A Scalar Field Theory for Dark Matter–Dark Energy Interaction

Pedro Miguel Greg´orioCarrilho

Disserta¸c˜aopara obter o Grau de Mestre em Engenharia F´ısicaTecnol´ogica

J´uri

Presidente: Professora Maria Teresa Haderer de la PeñaStadler Orientador: Doutor Jorge Tiago Almeida Páramos Orientador Externo: Professor Orfeu Bertolami Vogal: Professor JoséPedro Mimoso Vogal: Doutor Nuno Miguel Candeias dos Santos

September 2012

Acknowledgments

I would like to start by thanking my supervisors Orfeu Bertolami and Jorge Páramosfor their guidance and support throughout the development of this work, for their pacience with my mistakes and for their teachings, not only on physics but on scientific research in general. Secondly I thank my family not only for their support and for the freedom they gave me to choose physics, but also for the sacrificies they made so that I could have a better education. Next I would like to show my appreciation to my collegues and friends for the nice moments we spent together either having fun, learning physics, or both, and also for frequently stimulating me to improve myself. Also I would like to thank my friends who lived with me, for giving me a nice atmosphere to live in and for their pacience when listening to my ramblings about physics, even when they had little clue what I was talking about. Last, but certainly not least, I would like to thank Susana for her constant efforts in making me happy and for her expertise in casting away all the threats to my sanity.

Abstract

In this thesis we study the effects of an interaction between dark matter and dark energy. To fulfil this objective, we introduce two scalar fields φ and χ, and endow them with an interaction potential V (φ, χ) = e−λφP (φ, χ), where P (φ, χ) is a polynomial. This improves on standard cosmology, which is instead based on the cosmological constant and non-interacting cold dark matter. In this model, we demonstrate that the relevant features of the present Universe are reproduced for a large range of the bare mass of the dark matter field. We also study modifications of the potential, revealing important implications of the interaction, including the possibility of transient acceleration solutions. The original work presented in this thesis closely follows Ref. [1].

Keywords

Dark energy, Dark matter, Scalar ﬁeld, Interaction model, Transient accelerated expansion.

iii

Resumo

Nesta tese estudam-se os efeitos de uma intera¸cãoentre matériaescura e energia escura. Para cumprir esse objectivo, sãointroduzidos dois campos escalares φ e χ, que sãodotados de um potencial de interaçcão V (φ, χ) = e−λφP (φ, χ), em que P (φ, χ) éum polinómio. Este modelo complementa a cosmologia padrão,que ébaseada na constante cosmológicae na matériaescura fria sem interaçcões. Neste modelo, demonstra-se que as propriedades relevantes do presente estado do Universo sãorepro- duzidas para um grande conjunto de valores da massa livre do campo de matériaescura. Estudam-se tambémmodifica¸cões do potencial, revelando implica¸cõesimportantes da interaçcão,incluindo a pos- sibilidade de solu¸cõescom acelera¸cãotransitória. O trabalho original apresentado nesta tese segue a Ref. [1].

Palavras Chave

Energia escura, Matériaescura, Campo Escalar, Modelo de intera¸cão,Expansãoacelerada tran- sitória.

Contents

1 Introduction 1 1.1 Standard Cosmology...... 1 1.1.1 FLRW Cosmology...... 1 1.1.2 Dark Matter...... 3 1.1.3 The Cosmological Constant and Dark Energy...... 4 1.1.4 Inﬂation...... 6 1.2 Modifying the Standard Cosmological Model...... 6

2 Scalar Field Cosmology7 2.1 General Equations...... 8 2.2 Scalar Dark Energy...... 9 2.3 Scalar Dark Matter...... 11

3 Dark Matter – Dark Energy Interaction 13 3.1 Interaction Model...... 15 3.1.1 Basic equations...... 15 3.1.2 Interaction Potential...... 15 3.1.3 Average Evolution Equations...... 16 3.2 Results...... 17 3.2.1 Analytical Considerations...... 17 3.2.2 Numerical Solutions...... 19

4 Conclusions and Future Work 25

Bibliography 27

vii

List of Figures

3.1 log ρdm and log ρde for different values of the bare mass m...... 20 −60 3.2 log ρdm and log ρde for m = 10 and different values of the initial condition χi..... 20 3.3 Transient solution for m = 5.9 × 10−57, λ¯ = 9.5...... 21 −15 ¯ 3.4 log ρdm and log ρde for m = 10 and different values of λ...... 22 3.5 Transient solution for m = 10−15, λ¯ = 2.8...... 22 3.6 (λ,¯ m) parameter space and transient line...... 23 3.7 Effective potential for m = 10−15, λ¯ = 2.8...... 23

Abbreviations

CDM Cold Dark Matter

FLRW Friedmann-Lemˆaitre-Robertson-Walker

EOS Equation of State

DM Dark Matter

CMB Cosmic Microwave Background

DE Dark Energy

WIMPs Weakly Interacting Massive Particles

GCG Generalized Chaplygin Gas

VAMP VAriable Mass Particle

1 Introduction

1.1 Standard Cosmology

The Standard Cosmological Model is currently a widely accepted phenomenological description of the evolution of the Universe. The reasons for that lie in its simplicity and the fact that it reproduces a variety of observational results coming from independent sources. For this last reason it is also called the Concordance Model. It is based on three fundamental ingredients: inﬂation, Cold Dark Matter (CDM) and the cosmological constant (Λ), as well as on the Standard Model of Particle Physics and General Relativity. In this section of the introduction we review the most relevant results of this Concordance Model as well as some of its limitations.

1.1.1 FLRW Cosmology

We begin our review by pointing out a main ingredient of most cosmological models: the cosmological principle. This principle states that the properties of the Universe are the same for all observers, on suﬃciently large scales, which implies the Universe to be homogeneous and isotropic. This in turn constrains the metric tensor gµν to be the Friedmann-Lemˆaitre-Robertson-Walker (FLRW) metric, given by the line element:

dr2 ds2 = g dxµdxν = −dt2 + a2(t) + r2dΩ2 , (1.1) µν 1 − kr2

2 where a(t) is the cosmological scale factor, normalized in this work so that at present a(t0) = 1, dΩ is the line element for the 2-sphere and k is a constant proportional to the scalar curvature of the spatial section of space-time: for ﬂat space it equals zero, for a closed one it is positive and it is negative if space is open, i.e. hyperbolic. The dynamics of this space-time are related exclusively with the scale factor a(t), which is so far an arbitrary function. To ﬁnd the evolution equations one must input the metric (1.1) into Einstein’s

1 1. Introduction equations1 [2] 1 R − Rg = T , (1.2) µν 2 µν µν µν where Rµν is the Ricci tensor, R = g Rµν is the Ricci scalar and Tµν is the energy momentum tensor.

