Dynamical Dark Energy

Dynamical Dark Energy
Vincent Poitras
Physics Department
McGill University
Montreal, Quebec
2014
A thesis submitted in partial fulfilment
of the requirements for the degree of Doctor of Philosophy
Doctor of Philosophy
c Vincent Poitras, 2014. All rights reserved
Abstract
The ΛCDM model is considered to be the most successful cosmological model by
reason of its simplicity and of the quality of the fit that it provides to disparate types
of data (SNeIa, CMB, BAO, etc.). According to this model, the current composition
of the Universe is dominated by two fluids, the cold dark matter and the cosmological
constant Λ, a form of dark energy with a constant energy density. In spite of its
observational successes, the ΛCDM model is facing two difficulties, the cosmological
constant problem and the coincidence problem. The cosmological constant problem
is a discrepancy of about 120 orders of magnitude between the observed values of
dark energy density and the value expected from quantum field computations. The
coincidence problem relies on the observation that the energy densities of cold matter
and of dark energy are currently of the same order of magnitude, although this could
happen only during a narrow window of time in the ΛCDM model. A possible way
to solve these two problems is to replace the cosmological constant by a dynamical
form of dark energy whose density could vary in time. This thesis will be focusing on
various aspects and different models of dynamical dark energy. More specifically, in
chapters 2 and 3 we consider a model where dark energy and dark matter are coupled
through an interaction term of the form Q0 ρnΛ . We show that for n
3{2, the ratio
of the densities of dark energy and cold matter becomes constant at late time and
thus possibly provides a solution to the coincidence problem. However, the values of
the model parameters consistent with the observational constraints disfavoured this
solution. In chapter 4, we consider the proposed hypothesis that the dark energy
should be in thermal equilibrium with the cosmological horizon and find the form of
the interaction term Q required to maintain this equilibrium. Finally in chapter 5,
based on studies of particles production from vacuum energy in de Sitter spacetime,
we verify under which conditions this decay could affect the probability distribution
of the values of the cosmological constant obtained from a modified version of the
anthropic principle, the causal entropic principle.
ii
Résumé
Le modèle ΛCDM est considéré comme étant le meilleur modèle cosmologique en
raison de sa simplicité et de la qualité de la concordance avec une vaste gamme de
données observationelles (SNeIa, CMB, BAO, etc.). Selon ce modèle, la composition
de l’Univers est présentement dominée par deux fluides, la matière sombre froide et
la cosmologique Λ, une forme d’énergie sombre dont la densité est constante dans le
temps. Malgré ses impressionants succès observationels, le modèle ΛCDM fait face à
deux difficultés théoriques, le problème de la constante cosmologique et le problème de
coincı̈dence. Le premier consiste en une différence d’environ 120 ordres de grandeur
entre la valeur observée de l’énergie sombre et la valeur théorique attendue à partir de
calculs en théorie des champs quantique. Le second problème repose sur l’observation
que les valeurs actuelles des densités d’énergies de la matière froide et de l’énergie
sombre sont du même ordre de grandeur, or ceci n’est possible que pendant une très
brève période de temps dans le modèle ΛCDM. Une des avenues considérée pour solutionner ces problèmes consiste à remplacer la constante cosmologique par une forme
dynamique d’énergie sombre dont la densité peut varier dans le temps. Cette thèse
est consacrée à l’étude de différents aspects et modèles d’énergie sombre dynamique.
Plus spécifiquement, dans les chapitres 2 et 3, nous considérons un modèle où l’énergie
sombre et la matière sombre sont couplées à travers d’un terme d’interaction de la
forme Q0 ρnΛ . nous montrons que pour n
3{2, le ratio des densités d’énergie som-
bre et de matière froide devient constant pour les temps lointains. Ceci aurait pu
expliquer le problème de coincı̈dence, par contre, les contraintes observationelles sur
les valeurs des paramètres du modèles excluent cette explication. Dans le chapitre 4,
nous considérons l’hypothèse déjà proposée voulant que l’énergie sombre doivent être
en équilibre thermique avec l’horizon cosmologique et trouvons la forme du terme
d’interaction Q requise pour maintenir cet équilibre. Finalement, en se basant sur
des études du phénomène de création de particules à partir de l’énergies du vide dans
l’espace-temps de de Sitter, nous vérifions sous quelles conditions, cette désintégration
iii
peut affecter la probabilité de distribution de la valeur de la constant cosmologique
obtenue à partir d’une variante du principe anthropique, le principe entropique causal.
iv
DEDICATION
This thesis is dedicated to my two younger cousins, Carl-Phillipe and
Jean-Christophe, who have this year begun their studies in physics and
mathematics, respectively.
v
ACKNOWLEDGEMENTS
First and foremost, I want to thank my thesis adviser, James Cline, who has been
so understanding of my sometimes difficult personal situation. I would also like to
thank all the members of the high-energy physics group, even though my introverted
personality somewhat limited our exchanges. It is with a certain sense of regret that
I leave this department without having benefited to the full extent of their knowledge
and expertise. A thought also goes out to my office mates, both past (Aaron, Johanna,
Kyoumars, Nick, Nima, Paul) and present (Annabel, Benjamin, Benoı̂t, Jean-Bernard,
Jean-François, Matt, Rob, Sébastien, Sheir), with whom I shared many good times
through the past four years. I would also like to thank M. Bryant, P. Doane, V.
Errasti Dı́ez, B.-A. Gaulin, R. Guénette, M. Kearney, B. Lefebvre, C. McKernan,
J.-F. Rajotte, C.-C. Roussel, J. Roy and D. Ste-Marie. for their help with different
technical matters (proof reading, computer problems, etc.). Finally, I would like to
thank my friends Yonathan and Verónica for all the moral support they gave me as I
was finishing this thesis.
vi
Statement of Originality
This thesis is based on original research which has led to the publication of three
articles as single author in peer reviewed journals [1–3] and to an other one that is in
preparation.
• Chapter 2 is based on [1]. We consider a model where dark energy is decaying
according to ρ9 de
Qρnde in a flat Universe.
This law was first proposed in [4]
to study the decay of dark energy into radiation, but we consider rather the case
where dark energy decays into dark matter. In addition to finding analytical
solution for the case n
3{2 (which is interesting in regard of the coincidence
problem), we constrain the value of model’s free parameters using various observational data.
• Chapter 3 is based on [2] and is an extension of the work presented in the previous chapter for a Universe with spatial curvature. We show that unlike the
flat case, there are two attractor solutions at late-time for n
3{2.
As in the
previous case, we constrain the values of the parameters using observational
data.
• Chapter 4 is based on [3]. In this chapter we consider the hypothesis made
in [5] according to which the dark energy must be in thermal equilibrium with
the cosmological horizon. We find that this could be true for a model of coupled
dark energy only if the interaction term, Q, has a specific form. We compute
the explicit expression of Q for two different types of dark energy.
• In chapter 5, after having reviewed the entropic causal principle proposed in
[6] and the mechanism of particle production from vacuum energy presented
vii
in [7], we check under which conditions the latter could affect the probability
distribution of the value of ρΛ inferred by the former.
viii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Résumé
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
Statement of Originality . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
1.1
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3
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25
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Constraints on Λptq-cosmology with a power law interacting dark
sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.1
2.2
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1.2
1.3
2
FRW Cosmology . . . . . . . . . . . . . . . . .
1.1.1 Cosmological principle . . . . . . . . . .
1.1.2 FRW metric . . . . . . . . . . . . . . . .
1.1.3 Energy conservation (continuity equation)
1.1.4 Friedmann equations . . . . . . . . . . .
1.1.5 Cosmological constant . . . . . . . . . . .
ΛCDM model . . . . . . . . . . . . . . . . . . .
1.2.1 Overview . . . . . . . . . . . . . . . . . .
1.2.2 Dark Matter . . . . . . . . . . . . . . . .
1.2.3 Dark Energy . . . . . . . . . . . . . . . .
1.2.4 Inflation . . . . . . . . . . . . . . . . . .
1.2.5 Cosmological constant problem . . . . . .
1.2.6 Coincidence problem . . . . . . . . . . .
Dynamical dark energy . . . . . . . . . . . . . .
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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34
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Can ΩΛ remain constant to late times? . . . . . . . . . . . . . . . .
57
3.1
3.2
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2.3
2.4
2.5
3
3.3
3.4
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76
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Particle production from vacuum energy in the causal diamond
94
4.3
4.4
5.1
5.2
5.3
5.4
5.5
6
Introduction . . . . . . . . . . . . . . . . . . . . . . . .
Models . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Basic equations . . . . . . . . . . . . . . . . . .
3.2.2 r9m 0 during a Λm-dominated era . . . . . . .
3.2.3 r9m 0 and r9k 0 during a Λmk-dominated era
3.2.4 r9m 0 and r9k 0 during a Λmk-dominated era?
Results and discussion . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
Dark energy in thermal equilibrium with the cosmological horizon? 75
4.1
4.2
5
2.2.1 MatterDark energy dominated era (MΛD-era): n 3{2
2.2.2 Radiation dominated era (RD-era): n 3{2 . . . . . . . .
Observational Constraints . . . . . . . . . . . . . . . . . . . . .
2.3.1 Distance modulus µ of SNeIa and GRB . . . . . . . . . .
2.3.2 Observational H pz q data (OHD) . . . . . . . . . . . . . .
2.3.3 Baryon acoustic oscillation (BAO) . . . . . . . . . . . . .
2.3.4 Cosmic Microwave Background (CMB) . . . . . . . . . .
Results and discussion . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Case n 3{2 . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 General case . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . .
Dynamics . . . . . . . . . . . . . . . . .
4.2.1 Interacting fluids . . . . . . . . . .
4.2.2 Types of dark energy . . . . . . .
Thermodynamics . . . . . . . . . . . . .
4.3.1 Cosmological horizon temperature
4.3.2 Conditions for thermal equilibrium
Summary . . . . . . . . . . . . . . . . .
Introduction . . .
Causal diamond .
Entropy produced
Entropy produced
Summary . . . .
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from radiating dust . . . . . . . . . . . . . .
by particle production from vacuum energy
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95
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112
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
x
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A
Curvature tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
A.1
A.2
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
FRW metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
117
Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . .
118
B.1
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118
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120
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
B
Observational constraints . . . . . . . . . . . .
B.1.1 Distance modulus µ of SNeIa and GRB
B.1.2 Observational H pz q data (OHD) . . . .
B.1.3 Baryon acoustic oscillation (BAO) . . .
