, the corrections to z and DL are expected to vary from for the masses of the holes are tested: M ¼ 10 M (blue, solid) 15 one line of sight to another. As pointed out by e.g. and M ¼ 10 M (purple, dashed).
123526-16 INTERPRETATION OF THE HUBBLE DIAGRAM IN A ... PHYSICAL REVIEW D 87, 123526 (2013) a Swiss-cheese universe. The effect increases with z and 8 decreases with f. Our results differ from those obtained in Swiss-cheese 6 models with LTB solutions inside the holes. In the latter FL case, a source can be either demagnified if light mostly L FL D
L 4 propagates through underdense regions [54,55,59] (and if D L the observer is far away from a void, see Ref. [76]), or D magnified otherwise. It has been proven in Refs. [58,61,62] 2 that the global effect averages to zero when many sources are considered. Hence, LTB holes introduce an additional 0 0.0 0.2 0.4 0.6 0.8 1.0 dispersion to the Hubble diagram, but no statistically sig- z nificant bias. On the contrary, in the present study, light only propagates through underdense regions, because we FIG. 17 (color online). Evolution, with redshift z, of the rela- only consider light beams which remain far from the hole tive correction to the luminosity distance averaged over Nobs ¼ 200 random directions of observation. Error bars indicate the centers. This assumption has been justified in Sec. VA by h i an ‘‘opacity’’ argument. The bias displayed by our results dispersion DL around the mean correction DL .Asin 11 Fig. 16, we compare Swiss-cheese models with M ¼ 10 M is mostly due to the selection of the light beams which can 15 (blue, filled markers) and M ¼ 10 M (purple, empty markers). be considered observationally relevant. Our results also differ qualitatively from those obtained in the framework of the perturbation theory. In Ref. [31], From the statistical samples, we can compute the mean the probability density function Pð Þ of the weak lensing correction h D iðzÞ and its standard deviation ðzÞ, L DL magnification , due to the large scale structure, has whose evolutions are plotted in Fig. 17. been analytically calculated by assuming an initial The results displayed in Fig. 17 confirm the conclusions power spectrum with slope n ¼ 2. Just as for LTB of Sec. VI B. The distance-redshift relation in a Swiss Swiss-cheese models, the magnification shows no intrin- cheese is biased with respect to the one of a purely homo- sic bias (i.e. h i¼1),butitisshownthatPð Þ peaks at genous universe. This effect is statistically significant, we a value slightly smaller than 1. Hence, a bias of indeed estimate (empirically) that peak order peak h i, which is typically 1% at z ¼ 1,can h ið Þ ð Þ DL z 8 DL z : (6.3) emerge from observations because of insufficient statis- tics. However, this bias is far smaller than the one The bias slightly decreases with the mass parameter M. obtained in our Swiss-cheese model, of order 2 DL However, it can be considered quite robust because a 15% at z ¼ 1. variation of 4 orders of magnitude for M only induces a Besides, the dispersion around the mean magnification is variation of 10% for the bias. stronger for perturbation theory ( 10%) than for both The intrinsic dispersion of D , associated with , L DL LTB and Kottler Swiss-cheese models ( 2%). can be compared with the typical dispersion of the obser- vation. For instance, at z ¼ 1 the former is 1%, while the latter is estimated to be typically 10% [75]. It follows VII. COSMOLOGICAL CONSEQUENCES that the dispersion induced by the inhomogeneity of the Since the Hubble diagram is modified by the presence of distribution of matter remains small compared to the inhomogeneities, the resulting determination of the cos- observational dispersion. mological parameters must be affected as well. More precisely, consider a Swiss-cheese universe whose D. Summary and discussion FL regions are characterized by a set of cosmological This section has provided a complete study of the effect parameters ð m; K; Þ, called background parameters of inhomogeneities on the Hubble diagram, investigating in the following. If an astronomer observes SNe in this both the corrections to the redshift and luminosity distance inhomogeneous universe and constructs the resulting of standard candles. The Swiss-cheese models are made of Hubble diagram, but fits it with the usual FL luminosity- identical holes, defined by their mass M, and arranged on a redshift relation—that is, assuming that he lives in a strictly regular hexagonal lattice. The fraction of matter remaining homogeneous universe—then he will infer apparent cos- in FL regions defines the smoothness parameter f. For the mological parameters ð m; K; Þ which shall differ hexagonal lattice, fmin 0:26. from the background ones. Evaluating this difference is The effect on the redshift is negligible ( z < 10 5), the purpose of Secs. VII A, VII B, and VII C. while the correction to the luminosity distance is signifi- The natural question which comes after is, assuming that cant ( DL > 10% at high redshift). Compared to the ho- our own Universe is well described by a Swiss-cheese mogeneous case, sources are systematically demagnified in model, what are the background cosmological parameters
123526-17 FLEURY, DUPUY, AND UZAN PHYSICAL REVIEW D 87, 123526 (2013) 10 X472 ðz j ; Þ 2 2ð ; Þ i FL i m ; (7.1) CDM m 8 i¼1 i SC EdS where no longer denotes the magnification, but rather the 6 distance modulus associated with DL, so that Gpc L D D 4 5log L : (7.2) 10 10 pc 2 In Eq. (7.1), ðzi; iÞ is the ith observation of the simulated catalog. In order to make the analysis more realistic, we 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 have attributed to each data point an observational error bar z i estimated by comparison with the SNLS 3 data set [77]. Besides, ðzj ; Þ is the theoretical distance FIG. 18 (color online). Hubble diagram of a Swiss-cheese FL m 11 modulus of a source at redshift z, in a FL universe with universe (dots) with f ¼ fmin , M ¼ 10 M and EdS back- ground cosmology. For comparison, we also display the cosmological parameters m, , K ¼ 1 m . distance-redshift relations of purely FL universes, with EdS The results of this analysis for two mock Hubble dia- parameters (blue, solid) and ð ; ; Þ¼ð0:3; 0; 0:7Þ grams are shown in Fig. 19. An EdS background leads to m K (black, dashed). apparent parameters ð m; K; Þ¼ð0:5; 0:8; 0:3Þ, which are very different from (1,0,0). Thus, the positive shift of DLðzÞ—clearly displayed in Fig. 18—turns out to be mostly associated to an apparent spatial curvature, which best reproduce the actual SN observations? This rather than to an apparent cosmological constant. In this issue is addressed in Sec. VII D. case the apparent curvature is necessary to obtain a good fit ( ¼ 0 is out of the 2 confidence contour), because a A. Generating mock Hubble diagrams K spatially flat FL model does not allow us to reproduce both The Hubble diagram observed in a given Swiss-cheese the low-z and high-z behaviors of the diagram. The effect is universe is constructed in the following way. We first weaker for a background with ð m; Þ¼ð0:3; 0:7Þ, choose the parameters of the model: f, M, and the 3 which leads to ð m; K; Þ¼ð0:2; 0:2; 0:6Þ. background cosmology ( m, ¼ 1 m). We then fix arbitrarily the position of the observer in the FL region, and we simulate observations by picking randomly the 1.0 ~ line of sight d, the redshift z 2½0;zmax , and we compute ~ the associated luminosity distance DLðz; dÞ as in Sec. VI. In order to make our mock SNe catalog resemble the SNLS 3 data set [77], we choose z ¼ 1:4 and 0.5 K max 0 Nobs ¼ 472. An example of mock Hubble diagram, corresponding to 11 a Swiss-cheese model with f ¼ fmin , M ¼ 10 M and ð m; K; Þ¼ð1; 0; 0Þ is plotted in Fig. 18. As a com- 0.0 parison, we also displayed DLðzÞ for a homogeneous uni- verse with (1) the same cosmological parameters, and (2) with ð m; K; Þ¼ð0:3; 0; 0:7Þ. 0.5 B. Determining apparent cosmological parameters The apparent cosmological parameters m, and 0.0 0.2 0.4 0.6 0.8 1.0 K ¼ 1 m are determined from the mock Hubble diagrams by performing a 2 fit. The 2 is m defined by FIG. 19 (color online). Comparison between background parameters (crosses) and apparent parameters (disks) for two 15 Swiss-cheese models with f ¼ fmin and M ¼ 10 M . In blue, 3 ð m; Þ¼ð1; 0Þ leads to ð m; Þ¼ð0:5; 0:3Þ. In black, Recall that in the practical implementation of the theoretical results (see Sec. IV C), we assumed that K ¼ 0, so that the ð m; Þ¼ð0:3; 0:7Þ leads to ð m; Þ¼ð0:2; 0:6Þ. The 1 background cosmology of our Swiss-cheese models is com- and 2 contours are respectively the solid and dashed ellipses. pletely determined by or . Nevertheless, the apparent The solid straight line indicates the configurations with zero m curvature parameter K is a priori nonzero. spatial curvature.
123526-18 INTERPRETATION OF THE HUBBLE DIAGRAM IN A ... PHYSICAL REVIEW D 87, 123526 (2013) 1.0 ones, the apparent expansion history is almost the same— 0.8 at second order—as the background one. m Note that the results displayed in Figs. 20 and 21 are 0.6 consistent with each other. The apparent cosmological 15 0.4 constant is slightly smaller for M ¼ 10 M than for ¼ 11 K M 10 M , so that q is slightly larger. 0.2
0.0 2. Influence of the background cosmological constant Now consider a Swiss-cheese model with f ¼ fmin and 0.2 change its background cosmology. Figure 22 shows the 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 evolution of the apparent cosmological parameters versus Smoothness f the background cosmological constant . As it could have already been suspected from Fig. 19, the difference FIG. 20 (color online). Apparent cosmic parameters m (gray between apparent and background parameters decreases disks), K (red squares) and m (black diamonds) versus with , and vanishes in a de Sitter universe. This can smoothness parameter f, for a Swiss-cheese universe with EdS be understood as follows. The construction of a Swiss- background ð m; K; Þ¼ð1; 0; 0Þ. Solid lines and filled 11 cheese universe consists in changing the spatial distribu- markers correspond to M ¼ 10 M , dashed lines and empty markers to M ¼ 1015M . tion of the pressureless matter, while the cosmological constant remains purely homogeneous. Thus, the geometry of spacetime is less affected by the presence of inhomoge- C. Quantitative results neities if = m is greater. In the extreme case 1. Influence of the smoothness parameter ð m; Þ¼ð0; 1Þ, any Swiss cheese is identical to its background, since there is no matter to be reorganized. Consider a Swiss-cheese model with EdS background We also plot in Fig. 23 the difference between the cosmology. Figure 20 shows the evolution of the apparent apparent deceleration parameter q and the background cosmological parameters with smoothness f. As expected, one q ¼ =2 , as a function of q. Again, q does we recover ¼ when f ¼ 1, the discrepancy between m i i not significantly differ from q. This result must be com- background and apparent cosmological parameters being pared with Fig. 11 of Ref. [53], where ðq qÞ=q 100%. maximal when f ¼ fmin . Surprisingly, a Swiss-cheese universe seems progressively dominated by a negative spatial curvature for small values of f. 3. Comparison with other recent studies The apparent deceleration parameter q ¼ m=2 The impact of a modified luminosity-redshift relation— is plotted in Fig. 21 as a function of f. Interestingly, even due to inhomogeneities—on the cosmological parameters for f ¼ fmin , q remains almost equal to its background has already been investigated by several authors. In value q ¼ 1=2. Therefore, though the apparent cosmologi- Ref. [55], it has been suggested that a Swiss-cheese model cal parameters can strongly differ from the background
0.6
0.54 0.4 K K 0.2 0.52 0.0 q 0.50 0.2
m m 0.48 0.4
0.46 0.2 0.4 0.6 0.8 1.0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Smoothness f FIG. 22 (color online). Difference between apparent and back- FIG. 21 (color online). Apparent deceleration parameter q as a ground cosmological parameters m m (gray disks), function of smoothness parameter f, for Swiss-cheese models K K (red squares) and (black diamonds) versus with EdS background (q ¼ 1=2). Solid lines and filled markers background , for Swiss-cheese models with f ¼ fmin . Solid 11 11 correspond to M ¼ 10 M , dashed lines and empty markers to lines and filled markers correspond to M ¼ 10 M , dashed 15 15 M ¼ 10 M . lines and empty markers to M ¼ 10 M .
123526-19 FLEURY, DUPUY, AND UZAN PHYSICAL REVIEW D 87, 123526 (2013) parameter b much larger than the Schwarzschild radius r 0.04 S of the central object. Moreover, since the cosmological 0.02 constant has no effect on light focusing, we conclude that inside a hole, the evolution of the cross-sectional area of a 0.00 light beam behaves essentially as in Minkowski spacetime. q This conclusion is supported by the perturbative calcula-
q 0.02 W tion of the Wronski matrix K performed in Sec. IV B 6. 0.04 If both the observer and the source are located on the surfacep offfiffiffiffiffiffiffiffiffiffiffiffiffi a hole, their angular distance is therefore hole D 0.06 DA ¼ det vout vin, where v denotes the affine parameter. More generally, for a beam which crosses N 1.0 0.8 0.6 0.4 0.2 contiguous holes, we get
XN FIG. 23 (color online). Difference between apparent and back- holes DA vi; (7.3) ground deceleration parameters q q as a function of q, for i¼1 Swiss-cheese models with f ¼ fmin . Solid lines and filled 11 v v v markers correspond to M ¼ 10 M , dashed lines and empty where i out;i in;i is the variation of the affine 15 markers to M ¼ 10 M . parameter between entrance into and exit from the i th hole. Let us now evaluate vi. The time part of the with LTB holes and EdS background displays an apparent geodesic equation in Kottler geometry yields cosmological constant ¼ 0:4; but as already dt E r rS mentioned in Sec. VI D, such a claim was proven to be kt ¼ E ¼ constant; (7.4) dv AðrÞ inaccurate in Refs. [57,58,62], because it relies on obser- vations along a peculiar line of sight. When many random where E is the usual constant of motion. We conclude that directions of observation are taken into account, the mean t vi kout;i ti. Besides, the relations (2.16) and (4.11) magnification goes back to 1. Hence, contrary to our between FL and Kottler coordinates on the junction hyper- results, the apparent cosmological parameters of a Swiss- surface, together with AðrhÞ 1, lead to ti Ti and cheese model with LTB holes are identical to the back- t kout aout=a0. Finally, ground ones. This conclusion is in agreement with XN Z 0 Ref. [78], where similar studies are performed in various a Tobs aðT Þ holes out;i 0 cosmological toy models; and also with Ref. [31] in the DA Ti dT ; (7.5) a0 T a0 framework of perturbation theory. i¼1 However, it is crucial to distinguish those approaches where we approximated the sum over i by an integral. This (LTB Swiss-cheese models and perturbation theory) operation is valid as far as Ti remains small compared to from the one adopted in this article, because they do not the Hubble time. In terms of redshifts, we have address the same issue. The former share the purpose of Z z z0 evaluating the influence of inhomogeneities smoothed on holes d DA ðzÞ¼ 0 2 0 : (7.6) large scales, while we focused on smaller scales for which 0 ð1 þ z Þ Hðz Þ matter cannot be considered smoothly distributed. Thus, By construction, this formula describes the behavior of the our results must not be considered different, but rather angular distance when light only travels through Kottler complementary. regions. In order to take the FL regions into account, we SC write the distance-redshift relation DA ðzÞ of the Swiss D. An alternative way to fit the Hubble diagram cheese as the following (heuristic) linear combination Let us now address the converse problem, and determine SC holes FL the background cosmological parameters of the Swiss- DA ðzÞ¼ð1 fÞDA ðzÞþfDA ðzÞ; (7.7) cheese model that best reproduces the actual observations. where f still denotes the smoothness parameter defined in For that purpose, the simplest method would be to fit our FL Sec. VI A 1, and DA ðzÞ is the distance-redshift relation in a observed Hubble diagram using the theoretical luminosity- FL universe, given by Eq. (4.10). DSCðzÞ redshift relation L of a Swiss-cheese universe. Hence, A comparison between the above analytical estimation we need to derive such a relation in order to proceed. and the numerical results is plotted in Fig. 24. The agree- ment is qualitatively good, especially as it is obtained 1. Analytical estimation of the distance-redshift relation without any fitting procedure. Moreover, it is straightfor- of a Swiss-cheese universe SC FL ward to show that DA ðzÞ and DA ðzÞ are identical up to As argued in Sec. V, any observationally relevant second order in z. This is in agreement with—and some- light beam which crosses a Kottler region has an impact how explains—the numerical results of Secs. VII C 1 and
123526-20 INTERPRETATION OF THE HUBBLE DIAGRAM IN A ... PHYSICAL REVIEW D 87, 123526 (2013) d2DSC dlnH 2 dDSC 14 A þ þ A dz2 dz 1 þ z dz 12 3 H 2 10 ¼ m 0 ð1 þ zÞfDFLðzÞ; (7.9) 2 H A FL 8 L FL
D which is similar to Eq. (7.8) with ðzÞ¼f, except that the
L 6 FL SC D L right-hand side is proportional to DA instead of DA . D 4 Nevertheless, it turns out that such a difference has only 2 a very weak impact, in the sense that 0 DSCðzÞ DDRðzÞ; i:e:DSCðzÞ DDRðzÞ; (7.10) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 A A L L z if ðzÞ¼f. This appears clearly in Fig. 24, where the FIG. 24 (color online). Comparison between the approxi- dashed and solid lines are almost superimposed. In fact, it SC 2 SC is not really surprising, since both approaches rely on the mate luminosity-redshift relation DL ðzÞ¼ð1 þ zÞ DA ðzÞ in a Swiss-cheese universe (solid lines), simulated observations same assumptions: no backreaction, no Weyl focusing and DR (dots), and the Dyer-Roeder model DL ðzÞ with ðzÞ¼f an effective reduction of the Ricci focusing. (dashed lines). Three different values of the smoothness parame- Note however that this approach models the effect of the ter are tested, from top to bottom: f ¼ fmin 0:26, f ¼ 0:7, inhomogeneities on the mean value of the luminosity dis- f ¼ 0:9. tance but does not address the dispersion of the data.
VII C 2, where we showed that the apparent deceleration 3. Fitting real data with DSCðzÞ parameter q is the same as the background one q. L SC Note finally that the general tendency of our analytical The modified luminosity-redshift relation DL ðzÞ de- rived in the previous paragraph can be used to fit the relation is to overestimate DL at high redshifts. The main W observed Hubble diagram. We apply the same 2 method reason is that its derivation uses the behavior of K at as described in Sec. VII B, except that now (1) the triplets zeroth order in rS=b; that is, it neglects the effect of the central mass in the Kottler region. The first-order term in ðzi; i; iÞ are observations of the SNLS 3 catalog [77], W K—taken into account in the numerical results—tends and (2) FLðzj m; Þ is replaced by SCðzj m;fÞ, to lower the associated luminosity distance. where the background curvature K is fixed to 0 (so that ¼ 1 m). Hence, we are looking for the smoothness 2. Comparison with the Dyer-Roeder approach parameter f, and the background cosmological parameters, of the spatially Euclidean Swiss-cheese model which best Another widely used approximation to model the propa- fits the actual SN observations. gation of light in underdense regions was proposed by Dyer The results of the 2 fit are displayed in Fig. 25. First of and Roeder [10] in 1972. It assumes that (1) the Sachs all, we note that the confidence areas are very stretched equation and the relation vðzÞ are the same as in a FL spacetime—in particular, the null shear vanishes—and (2) the optical parameter 00 (see Sec. III B 2) is replaced by ðzÞ 00, where ðzÞ represents the fraction of matter intercepted by the geodesic bundle. In brief, the DR model encodes that light propagates mostly in underdense regions by reducing the Ricci focusing, while still neglecting the Weyl focusing. Under such conditions, the DR expression DR of the angular distance DA ðzÞ is determined by d2DDR dlnH 2 dDDR A þ þ A dz2 dz 1 þ z dz 3 H 2 ¼ m 0 ð1 þ zÞ ðzÞDDRðzÞ: (7.8) 2 H A
This attempt to model the average effect of inhomogene- ities, while assuming that the Universe is isotropic and FIG. 25 (color online). Fit of the Hubble diagram constructed homogeneous, has been widely questioned [79–82] and from the SNLS 3 data set [77], by using the luminosity-redshift SC recently argued to be mathematically inconsistent [8]. relation D ðzj m;fÞ of a spatially Euclidean Swiss-cheese SC L Interestingly, our estimation DA ðzÞ of the distance- model. The colored areas indicate (from the darkest to the redshift relation in a Swiss-cheese universe reads lightest) the 1 , 2 and 3 confidence levels.
123526-21 FLEURY, DUPUY, AND UZAN PHYSICAL REVIEW D 87, 123526 (2013) horizontally, so that the smoothness parameter f cannot be simulating Hubble diagrams for various Swiss-cheese reasonably constrained by the Hubble diagram. There are models, and by fitting them with the usual FL two reasons for this. On the one hand, we know from luminosity-redshift relation. In general, the resulting Sec. VII C 2 that f has only a weak influence on the ‘‘apparent’’ cosmological parameters are very different luminosity-redshift relation in a universe dominated by from the ‘‘background’’ ones which govern the cosmic the cosmological constant, which is the case here expansion, but in a way that leaves the deceleration pa- SC ( 0:7–0:8). On the other hand, since DL ðzÞ and rameter unchanged. Moreover, the discrepancy between FL 3 DL ðzÞ only differ by terms of order z and higher, apparent and background cosmological parameters turns one would need more high-redshift observations to out to decrease with , and is therefore small for a universe discriminate them. However, all the current SNe dominated by the cosmological constant. Finally, we have catalogs—including the SNLS 3 data set—contain mostly derived an approximate luminosity-redshift relation for low-redshift SNe. Swiss-cheese models, which is similar to the one obtained Besides, Fig. 25 shows that fixing a given value of f following the Dyer-Roeder approach. Using this relation to changes the best-fit value of m. In the extreme case of a fit the Hubble diagram constructed from the SNLS 3 data Swiss cheese only made of clumps (f ¼ 0) we get set, we have found that the smoothness parameter cannot m ¼ 0:3, while in the FL case (f ¼ 1) the best value is be constrained by such observations. However, turning m ¼ 0:24, in agreement with Ref. [77]. Such a discrep- arbitrarily f ¼ 1 into f ¼ 0 has an impact of order 20% ancy, of order 20%, is significant in the era of precision on the best-fit value of m, which is significant in the era of cosmology, where one aims at determining the cosmologi- precision cosmology (see Ref. [83] for further discussion). cal parameters at the percent level. Of course, our model is oversimplifying for various Let us finally emphasize that such a fit is only indicative, reasons. First, it does not take into account either the because it relies on an approximation of the luminosity- complex distribution of the large scale structures, or the redshift relation in a Swiss-cheese universe. presence of diffuse matter on small scales—such as gas and possibly dark matter. Second, it does not take strong VIII. CONCLUSION lensing effects into account, assuming that clumps are ‘‘opaque.’’ We can conjecture that this overestimates the In this article, we have investigated the effect of the actual effect of the inhomogeneities. Nevertheless, it shows distribution of matter on the Hubble diagram, and on the that their imprint on the Hubble diagram cannot be ne- resulting inference of the cosmological parameters. For glected, and should be modeled beyond the perturbation that purpose, we have studied light propagation in Swiss- regime. Note finally that several extensions are allowed by cheese models. This class of exact solutions of the Einstein our formalism. For instance, we could introduce different field equations is indeed very suitable, because it can kinds of inhomogeneities, in order to construct fractal describe a strongly inhomogeneous distribution of matter structures for which the smoothness parameter is arbi- which does not backreact on the global cosmic expansion. trarily close to zero. Additionally to the Hubble diagrams, The latter is entirely governed by the background cosmo- we could also generate the shear maps of Swiss-cheese logical parameters m, characterizing the FL regions models, and determine whether their combination allows of the model. The inhomogeneities are clumps of mass M, for better constraints on the various parameters. while the fraction of remaining fluid matter is f—called This work explicitly raises the question of the meaning smoothness parameter. The Swiss-cheese models are there- of the cosmological parameters, and of whether the values fore defined by two ‘‘dynamical’’ parameters ð m; Þ, we measure under the hypothesis of a pure FL background and two ‘‘structural’’ parameters ðf; MÞ. represent their ‘‘true’’ or some ‘‘dressed’’ values. Similar The laws of light propagation in a Swiss-cheese universe ideas have actually been held in other contexts [84,85], and have been determined by solving the geodesic equation and in particular regarding the spatial curvature [86,87]. We the Sachs equation. For the latter, we have introduced a new claim that the simplest Swiss-cheese models are good technique—based on the Wronski matrix—in order to deal models to address such questions—as well as the Ricci- more easily with a patchwork of spacetimes. Our results, Weyl problem and the fluid approximation—with their mostly analytical, have been included in a Mathematica own use, between the perturbation theory and N-body program, and used to compute the impact of the Swiss- simulations. cheese holes on the redshift and on the luminosity distance. For a light beam which crosses many holes, we have shown ACKNOWLEDGMENTS that the effect on the redshift remains negligible, while the luminosity distance increases significantly with respect to We thank Francis Bernardeau, Thomas Buchert, the one observed in a FL universe ( DL 10% for sources Nathalie Palanque-Delabrouille, Peter Dunsby, George at z 1), inducing a bias in the Hubble diagram. Ellis, Yannick Mellier, Shinji Mukohyama, Cyril Pitrou, The consequences of the bias on the inference of the Carlo Schimd, and Markus Werner for discussions. We also cosmological parameters have been investigated by thank Julian Adamek, Krzysztof Bolejko, Alan Coley,
123526-22 INTERPRETATION OF THE HUBBLE DIAGRAM IN A ... PHYSICAL REVIEW D 87, 123526 (2013) Pedro Ferreira, Giovanni Marozzi and Zdenek Stuchlı´k for Paris for hospitality during the later phase of this work. comments. P.F. thanks the E´ cole Normale Supe´rieure de This work was supported by French state funds managed Lyon for funding during his master and first year of Ph.D. by the ANR within the Investissements d’Avenir pro- H. D. thanks the Institut Lagrange de Paris for funding gramme under reference ANR-11-IDEX-0004-02 and the during her master’s and the Institut d’Astrophysique de Programme National Cosmologie et Galaxies.
[1] P. Peter and J.-P. Uzan, Primordial Cosmology (Oxford [23] E. Mortsell, arXiv:astro-ph/0109197. University, New York, 2009). [24] K. Bolejko, Mon. Not. R. Astron. Soc. 412, 1937 (2011). [2] G. F. R. Ellis and T. Buchert, Phys. Lett. A 347, 38 (2005). [25] R. Takahashi, M. Oguri, M. Sato, and T. Hamana, [3] G. F. R. Ellis and W. Stoeger, Classical Quantum Gravity arXiv:1106.3823. 4, 1697 (1987). [26] U. Seljak and D. E. Holz, Astron. Astrophys. 351, L10 [4] T. Buchert, Gen. Relativ. Gravit. 32, 105 (2000); 33, 1381 (1999). (2001);S.Ra¨sa¨nen, J. Cosmol. Astropart. Phys. 02 (2004) [27] R. B. Metcalf and J. Silk, Astrophys. J. 519, L1 (1999). 003; E. Barausse, S. Matarrese, and A. Riotto, Phys. Rev. [28] R. B. Metcalf and J. Silk, Phys. Rev. Lett. 98, 071302 D 71, 063537 (2005); E. W. Kolb, S. Matarrese, and A. (2007). Riotto, New J. Phys. 8, 322 (2006); S. Rasanen, J. Cosmol. [29] R. Kantowski, T. Vaughan, and D. Branch, Astrophys. J. Astropart. Phys. 11 (2006) 003; M. Kasai, Prog. Theor. 447, 35 (1995); J. A. Frieman, Commun. Astrophys. 18, Phys. 117, 1067 (2007). 323 (1996); Y. Wang, Astrophys. J. 531, 676 (2000); 536, [5] A. Ishibashi and R. M. Wald, Classical Quantum Gravity 531 (2000); J. Cosmol. Astropart. Phys. 03 (2005) 005. 23, 235 (2006); E. E. Flanagan, Phys. Rev. D 71, 103521 [30] C. Bonvin, R. Durrer, and M. A. Gasparini, Phys. Rev. D (2005); C. M. Hirata and U. Seljak, Phys. Rev. D 72, 73, 023523 (2006). 083501 (2005); S. R. Green and R. M. Wald, Phys. Rev. [31] P. Valageas, Astron. Astrophys. 354, 767 (2000). D 83, 084020 (2011). [32] N. Meures and M. Bruni, Mon. Not. R. Astron. Soc. 419, [6] E. W. Kolb, S. Matarrese, A. Notari, and A. Riotto, Phys. 1937 (2012). Rev. D 71, 023524 (2005); N. Li and D. J. Schwarz, Phys. [33] I. Ben-Dayan, M. Gasperini, G. Marozzi, F. Nugier, and G. Rev. D 76, 083011 (2007); 78, 083531 (2008);C. Veneziano, Phys. Rev. Lett. 110, 021301 (2013). Clarkson, K. Ananda, and J. Larena, Phys. Rev. D 80, [34] E. Di Dio and R. Durrer, Phys. Rev. D 86, 023510 (2012). 083525 (2009); O. Umeh, J. Larena, and C. Clarkson, J. [35] I. Ben-Dayan, M. Gasperini, G. Marozzi, F. Nugier, and G. Cosmol. Astropart. Phys. 03 (2011) 029; G. Marozzi and Veneziano, arXiv:1302.0740. J.-P. Uzan, Phys. Rev. D 86, 063528 (2012). [36] D. E. Holz and E. V. Linder, Astrophys. J. 631, 678 [7] C. Clarkson, G. Ellis, J. Larena, and O. Umeh, Rep. Prog. (2005). Phys. 74, 112901 (2011). [37] A. Cooray, D. Holz, and D. Huterer, Astrophys. J. 637, [8] C. Clarkson, G. F. R. Ellis, A. Faltenbacher, R. Maartens, L77 (2006); S. Dodelson and A. Vallinotto, Phys. Rev. D O. Umeh, and J.-P. Uzan, Mon. Not. R. Astron. Soc. 426, 74, 063515 (2006). 1121 (2012). [38] A. Cooray, D. Huterer, and D. Holz, Phys. Rev. Lett. 96, [9] Ya. B. Zel’dovich, Astron. Zh. 41, 19 (1964) [Sov. Astron. 021301 (2006). 8, 13 (1964)]. [39] D. Sarkar, A. Amblard, D. E. Holz, and A. Cooray, [10] C. C. Dyer and R. C. Roeder, Astrophys. J. 174, L115 arXiv:0710.4143. (1972); 180, L31 (1973). [40] A. R. Cooray, D. E. Holz, and R. Caldwell, J. Cosmol. [11] V.M. Dashevskii and V.I. Slysh, Astron. Zh. 42, 863 (1965) Astropart. Phys. 11 (2010) 015. [Sov. Astron. 9, 671 (1966)]. [41] A. Vallinotto, S. Dodelson, and P. Zhang, [12] B. Bertotti, Proc. R. Soc. A 294, 195 (1966). arXiv:1009.5590. [13] J. E. Gunn, Astrophys. J. 147, 61 (1967); 150, 737 (1967). [42] B. Me´nard, T. Hamana, M. Bartelmann, and N. Yoshida, [14] S. Refsdal, Astrophys. J. 159, 357 (1970). Astron. Astrophys. 403, 817 (2003). [15] S. Weinberg, Astrophys. J. 208, L1 (1976). [43] N. Dalal, D. E. Holz, X. Chen, and J. A. Frieman, [16] C. C. Dyer and R. C. Roeder, Gen. Relativ. Gravit. 13, Astrophys. J. 585, L11 (2003). 1157 (1981). [44] K. P. Rauch, Astrophys. J. 374, 83 (1991). [17] L. Fang and X. Wu, Chin. Phys. Lett. 6, 233 (1989);X. [45] D. E. Holz and R. M. Wald, Phys. Rev. D 58, 063501 Wu, Astron. Astrophys. 239, 29 (1990). (1998). [18] H. G. Rose, Astrophys. J. 560, L15 (2001). [46] R. W. Lindquist and J. A. Wheeler, Rev. Mod. Phys. 29, [19] T. W. B. Kibble and R. Lieu, Astrophys. J. 632, 718 432 (1957). (2005). [47] E. Gausmann, R. Lehoucq, J.-P. Luminet, J.-P. Uzan, and [20] V. Kostov, J. Cosmol. Astropart. Phys. 04 (2010) 001. J. Weeks, Classical Quantum Gravity 18, 5155 (2001);R. [21] E. V. Linder, Astron. Astrophys. 206, 190 (1988). Lehoucq, J. Weeks, J.-P. Uzan, E. Gausmann, and J.-P. [22] K. Tomita, Prog. Theor. Phys. 100, 79 (1998). Luminet, Classical Quantum Gravity 19, 4683 (2002);
123526-23 FLEURY, DUPUY, AND UZAN PHYSICAL REVIEW D 87, 123526 (2013) J. Weeks, R. Lehoucq, and J.-P. Uzan, Classical Quantum [69] P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Gravity 20, 1529 (2003); J.-P. Uzan, A. Riazuelo, R. Lenses (Springer, New York, 1992). Lehoucq, and J. Weeks, Phys. Rev. D 69, 043003 (2004). [70] Z. Stuchlı´k, Bull. Astron. Inst. Czech. 34, 129 (1983). [48] T. Clifton and P. G. Ferreira, Phys. Rev. D 80, 103503 [71] Z. Stuchlı´k, Bull. Astron. Inst. Czech. 35, 205 (1984). (2009); J. Cosmol. Astropart. Phys. 10 (2009) 026. [72] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, [49] J.-P. Bruneton and J. Larena, Classical Quantum Gravity Series, and Products (Academic, New York, 2007). 29, 155001 (2012); 30, 025002 (2013). [73] F. Combes, P. Boisse´, A. Mazure, and A. Blanchard, [50] K. Kainulainen and V. Marra, Phys. Rev. D 80, 127301 Galaxies et cosmologie (CNRS, Paris, 2003). (2009); 83, 023009 (2011). [74] N. Suzuki et al. (The Supernova Cosmology Project), [51] A. Einstein and E. G. Straus, Rev. Mod. Phys. 17, 120 Astrophys. J. 746, 85 (2012). (1945); 18, 148 (1946). [75] S. Perlmutter et al., Astrophys. J. 517, 565 (1999);A.G. [52] R. Kantowski, Astrophys. J. 155, 89 (1969). Riess et al., Astron. J. 116, 1009 (1998); J. L. Tonry et al., [53] C. C. Dyer and R. C. Roeder, Astrophys. J. 189, 167 Astrophys. J. 594, 1 (2003); R. A. Knop et al., Astrophys. (1974). J. 598, 102 (2003); A. G. Riess et al., Astrophys. J. 607, [54] N. Brouzakis, N. Tetradis, and E. Tzavara, J. Cosmol. 665 (2004). Astropart. Phys. 02 (2007) 013. [76] K. Bolejko, C. Clarkson, R. Maartens, D. Bacon, N. [55] V. Marra, E. W. Kolb, S. Matarrese, and A. Riotto, Phys. Meures, and E. Beynon, Phys. Rev. Lett. 110, 021302 Rev. D 76, 123004 (2007). (2013). [56] T. Biswas and A. Notari, J. Cosmol. Astropart. Phys. 06 [77] A. Conley et al., Astrophys. J., Suppl. Ser. 192, 1 (2011). (2008) 021. [78] K. Bolejko and P. Ferreira, J. Cosmol. Astropart. Phys. 05 [57] R. A. Vanderveld, E. E. Flanagan, and I. Wasserman, Phys. (2012) 003. Rev. D 78, 083511 (2008). [79] J. Ehlers and P. Schneider, Astron. Astrophys. 168,57 [58] N. Brouzakis, N. Tetradis, and E. Tzavara, J. Cosmol. (1986). Astropart. Phys. 04 (2008) 008. [80] M. Sasaki, Prog. Theor. Phys. 90, 753 (1993). [59] T. Clifton and J. Zuntz, Mon. Not. R. Astron. Soc. 400, [81] K. Tomita, H. Asada, and T. Hamana, Prog. Theor. Phys. 2185 (2009). Suppl. 133, 155 (1999). [60] W. Valkenburg, J. Cosmol. Astropart. Phys. 06 (2009) [82] S. Rasanen, J. Cosmol. Astropart. Phys. 02 (2009) 011. 010. [83] P. Fleury, H. Dupuy, and J.-P. Uzan, arXiv:1304.7791. [61] S. J. Szybka, Phys. Rev. D 84, 044011 (2011). [84] T. Buchert and M. Carfora, Phys. Rev. Lett. 90, 031101 [62] E. E. Flanagan, N. Kumar, I. Wasserman, and R. A. (2003). Vanderveld, Phys. Rev. D 85, 023510 (2012). [85] D. L. Wiltshire, New J. Phys. 9, 377 (2007); Int. J. Mod. [63] F. Kottler, Ann. Phys. (Berlin) 361, 401 (1918). Phys. D 17, 641 (2008); P. R. Smale and D. L. Wiltshire, [64] H. Weyl, Phys. Z. 20, 31 (1919). Mon. Not. R. Astron. Soc. 413, 367 (2011). [65] V. Perlick, Living Rev. Relativity 7, 9 (2004). [86] C. Clarkson, T. Clifton, A. Coley, and R. Sung, [66] W. Israel, Nuovo Cimento B 44, 1 (1966). arXiv:1111.2214. [67] W. Israel, Nuovo Cimento B 48, 463 (1967). [87] A. A. Coley, N. Pelavas, and R. M. Zalaletdinov, Phys. [68] R. Sachs, Proc. R. Soc. A 264, 309 (1961). Rev. Lett. 95, 151102 (2005).
123526-24 1.6. Article “Can all observations be accurately interpreted with a unique geometry?”
1.6 Article “Can all observations be accurately inter- preted with a unique geometry?”
54 week ending PRL 111, 091302 (2013) PHYSICAL REVIEW LETTERS 30 AUGUST 2013
Can All Cosmological Observations Be Accurately Interpreted with a Unique Geometry?
Pierre Fleury,1,2,* He´le`ne Dupuy,1,2,3,† and Jean-Philippe Uzan1,2,‡ 1Institut d’Astrophysique de Paris, UMR-7095 du CNRS, Universite´ Pierre et Marie Curie, 98 bis boulevard Arago, 75014 Paris, France 2Sorbonne Universite´s, Institut Lagrange de Paris, 98 bis boulevard Arago, 75014 Paris, France 3Institut de Physique The´orique, CEA, IPhT, URA 2306 CNRS, F-91191 Gif-sur-Yvette, France (Received 3 June 2013; published 29 August 2013) H The recent analysis of the Planck results reveals a tension between the best fits for ( m0, 0) derived from the cosmic microwave background or baryonic acoustic oscillations on the one hand, and the Hubble diagram on the other hand. These observations probe the Universe on very different scales since they involve light beams of very different angular sizes; hence, the tension between them may indicate that they should not be interpreted the same way. More precisely, this Letter questions the accuracy of using only the (perturbed) Friedmann-Lemaıˆtre geometry to interpret all the cosmological observations, regardless of their angular or spatial resolution. We show that using an inhomogeneous ‘‘Swiss-cheese’’ model to interpret the Hubble diagram allows us to reconcile the inferred value of m0 with the Planck results. Such an approach does not require us to invoke new physics nor to violate the Copernican principle.
DOI: 10.1103/PhysRevLett.111.091302 PACS numbers: 98.80.Es, 98.62.Py, 98.70.Vc, 98.80.Jk
H : : = = ; The standard interpretation of cosmological data relies 0 ¼ð67 3 1 2Þ km s Mpc on the description of the Universe by a spatially homo- : : geneous and isotropic spacetime with a Friedmann- m0 ¼ 0 315 0 017 (1) Lemaıˆtre (FL) geometry, allowing for perturbations [1]. The emergence of a dark sector, including dark matter at a 68% confidence level. It was pointed out (see H and dark energy, emphasizes the need for extra degrees Secs. 5.2–5.4 of Ref. [3]) that the values of 0 and m0 of freedom, either physical (new fundamental fields are, respectively, low and high compared with their values or interactions) or geometrical (e.g., a cosmological inferred from the Hubble diagram. Such a trend was solution with lower symmetry). This has driven a lot already indicated by WMAP-9 [6] which concluded H : = = of activity to test the hypotheses [2] of the cosmological 0 ¼ð70 2 2Þ km s Mpc. model, such as general relativity or the Copernican Regarding the Hubble constant, two astrophysical principle. measurements are in remarkable agreement. First, the The recent Planck data were analyzed in such a frame- estimation based on the distance ladder calibrated by work [3] in which the cosmic microwave background three different techniques (masers, Milky Way cepheids, H (CMB) anisotropies are treated as perturbations around a and Large Magellanic Cloud cepheids) gives [7] 0 ¼ FL universe, with most of the analysis performed at linear ð74:3 1:5 2:1Þ km=s=Mpc, respectively, with statisti- order. Nonlinear effects remain small [4] and below the cal and systematic errors. This improves the earlier con- constraints on non-Gaussianity derived by Planck [5]. The straint obtained by the Hubble Space Telescope (HST) H = = results nicely confirm the standard cosmological model of Key program [8], 0 ¼ð72 8Þ km s Mpc. Second, the a spatially Euclidean FL universe with a cosmological Hubble diagram of type Ia supernovae (SNe Ia) calibrated H constant, dark matter, and initial perturbations compatible with the HST observations of cepheids leads [9]to 0 ¼ with the predictions of inflation. ð73:8 2:4Þ km=s=Mpc. Other determinations of the Among the constraints derived from the CMB, the Hubble constant, e.g., from very-long-baseline interferom- Hubble parameter H0 and the matter density parameter etry observations [10] or from the combination of Sunyaev- m0 are mostly constrained through the combination Zel’dovich effect and X-ray observations [11], have larger h3 H h = = m0 , where 0 ¼ 100 km s Mpc. It is set by the error bars and are compatible with both the CMB and acoustic scale ¼ rs=DA, defined as the ratio between the distance measurements. sound horizon and the angular distance at the time of last Additionally, the analysis of the Hubble diagram of SNe : : scattering. The measurement of seven acoustic peaks Ia leads to a lower value of m0—e.g., 0 222 0 034 with enables one to determine with a precision better than the SNLS 3 data set [12]—compared to the constraint (1) H 0.1%. The constraints on the plane ( m0, 0) are presented by Planck. As concluded in Ref. [3], there is no direct in Fig. 3 of Ref. [3] and clearly show this degeneracy. inconsistency, and it was pointed out that there could be The marginalized constraints on the two parameters were residual systematics not properly accounted for in the then derived [3]tobe SN data. Still, it was stated that ‘‘the tension between
0031-9007=13=111(9)=091302(5) 091302-1 Ó 2013 American Physical Society week ending PRL 111, 091302 (2013) PHYSICAL REVIEW LETTERS 30 AUGUST 2013
CMB-based estimates and the astrophysical measurements large scales, it is dominated by the Ricci tensor. Both H of 0 is intriguing and merits further discussion.’’ situations correspond to distinct optical properties [23]. H Interestingly, the CMB constraints on ( m0, 0) are in In the framework of geometric optics, a light beam is excellent agreement with baryon acoustic oscillation described by a bundle of null geodesics. All the informa- (BAO) measurements [13], which allow one to determine tion about the size and the shape of a beam can be encoded a the angular distance up to redshifts of order 0.7. The in a 2 2 matrix D b called the Jacobi map (see Ref. [24] common point between the CMB and BAO measurements for further details). In particular, the angular and luminos- is that they involve light beams much larger than those ity distances read, respectively, involved in astronomical observations. Indeed, a pixel of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 Planck’s high-resolution CMB maps corresponds to 5 arc DA ¼ j det D bj;DL ¼ð1 þ zÞ DA; (2) min [14], while the typical angular size of a SN is 10 7 arc sec. This means that the two kinds of observa- where z denotes the redshift. The evolution of the Jacobi tions probe the Universe at very different scales. Moreover, map with light propagation is governed by the Sachs for both the CMB and BAO measurements the crucial equation [25,26] information is encoded in correlations, while SN observa- 2 d a a c D b ¼ R cD b; (3) tions rely on ‘‘1-point measurements’’ (we are interested in dv2 the luminosity and redshift of each SN, not in the correla- tions between several SNe). Because of such distinctions where v is an affine parameter along the geodesic bundle. a one can expect the two classes of cosmological observa- The term Rab, which controls the evolution of D b,isa tions to be affected differently by the inhomogeneity of the projection of the Riemann tensor called the optical tidal Universe, through gravitational lensing. matrix. It is defined by Rab R k k sa sb , where k The effect of lensing on CMB measurements is essen- is the wave vector of an arbitrary ray, and the Sachs basis s tially due to the large-scale structure, and it can be taken f a ga¼1;2 spans a screen on which the observer projects the into account in the framework of cosmological perturba- light beam. Because the Riemann tensor can be split into a tion theory at linear order [15] (see, however, Ref. [16] for Ricci part R and a Weyl part C , the optical tidal a discussion about the impact of strong inhomogeneities). matrix can also be decomposed as We refer to Ref. [17] for a description of the lensing effects ! ! on BAO measurements. Regarding the Hubble diagram, 00 0 Re 0 Im 0 ðRabÞ¼ þ ; (4) the influence of lensing has also been widely investigated 0 00 Im 0 Re 0 [18]. The propagation of light in an inhomogeneous uni- |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} verse gives rise to both distortion and magnification. Most Ricci lensing Weyl lensing images are expected to be demagnified because their lines = R k k = C with 00 ð1 2Þ and 0 ð1 2Þ of sight probe underdense regions, while some are ampli- s is k s is k ð 1 2 Þ ð 1 2 Þ . It clearly appears in Eq. (4) fied due to strong lensing. It shall thus induce a dispersion that the Ricci term tends to isotropically focus the light of the luminosities of the sources, that is, an extra scatter in beam, while the Weyl term tends to shear and rotate it. The the Hubble diagram [19]. Its amplitude can be determined behavior of a light beam is thus different whether it expe- from the perturbation theory [20] and subtracted [21]. riences Ricci-dominated lensing (large beams, e.g., CMB However, a considerable fraction of the lensing effects measurements) or Weyl-dominated lensing (narrow beams, arises from sub-arc-min scales, which are not probed by e.g., SN observations). shear maps smoothed on arc min scales [22]. This Ricci-Weyl problem can be addressed with differ- H The tension on ( m0, 0) may indicate that, given the ent methods. One possibility, a representative of which is accuracy of the observations achieved today, the use of a the Dyer-Roeder approximation [27], is to construct a (perturbed) FL geometry to interpret the astrophysical data general distance-redshift relation which would take into is no longer adapted. More precisely, the question that we account the effect of inhomogeneities in some average want to raise is whether the use of a unique spacetime way. However, such approaches are in general difficult to geometry is relevant for interpreting all the cosmological control [18] because they rely on approximations whose observations, regardless of their angular or spatial resolu- domain of applicability is unknown. An alternative possi- tion and of their location (redshift). Indeed, each observa- bility consists in constructing inhomogeneous cosmologi- tion is expected to probe the Universe smoothed on a cal models, with a discrete distribution of matter, and typical scale related to its resolution, and this can lead to studying the impact on light propagation. Several models fundamentally different geometrical situations. In a uni- exist in the literature: the Schwarzschild-cell method [28] verse with a discrete distribution of matter, the Riemann or the lattice universe [29], which are both approximate curvature experienced by a beam of test particles or solutions of the Einstein equations, and the Swiss-cheese photons is dominated by the Weyl tensor. Conversely, in models [30], which are constructed by matching together a (statistically spatially isotropic) universe smoothed on patches of exact solutions of the Einstein equations.
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This last approach is the one that we shall follow in this assuming that it is strictly homogeneous, then one under- Letter. estimates the value of m0. The error reaches a few per- Consider a Swiss-cheese model in which clumps of cent, which is comparable to other estimates in similar matter (modeling, e.g., galaxies), each of them lying at contexts [32]. Note, however, that in the case of Swiss- the center of a spherical void, are embedded in a FL cheese models with Lemaıˆtre-Tolman-Bondi patches spacetime. The interior region of a void is described by instead of Kottler voids, the effect of inhomogeneities the Kottler geometry—i.e., Schwarzschild with a cosmo- has a much smaller impact on the Hubble diagram [33]. logical constant—while the exterior geometry is the FL Thus, the systematic effect exhibited in Ref. [24] must be one. By construction, such inhomogeneities do not modify attributed to the discreteness of the distribution of matter. the expansion dynamics of the embedding FL universe, Simulating the mock Hubble diagrams for Swiss-cheese thus avoiding any discussion regarding backreaction. The universes with various values of its parameters, we inferred resulting spacetime is well defined, because the Darmois- a phenomenological expression for the luminosity distance D z ; ;H ;f Israel junction conditions are satisfied on the boundary of Lð ; m0 0 0 Þ, which is very close to the Dyer- every void. Compared to a strictly homogeneous universe, Roeder one. This expression was then used to fit the a Swiss-cheese model is therefore characterized by two Hubble diagram constructed from the SNLS 3 catalog additional parameters: the size of the voids (or equivalently [12]. Figure 25 of Ref. [24] shows that f influences the the mass of their central bodies) and the volumic fraction of result of the best fit on m0 that can shift from 0.22 for the remaining FL regions, which encodes the smoothness f ¼ 1 (in agreement with the standard FL analysis per- of the distribution of matter. It is naturally quantified by the formed in Ref. [12]) to 0.3 for f ¼ 0. h smoothness parameter Figure 1 shows the constraints in the plane ( , m0) V imposed by Planck on the one hand, and by the Hubble f lim FL ; (5) diagram on the other hand, whether it is interpreted in a V V !1 spatially flat FL universe (f ¼ 1) or in a spatially flat V where FL is the volume occupied by the FL region within Swiss-cheese model for which matter is entirely clumped a volume V of the Swiss cheese. With the definition (5), (f ¼ 0). The agreement between the CMB and the Hubble f ¼ 1 corresponds to a model with no hole (i.e., a FL diagram is clearly improved for small values of f, espe- f universe), while ¼ 0 corresponds to the case where cially regarding m0, while h is almost unaffected. H matter is exclusively under the form of clumps inside Note that SN observations alone cannot constrain 0, voids. because of the degeneracy with the (unknown) absolute Of course such a model cannot be considered realistic, magnitude M of the SNe. For the results of Fig. 1 the but neither does the exact FL geometry, used to interpret degeneracy was broken by fixing M ¼ 19:21, according the Hubble diagram. Both spacetimes describe a spatially to the best-fit value obtained by Ref. [12] with a fiducial statistically homogeneous and isotropic universe, and the Hubble constant h ¼ 0:7. Thus, the horizontal positions of former permits additionally the investigation of the effect the SN contours in Fig. 1 are only indicative. of a discrete distribution of matter. Since the FL universe is a particular Swiss-cheese model, this family of spacetimes therefore allows us to estimate how good the hypothesis of strict spatial homogeneity—with a continuous matter distribution at all scales—is. The propagation of light in a Swiss-cheese model has been comprehensively investigated in Ref. [24], generaliz- ing earlier works [31], with the key assumption that light never crosses the clumps. This ‘‘opacity assumption’’ can be observationally justified in the case of SN observations if the clumps represent galaxies (see Ref. [24] for a dis- cussion). Compared to the strictly homogeneous case, any light signal traveling through a Swiss-cheese model then experiences a reduced Ricci focusing. This leads [see Eqs. (2)–(4)] to an increase of the observed luminosity distance DL. The effect of Weyl lensing—i.e., here shear—is relatively small. This systematic effect, due to inhomogeneities, tends to FIG. 1 (color online). Comparison of the constraints obtained bias the Hubble diagram in a way that mimics the contri- h by Planck on ( m0, )[3] and from the analysis of the Hubble bution of a negative spatial curvature or a positive cosmo- diagram constructed from the SNLS 3 catalog [12]. The shaded logical constant. In other words, if one interprets the contour plots correspond to two different smoothness parame- Hubble diagram of a Swiss-cheese universe by wrongly ters. For f ¼ 1, the geometry used to fit the data is the FL one.
091302-3 week ending PRL 111, 091302 (2013) PHYSICAL REVIEW LETTERS 30 AUGUST 2013 H Alleviating the tension on 0 remains an open issue. [11] M. Bonamente, M. K. Joy, S. J. LaRoque, J. E. Carlstrom, Because inferring its value from SNe is a local measure- E. D. Reese, and K. S. Dawson, Astrophys. J. 647,25 ment, a promising approach consists in taking into account (2006). the impact of our close environment. It has been suggested [12] J. Guy et al., Astron. Astrophys. 523, A7 (2010). [34] that cosmic variance increases the uncertainty on [13] S. Cole et al., Mon. Not. R. Astron. Soc. 362, 505 (2005); D. J. Eisenstein et al., Astrophys. J. 633, 560 (2005);W.J. Hlocal and thus reduces the tension with HCMB. More 0 0 Percival et al., Mon. Not. R. Astron. Soc. 401, 2148 Hlocal >HCMB speculatively, 0 0 may be a hint that our local (2010); N. Padmanabhan, X. Xu, D. J. Eisenstein, R. environment is underdense [35]. Our conclusions on m0 Scalzo, A. J. Cuesta, K. T. Mehta, and E. Kazin, Mon. remain, however, unaffected by this issue. Not. R. Astron. Soc. 427, 2132 (2012). Our analysis, though relying on a particular class of [14] P. A. R. Ade et al. (Planck Collaboration), arXiv:1303.5062. models, indicates that the FL geometry is probably too [15] P. A. R. Ade et al. (Planck Collaboration), arXiv:1303.5077. simplistic to describe the Universe for certain types of [16] K. Bolejko, J. Cosmol. Astropart. Phys. 02 (2011) 025. observations, given the accuracy reached today. In the [17] A. Vallinotto, S. Dodelson, C. Schimd, and J.-P. Uzan, end, a single metric may not be sufficient to describe all Phys. Rev. D 75, 103509 (2007). the cosmological observations, just as Lilliputians and [18] C. Clarkson, G. F. R. Ellis, A. Faltenbacher, R. Maartens, O. Umeh, and J.-P. Uzan, Mon. Not. R. Astron. Soc. 426, Brobdingnag’s giants [36] cannot use a map with the 1121 (2012). same resolution to travel. A better cosmological model [19] R. Kantowski, T. Vaughan, and D. Branch, probably requires an atlas of maps with various smoothing Astrophys. J. 447, 35 (1995); J. A. Frieman, Commun. scales, determined by the observations at hand. Astrophys. 18, 323 (1996); Y. Wang, Astrophys. J. 531, Other observations, such as lensing [37], may help to 676 (2000). characterize the distribution and the geometry of voids [20] C. Bonvin, R. Durrer, and M. A. Gasparini, Phys. Rev. D [38], in order to construct a better geometrical model. 73, 023523 (2006); N. Meures and M. Bruni, Mon. Not. R. For the first time, the standard FL background geometry Astron. Soc. 419, 1937 (2012); P. Valageas, Astron. may be showing its limits to interpret the cosmological Astrophys. 354, 767 (2000). data with the accuracy they require. [21] A. Cooray, D. Holz, and D. Huterer, Astrophys. J. 637, We thank Francis Bernardeau, Yannick Mellier, Alice L77 (2006); S. Dodelson and A. Vallinotto, Phys. Rev. D Pisani, Cyril Pitrou, Joe Silk, and Benjamin Wandelt for 74, 063515 (2006); A. Cooray, D. Huterer, and D. E. Holz, Phys. Rev. Lett. 96, 021301 (2006); D. Sarkar, A. discussions. This work was supported by French state funds Amblard, D. E. Holz, and A. Cooray, Astrophys. J. 678, managed by the ANR within the Investissements d’Avenir 1 (2008); A. R. Cooray, D. E. Holz, and R. Caldwell, J. programme under reference ANR-11-IDEX-0004-02, the Cosmol. Astropart. Phys. 11 (2010) 015; A. Vallinotto, S. Programme National Cosmologie et Galaxies, and the ANR Dodelson, and P. Zhang, Phys. Rev. D 84, 103004 (2011); THALES (ANR-10-BLAN-0507-01-02). B. Me´nard, T. Hamana, M. Bartelmann, and N. Yoshida, Astron. Astrophys. 403, 817 (2003). [22] N. Dalal, D. E. Holz, X. Chen, and J. A. Frieman, Astrophys. J. 585, L11 (2003). [23] C. C. Dyer and R. C. Roeder, Gen. Relativ. Gravit. 13, *fl[email protected] 1157 (1981). †[email protected] [24] P. Fleury, H. Dupuy, and J.-P. Uzan, Phys. Rev. D 87, ‡[email protected] 123526 (2013). [1] P. Peter and J.-P. Uzan, Primordial Cosmology (Oxford [25] R. Sachs, Proc. R. Soc. A 264, 309 (1961). University Press, Oxford, England, 2009). [26] P. Schneider, J. Ehlers, and E. E. Falco, Gravitational [2] J.-P. Uzan, Gen. Relativ. Gravit. 39, 307 (2007); Lenses (Springer, New York, 1992). arXiv:0912.5452; Living Rev. Relativity 14, 2 (2011). [27] C. C. Dyer and R. C. Roeder, Astrophys. J. 174, L115 [3] P.A. R. Ade et al. (Planck Collaboration), arXiv:1303.5076. (1972); C. C. Dyer and R. C. Roeder, Astrophys. J. 180, [4] C. Pitrou, J.-P. Uzan, and F. Bernardeau, Phys. Rev. D 78, L31 (1973). 063526 (2008); C. Pitrou, J.-P. Uzan, and F. Bernardeau, [28] R. W. Lindquist and J. A. Wheeler, Rev. Mod. Phys. 29, J. Cosmol. Astropart. Phys. 07 (2010) 003; Z. Huang and 432 (1957); T. Clifton and P. G. Ferreira, Phys. Rev. D F. Vernizzi, Phys. Rev. Lett. 110, 101303 (2013). 80, 103503 (2009); J. Cosmol. Astropart. Phys. 10 [5] P.A. R. Ade et al. (Planck Collaboration), arXiv:1303.5084. (2009) 026. [6] G. Hinshaw et al., arXiv:1212.5226. [29] D. E. Holz and R. M. Wald, Phys. Rev. D 58, 063501 [7] W. L. Freedman, B. F. Madore, V. Scowcroft, C. Burns, A. (1998); J.-P. Bruneton and J. Larena, Classical Quantum Monson, S. Eric Persson, M. Seibert, and J. Rigby, Gravity 29, 155001 (2012); J.-P. Bruneton and J. Larena, Astrophys. J. 758, 24 (2012). Classical Quantum Gravity 30, 025002 (2013);K. [8] W. L. Freedman et al., Astrophys. J. 553, 47 (2001). Kainulainen and V. Marra, Phys. Rev. D 80, 127301 [9] A. G. Riess, L. Macri, S. Casertano, H. Lampeitl, H. C. (2009). Ferguson, A. V. Filippenko, S. W. Jha, W. Li, and R. [30] A. Einstein and E. G. Straus, Rev. Mod. Phys. 17, 120 Chornock, Astrophys. J. 730, 119 (2011). (1945); A. Einstein and E. G. Straus, Rev. Mod. Phys. 18, [10] M. J. Reid et al., arXiv:1207.7292. 148 (1946).