Having chosen the metric tensor, Eq. (1.1), the only ingredient yet unknown is Tµν . However, there is very little freedom in that choice due to our assumption of homogeneity and isotropy: the energy momentum tensor reduces to that of a perfect ﬂuid:

µ [Tν ] = diag(−ρ(t), p(t), p(t), p(t)) , (1.3) in which ρ is the energy density and p is the pressure of the fluid. What results from the substitutions of this and the metric in Eq. (1.2) are Friedmann and Raychaudhuri equations, respectively: 1 k H2 = ρ − , (1.4) 3 a2 a¨ 1 = − (ρ + 3p) , (1.5) a 6 with H =a/a ˙ the expansion rate, also called the Hubble parameter. Without further information about the specific fields originating the energy momentum tensor, these equations provide a complete description of the evolution of the Universe. What we mean by this is that the evolution of the energy density, ρ˙ + 3H(ρ + p) = 0 , (1.6) is not independent of the previous equations. It is, nevertheless, quite useful to determine ρ(t) as a function of the scale factor a(t). This is only possible, of course, if there exists an Equation of State (EOS) p = p(ρ) relating the pressure to the energy density. The EOS depends fundamentally on the type of matter one introduces into the theory. However, for cosmological purposes, a simple and important example is a fluid with a linear EOS, that is:

p = wρ , (1.7) in which the EOS parameter w is taken to be constant for the moment. We see immediately from Eq. (1.5) that the sign ofa ¨ is intimately related to the value of this parameter. In particular, the threshold for zero acceleration occurs for w = −1/3, meaning that only a negative pressure, smaller than this value, can accommodate for an accelerated expansion. Beyond that, with this EOS, the solution to Eq. (1.6) is simply −3(1+w) ρ(a) = ρ0a , (1.8) with ρ0 the energy density at present. The known fields of the Standard Model of Particle Physics can be described cosmologically by fluids with this EOS, separated in two extreme cases: non-relativistic matter, with wm = 0, and relativistic matter or radiation with wr = 1/3. As expected, the energy −3 −4 density varies with the volume a for non-relativistic matter (ρm), or with a for radiation (ρr) due to an additional contribution from the cosmological redshift. Other cases exist, such as ultra- −6 stiff matter, with w = 1 and ρs ∝ a and also the more exotic case with w = −1 that leads to a

1 Note that we use reduced Planck units, i.e. units in which ~ = c = 8πG = 1. For this reason every mass or energy scale will appear in units of the reduced Planck mass.

2 1.1 Standard Cosmology

constant energy density ρΛ and to an exponential expansion as is the case in a cosmological constant dominated Universe. These two are the limiting values for w if one chooses to respect the dominant energy condition. We close our discussion of the FLRW cosmology by introducing the density parameter Ω = ρ/3H2 and the so-called deceleration parameter, given by

aa¨ 1 q ≡ − = Ω (1 + 3w) , (1.9) a˙ 2 2 which can be generalized for multiple fluids with density parameter Ωi and EOS parameter wi, by P treating w as an effective parameter: w = i wiΩi/Ω. More than the actual geometry of space-time, the Concordance Model makes certain hypothesis about the components of the Universe. Specifically, it postulates the existence of cold dark matter and models the accelerated expansion through a cosmological constant Λ. In what follows, we will motivate the need for these two components and shortly mention the main observational results.

1.1.2 Dark Matter

Dark Matter (DM) was first discovered by the observations of Jan Oort in 1932 and confirmed shortly after by Fritz Zwicky in 1933 [3]. J. Oort found stars in the Milky Way that should not be bound, considering just the visible mass of the galaxy. As for F. Zwicky, he discovered an anomalous distribution of velocities of the galaxies in the Coma cluster. In both cases, the mass measurements derived from dynamics led to much larger values than the ones obtained by measuring the luminosities of the visible objects. Although some part of that matter was later discovered to be very hot gas, which only emitted X-rays, most of it was still completely invisible, i.e. did not seem to interact electromagnetically. These initial detections were only the first two pieces of a large body of evidence in favour of dark matter. The one to have the largest impact on the scientific community was the measurement of galaxies’ rotation curves by Rubin and Ford in 1970 [4,5]. On cosmological scales, indications about dark matter come from the mapping of the microwave sky. The anisotropies of the Cosmic Microwave Background (CMB) provide important information on several cosmological parameters [6].

In particular, one of the conclusions is that Ωdm is considerably larger than Ωb, the density of baryons. This measurement is also corroborated by the determination of the baryonic number density from nucleosynthesis [7] and from Baryon Acoustic Oscillations [8], thus showing dark matter to be, at least in the most part, non-baryonic. The actual composition of dark matter remains a mystery, although there exist some clues. Beyond being non-baryonic, there is evidence that it should be cold, i.e. be composed of non-relativistic particles. This conclusion comes from observations of structure at diﬀerent scales: only with cold dark matter can there be small scale structures like galaxies; hot dark matter cannot collapse at these scales, since it escapes gravitational attraction too easily due to its relativistic nature. Beyond that, the discovery of the bullet cluster (1E 0657-558) revealed that the interaction cross section of dark matter with itself is likely to be very small, almost negligible [9].

3 1. Introduction

These and other clues led the scientiﬁc community to believe in the existence of dark matter. Although it has not yet been directly detected, many candidates have been put forward (see Ref. [10] for a review), most under the label of Weakly Interacting Massive Particles (WIMPs). Most of these candidates are fermions, like those arising from SUSY, but there are also some bosonic candidates, such as the axion, which is also well motivated, or the phion, even though self-interacting [11, 12]. The likeliest scenario should be the existence of several particles that comply with the requirements for dark matter. The experimental community is highly focused in this research and many new detectors are being built to detect dark matter directly. The most recent excitement in this area came from the DAMA/LIBRA collaboration, which claimed a positive detection of dark matter particles [13]. How- ever, so far those results are not compatible with other such experiments, which casts doubts on their validity2.

1.1.3 The Cosmological Constant and Dark Energy

The discovery of the accelerated expansion of the Universe by the Supernova Cosmology Project [15] and by the High-z Supernova Search Team [16] in 1998 revolutionized our knowledge of the Universe. These two results showed that, besides dark matter, there seems to exist an unknown source of energy, more abundant today than the rest of the other constituents of the Universe and with an effective negative pressure. The general name for such a component is Dark Energy (DE)[17], although not all explanations for this result rely on a new component, i.e. it might hint at a more general gravity theory (see e.g. Ref. [18] and Refs. therein). The simplest solution was proposed by Einstein [19], although with the very different objective of obtaining a static cosmological solution. As mentioned earlier, such a static situation still requires a component with negative pressure, more precisely a combination with weff = −1/3. Being that an unusual equation of state, Einstein ruled it out as unphysical and chose instead to add his famous Cosmological Constant Λ to the gravitational action:

Z √ 1 d4x −g R + L − Λ, (1.10) 2 where g is the determinant of the metric and L is the Lagrangian of all the other fields. The variation of this action with respect to the metric leads to the modified field equations,

1 R − Rg + Λg = T . (1.11) µν 2 µν µν µν

Taking the term with Λ to the r.h.s. it is easily seen to be equivalent to a ﬂuid with w = −1, as remarked above. As was later discovered ﬁrst by Slipher and then by Hubble, the Universe is not static, but Einstein’s idea is now useful for explaining the accelerated expansion. Supposing this to be true, the observations already mentioned measured Λexp to be of the order of

−47 4 Λexp ∼ 10 GeV . (1.12)

2See for example Ref. [14] for more details on the compatibility of the DAMA result with other experiments.

4 1.1 Standard Cosmology

However, there are some problems with the cosmological constant. The way Einstein introduced it, the idea was for it to be a new parameter of the theory of gravity and thus could take any value.