B.1.4 Cosmic Microwave Background (CMB)
xi
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List of Figures
Figure
Page
Examples of an open (κ 0), a flat (κ 0) and a closed (κ ¡ 0) space
in two spatial dimensions. . . . . . . . . . . . . . . . . . . . . . . .
3
Evolution of the Hubble parameter H and of the density parameters
Ωr , Ωm , Ωk and ΩΛ as a function of the scale factor a for the ΛCDM
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Evolution of the scale factor a as a function of time for the ΛCDM
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.4
Decomposition of the rotation curve of the spiral galaxy NGC 3198.
15
1.5
Baryonic and dark matter in the Bullet cluster. . . . . . . . . . . . .
15
1.6
Illustration of the horizon problem. . . . . . . . . . . . . . . . . . . .
20
1.7
Slow-roll potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
1.8
Illustration of the coincidence problem. . . . . . . . . . . . . . . . . .
26
2.1
Curve ΩΛ0
. . . . . . . . . . . . . . . .
35
2.2
Ratio of the initial and the current dark energy density (ρΛi {ρΛ0 ) as a
function of ΩΛ8 ΩΛ0 . . . . . . . . . . . . . . . . . . . . . . . . .
35
Examples of the evolution of the dark energy density parameter ΩΛ ,
the matter density parameter Ωm and the modulus of the scale factor
|a| in absence of radiation and baryons for an expanding Universe
(H0 ¡ 0) with (a) ΩΛ0 ¡ ΩΛ8 , (b) ΩΛ0 ΩΛ8 1 and (c) ΩΛ8 ¡ 1.
36
Evolution of dark energy (ΩΛ ), matter (Ωm ) and radiation (Ωr ) density
parameter for different values of ΩΛ8 and with fixed values for ΩΛ0
and H0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
1.1
1.2
1.3
2.3
2.4
ΩΛ8 in the plane ΩΛ Q.
0
xii
2.5
Projection (a) in the plane ΩΛ0 ΩΛ8 , (b) in the plane ΩΛ0 H0 and
(c) in the plane ΩΛ8 H0 of the 1-σ and the 2-σ confidence region
obtained from four different types of observational data. . . . . . .
49
³z
Percentage of difference on the value of the integral 0 dz {pH pz qp1 z qq
evaluated for the ΛCDM model and for the ΛptqCDM model (n 3{2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
1-σ and the 2-σ confidence intervals for ΩΛ0 , H0 and Q{QBF V at
different value of n. . . . . . . . . . . . . . . . . . . . . . . . . . .
53
3.1
Late-time matter-to-dark energy ratio r̃m .
66
3.2
Examples of trajectories in the plane rm rk for q
2.6
2.7
3.3
3.4
. . . . . . . . . . . . . . .
q0 and q ¡ q0. .
Observational constraints on the parameters of the ΛptqCDM model.
67
69
Evolution of the energy densities (ΩΛ , Ωk , Ωm , Ωr ) and of the function
1 q in function of the time. . . . . . . . . . . . . . . . . .
minprm , rm
70
Observational constraints on the parameters of the ΛptqCDM model
projected in the plane rm0 r̃m . . . . . . . . . . . . . . . . . . . .
72
5.1
Comoving 3-volume Vc of the causal diamond. . . . . . . . . . . . . .
101
5.2
Entropy production rate (radiating dust) . . . . . . . . . . . . . . . .
103
5.3
Probability distribution over log ρΛ computed from the CEP (radiating
dust). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
104
Probability distribution over log ρΛ computed from the CEP (particle
production). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
Rpeak as a function of toff and t̃off as a function of mass. . . . . . . . .
111
3.5
5.4
5.5
xiii
List of Tables
Table
Page
1.1 Evolution of the energy density ρi and of the total energy Ei of a perfect
fluid as function of its effective EoS parameter wieff . . . . . . . . . .
6
1.2 Relation between the spatial curvature (κ), energy density (ρ) and
density parameter (Ω). . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3 Values of the EoS parameters and of the density parameters (from
Planck) for the different fluids of the ΛCDM model . . . . . . . . .
12
1.4 Sign of the derivative dΩk {da
19
. . . . . . . . . . . . . . . . . . . . . .
2.1 Observational values of H pz q used to constrain the model.
. . . . . .
43
2.2 Best fit values for the ΛCDM model parameters and the corresponding
χ2min in a flat spacetime. . . . . . . . . . . . . . . . . . . . . . . . .
47
2.3 Best fit values for the ΛptqCDM model parameters and the corresponding χ2min in a flat spacetime. . . . . . . . . . . . . . . . . . . . . . .
48
3.1 Best-fit values for the free parameters and the corresponding χ2min for
the ΛCDM and the ΛptqCDM models in a spacetime with spatial
curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
xiv
1
The core of this thesis (chapters 2, 3, 4 and 5) consists of four articles focusing
to the study of different aspects of dynamical dark energy models. By dark energy,
we mean any cosmological “fluid” (which includes the cosmological constant) whose
thermodynamical properties could lead to a late-time accelerated expansion of the
Universe. We qualify it as dynamical if its energy density is varying in time. The
aim of this chapter is to present the motivations and the basis needed to such a study
with more details than are given in the subsequent chapters. In section 1.1, we will
introduce the basic elements of FRW cosmology, on which most of the contemporary
cosmological models are based. Afterwards, in section 1.2, we will review the ΛCDM
model, which is currently considered as the “standard cosmological model”. Finally,
in section 1.3, we will give an overview of different types of dynamical dark energy.
Throughout this thesis, the mostly plus convention is used for the metric signature
p, , , q. In the tensorial notation, the indices represented by a Greek letter are
used for space and time components p0, 1, 2, 3q, and those by a Latin letter, for space
components p1, 2, 3q. In this chapter, we have set c ~ kB 1 but we have
explicitly keep the Newton constant G in all equations. This convention is also used
in chapters 2 and 3, but in chapters 4 and 5, we have also set G 1.
1.1
1.1.1
FRW Cosmology
Cosmological principle
The cosmological principle is one of the most important concept in modern cosmology. In simple language, it says that there is no special point in the Universe 1 .
More formally, it states that on large scales, the Universe is homogeneous (is the same
everywhere) and isotropic (looks the same in all directions). In a more constraining
version of this principle, dubbed perfect cosmological principle, it is also assumed that
the Universe must be invariant in time. Although it was abandoned in the first half
of the 20th century, the perfect cosmological principle remains important from an historical point of view. Indeed, it was in order to obtain a static Universe that Einstein
introduced, in 1917, a cosmological constant in the general relativity equations [8].
By fine-tuning its value, it was possible to obtain a time-independent solution. At
the time, Einstein claimed that was “required by the fact of the small velocities of the
stars” [8]. However, in 1929, Hubble showed, by measuring the redshift of various
galaxies receding from the Earth, that the Universe was not static but actually in
1
Although they are commonly used indistinctly, the cosmological principle and the
Copernican principle are two related, but not identical, ideas. The later states only
that Earth (or more generally a generic observer) does not have to occupy a privileged
position in a given system, but does not exclude the existence of such position, for
instance, that of sun in the Copernicus heliocentric model.
2
expansion [9]. According to Gamow, Einstein was referring to this episode as the
“biggest blunder that he [Einstein] made in his entire life” [10]. However, despite that
the assumption of staticity is now abandoned, this constant is now a central feature
of most of the current cosmological models. As for the assumptions of spatial isotropy
and spatial homogeneity, although they were initially proposed without observational
grounds, only to simplify the the mathematical analysis, they are now supported by
numerous observational evidences.
1.1.2
FRW metric
Mathematically, the notions of homogeneity and isotropy are respectively related
with the translational and the rotational invariance of a space. The FriedmannRobertson-Walker (FRW) metric2 is the most general metric which is consistent with
these two types of invariance and thus, with the cosmological principle. Its spatial
part is obtained by multiplying a static metric g̃ij , describing a generic maximallysymmetric 3-dimensional space, with a dimensionless and time-dependent scale factor
aptq:
~0
g00 ~0
1
gµν ~ t
~0t a2 ptqg̃ .
(1.1.1)
0 gij
ij
To have a more compact notation, we have defined ~0 p0 0 0q and ~0t as the corresponding transposed vector. In reduced-circumference polar coordinates pr, θ, φq, g̃ij
take the form
1
0
0
1κr 2
g̃ij 0
(1.1.2)
r2
0 .
2
0
0 r sin2 θ
Since the system of coordinates pr, θ, φq remains unaffected by the contraction (a9 0)
or the expansion (a9 ¡ 0) of the Universe, these coordinates are known as comoving
coordinates. As for the physical coordinates, they are given by par, θ, φq. The lineelement of the FRW metric is given by
ds
2
gµν dx
µ
ν
dx
dt
2
a ptq
2
dr 2
1 κr 2
2
2
r dΩ
,
(1.1.3)
where dΩ2 represents the line-element on a two-sphere (dΩ2 dθ2 sin2 θdφ2 ). The
curvature parameter κ has a dimensions of plengthq2 and it is usual to classify the
type of space described by g̃ij according to its sign. A space with negative spatial
curvature (κ 0) is called open, one without spatial curvature (κ 0), flat, and
2
This metric is also known as the Friedmann-Robertson-Walker-Lemaı̂tre (FRWL)
metric and as the Robertson-Walker (RW) metric.
3
one with positive spatial curvature (κ ¡ 0), closed. The simplest examples of such
spaces are, respectively for each case, the hyperboloid H 3 , the hyperplane R3 and the
3-sphere S 3 (at present, there is no evidence for spaces with non-trivial topology [11]).
The 2-dimensional analogous spaces (H 2 , R2 and S 2 ) are represented in figure 1.1.
The information about the curvature of a given space-time is encoded in the Riemann tensor (Rρ σµν ), from which two subsidiary quantities, the Ricci tensor (Rµν ) and
the Ricci scalar (R), are derived. These three objects are defined in the appendix A
and evaluated for the FRW space-time.
Figure 1.1: Examples of an open (κ 0), a flat (κ 0) and a closed (κ ¡ 0) space in
two spatial dimensions. This image was modified from [12].
1.1.3
Energy conservation (continuity equation)
The energy conservation is a fundamental principle in most fields of physics. However, in a cosmological context, the energy is not always conserved, or at least, not
in the way that it is usually understood. In special relativity, the conservation of
energy-momentum corresponds to the vanishing of the ordinary divergence of the
energy-momentum tensor (Bµ T µν =0). In general relativity, the analogous equation,
arising from the Bianchi identity, is obtained by replacing the ordinary divergence by
a covariant divergence
∇µ T µν 0.
