Testing Models of Cosmological Inflation

Christopher Michael Graham

Thesis submitted for the degree of Doctor of Philosophy

School of Mathematics and Statistics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom

April 2010 Dedicated to the memory of Norah Mander. Acknowledgements

This research was made possible through financial support from the Science and Technology Facilities Council and the School of Mathematics and Statistics. I’d like to say a big thank you to my fellow postgraduate students, companions on my PhD journey. From my 1st year, the M416 Merz Court crew, in particular Dave Ryan, who’s singing brightened many a gloomy afternoon and Sam James for helpful physics discussions and as a team leader in all things extra-curricular. More recently thanks to Andrew Baggaley, Stuart Cockburn (and his supervisor Nick Proukakis) for many helpful chats on thermalisation, and all of the other postgraduate students with whom I shared offices, borrowed books and stationary or conversed with over lunch. A big thank you to my second supervisor Andrew Fletcher who has always been available for useful discussions about my work, and to Anvar Shukurov and David Toms who stepped in to help with my viva arrangements in the last few months. Thank you to my external examiners also, Arjun Berera and Ed Copeland. A huge thank you to Antonia who endured the strains of living with me, and was always there to support me, and to my parents, my sister Catherine and my extended family. To my friends outside of academia too, particularly Andy and Darren who helped me juggle my life in recent months. And finally this thesis would not have been possible without my supervisor Ian Moss. Without Ian’s guidance, enthusiasm and seemingly infinite knowledge of the subject there would still be no end in sight and it is to him that I am most grateful. Abstract

Over the past two decades cosmological inflation has become an essential ingredient to any model of the early universe. Measurements of temperature fluctuations observed in the Cosmic Microwave Background are in remarkable agreement with the predictions of inflationary models, and for the first time these results have allowed these models to be intensively tested against observational data. This thesis considers the building and testing of inflationary models which can be consistent with the Cosmic Microwave Background data. In particular the consistency of warm inflation is tested, a model where density fluctuations arise from thermal, rather than quantum vacuum fluctuations, and the possibility of distinguishing this model from standard inflationary theories is considered. The decay channels for the production of low-mass particles during standard and warm inflation are analysed to yield new results for the particle production rates. The subsequent thermalisation of particles is investigated numerically for both cases. It is concluded that in some situations the radiation produced during inflation can remain in thermal equilibrium, allowing modified inflaton equations parameterised by a single temperature-dependent friction coefficient. New results are obtained for the power spectrum of fluctuations at the end of inflation for several models of warm inflation, results that will be very important when testing temperature-dependent models against Cosmic Microwave Background data. The primordial power spectrum of fluctuations alone is not enough to distinguish between different models of inflation. The bispectrum statistic of non-Gaussianity is calculated for temperature-independent warm inflation, and for comparison the curvaton model, to conclude that forthcoming Cosmic Microwave Background satellite data should be able to detect and distinguish any non-Gaussian signal for warm inflation from other models through its signal strength and the shape of the bispectrum. Preface

This thesis is based partially upon the following papers:

”Testing Models of Inﬂation with CMB non-Gaussianity” • Ian G. Moss, Chris M. Graham JCAP 0711:004,2007

”Particle Production and Reheating in the Inﬂationary Universe” • Ian G. Moss, Chris M. Graham Phys.Rev.D78:123526,2008

”Density Fluctuations from Warm Inﬂation” • Chris M. Graham, Ian G. Moss JCAP07(2009)013

i Contents

1 Introduction 1

2 Cosmological inflation 4 2.1 Motivation for inflation ...... 4 2.2 Inflationarydynamics ...... 9 2.3 Theslow-rollapproximation ...... 12 2.4 Reheating ...... 13 2.5 Inflationarymodels...... 13 2.6 Warminflation ...... 15

3 The source of density fluctuations 17 3.1 Comovingscales ...... 18 3.2 Vacuum fluctuations as the source of density fluctuations ...... 18 3.3 Thermal fluctuations as the source of density fluctuations ...... 21 3.3.1 Fluctuations in the case Γ = Γ (φ) ...... 22 3.3.2 Fluctuations in the case Γ = Γ (T, φ)...... 25 3.4 Numerical simulation of coupled fluctuations ...... 27 3.4.1 Numericalapproach ...... 27 3.4.2 Dependency on r and c ...... 29 3.4.3 Dampingeffects ...... 30

4 Reheating the universe 34 4.1 Particleproductionmethod ...... 35 4.1.1 Someusefuldeﬁnitions...... 35

ii Contents

4.1.2 Theparticlenumberdensity...... 36 4.1.3 Theparticleproductionrate ...... 37 4.2 Particle production calculations ...... 39 4.2.1 Oscillating ﬁelds by the scattering matrix method ...... 40 4.2.2 Oscillating ﬁelds by the general method ...... 41 4.2.3 Particle production in the adiabatic approximation ...... 43 4.2.4 Particle production in the 2-Stage decay model ...... 44 4.3 The thermalisation of particles ...... 46 4.3.1 Intheoscillatingphase ...... 49 4.3.2 Intheslow-rollphase ...... 50 4.3.3 Thermalisation with the full Boltzmann collision integral . . . . 52

5 The cosmic microwave background 57 5.1 Evolution of perturbations after inﬂation ...... 57 5.2 MeasurementoftheCMB ...... 59 5.3 CMBdataanalysis...... 60 5.4 Theangularpowerspectrum ...... 60 5.5 Power spectrum predictions using numerical code ...... 63

6 Testing models of inflation with CMB non-Gaussianity 66 6.1 Quantifyingthenon-Gaussianity ...... 67 6.2 TheCMBbispectrum ...... 67 6.2.1 Thecurvatonmodel ...... 69 6.2.2 Warminflation ...... 74 6.3 Numerical calculation of the bispectrum ...... 74 6.4 Distinguishing between inflationary models ...... 77 6.5 Non-Gaussianity tests on CMB data ...... 80

7 Conclusion 85

A Numerical methods 87 A.1 Numerical simulation of the stochastic ﬂuctuations ...... 87

iii Contents

A.2 ModiﬁcationstoCMBFAST...... 89 A.3 CalculatingtheFishermatrix ...... 92

B Reduction of the Collision Integral 95

C Bispectrum calculations 99 C.1 General expression for the angle-averaged bispectrum ...... 99 C.2 Warm inﬂation angle-averaged bispectrum ...... 100

iv Chapter 1

Introduction

The discovery and subsequent precise measurements of the Cosmic Microwave Back- ground (CMB) in the past two decades have transformed the subject that is Cosmo- logical Inflation. The early moments of the universe have long been studied, but it is with the measurement of the CMB that the earliest of these, the exponential expansion of the universe known as inflation at some 10−42 seconds old, can be studied in great depth. Indeed, the very title of this thesis, ’testing models of inflation’, is now a reality. It is very well known that towards the end of the Hot Big Bang, a very long time after inflation, photons and electrons in a plasma universe decouple, leaving photons to travel freely through space in the transparent universe that we see today. It is these photons released at the decoupling time, which after billions of years are observable as the microwaves of the CMB. While this is overwhelming evidence for the Hot Big Bang, even more astounding is that the imprint of inflation is left in the CMB in the form of small scale temperature anisotropies. At the onset of inflation, the universe is small enough that quantum vacuum fluctuations are important. Inflationary models predict that these quantum fluctuations are the seed for density fluctuations which are not only visible in the CMB temperature spectrum, but also in the distribution of large scale structure in the universe. With this snapshot of the universe at the decoupling time, measurements of the CMB are an extremely powerful tool for determining the physics of the early universe, both during inflation and beyond. Indeed, it would not be bold to say that the precise measurements of the COBE satellite in 1991 and WMAP satellite in 2001-2007 have transformed the subject of Cosmology. This thesis considers the building and testing of inflationary models which can be consistent with the CMB data. There are at present many many models that can be considered consistent and the quest now is to find a definitive model of inflation, a task

1 Chapter 1. Introduction that will require analysis after analysis of each model, and ultimately even more precise measurement of the CMB. In particular, this thesis considers the consistency of a model known as warm inflation, and the possibility of distinguishing this model from standard (cold) inflationary theories. In warm inflation, radiation is produced alongside the expansion of the universe, which solves very neatly one, perhaps undesirable, problem of standard inflationary models, that the universe cools as it expands, leaving it far below the temperatures required for the Hot Big Bang to continue. In standard theories this is solved by having a separate period of reheating at the end of inflation. In warm inflation this is bypassed as the universe makes a smooth transition to the post-inflation era. The production of radiation in warm inflation models has implications for the physics during inflation and therefore the observational consequences. In particular, density fluctuations arise from thermal, rather than quantum vacuum fluctuations in this scenario. This thesis introduces the ideas behind inflationary theory and follows the evolution of density fluctuations from their primordial source during the inflationary epoch to their signature in the Cosmic Microwave Background and subsequent measurement. The work carried out can be considered almost chronologically into the chapters that follow. Chapter 2 introduces inflationary theory, its motivation and formulation. The equations that govern the dynamics of inflation are defined. Some significant models of inflation are introduced, including warm inflation and the relevant equations and definitions required to examine this model closely in later chapters. The source of fluctuations in the inflationary epoch is considered in chapter 3, where the primordial power spectrum of fluctuations is defined in the case of cold models of inflation, and warm inflation, where the source of fluctuations is the coupling between the inflaton field and radiation. In this scenario the results obtained for the primordial spectrum of fluctuations will be new, and very important when testing these models against CMB data later. Chapter 4 concentrates on the period of reheating that must follow inflation in most models, or the alternative, which is particle production during warm inflation. The aim of this chapter is to consider both approaches with a common method, using thermal field theory techniques to calculate the rate of particle production. Towards the end of chapter 4 the subsequent thermalisation of particles is analysed in both cold and warm inflation models. Calculations are made to identify any departure from thermal equilibrium in either case which would have considerable ob-

2 Chapter 1. Introduction servational consequences. Following inflation, the imprint of density fluctuations, whether thermal or quantum in origin, is left as temperature fluctuations in the Cosmic Microwave Background. The radiation has evolved almost without incident for nearly 14 billion years 1 until first observed as microwaves in the 1960s. Today we have extensive data from satellites dedicated to measuring the temperature anisotropy to a very high degree of accuracy. In chapter 5 the measurement and analysis of CMB data is reviewed, introducing important considerations for the work that follows. The angular power spectrum of temperature fluctuations is calculated from the CMB data and compared to the predictions that can be calculated numerically from a given inflationary model and set of cosmological parameters. In chapter 6 two models of inflation are tested using the CMB bispectrum, a measure of the non-Gaussianity of the temperature anisotropies which promises to distinguish between models of inflation, if not with the current CMB satellite data, then with forthcoming even more precise measurements. The two models used here are warm inflation and a cold inflation curvaton model. Most promisingly, models can be distinguished not just by the extent of non-Gaussianity but also by the wavenumber-dependent shape of the bispectrum, which varies from model to model. In this chapter the bispectrum is calculated for both models and their separability considered in the event of forthcoming satellite experiments confirming a non-Gaussian signal in the CMB. Finally in chapter 7 the important conclusions are drawn from this work and the future of this area is considered.

1WMAP data gives the current age of the universe to be 13.75 ± 0.13 Gyr.

3 Chapter 2

Cosmological inﬂation

2.1 Motivation for inﬂation

Inflation was first proposed by Alan Guth in 1981 and gave a solution to several problems outstanding in the Hot Big Bang model. Whilst the theory has its roots in these problems it is important to recognise that inflation is an addition to the Hot Big Bang model and preserves the standard description of the expanding universe. For these reasons, and as a primer for discussing inflationary models, the following sections describe some of the features of the Hot Big Bang model. From today’s viewpoint we can see another motivation for inflation; that it can successfully generate the large scale irregularities that we observe in the form of galaxies and galaxy clusters through the freezing-in of primordial fluctuations in the inflationary epoch. Whilst this was not of primary importance in the initial work on the topic, the prediction of fluctuations in the CMB temperature has been the great success of inflation and is the subject of much of the work described here. The Hot Big Bang model of the expanding universe is based on observations by Hubble in 1924 that the redshift of galaxies was proportional to their distance. The assumption that this result is independent of our position and orientation, that is to say that on large scales the universe is both homogeneous and isotropic, tells us that the universe was once small and dense [1]. The existence of the CMB also confirms that the universe was initially very hot. The Hot Big Bang model has now become the standard model for the early universe and its evolution to the present time. Further- more, the ’Cosmological Principle’ of homogeneity and isotropy on large scale has now been supported with the overwhelming evidence of an almost identical CMB temperature across the entire sky. Leaving aside the inflationary theory, the Hot Big Bang model describes the universe from an age of around 1 second, after which the physical

4 Chapter 2. Cosmological inflation processes involved in the evolution of the universe are believed to be relatively well understood. The process of nuceosynthesis at around 180 seconds produces many of the lighter elements and sets their relative abundance; one of the most important predictions of the Hot Big Bang model. At this time, the temperature T 0.1MeV is at the ≈ required level for the nuclear fusion of protons and neutrons to produce heavier nuclei, including Hydrogen and Helium-4 through the intermediate production of Deuterium. At around 100 seconds the temperature of the universe becomes too cool for nuclear fusion, limiting the range of elements produced to be no heavier than Beryllium, and fixing the ratio of Hydrogen and Helium nuclei. Crucially these abundances depend critically on the baryon-to-photon ratio set earlier in the evolution, one of the key cosmological parameters. The era which follows nucleosynthesis is referred to as the radiation-dominated era. The universe is then an opaque plasma, in which photons are scattered off the free electrons and the energy density of the universe eventually becomes dominated by dark matter. As the universe continues to expand, so the temperature of the plasma cools. At a temperature T 3000K the so-called recombination of electrons and nuclei creates ≈ the first atoms; electrons and protons form hydrogen atoms whilst helium nuclei and electrons produce helium atoms. Following the binding of the electrons the photons are able to travel freely and the universe becomes transparent. This is referred to as the surface of last scattering and is the source of the CMB radiation we observe today, redshifted to microwave length-scales and cooled to a temperature of 2.7K. The formation of atoms also marks the beginning of a matter-dominated era which continues to the present time. The evolution of the universe from the Hot Big Bang to the present time is dependent on the cosmological parameters. In particular, this set of parameters describes the expansion and geometry of the universe and all of the matter contained within it. Many of these parameters have been assigned values for many years from observation of, for example, the distribution and velocities of galaxies or the behaviour of more distant astrophysical objects; however it is only in recent years that the values assigned to the parameters have been constrained to any real accuracy and even today there is progress to be made. The rate of expansion of the universe is described by the Hubble parameter H = a/a˙ , where the scale factor a(t) is defined to be the separation between two comoving observers. The present value of the Hubble parameter is H0 = 70.5 1.3km/s/Mpc, ± −1 as derived from WMAP data and other sources [2]. The Hubble distance H0 gives a rough estimate of the age of the universe and is comparable to the age of the oldest

5 Chapter 2. Cosmological inﬂation astrophysical objects. The evolution of the universe is based on the laws of General Relativity with the Friedmann-Robertson-Walker metric [3; 4; 5; 6]:

dr2 ds2 = dt2 + a2 + r2 dθ2 + sin2 θdφ2 , (2.1) − 1 kr2 − where the curvature of the universe depends on the parameter k which takes the normalised value 0 for a spatially-flat universe, +1 for a universe with positive curvature (open universe), or 1 with negative curvature (closed universe). Einstein’s field equa- − tions relating the spatial curvature of the universe to the matter contained within it are 1 R Rg + Λg = 8πGT , (2.2) µν − 2 µν µν µν where the Einstein summation convention is used and the speed of light is set to c = 1 hereafter [7]. The terms in the left side of the field equations describe the curvature, where the Ricci tensor Rµν and the Ricci Scalar R are derived from the metric (2.1). The cosmological constant labelled Λ is of some controversy, famously introduced and then dismissed by Einstein himself in later work. The right side of the field equations describe the matter content. Here Tµν is the stress-energy-momentum tensor which for a perfect fluid of 4-velocity uµ, density ρ and pressure p is given by

µ ν µν Tµν = (P + ρ) u u + pg , (2.3) where gµν is a metric tensor, T 00 ρ and T P δ . Using the conservation law ≡ ij ≡ ij µν T ;ν = 0 it can be shown that the time-dependence of ρ is determined by

∂ a˙ ρ = 3 (p + ρ) , (2.4) ∂t − a known as the continuity or ﬂuid equation [8]. Using this stress-energy-momentum relation, Einstein’s ﬁeld equations yield the Friedmann equations

a˙ 2 1 8πGρ k = Λ + , (2.5) a 3 3 − a2 a¨ 4πG Λ = (ρ + 3p)+ (2.6) a − 4 3 where the second is often known as the acceleration equation, G is Newton’s gravitational constant, often written in terms of the reduced Planck mass Mp using Mp = (8πG)−1/2 [9]. Once again the Friedmann equations are presented with the cosmological constant

6 Chapter 2. Cosmological inﬂation

Λ. Interest in Λ gained momentum after supernova observations showed that the expansion of the universe was accelerating [10; 11; 12], such that a non-zero value of Λ is required in equation (2.6). This is now also supported by CMB measurements [13]. The source of the Λ term is today of much debate in the cosmological community and extensive research is being applied to the problem from several different angles. The most common solution is that Λ is some form of exotic matter that we cannot observe, most often referred to as dark energy. This alters the right hand side (matter terms) of the field equations (2.2). Alternative approaches alter the left side of (2.2) with a modified gravity theory, where general relativity is assumed to be incorrect at Hubble length-scales. The third approach and perhaps the most radical is that the universe is inhomogeneous on the largest scales. This however would violate the Copernican principle that we live in an arbitrary unpreferred point in the universe, and observationally it is difficult to test. In the following sections Λ is referred to as dark energy and makes up roughly 3/4 of the total energy density. For the calculations made here its source is not important. Using the Friedmann equations, there is a critical density at which the universe is flat, given by 3H2 ρ = . (2.7) c 8πG

The density ρ is parameterised by Ωtot, deﬁned as Ωtot = ρ/ρc. This density parameter is equal to unity for a ﬂat universe and can be split into its constituent parts;

Ωtot = Ωb + Ωc + Ων + ΩΛ, (2.8) the first three contributions from baryon, cold dark matter and neutrino density are commonly referred to as Ωm, the matter energy density. The fourth contribution comes from the dark energy density. The accurate measurement of the relative weight of these constituent parts for the present density parameter has been one of the important aims of CMB observation. The Friedmann solutions can easily be solved for the scale factor a in both the radiation and matter-dominated eras of expansion with the assumption that Λ = 0. For a spatially flat universe (k = 0) it can be shown that a t1/2 for the radiation ∝ era and a t2/3 for the matter era [8]. Similarly, albeit with more complication, the ∝ equations can be solved for open universe scenarios and for Λ = 0, however the details 6 are omitted here for the sake of brevity. Although the Hot Big Bang was extremely successful in explaining the observational evidence of an expanding universe, it had several problems which motivated the first inflationary models:

7 Chapter 2. Cosmological inﬂation

The Flatness Problem; Observations show that the total energy density of the • universe lies very close to the critical density ρc, so that the density parameter today is Ω 1. Using Friedmann’s equation (2.5), the evolution of the density 0tot ≈ parameter is given by k Ω 1= (2.9) tot − a2H2 where the right side of the equation is increasing with time. There are two major

observations to make: One is that if Ω0tot = 1 precisely now then it has always been equal to one. The other is that if Ω 1 now then it was extremely close 0tot ≈ to 1 at the time of the Hot Big Bang. In fact, extrapolating back to the era of Planck scale energies then Ω = 1 1 10−60 at that time. This would require tot ± × an incredible degree of ﬁne tuning and in order to avoid anthropic arguments it would be necessary for some process to set this initial condition in the very early universe.

The Horizon Problem; The large scale homogeneity, as observed for example • in the almost uniform CMB temperature across the sky, poses difficulties for the Hot Big Bang model. The expansion of the universe would be required to be far quicker than was described earlier (where a t1/2 or t2/3) in order for photons ∝ in different regions of the sky to lie within their horizon size (the distance the photon could have travelled since the Hot Big Bang). In other words, different regions of the sky could not have been causally connected at the time of the Hot Big Bang.

The Monopole Problem; According to Grand Unified Theories of particle • physics the Hot Big Bang would produce a vast number of magnetic monopoles. The predicted length scales of their creation is such that the universe would be large enough at the GUT energy scale to contain a significant number, yet a magnetic monopole has never been observed despite many efforts to detect them [14].