Nevertheless, quantum field theory arguments predict the existence of a vacuum energy density ρV , associated with quantum fields, which should also contribute to the effective Cosmological Constant

Λeﬀ = Λ + ρV . A problem arose when such a vacuum contribution was estimated, for instance, by summing the energies of the zero modes of a quantum ﬁeld of mass m [20]:

Z M 2 4 2 2 4 2 1/2 4πk dk 1p 2 2 M m M m m e −1 ρV = 3 m + k = 2 + 2 + 2 log 2 + O(M ) , (1.13) 0 (2π) 2 16π 16π 64π 4M where M is a cutoff introduced to regularize the calculation, m is the mass of the field and e is Euler’s number. A typical argument to choose the value of M is to say it is the scale up to which one trusts quantum field theory. Usually that means choosing the Planck mass, which leads to a value roughly

120 orders of magnitude larger than Λexp. Other approaches exist, such as choosing different scales, but the discrepancy between the values obtained and experiment is never less than tens of orders of magnitude. For any such calculation to be true, one must assume that the gravitational part of Λeff cancels out the vacuum contribution up to an unnatural accuracy. This is the famous fine tuning problem of the cosmological constant and, as can be expected, this was a known issue for quite some time even before the accelerated expansion was discovered (see, for example, Weinberg’s review, Ref. [21], from 1989, or Ref. [22] for more recent work on the subject). The need for a non-zero Λ imposed by the current supernovae results, raises yet a different problem: the fact that its corresponding density is presently the same order of magnitude as the non-relativistic matter density. This so called coincidence problem can be posed in several different ways. The general question is usually “Why now?”, as it appears to be highly unlikely that the human race would develop at such a pace as to observe this phenomenon exactly during a transition between the epochs of matter and Λ domination. A rather different way to put it has to do with the value of Λ appearing to be within a narrow range that allows for a “good amount” of structure formation, something also considered improbable. Similarly, there is another coincidence in the time at which Λ starts to dominate and the time when structure formation becomes non-linear. All these issues are fundamentally linked with the current lack of knowledge on this mysterious constant: everything would be solved if its value was correctly calculated from first principles. Many attempts have been made to do so, but none with the desirable result. We emphasize however that there are two distinct problems: the fact that the vacuum energy does not gravitate and the origin of the accelerated expansion, linked with the mentioned coincidences. If the solution to the dark energy problem is indeed the cosmological constant then clearly both those issues are related. If not, and if, for instance, some symmetry was found to cancel the cosmological constant, one would have to find what drives the accelerated expansion. This last path has been extensively followed and it is the basis of this thesis. We will describe one of the ways to account for dark energy without the cosmological constant in chapter2, thus laying the foundations of our work.

5 1. Introduction

1.1.4 Inﬂation

Our introduction to standard cosmology would be incomplete without a discussion about inflation [23]. We start by quickly referring to the flatness problem and its relation to the curvature parameter k. Rewriting Eq. (1.4), using the density parameter Ω, one finds that

3k = (1 − Ω−1)ρa2 . (1.14)

The conclusion then is that the curvature depends very strongly on the value of Ω, specifically on how close to unity it is. Furthermore, the value of ρa2 is expected to decrease with time (unless the constituents of the Universe are exotic), meaning that |1 − Ω−1| must increase during the evolution of the Universe to keep k constant. However, cosmological measurements revealed Ω to be remarkably −1 −60 close to 1 at present, which meant that |1 − Ω | . 10 at the Planck Epoch, i.e. space was very flat at that time, apparently without any reason. Inflation presents a solution to this serious fine-tuning problem. This theory supposes the Universe to have suffered a very rapid, possibly exponential expansion shortly after the Big Bang. This way the curvature term in Eq. (1.4) becomes negligible after inflation, or, in other words, space becomes essentially flat from that point on. For this reason, in almost every discussion of cosmology at late times, k is set to 0, as will be the case in the present thesis. Other problems existed that were also solved by the theory of inflation. The so-called horizon problem is arguably the most important one, and is related to the observed homogeneity and isotropy of the Universe. The problem lies in the fact that regions that should not be causally connected, appear to have the same physical properties. Inflation clearly solves this issue, since it increases the size of small, causally connected regions by many orders of magnitude, which then originate regions the size of the observable Universe today. After the inflationary period, the temperature and the energy density of matter and radiation are effectively null. For that reason, every inflationary model must also explain how the Universe abandoned such a state and became dominated by hot matter and radiation. That is achieved through a process called reheating. After the inflationary phase, the field that powers inflation — the inflaton — begins oscillating coherently. These oscillations give rise to the quantum creation of particles, which eventually consume all the energy of the inflaton and reheat the Universe. Among the several models of reheating studied so far, one of the most efficient is the so-called preheating scenario [24], in which the oscillating frequency is variable, leading to parametric resonances that allow for a much faster production of particles.

1.2 Modifying the Standard Cosmological Model

The Standard Cosmological Model relies on cold dark matter and the cosmological constant, which is a rather incomplete description. There are a number of problems with the cosmological constant and dark matter, albeit better understood than dark energy, is still far from being a closed issue. This is the motivation that drives this study and the work of many to better motivate the Concordance Model and replace it for one that provides a deeper insight into the dark sector.

6 2 Scalar Field Cosmology

Scalar fields are extremely important in modern physics. Being invariant under coordinate trans- formations, they are the simplest tensor fields, with order 0. Examples are the recently detected Higgs field, which provides the mechanism to endow mass to the particles of the Standard Model of Particle Physics, or the inflaton, presumably the scalar field that drives inflation. The latter, in particular, gives origin to a dynamics similar to that of dark energy, since both give rise to an accelerated expansion. For that reason it is reasonable to presume that dark energy might also be described by a scalar field, instead of the cosmological constant. The first suggestions to go beyond the cosmological constant and replace it for a scalar field were made by Wetterich [25], and Ratra and Peebles [26], although there were some earlier attempts to study a variable cosmological term [27]. As can be deduced from their dates of publication, these attempts only tried to solve the fine tuning problem or to account for the missing energy in the Universe, since there was mounting evidence that the Universe was flat and regular matter was not enough to accomplish that. Only later did the motivation of the accelerated expansion become the dominant drive for these models. There has been a great activity in this area, motivated by explaining the accelerated expansion of the Universe without the problems of the cosmological constant. That is exactly what we aim for in this thesis, with the addition of another scalar field to represent dark matter and letting them interact with each other. For that reason, in this section we review the effects of introducing scalar fields in an FLRW Universe, with the objective of modeling both dark energy and dark matter.