(1.1.4)
This equation sets constraints on how the energy evolves, but unlike the case of special
relativity, this does not necessarily imply that energy is conserved. Conceptually, this
is not difficult to understand since according to the Noether theorem, for any conserved
quantity in a physical system, the Lagrangian description of the system must feature
a corresponding symmetry. In the case of energy conservation, the corresponding
symmetry is the invariance under time translation. For a non-static Universe (a9 0),
this invariance does not hold; consequently we cannot expect that energy should be
conserved. This will be illustrated more explicitly in a moment.
In a cosmological framework, it is usual to model the Universe content (baryonic
matter, dark matter, radiation, dark energy, etc.) as a perfect fluid, which is defined
as a fluid for which the energy-momentum tensor is given by
Tµν
pρ
pquµ uν
pgµν .
(1.1.5)
4
Here, the energy density and the pressure of the fluid
° are simply
° given by the sum
of the individual components of the mixture (ρ i ρi , p i pi ). In the frame
where the fluid is at rest with respect to the comoving coordinates, i.e. where the
four-velocity is uµ p1 0 0 0q, the energy-momentum tensor becomes
Tµν
ρ
~0t
~0
.
gij p
(1.1.6)
For this tensor, the evaluation of the zeroth component of eq. (1.1.4) yields
0 ∇µ T µ0
ρ
9
3H pρ
pq,
(1.1.7)
where we have introduced the Hubble parameter, defined as H
known as the continuity equation, can be rearranged as
dE
dt
a{a. This equation,
9
,
p dV
dt
(1.1.8)
where E ρa3 is the total energy contained in a volume element of size V a3 .
Obviously, except for the case of a pressureless fluid (p 0), the energy is not conserved in a non-static Universe. More specifically, during an expanding phase (a9 ¡ 0),
the total energy contained in the volume V will decrease for a fluid with a positive
pressure and increase for a fluid with a negative pressure. The converse is true during
a contracting phase (a9 0).
Since the total energy density
° and the
° total pressure are given by the sum of the
individual components (ρ i ρi , p i pi ), we can rearrange eq. (1.1.7) as
0 looooooooomooooooooon
ρ9 1 3H pρ1 p1 q
Q1
ρ
92
3H pρ2 p2 q
looooooooomooooooooon
Q2
ρ3 3H pρ3 p3 q
looooooooomooooooooon
9
...
(1.1.9)
Q 3
Here, we have introduced the functions Qi in order to obtain a separate continuity equation for each fluid composing the mixture. This is equivalent to setting
Qiuνpiq , where Tpµνiq is the energy-momentum tensor of the ith component
∇µ Tpµν
iq °
(T µν i Tpµν
iq ). Defining the equation of state (EoS) parameter, wi pi {ρi , we get
ρ9 i
3H p1
¸
Qi
wi qρi
0.
Qi ,
(1.1.10)
(1.1.11)
i
The EoS parameter of a single fluid i is generally assumed to be constant, but there
are some exceptions, such as a Chaplygin gas [13] which acts as unfied form of dark
energy and dark matter. In the case of a mixture of perfect fluids
wi ,
° with different
°
the EoS parameter is generally not constant since w p{ρ p i wi ρi q{p i ρi q. The
5
first term on the RHS of eq. (1.1.10) represents the variation of the energy density
caused by the expansion or the contraction of the Universe. For wi ¡ 1, assuming
that the energy density is positive (ρi ¡ 0), this term will contribute to decrease the
energy density during an expanding phase (dilution effect) and to increase it during a
contracting phase (concentration effect). For wi 1, the behaviour will be opposite.
Even if this is counter-intuitive, there are some forms of dark energy for which the
EoS parameter is allowed to be smaller than 1 (see e.g. [14]). For wi 1, only the
second term on the RHS of eq. (1.1.10) will contribute (of course, if it is non-zero).
This term, Qi , represents a variation of the energy density resulting from an energy
to a gain of energy
transfer between the fluids. A positive value (Qi ¡ 0) corresponds
°
and a negative value (Qi 0), to a loss. The condition i Qi 0 ensures that the
“energy budget” is closed. In the ΛCDM model, the interaction term Qi is zero for
all fluids, but it plays a central role in many models involving dynamical dark energy.
To take into account the combined effect of these two terms, it is useful to introduce
the effective EoS parameter wieff
wieff
ρ9 i
Qi
wi 3Hρ
,
(1.1.12)
i
3H p1
wieff qρi .
(1.1.13)
The preceding equation may be rearranged in the form of eq. (1.1.8)
dEi
dt
,
peffi dV
dt
(1.1.14)
eff
where Ei ρi a3 and peff
i wi ρi . The discussion of how the time evolution of ρi and
Ei depends on the effective EoS parameter is similar to that which has followed from
eqs. (1.1.8) and (1.1.10) and the main conclusions are summarized in table 1.1.
The most general solution to eq. (1.1.13) is given by
ρi ρi0 a3 exp
3
»
da
wieff
a
.
(1.1.15)
Here and throughout this thesis, the quantities with an index 0 are evaluated at a
reference time t0 ; it is usual to choose t0 tnow and to set a0 1. For most fluids,
the EoS parameter wi is constant, in this case the previous equation becomes
ρi ρi0 a3p1
In absence of interaction (Qi
wi
q exp
»
Qi
da .
aHρi
(1.1.16)
0), the exponential term is equal to one, which yields
ρi ρi a3p1 w q .
(1.1.17)
i
0
6
Energy density
Eq. (1.1.13)
eff
wi ¡ 0
ρ9 i 0
eff
wi 0
ρ9 i 0
eff
1 wi 0
ρ9 i 0
eff
wi 1
ρ9 i 0
eff
wi 1
ρ9 i ¡ 0
Total energy
Eq. (1.1.14)
E9 i 0
E9 i 0
E9 i ¡ 0
E9 i ¡ 0
E9 i ¡ 0
Table 1.1: Sign of the time derivative of the energy density ρi and of the total energy Ei (contained in a volume element V a3 ) of a perfect fluid i, as function of
its effective EoS parameter wieff , during an expanding phase (a9 ¡ 0). During a contracting phase (a9 0), the behaviour is opposite. In absence of interaction (Qi 0),
the effective EoS parameter reduces to the ordinary EoS parameter wieff wi . For
the perfect fluid modeling the total matter content of the Universe, the replacement
pρi, Ei, wieff q Ñ pρ, E, wq must be made.
To go further, we must determine how the scale factor aptq evolves in time. This is
the object of the next section.
1.1.4
Friedmann equations
The evolution of the scale factor aptq is determined by the Einstein equation
Gµν
Rµν 12 gµν R 8πGTµν .
(1.1.18)
For now, we have voluntarily omitted to include the cosmological constant in the
Einstein equation; this issue will be discussed in the next section. Since the Ricci
scalar may be expressed as R 8πGT , where T Tµµ is the trace of the energymomentum tensor, the Einstein equation may be rearranged as
Rµν
8πG
Rµν
1
gµν T
2
(1.1.19)
.
After some simple manipulations (for the tensor (1.1.6), T ρ 3p), we obtain
from the µν 00 and the µν ii components of eq. (1.1.28), the first and the second
Friedmann equation
H2
H9
2
a9
a
H2
8πG
3
¸
ρi i
¸
aa 4πG
p1
3
:
i
3κ
8πGa2
3wi qρi .
,
(1.1.20)
(1.1.21)
7
It is sometimes convenient to simplify the first equation by treating the contribution
of the spatial curvature as a fictitious fluid with an energy density ρk . If we compare
the term involving the curvature parameter κ in eq. (1.1.20) to eq. (1.1.15), we see
that is possible if we define
ρk0
3κ
,
8πG
wk
13 ,
Qk
0.
(1.1.22)
Here it is important to emphasize the fact that the interaction term associated with the
spatial curvature must be zero. Indeed, if the fictitious fluid was allowed to exchange
energy with the real fluids, that would imply that i Qi Qk 0 which would
violate the energy condition (1.1.11). Once the value Qk set to zero, wkeff wk 1{3
is the only value for which we recover the correct dependence on the scale factor
(9a2 ). It is also important to notice that, while the energy density of the real fluids
is always assumed to be positive, ρk could be either positive or negative depending
on the sign of κ. We can then write the first Friedmann equation as
°
H2
8πG
3
¸
pq
(1.1.23)
ρi ,
i k
where the notation ipk q, borrowed from Carroll [15], is used to keep in mind that the
summation is made on the real fluids piq and on the fictitious one associated with the
spatial curvature pk q. We can optionally make the replacement i Ñ ipk q in eq. (1.1.21),
since the term associated with the spatial curvature vanishes (1 3wk 0).
The continuity equation eq. (1.1.10) and the two Friedmann equations are the
main results of this section. Together, they describe how the scale factor a (or the
Hubble parameter H) and the energy density of the different cosmological fluids ρi
evolve with time. Actually, only two of these three equations are independent, hence
only two of them are needed to describe the cosmological evolution. In many cases,
it is more convenient to work with the continuity equation and the first Friedmann
equation.
Another way to express the first Friedmann equation is by the mean of the dimensionless density parameter, defined for each fluid as
Ωi
ρi
8πG
ρi 2
3H
ρ
.
(1.1.24)
Ωk .
(1.1.25)
crit
Dividing each side of eq. (1.1.23) by H 2 yields
1
¸
pq
i k
Ωi
Ω
8
°
The density parameter Ω i Ωi represents the contribution of all forms of matter
and energy (the real fluids) and Ωk , that associated with the spatial curvature. It
is then possible to determine the spatial curvature of the Universe by measuring the
value of Ω or equivalently, that of ρ and ρcrit (see table 1.2).
Type of Universe
Closed (κ ¡ 0)
Flat (κ 0)
Open (κ 0)
Energy density
ρ ¡ ρcrit
ρ ρcrit
ρ ρcrit
Density parameter
Ω¡1
Ω1
Ω 1
Table 1.2: The sign of the spatial curvature of the Universe can be determined by
comparing the energy density of the matter content ρ to the critical density ρcrit or
by comparing the density parameter of the matter content Ω to 1.
1.1.5
Cosmological constant
The cosmological constant is a term that can be added to the Einstein equation without affecting its consistency. This equation can be derived by noting that
∇µ Rµν 12 ∇µ gµν R and by using eq. (1.1.4). Combining these two expressions, we
get
1
µ
(1.1.26)
∇ Rµν gµν R ∇µ rTµν s,
2
which means that the expressions in the square bracket are related as follow
Rµν
12 gµν R αTµν
Fµν ,
(1.1.27)
where α is a constant and Fµν , an arbitrary tensor whose covariant derivative vanishes.