Although the universe is homogeneous on large scales, it is not diﬃcult to observe that on more moderate scales there lies irregularity in the form of galaxies and galaxy clusters. In considering the evolution of the irregularities seen in the CMB to what we see today it is important to consider the process of gravitational instability. Stated simply, matter that is not uniformly distributed draws in even more matter by gravitational attraction so that the distribution becomes less and less uniform over time. This tells us that to create the large scale structure we see today, we only require a small irregularity in the early universe and then allow gravitational instability to take its course.

8 Chapter 2. Cosmological inﬂation

There are a selection of models available which could produce the required initial irregularities. These include topological defects [15] and Friedmann models with a decaying cosmological constant [16] amongst others, however by far the most promising are inflationary models. As mentioned earlier, since this was not the primary motivation for inflation, the outcome that a wide range of inflationary models can also generate large scale structure has been an outstanding success for the theory.

2.2 Inﬂationary dynamics

Motivated to smooth out the problems of the Hot Big Bang model, inflation proposes that the universe underwent a period of exponential expansion in the first fraction of a second, at an energy close to the Planck scale [17]. More precisely, inflation corresponds to an accelerating scale factora> ¨ 0 during the inflationary epoch. The success of inflation is in the first case its solution to the problems of the Hot

Big Bang model. Consider the evolution of the density parameter Ωtot given by equation (2.9); an accelerating scale factor drives the parameter towards 1 without any fine-tuning of initial conditions. The smaller Hubble scale during inflation solves the horizon problem, since the universe can remain entirely causally connected, whilst the expansion also suppresses any monopole density that may have been present prior to inflation [18; 19]. Most importantly for this discussion, inflation can explain the observed inhomogeneities of large scale structure since the expansion could freeze in the quantum fluctuations present when the universe was very small. Using the Friedmann equation (2.5) with the continuity equation (2.4), the requirement for inflationa> ¨ 0 is ρ + 3p< 0. (2.10)

Since we consider the density of matter to be positive, it can be concluded that a negative pressure is required. Inflationary models are based on the assumption that at the onset of inflation the universe contains little more than a scalar field, the inflaton. The inflaton is one of a family of spin 0 particles which have the property that their equation of state effectively has negative pressure. Their existence however was not invented by cosmologists to explain inflation; in fact they had been predicted in particle physics theories for many years as the particles responsible for breaking the symmetry of the fundamental forces. The observation of spin 0 particles such as the Higg’s boson is one of the main goals of particle accelerators today. In the original models of inflation the universe supercools during the inflationary era, with the end of inflation marked by a period of reheating so that the important processes of baryogenesis and nucleosynthesis can take place and the universe can evolve through

9 Chapter 2. Cosmological inflation the radiation- and matter-dominated eras described in the Hot Big Bang model. In Guth’s original model, the inflaton φ is trapped in a local minimum of a potential V (φ), with inflation ending when the inflaton undergoes a phase transition, tunnelling to reach the absolute minimum of the potential [17]. Guth’s model however had its own problems. The formation of bubbles required for tunnelling created problems as they predicted the unwanted side-effects of either large-scale inhomogeneity or the formation of separate universes depending on their proximity to each other [20; 21]. Guth’s ideas were resurrected almost a decade later by La and Steinhardt [22] who solved the problems caused by these bubbles with their ’extended inflation’ model, however their results violated the laws of relativity and were finally ruled out by the results of the COBE satellite [23]. In 1982 a new model of inflation, ’new inflation’ was proposed by Linde, Albrecht and Steinhardt amongst others [24; 25; 26]. In this new model, inflation begins near the maximum of a relatively flat potential; the inflaton slowly rolls down the potential, with the end of inflation marked by oscillations at the minimum. We will see later that this idea of ’slow roll’ and the approximations which follow become important in solving the equations describing the inflationary epoch. This modification to Guth’s work represented the first viable model of inflation and most subsequent models are based on the ideas of new inflation, differing only in their choice of potential V or scalar field φ. The behaviour of the inflaton field is governed by a set of partial differential equations which can be derived using particle theory from the Lagrangian L for the scalar field, given by (neglecting coupling with other fields)

1 ∂φ ∂φ L = ηµν V (φ) (2.11) −2 ∂xµ ∂xν −

The Lagrangian and the stress-energy-momentum tensor (2.3) yield equations for the energy density and pressure of the ﬁeld [8]

1 ρ = φ˙2 + V (φ), (2.12) φ 2 1 P = φ˙2 V (φ). (2.13) φ 2 − These results can be used in the Friedmann equations with k = 0, which can either be assumed or will occur quickly under the arguments for solving the ﬂatness problem. This yields two further expressions for the evolution of the ﬁeld:

a˙ 2 8πG 1 = V (φ)+ φ˙2 , (2.14) a 3 2

10 Chapter 2. Cosmological inﬂation

φ¨ + 3Hφ˙ + V,φ = 0. (2.15) where V,φ is the derivative with respect to the field φ. The second expression, the equation of motion for the scalar field is analogous to that of a harmonic oscillator, where the damping term is given here by 3Hφ˙. In order to solve this set of equations we need to know how long the period of inflation must last in order for the problems of the Big Bang to be sufficiently solved. This would require at least 60 e-foldings, N of inflation where

a N(t)=ln tend (2.16) at is the log of the ratio of the scale factor at the end of inflation to the scale factor at time t (the log is for convenience since e60 1026) [8]. Any model of inflation must ≈ ensure that this requirement is met in order to be viable. Equations (2.12)-(2.15) can in theory be solved to fully describe the evolution of the inflaton field. In practice however this has only been done for the very simplest forms of the potential V (φ). Our connection between the inflationary models and our observation of the CMB lies in the primordial vacuum fluctuations resulting from the quantum nature of the scalar field. These fluctuations freeze into the expansion of the universe during inflation, ultimately generating large scale structure as the seeds for gravitational instability. We will consider the freezing in of perturbations in section 3, ultimately to find the primordial power spectrum Pζ of density fluctuations ζ which can be used to make powerful predictions for the temperature anisotropies observed in the CMB. In particular, the power spectrum is characterised by its scale dependence, quantified by the spectral index ns, with a dependence on the scale k of fluctuations given by,

d ln P (k) n =1+ ζ . (2.17) s d ln k

The value of ns obtained through CMB measurements has ruled out many models of inflation which could not be fitted. The spectral index has a value close to ns = 1, which represents a scale-invariant spectrum [27; 28], or in other words, scales are frozen in at almost the same time. It is useful to define the conformal time τ which is used as a coordinate of time when integrating from the decoupling time to the present day in chapters 5 and 6. The conformal time difference between decoupling td at the surface of last scattering and the present t0 is t0 a(t ) 0 dt (2.18) a(t) Ztd

11 Chapter 2. Cosmological inﬂation and the conformal distance is the speed of light multiplied by this time.

2.3 The slow-roll approximation

The equations for the inflaton field can be simplified using the slow-roll approximation. In the slow-roll approximation, the inflaton is assumed to roll down a large and flat potential, so that the highest time derivative may be dropped from equations (2.14) and (2.15), then 3Hφ˙ + V, 0, (2.19) φ ≈ 8πGV (φ) H2 . (2.20) ≈ 3 The accuracy of the slow-roll approximation is quantified through two slow-roll parameters, ǫ and η, 1 V, 2 ǫ = φ , (2.21) 16πG V 1 V, η = φφ . (2.22) 8πG V Both parameters must be small to justify the use of equations (2.19) and (2.20), although this is a necessary, not sufficient condition and an additional requirement for slow-roll inflation is that the relation in (2.19) is an attractor solution [29]. The limits ǫ,η << 1 can be applied for the purposes of using (2.19) and (2.20), and η < 1 corresponds precisely toa> ¨ 0, the condition for inflation itself. In the slow-roll approximation, the number of e-foldings N given by (2.16) can be obtained in terms of the potential V (φ) by

1 φ V N dφ (2.23) ≈ M 2 V ′ p Zφend where φend is the value of the field when the slow-roll approximation becomes invalid and inflation ends. It is straightforward to see that, without any major calculations, many inflationary models can be thrown out at this stage, simply because they do not provide enough e-foldings. In addition, it can be shown that the spectral index defined by (2.17) can be expressed in terms of the slow-roll parameters, so that

n 1 + 2η 6ǫ. (2.24) s ≈ −

Using the slow roll parameters it is possible to ﬁnd solutions for inﬂationary models

12 Chapter 2. Cosmological inflation with most forms of the potential V , and to test these solutions against well-known limits on, for example, N and ns. In addition, in models which do not have a process built in for ending inflation whilst still within the slow roll approximation, η = 1 will mark the end of inflation.

2.4 Reheating

After applying a period of rapid expansion we have seen that the inconsistencies of the Hot Big Bang model have been smoothed out. However in solving one problem another has been created. With acceleration,a ¨ > 0, the temperature of the universe falls exponentially, such that at the end of inflation the universe is in a supercooled state. As stated in section 2.1, the Hot Big Bang processes which follow inflation, such as the formation of atomic nuclei, occur at well-known far greater temperatures, and in order to remain consistent, all inflationary models must have a mechanism for defrosting the universe to this state. In most models of inflation this is achieved through an epoch of reheating at the end of inflation, during which the production of light particles heats the universe. The underlying physics for this process is reviewed extensively at the beginning of chapter 4. The picture of reheating has evolved over the last two decades and often acts simply an add-on to a favoured inflationary model, although this is not always the case (for a recent review see [30]) and the reheating process could have an important effect on the spectrum of gravitational waves [31]. One model which overhauls the idea of a separate reheating phase is warm inflation, introduced in the following section and considered extensively throughout this thesis. In this model, the production of radiation during inflation negates the need for a separate reheating phase and allows a smooth transition to the post-inflation Hot Big Bang processes.

2.5 Inﬂationary models

Having looked at some of the general characteristics of inflation we can now look at some specific inflationary models. There are at present a vast number of models which are yet to be ruled out by observational evidence; here only an overview of the most important models is presented. The required ingredients for an inflationary model are as follows:

13 Chapter 2. Cosmological inﬂation

A scalar field or fields φ. The scalar field is believed to be responsible for symme- • try breaking in particle physics and often the field is chosen to fit with a particular favoured particle physics model.

A form of the potential V (φ), determined by the speciﬁc behaviour of the ﬁeld • that the model requires.

A mechanism for reheating to allow nucleosynethesis and the subsequent evolution • of the universe to continue.

Following on from new inflation, models have become increasingly sophisticated in their choice of a field or potential, not for the sake of adding complication but motivated by underlying theories. Many of the modern inflationary models are now deeply connected to particle physics and often deal with theories such as supersymmetry, supergravity and superstring theory which are barely touched upon here; a complete review of all of the models to date would certainly be exhausting, although a good review is provided by [29]. Here only the most significant advances are discussed, often themselves giving rise to many subsets of models. Shortly after new inflation and motivated to obtain a model viable on Planck scales, Linde proposed a model of ’chaotic inflation’, in which the initial position of the inflaton would be arbitrary and vary from place to place [32]. Chaotic inflation is one of the most extensively studied single field models as it allows simple monomial potentials (those with a single power of φ), most simply V φ2. Single field models predict ∝ that gravitational waves may be observable in the data from the forthcoming Planck Surveyor satellite. More recent calculations made in supergravity theory have suggested that, at Planck scales, single field inflation may not be sustainable and favour multi- field models. The first of these, ’hybrid inflation’ was proposed in 1991; here a second scalar field is coupled to the inflaton field described in new inflation [33]. This allows the use of more sustainable flat potentials. There are several observational predictions that allow us to constrain inflationary models. The most important of these are the shape of the CMB temperature power spectrum, the ratio of gravitational to density perturbations and the non-Gaussianity of temperature fluctuations. Often the models that are studied in most depth have the quality that one of these is particularly interesting and makes the model distinguishable from others. One model considered throughout this thesis is warm inflation and it is useful to introduce some of its ideas and equations at this stage.

14 Chapter 2. Cosmological inﬂation

2.6 Warm inﬂation

Warm inflation refers to a model in which dissipative effects become important during the inflationary epoch. In the cold inflationary scenario we have seen that the production of radiation takes place after inflation. In warm inflation production of radiation occurs alongside the expansion, resulting in inflaton interactions which prolong inflation [34; 35; 36]. Decay mechanisms arising in supersymmetry theory for the production of particles have recently shown that warm inflation can be consistent with both hybrid and chaotic inflation [37; 38]. The requirement for production of radiation during inflation is that the radiation 1/4 density is such that ρr >H. In this scenario the inflaton equation of motion (2.15) contains a friction term Γ which contributes to the damping so that

φ¨ + (3H + Γ )φ˙ + V,φ = 0, (2.25) where Γ = Γ (φ, T ) depends on the temperature T in general, although the temperature- independent friction coeﬃcient Γ = Γ (φ) is also considered in places here. The ratio of thermal over expansion damping can be expressed as

Γ r = , (2.26) 3H where negligible r gives the standard cold inflation scenario and large r gives warm inflation [39]. Here a strong r >> 1 and weak r << 1 regime for warm inflation may also be identified; these regimes are considered separately in some of the related work. Warm inflation predicts the production of radiation, where the energy density of the universe, equation (2.12) gains an extra term,

1 ρ = φ˙2 + V (φ)+ ρ (T ), (2.27) 2 r for temperature T . Using equations (2.25) and (2.27) with the continuity equation (2.4), the production of radiation is described by [40]

2 ρ˙r + 4Hρr = Γ φ˙ . (2.28)

The equations for warm inﬂation can be simpliﬁed using the slow-roll approximation In this case, equation (2.19) becomes

3Hφ˙(1 + r)+ V, 0, (2.29) φ ≈

15 Chapter 2. Cosmological inﬂation

(2.20) holds, and from (2.28), 4Hρ Γ φ˙2. (2.30) r ≈ Here an extra slow-roll parameter is required

1 Γ, V, β = φ φ , (2.31) 8πG Γ V where β,η,ǫ << 1+ r are the constraints for warm inflation to be valid. The main motivation for considering warm inflation in this thesis is that recent calculations have shown that this model generates non-Gaussianity at a level that should be observable in the CMB temperature fluctuations by the forthcoming Planck satellite [41; 42]. The predictions are considered closely in chapter 6 where the warm inflation bispectrum is calculated, and the model is considered throughout this thesis as the evolution of fluctuations is followed from their primordial origin.

16 Chapter 3

The source of density ﬂuctuations

The great success of inflation can be attributed to the explanation of the distribution of large scale structure we observe around us, and the prediction of the temperature anisotropies in the CMB data. Both come about through the freezing in of fluctuations as the universe expanded during inflation. We will see in chapter 6 that the primordial power spectrum of density fluctuations, imprinted in the Cosmic Microwave Background, is in very good agreement with the predictions of inflation [43; 44; 45] and is a powerful tool for distinguishing between models. This chapter deals with the very source of density fluctuations, which in the standard inflationary theories lies in the quantum nature of the universe during inflation. An alternative proposed by warm inflation is that the fluctuations could be thermal in origin and directly related to the coupling of the inflaton field and surrounding radiation. The discriminating factor for these, and indeed any other model for the source of density fluctuations, lies in the form of the primordial power spectrum of the perturbations Pζ , which in chapter 5 we will see is closely related to the spectrum of temperature anisotropies in the CMB. Indeed it is possible to compute the theoretical spectrum from any given expression for the primordial power spectrum and so these calculations are an important check for consistency when building any inflationary model. In this chapter the primordial power spectrum of perturbations is calculated in the case that quantum vacuum fluctuations seed density perturbations, and in the case of warm inflation where the primordial power spectrum depends on the temperature dependence of the friction coefficient and the regime of warm inflation, parameterised by the variable r introduced in equation (2.26). Most of this chapter describes new work on thermal fluctuations done in collaboration with Professor Ian G. Moss [46].

17 Chapter 3. The source of density ﬂuctuations

3.1 Comoving scales

To understand the source of the density fluctuations we must consider the evolution of different scales in the early universe, and in particular whether a scale is larger or smaller than the horizon size or Hubble radius H−1. It is the solution to the causality problem of the Big Bang theory that is also the answer to generating inhomogeneity. During inflation the Hubble radius H−1 remains roughly constant. However compared to the fixed comoving scale of a fluctuation, the Hubble radius is shrinking. After inflation, H−1 increases more rapidly than the fixed comoving scale. Perturbations of the inflaton field are commonly considered by taking the Fourier transform such that each scale is associated with a wavenumber k and physical wavenumber k/a where a(t) is the scale factor. During inflation, perturbations of the inflaton field are generated inside the Hubble radius and grow until they leave the horizon when k = aH, the horizon crossing (or more precisely horizon exit). At this point perturbations effectively freeze as they lose causal contact and become density fluctuations. Some time after inflation ends they re-enter the horizon since the Hubble radius increases. The fluctuations can then evolve to large scales and, through gravitational instability, form the large scale structure we see today. The parameter z = k/aH proves to be convenient when carrying out a numerical analysis of the resulting expressions later in the chapter, such that horizon crossing occurs when z = 1.

3.2 Vacuum ﬂuctuations as the source of density ﬂuctua- tions

According to the standard theory of inflation, density fluctuations arise from quantum vacuum fluctuations during inflation [47]. Consider a scalar field φ with quantum perturbation δφ, then we can set

φ(x,t)= φ(t)+ δφ(x,t). (3.1)

During inﬂation it is useful to make a Fourier expansion in terms of the comoving wavenumber k, so that ik.x δφ(x,t)= δφk(t)e . (3.2) Xk

18 Chapter 3. The source of density ﬂuctuations

The evolution of perturbations follows

2 −2 δφ¨k(k,t) + 3Hδφ˙k(k,t)+ k a δφk(k,t) = 0, (3.3) and perturbations have a power spectrum Pφ(k) deﬁned through the 2 point correlation function, the ensemble average,

3 3 ′ δφkδφk′ = (2π) P (k)δ (k + k ). (3.4) h i φ

At horizon crossing the inflaton perturbations are frozen into the expansion of the universe and become density fluctuations. At this point the power spectrum is well known to be [48; 49] H 2 P (k = aH)= . (3.5) φ 2π In order to consider the evolution of the perturbation before and after horizon crossing it is useful to define the primordial curvature perturbation ζ, known as the Bardeen variable, that remains constant for scales outside the horizon [50], and then match this to the inflaton perturbation at the horizon crossing k = aH. In defining ζ, Bardeen took advantage of gauge choices, noting that although the slicing of space-time can make density perturbations appear very different, they must give the same results regardless of gauge. The most general FRW metric with perturbations is

ds2 = (1 + 2α)dt2 2aβ dtdxi + a2 g (1 + 2ϕ) + 2γ dxidxj (3.6) − − i ij i|j where a is the scale factor and a choice of the functions α, βi, γi|j and ϕ represents a gauge choice. For a general hypersurface with pressure p and density ρ, the curvature perturbation is deﬁned by

1 1 dρ ζ = ln(1 + 2ϕ)+ . (3.7) 2 3 p + ρ Z and it is useful to proceed using the uniform curvature gauge, where ϕ = 0, Using the slow roll approximation, (2.19) and (2.20) with ρ V (φ), (3.7) reduces to ≈ H ζ = dφ. (3.8) φ˙ Z The curvature perturbation in this case is related to the inﬂaton perturbation simply by

ζ = ζφδφ (3.9)

19 Chapter 3. The source of density ﬂuctuations

and the primordial power spectrum of the curvature perturbations Pζ (k) can be deﬁned as for (3.4) through

ζ(k )ζ(k ) = (2π)3P (k )δ3(k + k ), (3.10) h 1 2 i ζ 1 1 2 where, 2 Pζ (k)= ζφPφ(k). (3.11)

Using ζφ = H/φ˙ from (3.8) with (3.5), at horizon crossing the primordial power spectrum of density perturbations is

H 2 H 2 H4 P (k = aH)= = . (3.12) ζ ˙ 2π ˙ 2 φ (2πφ)

Following horizon crossing the evolution of perturbations depends simply on the dynamics of inﬂation and subsequently only on the cosmological parameters. The evolution of the curvature perturbation ζ to a generic perturbation g is given in linear theory by

gk = T (t, k)ζ(k), (3.13) where the transfer function T (t, k) is determined by the cosmological parameters. The ﬂuctuations ζ therefore are the seeds for the density perturbations that grow into large scale structure and are imprinted in the Cosmic Microwave Background radiation. The power spectrum is characterised by its scale dependence, quantiﬁed by the spectral index, d ln P (k) n =1+ ζ , (3.14) s d ln k with the simple power law behaviour,

P (k) kns−1. (3.15) ζ ∝

Latest measurements by the WMAP satellite [51] give a value for the spectral index, n = 0.963 0.014, slightly below the scale-invariant spectrum n = 1 for which scales s ± exit the horizon at almost the same time (this is better known as the Harrison-Zel’dovich spectrum after the authors [27; 28]). Inﬂationary models diﬀer in their prediction for ns and as this parameter is now known to very high accuracy, this is a powerful constraint on cosmological models. Indeed the wavenumber dependence of ns, known as the tilt or the running αs, where, dn α = s , (3.16) s d ln k can provide yet a further constraint on models. WMAP measurements give the con-

20 Chapter 3. The source of density fluctuations straint, α = 0.034 0.026, on this parameter [51]. s − ± The restrictions on these parameters alone have already ruled out whole classes of inflationary models. In particular for power-law inflation V (φ) φα, models with ∼ α 4 are ruled out to 3σ [52]. ≥ Similar equations also exist for the power spectrum Pgrav and spectral index ngrav of gravitational waves also predicted from vacuum fluctuations, although their treatment is not covered here.