091302-4 week ending PRL 111, 091302 (2013) PHYSICAL REVIEW LETTERS 30 AUGUST 2013
[31] R. Kantowski, Astrophys. J. 155, 89 (1969); C. C. Dyer S. J. Szybka, Phys. Rev. D 84, 044011 (2011);E.E. and R. C. Roeder, Astrophys. J. 189, 167 (1974). Flanagan, N. Kumar, I. Wasserman, and R. A. [32] K. Bolejko and P.G. Ferreira, J. Cosmol. Astropart. Phys. Vanderveld, Phys. Rev. D 85, 023510 (2012). 05 (2012) 003. [34] V. Marra, L. Amendola, I. Sawicki, and W. Valkenburg, [33] N. Brouzakis, N. Tetradis, and E. Tzavara, J. Cosmol. Phys. Rev. Lett. 110, 241305 (2013). Astropart. Phys. 02 (2007) 013; V. Marra, E. W. Kolb, S. [35] S. Jha, A. Riess, and R. Kirshner, Astrophys. J. 659, 122 Matarrese, and A. Riotto, Phys. Rev. D 76, 123004 (2007); (2007). T. Biswas and A. Notari, J. Cosmol. Astropart. Phys. 06 [36] J. Swift, Travels into Several Remote Nations of the World. (2008) 021; R. A. Vanderveld, E. E. Flanagan, and I. In Four Parts (Benjamin Motte, London, 1726). Wasserman, Phys. Rev. D 78, 083511 (2008);N. [37] K. Bolejko, C. Clarkson, R. Maartens, D. Bacon, N. Brouzakis, N. Tetradis, and E. Tzavara, J. Cosmol. Meures, and E. Beynon, Phys. Rev. Lett. 110, 021302 Astropart. Phys. 04 (2008) 008; T. Clifton and J. Zuntz, (2013). Mon. Not. R. Astron. Soc. 400, 2185 (2009);W. [38] G. Lavaux and B. D. Wandelt, Mon. Not. R. Astron. Soc. Valkenburg, J. Cosmol. Astropart. Phys. 06 (2009) 010; 403, 1392 (2010).
091302-5 1.7. Continuation of this work
1.7 Continuation of this work
Several improvements could be brought to this work, which is a mere manipulation of a toy model. For example, to be more realistic, it would be useful to study in detail the space distribution of the structures of the universe. What are the relevant masses? What fraction of the total volume do these structures represent? How lumpy the universe is? Introducing holes with different sizes could in particular be relevant. We could also envisage a similar study in which light is replaced by gravitational waves (playing the role of “standard sirens”) to check that conclusions about cosmology do not depend on the observables considered. In the continuity of our project, my collaborators P. Fleury and J. P. Uzan have realized several studies. For instance, in [78], they propose a comprehensive analysis of light propagation in a spatially anisotropic and homogeneous spacetime and in [75] P.Fleury compares the distance-redshift relation of a certain class of Swiss-cheese models to the one predicted by the so-called Dyer-Roeder approach ([61]). This kind of study is an illustration of what precision cosmology means because it is the description of the very fine structure of the universe that is questioned. Such approaches do not aim at proposing new models but rather at adding a level of description to the existing ones. In the present case, the novelty would consist in adapting the mapping of the universe to the scale considered instead of using homo- geneous and isotropic descriptions smoothed on large scales. Precision cosmology can also be developed by considering the large-scale structure of the universe. In the framework of perturbation theory, a common effort is indeed realized in order to go beyond the linear regime and to account for cosmic components that had been neglected so far. This is exactly the type of issues addressed in the remainder of this manuscript.
60 Chapter 2
A glimpse of neutrino cosmology
Neutrino phenomenology is extremely rich. Rather than getting into the details, details which are nevertheless captivating, I confine this chapter to elements that I consider relevant for familiarizing with neutrino cosmology and understanding what is at stake in the part of this thesis oriented towards neutrinos. A clear and comprehensive review of the role of neutrinos in cosmology can be found in [104].
2.1 Basic knowledge about neutrinos
In December 1930, Wolfgang Pauli suggested the existence of an unknown particle. This was motivated by the observation that, if it did not exist, conservation laws would be violated in β decay. For energy, momentum and angular momentum to be conserved, this particle had to be electrically neutral, extremely light and with a spin 1/2. Besides, to explain why it had never been detected, it has been postulated that it should interact very weakly with matter. At that time, its existence was purely hypothetical. Pauli’s idea was even considered by many as a desperate theorist’s attempt to keep up appearances. However, Enrico Fermi took it seriously and incorporated the new particle, that he named neutrino, in his theory of β decay in 1934. This theory was to become the theory of the weak interaction. Between 1953 and 1956, Clyde Cowan and Frederick Reines achieved the first neutrino detections thanks to a nuclear reactor of the Savannah River site ([41]).
61 2.2. Why are cosmologists interested in neutrinos?
For this finding, they received the Nobel Prize in 1995. Then it has been demonstrated that neutrinos are of several types, or flavors, i.e. one can distinguish between “electron neutrinos” (the firsts to be discovered), “muon neutrinos” (discovered in 1962 by Leon M. Lederman, Melvin Schwartz and Jack Steinberger, winners of the Nobel Prize for their research on neutrinos in 1988) and “tau neutrinos” (whose detection has been announced in 2000 by the DONUT collaboration at Fermilab). A remarkable fact is that, even before muon neutrinos were discovered, Bruno Pontecorvo suggested that, provided they are massive, neutrinos can change in na- ture during their time evolution (see his paper [138], which dates from 1957). As- suming a non-zero mass is a modification of the standard model of particle physics1, in which neutrinos are massless leptons. On the other hand, the absence of mass is a mere assertion, which is not justified theoretically by the model. Naturally, the detection of a second neutrino flavor strengthened B. Pontecorvo’s intuition ([139]). The existence of flavor neutrino oscillations, i.e. of transitions from one flavor to another, has been confirmed by the Homestake experiment thirty years later ([34]). It is a thrilling point since its theoretical justification demands to go beyond the standard model of particle physics, that is to say to invoke new physics. At the present time, thanks to experiments involving atmospheric and solar neutri- nos, there is no doubt that at least two of the three neutrino species are massive. Planetary atmospheres (when excited by cosmic rays) and stars are indeed powerful natural sources2 of neutrinos.
2.2 Why are cosmologists interested in neutrinos?
At ordinary energies, neutrinos interact so weakly with matter that they can go through a planet with no consequence. Such an insensitiveness is annoying for particle physicists. It makes the detection of neutrinos and the study of their prop- erties very challenging. But the universe is an amazing laboratory in which highly energetic phenomena occur naturally. By looking far into space, one accesses to the extreme physical conditions that characterized the young universe. Observa- tional astrophysics offers therefore an opportunity to get unhoped constraints on
1The standard model of particle physics is a scientific theory describing interactions between the elementary particles that constitute matter. It was developed throughout the latter half of the twentieth century. 2Among such sources, one can cite supernovae too.
62 2.2. Why are cosmologists interested in neutrinos? particles and fundamental interactions. This is what justifies the existence of the research field called astroparticle physics. In particular, when the observed object is the observable universe and the studied particles are neutrinos, it is referred to as neutrino cosmology. For the game to be fair, cosmologists too should enjoy the benefits of testing neutrino properties. And they definitely do! Indeed, the ambitions of precision cosmology require to develop a solid knowledge of the properties of each cosmic component involved in the dynamical evolution of the universe. As will be sketched in the next sections, neutrinos truly had a role to play in this evolution, their means of action being weak interaction and gravitation.
2.2.1 An equilibrium that does not last long
At the beginning, the particles of the universe are all maintained in equilibrium by processes involving very high energies. At that time, neutrinos are relativistic and interact continuously with the rest of the cosmic plasma. In the meantime the universe cools, which lowers the rate at which neutrinos interact. This decrease is faster than the one of the Hubble rate. Hence, there is a moment at which weak interactions are beaten by expansion: neutrinos decouple from electrons, positrons and photons and then propagate freely. So, in the same way as there exists a cosmic microwave background, there exists a cosmic neutrino background. Unfortunately, there is little chance for the latter to be detected directly someday because neutrinos interact very weakly, which is worsened by the fact that their interaction rate is largely suppressed by the cooling of the universe. However, some imprints left by neutrinos during different cosmological eras are present in several observables. As we will see, the role played by neutrinos is especially significant at late times, i.e. during the formation of the large-scale structure of the universe. Neutrino decoupling takes place much earlier than photon decoupling, when the universe is at most few-minute-old3 and has a temperature in the MeV range4. At this temperature, neutrinos are still relativistic, whence the appellation “hot relics” which is sometimes used. 3There is no single decoupling time because decoupling is not an instantaneous phenomenon. It is nevertheless possible to obtain a correct order of magnitude in the “instantaneous decoupling limit”. Neutrino decoupling is estimated to start about one-tenth of a second after the Big Bang. 4The typical temperature of neutrino decoupling is thus almost seven orders of magnitude higher than the one of photon decoupling (photon decoupling marks the start of the CMB propagation).
63 2.2. Why are cosmologists interested in neutrinos?
Thermodynamical considerations show that, when equilibrium is broken, the neutrino-to-photon temperature ratio, which used to be 1, becomes5
T 4 1/3 ν = . (2.1) T 11 γ Note that, as well as photons, neutrinos are exceptionally cold today. Indeed, the measurement of the present CMB temperature gives Tγ,0 = 2.725 K, whence T = 1.945 K ( 10 4 eV). As explained in [104], it means in particular that at ν,0 ∼ − least two of the three neutrino species are non-relativistic today.
2.2.2 The effective number of neutrino species, a parameter rele- vant for precision cosmology
Definition
After decoupling and as long as neutrinos are relativistic, the total energy density of relativistic species can be written (see also [104])
7 4 4/3 ρ = ρ 1 + N . (2.2) R γ 8 11 eff " #
ργ is the photon energy density. Neff is called effective number of neutrino species. It characterizes the neutrino abundance in the early universe, plus possibly extra relativistic species. In the instantaneous decoupling limit, provided that neutrinos are well described by Fermi-Dirac distribution functions with no chemical potential6 and that the only relativistic components of the universe are photons plus three neutrino species, Neff is equal to 3. So, as a first approximation, any deviation from Neff = 3 would be indicative of a need for a refinement of the cosmological model, hence the importance of measuring this parameter. Actually, the standard model of cosmology predicts Neff = 3.046, the difference from three being due to a small amount of entropy inherited by neutrinos from the electron/positron annihilation process (see [114]). The latest observational constraint, obtained by combining Planck data with other surveys, is N = 3.15 0.23 ([137]). As the rest eff ± of the Planck results, this estimation confers credibility to the cosmological scenario currently adopted.
5 It is valid only when T me, me being the electron mass. See e.g. [104] for a demonstration. 6See section 2.3.2 for discussions on the form of the phase-space distribution function of neutri- nos.
64 2.2. Why are cosmologists interested in neutrinos?
Impact of Neff on primordial abundances
It is also at a temperature of the order of 1 MeV that neutrons are trapped by protons for the first time, producing deuterium,
n + p 2H + γ, (2.3) ↔ and initiating subsequent formations of light elements, mainly 3He, 4He and 7Li. This key period of the history of the universe is called primordial nucleosynthesis, or Big Bang nucleosynthesis (see e.g. [156, 124, 174, 95, 140] for reviews on Big Bang nucleosynthesis). So, do neutrinos intervene in this? Equation (2.2) tells one that extra relativistic species (with respect to photons) enhance the primordial radiation energy density. Consequently, they affect the expansion rate of the universe before and during the Big Bang nucleosynthesis (see the Friedmann equations (1.9) and (1.10)). It changes in particular the freezing temperature of the neutron-to-proton ratio, and thus the 4 primordial abundance of He. More precisely, the larger Neff is, the quicker the universe expands so the earlier the neutron-to-proton ratio is frozen. It leads to a larger relic neutron abundance and eventually to a higher production of 4He. The theoretical impact of Neff on primordial abundances is depicted in figure 2.1. Yp characterizes the 4He abundance7. It is this abundance that is the most impacted but one can see that all primordial abundances of light elements are influenced by the effective number of neutrino species. The measurement of such abundances is an investigation tool widely exploited in precision cosmology. It is also part of the elements that favored historically the Big Bang model. One can find a recent update of the predictions related to primordial nucleosynthesis, based on the results of the Planck mission, in [36]. The dependence of primordial abundances on Neff illustrates the importance of including neutrinos in modern cosmological models.
Impact of Neff on the anisotropies of the cosmic microwave background
Given the richness of the information contained in CMB anisotropies, it seems crucial to determine if the presence of neutrinos makes a difference. Recombination marks the end of thermal equilibrium between baryons, elec-
7 n4He Yp = 4 , n standing for number densities and B for baryons. nB
65 2.2. Why are cosmologists interested in neutrinos?
Figure 2.1: Big Bang nucleosynthesis abundance predictions as a function of the baryon-to-photon ratio η, for Neff = 2 to 7. The bands show the 1σ error bars. All bands are centered on Neff = 3. Authors: Cyburt et al.,[47].
trons and photons. From this time, photons propagate freely and carry with them physical characteristics of their last scattering point. In particular, after decoupling, each photon can be associated with a local temperature T + δT (η, ~x, nˆ), wheren ˆ gives the direction of propagation. It is therefore possible to construct a CMB tem- perature map. The relevant information is contained in the angular scale, shape and amplitude of its angular correlation function. To be more precise, temperature anisotropies on the last scattering surface are usually expanded in the following way,
δT (ˆn) = a Y (ˆn). (2.4) T lm lm Xlm Then one defines the harmonic power spectrum Cl as
C = a a∗ m. (2.5) l h lm lmi ∀
66 2.2. Why are cosmologists interested in neutrinos?
This power spectrum is isotropic, i.e. it depends on l but not on m. Here, and throughout the manuscript, the symbol “ ” denotes ensemble averages of variables h i X depending on scalar stochastic8 fields, s(x). Those fields are random variables: they can be described with the help of distribution functions, which determine the probability for them to have given values at given positions. Explicitly,
X X(s , s , ...) (s , s , ...)ds ds ..., (2.6) h i ≡ 1 2 P 1 2 1 2 Z with (s , s , ...)ds ds ... the probability for s to have a value between s and P 1 2 1 2 1 s1+ds1 at position x1, between s2 and s2+ds2 at position x2, etc. When the number of variables is sufficient (which is usually not restrictive in cosmology, especially given the richness of the latest observation surveys), it is commonly assumed that ensemble averages are equivalent to space averages (ergodic hypothesis).
Isolating the theoretical impact of Neff on this power spectrum is a subtle task, described in full detail in [104]. The difficulty arises from the fact that a large variety of effects, originating from several species, are entangled in the Cl’s. On the one hand, at the background level, each cosmic component affects the time evolution of the scale factor and, on the other hand, perturbations of decoupled species disturb the growth of metric perturbations. Depending on the effect that one wants to bring to light, one decides to fix one or another set of parameters of the model. As an illustration, figure 2.2 highlights the effect of Neff for three different settings. The tuning of the parameters has been done so that the features of the upper curve can entirely be interpreted in terms of direct effects due to neutrino perturbations, whereas the lower curve is more realistic but mixes background and perturbative effects. More precisely, the rapid decrease at high l’s visible on the lower curve is a background effect due to diffusion damping, or Silk damping ([166]).
Indeed, varying Neff affects the typical length of the random walk that character- izes interactions between photons and electrons. In multipole space, this length is 9 hidden in the parameter lD and the envelope of the curve is controlled by a fac- tor exp[ (l/l )2]. Concerning the perturbative level, basically, metric fluctuations − D are reduced on scales at which neutrinos free-stream because they do not cluster
8The reason why statistics occupies inevitably a place in cosmology is briefly addressed in the item “Impact of Neff on the matter power spectrum” of the present subsection. 9 lD kD(η0 ηLS), where kD is the wavenumber associated with the comoving distance traveled ∼ − by a photon during Thomson diffusion, η0 is the present comoving time and ηLS is the last scattering comoving time.
67 2.2. Why are cosmologists interested in neutrinos? while free-streaming. One can refer to e.g. [93,9, 104] for analytic interpretations of the patterns of the curves. In the plot 2.2, neutrino masses have been fixed to zero for the contribution of the mass not to mix with the one of Neff (see section
2.2.3 for discussions on mass effects). Note chiefly that the suppression of the Cl’s that appears when one switches from a neutrinoless model to a model with three neutrino species is close to 20% in the case represented by the upper curve, which is non-negligible in precision cosmology.
Figure 2.2: CMB temperature spectrum for models with Neff = 3.046 divided by the spectrum of a model with Neff = 0. zeq is the redshift of equality between matter and radiation. zΛ is the redshift of equality between matter and dark energy. lD is a multipole parameter which characterizes diffusion damping. It has been fixed in the second case in order to get rid of diffusion damping effects. ISW stands for Integrated Sachs-Wolfe effect, whose contribution is tiny in this context. Authors: Lesgourgues et al.,[104].
Impact of Neff on the matter power spectrum
Strictly speaking, density perturbations are realizations of stochastic fields. Ac- cording to the inflation theory, stochasticity is present from the beginning because inhomogeneities originate from quantum fluctuations, whose evolution is not deter- ministic. Furthermore, the ignorance of the initial conditions and the existence of unobservable regions confirm the necessity of a statistical approach in cosmology.
68 2.2. Why are cosmologists interested in neutrinos?
Besides, it is very common to describe the observable universe in terms of vari- ables belonging to the reciprocal space (in particular when it comes to compute correlation functions). As usual, one switches from real space to reciprocal space with the help of Fourier transforms. In this thesis, the convention that we use is (for any field X)
d3k X(x) = X(k) exp(ik.x), (2.7) (2π)3/2 Z d3x X(k) = X(x) exp( ik.x). (2.8) (2π)3/2 − Z Since initial fluctuations of density emerge from a field which is free in the sense of quantum mechanics, they are usually assumed to be Gaussianly distributed (which is in very good agreement with the latest results of the Planck mission, [136]). In this context, all the needed information regarding initial time derives from two- point correlation functions (but the observables that one uses to study large-scale structures at a later time are in general multi-point correlation functions). It is a consequence of Wick’s theorem. More precisely, cosmologists introduced a quantity P (k), omnipresent in studies of large-scale structure formation and defined so that
δ (~k)δ (~k0) = P (k)δ (~k + ~k0), (2.9) h m m i D where δD is the Dirac delta function and δm is the matter density contrast, i.e. ρm δm = 1 withρ ¯m the spatial average of the matter density ρm. P (k) is called ρ¯m − matter power spectrum. According to Wick’s theorem, higher order initial-time correlations are zero if they involve an odd number of fields. In contrast, in case of an even number 2N of fields, then (2N 1)!! terms contribute to the correlators. − Those terms correspond to all different pairings of the 2N fields:
δ1 (~k )...δ2N (~k ) = δi (~k )...δj (~k ) . (2.10) h m 1 m 2N i h m i m j i all pairX associations pairsY (i,j) If one assumes, in addition, that initial conditions are adiabatic, all species share initially the same density contrast (with only a known species-dependent factor in front of it). It means that primordial power spectra of photons and neutrinos (and of any hypothetical other species, like dark energy) are proportional to the
69 2.2. Why are cosmologists interested in neutrinos? primordial matter power spectrum. More details about the meaning of adiabaticity will be given in section 4.3.3. We will see that it can generally be considered as a reasonable assumption. As in the case of the power spectrum of CMB anisotropies, it is not possible to disentangle totally the contributions of the various parameters to the matter power spectrum (again, I refer the reader to [104] for a thorough analysis). Illustratively, figure 2.3 shows how the phase and amplitude of BAO, imprinted in the matter power spectrum, are altered by the presence of massless neutrinos. The lower curve mixes neutrino perturbation effects with a background effect due to the fact that the 10 baryon density ωB is not fixed: a non-zero value of Neff increases ωB, leading to an earlier release of baryons. On the contrary, ωB is fixed in the case corresponding to the upper curve whereas the baryon-to-cold dark matter ratio varies, this latter variation altering the BAO amplitude. In both cases, neutrino perturbations damp the BAO amplitude and induce a phase shift in it. All these examples prove that even massless neutrinos would contribute to the development of the universe. Yet, to be in tune with reality, neutrino masses should not be ignored.
2.2.3 Neutrino masses: small but decisive
Current estimates
It is thanks to neutrino oscillation experiments that neutrino masses have been revealed. Such experiments are in fact interference experiments, which can only de- tect differences between squares of neutrino masses (see [119, 66, 21,8, 118, 130, 71] for precisions). The absolute neutrino mass scale being still out of reach11, many efforts realized in order to constrain neutrino masses consist in putting bounds on the total neutrino mass Mν = i mνi , where each value of i denotes a neutrino species. The number of studies carriedP out for that purpose is phenomenal. Predic- tions vary from one study to another, depending on the selected data and on the cosmological assumptions made to interpret them. Table 2.1 presents few examples of mass constraints (including the estimation from the 2015 release of the Planck
10 2 ωB is part of the six parameters of the standard model of cosmology. It is defined as ωB = ΩBh , 8πG with ΩB = 2 mNnB, mN being the average nucleon mass. 3H0 11It is nevertheless reasonable to hope that, in the near future, large-scale structure surveys will unveil the absolute neutrino mass scale.
70 2.2. Why are cosmologists interested in neutrinos?
Figure 2.3: Matter power spectra for models with Neff = 3.046 divided by the power 8πG spectrum of a model with Neff = 0. ΩM = 2 ρm,0, with ρm,0 the present energy 3H0 density of the matter fluid. ωC is the cold dark matter density and ωB is the baryon density. Authors: Lesgourgues et al.,[104].
mission, [137]) among many others.
Observables Year and reference Result CMB+BAO 2015, [137] Mν < 0.23 eV CMB+BAO+Lyman-α forest 2014, [125] Mν < 0.14 eV CMB+BAO 2014, [135] Mν < 0.23 eV Galaxy survey+CMB+BAO 2014, [147] Mν < 0.18 eV Galaxy survey+CMB+BAO+Supernovae 2010, [115] Mν < 0.33 eV
Table 2.1: Some 2σ-upper bounds on the total neutrino mass.
What cosmological repercussions can one expect from sub-eV masses?
Impact of Mν on the anisotropies of the cosmic microwave background
Assuming the existence of three equal-mass neutrino species and adjusting judi- ciously the other parameters, the authors of [104] highlight the impact of non-zero neutrino masses on the CMB temperature spectrum. It is presented in figure 2.4.
71 2.2. Why are cosmologists interested in neutrinos?
Figure 2.4: CMB temperature spectra for models with either M = 3 0.3 eV or ν × M = 3 0.6 eV divided by the spectrum of a model with three massless neutrinos ν × sharing the same temperature. lpeak, which is kept constant here, is a parameter fixing the l’s at which acoustic oscillations peak. Authors: Lesgourgues et al.,[104].
The leading effect, impacting the multipoles for which l < 20, is a background effect. At late times, dark energy becomes predominant, which makes metric fluctu- ations decay and therefore disturbs photons in their path between the last scattering surface and observers. This phenomenon, known as late integrated Sachs-Wolfe ef- fect12, is affected by massive neutrinos because the time of equality between matter and dark energy depends on neutrino masses. In figure 2.4, the integrated Sachs- Wolfe effect is controlled by the parameter ISW, which has been fixed on the upper curves to differentiate ISW from other phenomena. The rise visible at l > 500 is also a background effect, corresponding to the Silk damping induced by the neutrinos that are already non-relativistic at decoupling. Between l = 20 and l = 500, features of the two lower curves are mainly at- tributed to the early integrated Sachs-Wolfe effect, which is simply the disclosure of the impact of neutrino masses on metric perturbations after decoupling. This perturbation effect, in the order of 3% for 0.3 eV masses, can seem tiny but it is
12The Sachs of the Sachs-Wolfe effect, Rainer K. Sachs, is the same as the Sachs of the Sachs equation discussed in chapter1.
72 2.2. Why are cosmologists interested in neutrinos? within the range of precision targetted today in cosmology.
Impact of Mν on the matter power spectrum
As already mentioned, in cosmological models, one usually assumes that initial con- ditions are adiabatic. So, as long as free-streaming can be neglected, assigning a mass to neutrinos does not affect the matter power spectrum13 because such mas- sive particles behave as cold dark matter. However, the free-streaming of massive neutrinos can not be mimicked by any other species. During free-streaming, mas- sive neutrinos increase the expansion rate of the universe without clustering under the influence of gravitational instability. Such a behavior slows down the growth of structure, or equivalently damps the matter power spectrum on small scales. This is well illustrated in figure 2.5.
Figure 2.5: Steplike suppression of the matter power spectrum due to neutrino mass. The power spectrum of a standard cosmological model with two massless and one massive species has been divided by that of a massless model, for several values of mν between 0.05 eV and 0.50 eV, spaced by 0.05 eV. All spectra have the same primordial power spectrum and the same parameters (Ωm, ωm, ωB). kNR is the free-streaming wavenumber at the non-relativistic transition (k 5.10 3h/Mpc). NR ∼ − Authors: Lesgourgues et al.,[104].
13It is true provided that the total matter density and the primordial power spectrum are un- changed.
73 2.3. How cosmic neutrinos are studied
An analytic justification, based on linear perturbation theory, can be found in [104] (see also references herein). In fact, while particularly enriching, adopting a linear approach to model this kind of phenomenon is not sufficient. Indeed, on small scales and at small redshifts, the evolution of cosmological perturbations is not linear14. Actually, the scales that are the most sensitive to neutrino masses (roughly k > 0.1h/Mpc) correspond to the nonlinear part of the matter power spectrum. Hence the importance of modeling the behavior of massive neutrinos beyond the linear regime when trying to describe the formation of the large-scale structure of the universe. Another argument along this is that a large amount of cosmological data would be wasted if the absence of theoretical counterpart made one skip the observations corresponding to nonlinear scales. Besides, it has been shown in several studies that neglecting the nonlinear corrections induces a spurious behavior on small wavenumbers due to a violation of momentum conservation (see [82, 91, 126, 27]). So a common effort is currently carried out to investigate the nonlinear regime (see section 2.3.2), in particular in the field of numerical simulations. Achieving this objective is to a large extent what motivates the work undertaken for my PhD thesis, during which I developed a new analytic description of the nonlinear evolution of massive neutrinos (see chapters4,5 and6).
2.3 How cosmic neutrinos are studied
2.3.1 Observation
It is fortunate that there exists a great variety of astrophysical sources because all are not sensitive to the same properties. The recent explosion in the number of high precision observational projects allows to multiply the possible combinations of data, increasing dramatically the relevance of the constraints. CMB, BAO and large galaxy surveys have already been mentioned as observables useful for neutrino cosmology but the list can be extended. One can cite for instance quasars, which are objects particularly luminous and distant. In a range of frequencies called Lyman-α forest, their spectra are good tracers of hydrogen density fluctuations on mildly nonlinear scales, and thus indi- rectly of the matter perturbations that govern the matter power spectrum (see e.g.
14Density perturbations grow with time and only extremely small perturbations can be treated at linear level (see chapter3 for precisions).
74 2.3. How cosmic neutrinos are studied
[186, 188, 152] for recent studies in which the Lyman-α forest is put at the service of neutrino cosmology). Gravitational lensing, i.e. distortion of galaxy images due to density fluctuations along the line of sight, is also useful in this context ([187, 178]). Note that the upcoming Euclid mission will give particular importance to the examination of gravitational lensing. Even cosmic voids contribute to the exploration of neutrino properties ([188]). Giving an exhaustive list and dissecting the advantages and drawbacks of each kind of observation is not the purpose here (an enlightening overview adapted to neutrino cosmology is nevertheless given in [104]). The intent was rather to ex- emplify the richness of observational cosmology, which is a direct gateway between infinitely large and infinitely small.
2.3.2 Modeling
The Boltzmann hierarchy
The standard way of modeling massive neutrinos is to consider them as a non-cold fluid, i.e. a fluid made of particles that are still relativistic when they decouple. As soon as the fluid is out of equilibrium, a kinetic approach is necessary to depict what happens. It is therefore convenient to describe the fluid with its phase-space distri- µ µ 15 bution function, f(x , pµ), where x is a space coordinate and pµ is the conjugate momentum of xµ, i.e.
p = mu with u = g dxν/ ds2. (2.11) µ µ µ µν − p m is the particle mass so, if neutrino masses are not degenerate, m is different for each species. Using standard results of general relativity, it is straightforward to show that p satisfies the mass-shell condition pµp = m2. µ µ − The starting point is that, in the absence of interactions, the Liouville operator,
df (f) (xµ(λ), p (λ)) , (2.12) L ≡ dλ µ is zero16. In this equation, λ is an arbitrary parameter. Note that the possibility
15Actually, there are several ways to define a phase-space momentum but only the conjugate 3 i 3 momentum is defined so that d x d pi is the phase-space unit volume and that the number of 6 i 3 i 3 particles in it is d N = f(x , pi)d x d pi (see the reference paper [111]). 16 To demonstrate it, one can e.g. write the geodesic equation for uµ and use the fact that f is
75 2.3. How cosmic neutrinos are studied of interacting neutrinos has been explored in many non-standard models (see the references given in [104]) and that the eventuality of self-interacting neutrino fluids is disfavored by observations ([51,4]). The time evolution of f is then governed by an equation called collisionless Boltzmann equation or Vlasov equation. In terms of the conformal time η, it reads:
i ∂f dx ∂f dpi ∂f + i + = 0. (2.13) ∂η dη ∂x dη ∂pi During the equilibrium stage, neutrinos are expected to be characterized by a relativistic Fermi-Dirac distribution in which the chemical potential is negligible,
q 1 f(q) = 1 + exp − , (2.14) aT where T is the temperature and q is the physical momentum17. The absence of chemical potential is justified theoretically, thanks to quantum field theory consid- erations, in [104]. Besides, this assertion can be tested experimentally. When equilibrium breaks down, the previous description is not valid anymore. However, the Boltzmann equation can be studied perturbatively. To that aim, f is decomposed into a homogeneous part f0 (given by equation (2.14)) and a perturbation f0Ψ,
i i f x , q = f0 (q) 1 + Ψ x , q . (2.15)
Note that, in the relativistic regime, the expression of Ψ is known. Indeed one can show that18, before the non-relativistic transition, f keeps a Fermi-Dirac form except that it exhibits local fluctuations of temperature:
1 q − f xi, q = 1 + exp in the relativistic regime. (2.16) a (T + δT (xi)) conserved along geodesics when particles do not interact. 17The physical momentum is defined so that the energy measured by a comoving observer is 2 2 2 i = m +(q/a) . Explicit relations between pi and q (in different metrics) are given in the papers presented in sections 4.4 and 5.4. 18 The theoretical argument is the writing of the geodesic equation for qi. Indeed, provided that q m, the momentum shift induced by metric perturbations does not depend on the momentum itself (see [104]).
76 2.3. How cosmic neutrinos are studied
Hence, one has
δT (xi) d ln f Ψ xi, q = 0 in the relativistic regime. (2.17) − T d ln q More generally, one can derive an equation governing the time evolution of Ψ. Its form depends on the gauge which is chosen, i.e. on the form of the metric. As will be explained in chapter3, a convenient way to take inhomogeneities into account is to add spatial perturbations in the homogeneous and isotropic Friedmann-Lemaˆıtre metric (1.4). In the gauge called conformal Newtonian gauge, those perturbations are denoted φ and ψ and the metric reads (assuming a flat universe)
ds2 = a2 (η) (1 + 2ψ) dη2 + (1 2φ) dxidxjδ . (2.18) − − ij Calculations are usually made in the linear regime, which means that all couplings between any perturbations are neglected. In this framework, qi = a2(1 φ)pi (see dxi pi − [57]). Noticing that = and writing the geodesic equation in the metric dη p0 dp (2.18) to compute i , one gets from equation (2.13), dη q d ln f (q) a ∂ Ψ + nˆi∂ Ψ + 0 ∂ φ nˆi∂ ψ = 0, (2.19) η a i d ln q η − q i wheren ˆ gives the direction of the physical momentum. Besides, in momentum space, the only dependence on k is through its angle with nˆ. Hence, one can define α k.ˆ nˆ and rewrite the linearized Boltzmann equation ≡ as q a ∂ Ψ˜ + iαk Ψ˜ + ∂ φ iαk ψ = 0, (2.20) η a η − q 1 d log f (q) − where Ψ˜ 0 Ψ. ≡ d log q The standard treatment is then to expand Ψ˜ in Legendre polynomials,
Ψ˜ = ( i)`Ψ˜ P (α), (2.21) − ` ` X` where P`(α) is the Legendre polynomial of order `. By plugging this expansion into
77 2.3. How cosmic neutrinos are studied the Boltzmann equation (2.20), one obtains the Boltzmann hierarchy,
qk ∂ Ψ˜ (η, q) = Ψ˜ (η, q) ∂ φ(η) (2.22) η 0 −3a 1 − η qk 2 ak ∂ Ψ˜ (η, q) = Ψ˜ (η, q) Ψ˜ (η, q) ψ(η), (2.23) η 1 a 0 − 5 2 − q qk ` ` + 1 ∂ηΨ˜ `(η, q) = Ψ˜ ` 1(η, q) Ψ˜ `+1(η, q) (` 2). (2.24) a 2` 1 − − 2` + 3 ≥ − It should be mentioned that the Boltzmann hierarchy is often written in a slightly different form, obtained by replacing the decomposition (2.21) by Ψ˜ = ( i)`(2l+ ` − ˜ 1)Ψ` P`(α). As highlighted in [111], the interest of the introductionP of a factor (2l + 1) is that it makes more natural the writing of the harmonic coefficients Ψ˜ `’s in terms of spherical Bessel functions when neutrinos free stream. It is useful in particular for the formulation of truncation schemes (see the next section). It is from the integration of the Boltzmann hierarchy that the relevant physical quantities are computed. Note that this approach is very general. In particular, it applies to photons provided that the Fermi-Dirac distribution is replaced by the Bose-Einstein one. It means that there exists a CMB Boltzmann hierarchy too. Codes dedicated to the numerical integration of such hierarchies are called Boltz- mann codes (see below). The Boltzmann approach is very powerful. However, the fact that it is a linear theory is too restrictive to be considered totally satisfactory.
Boltzmann codes
The Boltzmann hierarchy is infinite. It can nevertheless be handled by numerical codes since it is convergent. The first scheme that comes to mind to integrate it numerically consists in finding theoretical arguments to perform a sharp truncation ˜ at a given lmax, i.e. to set a Ψlmax brutally to zero. In this context, the multipoles ˜ of interest must be i) of a lower order and ii) sufficiently far from Ψlmax in the hierarchy for their evolution not to be impacted by the truncation. In practice, the only multipoles whose integration over phase-space momenta gives macroscopic quantities useful for cosmology correspond to l = 0, 1, 2. They allow to compute (1) (1) respectively the linear density perturbation ρν , velocity divergence θν and shear (1) stress σν :
78 2.3. How cosmic neutrinos are studied
f (q) d log f (q) ρ (1)(η) = 4π q2dq 0 0 Ψ˜ (η, q), (2.25) ν a3 d log q 0 Z 4π f (q) d log f (q) q (ρ (0) + P (0))θ (1)(η) = q2dq 0 0 Ψ˜ (η, q), (2.26) ν ν ν 3 a3 d log q a 1 Z 8π f (q) d log f (q) q2 (ρ (0) + P (0))σ (1)(η) = q2dq 0 0 Ψ˜ (η, q), (2.27) ν ν ν 15 a3 d log q a22 2 Z where superscripts (0) denote background quantities and P stands for pressure. As explained in [104], the “sharp truncation method” has proven to be accurate when lmax is sufficiently large. The problem is that, despite the truncation, the number of equations that one needs to integrate is huge. Indeed, lmax must be chosen well above19 2 and the hierarchy is coupled to other equations describing baryons and dark matter. Moreover, each calculation should be performed for a large number of wavenumbers and, in the case of neutrinos, several momenta. In practice, the resulting numerical cost is prohibitive. A more subtle procedure is proposed in [111]. The authors show in particular that, for free-streaming species, the harmonic coefficients of the Boltzmann hierar- chy are easily expressible in terms of the spherical Bessel functions (especially when a factor (2l + 1) has previously been introduced in the harmonic decomposition
(2.21)). Recurrence relations can thus be used to relate Ψ˜ l+1 to Ψ˜ l and Ψ˜ l 1, which − allows to control the impact of a truncation down to l 2. In usual Boltzmann max − codes, it is this strategy that is adopted. In this framework, experience shows that choosing l 10 has very little impact on the three first multipoles. Typically, max ∼ depending on the desired level of precision, lmax is at most 20 in usual codes. For instance, in the numerical tests performed in the study that I will present in chapter
4, we truncated the Boltzmann hierarchy at lmax = 6. The main reference Boltzmann codes are CMBFAST ([162]), CAMB ([107]), CMBEASY ([55]) and CLASS ([28]). Note that, for codes which include massive neutrinos, it is the integration of massive neutrino equations that is the most time-consuming part because of the need to sum over momenta (see [106] for the presentation of a method, called quadrature approach, allowing to keep the number of momenta reasonable).
19 In [104], the authors give kηnr as typical value of lmax for massive neutrinos, ηnr being the conformal time at the non-relativistic transition.
79 2.3. How cosmic neutrinos are studied
Modeling the nonlinear evolution of neutrinos
So far, most descriptions of the nonlinear evolution of neutrinos rely on N-body simulations. One can find a presentation of the algorithms at play in such simu- lations in [104]. In this context, one of the most powerful code is the one called GADGET ([170]). Interestingly, the numerical works [29, 186, 23, 88, 189] conclude that, in the nonlinear regime, the steplike suppression induced by massive neutrinos on the matter power spectrum is replaced by a “spoon-shaped” suppression, see figure 2.6. Since the total matter density is fixed in figure 2.6, it is not surprising that non- zero neutrino masses induce an extra suppression of the nonlinear matter power spectrum. Indeed, in that case, the density perturbations of cold dark matter evolve more gently, which postpones the nonlinear clustering that enhances the matter power spectrum. Upcoming large-scale surveys should exhibit features of this subtle scale dependence of the suppression of the matter power spectrum.
Figure 2.6: Spoon-shaped suppression of the matter power spectrum due to neu- trino masses. The solid lines show the power spectrum of a standard model with three massive species of total mass Mν = 0.6 eV, divided by that of a massless model with the same parameters (Ωm, ωm). Baryons have been neglected in these simulations. In each plot, the two solid lines correspond to two different N-body simulations with different resolution scales. The step-shaped dashed line shows the linear predictions whereas the spoon-shaped dashed curve represents the nonlinear prediction accounting for massive neutrino effects. Authors: Bird et al.,[23].
Such results, obtained from N-body simulations, are reliable and very enlight-
80 2.3. How cosmic neutrinos are studied ening. However, the successes of numerical cosmology do not exclude the need for analytic models, in particular to avoid heavy calculations and to be able to explore a wider variety of cosmological parameters and scales. Note that the recent theo- retical study [27] manages to explicitly predict an extra suppression of the matter power spectrum due to nonlinearities thanks to nonlinear cosmological perturbation theory. Still on the analytic side, a natural idea is to extend the Boltzmann hierarchy to the nonlinear regime. It has been done in [184], in which the moments of f,
qi qj qk f Aij...k d3q ... , (2.28) ≡ a a a a3 Z are studied without performing any linearization. In the conformal Newtonian gauge (2.18), the nonlinear hierarchy resulting from the collisionless Boltzmann equation is
∂ Ai1...in + ( ∂ φ) (n + 3)Ai1...in (n 1)Ai1...injj η H − η − − n n i1...im 1im+1...i n i1...im 1im+1...injj + (∂im ψ)A − + (∂im φ)A − m=1 m=1 X X +(1 + φ + ψ)∂ Ai1...inj + [(2 n)∂ ψ (2 + n)∂ φ] Ai1...inj = 0. (2.29) j − j − j
ij...k For physical interpretation, a non-trivial mapping between the A ’s and the Ψl’s is provided in [184]:
f A = 4π q2dq 0 (1 + Ψ ), (2.30) a3 0 Z q2 f Aii = 4π q2dq 0 (1 + Ψ ), (2.31) a22 a3 0 Z 4π q f ik Ai = q2dq 0 Ψ , (2.32) i 3 a a3 1 Z 1 8π q2 f k2δ j + (ik )(ik ) Aij = k2 q2dq 0 Ψ , (2.33) 3 i i j 15 a22 a3 2 Z 1 8π q4 f k2δ j + (ik )(ik ) Aijkk = k2 q2dq 0 Ψ , (2.34) 3 i i j 15 a44 a3 2 Z 1 16π q3 f k2δ j + (ik )(ik ) (ik )Aijk = k3 q2dq 0 Ψ ... (2.35) 3 i i j k − 45 a33 a3 1 Z
81 2.3. How cosmic neutrinos are studied
i (0) One can see in particular from (2.25) that A = ρν, from (2.26) that ikiA = (ρν + 1 P (0))θ and from (2.27) that k2δ j + (ik )(ik ) Aij = k2 ρ (0) + P (0) σ . Be- ν ν 3 i i j ν ν ν sides, Aii is nothing but three times the pressure. Unfortunately,h the facti that the nonlinear hierarchy (2.29) is multi-dimensional makes its numerical integra- tion cumbersome, especially because developing a satisfying truncation scheme is extremely difficult. Independently on this attempt, and as will be explained in chapter3, there exist robust analytic approaches that study the behavior of the nonlinear power spectrum. Originally, neutrinos were not included in such works. Then some extensions have been proposed to account for a neutrino component (see in particular [198, 154, 105]). The limitation is that massive neutrinos are treated at the linear level in those studies, which are furthermore valid in a very small range of scales (up to k 0.2h/Mpc for z > 2 and k 0.15h/Mpc for z = 1). Some recent improvements ∼ ∼ will also be presented in chapter3. One can cite in particular the studies [27] and [81].
82 Chapter 3
Refining perturbation theory to refine the description of the growth of structure
I included a glimpse of neutrino cosmology in my manuscript because, to understand the interest of my work, one has to be aware of the crucial role played by neutrino cosmology in precision cosmology. However, studying neutrino phenomenology was not the objective of my thesis. My everyday task consisted rather in very gen- eral thoughts regarding cosmological perturbation theory beyond the linear regime and beyond the Newtonian approximation. The link with neutrinos is that such considerations suit perfectly for the study of the impact of neutrinos on the nonlin- ear matter power spectrum. In this chapter, I introduce all the tools of nonlinear cosmological perturbation theory that I used in the studies presented in the three following chapters. Since physical conditions evolve drastically over the cosmological eras, the sim- plifying hypotheses that can be formulated in the models evolve accordingly. Some stages of the history of the universe are thus more difficult to describe than others. In particular, modeling analytically the late-time evolution of perturbations (dur- ing which the large-scale structure emerges) is very challenging. In this context, numerical cosmology is very helpful. Indeed, advanced numerical simulations such as the Horizon-AGN simulation ([56]) or the Millennium simulation ([171]) provide insightful pictures of the construction of the cosmic web. On the theoretical side, important advances have been realized since the 2000s.
83 3.1. A brief presentation of cosmological perturbation theory
The initiating element has been standard perturbation theory then several refine- ments have followed. A review of standard perturbation theory is given in [13] and an outline of the subsequent developments can be found in [12]. In this chap- ter, I introduce the concepts of perturbation theory on which I based my work on neutrinos. The approach that I present in the next chapters is indeed a relativistic extension of the standard formalism, which had originally been designed to describe the emergence of gravitational instabilities in the cold dark matter component.
3.1 A brief presentation of cosmological perturbation theory
Cosmological perturbation theory is based on the observation that, on very large scales, the universe is quasi-homogeneous. More precisely, maps obtained by prob- ing all the directions of the sky exhibit relative spatial fluctuations in the order 5 of 10− . This is certified by the CMB temperature maps obtained by the Planck satellite, as illustrated in figure 3.1. Consequently, strictly speaking, the geometry of the universe can not be characterized by the Friedmann-Lemaˆıtremetric (1.8), even on large scales. However, it seems reasonable to assume that, on appropriate scales, the real metric differs little from the homogeneous and isotropic one. This is the reason why cosmologists usually consider perturbed Friedmann-Lemaˆıtremet- rics, that is to say metrics in which inhomogeneous quantities, assumed to be small compared with the background values, have been introduced (see the pioneer work [108], reviewed in [109, 193, 126, 101, 123, 63] and standardly used since then). A generic form of a perturbed Friedmann-Lemaˆıtremetric is
ds2 = a2(η) (1 + 2A) dη2 + 2B dxidη + (γ + h ) dxidxj , (3.1) − i ij ij where the metric perturbations are A, Bi and hij. As expected in general relativity, a given phenomenon can be described by different equations, their forms depending on the point of view adopted. In this context it is the form chosen for the metric perturbations, usually called choice of gauge, which is determining. The conformal Newtonian gauge, given by (2.18), is an example of gauge widely used in cosmology. It is characterized by the absence of time-space perturbations and by a diagonal space-space perturbation. Many studies have also been carried out using the Synchronous gauge, characterized by
84 3.2. The Vlasov-Poisson system, cornerstone of standard perturbation theory
the absence of time-time and time-space perturbations (i.e. by A = 0 and Bi = 0). The predictions of cosmological perturbation theory in those two gauges have been studied in full detail in [111]. Besides, a presentation of the most common gauges and of the transformations laws allowing to switch from one to another is given in [131]. In cosmology, a particular effort is devoted to the identification of quantities that are gauge-independent (see [6, 60]). Note that the quantities that can be observed, called observables, naturally satisfy this property. As described by the Einstein equations, such metric perturbations generate den- sity perturbations, which in turn affect the geometry of the spacetime, and so forth. Understanding the time evolution of density perturbations under the influence of gravity in an expanding universe is precisely the purpose of cosmological perturba- tion theory.
Figure 3.1: CMB temperature map obtained from the SMICA pipeline of the Planck satellite (February 2015). c EUROPEAN SPACE AGENCY - PLANCK
COLLABORATION.
3.2 The Vlasov-Poisson system, cornerstone of stan- dard perturbation theory
In standard perturbation theory, the large-scale structures observed in galaxy sur- veys are interpreted as the result of the clustering of collisionless cold dark matter, represented by a non-relativistic pressureless fluid. The particles of the fluid un- dergo gravitational interaction only and have a mass m. In this framework, the
85 3.2. The Vlasov-Poisson system, cornerstone of standard perturbation theory starting point of the description is again the Vlasov equation1
i ∂f dx ∂f dpi ∂f + i + = 0, (3.2) ∂T dT ∂x dT ∂pi where T denotes here the cosmic time. Because of the expansion of the universe, a distinction should be made between comoving distances ~x and physical distances ~r = a(T )~x. The same distinction ap- plies to velocities: peculiar velocities ~u must not be confused with physical velocities 2 ~v = ~u + ~vH, where ~vH is the velocity associated with the Hubble flow . For ~p to be the conjugate momentum of ~x in the Lagrangian describing the dynamics, it has to be defined as
~p = ma~u. (3.3)
Besides, the study is much simplified by the fact that cold species can be studied in the Newtonian approximation. In this context, it is straightforward to notice that
d~x ~p = (3.4) dT ma2 and
d~p = m~ Φ, (3.5) dT − ∇ Φ being the gravitational potential. It is convenient to define macroscopic fields from f and to derive their time evolution from the first moments of the Vlasov equation. Since the phase-space variables have been chosen so that d3~x d3~pf is the number of particles contained in a phase-space comoving unit volume, the density ρ of the cold dark matter fluid, i.e. the total mass contained in a physical unit volume, is given by
m ρ(~x,T ) = f(~x,~p,T )d3~p. (3.6) a3 Z 1The Vlasov equation can be applied because two-point interactions are neglected and the number of particles is conserved. d~x da 2~u = a and ~v = ~u + ~x. dT dT
86 3.3. The single-flow approximation and its consequences
The velocity field is defined as the average of the phase-space velocities:
1 pi 3 ui(~x,T ) = f(~x,~p,T )d ~p. (3.7) f(~x,~p,T )d3~p ma Z The velocity dispersion field σRij(~x,T ) is defined so that
1 pi pj 3 ui(~x,T )uj(~x,T ) + σij(~x,T ) = f(~x,~p,T )d ~p. (3.8) f(~x,~p,T )d3~p ma ma Z Finally, one introduces the densityR contrast field δ(~x,t):
ρ(~x,T ) δ(~x,T ) = 1, (3.9) ρ¯ − whereρ ¯ is the spatial average of the density field. In the Newtonian approximation, the Poisson equation then imposes3
∆Φ = 4πGa2ρδ.¯ (3.10)
Equations (3.2) and (3.10) form the Vlasov-Poisson system. They are the basis on which all the calculations of standard perturbation theory rely. From the Vlasov equation, it is easy to show that the fields previously defined satisfy the continuity and Euler equations4
∂δ(~x,T ) 1 + ∂ [(1 + δ(~x,T )) u (~x,T )] = 0, (3.11) ∂T a i i
∂u (~x,T ) 1 da 1 1 ∂ [ρ(~x,T )σ (~x,T )] i + u (~x,T )+ u (~x,T )∂ u (~x,T ) = ∂ Φ(~x,T ) j ij . ∂T a dT i a j j i −a i − aρ(~x,T ) (3.12)
3.3 The single-flow approximation and its consequences
Cold fluids do not experience much thermal motion. Consequently, in the early stages of gravitational instability, one expects velocity dispersion to be small com- pared with the velocity gradients induced by density fluctuations. It is only at more
3Here, it is assumed that the gravitational effect of other cosmic components is negligible. 4The Einstein convention applies in those equations.
87 3.3. The single-flow approximation and its consequences advanced stages that velocity dispersion becomes effective. Indeed, gravity guides the different flows in the same direction so there is a time at which they cross. This phenomenon, called shell crossing, is illustrated in figure 3.2.
Figure 3.2: Schematic illustration of the emergence of multi-flow regions under the influence of gravity. Distances and velocities are represented as one-dimensional quantities. Author: Bernardeau, [12].
It is during the late phase of virialization that astrophysical objects such as galaxies begin to form (see [22]). Unfortunately, as soon as shell crossing starts, the physical processes at play become very difficult to model analytically (there exist nevertheless attempts of post-shell-crossing modeling, see e.g. [87, 180, 37]). So far, what is known about this epoch relies mostly on N-body simulations. On the contrary, before shell crossing, analytic developments are possible be- cause velocity dispersion is negligible. More precisely, one can impose σij = 0 in the Euler equation, which makes the second term of the right hand side vanish. This approximation is known as the single-flow approximation. It means that, at given time and position, all the particles of the fluid have the same velocity. In the single-flow approximation, the Euler equation reads5
d [au (~x,T )] i = ∂ Φ(~x,T ). (3.13) dT − i
One can see from this equation that the velocity field ui(~x,T ) is a gradient (plus possibly a homogeneous function of time). Consequently, no vorticity can appear in the fluid in the absence of shell crossing (see [132, 182, 142] for more details). In this context, the behavior of the velocity field can entirely be described in terms of dX 1 5For any field X(~x,T ), = ∂ X + ∂ X∂ xi = ∂ X + ui∂ X. dT T i T T a i
88 3.3. The single-flow approximation and its consequences its divergence θ(~x,T ), defined as
1 θ(~x,T ) = ∂ u (~x,T ). (3.14) aH i i This property is very useful. In particular, it allows to rewrite the system of equa- tions on a compact form, much easier to manipulate. To that aim, it is convenient to move to reciprocal space, to use the conformal time as time variable and to introduce the doublet6
Ψ (k, η) (δ(k, η), θ(k, η))T. (3.15) b ≡ −
In this framework, after taking the divergence of the Euler equation to make the quantity θ appear7, the equations of motion can be grouped into a single equation8 (the convention that repeated Fourier arguments are integrated over applies here),
c cd a(η)∂aΨb(k, η) + Ωb (η)Ψc(k, η) = γb (k1, k2)Ψc(k1, η)Ψd(k2, η). (3.16)
The left-hand side is linear (i.e. there is no mode coupling at play in it) and the right hand side contains all nonlinear interactions between the fields. Note in particular that those couplings are quadratic without adding any approximation. c The explicit expression of Ωb (η) is
0 1 c − Ωb (η) = 3 ∂ , (3.17) Ω 1 + a aH −2 m H 8πGρa¯ 2 where Ω = . Note in particular that this matrix is scale-independent. m 3 2 H cd Besides, the non-zero elements of the symmetrized coupling matrix γb (k1, k2) are given by 2 22 k1 + k2 (k1.k2) γ2 (k1, k2) = δD(k k1 k2)| 2| 2 , (3.18) − − 2k1k2
6 The index b is either 1 (Ψ1 = δ) or 2 (Ψ2 = θ). − 7 aH Note that, in reciprocal space, ui(k, η) = i kiθ(k, η). − k2 1 8By definition, ∂ = ∂ . a aH T
89 3.4. Perturbation theory at NLO and NNLO
12 (k1 + k2).k2 γ1 (k1, k2) = δD(k k1 k2) 2 , (3.19) − − 2k2
21 (k1 + k2).k1 γ1 (k1, k2) = δD(k k1 k2) 2 . (3.20) − − 2k1 Note that, in this approach, this matrix does not depend on time. The compact equation presents the benefit of making possible the writing of a formal solution:
η c c de Ψb(k, η) = gb (η)Ψc(k, η0) + dη0gb (η, η0)γc (k1, k2)Ψd(k1, η0)Ψe(k2, η0), Zη0 (3.21) c η0 being the initial time and gb the Green function of the equation (3.16) (see the reference articles [159, 160, 45]). The first term of the right hand side of (3.21) is the linear solution, i.e. the solution of the linear equation
c a(η)∂aΨb(k, η) + Ωb (η)Ψc(k, η) = 0. (3.22)
Studying the properties of this linear limit is not problematic in standard per- turbation theory (see [13]). However, dealing with the nonlinear terms is more challenging. Several approaches aiming at going beyond the linear regime have nevertheless been proposed in order to gain accuracy.