Setting α 8πG (this is required to recover the correct Newtonian limit) and Fµν 0
yields the Einstein equation as expressed in eq. (1.1.18). However, since ∇µ gµν 0,
we have the possibility to set Fµν Λgµν . In this case, eq. (1.1.27) becomes
Rµν
12 gµν R
Λgµν
8πGTµν ,
(1.1.28)
which is identical to the previous form of the Einstein equation, but with an additional
term involving the cosmological constant Λ.
We can wonder if other terms than the cosmological constant can be consistently
added to eq. (1.1.18). To answer to this question, it is convenient to adopt a variational
approach to derive again the Einstein equation. In general relativity, we can separate
the action in two terms (S Sg Sm ), one associated with the gravitational sector
(Sg ) and one associated with the matter sector (Sm ). If we set Sg SEH , where the
9
Einstein-Hilbert action is defined by
SEH
1
16πG
»
a
|g|R,
d4 x
(1.1.29)
we recover, by applying the Euler-Lagrange equation, the original form of the Einstein
equation (eq. (1.1.18)). To obtain the equation involving the cosmological constant,
we must set Sg SEH SΛ , where
Λ
SΛ 8πG
»
a
d4 x
|g|.
(1.1.30)
According to the Lovelock theorem [16, 17], the action Sg SEH SΛ is the most
general one that can be built from the metric gµν in four spacetime dimensions, if
no higher than second order derivatives of the metric are allowed. Within these
restrictions, the cosmological constant is then the only term that can be added to the
original form of the Einstein equation.
In the previous section, we have derived the two Friedmann equations ignoring
the cosmological constant. However, the results obtained hold on even if we had
explicitly taken in account the constant since it is possible to express eq. (1.1.28) in
the exact same form as eq. (1.1.18) by treating the cosmological constant as a perfect
fluid. Indeed, by passing the term Λgµν to the RHS of eq. (1.1.28), we can define an
energy-momentum tensor for the cosmological constant
1
Λ
gµν .
Rµν gµν R 8πG Tµν 2
8πG loooomoooon
(1.1.31)
pΛq
Tµν
Since the energy-momentum tensor Tµν is the sum of each individual component
° piq
pΛ q
( i Tµν ), we just have to include Tµν in this summation to recover eq. (1.1.18). If
we compare the energy-momentum tensor of the cosmological constant to the generic
expression for a perfect fluid (eq. (1.1.5)), we see that the energy density and the EoS
parameter must have the form
ρΛ
Λ
,
8πG
wΛ
1.
(1.1.32)
piq
Since ρΛ is the energy density characteristic of an empty space (Tµν 0, for i Λ),
this term is naturally interpreted as the energy density of the vacuum. As it will
be shown in section 1.2.5, its theoretical value is estimated to be ρΛ Mp4 , where
1
Mp p8πGq 2 is the reduced Planck mass.
10
It is worth to notice that the EoS parameter (and the pressure) of the vacuum is
negative, while for ordinary matter, the EoS parameter range from 0 (non-relativistic
case) to 1{3 (ultra-relativistic case). This fact was crucial to Einstein when he modified
its equation in order to obtained a static Universe. Indeed, to be static, the expansion
velocity and the expansion acceleration must both vanish (a9 0, a
: 0). According
to the first Friedmann equation (eq. (1.1.20)), the condition a9 0 is fulfilled if the
spatial curvature is positive and ρk ρ, so here the introduction of the cosmological
is not required. However, according to the second Friedmann equation (eq. (1.1.21)),
the condition a
: 0 could not be fulfilled if only ordinary matter is involved since all
the terms summed on the RHS of the equation would be positive. The cosmological
constant is then needed to provide a negative term in the summation (1 3wΛ 2),
which could lead to a zero acceleration. For instance, if we consider the example where
the only matter field involved is a pressureless fluid (wm 0), it is straightforward
to verify that if its energy density and that associated with the spatial curvature are
given by ρm 2ρΛ and ρk 3ρΛ , this leads to a static solution. However, it can
also be verified that such solutions are unstable.
1.2
1.2.1
ΛCDM model
Overview
The ΛCDM model is named after the two constituents which currently dominate
the Universe composition: a form of dark energy provided by the cosmological constant (Λ) and cold dark matter (CDM). The dark matter refers to a non-baryonic
form of matter which does not emit light. In the ΛCDM model, the dark matter is
assumed to be “cold”, i.e. non-relativistic (wdm 0). The motivations that led to
include these two dark fluids in a cosmological model are reviewed in sections 1.2.2
and 1.2.3. If we exclude the very early times, the other relevant components are the
baryons (wb 0) and the radiation (photons and massless neutrinos3 , wr 1{3);
it is also assumed that there is no energy transfer between the different cosmological
fluids. Of course, this is not strictly true, for instance the stellar burning process
implies an energy transfer from the baryonic sector to the radiation sector, but the
amount of energy involved remains negligible from a cosmological point of view, so
we can set Qi 0. The solution to the equation of motion (1.1.10) is then given by
eq. (1.1.17), which becomes for each fluid (including spatial curvature)
ρΛ
3
ρΛ ,
0
ρk
ρk a2 ,
0
ρm
ρm a3,
0
ρr
ρr a4.
0
Here we simplify the picture, the neutrinos are actually massive.
(1.2.1)
11
Here we treat the non-relativistic matter as a single fluid (ρm ρb ρdm ). Together
with one of the two Freidmann equations, these four equations completely describe
the evolution of the Universe at the background level.
Despite its relative simplicity, the ΛCDM model provides a reasonably good fit
to a wide range of cosmological data. The first observational hint pointing toward
this model dates back to 1990. At the time, the CDM model, which is identical to
the ΛCDM model but without the cosmological constant, was the model usually accepted by the cosmological community. However, it was shown that the theoretical
prediction for large-scale angular correlation function of galaxies was in better agreement with the observational result if a small and positive cosmological constant is
introduced [18]. A more compelling evidence came in the late 90s when the ΛCDM
provided a successful explanation to the observed magnitude-distance relation of type
Ia supernovae (SNeIa) [19–21]. Since then, the model has been tested with additional
supernovae data sets [22–24] and using complementary types of standard candles such
as gamma ray bursts (GRB) [25] and Cepheid stars [26]. Other cosmological phenomena successfully described by the ΛCDM model include the cosmological microwave
background (CMB) and its temperature fluctuation [27–31], the baryon acoustic oscillation (BAO) [32–36], the abundance of galaxy clusters [37, 38], etc. Despite these
remarkable achievements, the ΛCDM model is facing some difficulties. Some of them
(flatness problem, horizon problem, monopoles problem) may be solved if the “standard” ΛCDM model is preceded by an early inflationary phase (section 1.2.4), but
others, namely the cosmological constant problem (section 1.2.5) and the coincidence
problem (section 1.2.6), are more puzzling.
To conclude this overview, we will look at how the composition of the Universe
is evolving in time and how this affects its expansion rate. The density parameter
of the different fluids and the Hubble parameter are shown as a function of the scale
factor in figure 1.2. The initial values (at t0 ) used to make this figure are derived
from the CMB temperature power spectrum data obtained by the Planck collaboration [39] (see table 1.3). We can note the presence of three different eras during
which the energy density of a given fluid is much larger than the other one (ρi " ρj ,
Ωi 1), with transitional phases between them (situated around Ωi Ωj ). The
are successively (starting with the earlier one) the radiation dominated era (RD-era,
Ωr 1), the matter dominated era (MD-era, Ωm 1) and the dark energy dominated
era (ΛD-era, ΩΛ 1). Since wm wk wΛ , it would have been possible to have a
curvature dominated era just before the ΛD-era, but this possibility is not supported
by the observational data (due to the flatness of the Universe, Ωk 0, the MD-era is
immediately followed by the ΛD-era). During these three eras, the Hubble parameter
evolves proportionally to the square root of the energy density of the dominating fluid
12
?
(H 9 ρi ) and thus, the scale factor as
a9
#
t 3p1 wi q ,
exp Ht,
2
if wi
if wi
1,
1
(1.2.2)
where wi is the EoS parameter of the dominating fluid. In figure 1.3, we see that
the Universe is experiencing a phase of accelerated expansion at late times. Before
the ΛD-era, the value of the scale factor increase proportionally to a positive power
of t, which suggests that the Universe has emerged from a singularity (a 0) at
t 0. This is however to be considered with caution since the energy scale involved
as a Ñ 0 becomes arbitrarily large and it is generally assumed that description of
nature in term of general relativity and quantum fields theory break down beyond the
Planck scale (Ep G1{2 1.22 1019 GeV).
Fluid
Cosmological constant
Spatial curvature
Non-relativistic matter
Dark matter
Baryons
Relativistic matter
Photons
Neutrinos (massless)
EoS parameter
wΛ
1
wk
1{3
wm
0
wdm
0
wb
0
wr
1{3
wγ
1{3
wν
1{3
Density parameter
ΩΛ0
0.686
Ωk0
5.00 104
Ωm0
0.306
Ωdm0
0.259
Ωb0
0.0478
Ωr0
9.05 105
Ωγ0
5.35 105
Ων0
3.70 105
Table 1.3: Values of the EoS parameters and current values of the density parameters
for the different fluids of the ΛCDM model. The values of the density parameters are
extracted from the results obtained by the Planck collaboration for the CMB temperature power spectrum data [39]. The value of ΩΛ0 , is explicitly given in table 2 of [39].
To compute Ωdm0 , Ωb0 and Ωm0 Ωdm0 Ωb0 , we have used the value of the current physical density of baryons and dark matter (Ωb0 h20 0.02207, Ωdm0 h20 0.1196,
table 2 of [39]) and that of the Hubble constant obtained for massless neutrinos,
H0 68.0 km s1 Mpc1 (eq. (16) of [39]). The density parameters of relativistic
and non-relativistic matter are related through Ωr0 Ωm0 aeq Ωm0 p1 zeq q1 ,
where the redshift of matter-radiation equality is (assuming massless neutrinos)
zeq 3386 (table 2 of [39]). The density parameters Ωγ0 and Ων0 are computed from
Ωr0 Ωγ0 Ων0 p1 Neff p7{8qp4{11q4{3 qΩγ0 , where we used Neff 3.046 for the
effective number of neutrino species. Finally, the density parameter associated with
the spatial curvature is given by eq. (68a) of [39].