3.3 Thermal ﬂuctuations as the source of density ﬂuctu- ations

It is possible to obtain results, consistent with CMB data, for which thermal rather than quantum fluctuations provide the source of density fluctuations [53; 54]. This is the prediction of the warm inflationary scenario [55]. It is likely that particles are produced continually during inflation, but their density would also rapidly diminish due to the expansion of the universe. If the net particle density is non-negligible then we have the warm inflation scenario and in this case the quantum statistical fluctuations in the particle number could be large enough to influence the classical inflaton field and produce density fluctuations [56; 57]. By considering a non-expanding situation first, standard thermal field theory methods can be used to reduce the description of the inflaton/radiation system to a pair of coupled stochastic equations, one for the inflaton and one for the radiation density [58; 59]. We will consider the case where the radiation remains close to thermal equilibrium and in the next chapter we will see that this is a valid assumption to make. In this case, the particle production rate, the dissipation in the inflaton equation of motion and the source terms in the inflaton equation are all related to the friction coefficient Γ defined in chapter 2 [58; 59; 60; 61]. The power spectrum of fluctuations in inflationary models is calculated first for a friction coefficient Γ depending on φ only, and then for the more general case with additional dependence on the temperature T . We shall see in chapter 4 that temperature dependence is a common outcome when making explicit calculations of the rate of particle production in this scenario. The temperature dependence of the friction coefficient can be parameterised by the coefficient c where,

TΓ c = T . (3.17) Γ

Worthy of particular mention are high-temperature models of warm inﬂation for which

21 Chapter 3. The source of density fluctuations c = 1 [62; 57; 63; 64] and two-stage decay models in which the inflaton decays − via an intermediate virtual boson [65], for which the friction coefficient is predicted to have c = 3 [63]. Most consistent warm inflationary scenarios now use the two- stage decay mechanism (see Ref. [55] for a review) where light radiation particles are produced via a virtual heavy boson. Although the high-temperature models are difficult to realise without producing large thermal corrections to the potential, making the models unstable, we will consider the predictions for fluctuations in both cases here and go on to give results for any temperature-dependent model of warm inflation. Individual models with temperature dependent friction coefficients have been exam- ined once before. The numerical code written for Ref. [56] solves coupled perturbation equations for the time evolution of the inflaton field, radiation and metric. Most no- tably, the model with Γ T −1 considered in [56] produced a distinctive oscillatory ≈ power spectrum, however it has not been possible until now to formulate general laws for the functional dependence of the power spectrum.

3.3.1 Fluctuations in the case Γ = Γ (φ)

Before considering the temperature-dependent case, in this section the calculation of inflaton fluctuations is reviewed in the case where the friction coefficient is independent of temperature, i.e. when Γ Γ (φ). (3.18) ≡ The power spectrum in this case is known and was calculated by [56]. This result will be useful in the later numerical simulation of fluctuations as a check for consistency. The interaction of the inflaton field with radiation can be analysed using standard techniques of non-equilibrium field theory [66; 67]. The inflaton field can be described simply by a stochastic system whose evolution is determined by a Langevin equation [68]. In flat spacetime, the Langevin equation takes the form

2 φ(x,t)+ Γ φ˙(x,t)+ V = (2Γ T )1/2ξ(x,t) (3.19) −▽ φ where 2 is the ﬂat spacetime Laplacian and ξ is a stochastic source with a predeter- ▽ mined probability distribution. Strictly speaking there should be an additional term

φ(x,t)ξ1(x,t) on the right hand side of (3.19) but since φ changes slowly this is considered to be absorbed into the ξ term. Assuming a weakly interacting radiation gas, the probability distribution of the source term can be approximated by using a Gaussian noise distribution with a correlation function [58; 61],

ξ(x,t)ξ(x′,t′) = δ3(x x′)δ(t t′). (3.20) − −

22 Chapter 3. The source of density ﬂuctuations

Restricting this analysis to a Gaussian noise approximation proves to be convenient when carrying out numerical work later. We now have a starting point for this analysis in flat spacetime, however to consider the evolution of perturbations we must rewrite (3.19) and (3.20) allowing for the expansion of the universe. The equivalence principle can be used to adapt the flat spacetime Langevin equation to an expanding universe during warm inflation when T >H. The Langevin equation retains its local form as long as the microphysical scales are small compared to the cosmological scale [55; 61]. However, in the rest frame of the radiation α there will be a non-zero 3-velocity vr with respect to the cosmological frame and an advection term must be included, such that (3.19) becomes,

¨ ˙ −2 α φ(x,t) + (3H + Γ )φ(x,t)+ Γ a vr ∂αφ(x,t)+ Vφ (3.21) a−2∂2φ(x,t) = (2Γ T )1/2ξ(x,t), − eff where a is the scale factor and ∂2 is the Laplacian in an expanding frame with coordinates xα. The correlation function for the noise expressed in terms of the comoving cosmological coordinates now becomes,

ξ(x,t)ξ(x′,t′) = a−3(2π)2δ3(x x′)δ(t t′) (3.22) − −

A new parameter Γeff has been introduced because at this stage the eﬀects of the expansion on the noise term cannot be predicted. We are now ready to introduce the inﬂaton perturbation δφ and follow the same approach to the case of vacuum perturbations using (3.1). In this case, δφ is now the linear response due to the source ξ and the velocity vr will be of the same order as

δφ. Terms due to δΓ through δΓ = Γφδφ are small in the slow-roll approximation and these terms will only appear in the next section where the friction coeﬃcient is temperature-dependent. Substituting φ φ + δφ into the Langevin equation and taking the Fourier trans- → form gives a full description for the evolution of δφ,

2 −2 1/2 δφ¨(k,t) + (3H + Γ )δφ˙(k,t)+ Vφφδφ(k,t)+ k a δφ(k,t) = (2Γeff T ) ξ(k,t). (3.23)

In addition, the inﬂaton will generate metric inhomogeneities. However, by choosing a uniform expansion rate gauge it can be shown that we can discard these on sub-horizon scales [69]. The analysis of the Langevin equation can be simpliﬁed by introducing the new

23 Chapter 3. The source of density ﬂuctuations time coordinate z = k/(aH) and by introducing the slow roll parameter ǫ deﬁned in (2.21). We are led to the equation for δφ(k, z)

(1 ǫˇ)2δφ′′ (3r + 2)(1 ǫˇ)z−1δφ′ +ǫ ˇ′δφ′ + δφ = (2Γ T )1/2(1 ˇǫ)1/2ξˆ (3.24) − − − eff − where a prime denotes a derivative with respect to the new variable z and the parameter ǫˇ = ǫ/(1+r). In addition, the slow roll approximation further simpliﬁes (3.24); keeping only leading terms reduces this to

δφ′′ (3r + 2)z−1δφ′ + δφ = (2Γ T )1/2ξ.ˆ (3.25) − eff

Whilst changing variable, the noise term has been rescaled so that the correlation function is now deﬁned by

ξˆ(k,t)ξˆ(k′,t′) = (2π)2δ3(k k′)δ(t t′)k−3. (3.26) − − D E The perturbation equation (3.25) with the above Gaussian noise approximation can be solved using the Green function techniques described in [39]. The solution is

∞ ′ ′ 1−2ν 1/2 ′ ′ δφ = G(z, z )(z ) (2Γeff T ) ξ(z )dz . (3.27) Zz where the retarded Green function G(z, z′) can be expressed in terms of Bessel functions, π G(z, z′)= zνz′ν (J (z)Y (z′) J (z′)Y (z)), (3.28) 2 ν ν − ν ν and for z < z′, 3 Γ + 3H ν = (1 + r)= . (3.29) 2 2H There are corrections due to the time dependence of ν, however these are similar in size to terms which we have already discarded in the slow roll approximation. We would now like to ﬁnd the functional form of the power spectrum in the case

Γ = Γ (φ). The inﬂaton power spectrum Pφ(k, z) is deﬁned by

′ 3 ′ δφ(k, z)δφ(k , z) = Pφ(k, z)(2π) δ(k + k ) (3.30)

Substituting the ﬁrst order inﬂaton perturbation (3.27) and the correlation function (3.28) gives ∞ −3 ′ ′ 2 ′ 2−4ν Pφ(k, z) = (2Γeff T )k dz G(z, z ) (z ) (3.31) Zz in the slow-roll approximation. This integral is treated analytically for large values of

24 Chapter 3. The source of density ﬂuctuations r in the appendix of [46], where it is shown that at horizon crossing, z = 1,

√π P (k, 1) k−3 (ΓH)1/2T. (3.32) φ ≈ 2

This is a familiar result also derived by [56]. An approximation which works remarkably well for both large and small values of r is given by [57]

√π P (k, 1) k−3 H1/2(Γ + 3H)1/2T. (3.33) φ ≈ 2

The accuracy of this approximation will be tested by the numerical approach in the following sections. We can match this result at horizon crossing using the deﬁnition of the curvature perturbation (3.7)-(3.9) to give

5/2 1/2 −3 √π H (Γ + 3H) T Pζ (k)= k . (3.34) 2 φ˙2

It is possible to test this approximation for the primordial power spectrum in the case of the friction coeﬃcient independent of temperature (c = 0) numerically. However ﬁrst a similar approach will be made to the temperature-dependent case.

3.3.2 Fluctuations in the case Γ = Γ (T,φ)

In this section the power spectrum of perturbations is calculated in the case where the friction coeﬃcient depends on temperature,

Γ Γ (T, φ). (3.35) ≡

The temperature dependence is parameterised by c, deﬁned in equation (3.17). The analysis follows much the same method as the temperature-independent case, calculating ﬁrst Pφ and matching at the horizon crossing to obtain the spectrum of density

ﬂuctuations Pζ . In this case however, a non-zero value for c gives additional terms in the ﬂuctuation equations since

δΓ = ΓT δT + Γφδφ. (3.36)

As in the temperature-independent case, the Γφδφ term can be dropped as it depends on the slow-roll parameter β defined in (2.31), and is first order in the slow-roll approximation. However the δT terms are leading order and result in a coupled set of equations for the perturbations of the inflaton and the radiation.

25 Chapter 3. The source of density ﬂuctuations

3.3.2.1 Radiation ﬂuctuations

The dominant source of fluctuations in the radiation is the inhomogeneous energy flux of the inflaton field which can be formulated using first order perturbation theory [56; 69; 70; 71]. A second source of radiation fluctuations due to the microscopic movement of particles described in [56] and [53] is rendered insignificant in all regimes of warm inflation [46]. Following the formulation of [69] and [71], the momentum flux J and energy flux δQ transferred from the inflaton field to the radiation are given by:

J = Γ φδφ,˙ (3.37) −

δQ = φ˙2δΓ. (3.38)

The energy and momentum equations for the radiation including these ﬂuxes from [71] are

δρ˙ + 3H(1 + w)δρ = ka−1(1 + w)v +ρ ˙ α +(1+ ω)ρ κ + δQ, (3.39) r r − r r r

(a4ρ v )˙= ka3ρ α + ka3(1 + w)−1(wδρ J), (3.40) r r r r − where w = p/ρ, and the metric perturbation κ and the function α depend on the gauge choice. The uniform expansion gauge again proves a convenient choice of gauge, for which κ and α are zero. Then,

δρ˙ + 3H(1 + w)δρ = ka−1(1 + w)v + δQ, (3.41) r r − r

(a4ρ v )˙= ka3(1 + w)−1(wδρ J). (3.42) r r r − Multiplying (3.41) by a5 and diﬀerentiating leads to:

2 2 − 2 −2 δρ¨r + (8 + 3w)δρ˙r + [3(1 + w)H˙ + 15(1 + w)H + wk a 2]δρr = k a J + 5HδQ + δQ.˙ (3.43)

We would like to write J and δQ in terms of the perturbations δρr and δρφ. Using the slow roll approximation (2.19)-(2.20), and treating Vφ as constant, the energy ﬂux is simply:

J = Vφδφ = δρφ (3.44)

Similarly, the energy ﬂux and its derivative can be expressed in terms of δρr using (3.17) and (3.37), together with the slow roll equation (2.30):

δQ = cHδρr, (3.45)

26 Chapter 3. The source of density ﬂuctuations

δQ˙ = cHδρ˙r, (3.46) where the latter is obtained using the slow roll approximation. Now setting w = 1/3 and collecting terms,

1 δρ¨ + (9 c)Hδρ˙ + (20 5c)H2 + k2a−2 δρ = k2a−2δρ . (3.47) r − r − 3 r φ

Here the term on the right with fluctuations δρφ from the inflaton field sources the radiation fluctuations δρr. Note that for c = 0 with δρφ = 0, (3.47) is the standard equation for radiation fluctuations in an expanding universe with a sound speed of √ω = 1/√3 [72]. This equation couples with one for the inflaton fluctuations which we shall now obtain.

3.3.2.2 Inﬂaton ﬂuctuations

We saw earlier that fluctuations in the inflaton field obeyed the Langevin equation

2 −2 1/2 δφ¨(k,t) + (3H + Γ )δφ˙(k,t)+ Vφφδφ(k,t)+ k a δφ(k,t) = (2Γ T ) ξ(k,t) (3.48) and we now have an additional term

δQ cHδρ δΓ φ˙ = = r (3.49) φ˙ φ˙ such that

2 −2 −1 1/2 δφ¨ + 3H(1 + r)δφ˙ + k a δφ + 3cr(Γ φ˙) δρr = (2Γ T ) ξ (3.50)

Equations (3.47) and (3.50) form 2 coupled equations to be solved in order to obtain the ﬂuctuation amplitude at freezeout and hence the power spectrum Pφ. The system was analysed using an analytic approach in [46] by using a matched asymptotic expansion technique. Here we will consider the numerical solution to this coupled system.

3.4 Numerical simulation of coupled ﬂuctuations

3.4.1 Numerical approach

To numerically solve equations (3.47) and (3.50) we ﬁrst introduce the variable z = k/aH and convert the equations in terms of the new variable to give

27 Chapter 3. The source of density ﬂuctuations

′′ 1 δρ (8 c)z−1δρ′ + (20 5c)z−2 + δρ + δρ = 0, (3.51) r − − r − 3 r φ ′′ δρ (3r + 2)z−1δρ′ + δρ 3crz−2δρ = V (2Γ T )1/2ξk−3/2, (3.52) φ − φ φ − r φ where a dash now denotes derivative with respect to z. Introducing the rescaled variables, 3/2 −1/2 −1 y = k (2Γ T ) Vφ δρφ, (3.53)

3/2 −1/2 −1 w = k (2Γ T ) Vφ δρr, (3.54) the perturbation equations become

1 w′′ (8 c)z−1w′ + (20 5c)z−2 + w + y = 0, (3.55) − − − 3 y′′ (3r + 2)z−1y′ + y 3crz−2w = ξ.ˆ (3.56) − − The normalised Gaussian variable ξˆ is obtained using the Box-Muller algorithm (see appendix A.1 for details) [73]. The coupled equations are solved using a fourth order Runge-Kutta routine and are evolved starting with initial conditions w = y = 1 at large negative z and ending at horizon crossing when z = 1 with 1000 steps in − ∼ z. All results are obtained by averaging over a large number of runs, nruns, which is taken to be 1,000 for the later analysis. A further loop over r in the program allowed eﬃcient analysis of the parameter space. The resolution in r varied with dr = 0.25 for c = 0, dr = 1 for c = 1, dr = 2 for c> 0. −

4 4

3 3

2 2

1 1 y 0 y 0

−1 −1

−2 −2

−3 −3

−4 −4 −40 −35 −30 −25 −20 −15 −10 −5 0 0 100 200 300 400 500 600 z frequency (a) (b)

Figure 3.1: The time evolution of the rescaled inﬂaton perturbation y is shown on the left from large z to to horizon crossing z = 1. The histogram on the right shows that the values of y at horizon crossing for 10,000 runs− are Gaussian distributed.

Figure 3.1(a) shows the time evolution of y for several runs from large z to horizon crossing with parameters c = 1 and r = 5. We will study the parameter dependence

28 Chapter 3. The source of density ﬂuctuations closely in the next section but of note here, 3.1(b) shows that the values of the perturbation at horizon crossing are Gaussian distributed.

3.4.2 Dependency on r and c

We are interested in obtaining the power spectrum Pφ given by

P = (2Γ T )k−3 y(z)2 (3.57) φ h i and the dependence on the relative size of Γ to H, parameterised by r and the temperature dependence of Γ parameterised by c. Several cases are considered within the allowed values of c including c = 0, no temperature dependence, and common cases in warm inﬂation motivated by particle physics models, with c = 1, c = 1 and c=3. In particular, the c = 1 case was solved − − numerically by [56] who found an oscillatory power spectrum. Figures 3.2-3.4 show how y(z)2 at horizon crossing depends on r for a given c. h i The theoretical curves plotted over the data points are given by

1 π y2 ( )1/2r−3cr3c−1/2 (3.58) h i≈ 4 3 c obtained from the analytic expression calculated in [46] and rescaled into the new variables. The parameter rc is the point at which the two terms corresponding to coeﬃcients Ac and Bc match. For c = 0 the rational approximation

√3π (1 + r)1/2 y2 (3.59) h i≈ 4 1 + 3r is a very good approximation to the numerical data and is consistent with the power spectrum obtained analytically in [46]. In the Γ T −1 case (c = 1) the density perturbation amplitude is diminished. ≈ − Moreover we have found no evidence for the oscillations in amplitude found previously by [56]. This behaviour is most likely averaged out over the many runs, indeed increasing nruns further brings deviations due to the stochastic nature of the code closer to −2 −3 the best fit curve. In this case the data is fitted to A−1r + B−1r where the best fit coefficients where found to be A = 5.30, B = 37.1. −1 −1 − In the positive c regime, data was obtained for c = 1, c = 2 and c = 3, where the 3c−1/2 3c−3/2 best fit curve is now Acr + Bcr . In these case there is an enhancement of the density perturbation amplitude. The best fit values for Ac and Bc are given in

29 Chapter 3. The source of density ﬂuctuations

0.8

0.7 c = 0 numerical c = 0 approximation 0.6

0.5 〉

2 0.4 y 〈

0.3

0.2

0.1

0 0 2 4 6 8 10 12 14 16 18 20 r

Figure 3.2: The best ﬁt curve for y(z)2 at horizon crossing as a function of r with c = 0. h i

c Ac Bc Range of validity 3 4.51 10−9 3.34 10−6 r> 100 × × 2 1.22 10−6 4.39 10−4 r> 20 × × 1 4.66 10−4 2.94 10−2 r> 10 × × 1 5.30 -37.1 r> 20 − 2 Table 3.1: Best fit coefficients Ac and Bc for the models of y at horizon exit in the case of c = 1, 1, 2 and 3. The range of r for which the coefficients remainh i a good fit is also indicated. − table 3.1, most reliably for c = 1 and c = 2 due to the rapid growth of y2 in the c = 3 h i case. These coefficients correspond to rc = 8.2, only 5% from the theoretical prediction of rc = 8.5 calculated in [46].