3.4 Perturbation theory at NLO and NNLO
In nonlinear perturbation theory, many achievements rely on the use of diagram- matic representations. Indeed, the form of the solution (3.21) quite naturally incites one to describe couplings in terms of diagrams (analogs of the Feynman diagrams of quantum field theory). As can be seen clearly in (3.21), the time evolution of a given field Ψb(k, η) is determined by the initial values of all fields. The statistical quantities that are relevant for cosmology are thus the correlation functions involv- ing those initial (Gaussian) fields. The simplest one is of course the one involving only two fields, i.e. the linear power spectrum
Ψ (k, η)Ψ (k0, η) = P (k, η)δ (k + k0). (3.23) h a b i ab D
90 3.4. Perturbation theory at NLO and NNLO
Its diagrammatic representation is in accordance with it, simple and linear. This level of description is called tree-level. Obviously, one can increase accuracy by increasing the number of contributing initial fields taken into account. Diagrammatically, it simply means adding loops in linear diagrams, those loops symbolizing pair associations9. Indeed, one can (l) introduce l-order power spectra Pab (l being also the number of loops) so that
∞ (l) Pab(k, η) = Pab (k, η), (3.24) Xl=0 with
2l+1 (l) (m) (2l+2 m) δ (k + k0)P (k, η) = Ψ (k, η)Ψ − (k0, η) . (3.25) D ab h a b i m=1 X The exponents in brackets are the numbers of interacting modes k1, k2, ... (satis- () fying k = k1 + k2 + ...) taken into account in Ψa (k, η). Usually, one-loop corrections are called next-to-leading-order (NLO) corrections and two-loop corrections are called next-to-next-to-leading-order (NNLO) correc- tions. The first calculations of loop corrections were performed in the nineties ([112, 97, 161]). More recent studies allow to reach the two-loop level ([15, 177]) or even the three-loop level ([26]). I do not present in more detail the associated formalism because it would be cumbersome and not particularly helpful to understand the work I did as a PhD student. Indeed, I did not directly use diagrammatic representations and renormal- ization techniques. Besides, all the needed information can be found in very detailed references. The two leading methods that make a fruitful use of diagrammatic rep- resentations thanks to renormalization group techniques are [45]10 and [117]. A particular formulation of such developments is the so-called time-flow equation ap- proach ([133]). Several improvements of such approaches have been achieved since then. One can cite e.g. the MPTBREEZE ([46]) or RegPT ([177]) approaches. Such enhancements of standard perturbation theory involving resummation techniques are necessary because the expansion of standard perturbation theory, order by or- der, is not convergent (see [15] and [26]).
9Again, only pair associations matter in virtue of Wick’s theorem. See section 2.2.2. 10This reference also explains in detail the procedure to be followed to draw diagrams and to interpret them.
91 3.5. Eikonal approximation and invariance properties
What is important here is rather to stress on the fact that all those developments, which are the cornerstones of the understanding of the nonlinear growth of structure, rely on the equation (3.16). For this reason, when starting my work on neutrinos, it was precisely in the relativistic generalization of this equation that I was interested. Succeeding in it would indeed mean that, in the long term, one could hope for comparable predictions without the need to restrict the studied cosmic fluid to cold dark matter.
3.5 Eikonal approximation and invariance properties
In the standard case of a cold pressureless fluid, a strategy has been developed to better understand how couplings between modes of very different wavelengths contribute to the nonlinear growth of structure. This strategy relies on the eikonal approximation, which in this context can be associated with the invariance prop- erties of the equation of motion (3.16). I will briefly present the principle of the reasoning in this section. Then it will be convenient to expose in the next chap- ters the way in which I started to extend those results to relativistic flows. Full details concerning the standard approach are to be found in [14] and [16]. Be- sides, the role of the so-called extended Galilean invariance is examined carefully in [25, 127, 100, 44, 43, 42].
3.5.1 Eikonal approximation
The notion of eikonal approximation intervenes in various fields of theoretical physics. The way it is used in cosmology is very similar to the method described in the quan- tum electrodynamics paper [1], i.e. highlighting the impact of given modes (in the QED case, those of soft photons) on the propagation of other modes (in the QED case, those of electrons). The starting point to apply the eikonal approximation to the equation of motion (3.16) is the fact that the amplitudes of the coupling terms (3.18), (3.19) and (3.20) k vary a lot with the wavenumber ratio 1 . Hence, it is convenient to split the k2 right hand side of equation (3.16) into two integration domains. The hard domain (characterized in the following by an index ) contains all modes whose wavelengths H are of the same order and the soft domain (characterized in the following by an index ) encompasses modes of very different wavelengths. S
92 3.5. Eikonal approximation and invariance properties
Assuming that the soft domain is obtained for k k , k k so that the 2 1 1 ≈ contribution corresponding to the soft domain appears simply as a corrective term in the linear equation of motion of the mode k,(3.22). Indeed, in this context, equation (3.16) can be rewritten
a∂ Ψ (k, η) + Ω c(η)Ψ (k, η) Ξ c(k, η)Ψ (k, η) a b b c − b c cd = γb (k1, k2)Ψc(k1, η)Ψd(k2, η) , (3.26) h iH with Ξ c(k, η) 2 d3q eik.γ cd(k, q)Ψ (q, η) . (3.27) b ≡ b d ZS The soft momenta q (i.e. q k) at play in equation (3.27) being integrated over, c Ξb (k, η) is a mere field, not a coupling matrix. When the contribution of the hard domain is negligible, the equation of motion reads
a∂ Ψ (k, η) + Ω c(η)Ψ (k, η) Ξ c(k, η)Ψ (k, η) = 0. (3.28) a b b c − b c
It can be interpreted as the equation of motion of the mode k evolving in a medium perturbed by large-scale modes. It is particularly interesting for cosmologists be- cause it is much simpler than the full equation while encoding the way in which long-wave modes alter the growth of structure. In the eikonal limit (i.e. when k k ), k k and k q so the predominant 2 1 1 ≈ 2 ≈ elements of the coupling matrix are
k.q γ 22(k, q) = γ 12(k, q) . (3.29) 2 1 ≈ 2q2 21 One neglects γ1 (k, q) because
1 k.q γ 21(k, q) . (3.30) 1 ≈ 2 2q2
c In this context, the NLO correction Ξb (k, η) simply reads
k.q Ξ c(k, η) δ c d3q θ(q, η) = δ c Ξ(k, η). (3.31) b ≡ − b q2 b ZS Note in particular that it does not directly depend on the density contrast field
93 3.5. Eikonal approximation and invariance properties
δ(k, η). Besides, it is a purely imaginary quantity because the velocity divergence c c is real and θ(q, η) = θ ( q, η) implies Ξ (k, η) = Ξ ∗( k, η). ∗ − b − b − The advantage of the equation of motion (3.28) is that it can easily be solved. b b Indeed, its Green functions ξa (k, η, η0) are related to the Green functions ga (η, η0) of the zeroth order equation (3.22) simply by
η c c ξb (k, η, η0) = gb (η, η0) exp dη0Ξ(k, η0) . (3.32) Zη0 Given the definition (3.31), the time integration of Ξ(k, η) is nothing but a displace- ment d projected along the direction k,
c c ξb (k, η, η0) = gb (η, η0) exp ik.d(η, η0) . (3.33) More precisely, the quantity d(η, η0) should be interpreted as the total displacement induced by the long-wave modes between times η and η0. Expression (3.33) shows that the only impact of long-wave modes is to shift the phase of the propagator of the background equation. Renormalized perturbation theory comes into play when one decides to construct nonlinear propagators associated with the general nonlinear equation of motion c (3.16). Such propagators, denoted Gb , are defined so that
Ψ (k, η) Ψ(0)(k, η) G c(k, η)δ (k k ) b − b . (3.34) b D 0 (0) − ≡ *Ψ (k0, η ) Ψc (k0, η )+ c 0 − 0 They must in particular take into account the perturbations induced by the soft domain, that is to say that one needs to compute (among others) the quantities
c c 1 2 2 ξ (k, η, η0) = g (η, η0) exp k σ (η, η0) , (3.35) b b −2 d
1 with σ2(η, η ) = d2(η, η ) . The idea developed in [45] is that, in diagrammatic d 0 3 0 representation, one-loop contributions to nonlinear propagators (then called renor- malized propagators) can be divided into sub-diagrams, whose resummations are connectable to physical quantities. For instance, the authors showed that
2 2 η η 3 1 lin σ (η, η0) = e e 0 d q P (q), (3.36) d − 3q2 Z where P lin is the two-point correlation function of initial linear fields. This key
94 3.5. Eikonal approximation and invariance properties result is the starting point of the computation of resummed propagators at NLO (see [45] for the “RPT method”, [46] for the “MPTBREEZE method” and [177] for the “RegPT method”). In this context, perturbative expansions of the nonlinear power spectrum contain renormalized propagators plus terms describing mode couplings. Interestingly (and independently on renormalized perturbation theory), one can show that introducing an eikonal correction in the linear equation of motion has no impact on the power spectra, provided that they involve fields evaluated at the same time. The reason is that the different phase shifts appearing in the corrected propagators (3.33) exactly cancel each other out. We will see now that it is directly related to the invariance properties of the equations of motion.
3.5.2 Invariance properties
It is easy to check that, in the single-flow approximation, the system of equations (3.11)-(3.12) is invariant under the change of coordinates (written here in terms of conformal time)
i i x˜ = x + Di(η), (3.37) η˜ = η, δ˜ = δ, dD (η) u˜ = u + i , i i dη 1 da dD (η) d2D (η) Φ˜ = Φ i xi i xi, − a dη dη − dη2 where Di(η) is an arbitrarily time-dependent uniform field. The relative motion between the two frames depending on time, this invariance is sometimes called ex- tended Galilean invariance. As shown in [44], it actually derives from the equivalence principle. It is a powerful property since it states that any acceleration of a given area of the universe does not affect the development of gravitational instabilities in it. The striking point is that such a change of frame affects the linear propagator c precisely in the same way as the eikonal correction Ξb (k, η) does. Indeed, according
95 3.5. Eikonal approximation and invariance properties to (2.8), transformations (3.37) impose
c c g˜ (˜η, η˜0) = g (η, η0) exp ik. D(η) D(η0) . (3.38) b b {− − } So the displacement Di at play in the transformations (3.37) can be related to the eikonal displacement di in such a way that
c c g˜b (˜η, η˜0) = gb (η, η0) exp ik.d(η, η0) . (3.39)
In other words, the perturbation of the linear propagator generated by long-wave modes, (3.33), can be wiped out by a mere change of coordinates. It explains why c equal-time statistical quantities such as power spectra are not sensitive to Ξb (k, η). Regarding unequal-time correlation functions, it is possible to derive consistency relations involving them. We have seen that velocity divergences (or equivalently density contrasts according to the continuity equation (3.11)) involving soft modes perturb the medium by generating displacement fields in the form
η 3 1 d(η, η0) = i d q q dη00 θ(q, η00). (3.40) q2 Z Zη0 Hence, taking the effect of a soft mode q into account is equivalent to considering that all other modes evolve in a perturbed medium, the perturbed fields being here nothing but the original fields transformed according to (3.37):
˜ ˜ δ(x1, η1)...δ(xn, ηn) p.m. = δ(x˜1, η˜1)...δ(x˜n, η˜n) , (3.41) h i u.m. D E where “p.m.” stands for “perturbed medium” and “u.m.” for “unperturbed medium”. In reciprocal space, to compute a correlation function involving a field evaluated at q, one can therefore compute the correlation function of all other fields (in the perturbed medium) and then correlate it with the disturber field:
δ(q, η)δ(k1, η1)...δ(kn, ηn) q 0, u.m. = δ(q, η) δ(k1, η1)...δ(kn, ηn) p.m. . h i → h i u.m. D E (3.42) In general, the form of the consistency relations (also called Ward identities) that can be derived from such considerations depends on the assumptions made to ex- press the perturbed density contrasts. Examples can be found in [100]. In the case of perturbations corresponding to “extended Galilean” transformations (3.37), one
96 3.6. Perturbation theory applied to the study of massive neutrinos
finds as a first approximation for the three-point correlator
δ(q, η)δ(k1, η1)δ(k2, η2) q 0 (3.43) h i →
q.(k1 + k2) lin η1 η η2 η = P (q; η) δ(k , η )δ(k , η ) e − e − , − q2 h 1 1 2 2 i − where P lin(q; η) is the linear power spectrum defined in (3.23). Note that, at equal times, δ(k , η )δ(k , η ) δ (k + k ) so that the right- h 1 1 2 1 i ∝ D 1 2 hand side of (5.29) is zero. In theory, consistency relations such as (5.29) are particularly interesting because they allow to gain one order in the knowledge of correlation functions. However, the fact that it involves unequal-time correlators only makes the measurement of such observables very challenging.
3.6 Perturbation theory applied to the study of massive neutrinos
The formalism described in the previous sections is valid only for cold fluids in which the velocity dispersion is negligible. It is tempting to extend it to other species than cold dark matter. For instance, multi-fluid approaches involving cold dark matter and baryons ([169, 17]) or dark energy ([155, 49,3]) have been realized. Including neutrinos is challenging because it is a hot species so neglecting velocity dispersion would not be realistic. Other simplifying assumptions are thus necessary. In [153, 198, 105, 181], mixtures of baryons, cold dark matter and massive neutrinos are considered at NLO but neutrino perturbations (whose effect is as- sumed to be particularly small) are kept at linear order. In [164, 106, 163], it is the time-dependence of the free-streaming length of neutrinos that is neglected. [27] proposes a hybrid approach: the fluid approach based on the Vlasov-Poisson system is applied at NLO at small redshifts and, at high redshifts, neutrinos are described using the linear Boltzmann hierarchy. Indeed, at small redshifts, a simple expression can be found for the velocity dispersion and the Newtonian approxi- mation holds whereas nonlinearities can be neglected at high redshifts. Note that [81] also recommends the use of hybrid approaches. Besides, it warns against the use of approximation schemes, invoking the emergence of spurious behaviors due to unrealistic assumptions.
97 3.6. Perturbation theory applied to the study of massive neutrinos
The work that I present in the next chapters is based on the idea that considering neutrinos as a collection of single-flow fluids instead of a single multi-flow fluid could allow one to get rid of velocity dispersion.
98 Chapter 4
A new way of dealing with the neutrino component in cosmology
We have seen that there exist two standard ways to account for neutrinos in cosmo- logical models. In the one based on the Vlasov-Poisson system, the main difficulty arises from the velocity dispersion term present in the Euler equation. Regarding the Boltzmann approach, it is rather the treatment of nonlinearities that appears challenging. As discussed in section 3.6, considering a hybrid approach that mixes those two methods might be a solution to get a satisfactory description of massive neutrinos beyond the linear regime. In this thesis, I propose an alternative which consists in decomposing neutrinos into several fluids labeled by their initial velocities and exploiting the nonlinear continuity and Euler equations of each flow. To extend its scope of application, this approach does not rely on the Newtonian approximation.
4.1 Principle of the method
It is clear that the single-flow approximation is decisive in the analytic modeling of the development of inhomogeneities, responsible for the formation of the large- scale structure, in cold pressureless fluids. The idea behind the article presented in section 4.4 is that one can take advantage of the single-flow approximation for a greater variety of species simply by splitting non-cold fluids into collections of flows.
99 4.1. Principle of the method
To be more precise, instead of considering a unique neutrino fluid with a con- tinuous phase space, one can discretize the momentum space at initial time and consider N fluids corresponding to the N initial momenta. This is illustrated in figure 4.1 for N = 11. Note that initial momenta are characterized not only by a norm but also by a sense and, in multi-dimensional spaces, by a direction. Further- more, by construction, such a description applies to one mass eigenstate at a time. What makes the decomposition possible is the fact that, after decoupling, neutri- nos free-stream. In other words, provided that initial conditions are chosen after neutrino decoupling, no interactions between cosmic fluids need to be taken into account. In the picture, variations of thickness represent fluctuations of density. It should be noticed that, at given positions, densities differ from one fluid to another on this snapshot. It is not surprising since we know that the number of particles will not be the same in each flow (we have seen in section 2.3.2 that phase-space distribution functions depend on p). However, one expects usual global quantities, which depend only on time and position, to be recovered after summing over all the flows (explicit summation formulae are given later in this chapter).
Figure 4.1: Discretized phase space at initial time for a collection of eleven flows. In each flow, momenta are initially homogeneous. Variations of thickness represent fluctuations of density. For simplicity, momenta and positions have been represented as one-dimensional quantities.
Although momenta are initially homogeneous, the unavoidable presence of fluc- tuations of density makes momentum gradients develop over time in each flow. Note that, for a given mass, flows of neutrinos are all the more sensitive to density fluctu- ations that the norm of their initial momentum is small because, in that case, they fall easily into potential wells. In particular, a fluid characterized by a zero initial momentum is expected to behave as cold dark matter. The development of spa- tial variations in momentum space is illustrated in figure6. Momentum gradients
100 4.1. Principle of the method are located on an inclined axis, characterizing the direction of the initial momenta (which is unique here since it is a one-dimensional representation). In the same way as for the cold dark matter component, one expects multi-flow regions to emerge eventually. It is clear in figure6 that it will occur much earlier in fluids with low initial velocities. However, before shell crossing starts, it is reasonable to assume that each fluid of neutrinos can be studied in the single-flow approximation. As for cold dark matter, describing shell crossing is beyond the scope of the study. Besides, it is worth noting that shell crossing occurs likely only after neutrinos have become sufficiently slow to be rightly considered as cold species.
Figure 4.2: Development of momentum gradients in the discretized phase space for a collection of eleven flows. This phenomenon is due to fluctuations of density, represented by variations of thickness.
As already mentioned, in this kind of description, the global behavior of neu- trinos should be recovered after summing results over all the single-flow fluids, provided that the number of flows is sufficient for the sample to be representative. Each flow being characterized by an initial momentum that I will denote ~τ, the total phase-space distribution function is given for instance by
tot f (η, x, p) = f~τ (η, x, p) (4.1) X~τ or, in the three-dimensional continuous limit,
tot 3 f (η, x, p) = d τf~τ (η, x, p), (4.2) Z where f~τ is the phase-space distribution function of the fluid labeled by ~τ. Symbolically, the choice of labeling fluids by their initial momenta and then deriving equations of motion specific to each of them can be seen as an analog of
101 4.2. Derivation of the equations of motion the choice of adopting a Lagrangian point of view to describe a flow. The difference is that, in a Lagrangian description, it is the initial position that is used as a label. Conversely, the standard description of neutrinos relying on the study of f tot(η, x, p) is more similar to an Eulerian approach.
4.2 Derivation of the equations of motion
Our first study concerning neutrinos (section 4.4), whose main goal was to test the reliability of the method described above, has been performed in the conformal Newtonian gauge (2.18). General relativity is indeed relevant to capture the effect of neutrino perturbations from the local universe to super-Hubble scales. The gauge has been chosen to facilitate comparison with the Newtonian equations of motion presented in chapter3. To be in tune with this formalism, we wished to derive nonlinear equations of motion governing the behavior of a density contrast field and of a velocity divergence field. This ambition maintains some freedom regarding the choice of variables. As we will see, the desired equations can be obtained from basic conservation laws. For simplicity, couplings between the metric perturbations φ and ψ are neglected throughout the analysis. It is important to have in mind that this action does not prevent one to depict the nonlinear growth of structure. Indeed, couplings involving density contrasts and/or velocity divergences are untouched and it is these nonlinear interactions that are relevant for the modeling of the formation of the large-scale structure of the universe.
4.2.1 Conservation of the number of particles
Once the flows have been defined at initial time, the number of neutrinos contained in each of them is constant. Indeed, only the initial momentum determines the flow to which each neutrino is assigned. The way in which one chooses the number of neutrinos assumed to have a given initial momentum will be discussed in section 4.3.3. Mathematically, the conservation of the number of particles is encoded in the continuity equation. In general relativity, it is given by the conservation of a quantity J µ called particle four-current1,
µ J ;µ = 0. (4.3)
1I recall that the symbol “ ; ” stands for the covariant derivative defined in section 1.3.3.
102 4.2. Derivation of the equations of motion
The four-current is the four-dimensional analog of the current density of classical physics, i.e. of a flow rate. It can thus be expressed in terms of a density and a velocity or momentum. Several possibilities concerning the choice of variables are explored in the article presented in section 4.4 and a continuity equation is given for each doublet. With hindsight, the variables that I retain are:
the comoving number density n (η, x), defined as the number of particles per • c comoving unit volume d3xi,
the comoving momentum field2 P (η, x). • i The latter choice will be argued in section 4.3.1 and definitively justified in chapter 5. In terms of the phase-space distribution function, the comoving number density is given by 3 i nc(η, x) = d pi f(η, x , pi) (4.4) Z and Pi(η, x) is nothing but the average of the phase-space comoving momenta
d3p f(η, xi, p )p P (η, x) = i i i . (4.5) i d3p f(η, xi, p ) R i i Besides, the four-current satisfies (seeR e.g. [18])
µ µ 3 1/2 p i J (η, x) = d p ( g)− f(η, x , p ), (4.6) − i − p0 i Z where g is the determinant of the metric tensor3 . The single-flow approximation is particularly helpful in this context. Indeed, at given time η and position x, all the particles of the fluid have the same momentum
Pi(η, x) whence
f(η, xi, p ) = n (η, x)δ (p P (η, x)). (4.7) i c D i − i It imposes directly
µ µ 4 P J = a− (1 ψ + 3φ) n . (4.8) − − P 0 c 2As the comoving phase-space momentum defined in (2.11), it satisfies the mass-shell condition µ 2 P Pµ = m . 3 − 2 α β I recall that the metric tensor gαβ is defined so that ds = gαβ dx dx . In the conformal 2 1/2 4 Newtonian gauge, ds is given by (2.18) whence ( g)− = a− (1 ψ + 3φ). − −
103 4.2. Derivation of the equations of motion
Putting this into the conservation equation (4.3), one finds eventually (φ and ψ being kept at linear order in the calculation) the succinct equation
P i ∂ n + ∂ n = 0. (4.9) η c i P 0 c This equation of motion contains two independent variables only because P 0 and P i are related via the mass-shell condition (the explicit relation is given in the article P i dxi of section 4.4). Note that is simply (see (2.11)). The form of the continuity P 0 dη equation (4.9) is thus particularly reminiscent of the continuity equation (3.11) of the cold pressureless fluid. It should be supplemented by a relativistic equivalent of the Euler equation (3.12).
4.2.2 Conservation of the energy-momentum tensor
In the single-flow approximation, deriving the equation of motion of the momen- tum field is straightforward. Indeed, in that case, the energy-momentum tensor is simply4 (see again [18])
T µν = P µJ ν. (4.10) − Given the conservation of the four-current, the conservation of the energy-momentum tensor,
µν T ;ν = 0, (4.11) reduces to
µ ν P ;νJ = 0. (4.12)
When applied to the spatial part of P µ, it results in
Pj PjPj ∂ηPi (1 + 2φ + 2ψ) ∂jPi = P0∂iψ + ∂iφ. (4.13) − P0 P0 It should be compared with the Newtonian Euler equation (3.12) taken in the single- flow approximation. In the article presented in section 4.4, variants of equation
4In the article presented in section 4.4, we omitted the minus sign. Fortunately, it is a harmless mistake.
104 4.2. Derivation of the equations of motion
(4.13) are proposed. They involve more concrete variables such as the physical velocity field or the energy field. Encoding the time evolution of neutrino perturbations in a system such as (4.9)- (4.13) is a novel point of view. It has been made possible thanks to the trick consist- ing in splitting the overall neutrino fluid into several flows. Those equations are very general because they have been derived from general relativity without neglecting field couplings. We will see in chapter5 that even more general equations of this type can be found if one decides not to specify the gauge. Concrete applications will be presented in the next chapters. In the present one, the focus is placed on the comparison with the standard Boltzmann approach presented in section 2.3.2.
4.2.3 An alternative: derivation from the Boltzmann and geodesic equations
In the case of a cold pressureless fluid, the continuity and Euler equations had been obtained from the first moments of the collisionless Boltzmann equation combined with Newton’s second law. The phase-space distribution functions of the different flows of neutrinos satisfying also the Vlasov equation, one expects the equations of motion of neutrinos to be re-derivable from comparable manipulations. It is useful to ensure that it is true. Indeed, recovering our equations from an alternative method would confer credibility to our results. So let’s write again the Vlasov equation:
∂ dxi ∂ dp f + ∂ f + i f = 0. (4.14) ∂η i dη ∂p dη i Note that this form is more general than the one given in section 2.3.2. It is thanks to the Hamiltonian evolution of the system that both formulations are valid. In the dxi non-relativistic case, had been expressed in terms of the phase-space variables dη dp thanks to the classic definition (3.4) and i had been related to the gravitational dη potential thanks to Newton’s second law (3.5). In general relativity, those relations are replaced respectively by
dxi pi = , (4.15) dη p0 which derives from the very definition (2.11) of the momentum pi, and by the
105 4.2. Derivation of the equations of motion geodesic equation
dpi pjpj = p0∂iψ + ∂iφ. (4.16) dη p0 Note that the definition of the macroscopic momentum field, (4.5), is equivalent to
3 i Pi(η, x)nc(η, x) = d pi f(η, x , pi)pi. (4.17) Z More generally, in the single-flow approximation, (4.7) imposes to any macroscopic field depending on P (η, x) the relation F i
[P (η, x)] n (η, x) = d3p f(η, xi, p ) [p ] , (4.18) F i c i i F i Z which gives a direct mapping between phase-space variables and macroscopic fields. In this framework, it is easy to show that the integration of equation (4.14) over phase-space momenta leads to
P i ∂ n + ∂ n = 0, (4.19) η c i P 0 c which is precisely our first equation of motion (4.9). Besides, by combining the geodesic equation (4.16) with (4.18), one finds5
Pj PjPj ∂ηPi (1 + 2φ + 2ψ) ∂jPi = P0∂iψ + ∂iφ, (4.20) − P0 P0 which is actually our second equation of motion (4.13). The system (4.9)-(4.13) ap- pears as a true relativistic generalization of the system (3.11)-(3.12). Indicatively, we also showed in our article that our approach allows to re-derive the nonlinear Boltzmann hierarchy (2.29). Finally, to close the system of equations, metric per- turbations should be related to the fields chosen as variables. In general relativity, such relations are given by the Einstein equations, which generalize the Poisson equation (3.10) of the Vlasov-Poisson system.
j 5 dpi p pj It requires to notice that = ∂ηpi + 0 ∂j pi = ∂ηpi (1 + 2φ + 2ψ) ∂j pi. dη p − p0
106 4.3. Comparison with standard results
4.3 Comparison with standard results
We have seen in section 2.3.2 that it is in the linear regime that the Boltzmann hierarchy is concretely exploitable. Indeed, the linearized Boltzmann hierarchy (2.22)-(2.23)-(2.24) allows to compute the linear perturbations of the first multipoles of the energy distribution (2.25)-(2.26)-(2.27) whereas the nonlinear Boltzmann hierarchy (2.29) is fruitless. To make a preliminary test of the accuracy of our approach, we thus expressed the linearized energy multipoles in terms of our fields and derived their time evolution from our equations of motion taken in the linear limit.
4.3.1 Linearized equations of motion
The equation of motion (4.13) shows that the zeroth order of the momentum field, (0) Pi , is constant. Besides, since the momentum field is initially homogeneous, P (η ) = P (0) (η )(η being the initial time). The label of each flow τ P (η ) i in i in in i ≡ i in is thus not only the initial momentum but also the homogeneous part of the mo- mentum at any time, i.e.
τ P (0). (4.21) i ≡ i Once linearized, the Euler equation (4.13) can thus be rewritten
dP (1) τ 2 i = ∂ τ ψ + φ , (4.22) dη i 0 τ 0 where
τ = δijτiτj (4.23) and p τ (η) = τ 2 + m2a2(η). (4.24) 0 − The form of the equation (4.22) is pleasantp because it is reminiscent of the form
(3.13), according to which the relevant characteristics of the velocity field ui are all contained in its divergence. This gives a first incitement to choose the momentum
field Pi as one of our variables. In our second study (see chapter5), we showed that
Pi keeps this attractive property in any gauge and at any order in perturbation theory. One can therefore definitely consider it as “the good variable”, especially
107 4.3. Comparison with standard results since none of the other fields that we tested presents this particularity. i It is legitimate to wonder what differentiates inherently Pi from P for instance. (1) A first clue may be the fact that Pi is a gradient in terms of its canonically conjugate space coordinates xi so one can expect P i(1) to be a gradient in terms of the space coordinates xi. Note moreover that it was already pointed out in [104] that lowering the momentum indices to work with the conjugate momenta of the comoving space coordinates is useful since the geodesic equation written in a homogeneous universe shows that P i(0) a 2 whereas P (0) is constant. ∝ − i After moving to Fourier space and introducing, in the spirit of the study of the cold pressureless fluid, a density constrast field
n(1)(x, η) δ (η, x) = c , (4.25) n (0) nc and a divergence field
(1) θP (x) = ∂iPi , (4.26) the linearized equations of motion take the form
τ τ 2 θ µ2τ 2 ∂ δ = iµk δ + ψ + φ 2 + P 1 (4.27) η n τ n − τ 2 τ − τ 2 0 0 0 0 and
τ τ 2 ∂ θ = iµk θ k2 τ ψ + φ , (4.28) η P τ P − 0 τ 0 0 with µ the Cosine of the angle between the wave vector k and the initial momentum direction, k τ µ = i i . (4.29) kτ In general, µ runs from 1 to 1 since the other number that characterizes the initial − momentum, τ, is defined in (4.23) as a positive quantity. Note nevertheless that the equations of motion impose
δ ( µ, k, τ) = δ (µ, k, τ) = δ∗(µ, k, τ) (4.30) n − n − n and the same for θP . Such relations would reduce the number of equations if one assumed that the real and imaginary parts of the fields are not independent.
108 4.3. Comparison with standard results
In the large mass limit, i.e. when τ 0 (and thus τ ma), one recovers → 0 → − the standard linear equations of cold dark matter (3.22)6. The additional terms, τ all proportional to , reflect the influence of a non-zero initial velocity. Their τ0 presence in the equations of motion affects the time evolution of the perturbations
δn and θP (in a different way for each flow). Such contributions characterize the free-streaming of neutrinos, i.e. they add in solutions a transitory period before the fall into potential wells. Usually, free-streaming is taken into account by assuming a non-zero (but approximate and hence valid only at given scales) velocity dispersion in the Euler equation and by taking the contribution of neutrinos into account in the Poisson equation (valid only on scales at which relativistic effects are negligible). The point of view adopted here is therefore less restrictive.
4.3.2 Linearized multipole energy distribution
Our study allows to track individually the behavior of each flow of neutrinos but what is of most interest for cosmology is of course the global effect of neutrinos. We were thus interested in reconstructing the global multipole energy distribution from individual features. In terms of the global energy-momentum tensor of neutrinos, the first global multipoles ρν, θν and σν read in Fourier space and in the linear regime (see [111]):
(1) ρ (1) T 0 , (4.31) ν ≡ − 0 (1) ρ (0) + P (0) θ (1) ikiT 0 , (4.32) ν ν ν ≡ i i j k k 1 (1) 1 (1) ρ (0) + P (0) σ (1) δ T i δi T k , (4.33) ν ν ν ≡ − k2 − 3 ij j − 3 j k with ([18]) µ ν µν 3 1/2 p p i T (η, x) = d p ( g)− f(η, x , p ). (4.34) i − p0 i Z A distinctive sign of our approach is that integrations over phase-space momenta are replaced by integrations over all the flows of the collection, or equivalently over
6 Indeed, in the non-relativistic regime, δn = Ψ1, θP = ma Ψ2 and ψ is the gravitational − H potential Φ.
109 4.3. Comparison with standard results all the initial momenta ~τ. More explicitly, since (4.2) and (4.7) impose
f tot(η, xi, p ) = d3τ n (η, x; ~τ)δ (p P (η, x; ~τ)), (4.35) i i c D i − i Z the mapping between the two approaches reads
d3p f tot(η, xi, p ) (p ) = d3τ n (η, x; ~τ) (P (η, x; ~τ)). (4.36) i i F i i c F i Z Z Eventually, in terms of our fields, the linearized multipole energy distribution is therefore given by
τmax 1 (1) 4π 2 (0) ρν = 4 e τ dτ dµ τ0 nc (µ, τ) (4.37) − a R 0 1 × Z Z− τ 2 µτ δ (µ, τ) + φ 3 + i θ (µ, τ) , n τ 2 − kτ 2 P 0 0 τmax 1 (0) (0) (1) 4π 2 (0) ρν + Pν θν = 4 e τ dτ dµ nc (µ, τ) iµkτ [δn(µ, τ) + 4φ] + θP (µ, τ) , a R 0 1 { } Z Z− (4.38)
τmax 1 (0) (0) (1) 4π 2 (0) ρν + Pν σν = 4 e τ dτ dµ τ0 nc (µ, τ) (4.39) a R 0 1 × Z Z− τ 2 1 τ 2 µτ 4 µτ µ2 δ (µ, τ) + φ 5 + i θ (µ, τ) i θ (µ, τ) , τ 2 − 3 n − τ 2 kτ 2 P − 3 kτ 2 P 0 0 0 0 where the “ e” operator takes the real part of the quantity it precedes. We give R alternative expressions, involving other fields, in our article. This is precisely in the comparison between this way of calculating the energy distribution and the standard one, (2.25)-(2.26)-(2.27), that we were interested.
4.3.3 Initial conditions
When the initial time is assumed to lie between the time of neutrino decoupling and the moment at which neutrinos become non-relativistic, an analytic expression of the global phase-space distribution function is known. It is given by equation (2.16):
1 q − f tot xi, q = 1 + exp at initial time, (4.40) a (T + δT (xi))
110 4.3. Comparison with standard results where qi = a2(1 φ)pi in the conformal Newtonian gauge (see [111]). − The mechanisms at play in the emergence of inhomogeneities (such as the lo- cal fluctuations of temperature δT ) are complex7. To simplify the description, a common practice is to assume that initial conditions are linear combinations of adiabatic (also called isentropic) and isocurvature modes. The former are the two solutions (a growing one and a decaying one) obtained by assuming that all species have been perturbed in the same way when homogeneity ended (see more details in [196]). It means in particular that relative entropies between flows, which are in general non zero after a thermodynamic process has occurred, have been neglected. The latter encompass all the combinations for which the spatial curvature remains homogeneous while two (and only two) cosmic fluids undergo density fluctuations. Hence, when considering N fluids (each characterized by two equations of mo- tion), solutions are a priori a superposition of one growing adiabatic mode, (N 1) − growing isocurvature modes and N decaying modes. Neglecting decaying modes is natural because they have become too small to be observable today. Besides, for simplicity, isocurvature modes are also neglected in most studies (see nevertheless [113] for an illustration of how to deal with isocurvature modes). Fortunately, the second simplifying hypothesis (that might appear arbitrary) is strongly supported by observations8. Following this trend, we assumed initial conditions to be purely adiabatic in our study. Actually, this option was really lucky for us. Indeed, our approach would have made the description of isocurvature solutions particularly heavy since we con- sider an increased number of fluids (see the next section for discussions concerning the requisite number of neutrino flows). As explained in our paper (and follow- ing some standard results), in the conformal Newtonian gauge, the adiabaticity hypothesis leads to (on appropriate scales)
i i δT ηin, x ψ ηin, x = . (4.41) T − 2 The physical meaning of equation (4.35) is that all the flows whose time evolu- tion is such that, at time η and location xi, the particles have a momentum equal tot i to pi contribute to f η, x , pi . However, at initial time, the only flow that con-
7The most robust theories describing this phenomenon are inflationary paradigms and models describing topological defects. 8For example, according to the 2015 Planck results [137], the isocurvature contribution repre- sents less than about 3% of the adiabatic contribution.
111 4.3. Comparison with standard results
tot i tributes to f η = ηin, x , pi is the one labeled by τi = pi, whence a perfect match tot i between the global initial distribution functions f ηin, x , pi = τi and the initial i comoving number densities nc ηin, x ; ~τ . 9 As demonstrated in the article of section 4.4, one finds eventually
i i ψ ηin, x i df0(τ) δn(ηin, x ; ~τ) = + φ(ηin, x ) , (4.42) 2 ! d log τ where f0 is the background distribution function given by (2.14). Besides, the definition
i Pi(ηin, x ; ~τ) = τi (4.43) dictates in each flow the initial condition
i θP (ηin, x ; ~τ) = 0. (4.44)
The expressions (4.42) and (4.44) give the leading behavior of the fields at initial time. They depend on τ but not on µ, which means that they describe isotropic fields. To be able to depict the early development of anisotropic pressure in the neutrino streams, we computed also the first early-time corrections obtained by expanding the fields in one-dimensional10 Legendre polynomials (whose variable is µ). The procedure is described in our article. In practice, the initial conditions that we implemented in our code are
1 θ k2 µk δ init(µ, τ) = δ + µ2 1 + (δ + φ + ψ) i (δ + φ + ψ) n 0 3 − 2 τ 2 2 0 − 0 H 0 H H (4.45) and
µk θ init(µ, τ) = θ i θ , (4.46) P 1 − 2 1 H ψ d log f τ where δ = + φ 0 and θ = k2 0 (φ + ψ). 0 2 d log τ 1 − H 9Note that, in our article, initial conditions are presented in terms of the proper number density whereas I consider here the comoving number density. 10It is possible to express the angular dependence with one-dimensional polynomials because the only variable that characterizes the direction of the initial momentum in the equations is µ.
112 4.3. Comparison with standard results
Note that the symmetries previously mentioned are preserved in those initial fields.
4.3.4 Numerical tests and discussion on the appropriateness of the multi-fluid approach
Once the analytic expression of the linearized multipole energy density distribu- tion, the equations of motion of the fields and the initial conditions are known, it is easy to check that the perspective we propose is consistent with the standard de- scription. The principle of the integration scheme, as well as the numerical results, are presented in the article of section 4.4. The main conclusions we drew are the followings.
For small wavenumbers, the most influential parameters are the numbers of • momenta taken into account in the discretized sums involved in the two for- mulations of the energy multipoles. On the contrary, the number of discrete
values of µ, Nµ, and the order at which the Boltzmann hierarchy is trun-
cated, lmax, are not determining at those scales. For example, with Nµ = 12,
`max = 6, k = 0.002h/Mpc and m = 0.05 eV, the mean relative errors decrease 2 4 from 10− to 10− when the number of discrete momenta (in each approach) is increased from 16 to 100. For the number of equations to remain reasonable
in our study, one can thus keep Nµ small as long as k . 0.01h/Mpc.
For larger wavenumbers, the numerical integration is more demanding because • the fields undergo rapid oscillations and also depend more significantly on τ and µ (which is visible on the equations (4.27) and (4.28) and is reminiscent from the fact that the pulsation of acoustic oscillations is expected to be proportional to k in neutrino fluids, see [104]). Illustratively, it is necessary
to go up to Nµ = 100 to get a percent accuracy for k = 0.1h/Mpc. Clearly, a concrete numerical exploitation of our results would therefore require to search for strategies allowing to reduce the numerical cost. It is something that we have not done yet.
Considering smaller masses also requires to improve the discretization scheme • since, in that case, the dependence on ~τ is increased.
At late times, all the flows behave as cold dark matter. Unsurprisingly, the • convergence towards this regime strongly depends on µ, τ and m. During the
113 4.3. Comparison with standard results
oscillatory phase, neutrino and cold-dark-matter fields are generally out of phase so, before the integration over µ has been performed, some velocity and density perturbations of neutrinos have a larger real part than the fluctuations experienced by the cold-dark-matter fluid. This can be misleading at first glance but global perturbations have actually the expected behavior.
Those numerical tests show unequivocally that both approaches are consistent 5 in the linear regime. We have proven it up to a 10− accuracy. Illustratively, a 3 10− accuracy (obtained for k = 0.01h/Mpc and m = 0.3 eV) is presented in figure 4.3, which shows the comparison of the behaviors of the first energy multipoles. The main drawback of our method is that, for the agreement to be satisfying, our number of equations must generally be larger than the one required in the Boltz- mann approach. However, our analytic model is valid beyond the linear regime. Besides, as will be illustrated in the next chapters, the great similarities that exist between our study and the study of the cold pressureless component inspire further developments.
114 4.4. Article “Describing massive neutrinos in cosmology as a collection of independent flows” L L in
Η 100 H Ψ
of 1 units in
H 0.01
10-4 Multipoles 10-6 0.001 0.01 0.1 1 10 100 1000
a aeq. L 4
- 5 10 of 0 units in H -5
-10 Residuals -15 0.1 1 10 100 1000
a aeq.
Figure 4.3: Time evolution of the energy density contrast (solid line), velocity divergence (dashed line) and shear stress (dotted line) of the neutrinos. The dot- dashed line is presented for comparison and corresponds to the density contrast of the cold-dark-matter component. Top panel: the quantities are computed with our multi-fluid approach. Bottom panel: residuals (defined as the relative differences) when the two methods are compared. Numerical integration has been done with 40 values of τ and q and 12 values of µ. k is set to 0.01h/Mpc, m is set to 0.3 eV and 3 lmax is set to 6. The resulting relative differences are of the order of 10− .