13
2
10
1
Ωi
0.8
0.6
0
10
0.4
0
√
−2
H∝
∼ ρr ∝ a
10
10
8
H/H0
10
√
H∝
∼ ρΛ = const
√
−3
H∝
∼ ρm ∝ a 2
6
a
0.2
−2
10
−4
10
10
4
10
2
10
0
10
−6
10
−6
10
−4
10
−2
10
a
Figure 1.2
0
10
2
10
1/2
a∝
∼t
−6
10
−4
10
2/3
a∝
∼t
−2
10
Time [Gyrs]
0
10
Ht
a∝
∼e
2
10
Figure 1.3
Figure 1.2: The upper panel shows the evolution of the density parameters Ωr (green),
Ωm (red), Ωk (black, 0) and ΩΛ (blue) as a function of the scale factor a for the
ΛCDM model. The values of Ωi0 and H0 used for the computation are those of
table 1.3. The lower panel shows the corresponding evolution the Hubble parameter
divided by its current value H {H0 . In both panel, we can distinguish an early era
dominated by radiation, an intermediate era dominated by matter and a late-time era
dominated by dark energy. The dashed line corresponds to the current value of the
scale factor a a0 1.
Figure 1.3: Evolution of the scale factor a as a function of time for the ΛCDM model.
The values of the parameters used for the computation are the same as in figure 1.2.
The time origin (t 0) is defined to be the time for which the Universe reaches the
singularity a 0. The time elapsed since then corresponds to the current age of the
Universe (t0 13.7 Gyrs, vertical dashed line). The first full vertical line (from the
left) corresponds to the time of matter-radiation equality (Ωm Ωr ) and the second
one, to the time of matter-dark energy equality (Ωm ΩΛ ).
14
1.2.2
Dark Matter
The notion of dark matter was introduced in the 1930s by Oort (1932) to explain
the observed stellar motion in the Milky Way [40] and by Zwicky (1933) to explain
the observed galactic motion in the Coma cluster [41] . In both case, the measured velocity of the astronomical objects studied (stars, galaxies) appeared to be larger than
what would be expected from the gravitational force provided by the visible mass of
their respective system (Milky Way, Coma cluster). These results suggested that a
significant fraction of the total mass of these systems should be present in a form of
an invisible matter. To refer to this invisible matter, Zwicky coined the expression
dark matter : “It is, of course, possible that luminous plus dark (cold) matter [dunkle
(kalte) Materie] together yield a significantly higher density” (translated in [42]). The
discrepancy between the visible and the dynamical mass (inferred from its gravitational effects) of such large astronomical objects has been more firmly established in
the 1970s when Rubin et al. made precise measurements of the rotation curves of 22
spiral galaxies [43, 44]. At a distance r from its centre, the rotational velocity of a
galaxy in a virial equilibrium is given by
v pr q GM prq
r
.
(1.2.3)
In absence of dark matter, the mass function of the galaxy beyond the disc radius
is expected to be approximately constant, M pr ¡ rdisc q M prdisc q. Consequently,
according to eq. (1.2.3),
the rotational velocity should be maximum at r rdisc and
?
then decreases as 1{ r beyond this point. However, this is not what was observed.
Rather the measured velocities were roughly constant beyond the point where the
light stops. These results are actually a characteristic feature of elliptical galaxies
and are easily explained if we postulate the existence of a dark matter halo in these
galaxies, as illustrated in figure 1.4.
Alternatively, it is possible to explained the profile of the rotational curves without the recourse of dark matter, by modifying the law of gravitation. The modified
Newtonian dynamics (MOND) theory was proposed in 1983 by Milgrom [45]. According to this theory, the gravitational force ranges from the usual inverse square
law on small scales (1{r 2 ) to an inverse law (1{r) on larger ones. The MOND, in its
original formulation, has been proven successful to fit the observed rotational curves
of galaxies, but was less consistent with other phenomena explained by the presence
of dark matter including, in particular, the gravitational lensing. The case of the
Bullet cluster is really explicit on that matter. By analyzing its X-ray emissions, it
is possible to determine the distribution of the hot gases (which account for the majority of the visible matter in a galaxy cluster). However, this distribution does not
match the mass distribution inferred from the gravitational lensing (see figure 1.5)
and this cannot be explained by the MOND theory [46]. To overcome this problem,
15
some relativistic generalizations of the MOND have been proposed such as the TensorVector-Scalar (TeVeS) theory, the Generalized Einstein-Æther (GEA) theory and the
bimetric MOND (BIMOND) theory (a description of these theories could be found
in [47]). However, even for these theories, the recourse to dark matter is sometimes
required. For instance in [48], the authors concluded that even if the scalar field of
the GEA model “may play a number of the roles that dark matter plays, it seemingly
cannot do all at once”, referring in this case to necessity of invoking dark matter to
determine the position of the acoustic peaks in the CMB temperature anisotropy.
Figure 1.4
Figure 1.5
Figure 1.4: Decomposition of the rotation curve of the spiral galaxy NGC 3198.
The rotational velocities inferred from the luminous disk alone are not sufficient to
account for the observed values and the additional contribution of a dark matter halo
is required. This image was taken from [50].
Figure 1.5: In the Bullet cluster (1E 0657-56), the distribution of the “visible” matter,
detected through x-rays emissions (colours), does not coincide with the total distribution of matter, detected through gravitational lensing (green contours). This is
usually interpreted as an indication of the existence of dark matter. This image was
taken from [46].
Up to now, the dark matter has been detected only through its gravitational
effects (there is however some hints of indirect detection, see for instance [49]) which
is little helpful to determine its nature. Among the few things that we know is that
most of the dark matter in the Universe must be non-baryonic 4 . We know this
4
In the context of the ΛCDM model, the distinction between the visible and the
dark (non-luminous gas, MACHOs) baryonic matter is not very relevant. In the
16
because the Big Bang nucleosynthesis (BBN) theory and the analysis of the CMB
provide tight constraints on the density of baryonic matter and the value obtained is
much smaller than that of the matter density based on calculations of the expansion
rate. For instance, according to the Planck results (table 1.3), the density of baryonic
matter accounts for only 15% of the total matter content of the Universe; the
remaining 85% must then be non-baryonic. An other thing that we know is that
most of the dark matter must be cold (or maybe warm, see e.g. [51]). Unlike to the
CDM, the hot dark matter (HDM) is ultra-relativistic (wdm 1{3) and the warm
dark matter (WDM) is an intermediate case between them (0 wdm 1{3). The
structures formation process takes different forms depending on the type of dark
matter considered. For CDM, the structures are said to form ”from bottom up“, i.e.
the smaller objects appear first and subsequently merge to form larger structures. In
contrast, for HDM, the structures form ”from top down“, i.e. the larger structures
form first and the smaller ones are expected to fragment out later. In both scenarios,
the density perturbations present at early times serve as seeds for the first structures
to appear. However, in HDM models the presence of relativistic particles in the early
Universe erases (due to free streaming) the density perturbations on scales larger than
those which lead, in the case of CDM models, to the formation of small structures.
Hence, in a HDM model, the large structures must form first. Numerical simulations
of the structures formation have shown that the CDM models are in better agreement
than the HDM models with the obsevational features of the Universe (for more details,
see e.g. [52]).
The theoretical physicists have proposed a large number of candidates for dark
matter. The WIMPs (weakly interacting massive particles) and the axions are among
the most popular ones. For the former, this is a consequence of the “WIMP miracle”:
if these particles were produced in the early Universe with a mass at the weak scale,
their current density is predicted to be of the same order as the dark matter density
(see e.g. [53]). As for the axions, they arise as a solution to the strong CP problem
in quantum chromodynamic (QCD) [54, 55]. An overview of other candidates may be
find in [56]. All these particles have specific properties, but to remain as general as
possible, we will only assume that the dark matter is cold (wdm 0) and non-baryonic.
following, we will use the expression “dark matter” specifically to refer to non-baryonic
dark matter.
17
1.2.3
Dark Energy
In 1998, based on observations of type Ia supernovae (SNeIa)5 , the High-Z Supernova Search Team [19, 20] and the Supernova Cosmology Project [21] have provided
evidence that the Universe has entered into a phase of accelerated expansion since
around a redshift of z 0.5. To reward this major discovery, the 2011 Nobel Prize
was granted to Saul Perlmutter, Brian P. Schmidt and Adam G. Riess, the leaders
of these two research teams. Type Ia supernovae occur when, in a binary system, a
white dwarf absorbs material from its companion star and gains sufficiently in mass
to exceed its Chandrasekhar limit, which results in its explosion. Ignoring some subtleties, the luminosity curves of these explosions are expected to be identical since they
correspond to the exact same phenomenon (explosion of white dwarfs with identical
masses). SNeIa can then be used as “standard candels”, i.e. we can use the measured
value of the apparent luminosity (at the peak of brightness) to determine the distance
of these objects. For historical reasons, this relation is usually expressed in terms of
the apparent magnitude, m, a quantity derived from the apparent luminosity,
mM
5 log dL
25.
(1.2.4)
Here the luminosity distance dL is expressed in Mpc and the absolute magnitude of
SNeIa at the peak of brightness is given approximately M 19. By measuring the
distance of these supernovae at different redshifts (z ptq a1 ptq 1), its possible to
reconstruct how the scale factor has evolved in the past. For the same redshift, the
luminosity distance should be larger in a Universe expanding at an accelerated rate
than for a Universe expanding at a constant or decelerating rate, and the observations
were consitent with the former case.
: ¡ 0) required that
In the context of general relativity, an accelerated expansion (a
w 1{3 (see (eq. 1.1.21)). However, cold matter (wm 0) and radiation (wr 1{3)
does not fulfill this condition. Thus, two main approaches have been considered to
explain the observed cosmic acceleration. One of them consists to keep the general
relativity and postulating the existence of a dark energy fluid, whose EoS parameter
fulfills the condition wde 1{3. The oldest and the simplest form of dark energy
is the cosmological constant (wΛ 1). Other forms of dark energy are considered
in section 1.3 of the introduction and in the remaining chapters of this thesis. The
other approach consists of replacing the Einstein gravity with a more general theory of
gravitation such as f pRq gravity [57–63], scalar-tensor theories [64–67], Gauss-Bonnet
5
The supernovae are classify according to their spectral features. The spectrum
of a type Ia supernova includes an absorption line of silicon but no spectral line from
hydrogen.
18
gravity [68–71], braneworld cosmology [72–75], etc. Alternatively, it has also been
proposed that the observed acceleration could be, at least partly, an artifact resulting
from the assumption that the spacetime is perfectly homogenous [76–80].