3.4.3 Damping eﬀects

Motivated by [72], it would be desirable to include damping effects in the radiation, as these may suppress fluctuations. Although the bulk viscosity should be very small, it is possible that shear viscosity η plays an important part. This would appear on the left hand side of equation (3.24) as a term 3ηz−2δφ [46]. Using the definition of the kinematic shear viscosity

νs = η/ρr, (3.60) and 4 η = ρ τ , (3.61) 15 r r

30 Chapter 3. The source of density ﬂuctuations

0.014

0.012 c = −1 numerical c = −1 approximation 0.01 〉

0.008 2 y 〈 0.006

0.004

0.002

0 20 25 30 35 40 45 50 55 60 r

Figure 3.3: The best ﬁt curve for y(z)2 at horizon crossing as a function of r with c = 1. h i − where τr is the relaxation time of the radiation [72]. The perturbation equation for the radiation (3.43) with the additional term becomes

1 δρ¨ + (9H + ηk2a−2)δρ˙ + (20H2 + k2a−2)δρ = k2a−2J + 5HδQ + δQ,˙ (3.62) r r 3 r where the eﬀectiveness of thermalisation can be parameterised in the second term by

Hτr [72]. As for the undamped case, analytic analysis of this equation in [46] predicts that

c6=0 P (k, 1) ′ φ r 3c c=0 ( ) , (3.63) Pφ (k, 1) ≈ rc where c′ is given by ′ 2c c = 1/2 . (3.64) 1 + (1 + 24cHτr/5)

It is interesting to see how the numerical results for y2 depend on Hτ . Follow- h i r ing the same numerical approach as the undamped case, in our rescaled parameters, equation (3.55) becomes

4 1 w′′ (8 c)z−1w′ + Hτ zw′ + (20 5c)z−2 + w + y = 0, (3.65) − − 15 r − 3 whilst equation (3.56) remains unchanged. Figure 3.5 compares the amplitude y2 h i for c = 3 and Hτr = 0 (no damping eﬀect), Hτr = 0.1 (equilibrium maintained) and

Hτr = 1.0 (marginal thermal equilibrium) plotted on a logscale for clarity. It is clear

31 Chapter 3. The source of density ﬂuctuations

70 c = 1 numerical c = 1 approximation 60

50 〉

2 40 y 〈

0 20 30 40 50 60 70 80 90 100 r

Figure 3.4: The best ﬁt curve for y(z)2 at horizon crossing as a function of r with c = 1. h i

′ that even with Hτr = 1.0, c is still large enough for significant enhancement of the fluctuations. To summarise, this chapter has considered the source of density fluctuations, and their evolution from a quantum or thermal source. The power spectrum of perturbations at horizon exit is well known in the standard inflationary scenario, and in later chapters we will see how this can be used to make predictions for the temperature spectrum of the CMB. For warm inflation the results for a temperature-independent friction coefficient have been carried out analytically in the past. However for the first time results have been established that can give the primordial power spectrum for arbitrary c and r, obtained through a standard perturbation approach to gain a coupled set of equations solved analytically in [46] and shown to work well here numerically. These results will be very important when making future comparisons of the CMB to warm inflation theories with Γ (φ, T ), and in the next section whilst looking in more depth at the particle production mechanisms that can create radiation in the warm inflation scenario, we will also consider the validity of the thermal equilibrium assumption used whilst making the calculations in this chapter.

32 Chapter 3. The source of density ﬂuctuations

15 10

H τ = 0.0 r H τ = 0.1 r 10 10 H τ = 1.0 r >) 2 5 10 log(

0 10

−5 10 −1 0 1 2 3 10 10 10 10 10 log(r)

Figure 3.5: Best ﬁt curves for the amplitude y2 at horizon exit with c = 3 and Hτ = 0 h i r (no damping eﬀect), Hτr = 0.1 (equilibrium maintained) and Hτr = 1.0 (marginal thermal equilibrium). It should be noted that the range of validity in the case of c =3 is r > 100 and for small values of r, equation (3.63) will not be very accurate.

33 Chapter 4

Reheating the universe

In chapter 2 it was shown that the universe supercools during inflation. One of the most important features of any inflationary model is a mechanism for reheating the universe to the hot radiation-dominated universe we know must follow, in other words a graceful exit to inflation. This chapter considers the possible solutions to this problem. Nearly all of this work is new research done in collaboration with Professor Ian G. Moss [60]. The details of a particular reheating mechanism lie in the interactions between the inflaton field φ(t) and the surrounding fields, although the underlying process is always particle production to fill the universe with radiation. The original work on reheating in the 1980s introduced particle production in an ad hoc fashion, assuming the rate of production was proportional to φ˙n [45; 74; 75]. A lot of work was done considering φ˙2 production but this was later superseded by calculations of the production rate for an inflaton field oscillating in the minimum of a steep potential well at the end of inflation [76; 77]. Rather than considering reheating entirely in a period at the end of inflation, a different approach considered particle production from an inflaton field with a small time derivative and calculated the additional friction term in the inflaton equation associated with particle production using linear response theory [64; 78]. Other approaches considered particle production using curved space field theory [79]. Following the initial flurry of work in this area, it became popular to focus not on the particle production, but to simply include it in effective field equations for the inflaton [80; 58]. This was until renewed interest was ignited by preheating, a nonperturbative process of inflaton decay though parametric resonance [81; 82; 83; 84; 85]), and this model has remained popular to this day. However, preheating is a result of oscillations with Planck scale amplitude which are perhaps an undesirable feature of standard models of inflation and preheating may or may not occur depending on the details

34 Chapter 4. Reheating the universe of the particle model; fermions, for example, are not produced through preheating. Perturbative reheating is still therefore a distinct possibility. Here a range of models are considered, including the two-stage decay model considered most consistent for warm inflation models. In two-stage models, heavy fields are considered as decay channels for the production of lighter fields, though the heavy fields are too massive to be produced directly. Those light fields can couple to the heavy intermediate field without developing large masses if their masses are protected by supersymmetry.

4.1 Particle production method

The production of particles is considered both during a separate oscillatory reheating phase as in standard inflationary theory, and in the case of warm inflation where radiation production occurs alongside inflation. A thermal field theory approach is required for the latter calculation and will be used for the former to check against particle production rates which are well known, so that a uniform method of calculating particle production can be presented. Many of the calculations are extremely lengthy and it is useful to first look at some useful definitions and identities used throughout this chapter.

4.1.1 Some useful deﬁnitions

Throughout the calculations that follow, the 3-dimensional momentum integral dk– is used with the shorthand, d3k dk– . (4.1) ≡ (2π)3 Z The Dirac δ-function commonly arises in these calculations. The relation

∞ f(x)δ(x a)dx = f(a), (4.2) − Z−∞ can be considered a deﬁnition of δ(x a). Other identities are − ∞ f(x)δ′(x a)dx = f ′(a), (4.3) − − Z−∞ ∞ f(x)δ′′(x a)dx = f ′′(a). (4.4) − Z−∞ The following integral arises many times here and yields the δ-function and a principle

35 Chapter 4. Reheating the universe value, t ′ i e±iω(t −t)dt′ = πδ(ω) πδ∓(ω) (4.5) ∓ ω ≡ Z−∞ where δ+ and δ− are used to represent the components of this result (which can be derived by using an ’iǫ’ prescription). The full integral yields,

∞ ′ e±iωt dt′ = 2πδ(ω). (4.6) Z−∞

The thermal distribution of bosons n(ω) = (eβω 1)−1 (where β = 1/T for tem- − perature T ) has several special properties including:

1+ n(ω)= n( ω) (4.7) − −

n′(ω)= n(ω) 1+ n(ω) (4.8) { } n(ω + ω )n( ω )= n(ω ) n(ω + ω ) n(ω ) (4.9) 1 2 − 2 1 { 1 2 − 2 } Throughout this chapter Boltzmann’s constant is set to 1.

Polylogarithmic functions arise in section 4.2.4, in particular the dilogarithm Li2 deﬁned by ∞ t Li (z)= z dt (4.10) 2 et z Z0 − which yields two very useful expressions:

∞ βk ∞ Li (1) = d(βk)= kn(k)dk, (4.11) 2 eβk 1 Z0 − Z0 ∞ βk ∞ Li (e−βp)= d(βk)= β2 kn(k + p)dk, (4.12) 2 eβk+βp 1 Z0 − Z0 where n is the thermal distribution.

4.1.2 The particle number density

Following the work of Morikawa and Sasaki [78], the particle creation and annihilation operators are defined for a fiducial free field which coincides with the interacting field φˆ and momentumπ ˆ at time t,

aˆ†(p,t) = ω φˆ( p,t) iπˆ(p,t) (4.13) p − − aˆ(p,t) = ω φˆ(p,t)+ iπˆ( p,t) (4.14) p −

36 Chapter 4. Reheating the universe

2 2 1/2 where the energy of a particle with momentum p and mass m is ωp = (p + m ) , which may depend on time. The transforms are deﬁned by

φˆ(p,t)= d3x φˆ(x,t)e−ip·x. (4.15) Z Assuming that the local number density is spatially homogeneous, the number density n(p) can be defined in terms of an ensemble average using the fiducial free field,

1 † 3 aˆ (p1,t)ˆa(p2,t) = (2π) δ(p1 p2)n(p1). (4.16) 2ωp h i −

This can be expressed in terms of the Wightman function G21(p,t1,t2), deﬁned by

φˆ(p ,t )φˆ(p ,t ) = (2π)3δ(p p )G (p ,t ,t ). (4.17) h 1 1 2 2 i 1 − 2 21 1 1 2

The expression for the number density is then

1 n(p,t)= (ω i∂ )(ω + i∂ )G (p,t ,t ) (4.18) 2ω p − t1 p t2 21 1 2 p

In the work that follows, the energy ωp refers always to the value at time t, unless stated otherwise.

4.1.3 The particle production rate

In theory, one could calculate the particle number density Eq. (4.18) at a certain time given some initial data by solving for the Wightman function directly. In reality, however, it is more useful to consider situations when the particles are produced at a rate depending only on local conditions, for example on the local ﬁeld values or the temperature. Indeed these are the situations that arise in reheating, both during and after inﬂation. In these cases it is useful to calculate the particle production raten ˙ arising from the process.

By letting d/dt = ∂t + ∂t1 + ∂t2 and taking the time derivative of equation (4.18), the particle production rate can be split into two terms,

n˙ =n ˙ mass +n ˙ int. (4.19)

The ﬁrst term proportional toω ˙ p represents particle production due to the changing

37 Chapter 4. Reheating the universe particle mass,

ω˙ 1 n˙ = p (ω i∂ ) + (ω + i∂ ) (ω i∂ )(ω + i∂ ) G (p,t ,t ) mass 2ω p − t1 p t2 − ω p − t1 p t2 21 1 2 p p (4.20) The second term contains everything else and represents particle production through interactions

i n˙ = (∂2 + ω2)(ω + i∂ ) (ω i∂ )(∂2 + ω2) G (p,t ,t ) (4.21) int − 2ω t1 p p t2 − p − t1 t2 p 21 1 2 p It is useful to introduce the self-energy Σ to put these equations into a more usable form. The self energy represents energy gained by the particle through its interactions with the surrounding ﬁelds. Since we are interested in the evolution of operators in a given initial state it is convenient to use the Schwinger-Keldysh ‘in-in’ formulism [66; 67]. Propagators carry two extra internal indices a and b, where the indices a and b can take the values 1 or 2 and the indices can be raised and lowered with a metric c = diag(+1, 1) following ab − Calzetta and Hu [68]. The Schwinger-Dyson equations for this formalism are,

∂2 + ω2(t ) G (p,t ,t ) = dt′G a(p,t ,t′)Σ (p,t′,t ) (4.22) t2 p 2 21 1 2 − 2 1 a1 2 Z ∂2 + ω2(t ) G (p,t ,t ) = dt′Σ a(p,t ,t′)G (p,t′,t ). (4.23) t1 p 1 21 1 2 − 2 1 a1 2 Z For the terms in (4.20) which have just one time derivative of the propagator, it is possible to use a simple trick of introducing a time integral, often used in LSZ formulism [86], where for a non-zero function F ,

t −iω(t−t2 ) F t2=t = dt2∂t2 e F (4.24) | −∞ Z h i Then (4.24) yields,

t (∂ iω )G (p,t ,t ) = dt e−iωp(t−t2)(∂2 + ω2)G (p,t ,t ) (4.25) t2 − p 21 1 2 |t2=t 2 t2 p 21 1 2 Z−∞ t (∂ + iω )G (p,t ,t ) = dt e−iωp(t1−t)(∂2 + ω2)G (p,t ,t ) (4.26) t1 p 21 1 2 |t1=t 1 t1 p 21 1 2 Z−∞ These allow the particle production rate to be expressed in terms of integrals of the propagator and the self-energy using (4.22) and (4.23). It is then possible to use perturbation theory to evaluate these expressions.

38 Chapter 4. Reheating the universe

Considering the leading order in perturbation theory, let

2 2 2 2 2 ωp = p + mσ + g φ (t) (4.27) where mσ is the constant mass of the σ particle and φ(t) is known, and consider the self energy Σ to be O(g4). This scenario could arise for example given an inﬂaton φ and an interaction Lagrangian density = g2φ2σ2/4. L Using Eq. (4.20) and Eqs. (4.22-4.26), to leading order

g2φφ˙ t e−iωp(t−t2) n˙ = Re dt (φ2(t) φ2(t ))G (p,t,t ) (4.28) mass ω 2 2ω − 2 21 2 ( p Z−∞ p ) where G21(p,t,t2) is a free Wightman function with the shifted mass (which need not be in the vacuum state). For the second term, taking the derivative of the Schwinger-Dyson equation (4.22) gives, 2 2 2 2 6 (∂t1 + ωp)(∂t2 + ωp)G21(p,t1,t2)= iΣ21(p,t1,t2)+ O(g ). (4.29)

Used together with Eqs. (4.21), (4.25) and (4.26) the second term then becomes,

t e−iωp(t−t2) n˙ = Im 2 dt Σ (p,t,t ) . (4.30) int 2 2ω 21 2 ( Z−∞ p ) where Σ21 is the self-energy of the σ-ﬁeld to leading order and this term represents the production rate associated with the imaginary part of the self-energy. The formulae for the particle production rates found here neglect the expansion of the universe. It was shown in [60] that to a reasonable degree of accuracy the expansion can be neglected if particle momenta are larger than the expansion rate.

4.2 Particle production calculations

In this section the particle production rate is obtained for three models using the formalism described in the previous chapter. All three models could be a possible solution to the problem of a supercooled universe at the end of inflation. In the first, particle production through oscillations in the minimum of the inflaton potential, the results are very well known and can be obtained using a scattering matrix method, this can be used first to check the consistency of the new approach.

39 Chapter 4. Reheating the universe

4.2.1 Oscillating ﬁelds by the scattering matrix method

Perturbative reheating theories for particle production after inflation predict particle production through small amplitude oscillations of the inflaton or other background field. In many inflationary models, this type of particle production would be eclipsed by preheating from large scale oscillations. This is not always the case however, so this simple example is of interest. In addition the particle production rate was calculated many years ago [76] and so this provides a useful test of the general method in the next subsection. The interaction Hamiltonian in this case is given by

1 H = gφσ2d3x, (4.31) I 2 Z where σ is a light particle ﬁeld and the inﬂaton, with mass mφ, is oscillating in the minimum of its potential such that

φ = φ0 cos(mφt). (4.32)

The decay channel φ σσ produces two σ particles with momenta p , p and → 1 2 energies ω1,ω2. This decay has the matrix element

p , p 0 = 0 a(p )a(p )T exp( i H dt) 0 . (4.33) h 1 2| i outh | 1 2 − I | i Z Using (4.31) and taking the leading order in g this becomes

p , p 0 = g d4x 0 a(p )a(p )σ2φ 0 dt. (4.34) h 1 2| i h | 1 2 | i Z After expressing σ in terms of particle creation and annihilation operators a(p) and a†(p), using commutation relations this reduces simply to

p , p 0 = gφ (2π)4δ(p + p )δ(ω + ω m ), (4.35) h 1 2| i 0 1 2 1 2 − φ where , the reduced matrix element, is deﬁned as M

= gφ . (4.36) M 0

Averaging over angles, the decay raten ˙ is given by the standard formula,

2 p n˙ = M | 1|. (4.37) 8π mφ

40 Chapter 4. Reheating the universe

In the centre of mass frame where p = p , | 1| | 2| 1 p2 = m2 m2 , (4.38) 1 4 φ − σ then 1 2 2 g2φ2 4 mφ mσ n˙ = 0 − . (4.39) 8π q mφ

Since g is typically very small, this type of perturbative reheating would take several Hubble times to take eﬀect, resulting in nonperturbative processes such as preheating often being favoured.

4.2.2 Oscillating ﬁelds by the general method

It is possible to consider the same problem using the general result forn ˙ mass given by equation (4.20). Using the free Wightman function for the vacuum state, given by

1 −iωp(t−t2) G21(t,t2)= e , (4.40) 2ωp and expressing the inﬂaton in terms of exponentials,

1 φ(t)= φ eimφt + e−imφt , (4.41) 2 0 equation (4.20) can be written as

ig2φφ˙ t n˙ = dt e−2iωp(t−t2) eimφt + e−imφt eimφt2 e−imφt2 . (4.42) 16ω3 2 − − p Z−∞ Using (4.5) to evaluate the time integral leaves a series of δ-functions and principle values. After manipulation using (4.2-4.4), equation (4.42) simpliﬁes to give

g2φ˙2π g2φ˙2φ2 m2 n˙ = δ(2ω m ) φ . (4.43) 8ω3m p − φ − 8ω3 ω (4ω2 m2 ) p φ p p p − φ ! The second term averages to zero and as we are interested only in the number density of particles after some certain time can be neglected for the time being. Using the deﬁnition, δ(p p0) δ(f(p)) = ′ − , (4.44) f (p0)

41 Chapter 4. Reheating the universe the delta function can be written,

ωp 1 2 2 δ(2ωp mφ)= δ p mφ mσ . (4.45) − p − r4 − !

It is now possible to seek the number density n(p,t) by integrating over time,

t d3p n(p,t)= dt′ n˙ (p), (4.46) (2π)3 Z0 Z which after carrying out the momentum integral gives

t g2φ˙2 1 n(p,t)= m2 m2 dt′. (4.47) 16πω2m 4 φ − σ Z0 p φ r All of the time dependence lies in φ˙ which integrates to

t 1 1 φ˙2dt′ = φ2t + φ2m cos(m t) sin(m t). (4.48) 2 0 2 0 φ φ φ Z0

Again the second term averages to zero and with ωp = mφ/2 the ﬁrst term gives

1 2 2 g2φ2t 4 mφ mσ n(p,t)= 0 − . (4.49) 8π q mφ

Not only is this result consistent with (4.39), but this method has also provided us with a more detailed insight into the time evolution of n(p,t), with two transient terms present as a result of evaluating the particle production instantaneously. The additional term arising in (4.48) is trivial, however the earlier term in (4.43) requires more work. The resulting integral can be evaluated using the Cauchy Principle Value, ∞ 2 2 2 2 m p π(8mσ m ) φ dp = − φ . (4.50) (p2 + m2 )2(4p2 + 4m2 m2 ) 2m2 m Z−∞ σ σ − φ φ σ Then in full the particle number density n(p,t) is,

1 2 2 m2 m2 2 2 2 2 g φ t 4 φ σ g φ (8mσ m ) n(p,t) = 0 − 0 − φ sin2(m t) cos2(m t) q 2 φ φ 8π mφ − 64π mφmσ 1 2 2 2 2 m mσ g φ0 4 φ − + cos(mφt) sin(mφt) (4.51) 8π q mφ

Over several oscillatory cycles, the production rate averages out to the scattering theory

42 Chapter 4. Reheating the universe result (4.39) and the general method has proven very eﬀective.

4.2.3 Particle production in the adiabatic approximation

In many ways, the opposite case to oscillatory particle production occurs in the ‘adiabatic’ limit when the inflaton has a small time derivative. This is a specialised form of particle production which does not usually occur at leading order in perturbation theory. An important exception occurs when the system starts out and remains close to thermal equilibrium. This type of particle production was first discovered by Hosoya and Sakagama [64] and by Morikawa and Sasaki [78]. This is the first model to consider in which particles are produced alongside inflation, leading to the Γ φ˙ friction term in the inflaton equation of motion. Although warm inflation theories now favour the two-stage decay model that we will consider in section 4.2.4, this is another model for which the particle production rate and corresponding form of the friction coefficient are very well known. The general equation for the particle production (4.20) is again evaluated, this time with φ(t) given by the adiabatic approximation

δφ2(t )= φ2(t) φ2(t ) 2φ(t)φ˙(t)(t t). (4.52) 2 − 2 ≈ 2 −

To proceed, the equations are rewritten by introducing the Fourier transforms,

∞ dω G (t t )= e−iω(t−t2)G (ω) (4.53) 21 − 2 2π 21 Z−∞ ∞ dω δφ2(t )= eiω(t−t2)δφ2(ω) (4.54) 2 2π Z−∞ where the thermal Wightman function G21 can be expressed in terms of a spectral function ρ and the thermal distribution function n (for example, see [55]),

G = i(1 + n)ρ. (4.55) 21 −

After integration, (4.20) reduces to

′ 4 2 ˙2 G21( ωp) n˙ = g φ φ −2 . (4.56) ωp

Inserting the Breit-Wigner form of the spectral function [59],

ρ = (ω2 ω2 2iωτ −1)−1 (ω2 ω2 + 2iωτ −1)−1 (4.57) − p − − − p

43 Chapter 4. Reheating the universe where τ is known as the relaxation time, (4.56) gives

′ 4 2 ˙2 τn (ωp) n˙ = g φ φ 3 . (4.58) − ωp where n′ is the derivative of n with respect to p. The formula for particle production in this case can also be derived using the methods of Ref. [78], and it is closely related to the work of Ref. [64]. It can be seen that the particle production is exponentially small for temperatures less than the σ particle mass due to the thermal factor n′(ω ). In fact, Eq. (4.56) − p vanishes at zero temperature due to a general property of the Wightman function [87]. The best way to view this type of particle production is as a type of transport phenomenon, similar to thermal or electrical conductivity. Particles are produced as the system responds to the disturbance of thermal equilibrium caused by the changing mass. Increasing the relaxation time τ therefore gives more time for the mass to change, which in turn drives the system further from equilibrium, increasing the particle production. Energy is balanced in the system by introducing the friction term Γ φ˙ into the inﬂaton equation. The total radiation energy is

d3p ρ = nω (4.59) r (2π)3 p Z

The time variation of ρr contains a term from the time variation of ωp, which relates to the change with time of the inflaton effective potential, and the Γ φ˙2 term in (2.28), where nω˙ d3p τn′(ω ) Γ = p = g4φ2 p . (4.60) φ˙2 − (2π)3 ω2 Z p The friction coefficient obtained in this way agrees with the previous results of [64] obtained using linear response theory.

