A Scalar Field Theory for

Dark Matter–Dark Energy Interaction

Pedro Miguel Gregorio Carrilho

Dissertacao para obter o Grau de Mestre em

Engenharia Fısica Tecnologica

Juri

Presidente: Professora Maria Teresa Haderer de la Pena StadlerOrientador: Doutor Jorge Tiago Almeida ParamosOrientador Externo: Professor Orfeu BertolamiVogal: Professor Jose Pedro MimosoVogal: Doutor Nuno Miguel Candeias dos Santos

September 2012

Acknowledgments

I would like to start by thanking my supervisors Orfeu Bertolami and Jorge Paramos for 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.

i

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 interacao entre materia escura e energia escura. Para

cumprir esse objectivo, sao introduzidos dois campos escalares φ e χ, que sao dotados de um potencial

de interaccao V (φ, χ) = e−λφP (φ, χ), em que P (φ, χ) e um polinomio. Este modelo complementa a

cosmologia padrao, que e baseada na constante cosmologica e na materia escura fria sem interaccoes.

Neste modelo, demonstra-se que as propriedades relevantes do presente estado do Universo sao repro-

duzidas para um grande conjunto de valores da massa livre do campo de materia escura. Estudam-se

tambem modificacoes do potencial, revelando implicacoes importantes da interaccao, incluindo a pos-

sibilidade de solucoes com aceleracao transitoria.

O trabalho original apresentado nesta tese segue a Ref. [1].

Palavras Chave

Energia escura, Materia escura, Campo Escalar, Modelo de interacao, Expansao acelerada tran-

sitoria.

v

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 Cosmology 7

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

3.2 log ρdm and log ρde for m = 10−60 and different values of the initial condition χi. . . . . 20

3.3 Transient solution for m = 5.9× 10−57, λ = 9.5. . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 log ρdm and log ρde for m = 10−15 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

ix

Abbreviations

CDM Cold Dark Matter

FLRW Friedmann-Lemaitre-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

xi

1Introduction

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 cosmolog-

ical 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-Lemaitre-Robertson-Walker (FLRW) metric,

given by the line element:

ds2 = gµνdxµdxν = −dt2 + a2(t)

(dr2

1− kr2+ r2dΩ2

), (1.1)

where a(t) is the cosmological scale factor, normalized in this work so that at present a(t0) = 1, dΩ2 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]

Rµν −1

2Rgµν = Tµν , (1.2)

where Rµν is the Ricci tensor, R = gµνRµν 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:

[Tµν ] = 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:

H2 =1

3ρ− k

a2, (1.4)

a

a= −1

6(ρ+ 3p) , (1.5)

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 of a 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

ρ(a) = ρ0a−3(1+w) , (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

density varies with the volume a−3 for non-relativistic matter (ρm), or with a−4 for radiation (ρr)

due to an additional contribution from the cosmological redshift. Other cases exist, such as ultra-

stiff matter, with w = 1 and ρs ∝ a−6 and also the more exotic case with w = −1 that leads to a

1Note that we use reduced Planck units, i.e. units in which ~ = c = 8πG = 1. For this reason every mass or energyscale 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

q ≡ − aaa2

=1

2Ω (1 + 3w) , (1.9)

which can be generalized for multiple fluids with density parameter Ωi and EOS parameter wi, by

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 be-

ing 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:∫d4x√−g 1

2R+ L − Λ, (1.10)

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,

Rµν −1

2Rgµν + Λgµν = Tµν . (1.11)

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

Λexp ∼ 10−47 GeV4 . (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]:

ρV =

∫ M

0

4πk2dk

(2π)3

1

2

√m2 + k2 =

M4

16π2+m2M2

16π2+

m4

64π2log

(m2e1/2

4M2

)+O(M−1) , (1.13)

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 chapter 2, 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

close to 1 at present, which meant that |1 − Ω−1| . 10−60 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

2Scalar 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:

Sφ =

∫d4x√−g−1

2gµν∂µφ∂νφ− V (φ)

, (2.1)

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:

δSφ =

∫d4x√−g−1

2gµν

(1

2gαβ∂αφ∂βφ+ V (φ)

)+

1

2gαµgβν∂αφ∂βφ

δgµν . (2.2)

Due to the choice of normalization for the gravitational action

SG =

∫d4x√−g 1

2R , (2.3)

the term in curly brackets equals half of the energy-momentum tensor, which is then given by:

Tµν = −gµν(

1

2∂αφ∂

αφ+ V (φ)

)+ ∂µφ∂νφ . (2.4)

The cosmological principle implies homogeneity, so all spatial derivatives vanish in a cosmological

model. For that reason the stress energy tensor takes the form:

Tµν = φ2δµ0 δν0 + gµν

(1

2φ2 − V (φ)

), (2.5)

where the dot represents a derivative with respect to time. Comparing it to the energy-momentum

tensor of a perfect fluid,

Tµν = (ρ+ p)UµUν + pgµν , (2.6)

one finds that the spatial part of the 4-velocity Uµ is null and that the energy density and pressure

are given by:

ρφ =1

2φ2 + V (φ) , (2.7)

pφ =1

2φ2 − V (φ) .