In addition to the supernovae observations, the acceleration of the Universe is
now supported by different observational evidences, of which the more important are
the baryon acoustic oscillation (BAO) data and the cosmic microwave background
(CMB) data. The baryons acoustics oscillation refers to the periodic fluctuations in
distribution of baryonic matter. Before the decoupling, baryons and photons were
forming an oscillating plasma. This oscillation has left an characteristic imprint in
the distribution of baryonic matter but also in the CMB. By measuring the scale of
this oscillation in the galaxy power spectrum at low redshift and comparing it to the
acoustic oscillation of the CMB at the decoupling epoch, we could obtain information
about the evolution of the scale factor. Other information can be extracted from the
CMB data, for instance, the shift parameter of the CMB, a quantity related to the
position of the first acoustic peak in the power spectrum of CMB, could be used to
constrain the value of the dark energy density. More details may be found in chapters
2 and 3 where the BAO and the CMB data will be used to constrain the parameters of
the usual ΛCDM and of an other cosmological model involving coupled dark energy.
1.2.4
Inflation
The presence of an early phase of accelerated expansion, known as inflation, has
been hypothesized by Alan Guth in 1981 [81] in order to solve the flatness problem,
the horizon problem and the monopole problem with single cosmological mechanism.
The inflation provides also a explanation the structure formation problem. A comprehensive introduction to the inflation may be found [82–84] and quite exhaustive review
of different models in [85]. The recent discovery of primordial gravitational waves in
the B-mode power spectrum by the BICEP2 collaboration [86] gives credence to the
hypothesis of inflation since their existence were predicted by inflationary cosmology.
It is however to be noticed that this prediction is not unique to inflationary models
and was also made, for instance, in the context of “String Gas Cosmology” [87]. In
the following, we will with more details the problems mentioned above and how they
can be solved by the presence of an early phase of inflation.
Flatness problem
According to observational data, the Universe is nearly flat today (Ωk 5 104 ,
[39]). In absence of inflation, this result is puzzling. To see this, it is useful to consider
the derivative of the curvature density parameter, Ωk κ{paH q2 , with respect to the
scale factor
dΩk
Ωk pΩk 1q
.
(1.2.5)
p
1 3w q
da
a
This expression vanishes for w 1{3 (which is however an unstable solution) as
well for Ωk 0 (exactly flat Universe) and for Ωk 1. Otherwise, the sign of the
derivative depends whether w is larger or smaller than 1{3 (this is summarized in
19
table 1.4). During a radiation dominated era (w 1{3) or a matter dominated era
(w 0) the curvature density parameter tends to evolve away from Ωk 0. This
means that to be nearly flat today, the Universe should have been even flatter in the
past. For instance, if we look back up too the Planck time (tpl 5 1044 s), the value
of Ωk should be smaller than 1060 . This result could be obtained by fine tuning the
initial conditions, but the inflation provides an more satisfying mechanism to explain
the observed flatness. Indeed, during a phase of accelerated expansion (w 1{3) the
point Ωk 0 becomes an attractor. Hence, before the inflation, the spatial curvature
can be arbitrarily large, provided that the duration of the latter (which depends on
the specific model of inflation considered) is sufficiently long to flatten the Universe.
w
w
13
¡ 13
Ωk
1 1 Ωk 0
0
0
0
¡0
Table 1.4: Sign of the derivative dΩk {da. For w
any value of Ωk .
Ωk
0
¡0
¡0
0
Ωk
0
0
1{3, the derivative vanishes for
Horizon problem
The horizon problem relies on the observation that the Universe is nearly homogeneous and isotropic on very large scales which seem a priori to be causally disconnected. For instance, the temperature of the CMB is the same in all direction within
one part in 105 . The CMB photons that are reaching us today have been emitted at
the time of last scattering , tls . Since photons travel on null geodesics (ds2 0), this
corresponds to a distance
ℓp ptls q »t
0
tls
dt
a
3t0
1
tls
t0
1
3
.
(1.2.6)
Here, to simplify the picture, we have considered that the Universe is in a matter
dominated era (a pt{t0 q2{3 ). Two photons reaching us from antipodal points should
then have been separated by a distance 2ℓp ptls q at the time of there emission. However,
since there is a only finite amount of time before the time of decoupling, regions of
the Universe larger that 2ℓf ptls q, where the forward lightcone ℓf ptls q is defined by
ℓf ptls q »t
ls
0
dt
a
3t02{3tls1{3 ,
(1.2.7)
were not in causal contact at this time. The evolution of ℓp and ℓf are shown for
a generic time in figure 1.6. To explain the homogeneity of the CMB, the Universe
should have been homogeneous on scale equal or larger than ℓp ptls q at the time of
20
last scattering, which is much larger than the larger causal region at the time, ℓf ptls q
(t0 14 Gyr, tls 4 105 yr). A fine tunning of the initial conditions could explain this
result, however, as for the the flatness problem, the presence of an early inflationary
phase provides a more satisfying explanation. If during the inflation (t P rti , tf s),
the scale factor is given by a9eHt , then ℓf will exponentially increase during this
time interval as we can see on in figure 1.6. A region of the size ℓp ptls ) could now
be in causal contact (ℓ̃f ptls q ¡ ℓp ptls q) provided that the duration of the inflation is
sufficiently long.
t0
l og t
ℓp
tl s
tf
ℓf
ℓ̃f
ti
l og x
Figure 1.6: Illustration of the horizon problem. The thin lines represent the forward
lightcone in absence of inflation (full line, ℓf ) and in presence of inflation (dashed
line, ℓ̃f ). In absence of inflation, ℓf ptls q ℓp ptls q, which means that the region in
causal contact at the time of recombination was too small to explain the observed
homogeneity and isotropy of the CMB. In presence of inflation, it is possible to have
ℓ̃f ptls q ¡ ℓp ptls q (provided that the duration of the inflation is sufficiently long) and
thus to explain the observed homogeneity and isotropy of the CMB.
Monopoles problem
Following the successful unification of electromagnetic and weak interactions by
Glashow [88], Weinberg [89] and Salam [90] (which has led to the experimental detection of the W and the Z bosons), it is generally expected that the three forces
of the standard model (which also includes the strong interaction) should also be
unified in a single one at high temperature (expected to be TGUT 1028 K). A first
model of Grand Unified Theory (GUT) has been proposed by Georgi and Glashow in
1974 [91] and different versions have been proposed since (see [93] for a review). One
of the prediction of the GUT is that the Universe should have experienced a phase
transition when it was very young (t 1036 s) when the temperature has dropped
21
below TGUT . This kind of phase transition is generally accompanied by the creation
of topological defects and most models of GUT predict the formation of magnetic
monopoles. Given the expected mass, stability and density of these monopoles, they
should quickly dominate the Universe after their creation and remain dominant until
dark energy takes over. This is obviously not the case, the Universe is not dominated
by monopoles. There is no experimental evidence that the monopoles really exit,
but in this eventuality, the inflation provides an explanation to why none of them is
observed today. If the scale factor increases by several orders of magnitude during the
inflation, the number of monopoles will be diluted in a large volume and their density
will becomes negligible. A obvious problem with this solution is that, although this
process will get rid off of the unwanted monopoles, the same thing will happen to
ordinary matter. Fortunately the inflation provides also a mechanism (reheating) to
repopulate the empty Universe when inflation comes to its end.
Structures formation problem
In the Universe, some structures such as galaxies and clusters of galaxies have nonrandom correlation on large scales which seems a priori causally disconnected. This
could be explained by argument relatively similar to that used for the horizon problem.
However, additionally, the inflation provides a mechanism to explain the formation of
the structure from the quantum fluctuations and the observed scale-invariant power
spectrum [82, 92].
Slow-roll inflation
In the ΛCDM model, the cosmological constant, which is responsible for the latetime phase of accelerated expansion, cannot drive the primordial inflation. Indeed,
since ρΛ is a constant and the energy density of the other fluids are monotonic decreasing functions of time, this means that if the cosmological constant was driving
the inflation, the Universe would have remained in this phase for ever. Two distinct
mechanisms are then required to drive the early inflation and the late-time accelerated expansion. It would be possible to get out of an inflation if the density energy
associated with the mechanism driving it decreases with time.
We will consider, as an example, one of the simplest models of inflation, which
involves an single scalar field known as the inflaton. The action of this field involves
a canonical kinetic term φ and a self interacting term V , which yields
» ? 1
µν
4
Sφ d g
g Bµ φBν φ V pφq .
2
(1.2.8)
The energy density and the pressure associated with the inflaton field is obtained by
varying this action with respect to the metric
ρφ
12 φ
9
V pφq,
(1.2.9)
22
pφ
12 φ V pφq.
9
(1.2.10)
The corresponding EoS parameter is then given by
wφ
1 9
φ
2
1 9
φ
2
V pφq ,
V pφq
(1.2.11)
and the continuity equation (eq. (1.1.10)) of the inflaton by
3H φ9 φ:
dV
.
dφ
(1.2.12)
During the inflation the Hubble parameter becomes
H
2
8πG
3
V pφq .
19
φ
2
(1.2.13)
Eq. (1.2.12) is analogous to the equation of motion of a particle accelerated by a
force dV {dφ and subject to a friction term 3Hφ. If for a certain period of time,
the acceleration is much smaller than the friction term and the kinetic energy much
smaller that the potential energy (φ9 ! V pφq), this will give rise to the slow-roll
inflation. These conditions can be expressed in terms of the slow-rolling parameters,
|ǫ| ! 1 and |η| ! 1, where
ǫ
1
2
2
V
V
1
,
η
VV
2
.
(1.2.14)
A prime denotes an derivative with respect to φ. During the slow-rolling phase the
EoS parameter of the inflaton approaches -1, which means that the Universe is exponentially expanding. Depending on the form of the potential considered (an example
is shown in figure 1.7), the slow-rolling conditions will eventually break down and put
a end to the inflation. At the end of the inflation, the Universe should be cold and
nearly empty of ordinary matter, however at this moment the large potential energy
of the inflaton field decays into particles through a mechanism know as reheating [94].
1.2.5
Cosmological constant problem
In section 1.1.5 we have identified the energy density associated with the cosmological constant to the the energy density of the vacuum. However, if other sources
contribute to the vacuum energy (ρother ), it will be impossible to distinguish them
from the contribution of the bare cosmological constant (ρΛ ). Then, it is useful to
define an effective cosmological constant whose energy density is given by
ρΛeff
ρΛ
ρother .