4.2.4 Particle production in the 2-Stage decay model

In the 2-stage decay model of particle production, the inflaton decays into light thermal scalar particles through an intermediate virtual boson. This model was introduced in the context of warm inflation [88], but the set-up can occur quite easily in models which contain both heavy and light particles. This is another ’adiabatic’ process, but this time the low temperature behaviour is suppressed by a power law instead of the exponential suppression that was found in section 4.2.3. The particles produced have a fixed (small) mass. This removes some ambiguity that arises in the definition of the particle number for massless particles, and it leads

44 Chapter 4. Reheating the universe to a ’cleaner’ result that we will be able to evaluate numerically in section 4.3. The interaction Lagrangian is,

1 1 1 = g2φ2χ2 hmσ2χ λσ4, (4.61) LI −4 − 2 − 4! where g,h,λ and m are constants and m = gφ0 can be chosen. The mass of the intermediate virtual boson is also considered to be greater than the inﬂaton mass. The third term represents the self-interaction of the particles, which allows them to come to thermal equilibrium. The initial state is taken to be in thermal equilibrium with temperature T <

iΣ (p,t,t ) = g4h2m2 dk– dω– dω– dω– eiω2(t−t2)G (p k,t t )G a(k,ω ) 21 2 1 2 3 σ21 − − 2 χ2 1 Z φ2(ω ω )G b(k,ω )φ2(ω ω )G (k,ω ), (4.62) × 1 − 3 χa 3 3 − 2 χb1 1 where Gσ is the σ-propagator and Gχ is the χ-propagator. Taking the Fourier transform, the self-energy becomes

m2φ2 d3k dω iΣ (p,t,t ) = 4g4h2 φ˙2 e−iω(t−t2 )G (p k,t t ) (α(ω)[1 + n(ω)])′′ . 21 2 m8 (2π)3 2π σ21 − − 2 χ Z (4.63) To proceed, the deﬁnition of the thermal Wightman function for the ﬁeld σ with particle distribution n can be used with the low energy approximation k,ω ,m , so ≪ χ that (see (4.55) and (4.57))

iα k 2 k Gχ21( ,ω2)= Gχ1 ( ,ω1) 4 (1+ n(ω1)) , (4.64) ≈ mχ

1 k Gχ11( ,ω2) 2 , (4.65) ≈ mχ 1 2 k Gχ2 ( ,ω2) 2 , (4.66) ≈ mχ where α = 4ω /τ h2m2 and τ is the heavy particle decay width. In general α is a p ∝ function of energy and momentum and was calculated explicitly by [63] for non-zero temperatures. After inserting these 3 expressions into (4.21), and with further integration, the

45 Chapter 4. Reheating the universe particle production is

m2φ2 d3k α n˙ = 4g4h2 φ˙2 n′′( ω ω )(1 + n(ω )) . (4.67) int − ω m8 (2π)3 ω − k − p k p χ Z k There are no assumptions made for the distribution function n at this stage. This general result will be evaluated numerically in section 4.3, however further analysis can be made by assuming the thermal distribution with identities (4.7) and (4.8).

Taking the simplest case where mσ = 0 sets ωk = k and ωp = p. Then the particle production can be written, g2h2 m4φ˙2 n˙ int = 2 8 F (p), (4.68) 2π τmχ where m = gφ and

∞ ′′ 1 F (p)= T 2 n(ω ) 1 e−nωp/T . (4.69) p n2 − ( n=1 ) X The function F (p) is plotted in figure 4.2 and is compared to the thermal distribution. Through this approximation it is clear that particles are produced with a spectrum far from thermal equilibrium. This raises the important question of whether the particles can thermalise in the timescale set by inflation, or if the radiation ultimately imprinted in the Cosmic Microwave Background, was out of thermal equilibrium. This will be explored in depth in the next section. The reheating coefficient Γ can be obtained by integrating over the momentum,

nω˙ Γ = d3p. (4.70) φ˙2 Z This gives (in the small mass limit),

2 2 4 αg h m 3 Γ = 4 8 T , (4.71) 4π mχ which agrees with the friction coefficient calculated from the effective field equations in [63].

4.3 The thermalisation of particles

The particle production rates calculated in the previous section can now be used to take a closer look at the evolution of a system which may be evolving and departing from thermal equilibrium.

46 Chapter 4. Reheating the universe

Figure 4.1: The two vertex loop Feynman diagram contributing to the imaginary part of the σ self energy Σ21.

Thermalisation has been considered in the context of reheating through classical non-linear ﬁeld theory by [89; 90; 91; 92] and is a common problem in several other areas, in particular within the subject of Bose Einstein Condensates, from which some of the approach used here is adapted. Here, the analysis begins with a relaxation time approach to thermalisation and goes on to make the far more complex full treatment using the Boltzmann collision integral in an expanding universe. In this section a pseudo-particle approximation is used, where the Wightman function takes a thermal form, such that the results of section 4.2 can be used, but with a distribution function n(p,t) that is allowed to depart from the thermal distribution, or even begin far away from it. The net change in particle number depends on several terms,

n˙ = + + , (4.72) Sp Sr Sc where represents the particle production through one of the mechanisms in sections Sp 4.2.3–4.2.4 , represents the dilution of particles due to the expansion of the universe, Sr and is the Boltzmann collision term which thermalises the distribution. Sc

47 Chapter 4. Reheating the universe

Figure 4.2: The momentum dependence of the particle production rate for small φ˙ with the two-stage decay model. The function p2F (p) is plotted. A thermal distribution p2n is shown for comparison in blue.

The expansion of the universe causes the stretching of the physical wavelengths of the modes by the scalar factor a and is represented by the term. In the absence of Sr collisions, n is a function of the comoving wavenumber p/a such that n(p,t)= n(p/a) andn ˙ = Hp∂pn, where H is the expansion rate. Therefore the redshift term is

= Hp∂ n. (4.73) Sr p

Note that whilst this dilutes the particle number without eﬀecting the distribution shape, it also pulls the peak of the distribution to low p, an eﬀect we have already seen in the term for the 2-stage decay model. This will add to the extra care that is Sp required when computing this term. The collision term for four particle interactions in Eq. (4.61) is

λ2 = dµ dµ dµ (2π)4δ(p + p p p )δ(ω + ω ω ω ) Sc 2ω 2 3 4 2 − 3 − 4 p p2 − p3 − p4 p Z [(1 + n)(1 + n )n n nn (1 + n )(1 + n )] , (4.74) × 2 3 4 − 2 3 4 where dµ = d3p/(16π3ω ). This represents the dominant 2 2 particle scattering. p → Additional smaller contributions from eg 3 1 scattering or quasi-particle annihilation → are not considered here, and indeed they do not conserve particle number making their treatment more complex.

48 Chapter 4. Reheating the universe

The Boltzmann collision term conserves energy, and as a consequence the integral of the Boltzmann equation gives

ρ˙r + 4Hρr = S, (4.75) where the source term S is d3p S = ω . (4.76) (2π)3 Sp p Z 2 In the oscillating case S = Γ ρφ and in the slowly evolving limit S = Γ φ˙ . These can be combined into the energy conservation,

ρ˙r + 4Hρr = Γ (ρφ + pφ), (4.77) where pφ is the average pressure term. The simplest way to analyse the Boltzmann term in (4.72) is to take a close to equilibrium approximation, introducing a thermal distribution function nT and a relaxation time τr. Defining the effective temperature by matching the energy density gives, d3p (n n )ω = 0. (4.78) (2π)3 − T p Z It is then possible to use the thermal particle production rates, such that (4.72) becomes, n˙ = RF (p)+ Hp∂ n τ −1(n n ), (4.79) p − r − T where R is the combination of pre-factors in, for example, equation (4.68): Equation (4.79) can be evaluated using a fourth order Runge-Kutta scheme for the time derivatives and second order differences for the momentum derivatives. This numerical procedure is both fast and stable, with most of the results by this method taking less than one second on an ordinary desktop computer. The 2-decay model is considered throughout the following sections, for which we are interested in how far the distribution departs from equilibrium and therefore its consistency as an alternative model to a separate reheating phase.

4.3.1 In the oscillating phase

Here the main phase of reheating during the oscillating phase is considered, which occurs whilst ρ >> ρ , and ends, according to Eq. (4.77), when H Γ . During φ r ≈ this period, the pressure averages to zero over the oscillation period and the universe

49 Chapter 4. Reheating the universe expands like a pressure free cosmological model, with a t2/3 and ∝ 3 −1 H(t) = H(0) 1+ H(0)t , (4.80) 2 3 −2 R(t) = R(0) 1+ H(0)t . (4.81) 2 where R is the pre-factor to F (p) in equation (4.68) and the second equation follows from R φ˙2 ρ . ∝ ∝ φ Some numerical results for the momentum distribution obtained from Eq. (4.79) are shown in figure 4.3 for two different relaxation times. The distribution thermalises, and does so more quickly with smaller relaxation times as might be expected. The momentum distribution of the source term shows up clearly in the plots at early times, before the relaxation takes effect, returning the distribution to thermal equilibrium. The initial temperature for the numerical solutions has been set equal to the de Sitter temperature H(0)/2π, which is consistent with the assumptions used in the particle production calculations. The evolution of the temperature is shown in figure 4.4. After a sharp rise to a maximum, the temperature falls off as t−1/4. This agrees with the analytic solution to the integrated Boltzmann equation (4.77) [75; 93].

4.3.2 In the slow-roll phase

In the slow-roll phase of inflation it is typical to expect small values of φ˙. The particle production rates calculated in sect 4.2.4 can be applied, provided we can justify the thermal hypothesis which was used. During the slow-roll phase of inflation, both H and Γ vary very little over several Hubble times, and they can be treated as constants in the Boltzmann equation (4.79). The numerical solutions show the existence of an attractor with non-zero temperature and a spectrum close to thermal equilibrium. The final momentum distribution does not show any dependence on the initial distribution, but it is dependent on the relaxation time, as shown in figure 4.5. Small relaxation times, corresponding to relatively large values of the self-coupling λ, lead to a final distribution very close to thermal. The parameters for the numerical solution were chosen to place the temperature in the range T >H required for the consistency of the particle production calculations. The time evolution of the temperature agrees very well with the analytic solution to the energy equation Eq. (4.77), which has the form

T = T 1 e−Ht , (4.82) ∞ −

50 Chapter 4. Reheating the universe

(a)

(b)

Figure 4.3: The stationary momentum distribution for the oscillatory phase in the two-stage decay model using the relaxation-time approximation. The relaxation times are τr =0.1/H(0) (top) and τr =0.05/H(0) (bottom). As might be expected, shorter relaxation times produce a spectrum which is closer to thermal equilibrium. The constant R(0) = 50H(0) and mφ =2H(0).

51 Chapter 4. Reheating the universe

Figure 4.4: The time evolution of the eﬀective temperature for the oscillatory phase with the two-stage decay model using the relaxation-time approximation. The de Sitter temperature H/2π is shown for comparison. The relaxation time is τr = 0.1/H(0), the constant R(0) = 50H(0) and mφ =2H(0). when Γ T 3. ∝

4.3.3 Thermalisation with the full Boltzmann collision integral

While the full treatment of the collision integral is expected to give similar results to the relaxation time approach, the thermalisation could be very diﬀerent as the full Boltzmann equation also has stable attractor solutions with a non-zero chemical potential µ of the form 1 n(p)= . (4.83) e(ωp−µ)/T 1 − While the collision term describes 2 2 particle scattering, there are neglected terms, → for example responsible for 3 1 scattering, which give zero contribution with conser- → vation of energy. In order to obtain charge neutral solutions with µ = 0 these processes would have to be considered. However as the distribution is expected to remain relatively close to equilibrium in these simulations, the chemical potential should stay close to zero. Following a similar method to [94] and [95] the integral can be reduced analytically from 9 to 2 dimensions. This method is detailed in appendix B. The result is,

D = θ(ω + ω ω m )min(p,p ,p ,p ) Sc ω p p3 p4 − p − σ 2 3 4 × p Z [(1 + n)(1 + n )n n nn (1 + n )(1 + n )] dω dω , (4.84) 2 3 4 − 2 3 4 p3 p4

52 Chapter 4. Reheating the universe

(a)

(b)

Figure 4.5: The stationary momentum distribution for diﬀerent relaxation times in the two- stage decay model using the relaxation-time approximation. The relaxation times are τr = 0.2/H (left) and τr =0.05/H (right). As might be expected, shorter relaxation times produce a spectrum which is closer to thermal equilibrium. The constant R = 15H and mσ =0.25H.

53 Chapter 4. Reheating the universe

Figure 4.6: The time evolution of the effective temperature for different initial conditions with the two-stage decay model using the relaxation-time approximation. In each case, the momentum distribution reaches a stationary state with the effective temperature shown. The relaxation time τr =0.1/H, the constant R = 15H and mσ =0.25H.

where p2 is obtained from ωp2 through energy conservation and a new constant is introduced, D = λ/64π3. Equation (4.79) is solved numerically with the full collision term for , once again Sc for the two stage decay model. A fourth order Runge-Kutta scheme is used for time derivatives and second order differences for the momentum derivatives. A mesh of 100 grid points in p and 200 1000 in time (depending on the model parameters) is − used. The high resolution in time is to evolve the thermalisation slowly and thus avoid numerical instabilities, although rather than a fixed grid size in time the program ends when the distribution stabilises (such that specific criteria are matched). The collision term itself is solved using a 2D Simpsons integrator. In order to avoid instability at low momenta the source term is damped with a factor p2/(p2 + H2), as the calculation of the source term cannot be trusted for p less than H anyway. In addition it is required to avoid a mesh so fine that would bring in grid points at very low momenta. Solving the full collision term is computationally far more demanding than using a relaxation approximation, however this final point regarding the mesh size also helps to keep the procedure time down and the results to H(0)t 5 are still obtained in well ∼ under an hour. Numerical results for the full collision term with the two stage decay model are shown in figures 4.7 and 4.8, and obtained using the same values for constants R

54 Chapter 4. Reheating the universe

and mσ. The distribution reaches a stable non-zero temperature as expected and is consistent with the ﬁndings using the relaxation time approach. A visible inspection of the results suggests that the constant D τ −1 as an order of magnitude estimate. ∼ r

Figure 4.7: The stationary momentum distribution in the two-stage decay (model D) using the full collision integral gives a good check for consistency with ﬁgure 4.5. The constant R = 15H and mσ = 0.25H, here the constant D = 10H, which visually is comparable to a relaxation time τr =0.1/H.

In this chapter we have made a uniform approach to the problem of reheating the universe, either through a separate reheating phase or through a coupling with the inflaton field during inflation. Most importantly it is clear that the thermalisation of particles in the case of 2-stage decay will allow the particle distribution n(p,t) to reach equilibrium in many cases. This is good news for warm inflation as a consistent inflationary model and the results give additional support to the 2-stage decay model, already considered to be the most reliable particle model yet for warm inflation. The assumption of thermal equilibrium made in chapter 3 whilst calculating the primordial power spectrum of fluctuations for temperature-dependent warm inflation models can now be considered sound. Equally there are cases in which the finite relaxation time of the distribution could have observational consequences if indeed thermal fluctuations are the source of density fluctuations in the Cosmic Microwave Background. This could lead to distortions in the spectrum of fluctuations and further work considering this scenario could be interesting.

55 Chapter 4. Reheating the universe

Figure 4.8: The time evolution of the effective temperature for different initial conditions with the two-stage decay (model D) using the full collision integral. As with the relaxation time approximation, the momentum distribution reaches a stationary state with the effective temperature shown. The two curves represent the same set of parameters but with starting values of T/H(0) = 1 and 3. The constant D = 10H, with R = 15H and mσ =0.25H.

56 Chapter 5

The cosmic microwave background

There are few examples of observations or discoveries that have completely reshaped a subject in the way that the Cosmic Microwave Background (CMB) has cosmology. Indeed it is this snapshot of the universe, at a mere 400,000 years old, that allows the work in this thesis to be testable and ultimately useful. In this section the angular power spectrum of temperature fluctuations will be calculated, with it’s characteristic peaks and troughs which depend closely on the cosmological parameters and the model of inflation at the source of the fluctuations. Although none of the work presented in this chapter is new, the methods used to process and analyse CMB data will be very useful in chapter 6 when the bispectrum of the CMB temperature anisotropies is considered. To begin quantifying the link between the primordial fluctuations and observable temperature anisotropies we will continue chronologically from the end of the last chapter, at the end of inflation.

5.1 Evolution of perturbations after inﬂation

In chapter 3 we followed the evolution of perturbations from primordial vacuum fluctuations to the horizon exit, when the curvature perturbation becomes a classical quantity. Although the region of interest for constraining inflationary models lies firmly inside this era, in order to make any analysis of the CMB anisotropy we must consider the subsequent evolution of perturbations to the last scattering surface when the CMB photons were released and on to the present day.

57 Chapter 5. The cosmic microwave background

At horizon exit, the curvature perturbation ζ evolves to recombination, the last scattering surface at which point photons decouple and are left to freely move through space, carrying with them the temperature fluctuations. If the probability of a photon colliding after recombination is considered to be negligible, this surface can be thought of as the surface of an almost perfect sphere and the evolution of the photons follows the collisionless Boltzmann equation [8]. Although the photons can travel long distances without impact, in reality collisions do occur and in addition there are several processes which may alter the spectrum of anisotropies by the time we view them. Three of these which may be of importance are reionization, gravitational lensing and the Sunyaev- Zel’dovich effect. The possible reionization of the universe at a point well after recombination is a nonlinear process that should be considered. Observations of absorption lines in the light coming from distant objects such as quasars suggest that the formation of large scale structure could provide the energy for reionization, and give a rough estimate on the age of the universe at that time [96]. Reionization would once again create free electrons and, in a repeat of the processes leading up to recombination, the photons would scatter off the free electrons creating a new last scattering surface. A new cosmological parameter, the optical depth for scattering τ, quantifies the reionization:

t0 τ = σ n(t)dt (5.1) Zt where σ is the Thomson scattering cross-section and n is the number density of free electrons [8]. The probability P that a photon has scattered a second time after the last scattering surface for a given τ is then

P = 1 e−τ(tlss) (5.2) − and the expected power spectrum can be recalculated appropriately; in fact this only effects scales small enough to be inside the horizon size at the new last scattering surface. Gravitational lensing, the redirection of photons due to the gravitational pull of in- tervening galaxies, may shift some of the power associated with the characteristic peaks in the CMB power spectrum into the troughs, resulting in a spreading out of the spectrum [97]. The total power however is still conserved and calculation of cosmological parameters is not significantly effected . In the Sunyaev-Zel’dovich effect [98], the CMB photons scatter off the hot gas in galaxy clusters. This tilts the black body spectrum over the frequency range; increasing the intensity slightly at high frequency and decreasing it slightly at low frequency. The

58 Chapter 5. The cosmic microwave background very large frequency range used in forthcoming satellite experiments will measure this to an extremely high accuracy and any corresponding eﬀect on the anisotropies can then be removed from the observations.