4.4 Article “Describing massive neutrinos in cosmology as a collection of independent flows”
115 JCAP01(2014)030 hysics P le ic t ar 10.1088/1475-7516/2014/01/030 doi: strop A [email protected] , osmology and and osmology C
1311.5487 cosmological neutrinos, neutrino masses from cosmology A new analytical approach allowing to account for massive neutrinos in the non- [email protected] rnal of rnal ou An IOP and SISSA journal An IOP and 2014 IOP Publishing Ltd and Sissa Medialab srl Institut de Physique CEA, Th´eorique, IPhT,CNRS, F-91191 URA Gif-sur-Yvette, 2306, F-91191 Gif-sur-Yvette, France E-mail:
c J of large-scale structure growth. Keywords: ArXiv ePrint: the particular case of adiabaticresolution initial conditions, of we this explicitly check newapproach that, system based at linear of on order, equations the the in reproduces collisionless this Boltzmann the paper hierarchy. results allowsdepending obtained Besides, to on in the show initial the approach how velocities.towards standard advocated each a Although fully neutrino non-linear valid flow treatment up of settles to the into dynamical shell-crossing evolution the only, of neutrinos it cold in is dark the a framework matter further flow step the overall neutrino fluidflows. can Starting either be from equivalentlyof elementary decomposed conservation the equations into or phase-space a fromderive the distribution collection the evolution function of equation two independent and non-linearequations assuming describe motion the that equations evolutionbetween there that of the each is collection macroscopic of of no fields. flows these shell-crossing, we We flows defined explain we and satisfies. in the detail standard Those the massive fluid neutrino connection fluid. Then, in Accepted January 9, 2014 Published January 17, 2014 Abstract. linear description of the growth ofthe the standard large-scale approach structure in of which the neutrinos universe are is described proposed. as a Unlike unique hot fluid, it is shown that Received November 29, 2013 independent flows Dupuy andH´el`ene Francis Bernardeau Describing massive neutrinos in cosmology as a collection of JCAP01(2014)030 4 7 5 5 7 8 4 9 6 1 3 9 16 13 12 14 21 22 10 16 14 15 16 11 17 12 21 19 n i P ) i ⌧ ; ) x i ⌧ , ; ] crown three decades of observational and in x 2 ⌘ , ( , n in 1 –1– ⌘ ( i P 4.4.1 Method 4.4.2 Results 4.3.1 Initial4.3.2 momentum field Initial4.3.3 number density field Early-time behavior 2.2.1 Evolution2.2.2 equation of the Evolution proper2.2.3 equation number of density the Other momentum formulations of the momentumfunction conservation 2.3.1 The2.3.2 non-linear moments of The the single-flow Boltzmann equations equation from the moments of the Boltzmann equation A.1 A BoltzmannA.2 hierarchy from A tensor Boltzmann field hierarchy expansion from harmonic expansion 4.4 Numerical integration 4.1 Specificities4.2 of the multi-fluid The description 4.3 multipole energy distribution Initial in conditions the linear regime 3.1 The3.2 zeroth order behavior The3.3 first order behavior The system in Fourier space 2.3 The single-flow equations from the evolution of the phase-space distribution 2.1 Spacetime2.2 geometry, momenta and energy The single-flow equations from conservation equations polarization are thus a very precious probe of the matter content of the universe. Although The recent resultstheoretical of investigations the on Planck theperturbations. mission origin, [ evolution Those and propertiescosmological statistical perturbations are — properties inflation governed is of notby the cosmological most matter only commonly itself. referred by explanation In theobservations — addition but mechanisms of to also that the the information produced Cosmic they give Microwave regarding Background inflationary (CMB) parameters, temperature anisotropies and A The Boltzmann hierarchies 1 Introduction 5 Conclusions 3 The single-flow equations in the linear regime 4 A multi-fluid description of neutrinos 1 Introduction 2 Equations of motion Contents JCAP01(2014)030 ] 3 ]). On 23 it allows an 1 ], and also on 16 – 13 ]. 14 ] or for the masses, even if small, ]. These physical interpretations 21 11 – 6 ], where the neutrino fluid is tentatively 22 –2– ], where the e↵ ect of massive neutrinos in the non- 20 ]). The consequences of these results are thoroughly – 4 17 ] for a recent review on the subject). The aim of this paper is 24 ], where the connection between neutrino masses and cosmology - in the 5 ], which have witnessed a renewed interest in the last years [ One can mention another alternative that has been proposed to study the e↵ect of The experiments on neutrino flavor oscillations demonstrating that neutrinos are indeed 12 Except the e↵ects of lensing, which reveal non-linear line-of-sight e↵ects. 1 valuable and robust (seethus [ to set the stagedescribing for the further neutrino theoretical analyses perturbation by growth,species presenting from a to super-Hubble complete those perturbations set of ofinterested of equations relativistic in non-relativistic deriving speciessystem equations within describing from the dark matter local which particles universe. the is convenient. connection In with particular the we standard are non-linear designing tools to explorethe the density impact perturbation ofmassive growth. massive neutrinos. neutrinos One Little within ofequations has the the of been non-linear reasons the obtained for regime neutrinothorough such species of in investigations a are this of limitation a context the isthe priori non-linear in that cumbersome other the hierarchy and presence hand non-linear equations di cult Perturbation of evolution to are Theory handle to applied (the be to most found pure in dark [ matter systems has proved very matter and energy contenttrue of for the instance universe for onof the cosmological the dark neutrino perturbation energy species, properties. equation as This of shown is in state numerical [ neutrinos experiments on [ thedescribed large-scale structure as growth, a [ perfect fluid. In the present study we are more particularly interested in theoretical investigations such aslinear [ regime is investigatedthat with it the is help potentiallytheory of possible o↵er. Perturbation to Observations Theory. getsensitive of to An better the the important constraints large-scale non-linear point thane↵ects growth structure of is what on within structure the the this and predictions local thus growth. also of universe to linear are Such the indeed a impact of coupling mode-coupling is expected to strengthen the role played by the the neutrino mass spectrum.exploiting To these a large dependences extent to currentof put and massive constraints future neutrinos on cosmology on the projectssuch the neutrino aim constraints structure masses. at growth possible, Indeed, hasare as the proved based impact to shown on be forref. significative numerical instance enough experiments, [ in to the make [ early incarnations dating back from the work of are properly treated in the(see linear theory also of gravitational its perturbationspresented dates companion in back paper from ref. ref. ref. [ [ standard [ case of threeanisotropies neutrino species are - indirectly isstructure sensitive investigated growth to in rate, full via massive detail. its neutrinos time It whereas and is scale the shown dependences, that late-time o↵ers CMB a large-scale much more direct probe of small to leave an imprint on the recombinationmassive physics. are thus ofof crucial those importance masses and ontriggered it various a is cosmological considerable necessary e↵ort observables.the in to Understandably, consequences theoretical, examine such on numerical minutely the a and cosmic the discovery observational structure cosmology impact has growth. to The infer first study in which massive neutrinos this observational window ismetric confined or to density theexquisite perturbations time determination of are of recombination, several allthem in fundamental remain deeply a cosmological elusive. rooted parameters. regime where in This However, the some is the of in linear particular regime, the case of the neutrino masses if they are too JCAP01(2014)030 , is 3 ⌘ = 0, where ⌘ f d d could be considered as 2 , the geometric context in which the ]): neutrinos are considered as a single 2 5 – 3 is devoted to the description of the specific 4 –3– . Calculations are performed in a perturbed spacetime. Friedmann-Lemaˆıtre f The paper is organized as follows. In section The possibility we explore in the present work is that neutrinos The strategy usually adopted to describe neutrinos, massive or not, is calqued from that As a matter of fact, neutrinos of each mass eigenstate. 2 In this section we present thefluid derivation of of the particles, non-linear equations relativisticstandard of approach or motion for based not. a on single-flow the the A present use confrontation section. of of the our Vlasov findings equation with is those presented of in a the more last part of the system ofindependent equations, fluids. when theintegration Results whole scheme are neutrino based explicitly fluid oneach compared is the flow to settles discretized Boltzmann into those into hierarchy. the obtained a cold Finally from finite dark we matter sum the give component. some of standard hints2 on how Equations of motion then derive the non-linearwe equations make the of comparison motionwe with associated describe the with the first eachconstruction linearized moments fluid of of system. of a the neutrinos multi-fluidearly-time Boltzmann Section system and number hierarchy. of density single In and flows.This section velocity We section fields present ends in corresponding by particular to the the adiabatic initial presentation initial and of conditions. the results given by the numerical integration of breaks down for the sameshell reason: crossings. an initially But single-flow thisbeyond fluid regime can the corresponds form scope multiple to of streams theinto after Standard account late-time Perturbation in evolution Theory the of calculations the following. so fields, no which shell-crossing is iscalculations taken are performed is specified as well as some physical quantities of interest. We of the fact thatand neutrinos they are actually do free notto streaming: interact distinguish they with the do matter notvelocity. fluid particles. interact elements The with We of will one complete another thecontributions see neutrino of in collection fluid each particular by behavior fluid that labeling isdescription element it then is over each is the actually naturally of possible initial very obtained them velocity similar by with distribution. to summing an As that the initial we of will dark see, matter this from the very beginning. It also super or sub-Hubbleto scales, the the metric motion fluctuations.late-time equations non-linearity We are will of derived here thecouplings at also large-scale but restrict structure linear to growth ourselves order the is to non-linear with not this growth respect due approximation of to as the direct the a density metric-metric collection contrasts of and single-flow velocity fluids divergences. instead of a single multi-flow fluid. We take advantage here The key equation is thein Boltzmann the equation. collisionless For limit neutrinos,time since contrary to neutrinos of radiation, do recombination it not nor isconservation interact taken of after). with the ordinary number It matter ofa (neither leads particles time at to applied coordinate. the to theare The a Vlasov computed Hamiltonian di↵erent equation, system, in terms which particular participating derives in with from the the the expanded help form of of the this geodesic equation equation. In practice, whether at used to describe thehot radiation multi-stream fluid fluid (see whose e.g. evolutionin is refs. phase-space dictated [ by the behavior of its distribution function JCAP01(2014)030 µ ). p 2.1 (2.2) (2.3) (2.4) (2.5) (2.6) (2.1) ), which )(the zero 2.1 x ⌘, ,isdrivenby ( a i P , field ⇤ 3 ij j . 2 x are the metric perturbations. s d i d x will be taken into account in all )d p and . / . 2 j ⌫ 2
2 q x ✏ i . a and d i q 2 ) p ij ) µ⌫
⌘ g +(1 q/a 0 ) and its momentum = = 2 p x 0 j ⌘ µ +( p p (1 ⌘, i u 2 2 )d ( –4– p 00 a n g 3) are the Cartesian spatial comoving coordinates,
m ij , g = 2 = = , i measured by an observer at rest in metric ( = 0 2 q ✏ where ✏ p i (1 + 2 0 p =1 i p µ ,i.e. i p ( . In the following we will also make use of the momentum ⇥ µ 2 i ) x mu ⌘ x m ( = 2 a µ is the Kronecker symbol and = p ij = µ 2 p s µ d ) rather than the momentum field. p x ⌘, ( i V is the conformal time, ⌘ We will consider massive particles, relativistic or not, freely moving in space-time ( Following the same idea, in the rest of the paper metric perturbations will be considered ) is the scale factor , We use an uppercase to distinguish it from a phase-space variable and we use sometimes in this paper 3 ⌘ , defined as ( i the velocity field starting from elementary conservationtwo fields, equations. its local Such numericalcomponent a density can field fluid be is deducedapproach entirely from contrasts characterized the with by spatial aintroduce ones the description using whole of the velocity the on-shell distribution. complete mass constraint). neutrino fluid This for which one has to It satisfies 2.2 The single-flowWe equations now from proceed conservation equations to the derivation of the motion equations satisfied by a single-flow fluid Another useful quantity is theis energy such that It obviously implies q in such a way that presented in the introduction,the only derivations linear that terms follow, in in particular in theTheir motion kinematic equations properties we are willas given derive. by the their conjugate momenta momentum so of we introduce the quadri-vector a The expansion historythe of overall the matter universe, andpractical encoded energy calculations in content we the of adopt time the the numerical universe. dependence values of of Itas the is concordance known, supposed model. to determined be by known the and Einstein for equations. Furthermore, following the framework this work the conformalmotion Newtonian equations gauge, of which non-relativisticis makes species, given the by the comparison Vlasov-Poisson with system, the easier. standard The metric where 2.1 Spacetime geometry, momentaIn and order energy towe describe consider the a impact spatiallyonly. of flat spacetime Units Friedmann-Lemaˆıtre massive with are neutrinos scalar chosen metric on so perturbations the that evolution the of speed inhomogeneities, of light in vacuum is equal to unity. We adopt in JCAP01(2014)030 i P ) (2.9) (2.7) (2.8) (2.11) (2.10) .The 2 ⌫ J µ 4 (1 P = 0. Thus 2 , a i = i 0 U P = P µ⌫ ) , i T P i is a vector tangent @ =0 )). The fact that the i µ 1 and J U . )
2.29 i ◆ n i @ i 0 = @ (2 P 3 P µ n is given by i U , @ µ ,where ) can thereby be rewritten
µ⌫ )+ i µ U + T @ U =0 2.8 µ . ⌫ ; H i J +( ⌫ P H 0 J a/a ⌘ µ J = ⌘ ,eq.( , @ ) ⌘ @ P i ( 0 @ n P n P ⌘ = + ✓ =0 0 @ ⌫ 3 J n µ H ; =3 –5– J µ ⌫ = ) that appears in the second term of this equation could be ; J ◆ µ ), by =3 i
n x P J ⌘ i 0 ◆ @ +2 P ⌘, P n = ( i 0 + ✓ ⌫ n ; P i P µ⌫ H @ ✓ ), ) T i
@ 2.1 +(4 i + +2 J . Given that . It leads to, n i 0 0 ⌘ @ J @ P ) + )
0
(1 + 2 J ⌘ @ (1 + n (1 + 2 a , expressed with covariant indices, thanks to the relations ⌘ i 2 @ P a = is the particle four-current and where we adopt the standard notation; to indicate is the conformal Hubble constant, 0 µ = U H J 0 0 The four-current is related to the number density of neutrinos as measured by an P J We note however that the factor (1 + 2 4 = we however keep such factor here and in similar situations in the following. same momentum so thatconservation of the this energy tensor momentum then tensor gives dropped as it is multiplied by a gradient term that vanishes at homogeneous level. For the sake of consistency where a summation is stillwe implied consistently on repeated keep indices. all contributions Note to that in linear all order2.2.2 these in transformations the metric Evolution perturbations. equationThe of second the motion momentum equationobservation expresses that, the for momentum conservation. a It single-flow is fluid, obtained all from particles the located at the same position have the momentum and We signal here thatwith this the number comoving density numberright-hand is density side the that of we proper itsthe will evolution number expansion equation define density, of is later not the non-zero (eq. to universe. is ( be This thus relation not confused can surprising. alternatively It be simply written reflects with the help of the where observer at restto in the metric worldlinen ( of this observer. The latter satisfies where a covariant derivative. It is easy to show that, in the metric we chose, this relation leads to, evolve in the same gravitational potential.the Thus neutrinos in have the initially following the wecreated consider same fluids velocity. nor in In which annihilated all sucheach physical nor systems, flow di↵used neutrinos obey are an through neither elementary collision conservation law, processes so neutrinos contained in 2.2.1 Evolution equationThe of key idea the is proper tois consider number a only density set one of velocity neutrinossatisfies that (one this form modulus a condition, and single flow, it one i.e. direction) will a fluid at continue in a to which given there do position. so If afterwards a since fluid the initially neutrinos it contains JCAP01(2014)030 (2.14) (2.15) (2.16) (2.17) (2.18) (2.12) (2.13) . )=0
5 + = 0. It eventually . ⌫ ( j J @ ⌫ . ; j i instead of contravariant 3 i , i )=0 ) V @ P i P j x . ⌘ 0 V P ) @ ⌘, j P ), defined by (
P 0 i x P V + ⌘, ) H j ( ( @
✏ 2 j i (1 + 2 . ) is our second non-linear equation of a ) 2 @ V . 0 a ✏V with the help of the relation
) 2 P i (1
+ 2.13 + ). We can easily show that, m =0 P 0 = +
⌫ i i P = J @ 0 P )+ i ⌫ j ; P 2
µ ( @ ) (1 + ✏V / j 0 ) and reads P x i 0 i –6– P P +2 +(1+ P P + P ⌘, ) can be obtained by introducing the physical ve- = ) i ( ✏ i 2.13
i ⌫ P is arbitrary but we will see that it is crucial when it ; @ m V @ i i 2 µ⌫ = 2.13 q +2 i T V V = ) 1 ) leads to (1 + 2 V + 2 2
i p
2.7 V P i + @ ), = (1 + 2 )= 2 0 )+ x 2 P 2.13 i ⌘, V ( P ✏ ⌘ +(1+ @ ✏ ⌘ )(1 @ ) (expressed in units of the speed of light). This velocity is along the is a sign convention that we use in all this paper. x ⌘ ✏ ) @ and is such that ⌘,
( can entirely be expressed in terms of i i i P V H ( (1 + V i a V + = i Similarly, the evolution equation of the energy field We have now completed the derivation of our system of equations. It is a generaliza- This relation should be valid in particular for spatial indices, 0 P V or a combination of both such as 5 ⌘ i @ can be deduced from eq. ( We will now compare thesewhich field is equations based to on those the obtained evolution from equation the of Boltzmann the approach, phase-space distribution function. of non-relativistic particles. Its evolution equation derives from eq. ( From this equation, it is straightforward to recover the standard Euler equation in the limit Note that equations in terms of the proper velocity field. 2.2.3 Other formulationsAn of alternative the representation momentum oflocity conservation eq. field ( momentum P comes to actually solve this system in thetion linear of regime, the see standard section particles. single-flow equations The of a latterto pressureless is unity fluid obtained composed (while simply of keeping non-relativistic by its imposing gradient large). to the To velocity see to it be more easily, small let compared us express the motion imposes, on the covariant coordinates of themotion. momentum. At Eq. this ( stage the fact that we choose covariant coordinates which combined with equation ( JCAP01(2014)030 ). of i 2.3 p 3 (2.26) (2.21) (2.22) (2.23) (2.24) (2.25) (2.19) (2.20) d i and the x i 3 x . =0 j defined in eq. ( @ 2 i ✏ 2 q 2 q a ⌘ , , j i we have, , ij @ q i 2 q q a✏ =0 . ) . j . ˆ ij ◆ n ) i f =0 ˆ
i n =0 . ⌘ , the comoving positions f p ⇣ i =0. To compute this total derivative, + 3 i ⌘ d j d ⌘ ) can be simplified into, a ✏f ˆ @ n + @f @q ✓ d i @p i i i q i n ⌘ q ⌘ ⌘ 3 . f/ p @ 2.19 d i d d @ d @p d q (1 + q +(ˆ and the energy-momentum tensor are related i i q a✏ + Z + + ⇢ q a✏ = i ⌘ f –7– + i ◆ @ i = i @f ˆ n f @x @ )=
q i i i i 0 i x ⌘ ⌘ x ⌘ p p x @ can thereby be expressed in terms of the distribution x + d d d d d d ⌘, ⇢ , which remains in any case the number of particles per (
= ✓ i ⇢ f + ], + i i i 4 ⌘ @ x f d @f @q d a✏ @ @⌘ @f + @ @⌘ a✏ f = + @ @⌘ i are a priori space and momentum dependent functions. Because is the unit vector along the direction f ⌘ q i i d ⌘ d @ n d i . The particle conservation implies that / q a✏ i i p p ) and ˆ
), defined as the number of particles per di↵erential volume d satisfies a Liouville equation, d i j + q f ,p i i and d q ij ⌘ . As demonstrated in [ ⌘, x d 0 ( 0 / = i f T ). On the basis of previous work, we adopt here the variable x 2 i +(1+ q p function 3 By definition, the proper energy density = @⌘ @f d i ⇢ x 3 by function, where The resulting Vlasov (or Liouville) equation takes the form, On the other hand the geodesic equation leads to, We are notCartesian interested coordinate in description. deriving From the a very multipole definition of hierarchy at this stage so we keep here a In other words, there is some freedoma↵ect about the the physical choice interpretation of of d the momentum variable (butIn this this choice context, does the not chain rule gives for the Liouville equation, where d of the Hamiltonian evolution of the system, eq. ( The Boltzmann approach consistsfunction in studying thethe evolution of phase-space, the with phase-spaceconjugate respect distribution momenta to the conformal time 2.3 The single-flow equations from the evolution of2.3.1 the phase-space distribution The non-linear moments of the Boltzmann equation JCAP01(2014)030 ) with 2.10 (2.29) (2.30) (2.31) (2.32) (2.33) (2.27) (2.28) (2.34) i q 3 . i p ) i , ,p i 3 , a )=0 )) ⌘, x ). This comparison ( x f , the relation between ⌘, i . i 2.25 ( ] q p i
i 3 3 ( P p i d [ )d )withrespecttod @ . i F Z i 3 A p ) a ✏f i ( . 2.25 . ) D ) )= ) +2 ,p i x k i i x i ) a✏ q ,p ⌘, ,p x A i ⌘, i ( i 3 ( ⌘, x =(1+3 c ... @ ⌘, a ( ), which is such that the proper energy n n ) ( i j ) f c ⌘, x ) ⌘, x x q
q a✏ ( ( x i 3 n x i f f p ⌘, + ( q ⌘, 3 a✏ i ⌘, i ( ( d n p i 3 q )= 3 associated with this fluid thanks to i i 3 a P q d Z d f 3 ,p –8– i d Z or Z ), reads 1+3 )= Z x i ⌘, x x ) can be defined as the average of the phase-space )+(1+ ( p )= )= x ) ⌘, ⌘ ii ⌘, i x ( )= x ( ) ✏ A ⌘, flow c x , a complete hierarchy giving the evolution equations of ) ,p ( ⌘, x ). Given that d ⌘, i i n ), i.e. the number of particles per comoving unit volume ( x A + ( ⌘, c x P )], ⌘, ( )] n ⇢ n ( one ⌘, number density x n x ⌘, x ( 2.26 ⌘, f ( ( n ⌘, )(3 c ⌘, f are defined as, ( ( ij...k i n i i ⌘ )= A P p i P @ )= [ 3 [ proper x d F ,p ij....k . Using the distribution function to compute this mean value, one i F i ⌘, A ), Z p ( H x ⇢ ⌘, x ) is therefore ( x ⌘, +( )= f ( i ⇢ x ⌘, ⌘ ( P @ ⌘, n ( c number density is the Dirac distribution function. As a result we have, for any macroscopic field n ), can alternatively be obtained from the Vlasov equation ( ) x D ) and can be obtained following the same idea. ⌘, x 2.13 In the particular case explored in this section, fluids are single flows so, for each of them, Let us start with the number density of particles. By definition, for any fluid, the ( , is related to the distribution function i i ⌘, P x ( ij...k c 3 where depending on comoving momenta thus has to be inn agreement with eq. ( Similarly, the momentum field On the other hand,density the is given by to explicitly relate them. comoving d A 2.3.2 The single-flowThe equations aim of from this the paragraphand moments is ( to of show the thatrequires Boltzmann the to motion equation precise equations the we physical derived meaning previously, of ( the quantities defined in both approaches in order where the quantities As explicitly shown in appendix Its evolution equation isproper obtained weight, by integrating equation ( JCAP01(2014)030 ) )) x ⌘, (3.3) (3.1) (3.2) ) are ( 2.34 (2.35) (2.36) (2.37) (2.38) i x P ⌘, following ( n n is constant. , integrated 3 ...i 1 , i (0) . i A f/a P ). etc. )=0 ) from our equations , 2.16 and ) ) in the linear regime. x 3 ) gives A.5 ⌘, /a
( 2.13 )–( ( i . 2.19 A 0 @ ) )–( i p ) gives the following evolution A.3 x ) i ⇢V . ⌘, )multipliedby ) 2.10 ( , defined as ,p 2.18 , q i is expressed in terms of j i )), ( 3 ) ⌧ , ✏ c V a ) in 2.27 ) n )+2 q , ⌘ i , ⌘, x x 2.13 ( 3 x =0 ( a ⌘, (0) f ) and ( ⌘, ( ⇢V . ◆ (0) n ( f ( i c i i i q i n 2.9 H P @ a good variable to label each flow. To take 3 q V of the equation ( ), after i ) 3 0 =0 3 ) defined previously as a function of d i d
P P x p (0) 2.9 i )= ), i.e. eq. ( R (0) 3 Z )= ✓ i + –9– = ⌘ ⌘, P i x ( ( P is (see eq. ( @ 0 ⌘ )= i ⌘, 2.27 ) allow to recover the equations of motion we derived (0) )= @ x P (0) ( P + i n x ⌘, ij ⌘ c P ( @ ⌘, n 2.34 n 6 ( ) ⌘ ⌘ ), n x @ ,A i ) )+(1+ x ) ⌧ ⌘, 2 x ( x ⌘, ✏ V ( ⌘, ) directly gives the second equation of motion when expressed ) and ( ⌘, ✏ ) and that, for a single-flow fluid, the fields ( ( ) 0 x x A P )is . It is then a simple exercise to check that the subsequent equations 2.23 2.32 ) ⌘, )(3 + ⌘, ( n x ( ). ⇢ ), ( 2.10 n ⌘ )satisfies ⌘, x @ ( i 2.13 ⌘, 2.29 ( )= V )= ✏ x x )by H ( x ⌘, ⌘, )= ⇢ ( ( ,eq.( x i ⌘, ⇢ A ( + P ⌘, A ⇢ ( ⌘ and divided by i ) is exactly the average of eq. ( ). Finally, it is also straightforward to show that the average (as defined by eq. ( @ i A q 3 Note that conversely, it is possible to derive the hierarchy ( 2.37 2.31 The energy field 6 As a result the numberThis density of latter particles result is is simplyadvantage decreasing of attractive as this as 1 property, it we makes introduce a new variable, Let us start withorder contribution the of homogeneous ( quantities. It is straightforward toand that see the that unperturbed the equation zeroth for 3 The single-flow equationsIn in this section the we linearIt explore regime the is system useful of insystem. motion particular equations in ( order to properly set3.1 the initial conditions The required zeroth to order solve behavior the eq. ( over d of the hierarchy can be similarly recovered with successive uses of eq. ( equation for Given that related to which is exactly the first equationeq. of ( motion, ( of the geodesic equationin ( terms of of motion. For instance, the combination of eqs. ( It is then easy to see that integration over d Taking advantage of this, we proceedwith to the show relations that ( the Vlasovpreviously. and geodesic We equations first together noteis that nothing the but field JCAP01(2014)030 (3.8) (3.9) (3.4) (3.5) (3.6) (3.7) (3.13) (3.10) (3.11) (3.12) (0) n ! (1) 0 is constant does not 2 0 P ⌧ i . @ (0) i i (1) ⌧ P j X v i , given by i @ ⌧ . v i 2 , (1) v ij i . i 0 ⌧ ⌧ a P + i (1) = @ ⇠ X (1) 2 . i
i . @ X 2 @ (0) i ⌘ + a 2 0 ,v . 2 @ (0) . (0) ⌧ ⌧ P i 0 2 i j 0 . ⌧ , (0) ⌧ m ⌧ v i = i P i P + n v ⌧ (0) i + 0 i
ij ⌧ + ⌧ (1) i 2 mv 1 = V ) @ ⌧ X p 0 2 i (1) p = ⌧ i / – 10 – p @ 1 @ i X = i 0 ) = = ⌘ v ⌧ ⌧ 2 i 7 @ ⌧ a ⌧ ⌧ = i
(1) = ⌘ 0 i + i ⌧ ⌧ d constant and @ P 2 (1) a d (1) ⌘ ⇠ 2 X d ⌘ X +(2 m @ d . Note that this quantity is not constant over time. Given the (0) ( i ]). (1) = P (0) = n 0 25 i P H v (1) ⌘ 3 d X adopted in this paper, it satisfies as is the comoving momentum so the fact that d 0 0 i ⌧ P P (0) n ⌘ @ =3 is the initial time. We also introduce the norm of in (1) ⌘ ⌘ n d By definition, d The behavior of the momentum variables with respect to the scale factor can also be deduced from the 7 geodesic equation (see e.g. ref. [ and that for the momentum is given by, With this notation the first order equation of the number density reads, total first order derivative operator as, It can be written alternatively, or alternatively, 3.2 The firstWe order focus behavior now on the first order system. In order to simplify the notations, we introduce the It satisfies other massive particles -At are this smoothed stage, we with can time already because note of that, the expansion of the universe. To be more comprehensive regarding notations,be let labeled us mention by that the the flows zeroth can order alternatively velocity, denoted sign convention for conflict with the fact that the physical velocities of neutrinos - as well as the velocities of any where Similarly, we define JCAP01(2014)030 ) 3.12 (3.20) (3.15) (3.16) (3.17) (3.18) (3.19) (3.14) .Itis 0 .Thisis i /⌧ v 2 ⌧ , which gives µ is potential at and + P i ), let us move to
P (1) 0 i ) ⌧ P x 3.14 ( n ◆ )–( 2 2 0 ⌧ ⌧ 3.13 1+ . , ) )–( 2 x ✓ . k . k 2 0 0 3.12 ⌧ ⌧ ⌧ ⌧ 0 . ⌧ , ) ) k µk + x i , ( (
)exp(i P ) form a closed set of equations describing 2 (1) ✓ i k (1) 2 k 0 i ( P 0 ◆ ⌧ n ⌧ k i 2 2 ⌧ i F @ 3.14 k µ 2 + (0) 1 / 2 2 0 n 3 k , which is a combination of ⌧ ⌧ ) P 3 )= – 11 – (1) i ✓ 0 ⇡ d (1) x )= ⌧ 0 P is entirely characterized by its divergence, i ( ⌧ )= i k (2 ⌧ 1 ( ⌧ P P x ✓ ( ✓ (1) Z µk i (1) 8 n i = 0 P P , such a mode is expected to be diluted by the expansion P ⌧ ✓ =i (1) 0 )= behavior is dictated by the gradient of (1) i + P x P P ( ✓ ) by its divergence, one finally obtains from equations ( (1) ⌘ i ⌘ F @ P @ 3.13 +3 n at first order, ) associated with relation ( 0 ⌧ ⌧ 0 P 3.13 µk =i remains e↵ectively potential. At linear order, this property should be rigorously n ) and ( ⌘ (1) ) i @ P form a gradient field. As a consequence, although one cannot mathematically exclude 3.12 3.13 As a consequence, the We close the system thanks to the on-shell normalization condition of (1) i But there is no guarantee it remains true to all orders in Perturbation Theory. 8 P and After replacing equation ( and ( of the divergence field, and of thelinear potentials. order. We It can indeed implies here that take full advantage of the fact that We will consider the Fourier transforms of the density contrast field 3.3 The systemTo in explore Fourier the space propertiesFourier of space. the Each solution fieldfor of the is the Fourier decomposed system transform, into ( Fourier modes using the following convention the expression of Eqs. ( the first order evolution of a fluid of relativistic or non-relativistic particles. exact for adiabatic initial conditions. not the case ofthe other reason variables we such preferably as write the motion equations in terms of this variable. The latter equation exhibits a crucialof property: it shows that the sourcethe terms existence of the of evolution a curlso mode that in JCAP01(2014)030 , v (3.21) (3.22) (3.23) , , ⇤ v
and k 2 v 1+ ⇥ µkv +i P ✓ ma . 2 µ .v 2 or alternatively between kv k . v ⌧ =
1 .⌧ + and k⌧ 2 – 12 – k v 2 k = v µ 1 2 k p 2 v ma ⌘ 1 @ p for illustration purpose). More precisely, the overall distribution + +3 1 P n µkv µkv ✓ i i = = n P ✓ ⌘ ⌘ . Sketch of 1D phase-space evolution. The top panel shows flows with initially no velocity gives the relative angle between @ @ µ into account (see figure fluid of neutrinos.neutrino Note that fluid. this If constructionthree is eigenstates. the valid masses for are any not given mass degenerate,4.1 eigenstate it of should the therefore Specificities be ofIn repeated the a multi-fluid for multi-fluid description each approach, the overall distribution function is obtained taking several fluids tion of a fluidbe of implemented initially numerically. relativistic species. In the following, we4 explicitly show how it A can multi-fluid descriptionIn of this neutrinos section, we explain how one can define a collection of flows to describe the whole This is this system that we encode in practice. As we will see, it provides a valid representa- where These equations can alternatively be written in terms of the zeroth order physical velocity gradients but density fluctuationspanel shows (illustrated how by the theflow flows develop thickness regions velocity variation after gradients of atflows. they later the experience It times. lines). is shell-crossing, Theyno expected in The can initial to ultimately bottom a velocity happen form way would multi- preferably behave similar to exactly to flows like what with a happens cold low dark initial to matter velocities. dark component. matter Note that a flow with Figure 1 JCAP01(2014)030 ]), (4.5) (4.6) (4.7) (4.1) (4.2) 4 . )) i ⌧ ; x , )) (4.3) )) (4.4) i i ⌧ ◆ ⌘, ⌧ ; ( f k ; , , k i i x x ◆ P ⌧ )) T i j ⌘, ⌘, ⌧ 2 ( k ( ; i j i i , i k i x k P p P 1 3 ( ( ( 3 . Before doing this, we will f ⌘, D F ( j F i . These quantities are directly 4 q 4.3 ) ) -fluids (i.e. a sum over all the ) ⌫ j j P i i a i i i ij ⌧ ⌧ k ⌧ ⌧ T ; ; ; i f, ✓ i x x q x ) 3 p ) i 3 ( i ⌘, ⌘, d q ⌘, j a q ( ( ( ( D q c c ( ◆✓ ✏ i n ✏ Z n ) n i q ij i 5 i i q ⌧ ⌧ a i ⌧ 3 3 ; 1 3 ⌧ 3 i d X d x q d 3 =i Z ⌘, Z d j Z ( i )= 2 c 0 k i = – 13 – i Z k ⌧ n T k 0 )= ; i 0 )= i i ⌧ i ✓ p i T 3 p ( ,p k ( d i 1 3 F F Z ) = =i = = ) ⌘, x i and the total shear stress i ( ⌫ ⌫ ⌫ ⌫ ,q ✓ ⇢ ,p i )= ✓ i i flow ⌘ ⌘ 3 ,p a ⌘, x (0) i (0) ⌫ ⌘, x ( ( ⌫ one P P f tot ⌘, x tot + ( f i f + ⌧ i i X are the density and pressure of the neutrino fluid at background level tot (0) q p (0) 3 ⌫ f 3 ⌫ ⇢ . For each flow, the evolution equations are known but we still have to set d d (0) ⇢ being assumed to describe a 3D continuous field. ⌫ ⇣ )= F i ⇣ i P Z Z ⌧ ,p i has to be reconstructed from the single flows labeled by and tot ⌘, x ( f (0) ⌫ ⇢ tot f It means in particular that the momentum integrations in phase-space used in the , the total energy flux dipole ⌫ and stands for the perturbed part of the quantity it precedes. where 4.2 The multipole energyOf distribution particular interest in to the compareputation linear our of the results regime overall to multipole energy those⇢ distribution. of the We will Boltzmann focus approach onrelated is the to total the energy the com- density phase-space distribution function thanks to the following relations (see [ for any function the initial conditions tocompute be the able multipole to energy distribution use associated them with in our practice, description. see or equivalently the parameter standard description (i.e. forshould a be single multi-flow replaced fluid)possible in to initial compute our momenta global description or physical velocities), by quantities a sum over the In the continuous limit, we thus have function JCAP01(2014)030 i . ⌧ )– and !# (4.8) (4.9) 4.9 ) (4.13) (4.12) (4.10) in , i ), where ⌧,µ ) ( 3 ) = 0. i ⌘ ⌘< ⌧ (1) ⌧,µ ⌧,µ ( ; ( V in (1) (1) ✏ . i ,⌘ ) ) ) x V ( i P k ⌧,µ ⌧,µ ) ⌧,µ ( ✓ ( ( ,definedbyeqs.( P ) (4.11) ◆ (0) ✓ k (1) ⌧,µ 1 3 n (1) ⌫ i ( i ⌧,µ V v ( i (0) ⌫ j . It obviously imposes )+ 2 i ⇢ P v ⌧ µ ✓ j µk and 3 + ) k ✓ ⌧,µ
( ) ) (1) ⌫ µk ⌧,µ i ✓ ) (0) ( ak k ✏ ⌧,µ ⌧,µ ) ( )yields . ( µv 2 i ⌧,µ 2 i within d v ⌧ v ( v ) ) 4.4 i ⌧,µ v . The initial conditions depend obviously ( ⌧ ) in , )= ma ⌧,µ ) (1) 1 ⌧,µ i ) ( ( ⌧ ) ⌧,µ n p i ; ( ) (0) ⌧ (1) 2 ⌧,µ x i ⌘ ⌘> ⌧,µ ; ⇢ ( ⇢ , (1) k – 14 – ( )= x " ⇢ 2 ) = 0 and consequently that ⌧,µ in are source terms generating the metric fluctuations , i (1) h ( v µ ⌘ µ ⌧ 2 in ( d µ ; d is equal to ⌧,µ i ⌘ µ⇢ (1) ⌫ ( i 1 d ( 1 1 in i P 1 d i v 1 mv 1 1 P ) 1 (1) P 1 ,⌘ Z Z 2 p ⇢ x Z v ⌧ ⌧ Z ( and d ⌧ d ⌧ 2 2 d (1) d )= ⌧ i 2 ⌧ (1) ⌫ 2 ⌧ ✓ P ⌧ Z , Z ⌧,µ Z ( ⇡ ⇡ Z i )=(1 ⌫ (1) 8 4 ⇢ ⇡ ⇡ (1) ✏ ⌧,µ ( is chosen so that the neutrino decoupling occurs at a time =4 =4 = (1) in i (1) ⌫ (1) (1) ⌫ ⌫ ⌘ ✓ ⇢ V ⌘ ⌘ . (0) (0) ⌫ ⌫ 4.4 P P + + At linear order these quantities can be expressed with the help of the linear fields ), to be zero but we still have some freedom in the way we assign each neutrino to one (0) (0) ⌫ ⌫ ⇢ ⇢ ⇣ 4.10 ⇣ It implies in particular that The description we adoptall is the the neutrinos following: whose at momentum initial time we assign to the flow labeled by adiabatic initial conditions. It( imposes to the quantities flow or to another.choice The respecting initial the conditions adiabaticity we constraint. present in the following correspond4.3.1 to a simple Initial momentum field in section 4.3 Initial conditions The initial time neutrinos become non-relativistic aton a the time cosmological model adopted. In this paper, we describe solutions corresponding to The physical quantities involved in the growth ofthe the predictions large-scale of structure the of multi-fluid the approach universe. with We those use of them the to standard compare Boltzmann approach For explicit calculation, note that and that introduced in our description. More precisely, eq. ( JCAP01(2014)030 2. / ) in (4.20) (4.16) (4.17) (4.18) (4.19) (4.14) (4.15) ,⌘ x (
, ) ]). We finally p at initial time, 4 . ( . i 0 ], after neutrino ) ⌧ )= f ⌧ in 25 ( )))] ⌧ = )] 0 ⌘ in . ( i f T ) )] p p ,⌘ p B . T /T ( x d log ) 0 B ( . ak ], on super-Hubble scales, f ) ))] both vanish at initial time d log in ( i 5 d log d x ak T ⌫ ( ,⌘ , ◆ p/ ,x ) ◆ x in + ) in p/ ( x ⌘ i ⌘ , ( T ( , ( in T ,x and ⌘ )] B exp[ T in ⌫ ( and no chemical potential (see e.g. T ⌘ ✓ 1+exp[ + ( ak +3 B ( T + ◆ ) ) / T i i 1 ) ak ( 1 i ) )) ( B ) ,⌧ ,⌧ x i ) is non zero only for in i i ) that 1 ,x , i q/ ⌧ . As explained in ref. [ ak i ; in 2 f ,⌘ in ,x ,x ( q ,x ⌘ x ⌘ x in in ( ( 4.10 , ( in q/ T ⌘ ⌘ ⌘
in ( ( is the norm - previously defined, see the geodesic ( – 15 – ⌘ )–( ), one thus finds ✓ ( ). Besides, the adiabaticity hypothesis imposes q (1) (0) in 1+exp[ n T ) f f in 4.8 i (1 + ⌘ ,⌘ ✓ )+ ( p / x ,x in 1+exp[ ( ) )= in (0) q T i
,⌘ ⌘ ( ⌧ / 2 ( /⇢ x ; 0 with a temperature i )= ) ) ( f T in 0 ,q ,p ) can be easily computed. It reads ,x i 1+exp[ f i ✓ x ,⌘ in )= , ⌘ ) and recalling that ,p x ,x / T i ( in ( in , it can be rewritten )=3 in i ) n ⌘ B i ⌘ p i ⌘ p ( ( ⌧ (1) ( 2.31 ; ⌘, x ⇢ ak f ,p ( (0) x (1) x , , f (1) /⇢ in ) )= in f ⌘ ⌘ )= in ( . Before decoupling, the background distribution of neutrinos is expected ( ,p ) and ( n i in f ,⌘ f ) - of the phase-space variable ⌘ ( x ,x ( ] for a physical justification of this assumption), 2.29 in /T ⌘ 2.23 25 (1) is the Boltzmann constant and ) ( ⇢ , in 5 B (1) – k ,⌘ f 3 Initial number densities are obviously strongly related to initial distribution functions x ( T with this choice of initial conditions. Besides, at super-Hubble scales,get the the neutrino following expression distribution for is the static linearized (see initial e.g. number [ density fluctuations, It can then easily be checked from eqs. ( The last step ofto the the calculation metric consists fluctuations inthe for relating adiabatic temperature the modes. local perturbation initial As of4 temperature mentioned the in fluctuations neutrino [ fluidequality between is the initial proportional density to contrastsphotons, of its all species. density constrast: Using the standard result that, for which can be reexpressed in the form, the initial numerical density contrast follows directly, The expression of In terms of the variable Given eqs. ( where equation ( decoupling, the phase-spaceDirac distribution distribution function but modified of by relativistic local neutrinos fluctuations is of temperature, still whence a Fermi- in phase-space to follow arefs. Fermi-Dirac [ law 4.3.2 Initial numberAlthough the density velocity field fields aredensity initially fields uniform, it as is we notat expect the linear case the order. of total the numerical individual numerical density field to depend on space coordinates JCAP01(2014)030 . . eq are are a k
(4.26) (4.24) (4.25) (4.27) (4.22) (4.23) and into Legendre P ✓ and n . ) (4.21) ) ) noting that µ µ , ( ( ` ` H 0 P P 3.23 ` ` P, ) i) i) ✓ ) are numerically integrated with am ⌧ )–( 3 ( ⌧ = 2 and the one of the momentum 0 2 at superhorizon scales for adiabatic ` / f ) 3.23 )( )( 1 3.22 a )
⌧ ⌧ 2 ; ; d log )] v + k k d log at leading order. ⌘, ⌘, ( ◆ ( ( ) and ( Mpc is the horizon crossing wave number at ` 2 +2 (1 a k ` n, ` P, h/
2 + ✓ 3.22 v – 16 – 1 scales like dependence of the fields 2 01
` 0 ` . n, 0 X 2 m X 0 µ P, ✓ 1 v n, ✓ H + ⇡ k [ p 3 )= )= . H k a 1 scales like k H 2 H eq k p ⌧,µ ⌧,µ ; ` n, = = =3 = = ; k k 1 0 1 2 0 = 1. The leading order behavior corresponding to these terms and n, n, n, P, P, ⌘, ` ⌘, ( ✓ ✓ ( and n P /a ,where . We know however that neutrinos develop an anisotropic pressure ✓ . µ +1 eq ` k a scales like 1 scales like H ` P, ✓ Once the equations of motion are decomposed into Legendre polynomials, one gets the help of a Mathematica program in which the time evolution of the metric perturbations expressed in units of matter-radiation equality. Similarly, theBesides, time convergence dependence tests will are be realized for expressed a in4.4.1 neutrino units mass of of 0.05 Method eV orThe 0.3 equations eV. of motion in Fourier space ( 4.4 Numerical integration This section aims atmulti-fluid description describing and the to numericalthe compare integration its Boltzmann scheme e ciency hierarchy. developed withusing All to that the of the deal WMAP5 the with plots cosmological standard a parameters. we integration will In of the present following, to the illustrate wave number our values findings are made where divergence multipoles up to can be obtainedconstant, easily from theinitial motion conditions. equations ( successively, and In order to properlythe compute expression the of source the terms of number the density Einstein multipoles equations, up one to needs to know these calculations in asub-leading numerical behavior code, at itarise initial is at time. therefore early time usefulpolynomials, To by to that decomposing examine aim, the in we more study detail how the higher order multipoles 4.3.3 Early-time behavior A remarkable property ofthey the do initial not fields dependwhich on we is just a computed source is term that of they the are Einstein isotropic, equations. i.e. For practical purpose, e.g. to implement JCAP01(2014)030 . µ )– eq N 5 k .In x 4.8 2 µ . (4.28) 05 eV. . and =( 4times. has to be / q =0 h µ =0 q on one side N , ). The main k N )betweenthe m , , µ 3 ⌧ nh )] N 5 N 4.27 + x and ( 1 . = 10 . 4 and )–( f x 2 and eq / eq q k ⌧ = 2 N . 4.23 . In practice, we divide N )+7 n 7 a/a 4 4 3 h 4, 4 = 6 give for x 1+ =0 / ( x ⌧ k and f correspond to particular cases ]). The initial conditions we N 10 2 5 10 2 10 max 2 ` 26 are conjugate complex numbers = 6, and on figure intervals - where 3 ) + 32 µ 3 3 3 1 µ = 10 max x ` by, ( N , we succinctly review the construction 10 10 eq f 5 10 is taken into account via the Legendre x A d and i and figure a/a 5 q and = 12 and 1 2 2 2 = 12, x µ 1 x µ q R µ ) + 12 16 40 100 is the order at which the Boltzmann hierarchy 2 N N 10 10 10 N x , ( – 17 – ⌧ f max ) ` N ! ( = 40, ⌧ ) + 32 N ⌧ 1 x N ( 16 40 100 f = [7 ) and q # h ( 45 2 N q necessary to compute the multipoles, the amplitude of accuracy. However, the relative performances of the two methods ⇡ i N 3 q x 3 , the largest magnitude of the relative error on either the density, the dipole ⌧ )d N x ( ). The angular dependence of on the other side, where f ranges into respectively and 5 x 1 q µ x A.15 max N Z ` 05 eV a 10 . )–( . and µ . Relative errors (averaged in time between and =0 q q A.10 , They show that the values computed in both descriptions can reach an extremely good In the multi-fluid description, integrals that appear in the distribution energy ( ) also involve a discretization on momentum directions, i.e. a discretization on m 4, to compute the integral. In such a scheme, which gives exact results when integrating N ⌧ / 1] for ) , 1 4.10 for which there are no or very few oscillations in the neutrino fluid (due to the relatively small and is truncated. Preliminary results are presented on table agreement. For example, and actually depend on wave numbers. Indeed, table Besides, since thewhen Fourier the modes initial computed gravitational potentials for [0 are real, we can restrict our calculations to4.4.2 the range Results We first compare the consistency of the two approaches by varying that is to sayx in using fivepolynomials discrete of values order regularly less spaced,the than i.e. 6, the errorare term multiples is of proportional to four - and we apply the integration scheme ( both approaches, all(Boole’s integrals rule) are which estimated consists using in approximating the third degree Newton-Cotes formula of the Boltzmanneqs. hierarchy. ( In thispolynomials approach, used energy to decompose multipolesof the are the distribution integration computed function on in thanksdiscretized d phase-space. to for Of numerical course, integration. because is given. It isfrom as usual a determined standard by theimplement Boltzmann correspond Einstein to equations code the but (the adiabatic inobjective expressions practice code appearing of simply in presented extracted this eqs. in ( are numerical [ identical experiment when isequations computed to of from check the multi-fluid that the description. the resolution In appendix multipole of energy the distributions Boltzmann hierarchy or from the Table 1 results obtained with the Boltzmanneach hierarchy value and of those obtainedor with the the shear multi-fluid is approach. given. For Calculations are made with JCAP01(2014)030 ⌧ . N 3 and and . ⇥ ⌧ eq µ k 1000 N 1000 ⇥ =2 100 k 100 10 ) for q . . eq eq N a a ê ê 1 a a 10 . . and eq ⌧ k ). Time integrations become 0.1 N ) allows to reach an exquisite 3 . 1 k . These tests show that the extra 4.28 do not appear as critical limiting 3 0.01
L H L L H H max 0.1 ` 4 4 0.001 1 5 5 0 - 10 15
- 10 100
0.01 - - 10
10 of units in Residuals in of units in Multipoles
h y
4 - and µ N for more details regarding the relative precision , to the quite large value of the neutrino mass). – 18 – 2 1 (with adapted values of 1000 1000 2 100 100 , is not dramatically large if 10 10 q = 12, the numerical scheme ( . . (or smaller neutrino masses), the field values experience rapid N eq µ eq a a ê ê k N ⇥ 1 a 1 a 05 eV (see table . = 8, it is still possible to reach 10 is set to 0.3eV. The resulting relative di↵erences are of the order of 10 max µ ` m =0 0.1 0.1 N intervals. With 16 values for each, only percent accuracy is reached for m q and . 0.01 0.01 or eq and
k
⌧ H L L H L L H H . eq . Time evolution of the energy density contrast (solid line), velocity divergence (dashed . Same as left panel of figure 5 0.001 . 4 6 k - 0.001 . 1 10 - - 0.1 2 10 eq 100 1000
0.01 .
0.001 10 10
k
For larger values of in in of units in Multipoles units in Multipoles of h y h y is set to k =0 =5 , direction, making necessary the improvement of the discretization scheme resolution. This cost of the useequations of instead our of representation, where neutrinos are described byoscillations a during set a of significant time 2 then interval particularly (as long seen as on theamplitudes figure system is depend becoming more sti↵er. significantly Furthermore, on at fixed the times, momentum, the field both on its amplitude and on its put in the k one can get).factors in Meanwhile, the the accuracy ofsmall. parameters the For numerical instance, integration, with providedaccuracy of and course with that they are not too Figure 3 k value of the waveIn number such and, a for situation, figure the main limitation in the relative precision is the number of points we Figure 2 line) and shear stressand (dotted corresponds line) to of the thecomputed density neutrinos. contrast with of The the the dot-dashed multi-fluid darkwhen line approach. matter the is component. two presented Right Left methods for panel:q panel: are comparison residuals the compared. (defined quantities Numerical are as integration the has relative been di↵erences) done with 40 values of JCAP01(2014)030 ). 6= 0. µ 4.12 is set to µ ). Similarly . eq k , the cosine of the 10 increases, a better µ ⇡ k k ), are derived at linear and ⌧ 2.13 . Unsurprisingly, for a fixed i k ]). Finally, let us remark that 5 ) and ( ). 2.10 dependence of the total energy density = 100. Clearly, as ⌧,µ ( . µ (1) eq ⌧⇢ d a/a , which show the convergence of the fluctuation 2 is related to the momentum divergence by eq. ( 6 ⌧ 5 – 19 – R and the wave vector is, the quicker the convergence takes place. This is ⇡ i ⌧ and v 5 ,wherethe = 100 for a percent resolution if 4 )=4 µ µ = 10 and ( . N (1) eq ⇢ 6= 0, but only partially visible on these plots as they give only µ , this convergence exhibiting non trivial patterns when a/a µ , the resulting number density and velocity divergence are largely is necessary (for both schemes). We leave for further studies a more , to be compared e.g. to figure 11 of [ µ 3 q is required (up to at fixed times µ 9 and on figure The convergence is also more rapid when, for a fixed momentum, the neutrino mass is 2 To finish, we illustrate the fact that the multi-fluid approach, by its specificity, allows to 10 The velocity divergence that appears on figure More precisely, it corresponds to 9 10 the linear regime, wea have proper shown choice precisely of howneutrino the a fluid. initial proper conditions choice These allow ofmetric initial to the perturbations. conditions recover single-flow are the fluids given physical and behavior explicitly of in the the overall case of initially adiabatic distribution function. Thelevel resulting with fluid respect equations, to ( density the fluctuations metric and perturbations velocity but divergences.resulting at They from full can non-linear the easily level standardmatter be with study particles. respect compared of to to In the the theyond both equations distribution the cases, function scope equations of of describe a this single-flow single fluids kind hot so of fluid shell-crossing study. of is After dark be- having considered in detail these equations in 5 Conclusions We have developed an alternative approachto to account the for method massive based neutrinostion, on in neutrinos the non-linear Boltzmann are cosmological hierarchy calculations. treatedin as In fluid a this equations new collection descrip- derived of single-flow from fluids conservation and laws their or behavior from is the encoded evolution of the phase-space suppressed compared to thosefigure of the CDMthe component convergence (as of the can velocityIt be divergence simply seen is illustrates more on the rapid factequations. the than that left the the panel one former of of acts the as a number density. source term of the latter in the motion dependent on the valueIn of that case, theexpected fluctuations to of be the outis neutrino of particularly flows phase, important for and inthe particular of real during the values the of CDM complex oscillatory componentafter Fourier periods. modes. are integration indeed A This over consequence phenomenon of this complex dependence is that, mass, the smaller the initialillustrated momentum on the left panelsamplitudes of of figures several neutrinozero. flows to those oflarger, the as dark illustrated matter by component theConcurrently, when di↵erence the between right the solid panels and show the dashed that lines the of way these figures. this convergence takes place is strongly thorough analysis of the numerical requirements. show the convergence offlow the to number the density ones contrastis of and characterized the of by Cold two the parameters: Dark velocityangle Matter its divergence between initial its (CDM) of momentum initial component. each modulus velocity Let vector us remind that each flow phenomenon is illustrated onis figure presented resolution in a better resolution in JCAP01(2014)030 . 0 . T eq B k ,the k . 1.0 eq 45 k . =10 1000 k = ranging from k 0.8 and µ . 100 eq k and 0.6 0 range from 0 . =5 T eq 3 eV neutrino mass. ⌧ a . k m B ê 10 , k a . 6 . eq 0.4 k =3 1 =2 ⌧ k , . 0.2 3 eV. The left panel corresponds to a eq .
k
0.1 ê L = H =0 k 0.0 m 0 5 4 3 2 1 1
-
1.0 0.8 0.6 0.4 0.2 0.0
= 100. The wave numbers associated with the
CDM v
Re q q density energy Flow . eq a/a – 20 – 1.0 = 0. Right panel is for 1000 µ 0.8 relative precision). An additional information exists in our 100 5 0.6 . m 05 eV neutrino mass and the dashed lines to a 0 eq . a ê 10 a (top lines) with 0.4 0 = 1 (bottom lines). The time evolution of the velocity divergence of each flow T B µ 1 k = 10 and the right panel to 0.2 . eq = 0. It is computed for a neutrino mass
µ a/a ê 0.1 ]. We leave for future work the examination of the importance of non-linear e↵ects ] or to incorporate the neutrino component at non-linear level in approaches such 0.0 . Time evolution of the velocity divergence. Left panel: values of . Angular dependence of the neutrino energy fluctuations. The amplitude is arbitrarily set 31 0.5 0.0 1.0 0.5 29
-
,
–
3 2 1 5 4
This representation opens the way to a genuine and fully non-linear treatment of the We then check that the two descriptions are equivalent at linear level through numerical density energy Flow CDM v q q 30 27 = 0 (top lines) to neutrino fluid duringcrossing the can late be stage neglectedincorporated of separately - into the since the large-scale equations theIn describing structure particular, two the it evolution growth non-linear should dynamics equations be -in of possible satisfied this as [ to by growth. apply long each resummation as techniquesas flow such [ shell- can as those be introduced on observables such as power spectra. can actually be accounted for by(in studying practice such a we collection of reachedapproach flows a with since an 10 it arbitrary also precision by describes showing the how physics individual oftheir neutrino each initial flows flow momentum converge separately. to and the of We illustrate the CDM neutrino this component mass as point at a play. function of µ is plotted in unitssolid of lines correspond the to dark a matter 0 velocity divergence. The wave number is setexperiments. to The conclusion is that the whole macroscopic properties of the neutrino fluid Figure 5 (bottom lines) to 9 Figure 4 to unity for time where solid, dashed, dotted and dot-dashed lines are respectively JCAP01(2014)030 (A.4) (A.1) (A.2) (A.3) , 1000 ,. . . defined =0 ij , weighted by , A i q , 3 100 @ i ) and in general jj A )=0 a✏ , A ( . / A eq + i a ê 10 q a 3
j ( @ i ij @ ✏/a i A , 1 density contrast. A 3 . 3 3 a ✏f +2 ✏/a )withrespecttod i
j k
A
@ i 0.1 a✏ q number ê L H @ ij 2.25 ) A ...
4 3 2 1 0 1 2 j
+
- -
+
q
a✏ CDM v Re d d i i q a✏ A@ q – 21 – 3 + d ij ), one can build hierarchies to describe the evolution )+(1+ 1000 A ⇢, Z ii j @ A ⌘ ⌘ 2.25 )
+ A 100 + A ij...k A )(3 . ) by adequate factors such as eq ⌘ a ê 10 @ a 2.25 ), integrations by parts directly give the desired hierarchy of equations. +(1+ . Same as the previous plot for the i a✏ H ( A ) / 1 n +( i ⌘ @ A . To that end, it is useful to introduce the tensorial fields ⌘ Figure 6 i ...q @ q a✏
)
0.1 ê H ]) a✏ ( 23 / it leads to, 2 0 6 4 2 it leads to 1 + 4(
i
-
i
i
CDM v q d d A A A 3 ⌘ @ For for After multiplying eq.✏/a ( A.1 A Boltzmann hierarchyA from first hierarchy tensor can field be expansion builtproducts by of integrating equation ( as (see [ A The Boltzmann hierarchies Starting from the Vlasov equationof ( the moments of the phase-space distribution function. There are several ways to do this. The authors are grateful toand Cyril Atsushi Pitrou, Taruya Jean-Philippe for Uzan, insightful Pierreof discussions Fleury, and the Romain encouragements. university Teyssier of FB Kyotothe also and thanks completion the the of RESCUE YITP of thisBS05-0002 the manuscript. of university the of This French Tokyo work for Agence is hospitality Nationale during partially de la supported Recherche. by the grant ANR-12- Acknowledgments JCAP01(2014)030 (A.7) (A.8) (A.9) (A.5) (A.10) (A.11) (A.12) =0 j and rewrite n ˆ n ...i . 1 i ˆ k. . In momentum 2) A n ⌘ ] ↵ j jj ` , n ( @ ) ...i n =0 ) +1 , ◆ m , ) into a homogeneous part i ) 1
(2 + q ⌘, q ⌘ i ( , ( @ =0 m i x
( +1 ˆ n ` ◆
...i ] for more details). It is based on f ,sowedefine )] (A.6) j q 1 ˜ q ⇤
i
a✏ and its direction ˆ a✏k n @ q ˜ 23 using Legendre polynomials thus , ) jj , A q ) a✏ q , ) n n x ↵ 5 +3 +1 ( – ...i ` . By plugging this expansion into the ◆ ` ↵k m 1 ⌘ ` 3 i ) i ` i 2 P @ @ ` A ( ) ✓ ˜ ⌘, q ⌘
1) ) ) ( ( ` =1 +[(2 n ⌘ q 2 ( )[1+ X j i) m ( q @ q ˜ ⌘ n
0 ( ⌘, q ✓ @ n f + ( 2 5 0 ( – 22 – ( ...i 1 f 1 n i ` d log ` ) X ...i ˜ A ) + n ˜ d log j
)= +1 @ q 1 q a✏ ...i ⌘, q ) m ⌘, q , 1 ( i ˜ i ( =
1 1 ` x 0 ↵k + ( A ˜ ` i
˜ is through its angle with ˆ +
m f 2 @ i ✓ k a✏ ...i , qk ˆ n 3 1 + 3) ` ˜ i +i q a✏ a✏ ˜ qk qk ⌘
a✏ n A ( @ ) ), one obtains the standard hierarchy, . The next step is to expand ⇥
) +(1 + 1 )= )= )= m + i A.8 ⌘ ⌘ ⌘ ). Once linearized, it is equivalent to the standard hierarchy of equa- @ @ ) @ ( ) leads to the following equation for , ⌘, q ⌘, q ⌘, q q ( ( ( ( q ` 0 =1 0 1 n f 2.13 X ˜ ˜ ˜ m
2.25 H ⌘ ( ⌘ ⌘ dlog + @ @ @ dlog f +( ) is the Legendre polynomial of order ⇣ n ↵ ) and ( ( ⌘ ...i ` 1 i ˜ P
2.10 A ⌘ @ we introduce the moments where Boltzmann equation ( space, the only dependence on the linearized Boltzmann equation as where and a decompositionequation of for the latter into harmonic functions. At linear order,where the the Vlasov local momentum is defined trough its norm that Boltzmann codes usuallya implement (see decomposition refs. of [ and the an phase-space inhomogeneous distribution contribution, function eqs. ( tions describing thegiven multipole below. decomposition of the distributionA.2 function perturbation A as BoltzmannWe hierarchy recall from here harmonic the expansion standard construction of the Boltzmann hierarchy, i.e. of the hierarchy Note that this hierarchy of coupled equations retains the same level of non-linearities as and in general it leads to the following equation, JCAP01(2014)030 ] ] , , ,ex- ] ` (A.13) (A.14) (A.15) ˜ ].
D 83 (2006) 307 SPIRE . ]. ) IN ]. ]. ][ 429 arXiv:1004.4105 [ ⌘, q arXiv:1112.4940 arXiv:1110.3193 [ ( ) , 2 SPIRE Phys. Rev. SPIRE CFHTLS weak-lensing ). SPIRE arXiv:0810.0555 ˜
, [ IN IN ) ⌘, q IN 2 ( ][ ][ 1 ⌘ ][ ⌘, q ˜ A.15
q ( a✏ 0 Neutrino masses and Phys. Rept. q (2010) 014 ⇣ a✏ )–( ˜ , Neutrinos in Non-linear
) ) ) 09 q q q (2012) 081101 (2009) 657 ( ( q q ]. ( q 0 0 A.13 0 astro-ph/9506072 f f [ f 500 d log d log D 85 JCAP d log SPIRE , Mixed dark matter in halos of clusters d log d log d log IN arXiv:1012.2868 ) ) [ arXiv:1210.2194 ) astro-ph/9308006 [ ][ q q [ q Neutrino constraints from future nearly all-sky 3 3 ( ( (1995) 7 3 ( 0 0 a a 0 a ✏f ✏f ]. ✏f q q 455 ]. q Euclid Definition Study Report Phys. Rev. d d Planck 2013 results. I. Overview of products and Planck 2013 results. XVI. Cosmological parameters d – 23 – 2 2 , 2 q q q (2011) 030 (1994) 22 SPIRE (2013) 026 Z Z SPIRE Z IN Neutrino mass constraint with the Sloan Digital Sky 03 Astron. Astrophys. [ IN ⇡ ⇡ 01 ⇡ , 3 429 15 4 8 ][ are computed from eqs. ( A Calculation of the full neutrino phase space inCosmological perturbation cold theory + in hot the synchronous and astro-ph/9509145 Massive neutrinos and cosmology ]. Astrophys. J. [ JCAP )=4 )= )= , JCAP (1) ⌫ , ]. ⌘ ⌘ ⌘ , ( ( ( The WiggleZ Dark Energy Survey: Cosmological neutrino mass (1) ⌫ (1) (1) ⌫ ⌫ SPIRE ✓ ⇢ IN ) and ) SPIRE Astrophys. J. ][ IN , (0) (0) ⌫ [ (1996) 102 ⌫ (1) ⌫ arXiv:1303.5062 P ✓ P , arXiv:1006.4845 ), + [ ), + 470 ⌘ ( (0) ⌫ ⌘, q (0) ⌫ collaboration, R. Laureijs et al., ( ⇢ ⇢ (1) ` ( collaboration, P.A.R. Ade et al., collaboration, P.A.R. Ade et al., ( ]. ]. ]. ]. ⇢ ˜
SPIRE SPIRE SPIRE SPIRE IN IN IN IN astro-ph/0603494 Since this hierarchy is infinite, it is of course necessary to truncate it at a given order [ constraints on the neutrino[ masses Astrophys. J. Structure Formation - The E↵ect on Halo Properties including theoretical errors spectroscopic galaxy surveys EUCLID [ (2011) 043529 constraint from blue high-redshift galaxies [ cosmological parameters from a Euclid-like survey: Markov Chain Monte Carlo forecasts conformal Newtonian gauges [ Survey power spectrum of luminous red galaxies and perturbation theory scientific results Planck arXiv:1303.5076 dark matter models Planck L. Kofman, A. Klypin, D. Pogosian and J.P. Henry, J. Brandbyge, S. Hannestad, T. Haugbøelle and Y.Y.Y. Wong, I. Tereno, C. Schimd, J.-P. Uzan, M. Kilbinger, F.H. Vincent and L. Fu, C. Carbone, L. Verde, Y. Wang and A. Cimatti, S. Riemer-Sørensen et al., B. Audren, J. Lesgourgues, S. Bird, M.G. Haehnelt and M. Viel, C.-P. Ma and E. Bertschinger, J. Lesgourgues and S. Pastor, S. Saito, M. Takada and A. Taruya, C.-P. Ma and E. Bertschinger, [9] [7] [8] [4] [5] [6] [2] [3] [1] [12] [13] [10] [11] Note that as thepressions numerical of integration of the Boltzmann hierarchy givesReferences access to for practical implementation.coe cients Finally, relevant physical quantities can be built out of the JCAP01(2014)030 , ] , , D ] D 80 ]. ]. SPIRE Phys. Rev. , IN Phys. Rev. , ][ SPIRE ]. , Cambridge ]. ]. arXiv:1209.3662 IN [ arXiv:0901.4550 [ ][ (2010) 089901] SPIRE SPIRE SPIRE (2013) 3375 IN IN IN ][ ][ ][ (2012) 063509 D 82 428 ]. (2009) 017 (2013) 043530 arXiv:1109.4416 Non-linear Power Spectrum including 06 D 85 [ SPIRE IN Neutrino Cosmology arXiv:0809.0693 [ ][ D 87 ]. JCAP , Erratum ibid. [ ]. Resummed propagators in multi-component Power spectra in the eikonal approximation arXiv:0801.0607 [ arXiv:0708.1367 arXiv:1110.1257 Phys. Rev. [ [ (2012) 2551 Neutrinos in Non-linear Structure Formation - a , SPIRE ]. IN SPIRE – 24 – (2008) 035 Phys. Rev. ][ 420 IN , 10 ][ Massive Neutrinos and the Non-linear Matter Power SPIRE Nonlinear power spectrum in the presence of massive Impact of massive neutrinos on nonlinear matter power ]. IN (2008) 617 (2012) 045 (2010) 123516 ][ astro-ph/0104371 Mon. Not. Roy. Astron. Soc. Testing quintessence models with large scale structure growth [ Renormalized cosmological perturbation theory (2008) 191301 , JCAP 02 A Closure Theory for Non-linear Evolution of Cosmological , An e cient implementation of massive neutrinos in non-linear 674 SPIRE ]. ]. ]. D 81 IN Massive Neutrinos in Cosmology: Analytic Solutions and Fluid 100 [ JCAP SPIRE SPIRE SPIRE arXiv:0806.0971 , [ IN IN IN astro-ph/0509418 ][ ][ ][ [ (2001) 083501 Phys. Rev. The evolution of the large-scale structure of the universe: beyond the linear Astrophys. J. arXiv:0907.2922 Higher order corrections to the large scale matter power spectrum in the , [ , Flowing with Time: a New Approach to Nonlinear Cosmological Perturbations D 64 CMBquick: Spectrum and Bispectrum of Cosmic Microwave Background (CMB) Mon. Not. Roy. Astron. Soc. (2008) 036 Phys. Rev. Lett. , ]. ]. , arXiv:1311.2724 ´ 10 Ecole Polytechnique & Institut de Physique CEA Th´eorique, Saclay, Gif-sur-Yvette , (2006) 063519 SPIRE SPIRE arXiv:1109.3400 IN arXiv:1003.0942 arXiv:1209.0461 IN JCAP Power Spectra cosmic fluids with the[ eikonal approximation with adiabatic and non-adiabatic[ modes University Press (2013). Astrophysics Source Code Library ascl:1109.009. 73 thesis, Cedex France (2012). regime spectrum Phys. Rev. Approximation [ neutrinos: perturbation theory approach,(2009) galaxy 083528 bias and parameter forecasts presence of massive neutrinos structure formation simulations [ Massive Neutrinos: the Time-RG[ Flow Approach Spectrum Simple SPH Approach M. Pietroni, A. Taruya and T. Hiramatsu, F. Bernardeau, N. Van de Rijt and F. Vernizzi, F. Bernardeau, N. Van de Rijt and F. Vernizzi, J. Lesgourgues, G. Mangano, G. Miele and S. Pastor, C. Pitrou, M. Crocce and R. Scoccimarro, N. Van de Rijt, Signatures of the primordial universe in large-scale structure surveys,F. Ph.D Bernardeau, K. Benabed and F. Bernardeau, M. Shoji and E. Komatsu, S. Saito, M. Takada and A. Taruya, Y.Y.Y. Wong, S. Saito, M. Takada and A. Taruya, Y. Ali-Haimoud and S. Bird, J. Lesgourgues, S. Matarrese, M. Pietroni and A. Riotto, S. Bird, M. Viel and M.G. Haehnelt, S. Hannestad, T. Haugbølle and C. Schultz, [30] [31] [28] [29] [25] [26] [27] [23] [24] [21] [22] [18] [19] [20] [16] [17] [14] [15] Chapter 5
Towards a relativistic generalization of nonlinear perturbation theory
In this chapter, the previously described multi-fluid approach is used to extend the scope of application of some standard results of nonlinear perturbation theory. In particular, I highlight analogies between the subhorizon limit of our nonlinear equations of motion and the equations describing cold pressureless fluids. I also show that it is possible to take inspiration of the Newtonian approach to identify and exploit invariance properties. Full details are given in the article of section 5.4.
5.1 Generic form of the relativistic equations of motion
In section 4.2.3, the equation reflecting the conservation of the number of particles in each flow, (4.19), has been derived without writing explicitly the metric. It is thus valid in any gauge. Besides, it does not rely on any approximation.
Similarly, the method presented in section 4.2.2 is valid in any metric gαβ. It is then easy to show that the universal equation of motion of the comoving momentum is 1 P ν∂ P = P σP ν∂ g . (5.1) ν i 2 i σν The system (4.19), (5.1) is actually an exact representation of the time evo- { } lution of a single-flow fluid, relativistic or not, provided that it interacts only grav-
141 5.2. Useful properties in the subhorizon limit itationally. In cosmology, it can thus describe the time evolution of any decoupled species as long as shell crossing can be neglected (whence the necessity of defining several neutrino species in our approach). It must be supplemented, on the one hand, with general initial conditions (the ones adapted to flows of massive neutri- nos are given in the paper of section 5.4) and, on the other hand, with the Einstein equations, which read in this context (in the absence of a cosmological constant, see the demonstration in section 5.4)
i PµPν Gµν(η, x ) = 8πG nc, (5.2) ( g)1/2P 0 speciesX and flows − 1 where Gµν is the Einstein tensor .