(1.2.15)
23
Figure 1.7: Example of potential for slow-roll inflation. Before the end of the inflation
( φend ), the kinetic energy of the inflaton is much smaller that its potential energy,
thus wφ 1 and the inflaton could drive the inflation. This image was taken
from [83].
An important contribution to ρother is expected to come from the zero-point energies
associated with vacuum fluctuations. For a bosonic field with a mass m, the total
zero-point energy is
1
ωk
1
E
p
k 2 m2 q 2 .
(1.2.16)
2
2 k
k
¸
¸
In the continuous limit, the energy density of this field, ρ E {V , becomes
ρ
1
2
»8
0
d3 k 2
p2πq3 pk
2
m
q 1
2
» 8 k dk
2
0
4π 2
pk2
m2 q 2 .
1
(1.2.17)
This expression exhibits an ultraviolet divergence. However, it is expected that the
quantum field theory is valid at most, up to the Planck scale, thus a cut-off kmax must
be set. Assuming that m ! kmax , eq. (1.2.17) becomes
ρ
4
kmax
.
16π 2
(1.2.18)
Evaluating this expression with the cut-off at the Planck scale (kmax Mpl ) yields
ρ 1074 GeV4 . This is approximately 121 orders of magnitude larger than the observed value of the dark energy ( 1047 GeV4 ). Even with the cut-off taken at the
QCD energy scale, ρ 103 GeV4 remains several orders of magnitude larger than
the observed value. This very severe discrepancy is know as the cosmological constant
problem. The expression given by eq. (1.2.18) represents only a partial contribution
to ρΛeff , however, it would be strange if the magnitude of total value was much smaller
24
than those of the individual component. By fine-tunning the value of the bare cosmological constant, its contribution to the vacuum energy could cancel that from
quantum fluctuations, however this means that two independent number must cancel
with an accuracy of 121 orders of magnitude, which seems very unlikely.
A possible way to get rid off the quantum fluctuation contributions could have been
provided by supersymmetric theories. These theories state that for each bosonic field,
there is a fermionic counter-part. For a fermionic field, the zero-point energy is given as
for a bosonic field by eq. (1.2.16) but with a minus sign. In this case, the contributions
from the vacuum fluctuations of the bosonic fields are exactly canceled by those of
the fermionic fields. However, we know that we are not living in a supersymmetric
Universe (if supersymmetry exists, it expected to be broken above 100 GeV) and thus,
the observed value of the vacuum energy remains intriguingly small.
An interesting approach to explain the small observed value of ρΛ relies on the
possibility that the cosmological constant could be an environmental variable taking
different values in different regions of spacetime. For instance, an other possible
contribution to the vacuum energy could come from the potential energy of a scalar
field. Since the latter may change as different regions of the Universe are passing
through phase transitions, the vacuum energy will also take different values in these
regions [95]. Based on somewhat different considerations, the membrane creation
mechanism proposed by Brown and Teitelboim [96, 97] leads to a similar situation.
More recently, it has been suggested that the string theory landscape admits as many
as 10500 vacua with different values of vacuum energy [98, 99]. Although the range
of accessible values could be quite large, the anthropic principle could be invoked to
explain why the observed value should be small. There are different versions of this
principle, but basically, it simply asserts that to measure the value of a given physical
parameter, their must be observers to perform the measurement. In [100], Weinberg
make only the simple assumption that galaxies are required for the emergence of life.
Thus, the observed value for the vacuum energy could lie only in a range compatible
with galaxy formation, which was estimated at the time
1 À ρρΛ À 550.
eff
(1.2.19)
m
The observed value for the ratio ρΛeff {ρm is of the order of unity and thus much
smaller than the upper bound, but this is far less severe than the discrepancy of 121
orders of magnitude mentioned above. Since the publication of the seminal paper of
Weinberg, observational data has been improved and the anthropic arguments has
been refined [101–103]. This is discussed with more details in chapter 5.
In this thesis, we will consider the possibility that dynamical dark energy could
solve the cosmological constant problem. This could be the case if its energy density
25
is allowed to evolve and reach the current observed value. Other approaches considered to solve the cosmological constant problem, including modified gravity, quantum
cosmology, holographic principle, etc. are reviewed in [104, 105].
1.2.6
Coincidence problem
In the ΛCDM model, the energy density of cold matter decreases as the Universe
is expanding (ρm 9a3 ) while that of the dark energy remains constant (ρΛ ρΛ0 ).
Their ratio is then given by
ρm
ρΛ
ΩΩm ΩΩm
0
Λ
a3 .
(1.2.20)
Λ0
It turn out that the energy density of dark matter and dark energy are currently
observed to be of the same order of magnitude (see table 1.3)
Ωm0
ΩΛ0
0.44.
(1.2.21)
Since Ωm {ΩΛ scales as a3 , the window of time during which the energy density of
dark matter and dark energy will be of the same order of magnitude is relatively short
in comparison to the whole Universe history (past and to come). This is illustrated
in figure 1.8, where the function R is defined as
R min
ΩΛ Ωm
,
Ωm ΩΛ
.
(1.2.22)
The fact that we are currently lying in the narrow period of time during which
R Op1q is known is the coincidence problem. This problem is considered crucial to understand by prominent physicists such as Carroll [106], Dodelson [107] or
Turner [108]. However, it can be argued from anthropic considerations that this coincidence is not so surprising. Indeed, it is not the whole Universe history which is
compatible with the emergence of observers. For instance, the Universe should not be
too young, since their must be at least some stars which have completed their time
on the main sequence to produce the heavy element necessary for life. Conversely,
it should not be too old since it seems unlikely that life could appear in absence of
an important source of energy such as that provided by stars, which will eventually
all die. If we compare the width of the peak in figure 1.8 with the interval of time
during which stars exist, this alleviates the severity of the coincidence problem. More
elaborated anthropic arguments may be invoked to explain that the likelihood to have
observers during this interval is not equal for all times and is, in particular, suppressed
at late time where only few stars are remaining. Such a discussion may be found for
instance in [110]. Although the anthropic explanation to the coincidence problem
seems quite reasonable, it is worth it to explore other possibilities since this could
provide indications on the nature of dark energy.
26
1
0.8
R
0.6
0.4
0.2
0
−5
10
−3
10
−1
10
1
10
3
10
5
10
7
10
t [ Gyr ]
Figure 1.8: Illustration of the coincidence problem. The energy density of cold matter
and dark energy are of the same order of magnitude when R Á 0.1 (this function is
defined in eq. (1.2.22)). The dashed line corresponds to the current time t0 . The
dotted lines represents the time at which appear the first stars (0.4 Gyr, [28]) and
die the last one ( 106 Gyr, [109]).
1.3
Dynamical dark energy
In the preamble of the current chapter, we have defined dark energy as a cosmological fluid able to drive a late-time phase of accelerated expansion. The simplest
form of dark energy is provided by Einstein’s cosmological constant whose energy
density remains constant in time. The concept of dark energy has paved the way to
that of inflation, which is an early phase of accelerated expansion. As we have seen
in section 1.2.4, inflation was proposed as a solution to some cosmological problems
(horizon, flatness and monopoles problems) and also provides a mechanism to explain
how the quantum fluctuations in the very early Universe seeded the structures formation. At some point, the phase of inflation should come to an end allowing the
Universe to enter a radiation-dominated era. This could be achieved with a dynamical energy density of the inflation field. (ρ9 φ 0). Thus, in its turn, the concept of
inflation gave credence to the hypothesis that dark energy could be a dynamical term
(ρ9 de 0).
Dynamical dark energy is desirable for many reasons. In sections 1.2.5 and 1.2.6,
we saw that the ΛCDM model is plagued with two theoretical difficulties, namely, the
coincidence problem and the cosmological constant problem. A cosmological model
involving a dynamical form of dark energy could be an interesting way to avoid these
two problems. For instance, dynamical dark energy could provide an explanation to
27
the cosmological constant problem if its energy density is allowed to decrease from
the initial large value expected from the theoretical computation to the smaller one
inferred from the observations. The matter to dark energy density ratio evolves differently for dynamical dark energy ρde {ρm 9aptqnptq . In comparison to the ΛCDM
model, the severity of the coincidence problem would be alleviated if |n| 3. We
consider an example of such a model in chapter 2 and 3 where dark energy and dark
3{2
matter are coupled through an interaction term Q0 ρΛ , which leads the late time
value of nptq to approach zero. Moreover, in the context of the ΛCDM model, the
inflation and cosmic acceleration occurring at late time are driven by two different
mechanisms. Dynamical dark energy could potentially provide an unified explanation
for both.
From the point of view of fundamental physics, a dynamical form of dark energy is
motivated by the quantum instabilities of the de Sitter spacetime discovered in [111–
113]. This spacetime is a vacuum solution of the Einstein equation, i.e. it describes a
Universe empty of any form of matter and radiation, but with a cosmological constant.
Although such a Universe is not realistic, its study is useful for determining the
properties of the cosmological constant. The instabilities of the de Sitter spacetime
could lead to a large production of particles from the vacuum energy. According to
the “energy conservation law”, ∇µ T µν 0, the value of the cosmological constant
should consequently decrease in time. In chapter 5, we will consider an example of
particle production from vacuum energy based on [7].
The application of these ideas to a more realistic Universe (containing matter
and radiation) is still in development. Meanwhile several models of dynamical dark
energy have been developed, each of them involving a certain part of phenomenology.
Following the example of inflation, it has been proposed that a late-time cosmic
acceleration could be obtained from a minimally coupled scalar field φ evolving in
a given potential [114, 115], dubbed quintessence in [116]. Some extension of the
quintessence models involve, for instance, non-canonical kinetic term (k-essence) [117–
119] or a phantom scalar field (wde 1) [14]. Another approach to study dark energy
is to parametrized the evolution of its EoS. The CPL parametrization [120,121] is one
of the most commonly used
wde paq w0
wa p1 aq,
(1.3.1)
The values of the parameters have been constrained by the Planck collaboration,
w0 1.040.72
0.69 and wa 1.32. In chapter 2, 3 and 4 we will consider models of
coupled dark energy, i.e where Qde 0. There is different motivations for the form of
the interaction term Qde . Our aim was to obtain a cosmology where ratio of ρm {ρde
becomes constant at late-time (chapter 2 and 3) and where the the cosmological
horizon and dark energy are in thermal equilibrium (chapter 4). Another interesting
proposal, that we will encounter in chapter 4, is the Chaplygin gas [13], which is a
28
dark fluid with an EoS parameter given by
wcg
ρcg8
ρcg
2
.
(1.3.2)
The constant ρcg8 is the value of the energy density of the Chaplygin gas at late time.