7 2. Scalar Field Cosmology

2.1 General Equations

We restrain our analysis to canonical scalar fields, which when applied to dark energy are usually referred to as quintessence models, whose general action reads: Z √ 1 S = d4x −g − gµν ∂ φ∂ φ − V (φ) , (2.1) φ 2 µ ν where we denote the field by φ and V (φ) is the potential. Other non-canonical models exist to describe dark energy but we shall not treat them here (See Ref. [28] for a review on dark energy, which includes these somewhat more complex approaches). We start by calculating the energy-momentum tensor, by varying the action with respect to the metric: Z √ 1 1 1 δS = d4x −g − gµν gαβ∂ φ∂ φ + V (φ) + gαµgβν ∂ φ∂ φ δg . (2.2) φ 2 2 α β 2 α β µν Due to the choice of normalization for the gravitational action Z √ 1 S = d4x −g R, (2.3) G 2 the term in curly brackets equals half of the energy-momentum tensor, which is then given by: 1 T µν = −gµν ∂ φ∂αφ + V (φ) + ∂µφ∂ν φ . (2.4) 2 α The cosmological principle implies homogeneity, so all spatial derivatives vanish in a cosmological model. For that reason the stress energy tensor takes the form: 1 T µν = φ˙2δµδν + gµν φ˙2 − V (φ) , (2.5) 0 0 2 where the dot represents a derivative with respect to time. Comparing it to the energy-momentum tensor of a perfect fluid, T µν = (ρ + p)U µU ν + pgµν , (2.6) one finds that the spatial part of the 4-velocity U µ is null and that the energy density and pressure are given by: 1 ρ = φ˙2 + V (φ) , (2.7) φ 2 1 p = φ˙2 − V (φ) . φ 2

It is clear that the EOS parameter wφ = pφ/ρφ satisfies −1 ≤ wφ ≤ 1, thus allowing scalar fields to describe a large range of fluids, in particular dark energy — something that is unique for this kind of fields. The equation of motion for φ can be equivalently derived from the Euler-Lagrange equations or from the divergence of the energy-momentum tensor. It reads dV φ¨ + 3Hφ˙ + = 0 . (2.8) dφ This equation is similar to the equation for a particle of unit mass in a potential V with a variable friction term proportional to H, which can simplify its analysis.

8 2.2 Scalar Dark Energy

2.2 Scalar Dark Energy

Having derived all the fundamental equations, we will now turn to the description of the components of interest, starting with dark energy. We begin by reviewing the case in which the scalar field is the only component of the Universe. In that situation the evolution equations are: 1 1 H2 = φ˙2 + V (φ) , (2.9) 3 2 a¨ 1 H˙ = − H2 = − φ˙2 . a 2 With such a set of equations one can easily determine φ(t) and the form of V (φ) if a(t) is known. Assuming a power law expansion a(t) ∝ tp, the solution is: r2 φ(t) = p2p ln(t/t ) ,V (φ) ∝ (3p − 1) exp − φ , (2.10) 0 p where t0 is an integration constant. The results indicate that an exponential potential can lead to an accelerated expansion if its slope is small enough, i.e. p > 1. This condition on the slope is well ˙2 understood by looking at the EOS parameter: it satisfies wφ < −1/3 if φ < V (φ), i.e. if the scalar field is rolling slowly down the potential. For that to happen the slope of the potential must be small in comparison to its height, so that the parameter 1 dV = − , (2.11) V dφ must be small (in natural units). For the exponential case, the condition translates into 2 < 2. In addition to having the possibility of driving the accelerated expansion, exponential models also possess scaling solutions when another fluid is present. In those solutions, the field closely mimics the evolution of the fluid: the two energy densities are proportional and their EOS parameters are equal. The best way to study scaling solutions is by finding the fixed points of the dynamical system constructed from the evolution equations 1 1 H2 = φ˙2 + V (φ) + ρ , 3 2 b dV φ¨ + 3Hφ˙ + = 0 , (2.12) dφ

ρ˙b + 3Hρb(1 + wb) = 0 , where ρb and wb are the energy density and the EOS parameter for the background ﬂuid. This dynamical system is more easily studied by using the variables √ 1 dφ V 1 dV dV −2 d2V N = ln a , x = √ , y = √ , = − , Γ = V , (2.13) 6 dN 3H V dφ dφ dφ2 and rewriting the evolution equations as ρ b + x2 + y2 = 1 , 3H2 dx r3 3 = −3x + y2 + x (1 − w )x2 + (1 + w )(1 − y2) , (2.14) dN 2 2 b b dy r3 3 = − xy + y (1 − w )x2 + (1 + w )(1 − y2) , dN 2 2 b b d √ = − 62x(Γ − 1) , dN

9 2. Scalar Field Cosmology where the last equation is just a consequence of the definitions of Γ and ; for an exponential potential is constant and Γ = 1, so the last equation is identically satisfied. Finding the fixed points is just a matter of solving the equations by setting the derivatives to zero. This yields five fixed points, given by [28]:

(a)(x, y) = (0, 0), Ωφ = 0, wφ = indef. ,

(b)(x, y) = (1, 0), Ωφ = 1, wφ = 1 ,

(c)(x, y) = (−1, 0), Ωφ = 1, wφ = 1 , (2.15) r ! 2 2 (d)(x, y) = √ , 1 − , Ωφ = 1, wφ = − 1 , 6 6 3

q 3 q 3 2  2 (1 + wb) 2 (1 − wb ) 3(1 + w ) (e)(x, y) = , , Ω = b , w = w .   φ 2 φ b

Points (a), (b) and (c) may be discarded as they lead to unrealistic situations. Point (d) is a scalar dominated solution, with accelerated expansion for 2 < 2, the same condition obtained before. Point (e) yields the previously discussed scaling solution. These last two are physically meaningful, but they 2 2 do not exist for all values of the parameters: point (d) only exists for < 6 and (e) for > 3(1+wb)— otherwise Ωφ would be larger than 1, which is not possible in a flat Universe, with all the components obeying the dominant energy condition. To evaluate the stability of each fixed point, one must find the eigenvalues of the Jacobian matrix of the right hand side of the dynamical equations: it is stable if the real part of both eigenvalues is negative. That only happens for points (d) and (e), for which the eigenvalues µ1 and µ2 are 1 (d) µ = 2 − 3(1 + w ), µ = − 6 − 2 , (2.16) 1 b 2 2 s ! 3 1 + wb 3(1 + wb) (e) µ1,2 = − (1 − wb) 1 ± 1 − 8 1 − 2 . (2.17) 4 1 − wb

2 However, (d) is stable only for < 3(1+wb), i.e. if (e) does not exist. As for (e), it is always stable (as long as it exists). An accelerated Universe implies point (d) with 2 < 2: however, that does not solve any of the problems of the cosmological constant. Solution (e), on the other hand, has the possibility of alleviating some of the coincidence problems, since its scaling properties provide a way to maintain

ρφ close to ρb, almost independently of the initial conditions, but does not accelerate the expansion of Universe. The ideal solution seems to be one that begins with scaling behavior and changes towards the accelerating solution at late times. Clearly this cannot be achieved through a potential with constant , but small changes can be made to the exponential model to accommodate that ideal. Several models exist with those properties, the simplest one being perhaps the two exponential model of Ref. [29]. Our interest will be centered, however, in models of the type ﬁrst discussed by Albrecht and Skordis in Ref. [30], which multiply the exponential with a polynomial. This kind of potentials is believed to arise in the low energy limit of M theory. A simple and very useful example is the following potential:

2 −λφ V (φ) = (A + (φ − φ0) )e , (2.18)

10 2.3 Scalar Dark Matter

where A, φ0 and λ are close to order unit parameters in reduced Planck units. This potential is useful for describing dark energy because it possesses a local minimum at

1 p φ = φ + 1 − 1 − Aλ2 , (2.19) m 0 λ if Aλ2 < 1. If the field reaches this minimum with sufficiently low kinetic energy, it will settle there and give rise to an accelerated expansion. Furthermore, the addition of the polynomial does not influence the scaling behavior while the field is away from the minimum. Most of the analysis shown relies on asymptotic solutions. However, one can argue that the Universe may not have reached these attractors yet. In that case, the present state of acceleration may be transient and end as the system is approaching the attractor solution. Such a solution is also useful for defining suitable asymptotic states free from future horizons in fundamental theories such as string theory. These transient solutions have been discovered in a large variety of models, including the exponential potentials discussed here [31], and also in models with two quintessence fields [32]. To summarize, dark energy can be successfully described using scalar fields, presenting a large variety of features that can be useful. In particular, contrarily to the cosmological constant, scalar fields are dynamical, something that has observable consequences. This type of research is thus important for motivating the astronomical searches for additional information, in order to shed some light on the nature of dark energy.