5.2 Measurement of the CMB

The CMB was first discovered in 1965 by Penzias and Wilson [99] who observed a black body spectrum at a temperature of 2.7K [100] across the entire sky. Early measurements of the CMB gave weight to the already popular Big Bang model, as the radiation had been predicted in work as early as 1947 as a relic of element formation [101]. Early measurements of the CMB were made through balloon-borne experiments, which in the 1980s observed the CMB dipole; this dominates the anisotropy and is subtracted to calculate the fluctuations from primordial sources. The dipole is a consequence of our relative motion against the CMB, and has been used to accurately calculate the velocity of our local group. The first major CMB measurements at small scales were made by the Cosmic Background Explorer (COBE) satellite, which launched in 1990 and returned data for 4 years. COBE had three separate apparatus [102; 103]: FIRAS (Far Infra-Red Abso- lute Spectrometer) made measurements of the frequency spectrum and confirmed that the CMB was a black body to 50 parts in a million [104], providing convincing evidence that the source of the radiation is primordial. DIRBE (Diffuse Infrared Background Experiment) measured the Cosmic Infrared Background (CIB) mapping the emissions of stars and galaxies including those from the very first stars, providing an important insight into the physics of the first star formations. The third apparatus DMR (Dif- ferential Microwave Radiometer) made measurements of the temperature anisotropy, confirming the presence of fluctuations for the first time [13]. The measurements made by COBE-DMR are now considered to be on a large scale with a beam width of 7◦ ≈ and a multipole moment range up to l 20 where l 180◦/θ. Although this resolution ≈ ≈ was not enough to measure the first of a series of acoustic peaks predicted to be observable in the angular power spectrum, the addition of ground based and balloon-borne experiments gave a reasonable estimate of the position of the first peak. More recently, measurements made by the Wilkinson Microwave Anisotropy Probe (WMAP) have probed small scales with spectacular success with an increased resolution to around 15 arcminutes and l up to 1000. Currently, balloon-borne experiments extend this range to l 1500. WMAP launched in 2001 and has remained in orbit in Lagrange ≈ L2, releasing data in 2003, 2006 and 2009. WMAP has ten differencing assemblies which

59 Chapter 5. The cosmic microwave background measure at ﬁve diﬀerent frequencies, spanning from 23-94GHz.

5.3 CMB data analysis

The calculation of the CMB bispectrum in chapter 6 uses data from the WMAP satellite, and in particular the 3rd year data release. The WMAP science team presents the data in a series of files, both as raw time-ordered data collected by the satellite, and using iterative algorithms [105] as sky maps of the temperature fluctuations ∆T/T for each differencing assembly and frequency1. Over time the archive of data has grown as skymaps are given for each single year, as well as coadded maps derived from data over several years. Any analysis of the CMB data begins here with one of these maps, however there are several caveats to consider. The most obvious is the contamination caused by the galactic plane and point sources. These problems are alleviated by effectively ignoring the galactic plane and the point sources stored in the WMAP point source catalog

[106]. The WMAP team provide a mask taking both into account to leave a factor fsky to analyse. Although the galactic plane is the most obvious problem and requires drastic mea- sures, one could expect that there would be foreground contamination in all directions. In fact there are several sources, namely synchotron, free-free and thermal dust emis- sion which are modelled to provide templates to subtract from the sky maps [107]. Although these templates are provided in the archive, the ’foreground-reduced’ maps are used here, which have been through this analysis already.

5.4 The angular power spectrum

The angular power spectrum of ﬂuctuations can be derived from the maps provided by the WMAP team. This can be combined with the results of other (e.g. balloon- borne) CMB experiments and thanks to the spectrum’s sensitive dependence on the cosmological parameters, compared to the predictions of early universe models. Here we obtain the angular power spectrum using a similar method to the WMAP team [107] and consider the subtleties involved with analysing CMB data. Measurement of anisotropy is made using an angular correlation function on the CMB temperature map, with the temperature T recorded at co-ordinates ~n. Expanding

1CMB data ﬁles are publically available on the Legacy Archive for Microwave Background Data (LAMBDA) website. http://lambda.gsfc.nasa.gov/

60 Chapter 5. The cosmic microwave background

T into spherical harmonics [102] gives

∆T (~n)= a Y (~n), (5.3) T lm lm Xl,m where the variance Cl of the observable coeﬃcients alm is 1 C = a 2 . (5.4) l 2l + 1 | lm| m X

In reality calculating the Cl spectrum from any one map, even a foreground-reduced map gives poor results due to masking and the noise which dominates the data above l 250. Fortunately the noise can be neglected when calculating the cross-spectrum ≈ deﬁned as 1 Cij = ai aj∗ . (5.5) l 2l + 1 lm lm m X where ij ij ij Cl = wl Cl + Nl δij, (5.6) and the noise Nl is uncorrelated between any two diﬀerencing assemblies, i.e. when i = j. 6 Equation (5.6) indicates that the true CMB signal Cl is obtained by dividing the ij observed power spectrum by a factor wl

ij i j 2 wl = blbl pl (5.7)

i known as the window function. The bl is the beam transfer function for map i which is provided in the WMAP data archive and describes the ﬁnite resolution of the antenna.

The pixel transfer function p accounts for the numerical scheme used to obtain the alm coeﬃcients from the temperature maps. Finally correcting for the factor fsky left by the mask yields the correct Cl for comparison to models, at least for l & 10.

The alm coefficients are most commonly obtained from the CMB data using the HEALPix package2. HEALPix (Hierarchical Equal Area isoLatitude Pixelisation) is a scheme for dividing a 2-sphere into equal areas for analysis. The WMAP temperature maps are formatted for use with the HEALPix software package which contains a number of routines for analysing the maps. The routine anafast obtains alm coefficients from a WMAP temperature map and can apply masking before output. The above method can then be followed using the coefficients from two maps to obtain Cl. This gives results consistent with the best fit Cls provided by the WMAP team, who follow a similar method, but average over every possible combination of two maps.

2HEALPIX is publically available software and can be downloaded at http://healpix.jpl.nasa.gov/

61 Chapter 5. The cosmic microwave background

6000

5000 ) 2 4000

(microK 3000 π l(l+1)/2 l 2000 C

1000

0 10 100 1000 l

Figure 5.1: The predicted angular power spectrum Cl for the ΛCDM +ns = 1 model is plotted obs here against the observed spectrum Cl measured by the WMAP satellite. The units are in µK2.

The angular power spectrum Cl is usually represented in a plot of l(l+1)Cl/2π; this factor is used as a flat spectrum would correspond to equal anisotropy at all angular sky scales. The CMB power spectrum, Cl calculated from the WMAP data via this method is plotted as points in figure 5.1, along with a predicted spectrum using the set of best-fit parameters for scale-invariant perturbations with a Λ-dominated matter content. The predicted spectrum is converted to µK2 to match the units of anisotropy presented by the satellite data. The observed spectrum contains two distinct regions.

The ﬁrst, approximately l < 100, is known as the Sachs-Wolfe plateau [108] and • corresponds to angular scales which were larger than the horizon size at the last scattering surface.

The second, 100

geometry of the universe, with a flat (Ωtot = 1) universe corresponding to a first peak at l 220. The height of the first peak depends on many of the cosmological ≈ parameters.

The low multipoles are dominated by the cosmic variance, which is sadly unavoidable in any observation since we have only one universe to observe. Cosmic variance is a problem at large angular scales since alm are random Gaussian variables giving width

62 Chapter 5. The cosmic microwave background to the distribution. The cosmic variance is given by [109],

1 ∆Cl = Cl, (5.8) 1 l + 2 q which increases at low l.

5.5 Power spectrum predictions using numerical code

The theoretical angular power spectrum for a given cosmological model is related to the power spectrum of primordial perturbations Pζ by

2 ∞ Ctheor = k2P (k) g (k,τ ) 2 dk, (5.9) l π ζ | T l 0 | Z0 where P kns−1 and the transfer functions g , defined below are a function of ζ ∝ T l conformal time (see 2.18) but are evaluated at the present τ0. The shape of the angular power spectrum can therefore be used as a powerful dis- riminator between cosmological models. The power spectrum predicted by a particular cosmological model can be computed using software such as CMBFAST for comparison to the observed spectrum Cl [110; 111]. One of the reasons that CMB observations have so remarkably transformed inflationary theory research has been the ability to calculate theoretical predictions for different models with a high degree of accuracy and very quickly. We are fortunate that the calculation of the power spectrum Cl involves only linear perturbation theory and avoids the complications of nonlinear physical processes, whilst still being sensitive to all of the cosmological parameters that we wish to determine. The CMBFAST software has been the most widely used code to carry out these calculations for some years and has spawned several similar tools 3. CMBFAST solves the Boltzmann equation, which describes the transport of photons, by splitting the time integration into a geometrical term which does not depend on the cosmological model used and a source term which varies slowly. This information is stored in the transfer functions gT l which at the present time τ0 for a flat geometry are calculated by CMBFAST using [110]

τ0 g (k,τ )= S(k,τ)j [k(τ τ)]dτ. (5.10) T l 0 l 0 − Z0 where jl are spherical Bessel functions.

3CMBFAST is publically available software and can be downloaded at http://www.cmbfast.org

63 Chapter 5. The cosmic microwave background

0.1 k=2x10-5 k=0.01 k=0.05 k=0.20 0.05 ) τ

0 Source function S(k, -0.05

-0.1 200 250 300 350 400 450 500 τ

Figure 5.2: The source function S, plotted as a function of time for several wavenumbers, varies only at the recombination epoch 300Mpc ≈

Here we consider only the scalar temperature anisotropies although similar equations apply to tensor components and polarization anisotropies, which CMBFAST can also calculate and output if required. The geometrical term jl needs only to be calculated once; it is stored in a matrix and can be used for any subsequent calculation regardless of cosmological model. The source function S contains all of the dynamical effects on CMB anisotropies; it does not depend on l and varies slowly with wavelength, thus requiring fewer sampling points than the geometrical term. It is made up of several components which can produce anisotropies, including major contributions from intrinsic anisotropy, the gravitational potential and the integrated Sachs-Wolfe effect, which is important after the recombination epoch. Also important are terms from photon polarization and Thomson scattering [110]. The source function is plotted in figure 5.2 as a function of the conformal time τ for various k values. The integration sampling carried out by CMBFAST is not required to be uniform as the recombination epoch provides the dominant contribution to the integral. The sampling can also vary with k, as for long wavelengths S varies slowly during recombination. The transfer functions gT l(k) os- cillate rapidly in k due to the spherical Bessel function. Figure 5.3 plots gT l(k) at l=200. CMBFAST can extend to non-flat models, in which the geometrical term becomes an ultraspherical Bessel function [111], this has the implication that CPU time is increased, however only flat geometries are considered here and are favoured by the WMAP data. In chapter 6 we shall see that several of the functions required for anal-

64 Chapter 5. The cosmic microwave background

0.01

0.005 (k) 0 T,200 g

-0.005

-0.01 0 0.02 0.04 0.06 0.08 0.1 k

Figure 5.3: The transfer function gT,200(k) plotted against wavenumber k. The oscillations are caused predominantly by the spherical Bessel function. ysis of the CMB bispectrum can be obtained using straight-forward manipulation of the CMBFAST code as the relevant equations closely resemble equation (5.9). This brings us to the present day situation for constraining inflationary models: The position of the acoustic peaks are now known to a high accuracy, confirming a geometry very close to flat, a large dark energy contribution and constraining all of the cosmological parameters to a good degree of accuracy. Although most inflationary models can be tuned to fit the angular power spectrum, using the WMAP data their predictions for the degree of non-Gaussianity and gravitational waves can begin to be tested, and forthcoming CMB experiments such as the Planck Satellite, which launched in 2009, promise to start eliminating models which cannot fit the data.

65 Chapter 6

Testing models of inﬂation with CMB non-Gaussianity

The angular power spectrum has provided us with a remarkable insight into the com- position and history of our universe. However most inflationary models can be tuned to fit the spectrum, and in our quest to find a definitive model of inflation we must turn to other ways to distinguish between models. Potentially, one of the most powerful tools to distinguish between inflationary models will be the existence (or otherwise) of non-Gaussianity in the temperature anisotropies. Calculations show that the statistics of the CMB anisotropies should de- viate only a small amount from purely Gaussian. This non-Gaussianity is made up in part by any non-linear processes occuring after recombination which are independent of inflation, and this sets a minimum value for the observable non-Gaussianity. These effects include the non-linear dynamics of gravity and foreground sources. The inflationary model will determine any non-Gaussianity arising from primordial processes and if the CMB anisotropies are indeed significantly non-Gaussian, a large number of single-field models would be eliminated and the data could powerfully discriminate between those that remain. With forthcoming satellite data expected to tell us for certain the extent of any non- Gaussianity, there is now much work in this area, making predictions for each model of inflation in preparation. Here the bispectra for warm inflation and the curvaton model are calculated, two models which predict a detectable non-Gaussian signal, however the method used here could equally be applied to many other models. This chapter contains new research done in collaboration with Professor Ian G. Moss [42].

66 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

6.1 Quantifying the non-Gaussianity

Earlier, we met the two-point correlation between ζ(k1) and ζ(k2) which describes a Gaussian distribution. Any non-Gaussianity is defined by non-vanishing higher correlation functions. Most studied are the three-point correlation function known as the bispectrum and the four-point correlation function known as the trispectrum. In the simplest models of inflation the non-Gaussianity is described by a single parameter fNL which shall be defined shortly. Currently the WMAP measurements only mildly constrain the non-Gaussianity to f < 114, however the Planck satellite is expected to either detect non-Gaussianity | NL| or to place the limit f < 5 at which non-primordial sources become significant [52]. | NL| Single field models of inflation predict f 1 through primordial sources, rendering | NL|≪ their investigation uninteresting here, however many multi-field or more exotic models do predict f 5. In addition each different model has a bispectrum with a different NL ≫ dependence on wavenumber, allowing the possibility of distinguishing between models, and the separability of models is also considered here.

6.2 The CMB bispectrum

The three-point correlation function is the most promising statistic for constraining the non-Gaussianity parameter. Indeed the importance of several non-Gaussian statistics was investigated in depth by [112], who showed that the bispectrum was by far the most powerful tool for detection. The aim of any bispectrum analysis of a given model is to predict its observable

CMB bispectrum, Bl1l2l3 and compare to the satellite data we have available in an eﬃcient way. Simply written, the theoretical bispectrum is given by the ensemble average Bm1m2m3 = a a a . (6.1) l1l2l3 h l1m1 l2m2 l3m3 i

Here we recall that alm are the coeﬃcients in the spherical harmonic expansion (equation 5.3). It is convenient to write the bispectrum in terms of the Wigner 3j coeﬃcients [113], l l l 1 2 3 , (6.2) m1 m2 m3 ! which take an integer or half-integer value. They can be computed in this form using the Racah formula [114], although we will only deal with the case where m1,m2,m3 = 0 which will be later evaluated in terms of Gamma functions.

67 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

The angle-averaged CMB bispectrum, used for comparison between theory and data, is deﬁned in terms of the Wigner 3j coeﬃcients,

l1 l2 l3 m1m2m3 Bl1l2l3 = Bl1l2l3 . (6.3) m1m2m3 m1 m2 m3 ! X

The primordial bispectrum for an inﬂationary model Bζ (k1, k2, k3) is deﬁned through the three point correlation function

3 3 ζ(k1)ζ(k2)ζ(k3) = (2π) B (k , k , k )δ (k1 + k2 + k3) , (6.4) h i ζ 1 2 3 where the shape of Bζ (k1, k2, k3) depends on the inflationary model and the normal- ization is specified through a different non-Gaussianity parameter f in each case. For this analysis, the primordial bispectrum will be rewritten as a Legendre poly- nomial expansion [115]

B (k , k , k )= f (k , k )P (k )P (k )P (kˆ1 kˆ2), (6.5) ζ 1 2 3 l 1 2 ζ 1 ζ 2 l · cyc X Xl where Pζ is the primordial power spectrum and ’cyc’ indicates the cyclic permutation.

Assuming that fl(k1, k2) are homogeneous, the bispectrum can be written as the double sum nl Bζ (k1, k2, k3)= fnlB (k1, k2, k3) , (6.6) Xn,l where nl n −n n −n B (k , k , k )= k k + k k P (k )P (k )P (kˆ1 kˆ2). (6.7) 1 2 3 1 2 2 1 ζ 1 ζ 2 l · cyc X nl B (k1, k2, k3) is related through the properties of the inﬂationary model of interest to the observed CMB bispectrum Bl1l2l3 [116]. The observed CMB bispectrum can now be expressed in terms of the transfer function gl(k) assuming that the curvature perturbation ζ decomposes into a linear and non-linear (which shall be denoted NL) part After lengthy analysis the bispectrum decomposes into a series of terms (see appendix C.1) [117]:

nl Bl1l2l3 = fnlBl1l2l3 , (6.8) Xn,l where the Bnl terms are [42]

∞ nl ′ ′ n −n NL 2 B = c(l l l; l1l2l3) b ′ (r)b ′ (r)b (r)r dr, (6.9) l1l2l3 1 2 l1l1 l2l2 l3 perm ′ ′ 0 X Xl1l2 Z

68 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

and ’perm’ is the permutation of l1,l2 and l3. The integral over r must be carried out from the recombination time to the present day and the calculation requires the evaluation of each of the bl functions over this range, for each permutation of l. This is a very time-consuming process but fortunately it is ′ ′ possible to calculate c(l1l2l; l1l2l3) quickly and use CMBFAST to calculate the functions bl which closely resemble the deﬁnition of Cl in equation (5.9). ′ ′ The coeﬃcients c(l1l2l; l1l2l3) are related to the Wigner 6j symbols [117] by

′ ′ ′ ′ (2l1 + 1)(2l2 + 1)(2l3 + 1) ′ ′ 2l+2l3+l1+l2−l1−l2 c(l1l2l; l1l2l3) = (2l1 + 1)(2l2 + 1)i (6.10) r 4π l′ l′ l l′ l l l′ l l l l l 1 2 3 1 1 2 2 2 1 3 . ′ ′ × 0 0 0 ! 0 0 0 ! 0 0 0 ! ( l1 l2 l )

These coeﬃcients have several properties which allow them to be calculated eﬃciently and as they vanish unless l′ lie in the range l l, they are calculated quickly for small i i ± values of l. Two new functions arise, which contain all of the dependence on the transfer function, and are later evaluated using the CMBFAST code,

∞ n 2 n+2 b ′ = k P (k)g (k)j ′ (kr)dk, (6.11) ll π ζ T l l Z0 2 ∞ bNL = k2g (k)j (kr)dk. (6.12) l π T l l Z0 We can now consider our two inﬂationary models in turn to obtain the angle- averaged bispectrum Bl1l2l3 in terms of these functions.