5.2 Useful properties in the subhorizon limit
Equations (4.19) and (5.1) have the advantage of being very general. Yet, they contain many coupling terms. In order to make them easier to manipulate (in the light of the handy equation (3.16)), we decided to eliminate the coupling terms that are not indispensable for the description of structure formation. More precisely, we used the Hubble radius2 as characteristic scale and focused on subhorizon scales, i.e. on scales smaller than the Hubble radius. Indeed, the nonlinear growth of structure being a late-time event, the terms that become evanescent on subhorizon scales are a priori not determining in this context. Those terms are easy to identify in reciprocal space by putting conditions on the highest power of k /k taken into H account, k being the inverse of the Hubble radius (see more details in our paper). H When calculations are performed in a generic perturbed Friedmann-Lemaˆıtremetric (3.1), the subhorizon limit of the equations of motion eventually takes the form3
1 1 α α Gµν = Rµν Rgµν , with Rµν the Ricci tensor (i.e. Rµν = R , with R the Riemann − 2 µαν µαν tensor defined in section 1.3.4) and R the scalar curvature, also called Ricci scalar (i.e. R = µν Rµν g ). 2The Hubble radius is defined as the inverse of Hubble’s constant. It gives the order of magnitude of the radius of the observable universe (whose boundary is called cosmological horizon). 3Here again, non-linearities involving the fields of interest are taken into account whereas cou- plings between metric perturbations are neglected.
142 5.2. Useful properties in the subhorizon limit
n + ∂ (V n ) = 0, (5.3) Dη c i i c 1 τjτk ηPi + Vj∂jPi = τ0∂iA + τj∂iBj ∂ihjk, (5.4) D − 2 τ0 with τi Pi τi τi τj(Pj τj) η = ∂η ∂i and Vi = − + −2 . (5.5) D − τ0 − τ0 τ0 (τ0) Those equations are very rewarding for several reasons.
Consequences on Pi
First, they enabled us to prove that the rotational part of the momentum field Pi is zero for adiabatic initial conditions. Indeed, as demonstrated in our article, the sole source of the rotational part Ωi of the momentum field is the rotational part itself on subhorizon scales4, which is initially zero because of adiabaticity. It can be seen by deriving Ωi’s evolution equation from that of Pi,(5.4). It means that, on subhorizon scales, Pi can be fully described in terms of its divergence even in the nonlinear regime and independently on the gauge. This is a major result of our study given the repercussion this property has when it concerns the velocity field of cold pressureless fluids. By analogy with (3.14), it is therefore convenient to introduce the velocity divergence in units of , H ∂ P (η, xi; ~τ) θ (η, xi) i i . (5.6) ~τ ≡ ma H Invariance properties
The form of equations (5.3) and (5.4) is favorable to a generalization of the in- variance properties given in (3.37). Indeed, one checks easily that the following transformations do not modify the system5
4Note that the general equation (5.1) shows that, in the linear regime, it is not necessary to drop any term to write Pi as a gradient. 5 i i The transformation law for Vi is equivalent to P˜i(˜η, x˜ ; ~τ) = Pi(η, x ; ~τ) τ0∂ηDi(η) τ0 − − τ 2 τ τ τiτj ∂ηDj (η). 0 − j j
143 5.2. Useful properties in the subhorizon limit
i i x˜ = x + Di(η), (5.7) η˜ = η, (5.8) ˜ i i δ~τ (˜η, x˜ ) = δ~τ (η, x ), (5.9) i i V˜i(˜η, x˜ ) = Vi(η, x ) + ∂ηDi(η), (5.10) where
n (η, xi; ~τ) δ (η, xi) = c 1 (5.11) ~τ (0) nc (~τ) − and with τi and τ0 left unchanged. Note that the metric perturbations A, Bi and 2 hi,j are not affected by such a transformation since the metric ds is an invariant in general relativity. Hence, the transformation of Φ (in the system (3.37)) is not the non-relativistic limit of our transformations of metric perturbations written in the conformal Newtonian gauge6. It is due to the fact that equations (3.11) and (3.12) contain additional terms compared to the non-relativistic limit of the relativistic equations when the latter are taken in the subhorizon limit. The momentum di- vergence, ∂iPi, is also preserved. We used this relativistic version of the extended Galilean invariance to derive consistency relations (see section 5.3) and to exploit the eikonal approximation in a study involving non-cold species (see chapter6). The existence of invariance properties was predictable according to the equivalence principle. As a matter of fact, we also identified transformation laws keeping the general equations (4.19) and (5.1) invariant (see the paper of section 5.4). I do not present them here because, in practice, it is the subhorizon limit of the equations that we handled.
Generalization of the standard compact equation
Universality is the principal interest of the system (5.3)-(5.4) . After decoupling, { } it distinguishes between baryons, cold dark matter and neutrinos only via their initial velocities (which set τi) and masses (which set τ0). Hence, introducing the
6 It is possible to find transformations of A, Bi and hi,j which have this property but they depend on ~τ, which does not make sense.
144 5.2. Useful properties in the subhorizon limit
2N-uplet
Ψ (k) = (δ (k), θ (k), . . . , δ (k), θ (k))T (5.12) b ~τ1 − ~τ1 ~τN − ~τN is sufficient to encode the physics of all free-streaming cosmic fluids (labeled by
~τ1, ..., ~τN ) in a unique equation (here again, the convention that repeated Fourier arguments are integrated over applies)
c cd a(η)∂aΨb(k, η) + Ωb (k, η)Ψc(k, η) = γb (k1, k2, η)Ψc(k1, η)Ψd(k2, η), (5.13) which is formally identical to (3.16) except that i) b, c and d run now from 1 to 2N, c cd ii) Ωb is now scale-dependent and iii) γb is now time-dependent. More precisely, the individual equations of motion read in Fourier space
µkτ ma µ2τ 2 a∂ i δ (k) 1 θ (k) = a − τ ~τ − τ − τ 2 ~τ H 0 0 0 ma d3k d3k α (k , k ; ~τ)δ (k )θ (k ), (5.14) τ 1 2 R 1 2 ~τ 1 ~τ 2 0 Z ∂ µkτ k2 1 + a aH + a∂ i θ (k) + (k) = a − τ ~τ ma 2 S~τ H H 0 H ma d3k d3k β (k , k ; ~τ)θ (k )θ (k ), (5.15) τ 1 2 R 1 2 ~τ 1 ~τ 2 0 Z where the source term (k) is given by S~τ
1 τiτj ~τ (k) = τ0A(k) + ~τ B~ (k) hij(k) (5.16) S · − 2 τ0 and where the kernel functions (which differ from one flow to another) are defined as
(k + k ) k ~τ α (k , k ; ~τ) = δ (k k k ) 1 2 k ~τ 2 · , (5.17) R 1 2 D − 1 − 2 k2 · 2 − τ 2 2 0
(k + k )2 k ~τk ~τ β (k , k ; ~τ) = δ (k k k ) 1 2 k k 1 · 2 · . (5.18) R 1 2 D − 1 − 2 2k2k2 1 · 2 − τ 2 1 2 0 Hence the non-zero matrix elements (for p an integer [1,N]), ∈
145 5.2. Useful properties in the subhorizon limit
2p 1 2p ma(η) γ − (k , k , η) = α (k , k ; ~τ ), (5.19) 2p 1 1 2 [p] R 1 2 p − −2τ0 (η) 2p 2p 1 ma(η) γ − (k , k , η) = α (k , k ; ~τ ), (5.20) 2p 1 1 2 [p] R 2 1 p − −2τ0 (η) ma(η) γ 2p 2p(k , k , η) = β (k , k ; ~τ ), (5.21) 2p 1 2 [p] R 1 2 p −τ0 (η)
[p] where an exponent means that one refers to the fluid labeled by ~τp, and
[p] 2p 1 [p] k τ Ω2p 1− = iµ , (5.22) − − τ [p] H 0 ma (µ[p])2(τ [p])2 Ω 2p = 1 , (5.23) 2p 1 [p] [p] 2 − τ0 − (τ0 ) ! ∂ k τ [p] Ω 2p = 1 + a a iµ[p] + ..., (5.24) 2p H [p] − τ0 2(p+n) 1 H H Ω2p − = ..., (5.25) Ω 2(p+n)(n = 0) = ..., (5.26) 2p 6 where the “...” denote contributions coming from the source term (5.16), which can be estimated by taking the subhorizon limit of the Einstein equations (5.2). Note that the standard matrix elements of section 3.3, which govern the cold-dark- matter evolution, are contained in these ones. They correspond to the two fields
Ψb labeled by ~τ = ~0. The difference here is that all gravitational contributions (including relativistic effects) are taken into account so the source term (which relates Ψ2p to all other fields of the collection) is not trivially connected to Ψ2p 1 − via the Poisson equation. Such a description paves the way for the implementation of resummation tech- niques, such as the ones enumerated in section 3.4, in the relativistic and nonlinear equation of motion (5.13). It therefore appears as a further step towards a relativis- tic generalization of nonlinear perturbation theory. Yet, I did not develop tools to put such an implementation into practice during my PhD thesis.
146 5.3. Consistency relations
5.3 Consistency relations
The fact that the relativistic system also satisfies an extended Galilean invariance makes easy the generalization of Ward identities. Taking inspiration of the results presented in section 3.5.2, we assumed in our paper that the density contrasts evolving in the medium perturbed by D~ are related to the unperturbed ones by (as long as the perturbation is small enough to be treated linearly)
δ˜ (k, η) = exp(ik.D~ )δ (k, η) (1 + ik.D~ )δ (k, η). (5.27) ~τ ~τ ≈ ~τ Moreover, in the light of (3.40), we related those displacements to the perturbative fields that generated them via
iq D~ (η, x) = d3q− eiq.xδ (η, q). (5.28) q2 adiab Z The index “adiab” means that the modes q that generated such displacements are assumed to be adiabatic, whence a displacement identical in each flow. Following the procedure depicted in section 3.5.2, we recovered eventually the standard consistency relations, i.e.
δ(q, η)δ(k1, η1)δ(k2, η2)...δ(kn, ηn) q 0 (5.29) h i → q.k = i P lin(q; η, η ) δ(k , η )δ(k , η )...δ(k , η ) , − q2 i h 1 1 2 2 n n i Xi where the wavenumber q is much smaller than all other wavenumbers ki and where lin P (q; η, ηi) is the linear unequal-time power spectrum. So, in our approach, it is still possible to relate (n + 1)-order correlation functions to n-order ones when the times at which the fields are computed are different. As already discussed, this property is in fact hardly operable. Nevertheless, it strengthens the consistency of our generalization work.
5.4 Article “Cosmological Perturbation Theory for streams of relativistic particles”
147 JCAP03(2015)030 hysics P le ic t cles and their properties ar uge — to the synchronous f adiabatic initial conditions ns written in a generic per- 10.1088/1475-7516/2015/03/030 yzed. We show in particular urbation Theory. Those equa- regimes, express the matter and valence principle is exploited to doi: f equations, extending that known nded Galilean invariance. b,a strop A [email protected] ) is always potential in the linear regime and remains so at , x ( i P osmology and and osmology and Francis Bernardeau C a,b 1411.0428v2 cosmic flows, cosmological perturbation theory
Motion equations describing streams of relativistic parti [email protected] rnal of rnal
ou An IOP and SISSA journal An IOP and 2015 IOP Publishing Ltd and Sissa Medialab srl Institut de Physique CEA Th´eorique, (CNRS:Orme URA des 2306) merisiers 91191 Gif-sur-Yvette,Institut France d’Astrophysique de Paris,98 UPMC bis (CNRS: boulevard UMR Arago 7095) 75014 Paris,E-mail: France b a c Received December 10, 2014 Accepted February 12, 2015 Published March 17, 2015 Abstract. ´leeDupuy H´el`ene Cosmological Perturbation Theory for streams of relativistic particles J
are explored in detailtions, in derived in the any framework metricmomentum of both conservation. Cosmological in In the Pert this linear— and context that nonlinear we extend was thegauge. initially setup performed o The in subhorizonturbed the metric limit Friedmann-Lemaˆıtre conformal is of Newtonianthat then the ga the derived nonlinear momentum and motion field anal equatio subhorizon scales in thehighlight invariance nonlinear properties regime. satisfied byfor Finally such streams a the system of equi o non-relativistic particles, namely the exte Keywords: ArXiv ePrint: JCAP03(2015)030 4 4 5 6 7 9 9 1 3 4 8 10 11 12 13 15 16 17 10 ive parti- ], the signature ion has evolved: 9 – 4 d Friedmann- rticles ontext and several N-body eed needs to ensure that the ay to improve our knowledge eferences [ icles is needed (see recent at- nation of the structure growth ical models to study their impact ale structure formation has often of massive neutrinos into account se of the difficulties encountered ot undermined by the presence of nt enough for those masses to be – 1 – (1) i P ]). Since neutrinos have been shown to be massive, the situat 3 – 1 eaır metric Lemaˆıtre Numerical approaches are obviously a precious tool in this c 5.4 Nonlinear5.5 equations in Fourier space The Ward identities 4.2 Linearized equations in Fourier space 5.1 Coupling5.2 structure at subhorizon The scales 5.3 no-curl theorem and The its extended consequences Galilean invariance 3.4 Link3.5 with the Einstein Invariance equations properties 4.1 A useful property of 3.1 Evolution3.2 equation of the Evolution comoving3.3 equation number of densities the Explicit momentum fields form for the momentum field in a generic perturbe cles it has become crucial toon include the neutrino masses late-time in growth cosmolog estimation of of structure. the On fundamental cosmological themassive parameters neutrinos. one is hand, On n one theof ind other hand, those doing enigmatic so particles is an because, efficient as w mentioned in many r 1 Introduction Entering in the era of precision cosmology requires an exami 6 Conclusions and perspectives A Adiabatic initial conditions for massive neutrinos 5 Perturbation Theory with relativistic streams in minute detail. Sobeen far, the overlooked, impact especially of inwhen neutrinos the on accounting large-sc nonlinear for regime, thetempts becau in gravitational [ dynamics of such part 4 Relativistic streams in the linear regime 3 Derivation of the nonlinear motion equations of relativistic mass Contents 1 Introduction 2 A multi-fluid description of non-interacting relativistic pa of neutrino masses onconstrained cosmological observationally. observables is significa simulations have been designed to take the nonlinear effects JCAP03(2015)030 ion is ]), this 18 Note that 1 explains how to cting relativistic rinos. In our first too restrictive to 5 as been put forward re- e.g. to reduce noise or to t the particles at play are otion equations in a more we recall the specificities of y cumbersome. mment the key properties of 2 ann hierarchy. It is possible ogy, as reviewed for instance ace distribution function. As t subhorizon scales have been he resulting equations. Ward al projects aiming at putting oposed led to the same results tions. It allows also to reach CDM) particles, neutrinos are and future data more relevant, r motion equations satisfied by nally, section of neutrinos is therefore difficult e detail the formalism associated s requires a good understanding e the standard study of a single mation is an effective approxima- s addressed in the linear regime sion into the nonlinear regime of ntary. For instance, Perturbation inear growth of structure, at least uids. Somehow it takes inspiration articular a global motion equation, . he following section is to bring out ion of flows evolving independently sented. What motivates this task is l simulations modeling the nonlinear power ion is not negligible and the Newtonian ] and its companion paper ref. [ ]. 17 20 – 2 – ]). From a theoretical point of view, this kind of investigat 16 – 10 2 ]. In this study neutrinos, or more generally any non-intera ]), the aim being to establish a similar description for neut 21 22 ], but this analysis is limited to the linear regime, which is 19 A new theoretical approach, which we further explore here, h The organization of the paper is the following. In section The only attempt we are aware of is described in [ The computational time is very problematic in some numerica 1 2 (see the recent works [ approach leads toto a examine hierarchy in of fullin equations detail called ref. the the [ impact Boltzm of such results on cosmol numerical simulations and analyticalTheory studies are can compleme bedescribe useful velocity to dispersion) testbetter at the precisions, play approximation which in makes schemesand some the (used to comparison N-body do with it simula present faster. a multi-fluid description.single-stream Then fluids we of derive relativistic thesome particles. remarkable fully The properties nonlinea aim related of to t the linear regime. Fi not straightforward. Indeed, contraryrelativistic at to horizon Cold crossing Dark soapproximation their Matter does velocity ( dispers not hold.to The describe. phase-space distribution Usually,only in by performing analytical a models,presented harmonic this in decomposition question some of i the standard phase-sp references (e.g. [ paper we demonstrated that,as at the linear standard level, one. the In methodwith the we the present pr article, approach we explore wegeneral in developed. framework, mor focusing In onidentities, particular resulting the we from symmetry derive those propertiesthe invariances, the fact of are m that t also carrying pre of out the Perturbation mode Theory coupling structure. calculation deal with relativistic streams infully Perturbation nonlinear Theory. but In p indropped, which the is terms presented that inthis are this system subdominant before section. a concluding We and then discussing present perspectives and co be in balanceconstraints with on the neutrino masses current arein surveys. all the sensitive mildly Indeed, to nonlinear thethe observation regime. nonl phase-space Problematically, harmonic the decomposition exten has proved to be ver or non-relativistic particles,from were one described another. asfree-streaming. a Such collect a This study ismulti-stream takes this fluid by advantage property that of of that theof a allows the collection fact CDM of to description tha single-stream replac (for fl tion, which the see single-stream e.g. approxi [ cently in ref. [ spectrum. JCAP03(2015)030 η (2.1) (2.2) (2.3) (2.4) and i x ). One also . i τ )) i ; i τ ace distribution ; . ,p x i . Each single-flow )) i i , ), we have τ η, τ i ( ; . η,x i c particles )) p i ( x ( P )) τ i n-interacting relativistic ; F η, τ − flow ( x ; i i − x veral flows in order to enter p P η, ng number densities and of ( denoted ( n equations in the conformal ( . The time evolution of each η, i one i D ( 4 F τ P f i δ ned as the collection of all the . The phase-space distribution is a constant. In the absence of shell-crossing, ) i ) P i i el through the same gravitational i n gle multi-flow fluid is replaced by city at the same time are particles − x space momenta usually performed τ p τ i ; − ; ds. A convenient way to distinguish e the phase-space distribution func- the conjugate momenta of ional form p i x alid for any given mass eigenstate. If the x to do so throughout the cosmological ( i p (we will use hereafter the two terms ) is computed by taking all the single- i p D ( η, η, δ 3 ( ( D ) c ,p c δ i i or each three eigenstates. n ) ) as labels, τ n i ; i in τ i τ x ; τ η η,x 3 X ( ( x ]. In the next section, we succinctly generalize i d η, (i.e. a sum over all the possible initial momenta f and position p ( η, i 21 c – 3 – ( Z τ η ), which is the value of the momentum of any )= n c i τ τ n ; ; i i i )= τ )= i 3 i p ,p τ d i ( η,x ; ( i i at time F Z P i η,x ,p ) describes a 3D continuous field, the continuous limit of this i i ( τ i τ ,p )= i i flow η,x ) then reads ( i − ,p τ i η,x are the comoving positions, ; ( i one flow i f f − x η,x ,p i i ( i p τ f one 3 X f d η,x ( Z )= i flow ), where ) shows that, to determine the time evolution of the phase-sp ,p − i ,η i one 2.3 is the comoving number density of the flow. ,p f η,x i c ( x n f ( The key idea is to split a relativistic multi-flow fluid into se Finally, the overall distribution function Eq. ( f In a homogeneous expanding universe, one can easily show that Note that, when applied to neutrinos, this construction is v 4 3 function, one needsthe to momentum study fields. the ANewtonian time derivation gauge of evolution was presented the of in corresponding detail the motio in comovi [ masses are not degenerate, it should therefore be repeated f 2 A multi-fluid descriptionWe of recall non-interacting here relativisti themassive particles method as we a developed collection to of describe streams a or fluid flows of no fluid considered in ourparticles multi-fluid that approach have at is initial therefore time defi the comoving momentum this derivation to an arbitrary spacetime. indistinctively). This approach requirestion to properly defin the conformal time. function in the field ofparticles application having of initially the the same single-flowtime. velocity approximation. will Indeed, continue in that case,that particles are that located have at the the samepotentials. same velo Those place sets so of such particlesbetween particles will are the trav thus flows single-flow flui is to use initial momenta flow itself is encoded in a phase-space distribution functio particle of the flow labelled by introduces the momentum field where flow fluids previously defined into account: expression is naturally Assuming that the parameter In this context, one canto see that compute the global integration physical overa quantities phase- sum associated over with the single-flow a fluids sin labelled by or velocities). It implies in particular that, for any funct JCAP03(2015)030 (3.1) (3.4) (3.3) (3.2) (3.5) (3.7) (3.6) from the d notation; to r at rest in the , ic massive par- f through the metric and i elated to its phase-space p er are conserved (they do . = 0 . servation laws such as d nor subjected to diffusion, as ption that the motion equations are . uid we have ) s at play at the time of interest). i , f rticles can besides be expressed as = 0 i η ,p p i , d = 0 d f i 0 i p p η x η,x c i d ( are related to d = n , ∂ i f , 0 i 0 ∂p . Note that this motion equation does not p i η τ p 2 3 p P P d d = 0 3 + d = 0 m i d µ τ x µ ; − i ; Z and – 4 – d µ d f µν Z ∂ i i J = ∂x T η p i x = d µ d ∂ i + )= ∂x P i η x c µ d d i n + P ∂ η,x c ∂ ( ∂x ∂η n c n + can be expressed in terms of momenta. More specifically, ∂ ∂η f ) and η i d ∂ / ∂η i ). Its evolution equation can be derived straightforwardly η,x x ( satisfied by the phase-space distribution function 3.4 ν 5 P µν g is not necessarily the number density measured by an observe ) = i c is the energy-momentum tensor and where we adopt the standar n η,x is the particle proper time and µν ( µ T τ P ticles The Vlasov equation derives from this equation under the assum 5 The only assumptionnot made decay here because is noIntegrating disintegration that over or momenta the scattering leads particles process to i we consid Note that metric (see subsection 3.1 Evolution equationThe of comoving the number comovingdistribution number density function densities of in a a very fluid, simple single-flow way, or not, is r 3 Derivation of the nonlinear motion equations of relativist where conservation equation Hamiltonian. For a single-flow fluid, d the on-shell mass constraint. As a result, for a single-flow fl indicate a covariant derivative. The conservation of the pa with rely on any perturbative expansion of the3.2 metric. Evolution equationA of fluid the in momentum which fields particles are neither created nor annihilate is the case with the fluids considered here, obeys general con where JCAP03(2015)030 , is a (3.9) (3.8) (3.10) (3.11) (3.14) (3.13) (3.12) dmann- elativistic , overning the time fluid, the energy- j x by d , i µ x P jk )d h the conservation of the h ij i ıtre metric. The metric we h ∂ the on-shell mass constraint, k ed of light in vacuum is equal n the time dependence of + se. The equation of motion for 0 P j ij P c. Besides, the time coordinate of in the previous equation, δ P derivative finally gives the following i . . orm of the motion equation satisfied 1 2 +( . i + η σν,i σν,µ , ∂ d j g g ν ∂x i . ν ν B i x J 0 i P P µ d ∂ P P i σ σ = 0 j P B P P ν P + ; − . Once again, this equation has been obtained 2 2 1 1 µ + – 5 – ∂ + 2 = P αβ ∂η 3) are the Cartesian spatial comoving coordinates, = = ν 2 g A , i η µν 2 P are respectively the time-time, time-space and space- ≡ and to the momentum field ∂ i,ν , µ,ν T 0 )d ij P ν η P d ν h P A d ν J = 1 as P − to spatial coordinates P i η ( µ and ) d i , whence i η / 0 x (1+2 ( B 2 J − = 0. Moreover, the energy-momentum tensor being symmetric, , a i 0 ν ) A P P = J η ( ν i ; is related to = 2 µ η ), it dictates the time evolution of a collection of massive r P a i d P d µν J = 3.5 T 2 s d gives ν J is the particle four-current. Noting that, for a single-flow µ ) can therefore be considered as the second motion equation g P µ is the conformal time, J − η eaır metric Lemaˆıtre 3.11 = ) is the scale factor and η µν ( Together with eq. ( Eq. ( where where we define the operator d particles evolving in an arbitrary metric evolution of the flow in the nonlinear regime. 3.3 Explicit form for the momentumAs an field illustration, we in presentby a in the this generic section momentum the perturbed field explicituse f in Frie reads a generic perturbed Friedmann-Lemaˆ Expressing the covariant derivative in termsmotion of the equation spatial where momentum tensor without performing any perturbative expansionthe of momentum the metri field being relatedone to can the restrict spatial the ones coordinates thanks to a driven by the overall matterthe and momentum energy is content then of the univer space metric perturbations.to Units unity are and chosen the so expansion that history the of spe the universe, encoded i the conservation offour-current the imposes energy momentum tensor combined wit T JCAP03(2015)030 = = 0 ions 0 i (3.21) (3.18) (3.17) (3.20) (3.16) (3.15) (3.19) U B ) simply 3.13 = 0 and A was formulated in ). For example, to ij ve in the synchronous = 0, one gets φδ , . If one calls the four- ) 2 . i so that eq. ( n i ) i − U ,p tion of one of the two equa- η,x = n to an arbitrary perturbed i ted has to set ( es the conformal time. m/a se-space distribution function ij . h η,x flow ta can be done straightforwardly ≪ c . ( − i . ombination because, when all the c n ach flow to the Einstein equations. f c used to express the number density n P 0 n ν one 0 µν 0 p 0 P T P p µ 2 P 2 / p ν 2 / A. 1 µ 2 i / 1 ) = 0 and . P ) and / 1 ) P g µ µ 1 ) i g A J g − P − ], which corresponds to the conformal New- 0 B − ) µ ( am∂ − − P ( 2 g , 21 ( X U / − ψ 1 − − (1 – 6 – ) ( = = = i 1, whence = = 00 p i n − g species and flows 3 η m/a P A flow d d − d = flow ( − = πG − µ Z − 0 one U µν one P µ µ = T J )= U ) = 8 , one indeed has i n i µ can also be computed very easily, U η,x η,x µ ( but the two approaches are of course equivalent. Those equat ( J µν c µν ]) T n G 23 , 18 from the constraint ). Furthermore, in the non-relativistic limit, we simply ha rather than ], the evolution equation of the number density of neutrinos 2 n / 21 1 3.13 − ) This equation is, on the one hand, a relativistic generaliza 00 g − allow to study explicitelycosmic the components Einstein are equations free-streaming, one after can rec write terms of velocity of such an observer Besides, the expression of theof particle four-current particles can as be seen by an observer at rest in the metric, deno This result is important sinceThe it particle gives four-current the contribution of e In ref. [ tonian gauge (in which, by definition, and conformal Newtonian gauges takes the form tions describing CDM metricFriedmann-Lemaˆıtre and, of on eq. the (2.13) other of hand, [ a generalizatio When considering single-flow fluids, integrationand over the momen resulting expression is write the motion equation in the synchronous gauge, one only It is of course nothing but Newton’s3.4 second law written using Link withIn the general, Einstein the equations energy-momentumthrough tensor (see is e.g. related [ to the pha Admitting that the four-velocity of such an observer satisfi in eq. ( ( JCAP03(2015)030 being (3.22) (3.29) (3.27) (3.26) (3.24) (3.23) (3.28) (3.25) in the G δ, η, = = , ) ˜ ˜ δ η η , ( om a change of variables i i j ing with the study of its δ v = = e cosmic fluid and principle. As pointed out in i j 0 i e sense that the displacement an be applied to relativistic as uences of coordinate transforms ticles (or, equivalently, the mo- hanges of coordinates and fields ed extended Galilean invariance of large-scale structure formation. ) is an arbitrarily time-dependent ssing. Together with the evolution c limit) satisfy. The corresponding ve the motion equations unchanged. η ( i d . . , ) fferent possible time dependencies of the spatial i ν µ . defined so that x x x ) d ( µ µ σ,ν η µ ξ ( ˜ x µ ν is the density contrast. Such an invariance i ˜ x ξ d = δ → 2 – 7 – 2 = η → µ d µ ν,σ d µ x ξ x µ , ξ − d˜ ) x , ξ i ) η ( x η i ) ( , v i ) η ( u η i ∂ ( ∂η , i d i ) d η η x )+ d η d ( η d i d ( d i ], the extended Galilean invariance plays a significant role v + + −H 22 i i =1+ = x A V i 0 0 , these equations form a closed set of equations. 0 i ξ ξ = = = P is the velocity field and i i ˜ A ˜ x ˜ i V and V ), c 6 n ] 24 So let us consider a transformation A priori, a whole set of transformation laws can be derived fr In the following we analyze this system in more detail, start In this study several particular cases, corresponding to di 6 In this study,that we are the interested standardtion in motion equations generalizing equations of the of thistransformation so-call paper non-relativistic laws written par are in the explicitly(see non-relativisti e.g. given [ by the following c computation of correlation functions involvedTo in generalize the it, study weclosely will connected explore to in special the Lorentz following transformations. the conseq translation, are explored. of the form invariance properties. 3.5 Invariance properties In this subsection, we present transformation laws that lea In particular, it is easy to check that It is an acceptable transformation provided that the sum being performed over all species participating in th Newton’s constant. Indeed, thewell formalism as we to are non-relativistic proposingequation species c of so no contribution is mi where we use the notations of this paper and where corresponds to ancan extension depend of on the time.recent Galilean studies, invariance It such in actually as th derives [ from the equivalence uniform field, JCAP03(2015)030 ). (4.1) 3.27 (3.30) (3.36) (3.31) (3.37) (3.32) (3.33) (3.34) (3.35) , 7 . roperty ( k . x i k are kept at linear ∂ v i ∂x ij u ) are invariant under δ + ... H 2 ∂ 3.13 and ∂η ) is given by i − )+ i are infinitesimal quantities i , v v )–( i τ l explore in more detail its ij i − ; x h u the consequences of such an ation of the extended Galilean that appears in the expression i 3.29 i mmetry property satisfied by i 3.5 formations. op a perturbation scheme that 0 v = metric perturbations = s given by P o the previous ones thanks to )–( i ) η,x + i and ij ˜ ( /P x ∂ u c ). η ˜ i h ∂ e quantities. Besides, the differential n in (2) v i i 3.28 ic streams in the linear regime. From + 2 η = ( P / P i i i 0 i 1 ˜ , η v P v i ( − P P i and ) u i )+ ≡ + g i P − v i i , τ − 0 τ 0 − ) ; + i ∂ i , η P P ). It is also the case for the Einstein equations ∂x i 0 ( ) ) B 1+ i 0 0 ) P P i g – 8 – i, i, η,x = ) c u v v ( i 3.37 + i i i n u 0 ˜ x x + )+ (1) B i )–( P i η + = i P − ( v ), it is straightforward to see that, in a homogeneous i v i c ( v d ˜ , + n − i 3.29 − i ) . x + τ 3.11 =( = (1+ = (1 = is a constant and 0 η do not depend on time. Considering that metric inhomo- i )–( ∂ ( i, i i 0 0 ∂η x i c ˜ v ˜ ˜ P (0) ˜ g P P i )= P n i i η = P d − 3.28 τ d x i i ; i ˜ i ) and ( x ˙ x v = )–( i and v since i − 3.5 i η,x i u 1 ( P τ i 3.26 ) are two arbitrarily time-dependent functions, obeys the p −H = P η = ( A i (0) i ) and ˜ η u P ∂ η = ∂ ( i ˜ A d η d d ) and η ( = i v i v In the rest of the paper, we will assume that both The invariance we are putting forward is clearly a generaliz It can then be easily checked that the motion equations ( By definition, 7 geneities are small compared to background values, we devel consists in expanding each relevant field with respect to the where with the transformations ( but the explicit verification is more involved. More explicitly, the change of frame corresponding to ( metric, the variables and finally the potentials transform the following way, The momentum components therefore transform as (once again and we will restrict alloperators calculations to associated linear with order the in thes new coordinates are related t invariance. We leaveinvariance. for In further thethe studies last motion the section, equations explorationconsequences. written we of will at subhorizon present a scales similar and sy we4 wil Relativistic streams inThis section the aims linear at highlighting regime the properties motion of relativist equations ( Interestingly, we can note that the combination ( of the energy-momentum tensor is invariant under such trans order) the transformation of the comoving numerical density field i JCAP03(2015)030 (4.2) (4.8) (4.4) (4.7) (4.3) (4.6) (4.5) 9 ], this number ], 21 21 onditions, proper to be sourced by . , ) jk (1) . i h 2 ) ,η 0 k P τ x τ and all the fields we ,η ( j ... . x τ k (0) c ( , (1) i 0 n i τ τ P )+ (1) c ! 2 i i y mentioned in [ , n τ ∂ A e notations as in [ ; − ges). Note that, to get this i , we expect (0) 0 i ij ij ) reads )= )= x P 0 er mode h e what matters in the nonlinear h η,x j τ j ( ,η ,η τ + ling between the fields of interest. τ . ave been taken into account. The x x 1 2 3.11 (2) c ( ( jk + n n P (1) µν,i − h i g . In the particular case of a perturbed B ) is potential at linear order. Besides, ν (0) 8 i (1) k )+ i is the gradient in terms of the potentials i − P (0) P (0) τ 0 ; η,x P A P i (1) (0) i (0)0 j ( µ i 0 i 0 P P τ τ τ P τ only. For instance, even at linear order, the (1) (0) i η,x 2 1 ( − P number density whereas it applied to the + P – 9 – (1) i i − 0 2 1 (1) c τ ], we explicitly write in this section the linearized j P j n , δ . τ Aτ 3 = B 0 21 i ) τ x + (1) (0) j )+ j ,η i (1) i η i , θ P comoving k P τ τ d P i ( ) (
d η + P B (1) i ( i (0) c θ ∂ i n P = 0 (1) k 2 i (0) τ i = 0 k P − (1) P 0 )= − i (1) P η i τ ≡ ; d P = i ) )= d η is sourced by a gradient term. For scalar adiabatic initial c ( ,η (1) does not derive from a potential η,x 0 k i ( τ ( (1) i c i 0 P n P (1) P P i P is sourced by a gradient in terms of the space coordinates , which applies here to the ]. n δ (1) i 21 P will therefore remain a potential field. Note that, as alread Since Except 8 9 (1) i which allows to express the source term whose one has expression, only linear termssame in approximation the will metric be perturbations usedregime h in is the not rest metric-metric of coupling this but paper the sinc nonlinear coup 4.2 Linearized equationsIn in light Fourier of space what is done in the study [ (for example in the conformal Newtonian and synchronous gau and It appears that study in Fourier space correspond to this mode. We use the sam Note that here we have used the fact that a gradient in terms of the space coordinates property is a specificity of the metric,Friedmann-Lemaˆıtre this variable equation takes the form equations in Fourier space. To this end, we introduce a Fouri momentum field density in [ 4.1 A useful property of P the on-shell mass constraint imposing When taking only linear perturbations into account, eq. ( JCAP03(2015)030 k is ρ δ (4.9) (4.10) (4.11) (4.13) (4.12) , where 2 2 τ 0 /k 2 τ µ H k ρ − δ 1 , )) 0 P . τ θ to study CDM thanks to and the initial momentum and the initial momentum lculations that we want to + tivistic species. The regime k 1 i to numerically solve the sys- k tice that metric perturbations omparing the wave number r ,µ,h,γ tant to keep in mind that the equations, in the source term B 0 − γ i ) scales like nsidered flow. o be small but can reach values ]) i 2 6 τ ik for the particular case of massive µ τ,τ e-scale structure formation, i.e. we 18 , ( 3 − died) with the horizon wave number 2 allow us to remove the subdominant i A ij κ i τ δ 0 γ P )) + τ 3 1 = + − 2 − h j τ j 2 B 2 k µ j i k ,µ,h,γ τ 1 2 k 0 + with 2 2 τ,τ – 10 – 0 A + ( τ τ i operator do not all have the same amplitude after 0 κ τ h τ i i ( j kτ k + ∂ 2 k 2 0 )= i k γ k k = 4 /P − µ i = − P ), moving to Fourier space and taking the divergence of P h θ ij ,µ,h,γ 0 h τ 0 3.5 − τ A τ,τ iµk ( + κ = n δ ( P ˙ θ 0 and that the relative velocity field ( τ τ 2 ), is explained in full detail in appendix are scalar modes defined so that (see e.g. [ /k iµk 2 γ H = k 4.10 ρ n ˙ δ δ is the Cosine of the angle between the wave vector and )–( ), one finally obtains µ h 4.9 The setting of the adiabatic initial conditions, essential 4.4 (defined as the inverse of the Hubble radius). First, let us no H direction, where Thus, after linearizing eq. ( subhorizon scales have been reached. 5.1 Coupling structureThe at identification subhorizon of scales the relevant coupling terms is made by c neutrinos. 5 Perturbation Theory withThe relativistic aim streams of thisPerturbation Theory section (PT) is can towe be will show investigate extended how is to the thewill the one formalism focus relevant study in here developed of on the subhorizon context rela coupling scales. of terms larg This from restriction the will nonlinear motion couplings equations. that It appearof is the for indeed Euler example impor equation in or the in Einstein the eq. ( where We can see in particular that the angle between the wave vecto and where and vector plays a significant role in the timetem evolution ( of the co (characterizing the scale at which thek field evolution is stu scale like the typical energy densitycomparable contrast. with The unity. latterexplore. is This assumed is t precisely this regime of PT ca JCAP03(2015)030 is (5.4) (5.1) (5.3) (5.6) (5.5) (5.7) (5.2) (5.8) (5.9) l,ij (5.10) (5.11) P l V tivistic case). , ) all the terms behaving , in practice we neglect . jk y vanishes since h via , i , i ∂ r. The objective here is to 3.13 = 0 l,i , k ∂ P ty. e source term of the Einstein ,j 0 τ ∂x l ) τ j . t, similarly to the velocity fields τ i sults on which rely the carrying 0 τ re computed at linear order from = 0 ) τ l,i τ j i 1 2 k,j P τ , l − Ω 2 P ) take the form − ) V − k j,k ( 0 1 and in ( k,i j τ k ∂ j τ ∂η V 3 W ( B P Ω , ) ≥ 3.13 i kij ( i 0 1 − ǫ = ∂ j ijk α τ kij j τ . W k ǫ η ( )–( ǫ τ + i 0 Ω τ D τ + = + ,j remains potential to all orders in Perturbation + + 3.5 = 0 i,i ) ,i i with , + V j,i A l,j m j,k P 2 i i i,i ), one obtains i α P – 11 – P τ P Ω ∂ + ) τ 0 l = Φ W l,i 0 5.2 = τ + V i ijk − /k k,i τ P ( ǫ 2 i P H 0 = Ω i,j a 1 i P k τ = i 2 by exploiting the relativistic Euler equation. Noticing ) = 0 P ( mil V c i − P ǫ − i m j n H Ω + i to eq. ( ∂ k = kij = q V j k into a potential and a non potential parts, ǫ i k ( V i − Ω i ∂ V l,i ∂ η + P P + = k kij D i + ǫ l,j 0 Ω P c τ V η η 2. In this limit, eqs. ( n η D D ≥ D α ] and references therein for demonstrations in the non-rela ) and 25 , with i,j 22 α ) /k H k ) all the terms behaving as ( Following the description of CDM fluids at subhorizon scales First, let us decompose The last term of the left hand side of this equation eventuall being the Levi-Civita symbol, or fully antisymmetric tenso H 3.5 k ijk The metric perturbations that appearthe in the Einstein Euler equation equation. a equation Note is that dominated in by the the fluctuations sub of horizon5.2 the limit, number th densi The no-curlIn theorem this and paragraph we its explicitlyout consequences demonstrate of Perturbation one Theory of calculations. theof What key non-relativistic we re flows, show the is momentum tha field Theory (see [ and One can then define the curl field, related to the momentum field with ǫ in ( as that derive the evolution equation of Ω with and applying the operator which is also symmetric in those indices. So finally we have symmetric in ( JCAP03(2015)030 i τ ) was (5.12) (5.21) (5.16) (5.13) (5.15) (5.14) (5.20) (5.17) (5.19) (5.18) 3.13 )–( 3.5 is given by . i ) τ η S ( j d present in the left hand of non-relativistic fluids, η omparable to the one of a ∂ µ j . ∂ i τ bing the scalar modes of the nal system ( µ i V the case for adiabatic initial y itself. Consequently, in the τ 0 P τ . j , , . present in the left hand side of . An immediate consequence is olution equations of the density ) τ . he time variable. Assuming that or − j i η jk 1 c flows. ( τ ∂ h roduce for each fluid labelled by 0 i = 0 = 0 i τ . By analogy, here we would like to − k d j − V i ,i 0 τ η ,ii ) 0 V τ j 2 P i 0 i ∂ )) τ j τ τ + τ τ i ) , τ τ i ; ) 1 2 , η i δ τ ) η − 2 )+ ( 0 i D ( −S i − ) i can be re-expressed as a transformation rule ,i /τ can be easily inverted, η d (0) c j η,x i ) j ( ( η,x n i (1 + i B τ η,x c V ( i + 1 j i,j V ( j d i n i i V τ τ τ η P – 12 – η, V δ x ∂ j ) and where the source term + 0 − V +( τ and = = )= 1 i i A i 5.4 i )= )= τ ˜ ), η 0 − i i +( ˜ δ x P τ − ˜ ˜ ) x x η i η,x 1 i,i ( = D η, η, 5.16 via ( i P ( ( i τ η i i i i τ η,x δ τ ˜ τ ( V P S D ˜ δ i P = , i i using ( τ ) between P i δ )= i P 5.4 ˜ x η, ( i ˜ P are unchanged, this can be obtained the following way is related to the field i 0 τ V -divergence fields form, together with the equations descri i It is then possible to write those equations on a form easily c We are here confronted to a situation very similar to the case P and i Note that the relation ( which means in particularabsence that of the curl suchconditions, field source no is terms curl only modes in sourced will b the be initial created in conditions, the as relativisti is The evolution equations then read invariant under transformations that preserved the operat metric fluctuations, a complete set of equations. In the following we explore in further5.3 detail this system. The extendedWhat Galilean are invariance the invariance properties of this system? The origi side of the equation describing the time evolution of non-relativistic pressureless fluid. To that aim, let us int the density contrast field where for the momentum field find transformations that preserve the operator The transformation rule for the velocity field in which curl modesthat, are in generated standard Perturbation afterand Theory shell-crossing calculations, only the ev the corresponding subhorizon equation,τ while preserving t JCAP03(2015)030 ma (5.24) (5.22) (5.27) (5.26) depend →− R 0 τ , β ) (5.23) ) 2 2 k k . ( ( and i i τ τ θ θ R ~τ ) ) · α 1 1 2 k k ( 2 ( k 0 i essed in recent papers , i τ ~τ τ τ δ · θ ations, the invariance of ) ) ~τ 1 i i in Fourier space, showing ) (5.25) · τ 2 τ 0 k k er all the cosmic fluids (see ; ; ge-scale structure formation. τ 2 ( 2 2 ion of the extended Galilean k − ij k k 2 ~τ , h , So let us introduce the velocity 1 1 j k uld be performed is given in section 4.1 . they propagate, those equations have a 0 − τ k · k i ctions of fields of that type to obey τ ) ( es in terms of the power spectra of ( 2 i τ 1 R τ R k k 2 1 ), ; β α 11 i 2 2 x · H ( )= )= − 2 k k ) would require to take the subhorizon i ) ) k k ) 3 3 τ 2 2 k η,x ( ( 2 2 d d ma k ( S ( ns of the fields are real since the fields are real in i i k k ). i k ( 1 1 τ τ ) read τ 2 2 2 i,i 1 k θ k k ~ S B + + S k k 3 3 P − 2 · ( 2 1 1 d d i 5.14 2 − 2 H ∗ ~τ k k τ 2 ( ( θ τ 0 – 13 – k Z Z 2 ) ) . τ )–( 10 i 2 2 ma . In the non-relativistic limit (i.e. when µ )+ )= 0 0 ) = ), V i k k ~τ τ τ i i k ma ma k − − ( ( 5.13 − − i ) 1 A τ η,x − − 1 1 k η,x θ 0 ma∂ ( ( ( i k k τ i i − τ τ τ θ 0 − − θ θ = τ ma , i )= ) and k k ( ( P k k 0 H i ( τ ∂ − i )+ ( τ i µkτ H k ∗ τ S Dirac Dirac i ( δ δ δ i τ − δ ) = a A remarkable property is that the kernel functions )= )= k ( 0 τ τ a∂ i τ ; ; τ δ 12 2 2 + µkτ H k k i )). , , H 1 1 − a H k k ) is the Fourier transform of the field ∂ a ( ( k 3.21 a R R ( a∂ i β α τ ]), this property has important consequences regarding lar S 1+ 26 ]. and all the potentials being unchanged under such transform , Because of the phase shifts that appear between the flows when In the non-relativistic limit, Details about the way in which the summation over the flows sho 21 11 10 12 24 i,i of [ P where on the flow considered via the variable When written in Fourier space, eqs. ( non-zero imaginary part.real However, space the and correlation thus functio In particular, one expects theWard unequal identities. time We correlation will fun derivethe them Fourier modes. for relativistic speci 5.4 Nonlinear equationsWe in complete Fourier this space workexplicitly by the presenting coupling a structuredivergence field of global in the motion units motion equation of equations. - The explicit computation of the source term the equation ofinvariance interest satisfied is by ensured. the([ CDM This flow. result For is CDM, an as extens it has been str and where the kernel functions are defined as limit of the Einsteinequation equations ( and to perform a summation ov JCAP03(2015)030 ) s ]). k ( i 28 τ ]. In- (5.28) (5.29) (5.31) S 28 r a CDM streams of encode the b a ion) for each mode ns that appear in the , cluding the scale-scale ) 2 , k T ( x elements Ω c ach introduced in [ onent. All these fluids obey , )) ) (5.30) ). Once their ensemble aver- ) onal procedure to implement k p p stein equations relate the po- icit computation of the linear )Ψ ( ,η 1 or the power spectra (see [ owing to distinguish them from n ,τ racting dark matter. We recall hink for example about making ,τ mponent system of pressureless 3 ) can formally be recast in the τ compass the standard Perturba- k 2 in which the source terms 2 h has already been advocated in k ( e structure in presence of massive g occurs. Formally, we consider k ( k b ,θ c , , ) y eliminated. 5.24 1 1 k )Ψ ) describes the nonlinear interactions es is adopted. k k ( )Ψ 2 2 ( ( n k k τ R ,η R ar regime at subhorizon scales. It can in , , 2 β α 1 1 k 0 k k 0 ( In the right hand side of this equation, it ) and ( τ ( ( b τ 2 ma ma bc bc ,...,δ a a ) − − )Ψ 13 5.23 . γ γ k n ( ,η -uplet elements. – 14 – 1 ), it is possible to compute the time derivative of 1 -uplet, τ )= )= n = 0 otherwise. Contrarily to the pure CDM case, k )= n 2 2 ( ,θ k k k a ) bc ( 5.29 , , a b matrix k 1 1 γ ( Ψ k k 1 ( ( b τ a p p δ 2 )orΨ 1 2 p . Remarkably though, they encode all the nonlinear coupling 1 2 p ,η − ) and 0 ′ 2 − p 1 τ γ )=( k 2 p ) + Ω k ( 2 ]). k k run from 1 to 2 b , γ ( ( 2 a 27 a b k )Ψ Ψ ( Ψ η ,η cb ∂ a k and γ ( ) is the main result of this paper. It encodes the evolution of a a streams. Each stream is single-flow. It corresponds to eithe symmetrized vertex 5.29 n ) = 2 k , 1 k 0), we recover the standard equations and the kernel functio ( matrix elements depend on time (and on the background evolut bc → a bc i γ a In this context, the motion equations ( It means that the time-dependent 2 We are now in position to write the full equation of motion, in Besides, the Equation ( τ γ The Einstein notation for the summation over repeated indic 13 CDM flow equation (see [ neutrinos. At this stagesuch however Perturbation we Theory do calculations. notuse To propose do of so, an the one operati so-called could t Time Renormalization Group (TRG) appro particular be used in the context of the growth of large-scal relativistic or non-relativistic particles in the nonline contains all the relevanttentials field to components. those fields Note thus that these the potentials Ein are eventuall component or a baryonic componentthe or very same a motion massive equations neutrinoone so comp another. there is no point in the foll and where the indices with Provided the truncation is properlytion made, Theory such calculations equations but en Green with function the is advantage necessary. that The no simplicity expl of this approac products such as Ψ is assumed that the wave modes are integrated over. The matri form deed, thanks to the motion equation ( nonlinear couplings in thethat presence these of equations colda and are hot collection valid non-inte of until the first shell-crossin can be re-expressed as a function of the 2 ages computed, one can get the coupled evolution equations f between different Fourier modes. Its components are given by of the system, whichfluids. is formally similar to that of a multi-co the linear theory couplings. They contain in particular the way through the time evolution of JCAP03(2015)030 tions (5.33) (5.35) (5.36) (5.34) (5.32) , i ) isplacement n aking use of . k . ) s on structure , n ) n n k η , k ( al as, in that case, n , n τ n . ) so that for a collection of flows , ...δ ) ,...,η ,η ) i 1 P ,...,η ) 1 . k ,q 1 ted to be proportional to ) k n ( ′ k i , k q , k τ n the quantity at play in eq. 1 , ng equal displacements in all 1 , δ cept those participating in the η 1 η ) η, n η,η ( ( η ( ( displacement fields to long-wave η d . n . ( 1 . ( vior). n a large-scale displacement field. τ sence of such displacement thanks n n can eventually derive the following k δ τ h being treated linearly). It can be ). ,...,τ g that the only dependence with a i adiab 1 adiab i ,...,τ . Note that the right hand side of the τ ! k δ i 1 P d ) i x ...δ P τ i k . (1+i 2 ) η q q P ) i i q P ( 1 . e ( ≈ ≪ k ,q ! i i q , ) = q 2 i 1 k q ) = (5.37) i η ,η − adiab ) n ( Dirac η,η i k d x q 1 δ ( k . ( . τ term because we are interested here in large wavelenghts. , X 3 , i i ′ – 15 – δ τ d n x k h η
. δ ( q ) i i Z adiab d − . X e P Dirac k adiab q δ ,...,η 2 . )= d i 1 q x ) k = k 1+i q , i η, 1
) ( i η, n ( X ,η = . k q ) = exp(i , i − ), and satisfying adiab ) n ], one can derive Ward identities that give consistency rela η, ,η q n d η ( adiab ( ( k 29 k . δ n ( n 14 , h i τ τ n ) the Fourier density contrast in presence of a large-scale d ). This relation is valid for ˜ δ η ,...,τ ( adiab q 1 ,η δ n ...δ k τ .,τ η, ( ] and [ ) ˜ ], where it is used to evaluate the impact of massive neutrino δ ( ˜ δ 1 . 1 26 k ... , adiab ) being the unequal time power spectrum of adiabatic modes. M , ) automatically vanishes when all the time variables are equ 1 ) ). One can then define the multi-point power spectra P adiab 1 η i ,q 24 δ ′ ( k k 1 5.37 , ) (with an arbitrary time dependence and i τ ), the correlator reads for an adiabatic displacement 1 η δ η,η ( η h P i ( ( . ( 1 d We denote ) and 5.33 τ ˜ To get this result, we have neglected the δ h 14 . Due to statistical homogeneity, such quantities are expec adiab Dirac i 5.36 expressed as a function ofto the Fourier a density simple contrast in phase ab shift, This relation gives explicitly theThe dependence Ward of identities each are mode then o modes. obtained by relating More such precisely, large the one flows, can denoted define adiabatic modes induci field τ P this context in [ δ where the ensemble average islarge-scale performed displacement over all perturbation. thelarge-scale modes adiabatic Finally, ex mode assumin is inrelation, the displacement field, one relation ( one expects the result to be proportional to growth (but restricting the neutrino fluid to5.5 its linear beha The WardLet identities us define unequal time correlators as between those quantities. It is obtained by( computing the average of the product betwee eq. ( This definition imposes Following [ JCAP03(2015)030 ], we 21 ]. We leave for es into account e way to address 31 , 27 the fact that fluids of ) satisfies an extended e invariances properties f the coupling structure 002 of the French Agence 5.29 relative motions that exist ived. This is the essence of hey capture all the relevant a power counting argument. e exhaustive. uge are specified. In [ ints we noticed. The first one ), which form a closed system mber density of particles and al number density of particles he resulting system gives very oupling mechanism. Indeed, as erturbation Theory calculations urther exploration of the mode se equations is entirely based on rmal Newtonian gauge. Here we e couplings are only quadratic in horizon scales. In this derivation, ow the long-wave modes and the nted in refs. [ posed into a collection of streams, 3.11 equations for streams of relativistic ). This is the main result of this pa- reams can entirely be described by in- ) and ( 5.29 3.5 ] that this decomposition is effective at the level – 16 – 21 ), appear as a slight extension of those describing flows 5.2 remains potential even in the nonlinear regime. It implies i )–( P 5.1 ], it is already the case for baryon-CDM mixtures. An effectiv as for CDM. As a result the overall motion equation, which tak 30 15 ). It has been explicitly shown in [ The last section of this study is devoted to the exploration o The key point allowing to make this construction sensible is Note also that, throughout the paper, we paid attention to th 2.4 This is true at subhorizon scales only. 15 that appears once the motionwe equations are retained restricted only to sub dominantThe nonlinear resulting coupling equations, terms ( based on that, similarly to non-relativistic ones, relativistic st troducing a two-component scalartheir velocity doublet divergence. containing The thethe second fields, key nu element is that th 6 Conclusions and perspectives We have presented a derivation of fullyparticles nonlinear in evolution an arbitraryconservation background. laws. The They derivation lead of to the the equations ( nonlinear couplings. Thepromising insights. exploration We of recallis the here that the properties the two of most momentum t important field po Galilean invariance. Interestingly,coupling it structure, paves and the particularlyshort-wave way for modes for a interact. a descriptionbetween f of the We different h expect streamsshown act in in as particular [ a that particularlythis efficient the issue c is to exploit the eikonal approximation, as prese of cold dark matter at subhorizon scales. However, we think t Acknowledgments This work is partially supported by the grant ANR-12-BS05-0 once the background is given. of the linear evolutionin of each stream the can wholecomputed be neutrino them computed fluid. for once adiabatic initialextend The initial the conditions initi conditions results and in to ga the the confo synchronous gauge in order to be mor all the streams, canper. be It recast provides in ainvolving the starting relativistic point formal species, for form such the ( as implementation neutrinos. of P further studies those calculations. Nationale de la Recherche. non-interacting particles, such aseach neutrinos, of can be them decom obeyingeq. the ( independent motion equations we der of the systems we studied. In particular, we showed that eq. ( JCAP03(2015)030 µ ], in p 32 (A.8) (A.2) (A.1) (A.5) (A.7) (A.4) (A.6) (A.3) within η<η i ) and τ µ q . 1 − defined so that the xplained in ref. [ . is equal to ]: at initial time we ) p q i p 21 ( , P ntum variable is defined (1) j 0 gy (and thus p f ))] j d log d thanks to the relation τ f neutrinos is expected to τ x ntum i , )) p , ) x in , η )+ (1) j ( in ented in [ p η j ( d to vary locally, whence δT τ and no chemical potential (see e.g. nction of neutrinos is still a Fermi- τ ernative respecting the adiabaticity , satisfies ,µ,h,γ + δT T ǫ 0 . . . . . The solutions we describe correspond ,µ,h,γ i 2 ) )+ T 0 + ) in τ ( 1 A τ,τ B T ( τ,τ ( κ q/a ( ) = 0 )= ak B i 2 2 i 0 κ ( τ η>η τ (1 + τ τ 2 ; ,µ,h,γ 0 ak +( ; τ q/ 0 τ in aǫ x 2 + , – 17 – − − ,η m τ,τ ) = 0 and consequently that in i T ( x τ η δT = τ ( = κ ( ; i P 0 2 0 2 τ θ in p P ǫ τ 1+exp[ − ,η − in terms of the metric perturbations depend on the gauge x ∝ being the four-velocity of the comoving observer. Thus ( τ all the neutrinos whose momentum )= ) µ j i 1+exp (1) = ,q τ U i ,p x q , P with a given temperature x δT/T , 0 , ∝ 0 p in is chosen so that the neutrino decoupling occurs at a time f 0 in ) η ( j η U in ( f η ,p − x (1) , = f in µ η p ( µ ] for a physical justification of this assumption). Then, as e f U 32 − , = 19 ) can thereby be re-expressed in terms of the variable ǫ – 17 Besides, before decoupling, the background distribution o In the synchronous and conformal Newtonian gauges, the mome A.4 . The initial time being the Boltzmann constant. The relation between the ener i τ 3 B so that follow a Fermi-Dirac law It implies in particular that A Adiabatic initial conditionsWe for revisit massive here neutrinos the setting of the initial conditions as pres d energy measured by an observer at rest in the metric, Eq. ( refs. [ to adiabatic initial conditions.constraint, We choose i.e. the simplest alt One can see in particular that assign to the flow labelled by after neutrino decoupling theDirac phase-space distribution, distribution fu that we express here in terms of the mome as well as the expression of and neutrinos become non-relativistic at a time k which gives chosen. Nonetheless, after decoupling, the temperature is expecte JCAP03(2015)030 ) ] ). in ] in ] η ,η ] eling ( (A.9) ]. x id (A.11) ( (0) the neu- ψ ]. 2 /ρ ) SPIRE − in IN ]. SPIRE ,η ] [ ]. )= . x IN ( in ] [ arXiv:1112.4940 η arXiv:1110.3193 [ . ( (1) arXiv:1311.0866 , ]. The second initial SPIRE [ ) ρ arXiv:0901.4550 CFHTLS weak-lensing [ SPIRE arXiv:0810.0555 τ (0) γ , [ IN ( 21 τ IN 0 (2010) 089901] ] [ /ρ ) = f ] [ ) in in η Neutrino masses and d log ( ,η D 82 d log al resolution of the linearized x .19) of [ (2014) 011 ( /T iel, ]. arXiv:1408.2995 (2009) 017 tion theory ) (2012) 081101 [ (2009) 657 ) Chain Monte Carlo forecasts (1) γ in 03 ent and L. Fu, ρ arXiv:1109.4416 Non-linear Power Spectrum including 06 [ ween the initial density contrasts of ,η Structure formation with massive 500 x SPIRE D 85 ( IN ,µ,h,γ JCAP arXiv:1012.2868 δT ] [ 0 [ JCAP arXiv:1210.2194 , [ , Erratum ibid. (2014) 039 Neutrino constraints from future nearly all-sky [ τ,τ 6 in the Synchronous gauge ( 11 in the Synchronous gauge, one thus finds ]. / κ (2012) 2551 ) 2 2 0 ii Euclid Definition Study Report 21 Phys. Rev. in τ τ h – 18 – , 3 2 ,η 420 (2011) 030 + x JCAP (2013) 026 − 2 in the conformal Newtonian gauge (A.10) ( Massive Neutrinos and the Non-linear Matter Power , / Neutrino mass constraint with the Sloan Digital Sky ii T 03 Astron. Astrophys. ) δT 01 h , in (2010) 123516 arXiv:1006.4845 − [ ,η − x Cosmology with massive neutrinos I: towards a realistic mod ( JCAP ]. )= JCAP , ψ D 81 Massive Neutrinos in Cosmology: Analytic Solutions and Flu in )= , The WiggleZ Dark Energy Survey: cosmological neutrino mass i η − τ ( ; SPIRE x ], on super-Hubble scales, the temperature perturbation of /T , )= ) IN 19 in in in ] [ η η (2011) 043529 ( ( ,η Phys. Rev. n x δ , /T ( ) in D 83 δT Mon. Not. Roy. Astron. Soc. collaboration, R. Laureijs et al., ,η , ]. ]. ]. ]. ]. x ( δT SPIRE SPIRE SPIRE SPIRE SPIRE IN IN IN IN IN arXiv:1003.0942 As mentioned in [ Survey power spectrum of luminous red galaxies and perturba spectroscopic galaxy surveys [ EUCLID [ [ [ Phys. Rev. constraint from blue high-redshift galaxies [ neutrinos: going beyond linear theory [ cosmological parameters from a Euclid-likeincluding survey: theoretical errors Markov constraints on the neutrino masses of the relation between matter, haloes and galaxies Spectrum Massive Neutrinos: the Time-RG Flow Approach Approximation [4] S. Saito, M. Takada and A. Taruya, [7] C. Carbone, L. Verde, Y. Wang and[8] A. Cimatti, [5] S. Riemer-Sorensen et al., [3] D. Blas, M. Garny, T. Konstandin and J. Lesgourgues, [1] J. Lesgourgues, S. Matarrese, M. Pietroni and A. Riotto, [6] B. Audren, J. Lesgourgues, S. Bird, M.G. Haehnelt and M. V [9] I. Tereno, C. Schimd, J.-P. Uzan, M. Kilbinger, F.H. Vinc [2] M. Shoji and E. Komatsu, [10] F. Villaescusa-Navarro et al., [11] S. Bird, M. Viel and M.G. Haehnelt, and In the conformal Newtonian gauge, we recover of course eq. (4 in the conformal Newtonian gauge and condition we need is therefore trino fluid is proportional to its density contrast: 4 References Besides, the adiabaticity hypothesis imposesall equality species. bet Using the standard results that, for photons, These relations are usefulmotion equations, in as order presented to in implement detail the in numeric [ JCAP03(2015)030 ]. ] , ] f , ucture SPIRE ]. nlinear le IN ] , Ph.D. ] [ ]. ]. (2006) 307 SPIRE IN ]. Perturbations rest with Massive ] [ 429 SPIRE ]. ]. SPIRE IN , ]. beyond the linear IN astro-ph/0112551 SPIRE ] [ arXiv:1209.3662 [ ture surveys [ ] [ IN SPIRE SPIRE ]. to a fluid description arXiv:1501.04546 ] [ (2015) 3361 IN , IN SPIRE arXiv:1302.0223 ] [ [ IN arXiv:1005.2416 ] [ [ Phys. Rept. 447 ] [ (2002) 1 Large scale structure of the , (2012) 063509 SPIRE ]. Semi-Analytic Galaxy Formation IN ] [ rro, 367 que, CEA Saclay, Saclay, France astro-ph/9506072 [ (2013) 043530 D 85 SPIRE Viel, arXiv:1203.5342 IN (2013) 031 [ Neutrino Cosmology arXiv:1310.7915 [ ] [ 05 D 87 ]. (2010) 083520 astro-ph/9308006 [ (1995) 7 arXiv:1302.0130 Power spectra in the eikonal approximation Resummed propagators in multi-component arXiv:1401.6464 [ [ Phys. Rept. arXiv:1110.1257 Phys. Rev. , [ Neutrinos in Non-linear Structure Formation — a , JCAP D 82 SPIRE 455 , (2012) L31 IN (2014) 011 – 19 – arXiv:1311.5487 [ ] [ Phys. Rev. (1994) 22 04 , Effects of the neutrino mass splitting on the non-linear 752 gr-qc/9801108 [ (2013) 514 ]. (2014) A79 Mon. Not. Roy. Astron. Soc. 429 (2012) 045 , Phys. Rev. Relative velocity of dark matter and baryonic fluids and the , JCAP Cosmological perturbation theory in the synchronous and A Calculation of the full neutrino phase space in cold + hot Describing massive neutrinos in cosmology as a collection o 567 02 Massive neutrinos and cosmology ]. , Astrophys. J. (2014) 030 SPIRE B 873 ]. ]. , . Symmetries and Consistency Relations in the Large Scale Str Ward identities and consistency relations for the large sca Galilean invariance and the consistency relation for the no IN [ 01 (1998) 1063 Astrophys. J. Small scales structures and neutrino masses SPIRE JCAP , SPIRE SPIRE arXiv:0806.0971 , IN [ 15 IN IN Astrophys. J. (2013) 1 ] [ , JCAP ] [ ] [ , 54 Nucl. Phys. Signatures of the primordial universe in large-scale struc The evolution of the large-scale structure of the universe: Suite of Hydrodynamical Simulations for the Lyman-Alpha Fo , Flowing with Time: a New Approach to Nonlinear Cosmological Dynamics of relativistic interacting gases: from a kinetic Astron. Astrophys. (2008) 036 , ]. ]. ]. ]. arXiv:1311.2724 10 , SPIRE SPIRE SPIRE SPIRE IN IN arXiv:1109.3400 IN astro-ph/0603494 arXiv:1409.6309 IN Thesis, Ecole Polytechnique &(2012). Institut de Physique Th´eori Neutrinos in Massive Neutrino Cosmologies structure with multiple species [ formation of the first structures Contemp. Phys. JCAP of the Universe [ [ with adiabatic and nonadiabatic modes Class. Quant. Grav. conformal Newtonian gauges [ matter power spectrum Simple SPH Approach [ [ [ squeezed bispectrum of large scale structure cosmic fluids with the eikonal approximation dark matter models independent flows universe and cosmological perturbation theory regime [21] H. Dupuy and F. Bernardeau, [15] F. Fontanot, F. Villaescusa-Navarro, D. Bianchi and M. [23] J.-P. Uzan, [24] A. Kehagias and A. Riotto, [25] F. Bernardeau, S. Colombi, E. Gaztanaga and R. Scoccima [30] D. Tseliakhovich and C. Hirata, [31] F. Bernardeau, N. Van de Rijt and F. Vernizzi, [20] N. Van de Rijt, [28] M. Pietroni, [29] M. Peloso and M. Pietroni, [14] S. Hannestad, T. Haugbolle and C. Schultz, [18] C.-P. Ma and E. Bertschinger, [12] C. Wagner, L. Verde and R. Jimenez, [16] F. Villaescusa-Navarro, [13] G. Rossi et al., [19] J. Lesgourgues and S. Pastor, [22] F. Bernardeau, [32] J. Lesgourgues, G. Mangano, G. Miele and S. Pastor, [26] M. Peloso and M. Pietroni, [27] F. Bernardeau, N. Van de Rijt and F. Vernizzi, [17] C.-P. Ma and E. Bertschinger, Chapter 6
A concrete application of the multi-fluid description of neutrinos
This chapter draws attention to the interdependence between eikonal approxima- tion, relative displacements of cosmic fluids and damping of the nonlinear matter power spectrum. Understanding it qualitatively is easy but the practical imple- mentation requires a long-term task. As a preliminary stage, an application of our multi-fluid description, whose goal is to get a first estimate of the spatial scales at which nonlinear couplings involving neutrinos are relevant, is outlined (see section 6.3 for a full article on this topic).