2.3 Scalar Dark Matter

Dark matter can also be described by a scalar field: one just has to find a potential for which wφ = 0 is a solution, at least from the age of matter domination, when structures formed, until the present. We have already seen that scaling solutions of the exponential model, V (φ) ∝ exp(−λφ), mimic the 2 background fluid. Thus, during matter domination, the field behaves as matter with Ωφ = 3/λ .

In order to get Ωφ = Ωdm ≈ 0.25, one must have λ ≈ 3.5. However, if the scaling attractor is reached during radiation domination, Ωφ ≈ 1/3 during that stage, which increases the expansion rate in comparison to standard cosmology. This is highly problematic for nucleosynthesis, as the freeze-out of weak interactions would occur sooner and hence a different amount of 4He would be produced. The current bounds on this were studied in Ref. [33] and found to be Ωφ ≤ 0.045 ⇒ λ ≥ 9.5, which clearly rules out the exponential model for dark matter. As can be expected, cosmologically, dark matter can be described by somewhat simpler potentials, since it just needs to represent a perfect fluid with vanishing pressure. In what follows, we review coherent scalar field oscillations as studied by Turner in Ref. [34], in order to argue that power-law potentials, given by V (φ) ∝ φn , (2.20) with n even, are useful for describing perfect fluids with constant EOS parameter, as is needed for dark matter. The actual potential is not required to be in this exact form: it must only have a minimum that can be suitably approximated by a power-law.

11 2. Scalar Field Cosmology

In a bounded potential, the ﬁeld will start oscillating around the minimum at some point in time. If the frequency of the oscillations ω ≈ φ/φ˙ is much larger than the expansion rate H, then only the averages over an oscillation are relevant. This can be shown as follows: one can always write φ˙2 as the sum of the average and the oscillation; taking into account that ρφ varies slowly, it is reasonable to write ˙2 φ = ρφ + pφ = (γ + γp)ρφ , (2.21)

˙2 where γ is the average and γp is the periodic part of φ /ρφ. The equation of motion for φ is

˙2 ρ˙φ = −3Hφ , (2.22) which can be formally solved as Z N Z t 0 0 ln(ρφ/ρφ0) = −3 γdN − 3 Hγpdt . (2.23) N0 t0 The last term can be integrated by parts, resulting in Z t Z t Z t Z 0 0 0 00 Hγpdt = H γpdt − H˙ γpdt dt . (2.24) t0 t0 t0 R 0 −1 Furthermore, the integral γpdt is O(ω ) due to its periodicity, so that the integrals above are roughly O(H/ω 1). Consequently, the last term in Eq. (2.23) is negligible when compared to the average term, meaning that φ˙2 can be substituted by its average over one cycle. ˙2 The factor γ is calculated by averaging φ /ρφ over a period T of the oscillations: 1 Z T φ˙2 γ = dt (2.25) T 0 ρφ

Using the fact that ρφ is nearly constant during an oscillation, one can substitute it by the value of the potential at the maximum of the oscillation, here denoted by VM = V (φM ). Using that, the identity pφ = ρφ − 2V , and changing the integration variable to x = φ/φM , the integrals turn into:

1 R (1 − xn)1/2 dx 2n γ = 2 0 = , (2.26) R 1 n −1/2 2 + n 0 (1 − x ) dx where the integration was made over half a cycle, in order for the variable change t → x to be invertible. Notice that this result is equivalent to the relation 1 n φ˙2 = hV (φ)i , (2.27) 2 2 which is the Virial theorem for power-law potentials, although here the averages are computed over one oscillation.

The case of importance for describing dark matter is n = 2, for which hwφi = γ − 1 = 0, which means that DM is well described by a massive scalar ﬁeld, with the potential being just a mass term: 1 V (φ) = m2φ2 . (2.28) 2 This treatment is equally valid if the mass is changing with time, as long as it varies in a time scale much larger than the period of the oscillations — as one can then calculate the cyclic averages as though the mass was constant. This happens for the model studied in this thesis, as presented in Chap. 3, with a varying dark matter mass, as well as for the axion whose mass varies with temperature.

12 3 Dark Matter – Dark Energy Interaction

There exist a large number of proposals to describe dark matter and dark energy, but most of them assume that these components are non-interacting and treat them as ﬂuids. However, since there are neither theoretical arguments forbidding such an interaction, nor is it ruled out by observations, it is natural to study the general situation in which dark matter and dark energy are coupled, and hopefully this feature may allow the gain of a deeper insight into the nature of these components. Moreover, the coincidence of their energy densities at present suggests a connection between them. A general way to describe the DM—DE interaction is to introduce an energy exchange term Q in the conservation equations as follows:

ρ˙de + 3H(ρde + pde) = Q, (3.1)

ρ˙dm + 3Hρdm = −Q.

One may phenomenologically study this interaction by withholding any assumptions about the nature of the dark sector and treat it straightforwardly as a two component fluid. The coupling Q is usually assumed to be of the form Q = δdeHρde +δdmHρdm, where H is the expansion rate and δi are coupling terms. This treatment is encountered, for instance, in the observational studies of Refs. [35, 36]. Unification models naturally connect the two dark components. A very well studied model of this kind is the Chaplygin gas model and its generalizations [37–39], where dark matter and dark energy are described by a single fluid, with EOS given by:

A p = − , (3.2) ρα where A is a positive parameter and 0 ≤ α ≤ 1 (α = 1 for the original Chaplygin model). The

13 3. Dark Matter – Dark Energy Interaction uniﬁcation is visible in the solution of the conservation equation:

1 B 1+α ρ(a) = A + , (3.3) a3(1+α) where B is an integration constant. At early times this expression is well approximated by ρ(a) ∝ a−3, thus describing non-relativistic matter, while at late times it approaches a constant, representing then dark energy. These two components can actually be separated into ρdm and ρde. Furthermore, an assumption about the equation of state of DE, leads to an explicit interaction between the dark components [40]:

ρ˙dm + 3Hρdm = −ρ˙de (3.4)

A map between the Generalized Chaplygin Gas (GCG) model and the interaction model discussed in the previous paragraph can be found in Ref. [35]. An alternative path to study the interaction assumes that dark energy is described by a scalar field φ, as introduced in section2, in interaction with a fluid, the so-called interacting quintessence model. The models of Refs. [41, 42], inspired by Brans-Dicke modifications of gravity, use such an interaction, in which the coupling is chosen to be ˙ Q = f(φ)φρb , (3.5) where f(φ) is a generic function of the field and ρb is the energy density of the background fluid. A similar mechanism is the so-called chameleon model [43]. In these cases the field interacts with every other component of the Universe, leading to observable effects in solar system tests of gravity. Nevertheless, they can easily be modified to an interaction between just quintessence and the DM

fluid, by substituting ρb with ρdm. Ref. [44] develops this approach, with the quintessence potential and the interaction term derived from a scaling solution resembling the one discussed in section 2.2. A more fundamental approach to tackle the interaction treats both DE and DM as fields, thus abandoning the need for fluids in the treatment of these components. Usually this is achieved through two new scalar fields, φ for DE and χ for DM [45–47]. For canonical scalar fields, their action is obtained by adding another kinetic term to Eq. (2.1), for χ, and introducing an interaction potential

Vint(φ, χ). Ultimately these models also lead to the coupling of the interacting quintessence scenario, as long as the interaction is in the form of a DM mass term. This is to be expected, since it was already stated that such a mass term correctly mimics the evolution of the dark matter energy density. This relation will be derived below when treating the average equations for motion for the dark matter field. These models are quite similar to the so-called VAriable Mass Particle (VAMP) models [48] in which a particle is introduced whose mass varies with the quintessence field. There are many advantages to this type of approach: one can find the full set of coupled equations from the action and consequently the functional form of the EOS parameter and the DE-DM coupling is fully determined. Furthermore, this is an elegant and straightforward way to link DM and DE with more fundamental physics models from which these components might stem from.