6.2.1 The curvaton model

The curvaton model predicts a level of non-Gaussianity that will be observable in forthcoming CMB experiments, and for this reason is chosen as a suitable model for comparison to warm inflation in this chapter. This multi-field model proposes that an inflaton φ is coupled to a curvaton σ such that the potential has two components [118]

1 1 V (φ, σ)= m 2φ2 + m 2σ2. (6.13) 2 φ 2 σ

The curvature perturbation ζ grows due to oscillations of the curvaton in the minimum of the second component. Contributions to ζ from the inﬂaton are considered negligible in this case, although more complicated models exist with a mixing of the two. The main feature of the curvaton model is the process under which the curvaton decays and

69 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

0.5

0.4

0.3

0.2 π

0.1 l(l+1)/2 l

b 0

-0.1

-0.2

-0.3 10 100 1000 l

(a) l(l + 1)bl(r)/2π plotted for various r values

6e-11

4e-11

2e-11 NL l b 0

-2e-11

-4e-11

10 100 1000 l NL (b) bl plotted for various r values

8e+07

6e+07

4e+07 π /2 2e+07 WI l b 4 l 0

-2e+07

-4e+07 10 100 1000 l 4 W I (c) l bl /2π plotted for various r values

NL W I Figure 6.1: This ﬁgure shows the behaviour of bl(r), bl (r) and bl (r), the terms used in calculating the CMB bispectrum for cold and warm inﬂation, as a function of r. In all cases the output is CMBFAST unnormalised

70 Chapter 6. Testing models of inflation with CMB non-Gaussianity in this case the amount of non-Gaussianity depends on the decay rate and the mass mσ of the curvaton. In the curvaton model, which we will denote with CI as a cold inflation model, the curvaton field generates density fluctuations with a bispectrum that has a dependence on wavenumber given by

CI Bζ (k1, k2, k3) = 2fCI Pζ (k1)Pζ (k2) (6.14) cyc X where ’cyc’ is the cyclic permutation of the momenta k [118]. The non-Gaussianity is parameterised by fCI which is related to the more commonly used non-linearity parameter fNL by 3 f = f . (6.15) CI 5 NL

This relationship can be used later to check results for fCI against the limits on fNL obtained by the WMAP team. In the curvaton model the angle-averaged bispectrum corresponds to l = n = 0 and ′ li = li. Then equation (6.9) becomes

∞ CI CI CI NL 2 Bl1l2l3 = c(l1l20; l1l2l3) bl1 (r)bl2 (r)bl3 (r)r dr, (6.16) perm 0 X Z where (2l1 + 1)(2l2 + 1)(2l3 + 1) l1 l2 l3 c(l1l20; l1l2l3)= , (6.17) r 4π 0 0 0 ! and there are two functions to evaluate. The ﬁrst is

2 ∞ bCI (r)= k2P (k)g (k)j (kr)dk, (6.18) l π ζ T l l Z0 and the second is given by equation (6.12). This notation follows the work of [112], although an alternative notation of α(r) and β(r) for these two functions is used in several papers, eg [115]. Both (6.12) and (6.18) can be computed using a modiﬁcation to the CMBFAST software, in particular a modiﬁcation of the subroutine that handles the equation for NL Cl. Particular care is required when calculating the bl terms due to the absence of Pζ which results in this function changing rapidly with r. The greatest contribution to the integral (6.16) is centred around the recombination epoch. Here, r = r = c(τ τ ) where τ corresponds to the present conformal time and ∗ o− d 0 τd the decoupling time. Figure 6.1 illustrates the behaviour of these functions for several r values in the range 13750Mpc < r < 14350Mpc, centred on r∗. The cosmological

71 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

Parameter Model −1 −1 Hubble Parameter, H0(kms Mpc ) 77.8 Matter energy density, Ωm 0.212 Dark energy density, ΩΛ 0.788 Optical Depth for reionization, τ 0.141 Spectral index, ns 1.0

Table 6.1: The cosmological parameters given by the WMAP best ﬁt to the ΛCDM + ns = 1 model. These parameters are used in the numerical work involving CMBFAST and the calculation of the CMB bispectrum.

model used here is the ﬂat Λ-CDM scale-invariant model with cosmological parameters given in table 6.1. These parameters give the present age of the universe to correspond to cτ0 = 14350Mpc and the age at recombination cτd = 291Mpc. NL The erratic behaviour of the bl function is due to a spike near to r = r∗. Indeed NL bl can be approximated by a delta function in the Sachs-Wolfe plateau, since using g j (kr )/5, T l ≈− l ∗ f 2 f δ(r r ) bNL NL k2j (kr )j (kr)dk = NL − ∗ (6.19) l ≈− 5 · π l ∗ l − 5 r2 Z ∗ Although δ(r r ) is commonly used in the literature, in fact the spike in bNL is not − ∗ l quite centred on r∗. At high l, the spike changes from downward to upward pointing, accompanied by a small shift in its position.

NL Figure 6.2 shows this behaviour of bl as a function of r for various l values. Panel (b) plots this for the range of r around the recombination epoch, centred on r 14000Mpc. Clearly this region dominates the integral (6.16), however it is sensible ∗ ≈ NL to investigate the region at low r also. To do this bl was computed at further points in the range 0

72 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

1e-10 l=2 l=50 l=250 l=500

5e-11 NL

l 0 b

-5e-11

-1e-10 0 2000 4000 6000 8000 10000 12000 r (a)

1e-10

l=2 l=50 l=250 l=500 5e-11 NL

l 0 b

-5e-11

-1e-10 13800 13900 14000 14100 14200 14300 r (b)

1e-10 l=2 l=50 l=250 l=500

5e-11 2 * /r

NL 0 l b 2 r

-5e-11

-1e-10 0 2000 4000 6000 8000 10000 12000 r (c)

NL Figure 6.2: bl as a function of r for various values of l and 0

73 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

integral can then be performed for the l permutation up to a desired lmax.

6.2.2 Warm inﬂation

The form for the bispectrum of warm inflation is based on calculations made by [41] who concluded there should be a detectable non-Gaussianity with a unique dependence on wavenumber. From these calculations, the bispectrum of primordial fluctuations for warm inflation is defined by:

WI WI −2 −2 B (k , k , k )= f (k + k )P (k )P (k )k1 k2. (6.20) ζ 1 2 3 1 2 ζ 1 ζ 2 · cyc X with a new parameter for this model, f WI. The angle-averaged bispectrum for warm inﬂation is derived in appendix C.2, it is

BWI = c(l l 0; l l l ) (6.21) l1l2l3 − 1 2 1 2 3 perm X∞ 1 1 ′ ′ dr l + l + b (r)bWI (r)bNL(r) cos θ + r2b (r)bWI (r)bNL(r) , × 1 2 2 2 l1 l2 l3 l1 l2 l3 Z0 where Edmond’s angle cos θ is introduced:

l (l +1)+ l (l + 1) l (l + 1) cos θ = 2 1 1 2 2 − 3 3 . (6.22) (2l1 + 1)(2l2 + 1)

WI The new function bl is deﬁned by:

2 ∞ bWI(r)= P (k)g (k)j (kr)dk. (6.23) l π ζ T l l Z0 This can also be computed by CMBFAST and is plotted in panel (c) of ﬁgure 6.1 for WI various values of r. In ﬁgure 6.3, Cl, bl(r∗) and bl (r∗) are plotted as a function of l at the recombination time r = r∗. For comparison, the Cl and bl have been scaled 2 WI −2 by l(l + 1)(2π∆ ) and the bl , which has dimensions of length by an additional (l 1)(l + 2)/r2, where ∆2 = 2.5 10−9 and r = c(τ τ ) = 14061Mpc. The 3 − ∗ × ∗ 0 − d functions are very similar in shape and amplitude at large l. In the Sachs Wolfe regime the approximation (6.24) is clearly visible for the bl function.

6.3 Numerical calculation of the bispectrum

Before carrying out a full numerical calculation of the bispectrum in the curvaton and warm inﬂation cases, initial calculations have been made to check for consistency with

74 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

4 WI ∆2 bl ( ) b (∆2) l ∆2 3 Cl( )

2 ) 2 1 π∆ /(2 l 0

-1 l(l+1)C -2

-3

-4 10 100 1000 l

W I Figure 6.3: Cl,bl(r∗) and bl (r∗) plotted as a function of l. Here the Cl and bl have been 2 W I 2 2 −9 scaled by l(l + 1)(2π∆ ) and the b by an additional (l 1)(l + 2)/r∗, where ∆ =2.5 10 l − × and r∗ = c(τ0 τ ) = 14061Mpc. − d

the work of Komatsu and Spergel [112]. This requires a diﬀerent set of cosmological parameters, using a scale-invariant cold dark matter model with no dark energy where

(h, ΩΛ, Ωm,ns) = (0.50, 0, 1.0, 1.0). These parameters give values of cτ0 = 11837Mpc and cτ = 235Mpc. Using the range r cτ the functions plotted in (6.18) and (6.12) d ∗ ± d are consistent, with the proviso that [112] used transfer functions defined from the Newtonian potential fluctuation and not the curvature fluctuation ζ. Although this model does not represent a viable cosmological model when compared to the WMAP data, it allows easy comparison of bl and Cl in the Sachs-Wolfe plateau, where, since gT l jl(kr)/5 ≈− 1 ∞ 1 C k2P g j (kr)dk = b . (6.24) l ≈−5 ζ T l l −5 l Z0 The cross sections of the integral (6.16) which are plotted for the curvaton model in figure 6.4 are also consistent with [112]. Full calculation of the bispectrum in the case of the curvaton and warm inflation models is carried out over the range 13750

75 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

0.3 l1=9,l2=11 l1=99,l2=101 l1=199,l2=201 l1=499,l2=501 0.2

0.1

-0.1

-0.2 10 100 1000 l3 (a)

0.3 l1=9,l2=11 l1=99,l2=101 l1=199,l2=201 l1=499,l2=501 0.2

0.1

-0.1

-0.2 10 100 1000 l3 (b)

Figure 6.4: Cross sections of the primary bispectrum integral. The top panel plots ∞ 2 2 2 2 3 3 NL 1 1 2 2 l (l + 1)l (l + 1) 0 r dr bl1 (r)bl2 (r)bl3 (r) /2π . The lower plots l (l + 1)l (l + ∞ 2 2 1) r dr b (r)b (r)bNL(r) /2π . The integrals are plotted as functions of l3 for 0 l1 l2 R l3 (l1,l2)=(9, 11),(99, 101),(199, 201) and (499, 501). In both plots f is set to 1. R NL

76 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity and correlation between models are extremely sensitive to this resolution. The spacing NL of the r values is chosen such that the bl data can be binned to the corresponding bl resolution for which the integral is calculated.

On obtaining the required functions, Bl1l2l3 can be calculated for each model. Cross- sections of the radical integrals for the curvaton model are plotted in ﬁgure 6.4, which show ∞ 2 NL 2 l2(l2 + 1)l3(l3 + 1) r dr bl1 (r)bl2 (r)bl3 (r) /2π , (6.25) Z0 ∞ 2 NL 2 l1(l1 + 1)l2(l2 + 1) r dr bl1 (r)bl2 (r)bl3 (r) /2π , (6.26) Z0 computed while ﬁxing l3 for (l1, l2) = (9, 11), (99, 101), (199, 201) and (499, 501). The factors in front of the integral are included to compensate for the behaviour in the

Sachs-Wolfe regime. Although the entire range of l3 is plotted, this is for illustrative purposes only as many of the multipole combinations will be thrown out by selection rules for the coeﬃcients of the Wigner-3j symbol in (6.17). The coeﬃcients l1, l2, l3 must satisfy the triangle equalities, that is l l + l and permutations. The | 1| ≤ | 2| | 3| Wigner-3j symbol can be evaluated numerically using:

1 1 1 1 2 l l l 1 Γ (g l1 + ) Γ (g l2 + ) Γ (g l3 + ) Γ (g + 1) 1 2 3 g − 2 2 2 2 = ( 1) (2π) − − − 3 0 0 0 − Γ (g l1 + 1) Γ (g l2 + 1) Γ (g l3 + 1) Γ (g + ) ! − − − 2 ! (6.27) where 2g = l1 +l2 +l3. This gives an extra constraint that the coeﬃcients l1, l2, l3 must be even, saving yet more computational time.

6.4 Distinguishing between inﬂationary models

Before applying the bispectra for our two models to experimental data we ﬁrst consider their distinguishability in an ideal experiment, where for example instrument noise, non-primordial sources and the galactic plane have no eﬀect. In this case the log- likelihood function obs 2 1 B fBl1l2l3 χ2 = l1l2l3 − (6.28) 6 Cl1 Cl2 Cl3 1 2 3 l X,l ,l obs is a function of the non-Gaussianity parameter f for a series of observations Bl1l2l3 . The estimator fˆ for the maximum likelihood is given by

obs 1 B Bl1l2l3 fˆ = l1l2l3 , (6.29) 6F Cl1 Cl2 Cl3 lX1l2l3

77 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

10000 Cold Inflation Warm Inflation

1000 f

∆ 100

1 10 100 1000 lmax

Figure 6.5: The Fisher information, plotted here as ∆f = F −1/2, for the warm inﬂation and curvaton models.

where 1 (B )2 F = l1l2l3 , (6.30) 6 Cl1 Cl2 Cl3 l1X,l2,l3 the normalisation factor is called the Fisher information. In an ideal experiment the parameter fˆ is the best estimate of f and the standard deviation ∆f is F −1/2. The Fisher information is calculated in the case of an ideal experiment by computing this sum. It is computationally more efficient to evaluate only the terms in the sum where l l l and disregard the factor of 1/6 in equation (6.30). In addition the 1 ≥ 2 ≥ 3 vectors l1, l2, l3 must satisfy the triangle equalities and from (6.27), the sum l1 + l2 + l3 must be even. These constraints cut down the time to evaluate the sum dramatically. Figure 6.5 plots the standard deviation F −1/2 in an ideal experiment for both the warm inflation and cold inflation (curvaton) models. Although the Fisher information for the curvaton model is well behaved, in the warm inflation model it is extremely sensitive the resolution of the integral, in particular with regards to the resolution used NL when computing the bl function as demonstrated in figure A.2. The features in the Fisher information in the form of the knee at l 100 and the dip ≈ at l 500 are caused mainly by squeezed triangles where l , l >> l . The predictions ≈ 1 2 3 made for the wavenumber dependence of the warm inflation bispectrum by [41] were not ideally valid for squeezed triangles which could explain the unusual features in the Fisher information. One way to reduce the effect of the squeezed triangles by carrying

78 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

1 2

r 0.1

0.01

0.001 10 100 1000 lmax

Figure 6.6: The square of the correlation, r2, between the warm inﬂation and curvaton model.

out the Fisher information calculation with l 10. ≥ i j If we consider the separability of two models with bispectra Bl1l2l3 and Bl1l2l3 due to their wavenumber dependence, then the Fisher information is a matrix

j 1 Bi B F = l1l2l3 l1l2l3 . (6.31) 6 Cl1 Cl2 Cl3 l1X,l2,l3 This can be computed using the same process and the separability of the two models can be quantiﬁed by the combination

F r = 12 , (6.32) −√F11F22 the correlation, which decreases as the separability increases. A value of r = 1 corresponds to two models that have no separability. The correlation between the two models of inflation described here is plotted in figure 6.6. At r2 0.5 at l = 500 the ≈ models are separable, although an even lower correlation would be desirable to separate the models definitively. This would require an analysis up to larger lmax which is not currently possible due to the noise limitations.

79 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

10000 200/800 resolution 400/1600 resolution 800/3200 resolution

1000 f

∆ 100

1 10 100 1000 lmax

Figure 6.7: The Fisher information, plotted here as ∆f for warm inﬂation with the number of sample points in the distance r, (nL,nNL) at (200, 800),(400, 1600) and (800, 3200) in the range 13750Mpc r 14350Mpc. ≤ ≤

6.5 Non-Gaussianity tests on CMB data

Obtaining the estimator for comparison to real CMB experiments is not quite as straightforward as the ideal case. As for the calculation of the angular power spectrum, detector noise is a problem, as are the galactic and point sources which must be masked out. In addition, masking solves one problem but creates another as it eﬀects the statistical properties of the alm. It was proposed by [119] that the estimator is the sum of the ideal estimator given ˆlin by 6.29 and a linear piece fi which corrects for the eﬀects of masking, such that.

ˆ ˆideal ˆlin fi = fi + fi (6.33)

The estimator for an ideal experiment with inﬂationary model i is

Bobs Bi ˆideal 1 l1l2l3 l1l2l3 fi = . (6.34) 6Fii Cl1 Cl2 Cl3 lX1l2l3 obs where Bl1l2l3 is observed in the sky and is given by

obs l1 l2 l3 Bl1l2l3 = al1m1 al2m2 al3m3 . (6.35) m1m2m3 m1 m2 m3 ! X

80 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

The alm are the spherical harmonic coeﬃcients that can be computed from sky maps using the HEALPix package. Computing this triple sum directly with this calculation of Bobs is unrealistic, however with some clever manipulation the sum can be evaluated m1m2m3 easily. Firstly by using the deﬁnition of the Gaunt integral Gl1l2l3 given in [115],

m1m2m3 2 Gl1l2l3 = d nˆYl1m1 (nˆ)Yl2m2 (nˆ)Yl3m3 (nˆ), (6.36) Z l1 l2 l3 = c(l1l20; l1l2l3) , m1 m2 m3 ! rearranging gives the replacement,

l1 l2 l3 −1 2 = c(l1l20; l1l2l3) d nˆYl1m1 (nˆ)Yl2m2 (nˆ)Yl3m3 (nˆ), (6.37) m m m 1 2 3 ! Z so that the estimator becomes

3 Bi îdeal 1 −1 2 l1l2l3 fi = c(l1l20; l1l2l3) d nˆ (alimi Ylimi (nˆ)) . (6.38) 6Fii Cl1 Cl2 Cl3 lXimi Z Yi=1 Although apparently still very difficult to compute, it is possible to move the summation inside the integral and define a series of sky maps σ(r, nˆ)

CI bl (r) σCI (r, nˆ)= almYlm(nˆ), (6.39) Cl Xl,m WI bl (r) σWI(r, nˆ)= almYlm(nˆ), (6.40) Cl Xl,m NL bl (r) σNL(r, nˆ)= almYlm(nˆ), (6.41) Cl Xl,m which can all be easily evaluated using HEALPix. The estimator for the local model then reduces simply to

1 fˆideal = d2nˆr2drσ σ σ . (6.42) CI F CI CI NL CI Z The calculation of the estimator for the warm inﬂation model follows in a similar fashion [42], this time however the terms in cos θ give rise to terms with the angular Laplacian D2 where D2Y (nˆ)= l(l + 1)Y (nˆ). (6.43) lm − lm

81 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

Then the estimator is

1 fˆideal = d2nˆdr (6.44) WI 2F WI Z (D2σ )σ σ + σ (D2σ )σ σ σ (D2σ ) 2σ′ σ′ σ r2 , × CI WI NL CI WI NL − CI WI NL − CI WI NL where σ′ is the r derivative of the respective sky map.

ˆlin The linear piece fi obtained by [115] is,

Bi ˆlin 1 l1l2l3 fi = Cl1m1l2m2 al3m3 . (6.45) −2N Cl1 Cl2 Cl3 lX1l2l3 where the normalisation N = fiifsky now has the factor fsky, the area of the sky remaining after masking. This can be evaluated in a similar way to the ideal estimator. It can be shown that to a reasonable approximation [42] ,

2l + 1 b b Y (nˆ)C Y M(nˆ) b2C (6.46) l1 l2 l1m1 l1m1l2m2 l1m1 ≈ 4π l l l1l2Xm1m2 Xl where the mask function M(nˆ) is 0 in the region being masked and 1 in the unmasked region. The linear parts of the estimators then reduces to a series of terms involving the sky maps (6.39-6.41)

1 fˆlin = d2nˆr2drM(2S σ + S σ ), (6.47) CI −N CINL CI CICI NL CI Z 1 fˆlin = d2nˆr2drM(S˜ σ +S(2) σ +S(1) σ′ +S(1) σ′ ), (6.48) WI N CIWI NL CIWI NL CINL WI WINL CI WI Z where the radial functions SAB(r) for A, B = CI,WI or NL are deﬁned in terms of the bl functions by A B 2l + 1 bl (r)bl (r) SAB(r)= , (6.49) 4π Cl Xl A B 2l + 1 bl (r)bl (r) S˜AB(r)= l(l + 1) , (6.50) 4π Cl Xl A ′ B (1) 2l + 1 bl (r) bl (r) SAB(r)= , (6.51) 4π Cl A ′ B ′ (2) 2l + 1 bl (r) bl (r) SAB(r)= . (6.52) 4π Cl

It is now possible to proceed by computing both parts of the estimator with the WMAP three-year data. The maximum value of l is taken to be 300, above which

82 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity detector noise becomes a problem, and foreground reduced W and V frequency band skymaps are used.

Using the relationship between fCI and fNL in (6.15) the obtained limits for cold and warm inﬂation models are

375

6.2 for the WMAP W and V foreground reduced maps. The bounds on fNL are in good agreement to those quoted by the WMAP team who include the effects of detector noise in their treatment. It is concluded that fWI < 0 at 94% C.L., which is in good agreement with the predictions of warm inflation [69]. The third row in table 6.2 gives the average of the estimator for 20 randomly generated Gaussian maps. This average is used to check that the linear piece correctly removes the effects of the mask, and without the linear piece the results are nonsensical. The analysis of the CMB bispectrum made here could be applied to any model of inflation for which the wavenumber dependence of the 3 point correlation function can be calculated. The CMB power spectrum can only go so far in constraining models of inflation and it will be this statistic which will be used to throw out many models when future CMB measurements are made. The bispectrum analysis for temperature-dependent models of warm inflation, where Γ = Γ (φ, T ), is yet to be carried out. Using the results from chapter 3 for the primordial power spectrum of fluctuations, this would provide important predictions in preparation for future CMB measurements.

83 Chapter 6. Testing models of inﬂation with CMB non-Gaussianity

Map Mask fsky l fCI ∆fCI fWI ∆fWI W kp0 0.765 300 24.9 20.5 -169 105 V kp0 0.765 300 33.7 20.5 -174 105 R kp0 0.765 300 -0.712 13.4 3.09 89.2

Table 6.2: The estimator and variance for the W and V WMAP data sets in the warm inﬂation (WI) and curvaton (CI) models.