6.1 Generalized definitions of displacement fields in the eikonal approximation
The compact equation of motion (3.16) having formally the same form with or without incorporating non-cold species, it is straightforward to generalize the use of the eikonal approximation. In the multi-fluid approach, the eikonal limit of the γ matrix elements (5.19), (5.20) and (5.21) becomes (for q k)
bc b c ma q ~τp γ2p (k, q) δ2p δ2p k q · ~τp , (6.1) ≈ − 2 [p] · − [p] 2 2q τ0 (τ0 ) !
168 6.1. Generalized definitions of displacement fields in the eikonal approximation
bc b c ma q ~τp γ2p 1(k, q) δ2p 1δ2p k q · ~τp , (6.2) − ≈ − − 2 [p] · − [p] 2 2q τ0 (τ0 ) ! which generalizes the result (3.29) obtained for a single cold-dark-matter fluid. What matters the most here is the fact that, still, all the non-zero matrix elements 2p 2p 1 are identical. Their form restricts Ξ2p and Ξ2p 1− , defined in (3.27), to integrals − 1 of the field θ~τp (q, η) . Consequently, it is still possible to interpret the eikonal eik. correction in terms of displacement fields dp , defined so that
η η 2p 2p 1 eik. Ξ2p (k, η00)dη00 = Ξ2p 1− (k, η00)dη00 = i k.dp (η0, η). (6.3) − Zη0 Zη0 The way one constructs them from the velocity divergence field of the fluid labeled by ~τp is necessarily
η eik. 3 ma q ~τp dp (η, η0) = i dη00 d q q · ~τp θ~τp (q, η00). (6.4) − 2 [p] − [p] 2 Zη0 Z q τ0 (τ0 ) !
For ~τp = ~0, one recovers of course (3.40). Note that a novelty of our approach is that, since the Ω matrix has become scale-dependent, one can also define background displacement fields, generated by the non-zero initial velocities. Indeed, the Ω and Ξ matrices have the same status in (3.28) when both are scale-dependent. By analogy with (6.3), such displacements 2p 2p 1 contribute to the time integrals of the scale-dependent parts of Ω2p and Ω2p 1− (0) − in the form i k.dp . According to equations (5.22) and (5.24), they read
η ~τ d(0)(η, η ) = dη p . (6.5) p 0 00 [p] − η (η )τ (η ) Z 0 H 00 0 00 Note that they do not impact on the time evolution of non-relativistic flows and (0) of flows in which ~τp is oriented in such a way that i k.dp = 0. In all other cases, (0) eik. dp ’s effect is dominant and superimposes to the perturbative one of dp . More precisely, in perturbation theory, one can write
(0) eik.(1) eik.(2) dp(η, η0) = dp (η, η0) + dp (η, η0) + dp (η, η0) + ... (6.6)
1 2p 2p 1 Similarly to what is done in section 3.5.1, the elements γ2p 1 − are neglected since they are reduced by a factor q/k in comparison with the other non-zero− matrix elements.
169 6.2. Impact on the nonlinear growth of structure
The role of such displacement fields with respect to structure formation is discussed in the next section.
6.2 Impact on the nonlinear growth of structure
6.2.1 Qualitative description
The case of baryons + cold dark matter
The eikonal approximation is useful when it comes to describe couplings between large-scale modes and small-scale modes. Such couplings have proven to impact noticeably on the structure-formation rate when they intervene in fluids made of baryons and cold dark matter, due to a non-zero relative velocity between those two species. Reference [179] was the first to examine this. It has then been supplemented by [17]. The relative velocity in question, vrel, emerges at recombination, when the speed of baryons starts to decline while the one of cold dark matter remains un- changed. Because of this, there is a transitory period during which baryons and cold dark matter evolve differently, which postpones the time at which baryons fall into the potential wells created by cold dark matter. Inevitably, it also postpones the time at which astrophysical objects made of baryons start to form under gravi- tational instability. In other words, relative motions between baryons and cold dark matter slow down the growth of structure. The scales affected by this phenomenon are the scales at which perturbations in the two fluids are advected relative to each other more rapidly than they grow, i.e. the scales at which k > / v2 1/2. In H h reli the equations of motion, it is encoded in couplings between small-scale modes and large-scale modes. In this particular context, “small scales” refers to scales at which baryonic objects begin to form, i.e. 10 kpc, and “large scales” stands for the typ- ∼ ical coherence length of the relative velocity field, i.e. few Mpc (see the calculation in [179]). One may expect relative displacements between cold dark matter and massive neutrinos to affect the nonlinear matter power spectrum in a similar way. Verifying it is the purpose of the article presented in section 6.3. The general reasoning is sketched in the next paragraphs.
170 6.2. Impact on the nonlinear growth of structure
The impact of adiabatic modes in the general case
In our multi-fluid approach, it is useful to notice that several modes q contribute to each eikonal displacement field, whose expression is given by (6.4). Some of them are perforce adiabatic modes. The corresponding perturbations, θ~τp (qadiab, η), contribute to the displacement field in the same manner in all flows. Hence, dp can be written as
dp = dadiab + δdp, (6.7)
2 with dadiab identical in each flow . Besides, as highlighted in section 3.5.1, the only effect of the eikonal correction is to shift the phase of the solution of equation (3.28).
It is directly related to the extended Galilean invariance, according to which δ~τp and
θ~τp are not affected by the introduction of a displacement field in the medium when they are written in real space (see section 5.2), and are thus only phase-shifted in reciprocal space. Hence, adiabatic displacement fields have no impact on equal- time power spectra because all the flows of the collection still evolve in phase when 3 δdp = ~0. Indeed, the equal-time ensemble average of adiabatically perturbed fields ψ˜ (k, η) = exp[ik d (η)]ψ (k, η) is i · adiab i
N ψ˜ (k , η)...ψ˜ (k , η) = ψ (k , η)...ψ (k , η) exp[i k d (η)] , (6.8) h 1 1 N N i h 1 1 N N i · adiab i Xi=1 whence ψ˜ (k , η)...ψ˜ (k , η) = ψ (k , η)...ψ (k , η) (6.9) h 1 1 N N i h 1 1 N N i because of statistical homogeneity (according to which ki = ~0). The adiabatic contributions to the displacement fields have therefore noP impact on the nonlinear matter power spectrum (at least in the standard case of a power spectrum involving equal-time fields). For non-relativistic species, it is known that the most growing mode is adiabatic. Consequently, we know that adiabatic modes will be dominant in neutrino streams at late time. However, non-adiabatic contributions can a priori play a significant role before the non-relativistic transition. Equation (6.6) recalls that their effects
2In section 5.3, I focused on the adiabatic contribution. 3 Those fields can be for instance δ~τp or θ~τp .
171 6.2. Impact on the nonlinear growth of structure
(0) are perturbative but they can take over when k.dp is small.
The impact of non-adiabatic modes in the general case
The non-adiabatic contributions, δdp, are specific to each flow. It means that, when such contributions are taken into account, the phases of the individual solutions of (3.28) vary from one flow to another. In this context, couplings between species are not effective mechanisms to enhance perturbations. The matter power spectrum is thus expected to be damped compared to what it is in the absence of δdp. This can be characterized by the statistical properties of relative motions between flows. In our study, we computed them explicitly for massive neutrinos by taking the cold pressureless fluid as benchmark.
6.2.2 Quantitative results in the case of streams of massive neu- trinos
To identify the scales at which relative displacements damp the matter power spec- trum, it is useful to compute the quantity
[k.(d d )]2 1/2 σ = h p − CDM i , (6.10) p k where the index “CDM” refers to the fluid labeled by ~τ = ~0. In this context, the most impacted scales are those for which k & 1/σp, i.e. the scales at which k.(d d ) is substantial4. p − CDM In the conformal Newtonian gauge, we found (the full calculation is given in the article presented in the next section):
1 (k.d )2 = 4πk2 dq P (q) T (q) 2 (6.11) h CDM i ψ 3 | CDM | Z and
1 2 2 [p] (k.(dp dCDM)) = 2πk dq Pψ(q) dα (6.12) h − i 1 × Z Z− 1 1 (µ[p])2 1 (α[p])2 T (0)(q) T (q) T (2)(q) 2 + (α[p])2(µ[p])2 T (2)(q) 2 . 2 − − | p − CDM − p | | p | h i h i [p] () α is the Cosine of the angle between ~τp and q. TCDM and Tp are transfer functions
4 c I recall that k.δdp is the non-adiabatic contribution to Ξb , see (6.3).
172 6.2. Impact on the nonlinear growth of structure defined so that
β η [p] ma τ (β) dη00 θ (q, η00) = T (η, η0, q)ψ (q), (6.13) − [p] [p] ~τp p init Zη0 τ0 τ0 ! η dη00 θCDM(q, η00) = TCDM(η, η0, q)ψinit(q), (6.14) Zη0 with ψinit the initial value of the metric perturbation ψ that appears in the per- turbed Friedmann-Lemaˆıtre metric written in the conformal Newtonian gauge,
(2.18). Pψ(q) is the initial power spectrum of ψ, i.e.
3 ψ (q)ψ (q0) = (2π) δ (q + q0) P (q). (6.15) h init init i D ψ
One can see from (6.12) that the amplitude of σp strongly depends on the flow considered, both via α[p], µ[p] and τ [p] (the latter being hidden in the transfer functions). We computed numerically the per mode contributions to σp (the setting of the cosmological parameters and initial conditions is precised in our article).
Clearly, there are fluids (those with a high initial velocity) in which σp is comparable [k.(d )]2 1/2 in amplitude with h CDM i . This is illustrated in figure 6.1. The top panel k (0) is of particular interest since the background displacement dp has no effect when µ = 0. In this context, relative displacements are the leading contributions to the damping of the nonlinear neutrino power spectrum. Modeling their impact is therefore crucial. As a preliminary result, we deduced from the plot 6.1 that the relevant scales correspond to wavenumbers larger than (or of the order of) about 0.2 to 0.5 h/Mpc,
σp being of the order of 2 to 5 Mpc/h (when considering neutrinos with a 0.3 eV mass). A much more thorough analysis is necessary to make testable predictions and go one step further in precision cosmology. In this exploratory study, our aim was rather to draw attention to the way in which relative displacements between neutrino streams and other fluids affect the nonlinear growth of structure.
173 6.3. Article “On the importance of nonlinear couplings in large-scale neutrino streams”
14
12 L
0 10 = Μ , q H 8 cdm , Ν 6 P q Π
3 4 4 2
0 0.001 0.005 0.01 0.05 0.1 0.5 1. q @hMpcD 14
12 L
1 10 = Μ , q H 8 cdm , Ν 6 P q Π
3 4 4 2
0 0.001 0.005 0.01 0.05 0.1 0.5 1. q @hMpcD
Figure 6.1: Power spectrum of the relative displacement as a function of the mode q for different neutrino flows. The quantities Pν,cdm are defined so that the right hand 2 sides of eqs. (6.11) and (6.12) read 4πk /3 dqPν,cdm(q). The values of τ range from 2.25 k T (bottom line) to 18 k T (top line). On the top panel µ is set to 0, B 0 B 0 R on the bottom panel µ is set to 1 and the neutrino mass is set to 0.3 eV. The gray dashed line represents the power spectrum of the cold dark matter displacement.
6.3 Article “On the importance of nonlinear couplings in large-scale neutrino streams”
174 JCAP08(2015)053 /Mpc h 2 . hysics =0 P k le ic t ar 10.1088/1475-7516/2015/08/053 doi: strop A [email protected] , osmology and and osmology C 1503.05707
cosmological neutrinos, cosmic flows, neutrino theory We propose a procedure to evaluate the impact of nonlinear couplings on the evolu- [email protected] rnal of rnal ou An IOP and SISSA journal An IOP and 2015 IOP Publishing Ltd and Sissa Medialab srl Universit´ePierre et Marie Curie, 98Institut bis de bd Physique Arago, CEA, Th´eorique, 75014 IPhT,CNRS, Paris, F-91191 URA France Gif-sur-Yvette, 2306, F-91191 Gif-sur-Yvette, France E-mail: Institut d’Astrophysique de Paris, UMR-7095 du CNRS,
c J ArXiv ePrint: relative displacements of di↵erentscales cosmic at streams which and the growth itredshift of specifies, zero, structure such for is couplings each a↵ectedfor can flow, by most nonlinear be the of couplings. spatial significant the for We neutrino found wavenumbers streams. that, as at small as Keywords: tion of massive neutrino streamscan in be the context described of byperspective, large-scale general which structure generalize growth. nonlinear the conservation Such conservation equationsThe streams equations, of relevance non-relativistic derived of pressureless from fluids. the amation nonlinear multiple-flow applied couplings to is the quantified subhorizon with limit the of help this of system. the eikonal It approxi- highlights the role played by the Received March 20, 2015 Revised June 26, 2015 Accepted August 6, 2015 Published August 28, 2015 Abstract. ´leeDupuy andH´el`ene Francis Bernardeau On the importance ofcouplings nonlinear in large-scale neutrino streams JCAP08(2015)053 ] 1 4 9 2 9 7 ]). 14 12 – 10 ]butthismethod 15 ]. Improvements upon 13 , 1 ]. 9 – 4 ], we proposed to describe massive ]). This is all the more unfortunate 17 3 , – 1 16 –1– ], which consists in a hybrid approach that matches the 3 /Mpc range, precisely correspond to the mildly nonlinear regime. To h 2 . 1-0 . In this paper, we are interested in exploiting those theoretical developments in order to Ideally, however, the fully nonlinear evolution of the neutrino fluid should be depicted. In the linear regime, the e↵ect of neutrinos is now well understood (see refs. [ distribution function to theturned out nonlinear to regime. be particularlyneutrinos This di cult as has to a handle. been superpositionbeing done In written [ in of in [ single-flow the fluids, nonlinear the regime. equationsidentify of the motion scales of at each which nonlinear of couplings them in the neutrino fluids are expected to play a is, for its part,order corrections a to systematic the perturbativetrack neutrino the expansion density contrast perturbed of are neutrino the computed momentum Vlasov without distribution. the equation explicit inA need which natural to high- way to dorelies so on would the be Boltzmann to equation, take and inspiration extend the of harmonic the decomposition standard of linear the description, phase-space which deal with this issue,neutrino several fluids strategies can had be alwaystroduced adopted. been in Until the treated recently, dark in inwith matter the analytic the description works, linear help only. the of regime, Thissuch the nonlinear nonlinear schemes Renormalization treatment have couplings Group been can being proposed time-flow befull in approach in- implemented [ Boltzmann [ hierarchy to an e ↵ective two-fluid description at an intermediate redshift. [ enough for those masses to be constrained observationallyThe [ need for nonlinearcosmological corrections observations in that their are equationsbers the of in most motion the sensitive has 0 to been neutrinos raised masses, because i.e. for the wavenum- as an e cient instrumentaccount of to the constrain role played cosmologicalin by massive parameters. the neutrinos is nonlinear In crucial.study or this So of far, quasilinear context, massive it a regime has neutrinosthat often careful because (see been cosmological recent overlooked of observations attempts can thesignature in fruitfully of technical improve [ neutrino our complexity masses knowledge specific on of cosmological to those observables is the particles. indeed The expected to be significant A Integral form of the overall equation of1 motion Introduction Statistical properties of the large-scale structure of the universe have long been proposed 3 The eikonal approximation 4 Relative displacements and power5 spectra: quantitative Conclusion results Contents 1 Introduction 2 Nonlinear equations of motion (multi-fluid description) JCAP08(2015)053 (2.4) (2.1) (2.2) (found to be i . Those fields ⌧ µ P ]) and had proved to be , denoted 18 i P 1 (2.3) , ) and momentum i . ⌧ c ; i =0 n , we recall the form of the nonlinear ) being the conformal Hubble constant) i 2 H being the metric, and ,µ ⌫ ⌧ ) ◆ g i ; H c ⌘, x i ⌫ ⌧ ( , µ⌫ n ma ( i g P i 0 P , (0) c i ⌘, x P P 2 H ( @ P n c ✓ 1 2 m ]. explains in detail how the eikonal approximation n i –2– 3 16 @ = @x = )= )= i i µ,⌫ µ + P c P ⌫ µ ⌘, x n ⌘, x ( ( P i P i @ ⌧ ⌧ @⌘ ✓ ) and , power spectra of the relative displacements between neutrino i 4 ⌘, x ( ⌫ P µ⌫ g ], it is now clear that any non-interacting relativistic fluid can be divided 17 , )= i 16 ⌘, x ( µ P We are interested here in equations i) involving linearized metric perturbations (but The article is structured as follows. In section ) having initially the same comoving momentum. They are entirely characterized by two and of the velocity divergence field (in units of initial conditions is described in detail in [ non-linearized fields) and ii)scales. rid They of can the be coupling written terms in terms that of are the subdominant number at density subhorizon contrast These equations directly ensueplied from to the each matter flow. andproperties momentum of At conservation the this equations, whole ap- stage,flows, fluid no each are perturbative of then them expansion inferredconstant being of at by labeled the zeroth examining by metric order an theeach is in appropriate flow initial Perturbation involved. number is value Theory). of constrained The of by The such its the initial field choice number of density initial of conditions. particles For in instance, the case of adiabatic with m coupled fields, namelyobey the the comoving following number general density equations, 2 Nonlinear equations ofFollowing motion [ (multi-fluid description) into several flows, eachcosmology, this of approach obviously them applies to evolvingIn massive this neutrinos then since framework, they independently each are flow until free-streaming. can first be shell-crossings. defined as the In collection of all the particles (with a mass can be implemented infields. those equations and Finally, emphasizes influids the section and key role the oflarge-scale cold relative structure displacement dark is matter then discussed component in are a quantitative presented. way. The impact on the growth of tions of motion.to Note develop that a this Perturbation Theory approximationable approach had to for already capture cold been the dark leading matter exploited coupling ([ in e↵ects. the literature equations of motion describing thewhen time using evolution a of multi-fluid non-interacting approach. fluids, Section relativistic or not, significant role. In order to do so, we apply the eikonal approximation to the nonlinear equa- JCAP08(2015)053 ) 2.6 (2.9) (2.5) (2.6) (2.7) (2.8) (2.10) (2.11) (2.12) (2.13) , . ) ) streams. 2 2 k k ( ( N i i ⌧ ⌧ ✓ ✓ ]) ) ) . 1 1 17 , k k ([ ⇤ ( ~⌧ ( i j i · ⌧ ⌧ k x 2 ✓ d ) ) 2 k i i 0 , i , ⌧ ) ⌧ ⌧ ~⌧ x ; 2 . ; · . 2 2 k )d T . ~⌧ 1 2 i ( k ) k · 2 ij ⌧ c 0 k )) , , k ⌧ h 2 1 1 k ( + ( k , k ) k ij + ( 1 ( ⌘ 2 N 2 ~⌧ h ⌧ R a k R 3 ij k j ( 2 ↵ 0 · ⌧ b ,✓ i 2 2 ⌧ ) For a generic perturbed Friedmann- 2 m 1 ⌧ k k k k ) k +( 1 3 3 ( 1 2 q 2 )= )= d d ⌘ · N k 1 1 k k 2 ⌧ d , ( ( ) ) i k k i i 1 ) 2 2 3 3 = ⌧ ⌧ x 2 2 k k k k ✓ d d d k S 0 ( ( i 2 2 2 1 ⌧ 2 ~ ◆ bc + + B k k B Z Z ,..., a 2 H 2 · 2 ) 1 1 2 ⌧ 0 k 0 0 k k k ~⌧ 2 –3– ⌧ and ⌧ ⌧ ( ( ( +2 ma ma ma 1 µ ], generalizes that of non-relativistic species. ) ) i 2 ⌧ )= 2 2 ⌧ ⌘ )+ 17 i k k k ,✓ ⌧ ( k ) ) )d 1 b ( k k = A ✓ ( ( A
1 1 i 1 0 2 b ⌧ k k ⌧ ⌧ a 0 ✓ ] for details in this context.). ⌧ ,⌧ ma i ◆ 19 ⌧ (1 + 2 k k 0 )= i k⌧ ( ( ⌧ k k )+⌦ )=( )+ ( ⇥ k µk⌧ H k i k ( ) i ⌧ ( ( = i a a ⌘ Dirac Dirac S ⌧ ( µ
2 z a a ◆ @ 0 a@ )= )= ], they are extensions of the kernel functions found for pressureless fluids = -uplet, ⌧ ⌧ ⌧ whose time variable is the conformal time 2 + ; ; 17 µk⌧ H N s 2 2 2 i has found to be potential in this regime. d k k i H , , a H ) is a source term given by P 1 1 @ a k k k ( a ( ( i a@ ⌧ R R S ✓ ) of all the flows can formally be recast in the form ↵ 1+ ✓ 2.7 In any practical implementation, it is necessary to consider a collection of This property, rigorously demonstratedUnits in are [ chosen soThe that Einstein the notation speed for of the light summation in over vacuum repeated is indices equal to is unity. adopted and, in the right hand side of 1 2 3 this equation, it is assumed that the wave modes are integrated over. Before shell-crossing, itbaryons) can as long incorporate asand all they ( the interact relevant only species via gravitation. (neutrinos, In dark this matter, context, the equations ( As mentioned in [ of non-relativistic species (see [ In that case,time-dependent the 2 general equation of motion can conveniently be written in terms of the These equations contain also the generalized kernel functions, adapted to relativistic flows, Besides, We have introduced here eaır metric, Lemaˆıtre one has in Fourier space for the mode characterized by the wave vector since the field JCAP08(2015)053 b a (2.14) (2.15) behavior k , significantly depend R ) describes the nonlinear , 2 ) , . The matrix elements ⌦ p k ) , p N and ,⌧ 1 2 ,⌧ k R 2 k ( , k ↵ bc 1 , a 1 k ( k ( R R ↵ 0 ) 0 -uplet elements. The left hand side of this p ]. Such studies illustrate the relevance of ) ⌧ p matrix ma N ) can be split into two integration domains. ⌧ ma 22 run from 1 to 2 2( ( ], is based on the observation that the ampli- , b 2.13 18 21 –4– (at decoupling), the wavelength of gravitational 2 and / = 0 otherwise. )= )= 1 2 2 i a bc k k a 2 rel , , v 1 1 h k k ( ( p p 2 12 p ) and developed in [ 1 2 p 1 2 symmetrized vertex p 4 k 2 p , 2 2 k>aH/ k ( and the indices cb a )= a@/@a 2 k ⌘ , ]. 1 k ]. In the same spirit, we propose here to investigate the impact of the relative 20 ( 24 bc @/@z a , ) can be re-expressed as a function of the 2 The main idea is that there are regimes in which the dominant coupling structure is in between the baryon flow and the one of cold dark matter is substantial. Because of k 23 In this context, the term refers to diagram resummations performed in quantum electrodynamic field ( 4 i ⌧ rel of the propagatorsin in [ case ofmotion a between single given pressurelesseikonal neutrino approximation. fluid, streams It reproducing and will the the allow us cold results to dark obtained infer matter theequations, amplitude fluid [ of with neutrino the coupling e↵ects. help the couplings between large scaleformation. modes For instance, and incoherence small the length case scale of of modes the mixtures relative ofthan in velocity baryons the field the and scales is cold framework of at dark(of of the matter, which order the structure the basic of typical order baryonic few of objects Mpc, which 10 start is to kpc). much form larger The under gravitational same clustering formalism can be used to obtain the large v this, for scalespotential at wells which is toodirection. small for Eventually, baryons this toThe phenomenon fall induces relative in a motion them dampingpower before between of spectrum being cold pushed the have dark towards matter been matter another power highlighted and spectrum. in baryons [ and its e↵ects on the matter functions, of the order of the wavelength ratio,the are large. soft domain.cold This dark is matter. thethe At case velocity the of for time cold of instance darkscales recombination, matter in (i.e. is the early-time not between baryon a↵ected fluids velocity the by containing drops decoupling. Silk baryons steeply It damping and whereas means length that, at and intermediate the sound horizon), the relative velocity on the ratio betweenfluids the and wave numbers we atto show play. the here This idea that has thatOne it been the is is observed right called for the hand the non-relativistic sideorder case hard of (and for domain eq. for relativistic and ( whichas fluids encompasses the the too. modes coupling soft functions whose domain. This are wavelengths It are always property is finite). of made leads of the The modes same other of one very is di↵erent wavelengths referred for to which the coupling 3 The eikonal approximation The eikonal approximation, tudes of the kernel functions describing mode couplings, with gather the linear couplings.S They containequation in is particular nothing the but wayterms. the in linear which More field evolution. the precisely,interactions source The the between right terms the hand Fourier side modes. contains It the is coupling given by where JCAP08(2015)053 ) ) ). k )is 3.3 .In z, k (3.6) (3.3) (3.4) (3.5) (3.1) (3.2) k ( 2.13 b z, a ), ( ( b a 3.2 It therefore , in eq. ( H 1 i k ) in the ⌅matrix, . . ) p ,z ⌧ ! ⌧ , q 2 , which vanishes in p , 2 )isthekeyelement. ~⌧ 0 p k ! k . In this framework, ( z ~⌧ p p 1 c ( ~⌧ ~⌧ 2 0 3.2 k p ) · p 2 ]). p ~⌧ 2 0 2 . ⌧ ) )
· ⌧ ( k 25 p ) in the non-relativistic limit 2 ,z ⌧ ! , 2 ( 1 k , p ,z ) means that the vertex values k ~⌧ b k 21 ) q a ( p ma ( 2 ,z 2 0 k b ~⌧ c ) k k ) · p 2 ( z,
b ⌧ ( ) k ) q a ( ,z b · ! 2 q 0
) can then be viewed as the equation , ) that the eikonal limit of the vertex k z k · 0 ) ) are integrated over so that ⌅ ( , k 0 ) p ( 1 k ) k p 3.1 q is the total displacement field induced by p ⌧ d k 0 bc 3.2 2.15 . ⌧ ( z, ) a
p ( ( k p bc ma d 2 2 b a . 0 ⌧ a eq. ( ) . It reads necessarily k ( ), the implicit convolution product excludes ⌅ p ma p 2 2 2 eik =i h ⌧ ⌧ k 0 ( ma 3.1 p –5– 2 q 2 ) and ( z c 2 = ) 3 q p k d 1 )d c 2 q equation describing the evolution of the mode k 3 S z, 2.14 p p , ( d Z 0 b b 2 2 b z 2 ( ), ( Z 1 and where b ) 0 a ⌘ k z linear ) at play in eq. ( ⌅ ) d k p 2.11 z z, 0 z 0 ,z ( z )= )= z b k Z 2 2 a ⌧ Z ), ( ( or 2 k k b , , a q ). It is a mere time and scale dependent matrix. The fact that p evolving in a medium perturbed by large-scale modes. k k ⌅ ( ( 2.10 k )=i )+⌦ 1 p z,k bc 0 2 and the contribution corresponding to the soft domain can be viewed k ( ) can be rewritten (i.e. p a . 1 bc 2 z, q z,z k ( eik . ( a 2.13 , are proportional to the velocity divergence of large-scale modes. Thus we p = 1 eik
d 1 are either 2 k p @ ) can be solved as a linear equation. This is precisely the eikonal approximation @z 2 b p 2 3.1 and ) that the two non-zero elements coming from the flow labeled by a and ⌅ 3.4 In the following, we exploit the consequences of this approximation in order to evaluate In practice, applying the eikonal approximation to ⌅ For convenience, let us assume that the soft domain is obtained for p 2 p 2 where the large-scale modes in the fluid labeled by the relevance ofand the ( coupling terms.⌅ As acan first write step, we can notice from eqs. ( This expression dependsthe on standard each non-relativistic flow equations. throughso that its Besides, one initial ( recovers momentum the expected formulae in this limit (see [ one deduces from eqs.elements is ( encodes the way ingiven, eq. which ( long-wave modesof alter the the global growth equation of of structure. motion. Oncethat ⌅ appear in this quantity have to be computed assuming is independent onthe integration domain is restrictedConversely, to in the the softthe wave right-hand numbers soft side in eq. domain of ( contribution eq. (i.e. of ( all the the hardof modes domain motion is of concerned negligible, the have mode comparable wavelengths). When the with The soft momenta We have thenas a mere correctiveother term words, in eq. the ( JCAP08(2015)053 ). k (3.8) (3.9) (3.7) ⌘, (3.12) (3.10) (3.11) ( a
)) ⌘ ( d · k , represents relative p d )=exp(i k , ⌘, ) (˜ , 0 ) a ˜ k
, z,z ), defined in such a way that (see . ]. In this paper, we show that such ( 0 0 k p z , z ; ( ) 22 0 , b d z d ) ( 0 i x ) ) d z,z p ( k p )+ ). Remarkably, such displacements have ⌘, ⌧ b ; 0 ~⌧ ( + ( a 0 a i g 3.1 H x z, z,z z ( 0 z,z z ( = = –6– b Z a i )= ˜ z g ˜ x ˜ adiab x d z, = (˜ )= a k )= ˜ (0) p
0 ] an reconsidered in [ d z, ( ]. Indeed, it has been shown that this system is invariant 21 a z,z 17 (
p d )in[ )) 2.7 2.7 )–( 2.6 denotes naturally the adiabatic part. The other part, ) and ( for details) 2.6 A adiab d More explicitly, it is known that the solution of the standard linear system can be fully When considering only the adiabatic part of the displacement field, the solution of the Knowing this, it is convenient to decompose displacement fields into The impact of the eikonal correction introduced in the equation of motion depends . The nature of the displacement is crucial in this context. In particular, one expects p are interested indisturbers making of the the large-scale medium. adiabatic displacement fieldsdescribed play with the the role helpappendix of of the its Green function, where the last transformationIt can means equivalently that be a written be homogeneous time-dependent re-absorbed displacement in which disrupts a the global medium phase can shift of the Fourier transforms of the fields. Here, we ance of the equationsfor of the motion, system well ( under established the for following non-relativistic transformations, species and uncovered motions between fluids. For non-relativisticis species, part it of isexpects known the that the adiabatic the most most modes. growing growingultimately. mode mode of Since each neutrinos fluid of become neutrinos non-relativistic to at becomeeikonal also equation late of an time, motion adiabatic is one mode easy to find. Its form is related to the extended Galilean invari- considerations can be extended to the study of relativistic fluids. where time (see the endparticular of displacements “adiabatic this displacements” section). in thedisplacements following. Using between species On the can the standard induce contrary, relative fact a language damping that of in species cosmology, power spectra. must webetween evolve call It them in is those to simply phase generate due (athighlighted a to least for the significant the during growth first of a time perturbations. small in period This [ of phenomenon has time) been for couplings on the wayd in which theglobal various displacement fields, large-scale whichshift modes a↵ect in contribute all species the tono in solution the a impact of displacement similar on the way, fields power to eikonal induce spectra, limit a provided mere of that phase ( the fields at play are evaluated at the same by the homogeneous momentum(see of the eqs. flow. ( This background displacement field is given by We will discuss this point in greater detail in the next section. Note that this displacement is superimposed on the zeroth order displacement field induced JCAP08(2015)053 . ) k cdm (4.3) (4.4) (4.1) (4.2) d (3.13) )), ]. First, p d 3.7 16 ( . is orthogonal k k , , ) ) q q ( ( . ⇤ init init . ij
) )) ) become proportional to ) j 0 q a q x , k , 0 d ... ) is small compared with the , 0 i z,z 0 a of the considered fluid and ( ) is small) or if x z 0 being defined in eq. ( z,z ( z,z )+ ~⌧ ( a z,z )d ( 0 ( )
z,z (0) p adiab ↵ ( 2 ( p cdm d d (0) ⌧ z,z . ( d D D (0) ⌧ k . d k (1) p as d ) )= )= +(1 ↵ q q ( p 2 )exp(i , , )+ 0 0 ⌘ k D 0 z z ; ( ( 0 )d p –7– ⌧ z,z
✓ ( and cdm z,z ✓ = 0. This is the reason why equal time spectra or ( ↵ (0) p b 0 a a ◆ d z cdm g 0 k d (1 + 2 ) D p a z p 0 ⌧ z ⌧ )= )= ⇥ ( P 0 Z ) k ✓ ; ⌘ 0 ( z,z 0 ( 2 ) p a p z,z d ⌧ ( of the corrected linear system (i.e. the linear system in which an ( ma b = a b 2 ⇠ 0 a s )], which is unity when the fields are computed at equal time. Indeed, ⇠ z a d d z ( z 0 z Z adiab two arbitrary times. Besides, when the displacement field is purely adiabatic, d 0 · z ), ensemble averages of any product of fields a k a and 3.11 z P Yet, we are interested in all contributions to the displacement field, including those let us define two transfer functions as a function ofCalculations are the performed angle in between the the conformal initial Newtonian gauge, momentum This choice will allow us to take advantage of the numerical work presented in [ Note also that the damping dueconsidered to flow non-homogeneous is corrections non-relativistic can (since beto in significant the that only zeroth if case the orderother contribution contributions). (since Hence, in in the that following, case we compute the expected values of total displacement field of coldrelative dark displacement matter fields and, of ondark the relativistic matter other component). flows hand, (with Note the power respect that,terms spectra in of to of perturbation di↵erent the the orders theory, a such (the displacements priori zeroth are dominant order sums contribution cold of 4 Relative displacements andAs power spectra: stressed in quantitativethat the results can previous cause section, aanother. in damping of the Hence, the eikonal in growth approximation, this of the structure section, displacement are we those fields compute that on di↵er the from one one fluid hand to the power spectrum of the baryons and cold darkbutions matter, that those develop relative in displacementsevolution the (i.e. of nonlinear those the regime) non-adiabatic power areear contri- spectrum. the growth leading rate, To contributions sketch we toinvolving the evaluate the them in impact nonlinear and the of we next massive compute section neutrinos the the on corresponding amplitude the power of nonlin- spectra. the relative displacements form ( exp[i statistical homogeneity imposes poly-spectra are not sensitive to the presence ofthat adiabatic induce displacements. large motions between species. As already mentioned, in early-time mixtures of the Green functions eikonal correction has been added) are related to those of the naked theory by A direct consequence of the symmetry property is that, after a transformation of the with JCAP08(2015)053 ), )in and and ) 4.9 (4.9) (4.5) (4.6) (4.7) (4.8) q k 4.9 ( . After cdm ~⌧ d cdm , . and is expected k D p 2 we find that | ~⌧ ) . and (1) ⌧ 2 .Ineq.( q / ) d q ( 1 k init · = 1 on the right q i ( 2
(2) p k k )) (0) µ ◆ p D and = 0) or for an initial | ) and 2 k D k q p | cdm is such that it does not µ , ~⌧ ~⌧ ) µ . 2 0 d ) 2 2 . k | q µ is close to zero. A precise µ ) ) µ ( q k q z,z
( + and ( p ( µ P 2 , for which d | q init ( (2) p ) ⌧ ) . 0 )(1 cdm d
q 2 k D k q ) ( D ( p (i.e. when µ / | q h ~⌧ + , (2) p 2 k 0 1 3 ), defined so that ) /k q D q q p ( ) . ( (1 ⌧ q p z,z ( (
1 2 ( ~⌧ =1 2
P q Dirac ⌧ µ d cdm qP 3 d . The statistical properties of those quantities ) d D ) 1 1 ⇡
= 0 on the left panel and q 2 of each flow when q Z , –8– q Z k 0 i and 2 ) µ orthogonal to (0) p =(2 q 2 ( p d / Mpc for the cosmological modes we used for which z,z i ⇡k 1 ~⌧ = 1).
( 1 ) i 0 , Cosine of the angle between k 2 )= q (0) p ) µ h =4 µ ( qP q ; d i D 0 2 2 init cdm ) q q Z
d . ) 2 z,z ✓ ( k q cdm i ( ( ⇡k d h . cdm k init (i.e. when /k ( d =2
h )= h k i depends on the angles between both q 2 =1 ; k )) 0 cdm z,z d cdm , we present the per mode contribution to the right hand side of eq. ( ( 1 d p 96. We can see that the resulting neutrino power spectra are comparable in Mpc,. This result depends of course on the flow considered and is actually of . d ), 1 0 is the initial value of the potential p is the Cosine of the angle between the initial momentum of the flow h d cdm ⇡ ( 6 3 eV and using the cosmological parameters derived from the Five-Year Wilkinson d . k init s . k µ
n ⇡ ( Denoting On figure Furthermore, one can notice that the displacement field of a relativistic flow along an Using the expressions of the transfer functions, the contribution of each mode to the h p is of the order of 2 to about 5 =0 d ⌧ cdm ( ⌫ . d d in mixtures of baryonsdamped for and wave cold numbers largerto dark than be matter, or finite. one comparable to expects Besides,due 1 to such the the a perturbation homogeneous non-adiabaticdetermination growth displacements damping of to is the be amplitude potentiallyevolution of larger of these than the e↵ects the system. would damping require We a leave this full for analysis a of future the study. nonlinear index amplitude to the cold dark matter one (represented by a thick dashed line). the order of the cold dark matter value. What does it mean? Similarly to what happens panel. The results havem been computed assuming aMicrowave single Anisotropy species of Probe neutrinospower (WMAP whose spectra mass 5) is have observations.we been adopt, obtained Besides, metric fluctuations under the are the values initially assumption of adiabatic and that, the characterized in initial by the the scalar cosmological spectral model where an integration isone can made notice over thatvanish the either dependence for of an theflow initial r.m.s. momentum momentum along with respect to the particular case of neutrino fluids for where arbitrary direction integration over thek other angles, one finds for the variances ofand respectively Besides, for the cold dark matter component, one simply has are entirely encoded in their initial power spectra displacement field of each relativistic flow reads where JCAP08(2015)053 ) ). ), 4.9 3.6 3 eV. 1. . 2.13 ), of the 0.5 (top line). k ) and ( ; 0 for di↵erent 0 T 4.8 q B k z,z ( b 0.1 a g D 0.05 Mpc ê h @ q ) can formally be written 0.01 2.13 (bottom line) to 18 0.005 0 T B
3 eV mass) are expected to induce a k L H .