14 3.1 Interaction Model

3.1 Interaction Model 3.1.1 Basic equations

We consider two interacting canonical real scalar ﬁelds φ and χ, whose action is given by

Z √ 1 S = d4x −g − gµν (∂ φ∂ φ + ∂ χ∂ χ) − V (φ, χ) , (3.6) d 2 µ ν µ ν which is just a two ﬁeld generalization of Eq. (2.1). Consequently, it leads to similar ﬁeld equations:

∂V φ¨ + 3Hφ˙ + = 0 , (3.7) ∂φ ∂V χ¨ + 3Hχ˙ + = 0 . (3.8) ∂χ

Again, from the stress energy tensor we obtain the usual expressions for the pressure and energy density:

1 1 ρ = φ˙2 + χ˙ 2 + V (φ, χ) , (3.9) d 2 2 1 1 p = φ˙2 + χ˙ 2 − V (φ, χ) . d 2 2

We also consider non-relativistic matter and radiation ﬂuids, which we assume to be uncoupled at late times, when interactions between radiation and matter are negligible. As a consequence, they evolve −3 −4 as was mentioned in the introduction: ρm ∝ a and ρr ∝ a , respectively. With these ingredients, the Friedmann equation reads

1 1 1 H2 = ρ + ρ + φ˙2 + χ˙ 2 + V (φ, χ) . (3.10) 3 m r 2 2

3.1.2 Interaction Potential

We are interested in analyzing the following potential:

1 V (φ, χ) = e−λφP (φ, χ) + m2χ2 , (3.11) 2 where P (φ, χ) is a polynomial in φ and χ and m is the dark matter bare mass. This model is inspired in exponential models that appear naturally in fundamental theories like string or M-theory, or N = 2 supergravity in higher dimensions [49–51]. Furthermore, the interaction of chiral superfolds in the context of N = 1 supergravity inﬂationary models [52, 53] has some similarities with the present model. One can separate the polynomial P (φ, χ) into the interacting and non-interacting terms, P (φ, χ) =

Pφ(φ) + Pint(φ, χ). For the non-interacting part, we choose the potential mentioned earlier in Eq. (2.18), which we rewrite here:

2 Pφ(φ) = A + (φ − φ0) .

As for the interacting term, an obvious choice is to require the χ ﬁeld to be equivalent to a ﬂuid of non-relativistic matter, i.e. with negligible pressure. We recall from section 2.3 that this is achievable

15 3. Dark Matter – Dark Energy Interaction by using a potential proportional to χ2. This leads to an eﬀective varying mass for DM, which depends on the value of φ. The full potential then reads

V (φ, χ) = Vde(φ) + Vdm(φ, χ) , (3.12) with

1 V (φ) = e−λφ A + (φ − φ )2 ,V (φ, χ) = M 2(φ)χ2 , de 0 dm 2 where the mass function M 2(φ) is given in our model by

M 2(φ) = m2 + 2P˜(φ)e−λφ , (3.13) and the polynomial for P˜ is written as:

P˜(φ) = B + Cφ + Dφ2 , (3.14) where B, C and D are order unit parameters in terms of the appropriate powers of the reduced Planck mass. Due to its φ dependent mass, our model leads to oscillations in χ with a variable frequency. Note however that, unlike preheating models (see for example Ref. [24]), which exhibit parametric resonance, the frequency here changes slowly. For that reason, our conclusions are unchanged, as explained in the end of section 2.3.

3.1.3 Average Evolution Equations

Since the frequency of the oscillations is very high, it is rather infeasible to solve the χ equation numerically. Thus, during the oscillation phase, it will be necessary to consider only averages of the ﬁeld. In particular, we will derive the equation for the dark matter density and work with that quantity instead. First, the dark matter density and pressure are deﬁned using the same separation as for the potential, Eq. (3.12),

1 1 ρ = χ˙ 2 + M 2(φ)χ2 , (3.15) dm 2 2 1 1 p = χ˙ 2 − M 2(φ)χ2 . dm 2 2

By design, the averages over an oscillation read:

2 2 2 hρdmi = χ˙ = M (φ) χ , hpdmi = 0 . (3.16)

˙ 0 Next, Eq. (3.8) is multiplied byχ ˙ and a term φVdm(φ) is inserted. We obtain d 1 ∂V χ˙ 2 + V (φ, χ) + 3Hχ˙ 2 − φ˙ dm = 0 . (3.17) dt 2 dm ∂φ

Taking the average yields

1 ∂M 2(φ) ρ˙ + 3Hρ − φ˙ χ2 = 0 , (3.18) dm dm 2 ∂φ

16 3.2 Results

where we have written hρdmi as ρdm and hρ˙dmi asρ ˙dm, as, to a good approximation, the density is not sensitive to the oscillations. This can be quickly shown by assuming the rapid oscillations of χ(t) are given by a sinusoidal function, and hence the density depends only on the amplitude of the oscillations, which is unaﬀected by a cyclic average. Substituting the average of χ2 given by Eq. (3.16), we ﬁnd

1 1 ∂M 2(φ) ρ˙ + 3Hρ = φ˙ ρ . (3.19) dm dm 2 M 2(φ) ∂φ dm

So, as previously mentioned, there is an equivalence relation between coupled quintessence and the ﬁeld theory approach. It is given by the relationship

1 ∂ ln M 2(φ) f(φ) = . (3.20) 2 ∂φ

Moreover, there is a formal solution of Eq. (3.19) as a function of φ, through

−3 ρdm(φ, a) = n0a M(φ) , (3.21) where n0 is an integration constant. Notice that this corresponds to stating that ρ = nM, the usual expression of the energy density as a function of the DM mass M and the number density n ∝ a−3. With this solution, the dynamics is reduced to a single differential equation for φ, which is obtained from Eq. (3.7), replacing V with an effective potential Veff given by

Veﬀ(φ, a) = Vde(φ) + ρdm(φ, a) . (3.22)

As we mentioned above, the validity condition of these equations is M 2(φ) H2, otherwise the oscillation regime is not relevant and we must also solve Eq. (3.8).

3.2 Results 3.2.1 Analytical Considerations

We start by noting from Eq. (3.13) that for a suﬃciently large value of φ we ﬁnd M 2(φ) ≈ m2.