84 Chapter 7

Conclusion

This thesis has dealt with the testing of models of inflation, in particular comparing the predictions of the standard inflationary theories with those of warm inflation. Ulti- mately it is the evolution of fluctuations generated during inflation, so sensitive to the physics at that time, which provide the means to distinguish between models. We have followed the evolution of fluctuations from their source in chapter 3 to their imprint and eventual observation as temperature fluctuations in chapter 5, and their statistical properties in chapter 6. New results have been obtained for the primordial power spectrum of fluctuations in the case of warm inflation with a temperature-dependent friction coefficient Γ . These results will form an essential part of any future comparison between warm inflation theory and CMB observations. In the past the power spectrum has only been known analytically for the temperature-independent case, despite warm inflation models favour- ing a two-stage decay mechanism where the inflaton decays into light radiation fields through an intermediate heavy boson, with a temperature-dependent friction coefficient. The particle production rate for the two stage mechanism has been calculated explicitly for the first time in chapter 4. In doing so the method used is a uniform description of particle production during the early universe, and the same method applied to standard reheating theory reproduces well known results. The calculation of two-stage decay shows that particles are produced with a low momentum spectrum, far from thermal equilibrium. The thermalisation of the particles has been described by solving the Boltzmann equation in an expanding universe with this source of radiation. The numerical simulations solve for the evolution of the momentum distribution for the radiation fields. In most cases the distribution approaches a thermal distribution, which is consistent with the assumptions made in calculating

85 Chapter 7. Conclusion the primordial power spectrum of fluctuations in chapter 3. Due to the finite relaxation time of the radiation, thermal equilibrium is not guaranteed however, and this could have significant observational consequences if indeed thermal fluctuations are the source of density fluctuations in the CMB. The bosonic decays we have considered in the two-stage decay model would likely be accompanied by fermionic decays in a supersymmetric model. The analysis of fermionic decay is not included here although the calculations are certainly possible and this would be a useful addition to this model. Measurements of the CMB by the satellites COBE and WMAP have transformed the subject of inflation and we are now in a position where models can be tested, constrained and even eliminated based on the CMB data. The method described in chapter 6 for calculating the CMB bispectrum will be one used again and again for different models as this statistic will be one of the most important tools in differen- tiating between models. We have concentrated here on two models well known to predict a significant non-Gaussian signal, the curvaton model and warm inflation, and placed constraints on the non-Gaussianity parameters in their respective cases and the wavenumber dependence of their bispectra. These results and others being calculated by groups on many other inflationary models will be very important when future CMB measurements are released. There is much analysis still to be done in this area. Most significantly, the work described here opens the door for a calculation of warm inflation with a temperature-dependent friction coefficient. The Planck satellite which launched in 2009 will break new ground to resolve the angular power spectrum of temperature anisotropies up to l 2000 [120] and will either ≈ detect a non-Gaussian signal or tell us that the primordial non-Gaussianity is so small that it cannot be distinguished from non-primordial sources. This will be an eagerly anticipated and groundbreaking step in the search for a definitive model of inflation.

86 Appendix A

Numerical methods

This appendix documents in additional detail some of the numerical methods used in this thesis.

A.1 Numerical simulation of the stochastic ﬂuctuations

In chapter 3 we solve coupled equations (3.55) and (3.56) for the inflaton and radiation density. The latter of these equations contains a stochastic source term ξˆ, a normalised Gaussian variable that can be obtained using the Box-Muller algorithm [73]. The Box-Muller algorithm is one of the simplest methods for generating Gaussian random variables. The method first takes 2 random integers, in this case using the intrinsic fortran subroutines RANDOM NUMBER and RANDOM SEED, and then scales the integers to 2 floating point numbers s and t in the interval (0, 1) such that the probability density p(s,t) = 1. Then the transform

x = 2 ln(s) cos(2πt), (A.1) − p y = 2 ln(s) sin(2πt), (A.2) − gives two random Gaussian variablespx and y distributed with mean 0 and variance 1. This can be shown by inverting (A.1) and (A.2), then

1 s = exp (x2 + y2) , (A.3) 2 1 y t = cot . (A.4) 2π x

87 Appendix A. Numerical methods

The probability density p(x,y) can be calculated using

∂(s,t) p(x,y)= p(s,t) , (A.5) ∂(x,y)

which by evaluating the Jacobian determinant gives,

1 2 1 2 p(x,y)= e−x /2 e−y /2 , (A.6) − √ √ 2π 2π the Gaussian distribution with mean 0 and variance 1. The normalisation of our random Gaussian variables can be made in this case using the expression for the correlation function of ξ in (3.22).

88 Appendix A. Numerical methods

A.2 Modiﬁcations to CMBFAST

In chapter 6 the CMBFAST software was modified to evaluate the integrals (6.18), (6.12), (6.23) and their r derivatives, required for the calculation of the Fisher information. In the modified code, the subroutine cmbflat.F calls a Bessel function routine to calculate jl(kr) for the specified r and each l multipole/k mode. This matrix element simply replaces one of the gT l elements in the calculation of Cl for each k in the calculation of (5.9). In addition the factor of k and Pζ (k) is changed according to the required function. There are several important points to consider when making these alterations:

1. The spherical Bessel functions are calculated using backward iteration; this has implications for the CPU time when carrying out calculations for large r, although for the region of interest r r there is no major problem. ≤ 0 NL 2. The absence of Pζ (k) in the bl function requires particular care. It is important to ensure that the time integral starts early enough to catch the behaviour at large k. This is done by lowering the parameter ’tmin’ in subroutine cmbﬂat.F

3. Care must be taken in the deﬁnition of the power spectrum used by CMBFAST.

The newest version 4.5.1 uses the Power spectrum Pζ (k) for curvature fluctuations defined by the Bardeen variable ζ rather than Newtonian potential fluctuations. This can create confusion when comparing to earlier work evaluating the bispectrum, for example using the latter we expect in the Sachs-Wolfe plateau g j (kr )/3, however using the former the factor is 1/5. T l ≈− l ∗

Initial results following the work of Komatsu and Spergel [112] were made using COBE normalisation, however subsequent output was made with CMBFAST’s COBE unnormalised option –with-cobe=no; which simplified the analytic checking of results and bypassed much of the cancellation of factors that occurs when calculating the Fisher information. In addition, output was made with the FITS option –with-fits=yes which allows easy manipulation of the data files and saves on file space. This was of particular consideration when the loop over r values was carried out; the FITS library program ’tabmerge’ allows FITS files to be merged whilst preserving the header information produced by CMBFAST. One other major advantage of the FITS format is that the data is compatible with the HEALPIX package, allowing any future work creating sky maps from the bispectrum results described here and comparing with the maps created from satellite data.

89 Appendix A. Numerical methods

′ Figure A.1: The approximation (dashed) and ’correct’ derivative bl(r) .

The derivatives in the warm inﬂation bispectrum were approximated using the expressions:

2 l + 1 b (r)′ = k2P (k)g (k) kj (kr) j (kr) dk, (A.7) l π T l l−1 − r l Z 2 l + 1 bWI(r)′ = P (k)g (k) kj (kr) j (kr) dk. (A.8) l π T l l−1 − r l Z As a check on the accuracy of these approximations, ﬁgure A.1 compares the approxi- ′ mation for bl(r) above (dashed line) to the derivative calculated by (solid line):

b(r + δr) b(r) b(r)′ = − , (A.9) δr

2 where r was chosen to be at r = r∗ = 14060.97 and δr = 11.43Mpc. l bl(r) is shown for clarity.

The loop over r is carried out for each of the 5 bl functions, along with the Cls required for calculating the Fisher matrix. This creates 6 FITS files ready for post- processing or input into HEALPIX. CMBFAST is also modified to produce a single text file with all of the details of the run, in particular the range and stepsize of r values. A further program reads this along with the data files and makes several plots including

90 Appendix A. Numerical methods the bispectrum functions at various r values for checking. The calculation of the Fisher matrix is carried out in a separate routine as this can be very time consuming for large l.

91 Appendix A. Numerical methods

A.3 Calculating the Fisher matrix

The main outcome of our calculations of the CMB bispectrum is to calculate the Fisher matrix F for comparison between inflationary models. Although we present the results for our 2 cosmological models in figure 6.5 using the range of r values described in that section, in reality the process in gaining this result was far more complicated and time-consuming. Original calculations followed the work of [112], however the results quickly indicated that the resolution in r was not sufficient to capture the behaviour of the NL bl (r) function (6.12). Although an increase in r resolution is clearly desired, the timescale for the calculations is also an issue with each of the bispectrum functions taking approximately 40 seconds for every r value. As a solution to this problem, NL subsequent runs were made with a ratio of 4:1 for the resolution of the bl (r) func- WI ′ WI ′ tion and the bl(r),bl (r),bl(r) and bl (r) functions; the integration was then car- NL ried out using binned bl data. A further increase in r values successfully resolved NL the features of bl (r) as plotted in figure 6.2, however the shape of the Fisher matrix, particularly for warm inflation, continued to change with further increases until 1 the results in figure 6.5 were reached. Figure A.2 plots ∆F = F − 2 for the warm inflation model with the number nL of r values for bl and the number nNL of r values for bNL given by (nL,nNL)=(200, 800),(400, 1600) and (800, 3200) in the range 13750Mpc r 14350Mpc. ≤ ≤ The features in the Fisher matrix in the form of the knee at l 100 and the dip ≈ at l 500 are caused mainly by squeezed triangles where l , l >> l . The predic- ≈ 1 2 3 tions made for the form of the bispectrum equations by [41] were not ideally valid for squeezed triangles which could explain the unusual features in the Fisher matrix. An alternative way to represent the results is to reduce the effect of the squeezed triangles by carrying out the Fisher matrix calculation with l 10. Figure A.3 shows a ≥ much better correlation between the different r resolution data, although there is still noticeable difference at l 500, such that presenting data at larger l would still be ≈ inaccurate. This scenario is similar for the correlation r between the two models. Figure A.4 plots the correlation squared for the same three resolutions investigated above. Here the correlation appears to be increasing slightly at large l and flattening out, corresponding to a slightly decreasing seperability between the two models. The effect on the Fisher matrix of including low and high r values was also in- NL vestigated. As described in the main text, the behaviour of bl (r) at very low r is negligible due to the factor of r2 in the integral. Further calculations in the large r range

92 Appendix A. Numerical methods

10000 200/800 resolution 400/1600 resolution 800/3200 resolution

1000 f

∆ 100

1 10 100 1000 lmax

Figure A.2: The Fisher matrix, plotted here as ∆F for warm inﬂation with the resolution in the distance r, (nL,nNL) at (200, 800),(400, 1600) and (800, 3200) in the range 13750Mpc r 14350Mpc. ≤ ≤

14350Mpc r 27000Mpc and an intermediate range 8750Mpc r 13750Mpc also ≤ ≤ ≤ ≤ had negligible eﬀect on the Fisher matrix, however the intermediate range did alter the behaviour of the correlation between the two models.

93 Appendix A. Numerical methods

10000 200/800 resolution 400/1600 resolution 800/3200 resolution 1000 f ∆ 100

1 10 100 1000 lmax

Figure A.3: The Fisher matrix, plotted here as ∆F and calculated with l 10 for warm inﬂation ≥ with the resolution in the distance r, (nL,nNL) at (200, 800),(400, 1600) and (800, 3200) in the range 13750Mpc r 14350Mpc. ≤ ≤

10 200/800 resolution 400/1600 resolution 800/3200 resolution

1 2

r 0.1

0.01

0.001 10 100 1000 lmax

Figure A.4: The correlation of our two models of inﬂation with the resolution in the distance r, (n ,n ) at (200, 800),(400, 1600) and (800, 3200) in the range 13750Mpc r 14350Mpc. L NL ≤ ≤

94 Appendix B

Reduction of the Collision Integral

The full collision integral for the dominant 2 2 particle scattering is → λ2 d3p d3p d3p = 2 3 4 (2π)4δ(p + p p p )δ(ω + ω ω ω ) Sc 2ω ω ω ω 2 − 3 − 4 p p2 − p3 − p4 p Z 2 3 4 [(1 + n)(1 + n )n n nn (1 + n )(1 + n )] , (B.1) × 2 3 4 − 2 3 4 where is the change in the distribution n = n(ω), i.e.n ˙ , λ is the collision cross- Sc section and ni = n(ωi). In the following derivation, the second line will be labelled with the shorthand F (n,n2,n3,n4). Computationally this has the potential to be extremely time-consuming, however the number of dimensions can be reduced using the δ-functions. Following the work of [94] and [95], ﬁrst make the Fourier transform,

d3ξ δ (p + p p p ) = ei(ξ·(p+p2−p3−p4)) 2 − 3 − 4 (2π)3 Z ξ2dξdΩ = ei(ξ·p)ei(ξ·p2)e−i(ξ·p3)e−i(ξ·p4) ξ , (B.2) (2π)3 Z and d3p = dφ d cos θ p2dp p2dp dΩ (B.3) i i i i i ≡ i i i

95 Appendix B. Reduction of the Collision Integral

Then the collision integral is

λ = δ (ω + ω ω ω ) F (n,n ,n ,n ) ξ2dξ ei(ξ·p)dΩ Sc (64)2π8ω p 2 − 3 − 4 2 3 4 p Z Z Z p2dp p2dp p2dp ei(ξ·p2)dΩ e−i(ξ·p3)dΩ e−i(ξ·p4)dΩ 2 2 3 3 4 4 , (B.4) × 2 3 4 ω ω ω Z Z Z 2 3 4 which can be written

λ = δ (ω + ω ω ω ) F (n,n ,n ,n ) Sc 64π3ω p p 2 − 3 − 4 2 3 4 p Z p2dp2 p3dp3 p4dp4 D(p,p2,p3,p4) , (B.5) × ω2 ω3 ω4 where

pp p p D(p,p ,p ,p ) = 2 3 4 ξ2dξ ei(ξ·p)dΩ ei(ξ·p2)dΩ 2 3 4 64π5 p 2 Z Z Z e−i(ξ·p3)dΩ e−i(ξ·p4)dΩ (B.6) × 3 4 Z Z The function D can be simpliﬁed using

4π e±iξ·pi dΩ = sin(ξp ). (B.7) pi ξp i Z i Then D reduces to

4 ∞ dξ D(p ,p ,p ,p )= sin(ξp) sin(ξp ) sin(ξp ) sin(ξp ) (B.8) 1 2 3 4 π ξ2 2 3 4 Z0 Integrating with respect to ξ then gives

4 D(p,p ,p ,p )= ( 1 s + s + s + s + s s s )p 2 3 4 π − − 1 2 3 4 5 − 6 − 7 +( 1 s1 s2 + s3 + s4 s5 + s6 + s7)p2 − − − − +( 1+ s + s + s s s s + s )p − 1 2 3 − 4 − 5 − 6 7 3 +( 1+ s + s s + s s + s s )p , (B.9) − 1 2 − 3 4 − 5 6 − 7 4

96 Appendix B. Reduction of the Collision Integral where

s = sgn(p + p p p ) 1 2 − 3 − 4 s = sgn(p p + p + p ) 2 − 2 3 4 s = sgn(p + p + p p ) 3 2 3 − 4 s = sgn(p + p p + p ) 4 2 − 3 4 s = sgn(p p p p ) 5 − 2 − 3 − 4 s = sgn(p p + p p ) 6 − 2 3 − 4 s = sgn(p p p + p ) (B.10) 7 − 2 − 3 4

2 2 2 2 which under conservation of momentum p + p2 = p3 + p4 makes the remarkable sim- pliﬁcation to

D(p,p2,p3,p4)= min(p,p2,p3,p4). (B.11)

Then our collision integral (B.1) has reduced from 9 to 3 dimensions:

λ = δ (ω + ω ω ω ) F (n,n ,n ,n ) Sc 64π3ωp p 2 − 3 − 4 2 3 4 Z p2dp2 p3dp3 p4dp4 min(p,p2,p3,p4) , (B.12) × ω2 ω3 ω4

We can reduce one further dimension using the δ-function to give

λ = θ(ω + ω ω m )F (n,n ,n ,n ) Sc 64π3ω p 3 4 − p − y 2 3 4 p Z p3dp3 p4dp4 min(p,p2(p,p3,p4),p3,p4) , (B.13) × ω3 ω4 where p2(p,p3,p4) is determined from momentum conservation. Re-writing the integral in terms of ω, we can use pidpi = dωi (B.14) ωi and

min(p,p2(p,p3,p4),p3,p4) = p(min(ωp,ω2,ω3,ω4)) 1/2 = min(ω ,ω ,ω ,ω )2 m2 . (B.15) p 2 3 4 − σ Then

λ = θ(ω + ω ω m )F (n,n ,n ,n ) Sc 64π3ωp 3 4 − p − σ 2 3 4 Z 1/2 min(ω ,ω ,ω ,ω )2 m2 dω dω . (B.16) × p 2 3 4 − σ 3 4 97 Appendix B. Reduction of the Collision Integral

It is worth noting that in general a full description of collisions will include 3 additional (and similar) terms to B.1 for 1 3, 3 1 and quasiparticle interactions. → → It is well known that 2 2 processes are the dominant term in the collision integral. → However, whereas 2 2 processes conserve particle number, these additional terms do → not, and as mentioned in the main text, to obtain charge neutral (µ = 0) distributions they must be included. In most cases the numerical results here show that the ﬁnal distribution has a relatively small value for µ and although the numerical code used can compute these terms we can justify their omission and therefore kept the computation time down.

98 Appendix C

Bispectrum calculations

This appendix includes some additional derivations required for the bispectrum analysis of chapter 6.

C.1 General expression for the angle-averaged bispectrum

Stated in (6.8), the angle-averaged bispectrum Bl1l2l3 decomposes into a series of terms such that, nl Bl1l2l3 = fnlBl1l2l3 . (C.1) Xn,l Using the 3 point correlation function (6.4) and the spherical harmonic coeffcients alm, defined in terms of the curvature fluctuations ζ by,

d3k a = 4π( i)l ζ(k)g (k)Y ∗ (k), (C.2) lm − (2π)3 l lm Z in equation (6.3), then

1 l l l Bnl = ( i)l1+l2+l3 1 2 3 (C.3) l1l2l3 − π3 m1m2m3 m1 m2 m3 ! X 3 d3k g (k )Y ∗ (kˆ ) Bnl(k , k , k )δ(k + k + k ) × i li i limi i 1 2 3 1 2 3 Z Yi=1

To obtain equation (6.9), following the work of [117], the Dirac delta function can ﬁrst be rewritten using

d3r δ(k + k + k )= eik1·reik2·reik3·r, (C.4) 1 2 3 (2π)3 Z

99 Appendix C. Bispectrum calculations where iki·r l ˆ ∗ e = (4π) (i) jl(kir)Ylm(ki)Ylm(ˆr). (C.5) m Xl X nl Equation (6.5) for B (k1, k2, k3) can also be rewritten in terms of spherical harmonics using l 4π P (kˆ kˆ )= Y (kˆ )Y ∗ (kˆ ) (C.6) l 1 · 2 2l + 1 lm 1 lm 2 mX=−l After manipulation using Wigner 3j symbol identities, and eliminating the momenta (see [112]) equation (C.3) takes the form given in (6.9).

C.2 Warm inﬂation angle-averaged bispectrum

Following [42], the warm inflation bispectrum is calculated from (6.9) and (6.10) with l = n = 1. The non-vanishing coefficients of (6.10) are c(l 1, l 1, 1 : l , l , l ) 1 1 1 1 1 ± 2 ± 1 2 3 = cos θ + 1 , c(l1l20; l1l2l3 −4 ± 2l1 + 1 ± 2l2 + 1 ± (2l1 + 1)(2l2 + 1) (C.7) c(l 1, l 1, 1 : l , l , l ) 1 1 1 1 1 ± 2 ∓ 1 2 3 = cos θ 1 , c(l1l20; l1l2l3 −4 − ∓ 2l1 + 1 ± 2l2 + 1 ± (2l1 + 1)(2l2 + 1) (C.8) where Edmond’s angle θ was defined in the main text in (6.22).

1 −1 Equation (6.9) contains four new functions bll±1 and bll±1 which can be written in CI WI terms of the bl (r) function (6.18) and the function bl (r) (6.23) using Bessel function identities: l ′ b1 = bCI (r) bCI (r), (C.9) ll+1 r l − l l + 1 ′ b1 = bCI (r)+ bCI (r), (C.10) ll−1 r l l l ′ b−1 = bWI(r) bWI (r), (C.11) ll+1 r l − l l + 1 ′ b−1 = bWI(r)+ bWI (r), (C.12) ll−1 r l l where the prime denotes the derivative of the function with respect to r. Substituting (C.9)-(C.12) into (6.9) then gives the angle-averaged bispectrum for warm inﬂation in equation (6.22).

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