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

φ+ 3Hφ+dV

dφ= 0 . (2.8)

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:

H2 =1

3

(1

2φ2 + V (φ)

), (2.9)

H =a

a−H2 = −1

2φ2 .

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:

φ(t) =√

2p ln(t/t0) , V (φ) ∝ (3p− 1) exp

(−√

2

pφ

), (2.10)

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

understood by looking at the EOS parameter: it satisfies wφ < −1/3 if φ2 < 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

ε = − 1

V

dV

dφ, (2.11)

must be small (in natural units). For the exponential case, the condition translates into ε2 < 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

H2 =1

3

(1

2φ2 + V (φ) + ρb

),

φ+ 3Hφ+dV

dφ= 0 , (2.12)

ρb + 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

N = ln a , x =1√6

dφ

dN, y =

√V√

3H, ε = − 1

V

dV

dφ, Γ = V

(dV

dφ

)−2d2V

dφ2, (2.13)

and rewriting the evolution equations as

ρb3H2

+ x2 + y2 = 1 ,

dx

dN= −3x+

√3

2εy2 +

3

2x[(1− wb)x2 + (1 + wb)(1− y2)

], (2.14)

dy

dN= −

√3

2εxy +

3

2y[(1− wb)x2 + (1 + wb)(1− y2)

],

dε

dN= −√

6ε2x(Γ− 1) ,

9

2. Scalar Field Cosmology

where the last equation is just a consequence of the definitions of Γ and ε; for an exponential potential

ε 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)

(d) (x, y) =

(ε√6,

√1− ε2

6

), Ωφ = 1, wφ =

ε2

3− 1 ,

(e) (x, y) =

√

32 (1 + wb)

ε,

√32 (1− w2

b )

ε

, Ωφ =3(1 + wb)

ε2, wφ = wb .

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 ε2 < 2, the same condition obtained before. Point

(e) yields the previously discussed scaling solution. These last two are physically meaningful, but they

do not exist for all values of the parameters: point (d) only exists for ε2 < 6 and (e) for ε2 > 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

(d) µ1 = ε2 − 3(1 + wb), µ2 = −1

2

(6− ε2

), (2.16)

(e) µ1,2 = −3

4(1− wb)

(1±

√1− 8

1 + wb1− wb

(1− 3(1 + wb)

ε2

)). (2.17)

However, (d) is stable only for ε2 < 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 ε2 < 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

ε, 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:

V (φ) = (A+ (φ− φ0)2)e−λφ , (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

φm = φ0 +1

λ

(1−

√1−Aλ2

), (2.19)

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

background fluid. Thus, during matter domination, the field behaves as matter with Ωφ = 3/λ2.

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 φ2 as

the sum of the average and the oscillation; taking into account that ρφ varies slowly, it is reasonable

to write

φ2 = ρφ + pφ = (γ + γp)ρφ , (2.21)

where γ is the average and γp is the periodic part of φ2/ρφ. The equation of motion for φ is

ρφ = −3Hφ2 , (2.22)

which can be formally solved as

ln(ρφ/ρφ0) = −3

∫ N

N0

γdN ′ − 3

∫ t

t0

Hγpdt′ . (2.23)

The last term can be integrated by parts, resulting in∫ t

t0

Hγpdt′ =

[H

∫γpdt

′]tt0

−∫ t

t0

H

(∫γpdt

′)dt′′ . (2.24)

Furthermore, the integral∫γpdt

′ is O(ω−1) 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 φ2 can be substituted by its average over one cycle.

The factor γ is calculated by averaging φ2/ρφ over a period T of the oscillations:

γ =1

T

∫ T

0

φ2

ρφdt (2.25)

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:

γ = 2

∫ 1

0(1− xn)

1/2dx∫ 1

0(1− xn)

−1/2dx

=2n

2 + n, (2.26)

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

2φ2

⟩=n

2〈V (φ)〉 , (2.27)

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 〈wφ〉 = γ − 1 = 0, which

means that DM is well described by a massive scalar field, with the potential being just a mass term:

V (φ) =1

2m2φ2 . (2.28)

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

3Dark 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:

ρde + 3H(ρde + pde) = Q , (3.1)

ρdm + 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:

p = − A

ρα, (3.2)

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:

ρ(a) =

[A+

B

a3(1+α)

] 11+α

, (3.3)

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]:

ρdm + 3Hρdm = −ρde (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 section 2, 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

Sd =

∫d4x√−g−1

2gµν(∂µφ∂νφ+ ∂µχ∂νχ)− V (φ, χ)

, (3.6)

which is just a two field generalization of Eq. (2.1). Consequently, it leads to similar field equations:

φ+ 3Hφ+∂V

∂φ= 0 , (3.7)

χ+ 3Hχ+∂V

∂χ= 0 . (3.8)

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

density:

ρd =1

2φ2 +

1

2χ2 + V (φ, χ) , (3.9)

pd =1

2φ2 +

1

2χ2 − V (φ, χ) .