At this regime the interaction is irrelevant, which can be veriﬁed by examining ρdm and its derivative with respect to φ. On the other hand, for small values of φ, the term with m2 can be neglected and the interaction is relevant for the dynamics. Considering the polynomial P˜(φ) to be of order unity, we 2 λφ can estimate the transition value φc between the two phases by setting m e = 1, which results in: 2 φ ≈ − ln m . (3.23) c λ

Thus, for φ < φc, the interaction is relevant, while it becomes negligible as the value of the scalar field increases. This work focusses mostly in the late time behavior of the Universe, close to the stage of accelerated expansion. We find then that it is important to estimate whether the interaction is relevant at late times. We can estimate the value of the DE field near the present φ(0) by assuming that ρde0 ≈ Vde(φ(0)) and that it is close to the critical density ρc0, which results in 1 φ(0) ≈ − ln ρ . (3.24) λ c0

17 3. Dark Matter – Dark Energy Interaction

Requiring that φc ≥ φ(0) yields a rather low bound for the bare mass,

√ −60 m . ρc0 ∼ 10 . (3.25)

This analysis suggests that, unless the DM bare mass is unnaturally small, the effects of the interaction will not be detected at the present. A way to modify this would be to change P˜(φ) to O(10s), which increased this estimate by roughly 2s orders of magnitude. However, this would only shift the naturalness problem to P˜(φ). Another important situation is the onset of the oscillatory phase: we must establish the φ field value 2 2 for which M (φ) & H . In order to obtain it, we recall from section 2.2 that exponential potentials lead to scaling solutions. Although our model is not a pure exponential, scaling is still valid before the field φ falls in the minimum of its potential. This is similar to what happens for the model of Ref.

[30], which we use for Vde, since the polynomial P (φ, χ) does not vary signiﬁcantly during the relevant stage. That being the case, we have

3(w + 1) 9(w + 1) Ω ≈ ⇒ V ∼ H2 , (3.26) de λ2 de λ2 where w is the eﬀective EOS parameter for the combination of all the components. The oscillation condition can then be rewritten as

9(w + 1) m2eλφ + 2P˜ 2 & 1 . (3.27) λ Pφ

We recall from our discussion of scalar dark matter that, if the scaling would occur during nucleosyn- ˜ thesis, then λ & 10 [33]. With such a value for λ and considering P ∼ Pφ, the l.h.s. is smaller than unity when the interaction is relevant, meaning that the field χ has not begun oscillating during that stage; on the other hand, one expects the oscillatory phase to begin as the interaction loses importance. A particular case is that for the threshold mass of Eq. (3.25), for which the field may not start oscillating until the present. The implication is that χ does not represent dark matter as such in the early Universe, which is clearly a problem for structure formation. With these issues in mind, the model is modified so to allow for a difference in the behavior of both exponentials, i.e. we generalize it to:

¯ M 2(φ) = m2 + 2P˜(φ)e−λφ , (3.28) with λ¯ 6= λ. The constraint on m is thus relaxed to the less strict condition:

¯ m ρλ/2λ . (3.29) . c0

The problem associated to the onset of oscillations is also evaded, as these start while the interaction is still relevant. Similarly to what we discussed above, discarding the assumption that the parameters B, C and D in

P˜(φ) are O(1) in terms of Mp, would also be a solution for these problems. However, to actually solve them it would require they must increase by several orders of magnitude, which is rather unnatural, since they are already at the Planck scale.

18 3.2 Results

3.2.2 Numerical Solutions

Similarly to the discussion of scaling solutions, we use the number of e-folds N = ln a as the “time” variable for our numerical analysis. In addition, for convenience, we use the rescaled variables of Ref. [54]: H φ˙ χ˙ H˜ = e2N , Φ˜ = e2N , X˜ = e2N , (3.30) H0 H0 H0 where H0 = 72 km/s/Mpc is the value of the expansion rate at present. Thus, Eqs. (3.7), (3.8) and (3.10) are now rewritten as 4N ˜ 2 N 1 ˜ 2 1 ˜ 2 e H = Ωm0e + Ωr0 + Φ + X + 2 V (φ, χ) , 6 6 3H0 4N ˜ ˜ 0 ˜ e ∂V H(Φ + Φ) + 2 = 0 , (3.31) H0 ∂φ 4N ˜ ˜ 0 ˜ e ∂V H(X + X) + 2 = 0 , H0 ∂χ where the primes denote derivatives with respect to N. These changes improve the numerical robust- ness of the system, since the new variables take values in a shorter range. After the oscillatory phase is established, the averaged equations are used instead, which become 4N ˜ 2 N 1 ˜ 2 e H = Ωm0e + Ωr0 + Φ + 2 Veff(φ) , (3.32) 6 3H0 4N ˜ ˜ 0 ˜ e ∂Veff H(Φ + Φ) + 2 = 0 . H0 ∂φ The equations are integrated from N = −70 to N = 5, ranging from the Planck epoch to some future time. Our results are presented in terms of log a instead of N, since it is easier to translate the former into redshift z = 1/a − 1. Nucleosynthesis occurs around log a = −10. We present our results starting the discussion on the dependence on the mass m and then on the change of the exponent λ¯. We choose to fix the parameters of the DE potential at A = 0.01,

φ0 = 28.6 and λ = 9.5. This choice is such that there is a local minimum of the DE potential at the present epoch, while simultaneously complying with the nucleosynthesis limits for λ. As long as they verify these conditions, they can be changed without inﬂuencing the qualitative results. We keep the parameters of the polynomial P˜(φ) ﬁxed at B = C = D = 1 in all the solutions presented, since no relevant change occurs when they are varied; in fact, even an increase by several orders of magnitude is similar to a change in m or λ¯.

Figs. 3.1a and 3.1b show the effect of the interaction on ρdm and ρde for the indicated values of the bare mass m. This is the range for which initial conditions can be found leading to suitable parameters for the Universe at present and after passing through a DM dominated phase. ρde is not significantly changed in the cases shown. Moreover, for the studied range of masses, there is no effect close to the present; the only visible differences in ρdm are in the far past and are irrelevant for the evolution of the Universe. Nevertheless, the solution for the lowest mass (m = 10−55) possesses an interesting feature: some form of scaling occurs for both DE and DM. However, for this value of the mass, the oscillations start rather late, already during the matter dominated era, which may be problematic for structure formation.

19 3. Dark Matter – Dark Energy Interaction

(a) (b)

−15 −35 Figure 3.1: Evolution of log ρdm (a) and log ρde (b) for m = 10 (dotted), m = 10 (dashed) and −55 m = 10 (dot-dashed). The solid line represents the background density, log(ρm +ρr), shown for comparison.

(a) (b)

−60 Figure 3.2: Evolution of log ρdm (a) and log ρde (b) for m = 10 with the initial conditions χi = 1 (dotted), χi = 2.609 (dashed) and χi = 10 (dot-dashed). The solid line represents the background density, log(ρm + ρr), shown for comparison.

20 3.2 Results

(a) (b)

−57 Figure 3.3: (a) Results for m = 5.9×10 showing (a) the evolution of the relative densities Ωde (solid), Ωdm (dashed), Ωr (dotted) and Ωm (dot-dashed) and (b) the evolution of the deceleration parameter q (dashed) and the DE EOS parameter wde (solid). Also shown is the eﬀect of the oscillations on the deceleration parameter before log a = −1.3; at that moment the oscillations are averaged and henceforth the evolution of the relevant quantities is obtained in terms of Eqs. (3.32).