8 6 4 2 0
0.001
14 12 10
3
cdm , n
1 , q P q = m
is set to 1 and the neutrino mass is set to 0 4 p k µ 1. –9– 0.5 range from 2.25 ⌧ are defined so that the right hand sides of eqs. ( Mpc. A detailed quantitative analysis of the consequences 0.1 cdm h/ D ⌫, 5 0.05 P . Mpc ê h @ q ). The values of 2to0 q . ( 0.01 is set to 0, on the right panel cdm k 0.005 ⌫, µ qP
d
L H R 3 6 4 2 0 8 . Power spectrum of the relative displacement as a function of the mode /
0.001
10 14 12
2 3
, cdm n P q 0 , q = m
⇡k 4 p In this paper, we evaluated the amplitudes of the nonlinear couplings and determined in an integral form. It requires the use of the associated Green operator, This work isNationale partially de la supported Recherche. by the grant ANR-12-BS05-0002A of the Integral French form Agence ofFor the a overall finite equation number of of flows, motion the overall equation of motion ( of this phenomenon isear yet couplings to in be thenumerical done dynamical studies but beyond evolution those the of findings linear neutrino regime. confirm fluids. the significance ThisAcknowledgments of sets nonlin- the stage for further the one associatedspectra with of cold the relative darkspectrum displacements matter between of neutrinos makes the and easy displacementthat cold of dark the couplings matter cold comparison involving and dark massive betweendamping the matter power of neutrinos the alone. the (with power We perturbationthe a found growth order 0 as in of) a neutrino about preliminary flows 0 result for wave numbers larger than (or of for each flow thegrowth. scales For at that which purpose, theytion we implemented are of the expected motion. eikonal to approximationare We impact into entirely the concluded driven significantly general by on that large-scale equa- The the the displacement comparison structure fields impact whose between of expressions the large-scale are displacement given modes field in on eq. associated ( an with arbitrary each mode flow of neutrinos and their impact on thecomplete growth set of of the equations cosmicor of structure. non-relativistic motion particles. Using that In thiswhich incorporate the approach, can subhorizon all one limit, be the indeed the easilyspecies. nonlinear gets system handled e↵ects a takes the of by form relativistic a of formalism eq. originally ( developed to depict non-relativistic On the left panel The gray dashed line represents the power spectrum of the5 cold dark matter displacement. Conclusion Describing neutrinos as a collection of single-stream fluids is an e cient strategy to infer Figure 1 neutrino flows. The quantities read 4 JCAP08(2015)053 (A.5) (A.1) (A.3) (A.4) ]. ] ]), which is SPIRE reads IN 28 b a , – ][ g ) 0 26 ,z 2 k ( d arXiv:0901.4550 [ (2010) 089901] ) 0 ,z , 1 ) D 82 k ) = 0 (A.2) k , ( , arXiv:1408.2995 ) k [ c 0 ; k ) these solutions, z 0 (2009) 017 , ( k Non-linear power spectrum including ) 0 ) 2 z ↵ Structure formation with massive z, ( b ( z,z 06 k , ( b ( c , b ) b ) 1 a ↵ c ( a k ) k g ( u ) is satisfied. k ) (2014) 039 z, Erratum ibid. ; cd ( k [ JCAP . This dependence is expected to gradually 0 b )= ) ; , k ↵ A.3 11 k z ( a ( ; ) ) z,z u c 0 0 0 ( a b – 10 – ,z ,z a ↵ 0 g X k Denoting z ,z,z ( JCAP ( b , k 5 b )+⌦ ( a )= k )= b g ; a k ) k 0 (2010) 123516 g 0 , 0 0 z, ( z z,z a d ]. ( ,z,z z,z b
z 0 Massive neutrinos in cosmology: analytic solutions and fluid ( D 81 a k ) are set so that ( z b a ( g k Z g b , a @ 0 @z g + SPIRE z ( IN ) ][ ↵ )= b ( c Phys. Rev. ,z , k ( a two arbitrary times. It is besides solution of the di↵erential equation
) the initial conditions. Many of the approaches developed in order to improve ]. 0 0 z ,z k ( ]). and a SPIRE arXiv:1003.0942 IN z As for the standard system of non-relativistic particles, the knowledge of the Green Studying in detail the Green operator of such a system is beyond the scope of this 30 approximation [ neutrinos: going beyond linear theory massive neutrinos: the time-RG[ flow approach M. Shoji and E. Komatsu, D. Blas, M. Garny, T. Konstandin and J. Lesgourgues, J. Lesgourgues, S. Matarrese, M. Pietroni and A. Riotto, – being the identity matrix. Formally, the Green function is the ensemble of all the inde- A priori the number of solutions is equal to twice the number of flows considered. 5 b 28 [2] [3] [1] a References with upon standard Perturbation Theorybeyond rely the on linear an([ regime. accurate description of This the is Green the functions purpose for instance of RPT and RegPT methods operator of the equationgiven of by motion allows to write a formal solution (see [ operator generally depends ondecay the over wave time mode andrelativistic. to At disappear this atdark stage, matter very the fluids. late situation time, is when then all identical the to flows the have one become of non- a collection of cold where the variables appendix. Su ce to note here that, unlike the case of a single pressureless flow, the Green with the condition pendent linear solutions of the system. with linear system. This operator satisfies JCAP08(2015)053 ]. ] ] ] JCAP Phys. SPIRE , ]. , (1969) 53 IN ,Ph.D. D 83 ][ 23 (2006) 307 SPIRE ]. IN ]. ][ 429 arXiv:1112.4940 [ arXiv:1110.3193 SPIRE Phys. Rev. CFHTLS weak-lensing arXiv:1209.3662 , SPIRE arXiv:0810.0555 [ , [ IN IN ]. ][ ][ Phys. Rev. Lett. arXiv:1005.2416 Neutrino masses and [ Phys. Rept. , Large-scale structure of the , (2012) 063509 (2002) 1. SPIRE IN ]. (2012) 081101 (2009) 657 ]. ][ 367 astro-ph/9506072 [ (2013) 043530 D 85 ]. 500 SPIRE D 85 SPIRE IN IN arXiv:1012.2868 D 87 [ ][ (2010) 083520 arXiv:1210.2194 ][ [ SPIRE Neutrino constraints from future nearly all-sky (1995) 7 Resummed propagators in multi-component Power spectra in the eikonal approximation IN Phys. Rep. ][ , Phys. Rev. , D 82 455 ]. ]. Euclid definition study report Phys. Rev. – 11 – , arXiv:1311.5487 [ Phys. Rev. (2011) 030 (2013) 026 , SPIRE SPIRE Neutrino mass constraint with the Sloan Digital Sky 03 Astron. Astrophys. IN IN Relativistic eikonal expansion 01 , (1994) 22. ][ ][ Phys. Rev. Relative velocity of dark matter and baryonic fluids and the arXiv:1412.2764 Memory of initial conditions in gravitational clustering arXiv:1411.0428 [ , [ Cosmological perturbation theory in the synchronous and Describing massive neutrinos in cosmology as a collection of Cosmological perturbation theory for streams of relativistic Massive neutrinos and cosmology A calculation of the full neutrino phase space in cold + hot dark ]. Astrophys. J. Higher-order massive neutrino perturbations in large-scale (2014) 030 429 JCAP ]. , JCAP , astro-ph/0509419 , The WiggleZ dark energy survey: cosmological neutrino mass [ 01 SPIRE SPIRE IN (2015) 046 IN (2015) 030 ][ JCAP ][ , 03 03 Signatures of the primordial universe in large-scale structure surveys Astrophys. J. arXiv:1006.4845 arXiv:0806.0971 [ [ , Flowing with time: a new approach to nonlinear cosmological perturbations (2006) 063520 collaboration, R. Laureijs et al., JCAP JCAP ]. ]. ]. ]. ]. , , D 73 (2008) 036 SPIRE SPIRE SPIRE SPIRE SPIRE IN IN arXiv:1109.3400 astro-ph/0603494 IN IN IN formation of the first structures with adiabatic and nonadiabatic[ modes Rev. Universe and cosmological perturbation theory [ independent flows particles cosmic fluids with the[ eikonal approximation structure thesis, Ecole Polytechnique & Institut de Physique CEA Th´eorique, Saclay, France (2012). conformal Newtonian gauges [ 10 constraints on the neutrino[ masses matter models cosmological parameters from a Euclid-likeincluding survey: theoretical errors Markov Chain Monte Carlo forecasts spectroscopic galaxy surveys EUCLID [ Survey power spectrum of(2011) luminous 043529 red galaxies and perturbation theory constraint from blue high-redshift galaxies [ F. Bernardeau, N. Van de Rijt and F. Vernizzi, M. Crocce and R. Scoccimarro, F. Bernardeau, S. Colombi, E. and Gazta˜naga R. Scoccimarro, H.D.I. Abarbanel and C. Itzykson, D. Tseliakhovich and C. Hirata, H. Dupuy and F. Bernardeau, F. Bernardeau, N. Van de Rijt and F. Vernizzi, F. and F¨uhrer Y.Y.Y. Wong, N. Van de Rijt, H. Dupuy and F. Bernardeau, C.-P. Ma and E. Bertschinger, J. Lesgourgues and S. Pastor, M. Pietroni, C.P. Ma and E. Bertschinger, I. Tereno, C. Schimd, J.-P. Uzan, M. Kilbinger, F.H. Vincent and L. Fu, C. Carbone, L. Verde, Y. Wang and A. Cimatti, S. Riemer-Sorensen et al., B. Audren, J. Lesgourgues, S. Bird, M.G. Haehnelt and M. Viel, S. Saito, M. Takada and A. Taruya, [9] [7] [8] [5] [6] [4] [22] [23] [19] [20] [21] [17] [18] [14] [15] [16] [11] [12] [13] [10] JCAP08(2015)053 D Phys. , Phys. Rev. , (2012) 103528 Mon. Not. Roy. , ]. D 86 SPIRE IN ][ ]. The onset of nonlinearity in Phys. Rev. , RegPT: direct and fast calculation of ,in ]. SPIRE IN ][ SPIRE ]. Cosmic propagators at two-loop order IN ][ arXiv:1006.4656 Multi-point propagators for non-gaussian initial [ SPIRE – 12 – IN ][ Renormalized cosmological perturbation theory astro-ph/9711187 (2010) 083507 ]. arXiv:1211.1571 [ . D 82 SPIRE IN astro-ph/0509418 ][ [ (1998) 1097 [ Transients from initial conditions: a perturbative analysis A new angle on gravitational clustering The evolution of the large-scale structure of the universe: beyond the linear 299 Phys. Rev. (2014) 023502 , , J.N. Fry et al. eds., New York Academy Sciences Annals volume 927, U.S.A. (2001). arXiv:1311.2724 , D 89 (2006) 063519 arXiv:1208.1191 regularized cosmological power spectrum at[ two-loop order cosmology 73 Rev. conditions regime Astron. Soc. A. Taruya, F. Bernardeau, T. Nishimichi and S. Codis, M. Crocce and R. Scoccimarro, F. Bernardeau, A. Taruya, and T. Nishimichi, F. Bernardeau, R. Scoccimarro, R. Scoccimarro, F. Bernardeau, M. Crocce and E. Sefusatti, [30] [28] [29] [25] [26] [27] [24] Conclusions and perspectives
This thesis, “Precision cosmology with the large-scale structure of the universe”, provides innovative results of different types. What they have in common is the quest for precision in the description of the physical phenomena at work in the universe. First, a toy model mimicking the propagation of light in an inhomogeneous spacetime has been presented. In this study, we chose a traditional Swiss-cheese representation. Often used in the litterature, such models offer the advantage of dealing with exact solutions of the Einstein equations, which do not affect the global dynamics of the universe while making it strongly inhomogeneous. A new method, mostly analytic, is proposed by us to investigate the impact of brutal fluctuations of density on Hubble diagrams. It relies on the introduction of Wronski matrices, which permit to determine the cumulative effect of several holes without difficulty when solving the geodesic and Sachs equations. We also derived an approximate luminosity-redshift relation for Swiss-cheese models, which turned out to be simi- lar to the formula used in Dyer-Roeder approaches. By generating mock Hubble diagrams corresponding to the patchy spacetime, we exemplified how initial pre- sumptions, such as the cosmological principle, can alter scientific conclusions, such as the estimation of cosmological parameters from Hubble diagrams. In general, the bias that an insertion of holes causes in Hubble diagrams can be large, due to a high sensitivity of luminosity distances with respect to the geometry of the medium. However, this effect has proven to decrease a lot with the cosmologi- cal constant, which is fortunate since observations clearly lean towards a universe content dominated by dark energy. By its extreme simplicity, our model (whose fluctuations of density are exaggeratedly abrupt), probably overestimates the bias. Indeed, large-scale structures, intergalactic gas, dark matter, etc. are likely to act as “smoothers” in the real universe. Compensation effects are undeniable. Yet
188 we are convinced that, in precision cosmology, it is worth examining this question minutely instead of crudely ignoring the mismatch between the scales at which the cosmological principle is valid and the scales probed by the standard candles used to build Hubble diagrams. Let us not forget that cosmological parameters are highly model-dependent so measuring them with an exquisite precision is not sufficient to attest that their values are representative of reality. In the same spirit, I have presented in this manuscript a study in which Planck results on the one hand and observations of the SNLS 3 catalog on the other hand are interpreted assuming that the universe has a Swiss-cheese geometry. What mo- tivated this investigation is the fact that the estimation of cosmological parameters from CMB and/or BAO (for instance observed by Planck) are not always con- sistent with that obtained from observational Hubble diagrams (for instance built from the SNLS 3 catalog). We have shown that, if one considers spacetimes that are particularly lumpy, the estimation of the cosmological parameter Ωm from Hubble diagrams is very different from what it is when a model with a Friedmann-Lemaˆıtre spacetime is used. It can even be reconciled with the Planck estimation. Of course, such lumpy Swiss-cheese spacetimes are not realistic representations of the universe. What we wanted to point out is that the discrepancy might be due to the fact that those kinds of observations probe very different angular scales and are nevertheless interpreted with the same geometrical model. Estimations of the cosmological pa- rameters being now very precise, a geometrical modeling which adapts to the scales considered should be opportune, or even necessary. Our approach has scope for improvement in many ways: introducing holes with different sizes to get more liberty regarding the proportion of underdense regions, replacing photons by other particles, generating mock observations other than Hub- ble diagrams, making it more realistic by taking into account the real structure of the universe, etc. Actually, during my PhD, I specialized in another field of cosmol- ogy, namely the study of the large-scale structure of the universe with the help of perturbation theory. However, my collaborator Pierre Fleury devoted a major part of his PhD thesis (completed also this year) to light propagation in inhomogeneous and/or anisotropic spacetimes (see in particular [78], [75]). The major result exposed in this thesis is the proposition of a new way of deal- ing with the neutrino component in cosmology. The idea is to decompose neutrinos into several single-flow fluids in order to get rid of velocity dispersion in each of them. Nonlinear equations of motion have been derived from conservation laws
189 in a generic perturbed Friedman-Lemaˆıtremetric, as well as formulae allowing to recover the usual physical quantities by summing over all the flows of the collection. Using those formulae to deduce the global energy density, velocity divergence and shear stress of massive neutrinos from our equations, we proved that, in the linear regime, our approach is consistent with the integration of the Boltzmann hierarchy. It required in particular to infer the individual initial conditions from the knowledge of the global initial distribution function of neutrinos. We did it explicitly for adi- abatic initial conditions. Similarly to the standard approach describing the growth of perturbations in cold pressureless fluids, our method breaks down when shell crossing emerges. We argue that this phenomenon is likely to occur when neutrinos have become slow enough to be considered as cold. It means that, potentially, our work can also take advantage of the efforts realized in order to model shell crossing in cold fluids. We think that this multi-fluid description provides a more convenient basis to investigate the nonlinear behavior of neutrinos than the Boltzmann hier- archy. Moreover, it provides an additional information since the flows of neutrinos can be followed individually. It can be applied not only to massive neutrinos but, more generally, to any non-interacting species. In this context, cold dark matter appears as a mere fluid of the collection, whose particularity is to have a zero initial momentum. Any initial velocity (relativistic or not) and any mass can be handled by this formalism. On this basis, we carried out a study devoted to the exploration of the coupling structure of our equations on subhorizon scales. The interest of this approximation is that it makes useful properties emerge while preserving the coupling terms that are relevant to depict the nonlinear growth of perturbations involved in structure formation. In particular, on subhorizon scales, the momentum fields that we chose as variables can be written as gradients in any gauge and at any order in pertur- bation theory. This key result is the starting point of several developments. First, taking inspiration of the description of cold dark matter, we introduced one doublet velocity divergence/density contrast per flow. It allowed us to gather in a unique nonlinear equation the growth of perturbations in all non-interacting fluids. The great advantage of this equation is that it is formally the same as the one derived from the Vlasov-Poisson system (on which most of the results of nonlinear pertur- bation theory rely) but with a much wider range of application. For this reason, our approach will hopefully allow to make one further step towards a relativistic generalization of nonlinear perturbation theory. We deliberately worked in arbi-
190 trary gauges in order to facilitate the demonstration of invariance properties. We therefore easily managed to generalize the so-called extended Galilean invariance. As for cold fluids, it has been useful to derive consistency relations and to study couplings between short-wave modes and long-wave modes with the help of the eikonal approximation. We devoted a study specifically to the implementation of the eikonal approx- imation in our equations of motion. The aim was to write explicitly the eikonal corrections associated with each flow and to interpret them as sources of peculiar displacements. Taking the cold-dark-matter fluid as a benchmark, we showed that, for large wavenumbers and high initial velocities, the fields characterizing neutrinos can be phase-shifted with respect to the cold-dark-matter ones during a substantial amount of time. Eventually, this postpones the fall of neutrinos into potential wells and is therefore expected to damp the growth of structure. It would of course be interesting to examine carefully the impact it can have on the nonlinear matter power spectrum. However, doing so is a long-term project, which is still at its preliminary stage. In general, a considerable difficulty related to the study of the impact of nonlinear couplings on a given dynamics is the fact that it is not easy to forecast which couplings will cancel each other out. Thanks to our Lagrangian-like decomposition, there is no risk of missing cancellations of this type here. The most evident perspectives offered by the multi-fluid point of view that we adopted are the following. Formally, standard resummation techniques allowing to compute the matter power spectrum at NLO can be extended to the global equation that we derived. However, the associated formalism is very heavy so replacing brutally the usual 2 2 matrices by N N ones, which have moreover non-zero scale- × × dependent imaginary parts, would be prohibitive. The continuation will therefore consist in finding arguments to reduce the complexity of the problem. On the one hand, it will be crucial to examine the scales at which both nonlinearities and relativistic corrections must be taken into account. The last study presented in this manuscript is a first step to address this. In practice, since non-linearities emerge at late times whereas the highest velocities concern early times, many simplifications can be expected. On the other hand, determining the number of flows necessary for the description to be accurate (in the sense of precision cosmology), will be decisive. With Julien Lesgourgues, I started to implement the equations of motion of the multi-fluid approach in his code CLASS. The results we will obtain should be helpful to test efficiently the numerical requirements and to envisage well-controlled
191 simplifications.
192 Acknowledgments
Note: below, the same in French.
Working under the supervision of Francis has been a pleasure. First, I thank him for having trusted me three years ago. Beginning in research alongside him has been very insightful. In the future, I will try to take inspiration from his way of dealing with scientific issues, i.e. using intuition and not being scared by conceptual or technical difficulties. I also want to thank him for his patience towards my questions (which are frequent, especially because I often don’t manage to follow him!). Before starting my PhD, I had the privilege of having two other remarkable advi- sors. Alain Lecavelier des Etangs´ and Jean-Philippe Uzan have greatly contributed to my interest in research. Even today, they always make sure that everything goes well for me. Besides, they are available whenever I need them. I thank them warmly for that. By the way, I thank Fr´ed´ericDaigne for having suggested to Alain, and then to Jean-Philippe, to offer me an internship and for his kindness. I am particularly grateful to Christian Fr`ere,who has so much spoken about astrophysics when I was his student that this topic ended up really intriguing me. I thank also the members of the astronomy group of the Palais de la d´ecouverte. Thanks to them, I had the opportunity to practice scientific dissemination in an amazing environment. I enjoyed preparing the talk “An overview of modern cos- mology”, presenting it and answer questions of curious people who came to listen to me. I have the good fortune to add to the list of thanks the L’Oreal fundation. In 2014, I have indeed been awarded the L’Or´eal-UNESCOFor Women in Science prize. In addition to financial support, this made me i) meet highschool students in order to promote science amongst them, ii) have my academic carrer highlighted in the media, iii) meet Claudie Haigner´e,the first French woman in space, iv) have
193 breakfast at the Ministry in the company of brilliant scientific... and even be elected Picard of the year! I thank the people I worked with during my saty at CERN for their warm welcome, for their interest in my work and for giving me plenty of scientific avenues to be explored. I thank the PhD students with whom I became friend (especially Lucile and Julien, of course), who made those three years even more enjoyable. I am grateful to my very-stimulating family members. I thank especially my two brothers (who think it’s fun to see their sister becoming a cosmologist), Christine and Eric´ (who accept to have a daughter-in-law who sometimes makes calculations while chatting with them) and my parents (who have encouraged me since the beginning, I am delighted to make them proud of me). Out of category, Jordane. Marrying a PhD student is brave! Leaving work to follow her when she gets a postdoctoral position too. Thank you for that and for everything else. Finally, I thank the members of the jury for having added the examination of my PhD thesis to their busy schedule.
194 Remerciements
Travailler sous la direction de Francis a ´et´eun plaisir. Premi`erement, je le remercie de m’avoir fait confiance il y a trois ans. D´ebuterdans la recherche `ases cˆot´esa ´et´e tr`esinstructif pour moi. J’essaierai par la suite de m’inspirer de sa fa¸cond’aborder les probl`emesscientifiques, `asavoir faire preuve d’intuition et ne pas ˆetreeffray´e par les difficult´esconceptuelles ou techniques. Je tiens aussi `alui dire merci pour sa patience face `ames questions, fr´equentes (notamment parce que j’ai souvent du mal `ale suivre !). Avant de d´ebuter ma th`ese, j’ai eu le privil`eged’avoir deux autres remarquables encadrants. Alain Lecavelier des Etangs´ et Jean-Philippe Uzan ont tr`eslargement contribu´e`amon attraˆıt pour la recherche. Encore aujourd’hui, ils veillent `ace que tout se passe bien pour moi et se montrent tr`esdisponibles `achaque fois que j’ai besoin d’eux. Je les en remercie chaleureusement. Merci au passage `aFr´ed´eric Daigne pour avoir sugg´er´e`aAlain puis `aJean-Philippe de me proposer un stage et aussi pour sa bienveillance. Je suis particuli`erement reconnaissante envers Christian Fr`ere,qui m’a tellement parl´ed’astrophysique lorsque j’´etaisson ´el`eve que cette discipline a v´eritablement fini par m’intriguer. Un merci aussi pour les membres de la section d’astronomie du Palais de la d´ecouverte. Grˆace`aeux, j’ai pu m’essayer `ala vulgarisation scientifique dans un cadre exceptionnel. J’ai pris ´enorm´ement de plaisir `apr´eparerl’expos´e Un aper¸cu de la cosmologie moderne, `ale pr´esenter et `ar´epondre aux questions de tous les curieux qui sont venus m’´ecouter. J’ai la chance de pouvoir ajouter `ala liste des remerci´esla fondation L’Or´eal. J’ai en effet re¸cuen 2014 la bourse L’Or´eal-UNESCOpour les femmes et la science. En plus de l’aide financi`ere,cette r´ecompense m’a permis d’aller `ala rencontre des coll´egienset lyc´eenspour promouvoir la science aupr`esd’eux ; de voir mon parcours
195 mis en avant dans les m´edias; de rencontrer Claudie Haigner´e,premi`erefemme fran¸caise `aˆetreall´eedans l’espace ; de participer `aun petit-d´ejeunerau minist`ere en compagnie de brillantes scientifiques... et mˆemed’ˆetre´elue Picard de l’ann´ee ! Merci aux personnes avec lesquelles j’ai eu l’occasion de travailler lors de ma visite au CERN pour leur accueil chaleureux, pour l’int´erˆetqu’ils portent `amon travail et pour m’avoir donn´eun tas de pistes de recherche `aexplorer. Je remercie les doctorants avec lesquels je suis devenue amie (en particulier Lucile et Julien ´evidemment), qui ont rendu ces trois ann´eesencore plus agr´eables. Bien sˆurje suis tr`esreconnaissante envers les tr`esstimulants membres de ma famille. Merci notamment `ames deux fr`eres(qui rigolent bien de voir leur sœur devenir cosmologiste), `aChristine et Eric´ (qui acceptent d’avoir une belle-fille qui fait parfois des calculs en mˆemetemps qu’elle discute avec eux) ainsi qu’`ames parents (qui m’encouragent depuis le d´ebutet que je suis extrˆemement heureuse de rendre fiers). Hors cat´egorie,Jordane. Se marier avec une doctorante, c’est courageux ! Quitter son travail pour la suivre lorsqu’elle part en post-doctorat aussi. Merci pour cela et pour tout le reste. Enfin, merci aux membres du jury pour avoir ajout´el’examen de ma th`ese `a leur emploi du temps d´ej`abien charg´e.
196 Bibliography
[1] Henry D.I. Abarbanel and Claude Itzykson. Relativistic eikonal expansion. Phys.Rev.Lett., 23:53, 1969.
[2] R. Amanullah, C. Lidman, D. Rubin, G. Aldering, P. Astier, K. Barbary, M. S. Burns, A. Conley, K. S. Dawson, S. E. Deustua, M. Doi, S. Fabbro, L. Fac- cioli, H. K. Fakhouri, G. Folatelli, A. S. Fruchter, H. Furusawa, G. Garavini, G. Goldhaber, A. Goobar, D. E. Groom, I. Hook, D. A. Howell, N. Kashikawa, A. G. Kim, R. A. Knop, M. Kowalski, E. Linder, J. Meyers, T. Morokuma, S. Nobili, J. Nordin, P. E. Nugent, L. Ostman,¨ R. Pain, N. Panagia, S. Perl- mutter, J. Raux, P. Ruiz-Lapuente, A. L. Spadafora, M. Strovink, N. Suzuki, L. Wang, W. M. Wood-Vasey, N. Yasuda, and T. Supernova Cosmology Project. Spectra and Hubble Space Telescope Light Curves of Six Type Ia Su- pernovae at 0.511 < z < 1.12 and the Union2 Compilation. ApJ, 716:712–738, June 2010.
[3] S. Anselmi, D. L´opez Nacir, and E. Sefusatti. Nonlinear effects of dark energy clustering beyond the acoustic scales. JCAP, 7:13, July 2014.
[4] M. Archidiacono, E. Calabrese, and A. Melchiorri. Case for dark radiation. Phys. Rev. D, 84(12):123008, December 2011.
[5] E.´ Aubourg, S. Bailey, J. E. Bautista, F. Beutler, V. Bhardwaj, D. Bizyaev, M. Blanton, M. Blomqvist, A. S. Bolton, J. Bovy, H. Brewington, J. Brinkmann, J. R. Brownstein, A. Burden, N. G. Busca, W. Carithers, C.-H. Chuang, J. Comparat, A. J. Cuesta, K. S. Dawson, T. Delubac, D. J. Eisenstein, A. Font-Ribera, J. Ge, J.-M. Le Goff, S. G. A Gontcho, J. R. Gott, III, J. E. Gunn, H. Guo, J. Guy, J.-C. Hamilton, S. Ho, K. Honscheid, C. Howlett, D. Kirkby, F. S. Kitaura, J.-P. Kneib, K.-G. Lee, D. Long, R. H.
197 Bibliography
Lupton, M. Vargas Maga˜na,V. Malanushenko, E. Malanushenko, M. Man- era, C. Maraston, D. Margala, C. K. McBride, J. Miralda-Escud´e,A. D. My- ers, R. C. Nichol, P. Noterdaeme, S. E. Nuza, M. D. Olmstead, D. Oravetz, I. Pˆaris,N. Padmanabhan, N. Palanque-Delabrouille, K. Pan, M. Pellejero- Ibanez, W. J. Percival, P. Petitjean, M. M. Pieri, F. Prada, B. Reid, N. A. Roe, A. J. Ross, N. P. Ross, G. Rossi, J. A. Rubi˜no-Mart´ın,A. G. S´anchez, L. Samushia, R. Tanaus´uG´enova Santos, C. G. Sc´occola,D. J. Schlegel, D. P. Schneider, H.-J. Seo, E. Sheldon, A. Simmons, R. A. Skibba, A. Slosar, M. A. Strauss, D. Thomas, J. L. Tinker, R. Tojeiro, J. A. Vazquez, M. Viel, D. A. Wake, B. A. Weaver, D. H. Weinberg, W. M. Wood-Vasey, C. Y`eche, I. Ze- havi, and G.-B. Zhao. Cosmological implications of baryon acoustic oscillation (BAO) measurements. ArXiv e-prints, November 2014.
[6] J. M. Bardeen. Gauge-Invariant Cosmological Perturbations, page 563. 1988.
[7] J. M. Bardeen, P. J. Steinhardt, and M. S. Turner. Spontaneous creation of almost scale-free density perturbations in an inflationary universe. Phys. Rev. D, 28:679–693, August 1983.
[8] V. Barger, D. Marfatia, and K. Whisnant. Progress in the Physics of Massive Neutrinos. International Journal of Modern Physics E, 12:569–647, 2003.
[9] S. Bashinsky and U. Seljak. Signatures of relativistic neutrinos in CMB anisotropy and matter clustering. Phys. Rev. D, 69(8):083002, April 2004.
[10] C. L. Bennett, D. Larson, J. L. Weiland, N. Jarosik, G. Hinshaw, N. Odegard, K. M. Smith, R. S. Hill, B. Gold, M. Halpern, E. Komatsu, M. R. Nolta, L. Page, D. N. Spergel, E. Wollack, J. Dunkley, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, and E. L. Wright. Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results. ApJS, 208:20, October 2013.
[11] A. Benoˆıt, P. Ade, A. Amblard, R. Ansari, E.´ Aubourg, S. Bargot, J. G. Bartlett, J.-P. Bernard, R. S. Bhatia, A. Blanchard, J. J. Bock, A. Boscaleri, F. R. Bouchet, A. Bourrachot, P. Camus, F. Couchot, P. de Bernardis, J. De- labrouille, F.-X. D´esert,O. Dor´e,M. Douspis, L. Dumoulin, X. Dupac, P. Fil- liatre, P. Fosalba, K. Ganga, F. Gannaway, B. Gautier, M. Giard, Y. Giraud- H´eraud,R. Gispert, L. Guglielmi, J.-C. Hamilton, S. Hanany, S. Henrot-
198 Bibliography
Versill´e,J. Kaplan, G. Lagache, J.-M. Lamarre, A. E. Lange, J. F. Mac´ıas- P´erez,K. Madet, B. Maffei, C. Magneville, D. P. Marrone, S. Masi, F. Mayet, A. Murphy, F. Naraghi, F. Nati, G. Patanchon, G. Perrin, M. Piat, N. Pon- thieu, S. Prunet, J.-L. Puget, C. Renault, C. Rosset, D. Santos, A. Starobin- sky, I. Strukov, R. V. Sudiwala, R. Teyssier, M. Tristram, C. Tucker, J.-C. Vanel, D. Vibert, E. Wakui, and D. Yvon. The cosmic microwave back- ground anisotropy power spectrum measured by Archeops. A&A, 399:L19– L23, March 2003.
[12] F. Bernardeau. The evolution of the large-scale structure of the universe: beyond the linear regime. ArXiv e-prints, November 2013.
[13] F. Bernardeau, S. Colombi, E. Gazta˜naga,and R. Scoccimarro. Large-scale structure of the Universe and cosmological perturbation theory. Phys. Rep., 367:1–248, September 2002.
[14] F. Bernardeau, M. Crocce, and R. Scoccimarro. Constructing regularized cosmic propagators. Phys. Rev. D, 85(12):123519, June 2012.
[15] F. Bernardeau, A. Taruya, and T. Nishimichi. Cosmic propagators at two- loop order. Phys. Rev. D, 89(2):023502, January 2014.
[16] F. Bernardeau, N. van de Rijt, and F. Vernizzi. Resummed propagators in multicomponent cosmic fluids with the eikonal approximation. Phys. Rev. D, 85(6):063509, March 2012.
[17] F. Bernardeau, N. Van de Rijt, and F. Vernizzi. Power spectra in the eikonal approximation with adiabatic and nonadiabatic modes. Phys. Rev. D, 87(4):043530, February 2013.
[18] J. Bernstein. Kinetic theory in the expanding universe. 1988.
[19] B. Bertotti. The Luminosity of Distant Galaxies. Royal Society of London Proceedings Series A, 294:195–207, September 1966.
[20] M. Betoule, R. Kessler, J. Guy, J. Mosher, D. Hardin, R. Biswas, P. Astier, P. El-Hage, M. Konig, S. Kuhlmann, J. Marriner, R. Pain, N. Regnault, C. Balland, B. A. Bassett, P. J. Brown, H. Campbell, R. G. Carlberg, F. Cellier-Holzem, D. Cinabro, A. Conley, C. B. D’Andrea, D. L. DePoy,
199 Bibliography
M. Doi, R. S. Ellis, S. Fabbro, A. V. Filippenko, R. J. Foley, J. A. Frieman, D. Fouchez, L. Galbany, A. Goobar, R. R. Gupta, G. J. Hill, R. Hlozek, C. J. Hogan, I. M. Hook, D. A. Howell, S. W. Jha, L. Le Guillou, G. Leloudas, C. Lidman, J. L. Marshall, A. M¨oller,A. M. Mour˜ao,J. Neveu, R. Nichol, M. D. Olmstead, N. Palanque-Delabrouille, S. Perlmutter, J. L. Prieto, C. J. Pritchet, M. Richmond, A. G. Riess, V. Ruhlmann-Kleider, M. Sako, K. Schahmaneche, D. P. Schneider, M. Smith, J. Sollerman, M. Sullivan, N. A. Walton, and C. J. Wheeler. Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples. A&A, 568:A22, August 2014.
[21] S. M. Bilenky, C. Giunti, J. A. Grifols, and E. Mass´o. Absolute values of neutrino masses: status and prospects. Phys. Rep., 379:69–148, May 2003.
[22] J. Binney and S. Tremaine. Galactic Dynamics: Second Edition. Princeton University Press, 2008.
[23] S. Bird, M. Viel, and M. G. Haehnelt. Massive neutrinos and the non-linear matter power spectrum. MNRAS, 420:2551–2561, March 2012.
[24] T. Biswas and A. Notari. ’Swiss-cheese’ inhomogeneous cosmology and the dark energy problem. JCAP, 6:21, June 2008.
[25] D. Blas, M. Garny, and T. Konstandin. On the non-linear scale of cosmolog- ical perturbation theory. JCAP, 9:24, September 2013.
[26] D. Blas, M. Garny, and T. Konstandin. Cosmological perturbation theory at three-loop order. JCAP, 1:10, January 2014.
[27] D. Blas, M. Garny, T. Konstandin, and J. Lesgourgues. Structure formation with massive neutrinos: going beyond linear theory. ArXiv e-prints, August 2014.
[28] D. Blas, J. Lesgourgues, and T. Tram. The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes. JCAP, 7:34, July 2011.
[29] J. Brandbyge, S. Hannestad, T. Haugbølle, and B. Thomsen. The effect of thermal neutrino motion on the non-linear cosmological matter power spec- trum. JCAP, 8:20, August 2008.
200 Bibliography
[30] R. Brandenberger, R. Kahn, and W. H. Press. Cosmological perturbations in the early universe. Phys. Rev. D, 28:1809–1821, October 1983.
[31] N. Brouzakis, N. Tetradis, and E. Tzavara. The effect of large scale inhomo- geneities on the luminosity distance. JCAP, 2:13, February 2007.
[32] N. Brouzakis, N. Tetradis, and E. Tzavara. Light propagation and large-scale inhomogeneities. JCAP, 4:8, April 2008.
[33] C. Clarkson, G. F. R. Ellis, A. Faltenbacher, R. Maartens, O. Umeh, and J.-P. Uzan. (Mis)interpreting supernovae observations in a lumpy universe. MNRAS, 426:1121–1136, October 2012.
[34] B. T. Cleveland, T. Daily, R. Davis, Jr., J. R. Distel, K. Lande, C. K. Lee, P. S. Wildenhain, and J. Ullman. Measurement of the Solar Electron Neutrino Flux with the Homestake Chlorine Detector. ApJ, 496:505–526, March 1998.
[35] T. Clifton and J. Zuntz. Hubble diagram dispersion from large-scale structure. MNRAS, 400:2185–2199, December 2009.
[36] A. Coc, J.-P. Uzan, and E. Vangioni. Standard big bang nucleosynthesis and primordial CNO abundances after Planck. JCAP, 10:50, October 2014.
[37] S. Colombi. Vlasov-Poisson in 1D for initially cold systems: post-collapse Lagrangian perturbation theory. MNRAS, 446:2902–2920, January 2015.
[38] A. Conley, J. Guy, M. Sullivan, N. Regnault, P. Astier, C. Balland, S. Basa, R. G. Carlberg, D. Fouchez, D. Hardin, I. M. Hook, D. A. How- ell, R. Pain, N. Palanque-Delabrouille, K. M. Perrett, C. J. Pritchet, J. Rich, V. Ruhlmann-Kleider, D. Balam, S. Baumont, R. S. Ellis, S. Fabbro, H. K. Fakhouri, N. Fourmanoit, S. Gonz´alez-Gait´an,M. L. Graham, M. J. Hudson, E. Hsiao, T. Kronborg, C. Lidman, A. M. Mourao, J. D. Neill, S. Perlmutter, P. Ripoche, N. Suzuki, and E. S. Walker. Supernova Constraints and Sys- tematic Uncertainties from the First Three Years of the Supernova Legacy Survey. ApJS, 192:1, January 2011.
[39] A. Cooray, D. E. Holz, and D. Huterer. Cosmology from Supernova Magnifi- cation Maps. ApJL, 637:L77–L80, February 2006.
201 Bibliography
[40] L. Cosmai, G. Fanizza, M. Gasperini, and L. Tedesco. Discriminating different models of luminosity-redshift distribution. Classical and Quantum Gravity, 30(9):095011, May 2013.
[41] C. L. Cowan, Jr., F. Reines, F. B. Harrison, H. W. Kruse, and A. D. McGuire. Detection of the Free Neutrino: A Confirmation. Science, 124:103–104, July 1956.
[42] P. Creminelli, J. Gleyzes, L. Hui, M. Simonovi´c,and F. Vernizzi. Single-field consistency relations of large scale structure part III: test of the equivalence principle. JCAP, 6:9, June 2014.
[43] P. Creminelli, J. Gleyzes, M. Simonovi´c,and F. Vernizzi. Single-field con- sistency relations of large scale structure part II: resummation and redshift space. JCAP, 2:51, February 2014.
[44] P. Creminelli, J. Nore˜na,M. Simonovi´c,and F. Vernizzi. Single-field consis- tency relations of large scale structure. JCAP, 12:25, December 2013.
[45] M. Crocce and R. Scoccimarro. Renormalized cosmological perturbation the- ory. Phys. Rev. D, 73(6):063519, March 2006.
[46] M. Crocce, R. Scoccimarro, and F. Bernardeau. MPTBREEZE: a fast renor- malized perturbative scheme. MNRAS, 427:2537–2551, December 2012.
[47] R. H. Cyburt, B. D. Fields, and K. A. Olive. Primordial nucleosynthesis with CMB inputs: probing the early universe and light element astrophysics. Astroparticle Physics, 17:87–100, April 2002.
[48] N. Dalal, D. E. Holz, X. Chen, and J. A. Frieman. Corrective Lenses for High-Redshift Supernovae. ApJL, 585:L11–L14, March 2003.
[49] G. D’Amico and E. Sefusatti. The nonlinear power spectrum in clustering quintessence cosmologies. JCAP, 11:13, November 2011.
[50] V. M. Dashevskii and V. I. Slysh. On the Propagation of Light in a Nonho- mogeneous Universe. Soviet Astr., 9:671, February 1966.
[51] F. De Bernardis, L. Pagano, P. Serra, A. Melchiorri, and A. Cooray. Anisotropies in the cosmic neutrino background after Wilkinson Microwave Anisotropy Probe five-year data. JCAP, 6:13, June 2008.
202 Bibliography
[52] P. de Bernardis, P. A. R. Ade, J. J. Bock, J. R. Bond, J. Borrill, A. Bosca- leri, K. Coble, B. P. Crill, G. De Gasperis, P. C. Farese, P. G. Ferreira, K. Ganga, M. Giacometti, E. Hivon, V. V. Hristov, A. Iacoangeli, A. H. Jaffe, A. E. Lange, L. Martinis, S. Masi, P. V. Mason, P. D. Mauskopf, A. Melchiorri, L. Miglio, T. Montroy, C. B. Netterfield, E. Pascale, F. Piacen- tini, D. Pogosyan, S. Prunet, S. Rao, G. Romeo, J. E. Ruhl, F. Scaramuzzi, D. Sforna, and N. Vittorio. A flat Universe from high-resolution maps of the cosmic microwave background radiation. Nature, 404:955–959, April 2000.
[53] W. de Sitter. Einstein’s theory of gravitation and its astronomical conse- quences. Third paper. MNRAS, 78:3–28, November 1917.
[54] S. Dodelson and A. Vallinotto. Learning from the scatter in type Ia super- novae. Phys. Rev. D, 74(6):063515, September 2006.
[55] M. Doran. CMBEASY: an object oriented code for the cosmic microwave background. JCAP, 10:11, October 2005.
[56] Y. Dubois, C. Pichon, C. Welker, D. Le Borgne, J. Devriendt, C. Laigle, S. Codis, D. Pogosyan, S. Arnouts, K. Benabed, E. Bertin, J. Blaizot, F. Bouchet, J.-F. Cardoso, S. Colombi, V. de Lapparent, V. Desjacques, R. Gavazzi, S. Kassin, T. Kimm, H. McCracken, B. Milliard, S. Peirani, S. Prunet, S. Rouberol, J. Silk, A. Slyz, T. Sousbie, R. Teyssier, L. Tresse, M. Treyer, D. Vibert, and M. Volonteri. Dancing in the dark: galactic proper- ties trace spin swings along the cosmic web. MNRAS, 444:1453–1468, October 2014.
[57] H. Dupuy and F. Bernardeau. Describing massive neutrinos in cosmology as a collection of independent flows. JCAP, 1:30, January 2014.
[58] H. Dupuy and F. Bernardeau. Cosmological Perturbation Theory for streams of relativistic particles. JCAP, 3:30, March 2015.
[59] H. Dupuy and F. Bernardeau. On the importance of nonlinear couplings in large-scale neutrino streams. ArXiv e-prints, March 2015.
[60] R. Durrer. Gauge invariant cosmological perturbation theory. A general study and its application to the texture scenario of structure formation. 1994.
203 Bibliography
[61] C. C. Dyer and R. C. Roeder. The Distance-Redshift Relation for Universes with no Intergalactic Medium. ApJL, 174:L115, June 1972.
[62] C. C. Dyer and R. C. Roeder. Observations in Locally Inhomogeneous Cos- mological Models. ApJ, 189:167–176, April 1974.
[63] G. Efstathiou. Cosmological perturbations. In J. A. Peacock, A. F. Heavens, and A. T. Davies, editors, Physics of the Early Universe, pages 361–463, 1990.
[64] A. Einstein. Die Grundlage der allgemeinen Relativit¨atstheorie. Annalen der Physik, 354:769–822, 1916.
[65] A. Einstein and E. G. Straus. The Influence of the Expansion of Space on the Gravitation Fields Surrounding the Individual Stars. Reviews of Modern Physics, 17:120–124, April 1945.
[66] S. R. Elliott and P. Vogel. Double beta decay. Annual Review of Nuclear and Particle Science, 52:115–151, 2002.
[67] G. F. R. Ellis. Cosmology and Verifiability, page 113. 1990.
[68] G. F. R. Ellis and T. Buchert. The universe seen at different scales [rapid communication]. Physics Letters A, 347:38–46, November 2005.
[69] G. F. R. Ellis and R. Goswami. Variations on Birkhoff’s theorem. General Relativity and Gravitation, 45:2123–2142, November 2013.
[70] A. Enea Romano. Conditions for low-redshift positive apparent acceleration in smooth inhomogeneous models. ArXiv e-prints, June 2012.
[71] AbazajianK. N. et al.. Light Sterile Neutrinos: A White Paper. ArXiv e- prints, April 2012.
[72] G. Fanizza and F. Nugier. Lensing in the geodesic light-cone coordinates and its (exact) illustration to an off-center observer in Lema#523tre-Tolman- Bondi models. JCAP, 2:2, February 2015.
[73] G. Fanizza and L. Tedesco. Inhomogeneous and anisotropic Universe and apparent acceleration. Phys. Rev. D, 91(2):023006, January 2015.
204 Bibliography
[74] E.´ E.´ Flanagan, N. Kumar, I. Wasserman, and R. A. Vanderveld. Luminosity distance in “Swiss cheese” cosmology with randomized voids. II. Magnification probability distributions. Phys. Rev. D, 85(2):023510, January 2012.
[75] P. Fleury. Swiss-cheese models and the Dyer-Roeder approximation. JCAP, 6:54, June 2014.
[76] P. Fleury, H. Dupuy, and J.-P. Uzan. Can All Cosmological Observations Be Accurately Interpreted with a Unique Geometry? Physical Review Letters, 111(9):091302, August 2013.
[77] P. Fleury, H. Dupuy, and J.-P. Uzan. Interpretation of the Hubble diagram in a nonhomogeneous universe. Phys. Rev. D, 87(12):123526, June 2013.
[78] P. Fleury, C. Pitrou, and J.-P. Uzan. Light propagation in a homogeneous and anisotropic universe. Phys. Rev. D, 91(4):043511, February 2015.
[79] A. Friedmann. Uber¨ die Kr¨ummung des Raumes. Zeitschrift fur Physik, 10:377–386, 1922.
[80] J. A. Frieman. Weak Lensing and the Measurement of q0; from Type Ia Supernovae. Comments on Astrophysics, 18:323, 1996.
[81] F. F¨uhrerand Y. Y. Y. Wong. Higher-order massive neutrino perturbations in large-scale structure. JCAP, 3:46, March 2015.
[82] M. H. Goroff, B. Grinstein, S.-J. Rey, and M. B. Wise. Coupling of modes of cosmological mass density fluctuations. ApJ, 311:6–14, December 1986.
[83] J. E. Gunn. A Fundamental Limitation on the Accuracy of Angular Measure- ments in Observational Cosmology. ApJ, 147:61, January 1967.
[84] A. H. Guth. Inflationary universe: A possible solution to the horizon and flatness problems. Phys. Rev. D, 23:347–356, January 1981.
[85] A. H. Guth and S.-Y. Pi. Fluctuations in the new inflationary universe. Physical Review Letters, 49:1110–1113, October 1982.
[86] J. Guy, M. Sullivan, A. Conley, N. Regnault, P. Astier, C. Balland, S. Basa, R. G. Carlberg, D. Fouchez, D. Hardin, I. M. Hook, D. A. How- ell, R. Pain, N. Palanque-Delabrouille, K. M. Perrett, C. J. Pritchet, J. Rich,
205 Bibliography
V. Ruhlmann-Kleider, D. Balam, S. Baumont, R. S. Ellis, S. Fabbro, H. K. Fakhouri, N. Fourmanoit, S. Gonz´alez-Gait´an, M. L. Graham, E. Hsiao, T. Kronborg, C. Lidman, A. M. Mourao, S. Perlmutter, P. Ripoche, N. Suzuki, and E. S. Walker. The Supernova Legacy Survey 3-year sample: Type Ia su- pernovae photometric distances and cosmological constraints. A&A, 523:A7, November 2010.
[87] O. Hahn, R. E. Angulo, and T. Abel. The Properties of Cosmic Velocity Fields. ArXiv e-prints, April 2014.
[88] S. Hannestad, T. Haugbølle, and C. Schultz. Neutrinos in non-linear structure formation: a simple SPH approach. JCAP, 2:45, February 2012.
[89] S. W. Hawking. The development of irregularities in a single bubble infla- tionary universe. Physics Letters B, 115:295–297, September 1982.
[90] G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunk- ley, M. R. Nolta, M. Halpern, R. S. Hill, N. Odegard, L. Page, K. M. Smith, J. L. Weiland, B. Gold, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack, and E. L. Wright. Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results. ApJS, 208:19, October 2013.
[91] C. J. Hogan, N. Kaiser, and M. J. Rees. Interpretation of anisotropy in the cosmic background radiation. Royal Society of London Philosophical Trans- actions Series A, 307:97–109, October 1982.
[92] D. E. Holz and E. V. Linder. Safety in Numbers: Gravitational Lensing Degradation of the Luminosity Distance-Redshift Relation. ApJ, 631:678– 688, October 2005.
[93] W. Hu and N. Sugiyama. Small-Scale Cosmological Perturbations: an Ana- lytic Approach. ApJ, 471:542, November 1996.
[94] E. Hubble. A Relation between Distance and Radial Velocity among Extra- Galactic Nebulae. Proceedings of the National Academy of Science, 15:168– 173, March 1929.
206 Bibliography
[95] F. Iocco, G. Mangano, G. Miele, O. Pisanti, and P. D. Serpico. Primordial nucleosynthesis: From precision cosmology to fundamental physics. Phys. Rep., 472:1–76, March 2009.
[96] W. Israel. Singular hypersurfaces and thin shells in general relativity. Nuovo Cim., B44S10:1, 1966.
[97] B. Jain and E. Bertschinger. Self-similar Evolution of Gravitational Cluster- ing: Is N = -1 Special? ApJ, 456:43, January 1996.
[98] R. Kantowski. Corrections in the Luminosity-Redshift Relations of the Ho- mogeneous Fried-Mann Models. ApJ, 155:89, January 1969.
[99] R. Kantowski, T. Vaughan, and D. Branch. The Effects of Inhomogeneities on Evaluating the Deceleration Parameter Q 0. ApJ, 447:35, July 1995.
[100] A. Kehagias and A. Riotto. Symmetries and consistency relations in the large scale structure of the universe. Nuclear Physics B, 873:514–529, August 2013.
[101] H. Kodama and M. Sasaki. Cosmological Perturbation Theory. Progress of Theoretical Physics Supplement, 78:1, 1984.
[102] F. Kottler. Ann. Phys. (Berlin), 56, 401, 1918.
[103] G. Lemaˆıtre.Un Univers homog`enede masse constante et de rayon croissant rendant compte de la vitesse radiale des n´ebuleusesextra-galactiques. Annales de la Soci´et´eScientifique de Bruxelles, 47:49–59, 1927.
[104] J. Lesgourgues, G. Mangano, G. Miele, and S. Pastor. Neutrino Cosmology. February 2013.
[105] J. Lesgourgues, S. Matarrese, M. Pietroni, and A. Riotto. Non-linear power spectrum including massive neutrinos: the time-RG flow approach. JCAP, 6:17, June 2009.
[106] J. Lesgourgues and T. Tram. The Cosmic Linear Anisotropy Solving Sys- tem (CLASS) IV: efficient implementation of non-cold relics. JCAP, 9:32, September 2011.
[107] A. Lewis, A. Challinor, and A. Lasenby. Efficient Computation of Cosmic Microwave Background Anisotropies in Closed Friedmann-Robertson-Walker Models. ApJ, 538:473–476, August 2000.
207 Bibliography
[108] E. Lifshitz. On the Gravitational stability of the expanding universe. J.Phys.(USSR), 10:116, 1946.
[109] E. M. Lifshitz and I. M. Khalatnikov. Investigations in relativistic cosmology. Advances in Physics, 12:185–249, April 1963.
[110] A. D. Linde. New scenario of the inflationary universe: possible solution of the problem of horizon, flatness, inhomogeneity, isotropy and relic monopoles. In M. A. Markov, editor, Quantum Gravitation. Quantum Theory of Gravitation, pages 106–112, 1982.
[111] C.-P. Ma and E. Bertschinger. Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges. ApJ, 455:7, December 1995.
[112] N. Makino, M. Sasaki, and Y. Suto. Analytic approach to the perturbative expansion of nonlinear gravitational fluctuations in cosmological density and velocity fields. Phys. Rev. D, 46:585–602, July 1992.
[113] K. A. Malik and D. Wands. Cosmological perturbations. Phys. Rep., 475:1– 51, May 2009.
[114] G. Mangano, G. Miele, S. Pastor, T. Pinto, O. Pisanti, and P. D. Serpico. Relic neutrino decoupling including flavour oscillations. Nuclear Physics B, 729:221–234, November 2005.
[115] A. Mantz, S. W. Allen, and D. Rapetti. The observed growth of massive galaxy clusters - IV. Robust constraints on neutrino properties. MNRAS, 406:1805–1814, August 2010.
[116] V. Marra, E. W. Kolb, S. Matarrese, and A. Riotto. Cosmological observables in a Swiss-cheese universe. Phys. Rev. D, 76(12):123004, December 2007.
[117] S. Matarrese and M. Pietroni. Resumming cosmic perturbations. JCAP, 6:26, June 2007.
[118] R. D. McKeown and P. Vogel. Neutrino masses and oscillations: triumphs and challenges. Phys. Rep., 394:315–356, May 2004.
[119] A. Melchiorri, G. L. Fogli, E. Lisi, A. Marrone, A. Palazzo, P. Serra, and J. Silk. Constraints on the Neutrino Mass from Cosmology and their impact
208 Bibliography
on world neutrino data. Nuclear Physics B Proceedings Supplements, 145:290– 294, August 2005.
[120] C. W. Misner, K. S. Thorne, and J. A. Wheeler. Gravitation. 1973.
[121] M. P. Mood, J. T. Firouzjaee, and R. Mansouri. Gravitational lensing by a structure in a cosmological background. Phys. Rev. D, 88(8):083011, October 2013.
[122] V. F. Mukhanov and G. V. Chibisov. Quantum fluctuations and a nonsingular universe. Soviet Journal of Experimental and Theoretical Physics Letters, 33:532, May 1981.
[123] V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger. Theory of cos- mological perturbations. Phys. Rep., 215:203–333, June 1992.
[124] K. A. Olive, G. Steigman, and T. P. Walker. Primordial nucleosynthesis: theory and observations. Phys. Rep., 333:389–407, August 2000.
[125] N. Palanque-Delabrouille, C. Y`eche, J. Lesgourgues, G. Rossi, A. Borde, M. Viel, E. Aubourg, D. Kirkby, J.-M. LeGoff, J. Rich, N. Roe, N. P. Ross, D. P. Schneider, and D. Weinberg. Constraint on neutrino masses from SDSS- III/BOSS Ly$ alpha$ forest and other cosmological probes. ArXiv e-prints, \ October 2014.
[126] P. J. E. Peebles. The large-scale structure of the universe. 1980.
[127] M. Peloso and M. Pietroni. Galilean invariance and the consistency relation for the nonlinear squeezed bispectrum of large scale structure. JCAP, 5:31, May 2013.
[128] A. A. Penzias and R. W. Wilson. A Measurement of Excess Antenna Tem- perature at 4080 Mc/s. ApJ, 142:419–421, July 1965.
[129] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Cas- tro, S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Wal- ton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter,
209 Bibliography
N. Panagia, H. J. M. Newberg, W. J. Couch, and T. S. C. Project. Measure- ments of Ω and Λ from 42 High-Redshift Supernovae. ApJ, 517:565–586, June 1999.
[130] S. T. Petcov. The Nature of Massive Neutrinos. ArXiv e-prints, March 2013.
[131] Patrick Peter and Jean-Philippe Uzan. Primordial cosmology. Oxford Grad- uate Texts. Oxford Univ. Press, Oxford, 2009.
[132] C. Pichon and F. Bernardeau. Vorticity generation in large-scale structure caustics. A&A, 343:663–681, March 1999.
[133] M. Pietroni. Flowing with time: a new approach to non-linear cosmological perturbations. JCAP, 10:36, October 2008.
[134] Planck Collaboration, R. Adam, P. A. R. Ade, N. Aghanim, Y. Akrami, M. I. R. Alves, M. Arnaud, F. Arroja, J. Aumont, C. Baccigalupi, and et al. Planck 2015 results. I. Overview of products and scientific results. ArXiv e-prints, February 2015.
[135] Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Ar- naud, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. J. Banday, and et al. Planck 2013 results. XVI. Cosmological parameters. A&A, 571:A16, November 2014.
[136] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, F. Arroja, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, and et al. Planck 2015 results. XVII. Constraints on primordial non-Gaussianity. ArXiv e-prints, February 2015.
[137] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, and et al. Planck 2015 results. XIII. Cosmological parameters. ArXiv e-prints, February 2015.
[138] B. Pontecorvo. Inverse beta processes and nonconservation of lepton charge. Sov.Phys.JETP, 7:172–173, 1958.
[139] B. Pontecorvo. Neutrino Experiments and the Problem of Conservation of Leptonic Charge. Sov.Phys.JETP, 26:984–988, 1968.
210 Bibliography
[140] M. Pospelov and J. Pradler. Big Bang Nucleosynthesis as a Probe of New Physics. Annual Review of Nuclear and Particle Science, 60:539–568, Novem- ber 2010.
[141] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical recipes in FORTRAN. The art of scientific computing. 1992.
[142] S. Pueblas and R. Scoccimarro. Generation of vorticity and velocity dispersion by orbit crossing. Phys. Rev. D, 80(4):043504, August 2009.
[143] M. Redlich, K. Bolejko, S. Meyer, G. F. Lewis, and M. Bartelmann. Probing spatial homogeneity with LTB models: a detailed discussion. A&A, 570:A63, October 2014.
[144] S. Refsdal. On the Propagation of Light in Universes with Inhomogeneous Mass Distribution. ApJ, 159:357, January 1970.
[145] C. L. Reichardt, L. Shaw, O. Zahn, K. A. Aird, B. A. Benson, L. E. Bleem, J. E. Carlstrom, C. L. Chang, H. M. Cho, T. M. Crawford, A. T. Crites, T. de Haan, M. A. Dobbs, J. Dudley, E. M. George, N. W. Halverson, G. P. Holder, W. L. Holzapfel, S. Hoover, Z. Hou, J. D. Hrubes, M. Joy, R. Keisler, L. Knox, A. T. Lee, E. M. Leitch, M. Lueker, D. Luong-Van, J. J. McMa- hon, J. Mehl, S. S. Meyer, M. Millea, J. J. Mohr, T. E. Montroy, T. Natoli, S. Padin, T. Plagge, C. Pryke, J. E. Ruhl, K. K. Schaffer, E. Shirokoff, H. G. Spieler, Z. Staniszewski, A. A. Stark, K. Story, A. van Engelen, K. Vander- linde, J. D. Vieira, and R. Williamson. A Measurement of Secondary Cosmic Microwave Background Anisotropies with Two Years of South Pole Telescope Observations. ApJ, 755:70, August 2012.
[146] A. Rest, D. Scolnic, R. J. Foley, M. E. Huber, R. Chornock, G. Narayan, J. L. Tonry, E. Berger, A. M. Soderberg, C. W. Stubbs, A. Riess, R. P. Kirshner, S. J. Smartt, E. Schlafly, S. Rodney, M. T. Botticella, D. Brout, P. Challis, I. Czekala, M. Drout, M. J. Hudson, R. Kotak, C. Leibler, R. Lunnan, G. H. Marion, M. McCrum, D. Milisavljevic, A. Pastorello, N. E. Sanders, K. Smith, E. Stafford, D. Thilker, S. Valenti, W. M. Wood-Vasey, Z. Zheng, W. S. Burgett, K. C. Chambers, L. Denneau, P. W. Draper, H. Flewelling, K. W. Hodapp, N. Kaiser, R.-P. Kudritzki, E. A. Magnier, N. Metcalfe, P. A. Price, W. Sweeney, R. Wainscoat, and C. Waters. Cosmological Constraints from
211 Bibliography
Measurements of Type Ia Supernovae Discovered during the First 1.5 yr of the Pan-STARRS1 Survey. ApJ, 795:44, November 2014.
[147] S. Riemer-Sørensen, D. Parkinson, and T. M. Davis. Combining Planck data with large scale structure information gives a strong neutrino mass constraint. Phys. Rev. D, 89(10):103505, May 2014.
[148] A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Lei- bundgut, M. M. Phillips, D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith, J. Spyromilio, C. Stubbs, N. B. Suntzeff, and J. Tonry. Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. AJ, 116:1009–1038, September 1998.
[149] A. E. Romano. Inhomogeneous Cosmological Models and H0 Observations. International Journal of Modern Physics D, 21:50085, December 2012.
[150] A. E. Romano. Can smooth LTB models mimicking the cosmological constant for the luminosity distance also satisfy the age constraint? General Relativity and Gravitation, 45:2529–2544, December 2013.
[151] A. E. Romano and P. Chen. Low-redshift formula for the luminosity distance in a LTB model with cosmological constant. European Physical Journal C, 74:2780, April 2014.
[152] G. Rossi, N. Palanque-Delabrouille, A. Borde, M. Viel, C. Y`eche, J. S. Bolton, J. Rich, and J.-M. Le Goff. Suite of hydrodynamical simulations for the Lyman-α forest with massive neutrinos. A&A, 567:A79, July 2014.
[153] S. Saito, M. Takada, and A. Taruya. Impact of Massive Neutrinos on the Nonlinear Matter Power Spectrum. Physical Review Letters, 100(19):191301, May 2008.
[154] S. Saito, M. Takada, and A. Taruya. Nonlinear power spectrum in the pres- ence of massive neutrinos: Perturbation theory approach, galaxy bias, and parameter forecasts. Phys. Rev. D, 80(8):083528, October 2009.
[155] F. Saracco, M. Pietroni, N. Tetradis, V. Pettorino, and G. Robbers. Nonlinear matter spectra in coupled quintessence. Phys. Rev. D, 82(2):023528, July 2010.
212 Bibliography
[156] S. Sarkar. Big bang nucleosynthesis and physics beyond the standard model. Reports on Progress in Physics, 59:1493–1609, December 1996.
[157] R. Scaramella, Y. Mellier, J. Amiaux, C. Burigana, C. S. Carvalho, J. C. Cuil- landre, A. da Silva, J. Dinis, A. Derosa, E. Maiorano, P. Franzetti, B. Garilli, M. Maris, M. Meneghetti, I. Tereno, S. Wachter, L. Amendola, M. Crop- per, V. Cardone, R. Massey, S. Niemi, H. Hoekstra, T. Kitching, L. Miller, T. Schrabback, E. Semboloni, A. Taylor, M. Viola, T. Maciaszek, A. Ealet, L. Guzzo, K. Jahnke, W. Percival, F. Pasian, M. Sauvage, and the Euclid Collaboration. Euclid space mission: a cosmological challenge for the next 15 years. ArXiv e-prints, January 2015.
[158] E. Schucking. The Schwarzschild line element and the expansion of the uni- verse (in german). Zs. f. Phys. 137, 595, 1954.
[159] R. Scoccimarro. Transients from initial conditions: a perturbative analysis. MNRAS, 299:1097–1118, October 1998.
[160] R. Scoccimarro. A New Angle on Gravitational Clustering. In J. N. Fry, J. R. Buchler, and H. Kandrup, editors, The Onset of Nonlinearity in Cosmology, volume 927 of Annals of the New York Academy of Sciences, pages 13–23, 2001.
[161] R. Scoccimarro and J. Frieman. Loop Corrections in Nonlinear Cosmological Perturbation Theory. ApJS, 105:37, July 1996.
[162] U. Seljak and M. Zaldarriaga. A Line-of-Sight Integration Approach to Cosmic Microwave Background Anisotropies. ApJ, 469:437, October 1996.
[163] M. Shoji and E. Komatsu. Third-Order Perturbation Theory with Nonlinear Pressure. ApJ, 700:705–719, July 2009.
[164] M. Shoji and E. Komatsu. Erratum: Massive neutrinos in cosmology: An- alytic solutions and fluid approximation [Phys. Rev. D 81, 123516 (2010)]. Phys. Rev. D, 82(8):089901, October 2010.
[165] J. L. Sievers, R. A. Hlozek, M. R. Nolta, V. Acquaviva, G. E. Addison, P. A. R. Ade, P. Aguirre, M. Amiri, J. W. Appel, L. F. Barrientos, E. S. Battistelli, N. Battaglia, J. R. Bond, B. Brown, B. Burger, E. Calabrese, J. Chervenak,
213 Bibliography
D. Crichton, S. Das, M. J. Devlin, S. R. Dicker, W. Bertrand Doriese, J. Dunk- ley, R. D¨unner,T. Essinger-Hileman, D. Faber, R. P. Fisher, J. W. Fowler, P. Gallardo, M. S. Gordon, M. B. Gralla, A. Hajian, M. Halpern, M. Has- selfield, C. Hern´andez-Monteagudo, J. C. Hill, G. C. Hilton, M. Hilton, A. D. Hincks, D. Holtz, K. M. Huffenberger, D. H. Hughes, J. P. Hughes, L. In- fante, K. D. Irwin, D. R. Jacobson, B. Johnstone, J. Baptiste Juin, M. Kaul, J. Klein, A. Kosowsky, J. M. Lau, M. Limon, Y.-T. Lin, T. Louis, R. H. Lup- ton, T. A. Marriage, D. Marsden, K. Martocci, P. Mauskopf, M. McLaren, F. Menanteau, K. Moodley, H. Moseley, C. B. Netterfield, M. D. Niemack, L. A. Page, W. A. Page, L. Parker, B. Partridge, R. Plimpton, H. Quintana, E. D. Reese, B. Reid, F. Rojas, N. Sehgal, B. D. Sherwin, B. L. Schmitt, D. N. Spergel, S. T. Staggs, O. Stryzak, D. S. Swetz, E. R. Switzer, R. Thornton, H. Trac, C. Tucker, M. Uehara, K. Visnjic, R. Warne, G. Wilson, E. Wollack, Y. Zhao, and C. Zunckel. The Atacama Cosmology Telescope: cosmological parameters from three seasons of data. JCAP, 10:60, October 2013.