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

as was mentioned in the introduction: ρm ∝ a−3 and ρr ∝ a−4, respectively. With these ingredients,

the Friedmann equation reads

H2 =1

3

(ρm + ρr +

1

2φ2 +

1

2χ2 + V (φ, χ)

). (3.10)

3.1.2 Interaction Potential

We are interested in analyzing the following potential:

V (φ, χ) = e−λφP (φ, χ) +1

2m2χ2 , (3.11)

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:

Pφ(φ) = A+ (φ− φ0)2 .

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

Vde(φ) = e−λφ(A+ (φ− φ0)2

), Vdm(φ, χ) =

1

2M2(φ)χ2 ,

where the mass function M2(φ) is given in our model by

M2(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),

ρdm =1

2χ2 +

1

2M2(φ)χ2 , (3.15)

pdm =1

2χ2 − 1

2M2(φ)χ2 .

By design, the averages over an oscillation read:

〈ρdm〉 =⟨χ2⟩

= M2(φ)⟨χ2⟩

, 〈pdm〉 = 0 . (3.16)

Next, Eq. (3.8) is multiplied by χ and a term φV ′dm(φ) is inserted. We obtain

d

dt

(1

2χ2 + Vdm(φ, χ)

)+ 3Hχ2 − φ∂Vdm

∂φ= 0 . (3.17)

Taking the average yields

ρdm + 3Hρdm −1

2φ∂M2(φ)

∂φ

⟨χ2⟩

= 0 , (3.18)

16

3.2 Results

where we have written 〈ρdm〉 as ρdm and 〈ρdm〉 as ρdm, 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

ρdm + 3Hρdm =1

2φ

1

M2(φ)

∂M2(φ)

∂φρdm . (3.19)

So, as previously mentioned, there is an equivalence relation between coupled quintessence and the

field theory approach. It is given by the relationship

f(φ) =1

2

∂ lnM2(φ)

∂φ. (3.20)

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

ρdm(φ, a) = n0a−3M(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 M2(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 M2(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

can estimate the transition value φc between the two phases by setting m2eλφ = 1, which results in:

φc ≈ −2

λlnm . (3.23)

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

φ(0) ≈ − 1

λln ρc0 . (3.24)

17

3. Dark Matter – Dark Energy Interaction

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

m .√ρc0 ∼ 10−60 . (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

for which M2(φ) & H2. 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

Ωde ≈3(w + 1)

λ2⇒ Vde ∼

9(w + 1)

λ2H2 , (3.26)

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)

λ2

m2eλφ + 2P

Pφ& 1 . (3.27)

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 impor-

tance. 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:

M2(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/2λc0 . (3.29)

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 =H

H0e2N , Φ =

φ

H0e2N , X =

χ

H0e2N , (3.30)

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

H2 = Ωm0eN + Ωr0 +

1

6Φ2 +

1

6X2 +

e4N

3H20

V (φ, χ) ,

H(Φ′ + Φ) +e4N

H20

∂V

∂φ= 0 , (3.31)

H(X ′ + X) +e4N

H20

∂V

∂χ= 0 ,

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

H2 = Ωm0eN + Ωr0 +

1

6Φ2 +

e4N

3H20

Veff(φ) , (3.32)

H(Φ′ + Φ) +e4N

H20

∂Veff

∂φ= 0 .

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)

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

(a) (b)

Figure 3.2: Evolution of log ρdm (a) and log ρde (b) for m = 10−60 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)

Figure 3.3: (a) Results for m = 5.9×10−57 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) andthe DE EOS parameter wde (solid). Also shown is the effect of the oscillations on the deceleration parameterbefore log a = −1.3; at that moment the oscillations are averaged and henceforth the evolution of the relevantquantities 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 dom-

inated 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)

Figure 3.4: Evolution of log ρdm (a) and log ρde (b) for m = 10−15 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)

Figure 3.5: Results for m = 10−15 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 parameterq (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 andthe line were M2(φ0) = H2

0 , 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) andN = −0.23 (dash-dotted) as compared to the case with no interaction i.e. Veff = Vφ (solid). Here we exageratethe minimum for visual purposes by setting A = 0.001, which would not change the results. Plot (a) displaysa large range of φ, while (b) focusses on the region of the minimum, showing how the interaction raises theminimum 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 inter-

action 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

4Conclusions 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 Chapter 1. The basic results regarding scalar

fields in cosmology were derived in Chapter 2 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 Chapter 3. 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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