For much smaller masses, such as the one predicted by Eq. (3.25)(m = 10−60), the problems are more dramatic, as shown in Figs. 3.2a and 3.2b. There are two types of solutions, both unrealistic, which have an accelerated expansion at present. The first case does not possess a dark matter dominated epoch (dotted case, Fig. 3.2a), while in the second there is χ domination, but that field does not oscillate around the minimum of its potential (dot-dashed case, Fig. 3.2a). In this second type of solution, the χ field slowly rolls down its potential and oscillations never start. As a consequence, it behaves similarly to DE and gives rise to an accelerated expansion. Furthermore, there is a special solution of the first type (dashed) with transient accelerated behavior, although again not presenting a DM dominated phase and not complying with observations. There is a threshold case, for m = 5.9×10−57, which establishes a frontier between the two regimes, m ≥ 10−55 and m = 10−60. For that value of the mass, the solution still agrees with the observational constraints at present and exhibits a period of transient accelerated expansion, as shown in Figs. 3.3a and 3.3b. This solution presents other features, besides being transient: the onset of oscillations is visibly very close to the present — which can impair structure formation; both Ωdm and Ωde are non- negligible during nucleosynthesis, but even combined, they still comply with the bounds in Ref. [33],

Ωr & 0.95. The averaging of the oscillations is also visible in Fig. 3.3b, around log a = −1.3. We proceed now to the case λ¯ 6= λ, for which we fixed the mass at m = 10−15 ∼ 1 TeV. Our results are somewhat similar to those already found for masses m > 10−55 for a large range of λ¯, as seen in Figs. 3.4a and 3.4b. As previously, no relevant effects are found at the present epoch: only the DM density differs as a function of λ¯ at early times. However, for λ¯ = 2.8, a special solution is found again, presenting transient acceleration, as shown in Figs. 3.5a and 3.5b. This solution does not present the problem found previously in the other solution of this kind, nor does it require an unnaturally small mass. This is obviously a result of our modified model with λ¯ 6= λ. The only slightly anomalous result is that Ωdm starts increasing around log(a) = −8, an effect of the interaction, which does not occur in standard cosmology. We also found the same transient result for other pairs of (λ,¯ m) and we plot their values in Fig.

21 3. Dark Matter – Dark Energy Interaction

(a) (b)

−15 Figure 3.4: Evolution of log ρdm (a) and log ρde (b) for m = 10 and λ¯ = 9.5 (dotted), λ¯ = 6.5 (dashed), λ¯ = 4.5 (dot-dashed) and λ¯ = 2.8 (double-dot-dashed). The solid line represents the background density, log(ρm + ρr), shown for comparison.

(a) (b)

−15 Figure 3.5: Results for m = 10 and λ¯ = 2.8 showing (a) the evolution of the relative densities Ωde (solid), Ωdm (dashed), Ωr (dotted) and Ωm (dot-dashed) and (b) the evolution of the deceleration parameter q (dashed) and the DE EOS parameter wde (solid).

22 3.2 Results

Figure 3.6: Plot of the parameter space singling out the line where transient acceleration solutions occur and 2 2 the line were M (φ0) = H0 , the limit for oscillations.

(a) (b)

Figure 3.7: Eﬀective potential for m = 10−15 and λ¯ = 2.8 at N = −6 (dashed), N = −3 (dotted) and N = −0.23 (dash-dotted) as compared to the case with no interaction i.e. Veﬀ = Vφ (solid). Here we exagerate the minimum for visual purposes by setting A = 0.001, which would not change the results. Plot (a) displays a large range of φ, while (b) focusses on the region of the minimum, showing how the interaction raises the minimum and lowers the potential barrier.

3.6. The expression found for the “transient line” is

λ¯ = −0.1625 log(m) + 0.3706 . (3.33)

This expression shows similarities to Eq. (3.23), changing λ to λ¯, since the slope is −2 ln 10/φ(0) ≈

−2 ln 10/φ0 = −0.16102. This is expected, since in these transient solutions we anticipate the interaction to last just until the present, before becoming irrelevant. Therefore, it is not surprising that if the mass decreases, a larger value of λ¯ is required if a transient solution is to exist. Still, not all of those (λ,¯ m) pairs are equally interesting, since in situations with mass smaller than 10−15, such as m = 10−20, the anomaly we mentioned above worsens: the DM density parameter starts growing too soon and can exceed the mentioned bounds from nucleosynthesis. However, for the lowest of masses the transient solution does not exhibit this problem, as is attested by the case for m = 5.9 × 10−57.

The interaction is the root of this transient behavior: the addition of ρdm to the eﬀective potential

23 3. Dark Matter – Dark Energy Interaction

raises the minimum of Vde. The field is then allowed to escape and continue to roll down the exponential. This effect is shown in Fig. 3.7a, where we plot the effective potential for several values of N. However, for parameter values below the “transient line” of Fig. 3.6, i.e. for smaller λ¯ or m, only unrealistic scenarios are achieved. For those parameter values the interaction is too strong: therefore either the solution presents a very low DM density at present, thus lowering the effect of the interaction, or the accelerated phase does not occur, since the field does not settle in the minimum for an adequate period of time. Both of these scenarios are in conflict with observational data. Albeit attractive, this transient scenario requires some degree of fine tuning, as the minimum of the potential needs to be “dissolved” for the field to run, but still allow for φ to slow down enough to create an accelerated expansion phase. As stated at the end of section 2.2, such solutions have been found previously in the absence of interaction [30, 31], for a similar degree of fine-tuning in Vde. The results presented in this thesis show that these transient solutions can be achieved in our model, through an alternative path, similar to the two-field quintessence approach of Ref. [32]. The advantage of the present proposal is that it requires tuning only of λ¯ or m, the initial condition of χ fixed to reproduce the present dark matter density.

24 4 Conclusions and Future Work

In this work we have studied a model with two coupled scalar fields, in which one plays the role of dark matter and the other of dark energy at late times. This study aims to address some of the shortcomings of the Concordance Model, presented in Chapter1. The basic results regarding scalar fields in cosmology were derived in Chapter2 and we concluded that besides describing dark matter and dark energy, scalar fields possess interesting properties such as scaling and transient solutions. The interaction model was presented in Chapter3. Through its study, we found solutions that fit the current observational constraints and showed that the interaction is irrelevant at the present epoch in the case λ = λ¯ (cf. Eqs. (3.1.2) and (3.28)) for masses larger than m = 5.9 × 10−57 ∼ 10−29eV. For this last case, the interaction originates a transient stage of acceleration and may be excluded, in particular by its effects on structure formation (see e.g. Refs. [35, 55, 56]). We were led to find similar behavior for the case λ 6= λ¯, for a wide range of values for that parameter, with an irrelevant contribution of the interaction at late time. However, for specific values in a line of the parameter space of (λ,¯ m), different solutions appear leading to a transient stage of acceleration, which is absent in the other cases. Such solutions are compatible with observational data and supply a way out to the accelerating regime — a useful property for defining suitable asymptotic states free from future horizons in fundamental theories such as string theory (see Ref. [32] and references therein). We also found that, as expected, the “transient line” divides the parameter space into the regions with or without accelerating solutions. As an outlook, we point out that future work should focus on the study of cosmological pertur- bations and structure formation, since the interaction may influence the gravitational colapse of dark matter, even in cases for which we found that it is irrelevant at present. This would be important to further constrain the parameter space of the viable models.

25 4. Conclusions and Future Work

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