[166] J. Silk. Cosmic Black-Body Radiation and Galaxy Formation. ApJ, 151:459, February 1968.
[167] G. F. Smoot, C. L. Bennett, A. Kogut, E. L. Wright, J. Aymon, N. W. Boggess, E. S. Cheng, G. de Amici, S. Gulkis, M. G. Hauser, G. Hinshaw, P. D. Jackson, M. Janssen, E. Kaita, T. Kelsall, P. Keegstra, C. Lineweaver, K. Loewenstein, P. Lubin, J. Mather, S. S. Meyer, S. H. Moseley, T. Murdock, L. Rokke, R. F. Silverberg, L. Tenorio, R. Weiss, and D. T. Wilkinson. Struc- ture in the COBE differential microwave radiometer first-year maps. ApJL, 396:L1–L5, September 1992.
[168] G. F. Smoot, M. V. Gorenstein, and R. A. Muller. Detection of anisotropy in the cosmic blackbody radiation. Physical Review Letters, 39:898–901, October 1977.
[169] G. Somogyi and R. E. Smith. Cosmological perturbation theory for baryons and dark matter: One-loop corrections in the renormalized perturbation the- ory framework. Phys. Rev. D, 81(2):023524, January 2010.
[170] V. Springel. The cosmological simulation code GADGET-2. MNRAS, 364:1105–1134, December 2005.
214 Bibliography
[171] V. Springel, S. D. M. White, A. Jenkins, C. S. Frenk, N. Yoshida, L. Gao, J. Navarro, R. Thacker, D. Croton, J. Helly, J. A. Peacock, S. Cole, P. Thomas, H. Couchman, A. Evrard, J. Colberg, and F. Pearce. Simulations of the formation, evolution and clustering of galaxies and quasars. Nature, 435:629–636, June 2005.
[172] A. A. Starobinsky. A new type of isotropic cosmological models without singularity. Physics Letters B, 91:99–102, March 1980.
[173] A. A. Starobinsky. Dynamics of phase transition in the new inflationary universe scenario and generation of perturbations. Physics Letters B, 117:175– 178, November 1982.
[174] G. Steigman. Primordial Nucleosynthesis in the Precision Cosmology Era. Annual Review of Nuclear and Particle Science, 57:463–491, November 2007.
[175] K. T. Story, C. L. Reichardt, Z. Hou, R. Keisler, K. A. Aird, B. A. Benson, L. E. Bleem, J. E. Carlstrom, C. L. Chang, H.-M. Cho, T. M. Crawford, A. T. Crites, T. de Haan, M. A. Dobbs, J. Dudley, B. Follin, E. M. George, N. W. Halverson, G. P. Holder, W. L. Holzapfel, S. Hoover, J. D. Hrubes, M. Joy, L. Knox, A. T. Lee, E. M. Leitch, M. Lueker, D. Luong-Van, J. J. McMahon, J. Mehl, S. S. Meyer, M. Millea, J. J. Mohr, T. E. Montroy, S. Padin, T. Plagge, C. Pryke, J. E. Ruhl, J. T. Sayre, K. K. Schaffer, L. Shaw, E. Shirokoff, H. G. Spieler, Z. Staniszewski, A. A. Stark, A. van Engelen, K. Vanderlinde, J. D. Vieira, R. Williamson, and O. Zahn. A Measurement of the Cosmic Microwave Background Damping Tail from the 2500-Square- Degree SPT-SZ Survey. ApJ, 779:86, December 2013.
[176] S. J. Szybka. Light propagation in Swiss-cheese cosmologies. Phys. Rev. D, 84(4):044011, August 2011.
[177] A. Taruya, F. Bernardeau, T. Nishimichi, and S. Codis. Direct and fast calculation of regularized cosmological power spectrum at two-loop order. Phys. Rev. D, 86(10):103528, November 2012.
[178] I. Tereno, C. Schimd, J.-P. Uzan, M. Kilbinger, F. H. Vincent, and L. Fu. CFHTLS weak-lensing constraints on the neutrino masses. A&A, 500:657– 665, June 2009.
215 Bibliography
[179] D. Tseliakhovich and C. Hirata. Relative velocity of dark matter and baryonic fluids and the formation of the first structures. Phys. Rev. D, 82(8):083520, October 2010.
[180] C. Uhlemann, M. Kopp, and T. Haugg. Schr¨odingermethod as N-body double and UV completion of dust. Phys. Rev. D, 90(2):023517, July 2014.
[181] A. Upadhye, R. Biswas, A. Pope, K. Heitmann, S. Habib, H. Finkel, and N. Frontiere. Large-scale structure formation with massive neutrinos and dynamical dark energy. Phys. Rev. D, 89(10):103515, May 2014.
[182] P. Valageas. Using the Zeldovich dynamics to test expansion schemes. A&A, 476:31–58, December 2007.
[183] W. Valkenburg. Swiss cheese and a cheesy CMB. JCAP, 6:10, June 2009.
[184] N. Van de Rijt. Signatures of the primordial universe in large-scale structure surveys. PhD thesis, Ecole´ Polytechnique, 2012.
[185] R. A. Vanderveld, E.´ E.´ Flanagan, and I. Wasserman. Luminosity distance in “Swiss cheese” cosmology with randomized voids. I. Single void size. Phys. Rev. D, 78(8):083511, October 2008.
[186] M. Viel, M. G. Haehnelt, and V. Springel. The effect of neutrinos on the matter distribution as probed by the intergalactic medium. JCAP, 6:15, June 2010.
[187] F. Villaescusa-Navarro, J. Miralda-Escud´e,C. Pe˜na-Garay, and V. Quilis. Neutrino halos in clusters of galaxies and their weak lensing signature. JCAP, 6:27, June 2011.
[188] F. Villaescusa-Navarro, M. Vogelsberger, M. Viel, and A. Loeb. Neutrino signatures on the high-transmission regions of the Lyman α forest. MNRAS, 431:3670–3677, June 2013.
[189] C. Wagner, L. Verde, and R. Jimenez. Effects of the Neutrino Mass Splitting on the Nonlinear Matter Power Spectrum. ApJL, 752:L31, June 2012.
[190] Y. Wang. Flux-averaging Analysis of Type IA Supernova Data. ApJ, 536:531– 539, June 2000.
216 Bibliography
[191] Y. Wang. Supernova Pencil Beam Survey. ApJ, 531:676–683, March 2000.
[192] Y. Wang. Observational signatures of the weak lensing magnification of su- pernovae. JCAP, 3:5, March 2005.
[193] S. Weinberg. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. July 1972.
[194] S. Weinberg. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. American Journal of Physics, 41:598–599, April 1973.
[195] S. Weinberg. Apparent luminosities in a locally inhomogeneous universe. ApJL, 208:L1–L3, August 1976.
[196] S. Weinberg. Adiabatic modes in cosmology. Phys. Rev. D, 67(12):123504, June 2003.
[197] H. Weyl. Phys. Z., 20, 31, 1919.
[198] Y. Y. Y. Wong. Higher order corrections to the large scale matter power spectrum in the presence of massive neutrinos. JCAP, 10:35, October 2008.
[199] Y. B. Zel’dovich. Observations in a Universe Homogeneous in the Mean. Soviet Astr., 8:13, August 1964.
[200] Z.-S. Zhang, T.-J. Zhang, H. Wang, and C. Ma. Testing the Copernican principle with the Hubble parameter. Phys. Rev. D, 91(6):063506, March 2015.
217 R´esum´ede la th`eseen fran¸cais
Cosmologie de pr´ecisionavec les grandes structures de l’univers
Introduction
Deux th´ematiques,chacune ancr´eedans l’`erede la cosmologie de pr´ecision, sont abord´eesdans cette th`ese.Dans un premier temps, c’est le principe cosmologique qui est questionn´e.Il s’agit de l’hypoth`eseselon laquelle, `atr`esgrande ´echelle1 (& 100 Mpc), la distribution de mati`eredans l’univers est homog`eneet isotrope. En cosmologie moderne, la mani`ere dont on interpr`eteles donn´eesobservation- nelles est grandement influenc´eepar ce principe de base. En collaboration avec Jean-Philippe Uzan et son ´etudiant Pierre Fleury, nous avons souhait´een savoir plus sur les limites de cette hypoth`ese.Plus pr´ecis´ement, nous avons test´el’effet induit par la pr´esencede structures dans l’univers en simulant des observations de supernovae dans un univers fictif non homog`ene.Nous avons en particulier attir´e l’attention sur le fait que, dans le contexte de la cosmologie de pr´ecision,il pourrait ˆetrepertinent d’adapter la mod´elisationg´eom´etrique de l’espace-temps `al’´echelle spatiale consid´er´ee.En effet, les ´echelles caract´eristiquesexplor´eesen cosmologie varient consid´erablement d’une observable `aune autre. Or la g´eom´etriede l’univers r´eeln’est pas la mˆeme`atoute ´echelle. Le second aspect de la cosmologie de pr´ecisiontrait´edans cette th`eseest le d´eveloppement de la th´eoriedes perturbations cosmologiques, notamment via l’ex- ploration des r´egimesnon lin´eaireet relativiste. Il s’agit de la tˆache `alaquelle j’ai consacr´ela grande majorit´ede mon temps pendant ma th`ese.Avec mon directeur de th`ese,Francis Bernardeau, j’ai propos´eune nouvelle approche analytique visant
1pc est le symbole de parsec, qui v´erifie 1 pc 3.1 1016m. ≈ ×
218 `ad´ecrireefficacement la croissance non lin´eairedes perturbations dans des fluides o`ula dispersion en vitesse n’est pas n´ecessairement n´egligeable.Cette m´ethode, bas´eeessentiellement sur une d´ecomposition des fluides en ensembles de flots, est particuli`erement adapt´ee`al’´etudedu rˆolejou´epar les neutrinos sur la formation des grandes structures de l’univers. La ph´enom´enologiedes neutrinos (dans un contexte cosmologique) est donc ´egalement un domaine auquel je me suis int´eress´eependant ma th`ese.
Un exemple illustrant les enjeux de la cosmologie de pr´ecision
La cosmologie est une discipline atypique, notamment parce que l’univers est un objet d’´etudeparticulier. Il ne peut ˆetreni ´etudi´ede l’ext´erieurni compar´e`ades syst`emessimilaires. De plus, on l’observe depuis une partie restreinte de celui-ci (la Terre et ses environs) pendant une dur´eelimit´ee(p´eriode durant laquelle des ˆetreshumains sont pr´esents sur Terre). La distance `alaquelle on peut observer est en outre finie car il faut du temps `ala lumi`erepour se propager d’un point `aun autre et, selon le mod`ele du Big Bang, l’univers n’a pas toujours exist´e.On parle d’horizon cosmologique pour d´efinir la fronti`erede l’univers observable. En raison de ces sp´ecificit´es,les mod`elescosmologiques contiennent n´ecessairement des hypoth`esesinv´erifiables,dont le principe cosmologique fait partie. En relativit´e g´en´erale,l’application de ce principe se traduit par l’utilisation d’une m´etriqueho- mog`eneet isotrope, connue sous le nom de m´etriquede Friedmann-Lemaˆıtre. L’univers r´eeln’est bien sˆurpas parfaitement homog`ene.S’il l’´etait,la croissance des grandes structures par instabilit´egravitationnelle n’aurait d’ailleurs pas ´et´epos- sible. L’explication actuellement privil´egi´eepour d´ecrirel’amor¸cagede la croissance des structures est en effet la pr´esencede gradients de densit´edans l’univers primor- dial. Plus pr´ecis´ement, selon la th´eoriede l’inflation, un champ scalaire existait au d´epartet ses fluctuations quantiques (in´evitablement pr´esentes) seraient deve- nues macroscopiques en raison d’un ´episode d’expansion extrˆemement rapide (voir [7, 85, 89, 173, 30, 122]). Cela aurait engendr´edes fluctuations de m´etriqueet donc in´evitablement des fluctuations de densit´e(car en relativit´eg´en´erale la g´eom´etriede l’espace-temps agit sur la mati`ere,et inversement). Le processus d’instabilit´egravi- tationnelle ´etaitalors lanc´e.L’´etudede l’´evolution de telles perturbations au cours
219 du temps est g´en´eralement r´ealis´ee`al’aide de la th´eoriedes perturbations cosmolo- giques. Dans ce contexte, la m´etriquede Friedmann-Lemaˆıtre est remplac´eepar une m´etriquede Friedmann-Lemaˆıtreperturb´ee (c’est `adire une m´etriquecontenant des corrections infinit´esimalesqui brisent l’homog´en´eit´e). Dans la formulation la plus ´el´ementaire du mod`elestandard de la cosmologie (en particulier, en l’absence de neutrinos), les caract´eristiquesde l’univers sont r´esum´eespar les valeurs de six param`etres,que l’on appelle param`etrescosmolo- giques. Ces valeurs ne peuvent ˆetred´etermin´eesqu’observationnellement. Plusieurs sources offrent cette possibilit´e.Parmi elles, les plus largement utilis´eessont les su- pernovae de type Ia (SNIa), les oscillations acoustiques de baryons (BAO) et le fond diffus cosmologique (CMB). Une supernova est un objet astrophysique r´esultant de l’explosion d’une ´etoile.Les SNIa, qui sont des supernovae dont le spectre contient du silicium mais pas d’hydrog`ene,sont utiles notamment pour retracer l’histoire de l’expansion de l’univers. Plus pr´ecis´ement, elles donnent des informations au sujet de l’´equationd’´etatde l’´energienoire (voir [38, 146]). Les BAO sont des os- cillations p´eriodiques de la composante baryonique2 de l’univers. La connaissance de leurs propri´et´esest ´egalement tr`esutile `ala cosmologie (voir notamment [5]). Le CMB est un rayonnement ´emisenviron 380 000 ans apr`esle Big Bang, lorsque la temp´eraturede l’univers est devenue suffisamment basse pour que les photons puissent se propager librement. Les observations du CMB fournissent des estima- tions extrˆemement pr´ecisesdes param`etrescosmologiques, en particulier lorsqu’elles sont combin´ees`ad’autres donn´eesastrophysiques ([137]). La th´eoriedes perturbations cosmologiques appliqu´eeau mod`elestandard de la cosmologie donne des r´esultatsen tr`esbon accord avec la r´ealit´e. Cela est en fait assez surprenant compte tenu de la simplicit´edu mod`ele et de la qualit´edes moyens mis en œuvre pour explorer minutieusement l’univers observable. Par exemple, une telle mod´elisationpr´esuppose que la mati`ereest distribu´eede fa¸concontinue, ce qui sous-entend l’existence d’une ´echelle de lissage, non pr´ecis´eedans le mod`ele([68]). Cela est d’autant plus probl´ematiqueque certaines observables cosmologiques sont des sources quasi-ponctuelles (d’un point de vue cosmologique), dont les faisceaux tr`es´etroitsparcourent l’univers en sondant des ´echelles trop petites pour que le principe cosmologique soit applicable et pour que l’hypoth`esede continuit´esoit valable. En particulier, l’´echelle caract´eristiqueassoci´ee`ala largeur des faisceaux
2Baryonique signifie fait de baryons. En particulier la mati`erestandard, faite d’atomes, est baryonique.
220 de SNIa est de l’ordre de l’unit´eastronomique3. L’utilisation non ´eclair´eedu principe cosmologique pourrait donc conduire `ades erreurs d’interpr´etation(voir [33], parmi d’autres r´ef´erences). Ce genre de probl´ematiqueest caract´eristiquede l’`erede la cosmologie de pr´ecision. D´ej`adans les ann´eessoixante, le paradoxe entre la structure r´eellede l’univers et sa repr´esentation lisse avait ´et´emis en ´evidence.Des tentatives d’am´eliorationde la mod´elisationavaient alors ´et´epropos´ees([199, 50, 19, 83, 98, 144]). Plus r´ecemment, l’effet des fluctuations de densit´esur les diagrammes de Hubble, qui sont des dia- grammes construits principalement `apartir de l’observation de SNIa et tr`esutilis´es en cosmologie moderne, a ´et´e´etudi´eattentivement ([99, 80, 191, 190, 192]). Des approches reposant sur la th´eoriedes perturbations cosmologiques ont ´egalement ´et´epropos´ees([92, 39, 54]), sans toutefois que l’hypoth`ese de continuit´esoit remise en question. De tels mod`elesutilisent des ´echelles de lissage de l’ordre de la minute d’arc4 alors que l’on s’attend `ace que l’essentiel de la dispersion induite par les structures dans les diagrammes de Hubble provienne d’´echelles inf´erieures(voir par exemple [48]). De plus, il semble inappropri´ede supposer que les fluctuations de densit´erencontr´eespar les faisceaux de SNIa soient repr´esentatives des fluctuations moyennes qui existent dans l’univers puisque, du fait de leur ´etroitesse,ces faisceaux se propagent beaucoup plus longuement dans des r´egions sous-denses que dans des r´egionssur-denses. Lorsqu’il s’agit d’interpr´eterles diagrammes de Hubble, il serait donc utile de tenir compte d’effets de s´election.Ces difficult´essont discut´eesde mani`ereapprofondie dans [33]. Les simulations num´eriquessont des alternatives aux mod`eles analytiques qui permettent de travailler en milieu non homog`ene.Cependant, leur r´esolutionn’est pas infinie donc, mˆeme num´eriquement, la structure `apetite ´echelle ne peut pas ˆetremod´elis´eeavec pr´ecision.Dans [33], la limite qui est donn´eeest de quelques dizaines de kiloparsecs, ce qui est nettement sup´erieur`ala largeur caract´eristique d’un faisceau de supernova. La premi`ere´etudepr´esent´eedans cette th`eseest une mod´elisationanalytique de la propagation de la lumi`eredans un espace-temps de type Swiss cheese. Il s’agit d’un espace-temps obtenu en enlevant des sph`eresde mati`ere`aun milieu initiale- ment homog`eneet isotrope et en rempla¸cant celles-ci par des masses ponctuelles, identiques aux masses enlev´ees(voir la figure2 et les r´ef´erences[65, 158]).
3Une unit´eastronomique vaut environ 1.5 1011 m. × 4La minute d’arc est une unit´ede mesure angulaire correspondant `aun soixanti`emede degr´e.
221 Figure 2 – Illustration repr´esentant un espace-temps de type Swiss cheese. Le fond est homog`ene. Chaque sph`ereest vide sauf en son centre, o`uune masse ponctuelle est pr´esente. La masse ponctuelle est ´egale`ala densit´edu fond multipli´ee par le volume de la zone blanche qui l’entoure.
Notre mod`elen’a pas pour vocation d’ˆetrer´ealistemais de tester l’impact de variations de densit´ediscontinues sur la propagation de la lumi`ere.D’un point de vue th´eorique,il est autant justifi´eque la description standard car il s’agit d’une solution exacte de la relativit´eg´en´eralequi a la mˆemedynamique globale que la m´etriquede Friedmann-Lemaˆıtre.Dans un premier temps, nous avons d´etermin´eanalytiquement la mani`eredont la propagation de la lumi`ereest affect´eepar la travers´eed’un trou. Apr`esavoir impl´ement´ele r´esultatdans un programme Mathematica, nous avons pu facilement pr´edirel’impact de la travers´eed’un plus grand nombre de trous. Il a alors ´et´epossible de construire des diagrammes de Hubble th´eoriquesen imaginant des faisceaux de supernovae se propageant dans un tel milieu. Pour y parvenir, nous nous sommes inspir´esde donn´eesr´eellesprovenant du catalogue SNLS 3 (voir [38]). Le fait de recourir `ades mod`elesde type Swiss cheese pour tester les limites du principe cosmologique n’est pas une id´eenouvelle. La particularit´ede notre approche est qu’il ne s’agit ni d’une solution moderne de type Lemaˆıtre-Tolman- Bondi, dans laquelle les masses ponctuelles sont remplac´eespar des gradients de densit´econtinus (voir [31, 116, 24, 185, 32, 35, 183, 176, 74, 200, 72, 73, 143, 151, 150, 121, 40, 149, 70]), ni d’un retour aux anciens mod`elesde type Swiss cheese ([98] et [62]) car le mod`elecosmologique que nous utilisons est totalement ancr´e dans la cosmologie moderne. Partant d’un observateur occupant une position quelconque de la partie non vide du Swiss cheese, nous avons reconstruit pas `apas les trajectoires suivies par les photons. La premi`ere ´etape consiste en l’identification des coordonn´eesde l’´ev´enement correspondant `ala sortie du dernier trou travers´e. En pratique, cela revient `ar´esoudrel’´equationdes g´eod´esiquesdu genre lumi`eredans la m´etriquede Friedmann-Lemaˆıtre. A` l’int´erieur de ce trou, la g´eom´etrien’est pas la mˆeme.Elle est d´ecritepar une m´etriqueappel´eem´etriquede Kottler. Nous avons donc ensuite
222 r´esolules ´equationsdu mouvement dans la m´etriquede Kottler, afin d’identifier l’´evenement correspondant `al’entr´eedu dernier trou travers´e,puis sommes reve- nus `ala m´etriquede Friedmann-Lemaˆıtre.L’impact de la pr´esencede trous sur l’´evolution de la largeur des faisceaux lumineux, et donc sur la luminosit´efinale- ment re¸cuepar l’observateur, a quant `alui ´et´e´etudi´een r´esolvant `achaque ´etape l’´equationde Sachs. Il s’agit d’une information cruciale car c’est cette luminosit´e qui figure en ordonn´eedes diagrammes de Hubble. La proc´edured´ecriteci-dessus peut ˆetrerenouvel´eeun nombre arbitraire de fois, jusqu’`ace que l’on d´ecideque la source a ´et´eatteinte. En abscisse des diagrammes de Hubble, on trouve le d´ecalage vers le rouge cosmologique associ´e`ala source. Il a donc ´egalement ´et´en´ecessairede d´ecrirel’´evolution du nombre d’onde afin de comparer les valeurs `al’´emissionet `a la r´eceptionet d’en d´eduirela valeur du d´ecalagevers le rouge de chaque SNIa. Dans un premier temps, nous avons trait´enos diagrammes comme des donn´ees observationnelles standard, c’est `adire que nous avons utilis´ele mod`elehomog`ene et isotrope pour d´eterminer(par le test du χ2) les valeurs des param`etrescosmolo- giques `autiliser dans les ´equationspour reproduire au mieux ces donn´ees.Utilisant des barres d’erreur r´ealistes(car semblables `acelles du catalogue SNLS 3), nous avons trouv´eque l’estimation finale des param`etrescosmologiques peut ˆetretr`es ´eloign´eedes valeurs r´eellesutilis´eespour construire notre milieu de propagation. Autrement dit, nous avons illustr´ela fa¸condont l’hypoth`esed’homog´en´eit´eaffecte l’interpr´etationdes donn´ees.Le fait que l’estimation des param`etrescosmologiques d´epende `ace point du mod`elen’est pas anodin tant l’id´eeque l’on se fait de l’univers en cosmologie moderne repose sur cette estimation. Dans notre ´etude,nous avons ´egalement analys´ede vraies observations en sup- posant que l’univers r´eelavait une g´eom´etriede type Swiss cheese. Nous avons pour cela dˆud´eriver une formule analytique reliant la luminosit´edes SNIa `aleur d´ecalagevers le rouge cosmologique dans des espaces-temps de type Swiss cheese. L`aencore, pour un mˆemecatalogue de donn´ees,les r´esultatsobtenus peuvent net- tement diff´ererde ce qu’ils sont lorsque l’on suppose que la g´eom´etriede l’univers est bien d´ecritepar la m´etriquede Friedmann-Lemaˆıtre.Ce travail a donn´elieu `a une publication dans Physical Review D en 2013, [77]. L’enjeu est important puisque les r´esultatsparus en 2013 de la mission Planck,
[135], avaient r´ev´el´eun d´esaccord entre l’estimation des param`etres h et Ωm `a l’aide d’observations du CMB et leur estimation `apartir d’autres observables (en particulier des SNIa). Dans une deuxi`eme´etudepubli´eedans Physical Review
223 Letter en 2013, [76], nous d´emontrons que le mod`eledu Swiss cheese permet de rendre compatibles les estimations du param`etreΩm. Ceci est illustr´esur la figure3. Par ailleurs, les r´esultats2015 de la mission Planck, [137], montrent que l’estimation de h par Planck est en fait compatible avec le catalogue r´ecent de SNIa dont il est question dans [20], qui se nomme Joint Light-curve Analysis.
Figure 3 – Comparaison des contraintes sur (Ωm, h) obtenues par Planck ([135]) et par le diagramme de Hubble construit `apartir du catalogue SNLS 3 ([86]). f est un param`etreindiquant la fraction du volume qui n’est pas sous forme de trous. Par exemple, f = 1 correspond `aune m´etriquede Friedmann-Lemaˆıtre.
Ces travaux illustrent les enjeux de la cosmologie de pr´ecision puisque c’est la structure tr`esfine de l’univers qui est ´etudi´ee.Le but n’est pas de proposer de nou- veaux mod`elesmais plutˆotd’ajouter un degr´ede pr´ecisionaux mod`elesexistants. Dans le cas pr´esent, la nouveaut´eserait d’adapter la repr´esentation g´eom´etriquede l’univers `al’´echelle consid´er´eeau lieu d’utiliser des mod`eleshomog`eneset isotropes, qui ne refl`etent que l’univers liss´e`agrande ´echelle. On peut ´egalement d´evelopper la cosmologie de pr´ecisionen travaillant avec les grandes structures de l’univers. Dans le cadre de la th´eoriedes perturbations cosmologiques, l’effort est actuelle- ment concentr´esur le d´epassement de la th´eorielin´eaireet sur la prise en compte de constituants de l’univers n´eglig´esjusqu’alors dans les mod`eles.Il s’agit pr´ecis´ement du type d’activit´eentrepris pendant le reste de ma th`ese.
224 Une nouvelle fa¸cond’´etudierles neutrinos en cosmologie
En th´eoriedes perturbations dite standard (voir la revue [13]), on mod´elisela croissance des structures en d´ecrivant l’effondrement gravitationnel de la mati`ere noire froide, trait´eecomme un fluide non relativiste sans pression. Dans ce contexte, la physique newtonienne suffit. Un ´el´ement-cl´ede la description est l’hypoth`esede flot unique. En effet, l’agitation thermique est minime dans les fluides froids si bien que l’on s’attend `ace que la dispersion en vitesse soit faible tant que les fluctuations de densit´en’induisent pas des gradients de vitesse trop importants. C’est seulement `ades stades plus avanc´esque la dispersion en vitesse devient cons´equente. En effet, la gravit´eguide les particules vers les puits de potentiels, et ce quelles que soient leurs vitesses initiales, donc les particules sont amen´ees`ase croiser au bout d’un certain temps. Ce ph´enom`ene,appel´ecroisement de coquilles, est illustr´esur la figure4.
Figure 4 – Illustration sch´ematiquede l’´emergencede r´egionsmulti-flots (croi- sement de coquilles) sous l’effet de la gravit´e.Les distances et les vitesses sont repr´esent´eescomme des quantit´esunidimensionnelles. Auteur : Bernardeau, [12].
C’est durant cette phase tardive de virialisation que les objets astrophysiques tels que les galaxies commencent `ase former (voir [22]). Malheureusement, d`eslors que le croisement de coquilles est amorc´e,l’´evolution du syst`emedevient tr`esdifficile `a mod´eliseranalytiquement. Actuellement, les connaissances `ace sujet reposent donc majoritairement sur des simulations num´eriques`a N corps. Au contraire, avant l’apparition de croisements de coquilles, le fait de n´egliger la dispersion en vitesse dans les ´equationsdu mouvement rend possible de nombreux d´eveloppements analytiques. On parle d’hypoth`esede flot unique. Elle implique qu’`aun instant et une position donn´es,toutes les particules du fluide poss`edent la mˆemevitesse.
225 En cosmologie de pr´ecision,on cherche `ad´ecrirela croissance des grandes struc- tures de la mani`erela plus rigoureuse possible. Afin de pouvoir mod´eliseranalyti- quement le rˆolejou´epar des esp`ecesautres que la mati`erenoire froide, nous avons cherch´eune astuce permettant d’´etendrele champ d’application de l’hypoth`esede flot unique. L’id´eeque nous proposons est tout simplement de consid´ererchaque esp`ecerelativiste non pas comme un fluide `aplusieurs flots mais comme un ensemble de fluides `aun flot. Plus pr´ecis´ement, `al’instant initial, nous consid´eronscomme un fluide `apart enti`erechaque ensemble de particules dont la vitesse est identique puis nous ´etudionss´epar´ement l’´evolution des N flots correspondant aux N vitesses initiales. La discr´etisationde l’espace des phases est illustr´eesur la figure5 pour N = 11. Les impulsions initiales sont caract´eris´ees non seulement par une norme mais aussi par un sens et, dans les espaces `aplusieurs dimensions, par une direc- tion. Dans le cas des neutrinos, une telle d´ecomposition ne pose pas de probl`eme puisque, apr`esleur d´ecouplage,les neutrinos n’interagissent plus entre-eux ou avec les autres esp`eces.Par cons´equent, si les conditions initiales sont choisies apr`esle d´ecouplagedes neutrinos, aucune interaction non gravitationnelle n’a besoin d’ˆetre prise en compte. Sur l’image, les variations d’´epaisseurdes traits repr´esentent des fluctuations de densit´e.
Figure 5 – Espace des phases discr´etis´e`al’instant initial pour un ensemble de onze flots. Dans chaque flot, les impulsions sont initialement homog`enes.Les varia- tions d’´epaisseurrepr´esentent des fluctuations de densit´e.Par souci de simplicit´e, les impulsions et les positions ont ´et´erepr´esent´eescomme des quantit´esunidimen- sionnelles.
Malgr´el’homog´en´eit´edes champs de vitesses initiaux, la pr´esencede fluctua- tions de densit´efait apparaˆıtredes gradients de vitesse. Pour une masse donn´ee, les particules sont d’autant plus sensibles `aces fluctuations que la norme de leur vitesse initiale est petite puisque, dans ce cas, elles tombent facilement dans les
226 puits de potentiel. En particulier, le flot caract´eris´epar une impulsion initiale nulle se comporte pr´ecis´ement comme le fluide de mati`erenoire froide. Le d´eveloppement de fluctuations spatiales dans l’espace des impulsions est illustr´esur la figure6. Comme pour la mati`erenoire froide, on s’attend `ace que des r´egionsmulti-flots finissent par apparaˆıtre.On voit clairement sur la figure6 que cela se produira beau- coup plus tˆotdans les fluides caract´eris´espar une faible vitesse initiale. Avant ces premiers croisements de coquilles, il est raisonnable de supposer que chaque fluide peut ˆetre´etudi´edans l’hypoth`ese de flot unique. Remarquons par ailleurs qu’il est probable que ces croisements surviennent uniquement lorsque les neutrinos seront devenus suffisamment lents pour pouvoir ˆetred´ecrits comme des esp`ecesfroides.
Figure 6 – D´eveloppement de gradients d’impulsion dans l’espace des phases discr´etis´epour un ensemble de onze flots. Ce ph´enom`eneest g´en´er´epar les fluc- tuations de densit´e,repr´esent´eesici par des variations d’´epaisseurdes traits.
Dans une telle approche, le comportement global des neutrinos est retrouv´een sommant les contributions des diff´erents flots de la collection, `acondition que le nombre de flots soit suffisant pour que l’´echantillon soit repr´esentatif. Par exemple, si on caract´erisechaque flot par une vitesse initiale appel´ee ~τ, la fonction de distri- bution globale de l’espace des phases s’obtiendra de la fa¸consuivante,
tot f (η, x, p) = f~τ (η, x, p), (16) X~τ ce qui donne dans la limite continue
tot 3 f (η, x, p) = d τf~τ (η, x, p), (17) Z avec f~τ la fonction de distribution du fluide caract´eris´epar ~τ. Symboliquement, le fait d’identifier les fluides par leurs vitesses initiales puis
227 de d´eriver des ´equationsdu mouvement sp´ecifiques`achacun est analogue au choix d’adopter un point de vue lagrangien pour d´ecrirel’´evolution d’un fluide. La diff´erence est que, dans la description lagrangienne, c’est la position initiale qui est utilis´ee comme label. Inversement, la description standard des neutrinos reposant sur l’´etude de f tot(η, x, p) se rapproche d’une description eul´erienne. Une fois les flots d´efinis`al’instant initial, le nombre de particules contenues dans chacun d’eux est constant. Math´ematiquement, la conservation du nombre de particules se traduit par une ´equationde continuit´e.En relativit´eg´en´erale,elle est donn´eepar la conservation de la quantit´e J µ, appel´eequadri-courant,
µ J ;µ = 0. (18)
Le quadri-courant est l’analogue quadri-dimensionnel de la densit´ede courant de la physique classique. Il s’agit donc d’un d´ebit.Par cons´equent, il peut s’exprimer en termes d’une densit´eet d’une vitesse ou d’une impulsion. Les variables que nous avons choisi d’utiliser sont les suivantes :
la densit´enum´eriquecomobile n (η, x), d´efiniecomme le nombre de particules • c par unit´ede volume comobile d3xi,
le champ d’impulsion comobile P (η, x). • i La densit´enum´eriquecomobile peut facilement ˆetrereli´ee`ala fonction de distribu- tion dans l’espace des phases. En effet, par d´efinition,
3 i nc(η, x) = d pi f(η, x , pi) (19) Z et Pi(η, x) est tout simplement la valeur moyenne des impulsions comobiles de l’espace des phases, d3p f(η, xi, p )p P (η, x) = i i i . (20) i d3p f(η, xi, p ) R i i Par ailleurs, le quadri-courant satisfaitR la relation (voir par exemple [18])
µ µ 3 1/2 p i J (η, x) = d p ( g)− f(η, x , p ), (21) − i − p0 i Z o`u g est le d´eterminant du tenseur m´etrique. L’approximation de flot unique est particuli`erement utile dans ce contexte. En effet, `aun instant η et une position x donn´es,toutes les particules du fluides ont
228 une impulsion Pi(η, x), d’o`u
f(η, xi, p ) = n (η, x)δ (p P (η, x)). (22) i c D i − i L’´equationde continuit´e(18) impose alors
P i ∂ n + ∂ n = 0. (23) η c i P 0 c Dans l’hypoth`esedu flot unique, le tenseur ´energie-impulsion v´erifie (voir encore [18])
T µν = P µJ ν. (24) − En combinant la conservation du quadri-courant `acelle du tenseur ´energie-impulsion, on trouve finalement
µ ν P ;νJ = 0, (25) d’o`unotre seconde ´equationdu mouvement
1 P ν∂ P = P σP ν∂ g . (26) ν i 2 i σν Notre approche permet de suivre individuellement le comportement des diff´erents flots de neutrinos mais, ce qui est int´eressant pour la cosmologie, c’est surtout leur comportement global. Nous avons donc exprim´eles premiers multipˆolesde la dis- tribution globale d’´energieen fonction des variables individuelles que nous avions choisies. De mani`eretr`esg´en´erale,ces multipˆolessont donn´espar (dans l’espace r´eciproque, voir [111]) :
ρ T 0 , (27) ν ≡ − 0 ρ¯ + P¯ θ ikiT 0 , (28) ν ν ν ≡ i kikj 1 1 ρ¯ + P¯ σ δ T i δi T k , (29) ν ν ν ≡ − k2 − 3 ij j − 3 j k o`u X¯ d´esignela partie homog`enede la quantit´e X et avec ([18])
µ ν µν 3 1/2 p p i T (η, x) = d p ( g)− f(η, x , p ). (30) i − p0 i Z 229 Dans notre approche, les int´egralessur les impulsions de l’espace des phases sont `a remplacer par des int´egralessur les diff´erents flots, ou de mani`ere´equivalente sur les impulsions initiales ~τ. D’apr`es(17) et (22), on a
f tot(η, xi, p ) = d3τ n (η, x; ~τ)δ (p P (η, x; ~τ)), (31) i i c D i − i Z d’o`u(pour toute fonction des impulsions) F
d3p f tot(η, xi, p ) (p ) = d3τ n (η, x; ~τ) (P (η, x; ~τ)). (32) i i F i i c F i Z Z Cette formule nous a permis d’exprimer sans difficult´esles multipˆolesen fonction de nos variables. Habituellement, la seule mani`erede les calculer est d’int´egrerla hi´erarchie de Boltzmann, obtenue en calculant les moments successifs de l’´equation de Boltzmann. En pratique, il est n´ecessairede lin´eariser et de tronquer cette hi´erarchie pour qu’elle soit int´egrablenum´eriquement. Afin de tester la validit´ede notre approche, nous avons donc calcul´enum´eriquement l’´evolution temporelle de nos multipˆolesdans le r´egimelin´eaireet compar´enos r´esultatsaux r´esultatsstan- dard. Ces calculs ont ´et´efaits en imposant des conditions initiales adiabatiques, choisies lorsque les neutrinos sont d´ej`ad´ecoupl´esdes autres esp`ecesmais encore re- lativistes. En effet, dans ce contexte, la fonction de distribution initiale des neutrinos est connue et l’hypoth`esede flot unique permet d’en d´eduireles valeurs initiales de nos champs. Les conclusions que nous avons tir´eesde ce test num´eriquesont les suivantes.
A` grande ´echelle, c’est la discr´etisationdes normes des impulsions qui est • d´eterminante pour que l’accord entre les deux m´ethodes soit satisfaisant. En revanche, la discr´etisationdes directions initiales importe peu. Par exemple, en prenant 12 directions diff´erentes, en tronquant la hi´erarchie de Boltzmann `al’ordre 6 et en consid´erant des neutrinos de masse m = 0.05 eV, les erreurs 2 4 relatives moyennes passent de 10− `a10− quand le nombre de normes d’im- pulsions (dans chaque approche) passe de 16 `a100 lorsque k = 0.002h/Mpc.
Pour k 0.01h/Mpc, l’int´egrationnum´erique est plus ardue car les ´equations • & du mouvement des flots de la collection deviennent tr`esdiff´erentes les unes des autres (autrement dit, d´ependent plus fortement des vitesses initiales). Il faut par exemple utiliser 100 directions diff´erentes pour obtenir une pr´ecision de l’ordre du pourcent lorsque k = 0.1h/Mpc.
230 En choisissant des masses plus petites, il faut ´egalement affiner la discr´etisation • puisque, dans ce cas, la d´ependance des champs de vitesses et de densit´esvis- `a-visde ~τ est plus grande.
Aux temps longs, tous les flots se comportent comme la mati`erenoire froide. • Les tests num´eriquesmontrent sans ambigu¨ıt´eque les deux approches sont com- 3 patibles dans le r´egime lin´eaire.Par exemple, un accord avec une pr´ecisionde 10− (obtenu pour k = 0.01h/Mpc et m = 0.3 eV) est pr´esent´esur la figure7, qui compare les comportements des premiers multipˆoles´energ´etiques.L’inconv´enient majeur de notre m´ethode est que, pour que l’accord soit satisfaisant, le nombre d’´equations`ar´esoudredoit g´en´eralement ˆetreplus important dans notre approche que dans l’approche de Boltzmann. Toutefois, notre mod`eleanalytique est valable au-del`adu r´egimelin´eaire,ce qui n’est pas le cas de l’approche de Boltzmann. De plus, les grandes similitudes qui existent entre notre ´etudeet celle de la mati`erenoire froide permettent d’esp´ererd’importants d´eveloppements bas´essur nos ´equations. Cette ´etude`a´et´epubli´eedans JCAP en 2014, [57].
231 L L in
Η 100 H Ψ
of 1 units in
H 0.01
10-4 Multipoles 10-6 0.001 0.01 0.1 1 10 100 1000
a aeq. L 4
- 5 10 of 0 units in H -5
-10 Residuals -15 0.1 1 10 100 1000
a aeq.
Figure 7 – Evolution´ temporelle du contraste de densit´ed’´energie(ligne continue), de la divergence des vitesses (tirets) et de la pression anisotrope (points) des neu- trinos. Dans la partie haute, les quantit´essont calcul´ees`apartir de nos ´equations. La partie basse montre les erreurs relatives entre les deux approches. L’int´egration num´eriquea ´et´er´ealis´eeen utilisant 40 normes d’impulsions et 12 directions ini- tiales. k est fix´e`a0.01h/Mpc et m `a0.3 eV.
Vers une extension de la th´eoriedes perturbations cos- mologiques au r´egimerelativiste
Par la suite, nous avons utilis´el’approche multi-fluides pr´ec´edente pour g´en´eraliser au cas relativiste certains r´esultatsmajeurs de la th´eoriedes perturbations non lin´eaires.Afin de rendre les ´equationsdu mouvement plus faciles `amanipuler, nous avons choisi d’´eliminerles termes de couplage qui ne sont pas indispensables pour d´ecrirela formation des structures. Plus pr´ecis´ement, nous avons utilis´ele rayon de Hubble5 comme ´echelle caract´eristiqueet nous nous sommes concentr´essur les ´echelles subhorizon, c’est `adire les ´echelles inf´erieuresau rayon de Hubble. En effet,
5Le rayon de Hubble donne l’ordre de grandeur du rayon de l’univers observable.
232 la croissance non lin´eairedes structures est un processus r´ecent dans l’histoire de l’univers donc les termes qui deviennent petits aux ´echelles subhorizon ne sont a priori pas d´eterminants dans ce contexte. Le premier r´esultatque nous avons pu g´en´eraliseren travaillant dans la limite subhorizon est le fait que, `atout ordre de la th´eoriedes perturbations, la partie rotationnelle des champs d’impulsions est n´egligeable.Cela signifie que ces variables sont enti`erement caract´eris´eespar leurs divergences. Or cette propri´et´e,´egalement satisfaite par le champ de vitesses du fluide de mati`erenoire froide, est capitale dans la description standard newtonienne (voir [12]). L’int´erˆetprincipal de notre approche est l’universalit´edes ´equations.En effet, apr`esle d´ecouplage,les seuls ´el´ements permettant de distinguer les baryons, la mati`erenoire froide et les neutrinos sont leurs masses et vitesses initiales. En introduisant un 2N-uplet
Ψ (k) = (δ (k), θ (k), . . . , δ (k), θ (k))T , (33) b ~τ1 − ~τ1 ~τN − ~τN o`uchaque δ est un contraste de densit´eet chaque θ une divergence de vitesse, on peut donc encoder le comportement de tous les fluides cosmiques d´ecoupl´esdans c cd une ´equationunique (les expressions des matrices Ωb et γb sont donn´eesdans la th`ese):
c cd a(η)∂aΨb(k, η) + Ωb (k, η)Ψc(k, η) = γb (k1, k2, η)Ψc(k1, η)Ψd(k2, η). (34)
Cette ´equationest formellement identique `al’´equation-cl´esur laquelle repose la th´eoriedes perturbations standard dans le r´egime non lin´eaire.La diff´erencecapi- tale est que tous les fluides, relativistes ou non, sont ici pris en compte alors que l’´equationstandard ne mod´eliseque l’´evolution de la mati`erenoire froide. Comme expliqu´edans la th`ese,nous avons ´egalement mis en ´evidencedes propri´et´esd’in- variance de mani`ere`ag´en´eraliser certaines relations de consistance permettant de relier des fonctions de corr´elationd’ordre n `ades fonctions de corr´elationd’ordre n+1. Ce travail pr´eparele terrain pour l’impl´ementation de techniques de resomma- tion tr`esutiles `ala description de la croissance non lin´eairedes grandes structures de l’univers (voir [45, 117, 133, 46, 177] pour la pr´esentation de telles techniques). Cela a d´ebouch´esur une publication dans JCAP en 2015, [58].
233 Une application concr`etede la description multi-fluides des neutrinos
En cosmologie, le spectre de puissance de la mati`ereest une quantit´ecruciale per- mettant de d´ecrire la distribution statistique de la mati`eredans l’univers. On le d´efinitcomme la fonction de corr´elation`adeux points des contrastes de densit´ede mati`ere.La mod´elisationanalytique de l’effet des neutrinos sur ce spectre de puis- sance dans le r´egimenon lin´eaireest un travail de longue haleine. Pour atteindre cet objectif, il est utile dans un premier temps d’identifier les ´echelles auxquelles il sera pertinent de tenir compte des couplages non lin´eairesdans les ´equationsdu mouve- ment impliquant les neutrinos. Dans ma th`ese,j’ai utilis´el’approche multi-fluides d´ecriteci-dessus dans ce but. La forme de l’´equation (34) permet l’impl´ementation de l’approximation ei- konale, d´ecritedans [14] et [16] dans le cas non relativiste, qui consiste en la d´ecomposition du membre de droite de (34) en deux domaines appel´es hard domain et soft domain puis en l’´eliminationdu hard domain. Le hard domain contient les couplages entre modes de longueurs d’onde comparables tandis que le soft domain d´ecritles couplages entre modes de grande et de petite longueurs d’onde. La cor- rection apport´eepar le soft domain par rapport `al’´equationlin´earis´ee(c’est `adire l’´equation(34) dans laquelle le membre de droite est fix´e`az´ero)peut ˆetrevue comme une perturbation du milieu. G´en´eralement, cette perturbation est d´ecrite par un champ de d´eplacement, dont l’expression g´en´eralis´eeau r´egime relativiste est explicitement donn´eedans ma th`ese.Cela d´ecouledirectement des propri´et´esd’in- variance que nous avons mises en ´evidencedans [58]. Comme expliqu´edans [179] et [17], lorsque les champs de d´eplacement ne sont pas les mˆemesdans tous les fluides (on parle alors de d´eplacements non adiabatiques), les perturbations de chacun ne sont plus en phase les unes avec les autres. Cela ralentit la chute des esp`ecesautres que la mati`erenoire froide dans les puits de potentiel, et donc finalement la crois- sance des structures. Nous avons donc calcul´eles champs de d´eplacement relatifs entre mati`erenoire froide et neutrinos et trac´eles r´esultatsen fonction de k afin de d´eterminerles ´echelles auxquelles l’amortissement du spectre de puissance dˆuaux couplages entre modes de grande et de petite longueurs d’onde peut ˆetrecons´equent. L’estimation pr´eliminaireque nous avons trouv´eecorrespond aux ´echelles dont les nombres d’ondes sont sup´erieurs(ou de l’ordre de) 0.2 `a0.5 h/Mpc pour des masses de 0.3 eV. Cependant, une ´etudenettement plus approfondie est n´ecessairepour
234 pouvoir tirer des conclusions solides `ace sujet. Le but de notre travail ´etaitplutˆot d’attirer l’attention sur la possibilit´ede g´en´eralisationdu lien entre approximation eikonale, d´eplacements relatifs entre fluides cosmiques et amortissement du spectre de la mati`eredans le r´egimenon lin´eaire. Les r´esultatscorrespondants ont ´et´epu- bli´esdans JCAP en 2015, [59].
Conclusions et perspectives
Cette th`esefournit des r´esultatsinnovants de plusieurs types. Leur point commun est la quˆetede pr´ecisiondans la description des ph´enom`enesphysiques `al’œuvre dans l’univers. D’abord, un mod`ele-jouetsimulant la propagation de la lumi`eredans un espace- temps non homog`eneest pr´esent´e.Dans cette ´etude,nous avons opt´epour la tra- ditionnelle repr´esentation de type Swiss cheese. Souvent utilis´eedans la litt´erature, elle permet de travailler avec des solutions exactes de la relativit´eg´en´erale,qui n’alt`erent pas la dynamique globale de l’univers tout en le rendant fortement non homog`ene.Nous avons illustr´ela fa¸condont les hypoth`esesde base, telles que le principe cosmologique, peuvent affecter les conclusions scientifiques, telles que l’es- timation des param`etrescosmologiques `apartir des diagrammes de Hubble. En g´en´eral,le biais caus´epar le fait d’introduire des trous dans un espace homog`ene et isotrope peut ˆetregrand en raison de la grande sensibilit´ede la luminosit´edes sources vis-`a-visde la g´eom´etriedu milieu travers´e.Nous avons montr´eque ce biais d´ecroˆıtavec la constante cosmologique, ce qui est une chance puisque les observa- tions penchent clairement vers un contenu de l’univers domin´epar l’´energienoire (et donc par la constante cosmologique dans le mod`elestandard). En raison de son extrˆemesimplicit´e, notre mod`ele(dont les fluctuations de densit´esont exag´er´ement abruptes) surestime probablement le biais. En effet, les grandes structures, le gaz intergalactique, la mati`erenoire, etc. agissent comme des lisseurs dans l’univers r´eel.Les effets de compensation entre r´egionsde densit´esinf´erieureet sup´erieure`a la moyenne sont ind´eniables. Nous sommes toutefois convaincus que, en cosmologie de pr´ecision,cela vaut la peine d’´etudierminutieusement ce genre de probl´ematique plutˆotque d’accepter qu’il y ait une incoh´erenceentre les ´echelles auxquelles le prin- cipe cosmologique est valide et celles sond´eespar les SNIa utilis´ees pour construire les diagrammes de Hubble. N’oublions pas que les param`etrescosmologiques sur lesquels repose la cosmologie moderne d´ependent fortement des mod`elesutilis´es
235 donc le fait de les mesurer avec une tr`esgrande pr´ecisionn’est pas suffisant pour affirmer que leurs valeurs refl`etent la r´ealit´e. Cette th`esepr´esente aussi une ´etudedans laquelle les r´esultatsde la mission Planck d’une part et les observations du catalogue de supernovae SNLS 3 d’autre part sont interpr´et´esen mod´elisant la g´eom´etriede l’univers par un espace-temps de type Swiss cheese. Nous avons montr´eque l’estimation du param`etrecosmo- logique Ωm `apartir des diagrammes de Hubble peut alors ˆetretr`esdiff´erente de ce qu’elle est dans le mod`elehomog`eneet isotrope, allant jusqu’`aune possible r´econciliationavec les r´esultatsde la mission Planck. Evidemment,´ les espaces- temps de type Swiss cheese ne sont pas des repr´esentations r´ealistesde l’univers. Ils permettent simplement de montrer que l’incompatibilit´eentre les diff´erentes ob- servables peut venir du fait qu’elles sondent l’univers `ades ´echelles tr`esdiff´erentes. Avec l’am´eliorationde la pr´ecisionde l’estimation des param`etrescosmologiques, un travail sur la mod´elisationg´eom´etrique de l’espace-temps pourrait ˆetrejudicieux, voire n´ecessaire. Notre approche peut ˆetream´elior´eede multiples fa¸cons.On pourrait notam- ment introduire des trous de tailles diff´erentes afin d’avoir davantage de libert´es concernant la proportion de r´egionsvides. L’´etudede donn´eesautres que les dia- grammes de Hubble, obtenues en rempla¸cant les photons par d’autres particules, constitue ´egalement une piste int´eressante. Une mod´elisationplus r´ealistede l’uni- vers, bas´eesur une ´etudeapprofondie de la structure r´eellede l’univers, pourrait aussi ˆetreb´en´efique.Toutefois, pendant ma th`ese, je me suis sp´ecialis´eedans une autre branche de la cosmologie, `asavoir l’´etudede la formation des grandes struc- tures de l’univers `al’aide de la th´eoriedes perturbations. Mon collaborateur Pierre Fleury a quant `alui consacr´eune grande partie de sa th`ese(soutenue ´egalement cette ann´ee)`al’´etudede la propagation de la lumi`ereen milieu non homog`eneet/ou non isotrope (voir notamment [78], [75]). Le r´esultatmajeur propos´edans cette th`eseest une nouvelle fa¸conde d´ecrire les neutrinos en cosmologie. L’id´eeest de d´ecomposer les neutrinos en plusieurs fluides `aun flot de mani`ere`ase d´ebarrasserde la dispersion en vitesse dans chacun d’eux. Les ´equationsdu mouvement correspondantes, non lin´eaires,ont ´et´ed´eriv´ees `apartir de lois de conservation dans une m´etriquede Friedmann-Lemaˆıtre per- turb´eequelconque, tout comme les formules permettant de retrouver les grandeurs physiques globales en sommant sur les diff´erents flots. En utilisant ces formules pour calculer la distribution multipolaire d’´energie,nous avons montr´eque notre
236 approche est totalement compatible avec l’int´egrationde la hi´erarchie de Boltzmann dans le r´egimelin´eaire.Nous pensons que notre approche fournit une base plus ap- propri´ee`al’´etudedu comportement non lin´eaire des neutrinos que la hi´erarchie de Boltzmann. Elle fournit de plus une information suppl´ementaire puisque les flots de neutrinos peuvent ˆetreexamin´esindividuellement. Cette m´ethode peut ˆetreap- pliqu´eeaux neutrinos et, plus g´en´eralement, `an’importe quelle esp`eceen chute libre. Dans ce cadre, la mati`erenoire froide apparaˆıtcomme un simple fluide de la collection, dont la seule particularit´eest d’avoir une vitesse initiale nulle. N’importe quelle vitesse (relativiste ou non) et n’importe quelle masse peuvent ˆetreg´er´eespar ce formalisme. Sur cette base, nous avons men´eune ´etuded´edi´ee`al’exploration de la struc- ture de couplage de nos ´equationsrestreintes aux ´echelles subhorizon. L’int´erˆetde cette hypoth`eseest de mettre en ´evidencedes propri´et´esutiles sans ´eliminerles termes qui sont pertinents pour d´ecrirela croissance non lin´eairedes structures. Nous avons ainsi pu r´esumeren une ´equationles processus en jeu dans l’´evolution des perturbations cosmologiques. L’avantage de cette ´equationest qu’elle est aussi simple que l’´equationde la th´eoriedes perturbations standard tout en ayant un champ d’application beaucoup plus large. C’est pourquoi on peut esp´ererque notre approche constitue un pas de plus vers une g´en´eralisationrelativiste de la th´eorie des perturbations non lin´eaire.Nous avons d´elib´er´ement travaill´edans une jauge quelconque de mani`ere`afaciliter la mise en ´evidencede propri´et´esd’invariance, ce qui nous a permis de d´eriver des relations de consistance et d’´etudierles couplages entre modes de grande et de petite longueurs d’onde `al’aide de l’approximation eikonale. L’un de nos articles a ´et´eenti`erement d´edi´e`al’impl´ementation de cette ap- proximation dans nos ´equationsdu mouvement. Nous avons ainsi pu monter que, pour des grands nombres d’ondes et des vitesses initiales importantes, les champs caract´erisant les neutrinos peuvent ˆetred´ephas´espar rapport `aceux de la mati`ere noire froide pendant une dur´eeimportante. Cela ralentit naturellement la croissance des grandes structures de l’univers. Les perspectives les plus ´evidentes offertes par ce point de vue multi-flots sont les suivantes. Formellement, les techniques de resommation standard permettant de calculer les corrections non lin´eaires du spectre de puissance de la mati`erepour- raient ˆetre´etendues `anotre ´equationglobale. Cependant, le formalisme en jeu est tr`eslourd donc le fait de remplacer brutalement les matrices 2 2 par des matrices ×
237 N N, qui poss`edent de plus des parties imaginaires d´ependantes d’´echelle, serait × r´edhibitoire.La suite exigera donc de trouver des arguments permettant de dimi- nuer la complexit´edu probl`eme.D’une part, il sera crucial d’identifier les ´echelles auxquelles `ala fois les non-lin´earit´eset les corrections relativistes doivent ˆetreprises en compte. La derni`ere´etudepr´esent´eedans la th`eseest un premier pas en ce sens. En pratique, puisque les non-lin´earit´esse d´eveloppent tardivement tandis que les plus grandes vitesses correspondent aux ´epoques les plus anciennes, on peut s’at- tendre `abeaucoup de simplifications. D’autre part, la d´eterminationdu nombre de flots n´ecessairepour que la description soit acceptable (au sens de la cosmologie de pr´ecision)sera essentielle. Avec Julien Lesgourgues, j’ai commenc´e`aimpl´ementer les ´equationsdu mouvement dans son code CLASS ([28]). Les r´esultatsque nous obtiendrons devraient se r´ev´elerutiles pour tester efficacement le coˆutnum´erique associ´eet envisager des simplifications bien contrˆol´ees.
238