This fluid behaves then as dark matter at early-time (wcg 0) and as dark energy at
late-time (wcg 1). A More completed review of different models and ideas on the
dynamical dark energy may be found in [104, 122–125].
Chapter 2
Constraints on Λptq-cosmology with
a power law interacting dark
sectors
The work presented in this chapter was originally published in the Journal of
Cosmology and Astroparticle Physics [1]. We are considering a cosmological model,
in a flat spacetime (κ
0), where dark energy and dark matter exchange energy
through an interacting term proportional to a power of the dark energy density. In
the next chapter, we will extend this study to the case of spacetime with spatial
curvature (κ 0).
Abstract
In order to find an explanation to the cosmological constant problem and the coincidence problem, we consider a cosmological model where the cosmological constant
Λ is replaced by a cosmological term Λptq which is allowed to vary in time. More
specifically, we are considering that this dark energy term interacts with dark matter
through the phenomenological decay law ρ9 Λ
QρnΛ . We have constrained the model
for the range n P r0, 10s using various observational data (SNeIa, GRB, CMB, BAO,
OHD), emphasizing on the case where n
3{2.
This case is the only one where the
late-time value for the ratio of dark energy density and matter energy density ρΛ {ρm
is constant, which could provide an interesting explanation to the coincidence problem. The observational constraints obtained on the model parameters allow solutions
29
30
where the severity of the coincidence problem is significantly decreased, but which fail
to provide a sensible explanation to the cosmological constant problem.
2.1
Introduction
It is now widely accepted by the cosmological community that our Universe is
currently experiencing a phase of accelerated expansion. The first significant evidences
came from the distance measurements of type Ia supernovae (SNeIa) [19–21] (see [22,
24] for an update) which has been later confirmed, among others, by the measurement
of the anisotropies of the cosmic microwave background (CMB) spectrum by the
Wilkinson Microwave Anisotropy Probe (WMAP) [27–30] and by the measurement
of the baryon acoustic oscillations (BAO) in the Sloan Sky Digital Survey (SDSS)
luminous galaxy sample [126, 127].
Normal matter satisfies the energy condition w
p{ρ ¥ 0 and cannot drive the
accelerated expansion of the Universe. Hence, the recourse to a cosmic fluid, known as
dark energy, characterized by a negative pressure (wx
0), is usually made to explain
the current accelerating expansion of the Universe. Alternatively to dark energy, other
approaches such as modified gravity [128–130] and inhomogeneous cosmology [80,131],
have been proposed to explain the (apparent) cosmic acceleration.
The simplest form of dark energy is provided by a cosmological constant, which has
led to the development of the ΛCDM model. In this model, the Universe is composed
1{3, baryon, wb 0), a pressureless
cold dark matter fluid (wc 0) and a cosmological constant Λ0 (wx wΛ 1). The
of, in addition to ordinary matter (radiation, wr
model provides a reasonably good fit to the current cosmological data but is however
is plagued by two serious theoretical difficulties. One of them is the cosmological
constant problem; the observed value of the cosmological constant is
120 orders
of magnitude smaller than what is expected from theoretical computations [105].
The other one relies on the observation that the values of the matter energy density
(ρm
ρb
ρc ) and of the dark energy density are currently of the same order of
magnitude. However, since the former is diluted proportionally to the volume of the
Universe as it is expanding (ρm 9a3 ) while the later remains constant (ρΛ
ρΛ ),
0
31
the period of time during which ρm {ρΛ
Op1q is predicted to be very narrow in
comparison to the whole Universe history. To currently lie in this narrow period, a
fine tuning of the initial conditions of the model is needed. This is the essence of the
coincidence problem.
A possible approach to alleviate the cosmological constant problem consists in
replacing the cosmological constant Λ0 by a cosmological term Λptq whose value varies
over time. This can also help to decrease the severity of the coincidence problem if, for
instance, the matter dilution caused by the cosmic expansion is compensated (at least
partially) by the matter produced through the dark energy decay. A cosmological
model with a dynamical Λ has been proposed as early as 1933 [132] and the idea
regained attention in the 1980s [4, 114, 133–135] following the publication of studies
[111–113,136–139] suggesting that the dynamical effects of quantum fields in de Sitter
space-time would lead to a decay of the effective value of the cosmological constant
(see refs. [125,140,141] for an update). In the absence of a complete theory of quantum
vacuum in curved spacetime, several decay laws, mostly based on empirical arguments,
have been proposed in the literature. A review of the first Λptq-models developed may
be found in ref. [123] and some more recent examples in refs. [142–155].
In this study, we will consider the phenomenological decay law
ρ9 Λ
QρnΛ
(2.1.1)
where Q is a parameter which can be constrained from observations and n an index
characterizing different models. This law has been proposed in ref. [4] to study the
decay of dark energy into radiation, but it was shown that the constraints on this
process are too tight to allow ρΛ to decay from an initial large value to that observed
today. The possibility that dark energy decays into ordinary matter is ruled out because the annihilation of matter and anti-matter would produce a γ-ray background
in excess of observed level [133]. In this paper, we will then only consider the interaction between dark energy and dark matter. Although, only the decay of dark energy
into dark matter (Q
¡ 0) is relevant to possibly explain the cosmological constant
32
problem and the coincidence problem, we will also consider the decay of dark matter
into dark energy (Q 0q for the sake of completeness.
The model with n
1 has already been studied as a peculiar case of two more
general models, one where the decay depends also on the cold dark matter density
(ρ9 Λ
QΛ ρΛ
Qc ρc ) [142, 143] and one where the equation of state parameter of
dark energy is not fixed to
1 (ρx Qρx) [144].
9
Another generalization, involving
this time a generic value of n, has been proposed in ref. [145] (ρ9 Λ
however the case where eq. (2.1.1) is recovered (nm
QρnΛ ρnm );
Λ
m
0) did not has been considered.
Although all these generalizations would be interesting to incorporate in our study,
the resulting analysis would be significantly complicated. We will thus restrict our
attention in this paper to the simple case of eq. (2.1.1).
The model with n 3{2 is particularly interesting with regard to the coincidence
problem since it is the only one where the ratio ρm {ρΛ is constant at late time. For
this reason, we will give it with a particular attention in the remainder of this article.
It is to be noticed that solutions found in refs. [146–150] where the ratio ρm {ρΛ is
constant for all time and where wx
wΛ 1 actually correspond to a subclass of
solutions of this model (see section 2.2.1 for more details).
2.2
Model
We consider a model where dark energy and dark matter interact with an energy
transfer rate Q̃Λ
Q̃c QρnΛ
in a spatially flat Friedmann-Robertson-Walker
(FRW) spacetime. The sign of the constant Q sets the direction of the transfer. The
interaction is a decay from dark energy to dark matter (Λ
Ñ DM) for Q ¡ 0 and a
decay from dark matter to dark energy (DMÑ Λ) for Q 0. The energy conservation
equations for radiation (wr
dark energy (wΛ
1{3), total matter (dark and baryonic, wm 0) and
1) read respectively
4Hρr ,
ρm 3Hρm
ρΛ QρnΛ ,
(2.2.1)
ρ9 r
9
9
QρnΛ ,
(2.2.2)
(2.2.3)
33
and the Friedmann equation takes the usual form
3H 2
8πG
If we set t0
ρr
ρm
(2.2.4)
ρΛ .
0 (the variables with the subscript 0 refer to their present value),
the solution to eq. (2.2.3) is given by
$
'
&ρ r1 s ,
ρ ptq '
%ρ e ,
t
tn
Λ0
Λ
Λ0
where tn is defined as
1
n 1
Qt
if n 1
if n 1
(2.2.5)
,
pn1q
ρ
tn rρΛ0 , Qs Λ0
pn 1qQ .
(2.2.6)
In the limit where Q approaches 0, the time tn becomes infinite and we recover the
ΛCDM model (ρΛ
ρΛ0 ). For the decay of dark energy (Q
¡
0), the time tn is
¡ 0) if n 1 and in the past (tn 0) if n ¡ 1. The converse
is true for the decay of dark matter (Q 0). Depending on its sign, tn constitutes, or
a past bound, either a future bound on the validity of eq. (2.2.5). Indeed, for n ¡ 1
situated in the future (tn
the model clearly breaks down when t approaches tn since ρΛ becomes infinite. For
n 1, the dark energy density is ρΛ
0 at t tn .
Hence for Q
¡ 0, tn corresponds
to the latest time when the interaction could be physically meaningful since the dark
energy would have completely decayed in dark matter at this point. Similarly, for
Q 0, tn corresponds to the earliest time that the interaction could be meaningful.
As mentioned in the introduction, the case where n
3{2 is the only one which
leads to a constant ratio of ρm {ρΛ at late times and could therefore provides an elegant
explanation to the coincidence problem. Indeed, for n 1 and |t{tn | " 1, we see from
1
eq. (2.2.5) that ρΛ becomes proportional to t n1 . Moreover, at late times ρm is by
hypothesis proportional to ρΛ and the radiation can be neglected in the Friedmann
1
equation, hence the Hubble parameter H becomes proportional to t 2n2 . It is easy
to verify from eq. (2.2.2) that the only value of n consistent with these expressions is
3{2 (otherwise, the term Hρm would not have the same time-dependence as the other
two). Because of this remarkable feature, we will consider in details the model with
34
n
3{2 and find analytical solutions in order to have a deeper understanding of its
characteristics. For that, we will divide the cosmological evolution into two eras: an
early one dominated by radiation (RD-era), and a late one dominated by matter and
dark energy (MΛD-era).
For all values of n (including n 3{2q, we will use numerical solutions to constrain
the models parameters. In that case, it will be more convenient to express the energy
conservation equations (2.2.1-2.2.3) as functions of the redshift (corresponding to
data)
dρr
dz
dρm
dz
dρΛ
dz
2.2.1
4 p1 ρr zq ,
(2.2.7)
n
3 p1ρm zq Q H p1ρΛ zq ,
n
Q H p1ρΛ zq .
(2.2.8)
(2.2.9)
MatterDark energy dominated era (MΛD-era): n 3{2
During the MΛD-era, we assume that ρr
man equation as
! ρm , ρΛ.
We can then write the Fried-
2
ρm
3H
8πG
ρΛ.
(2.2.10)
In order to have a solution for ρm , we must solve for H first. Using the preceding
equation to replace ρm and ρ9 m in eq. (2.2.2) leads to the equation
1 9
H
4πG
3
H2
8πG
ρΛ.
(2.2.11)
The solution for n 3{2 is given by
H ptq SQ
1 ρΛ
3
8πG
2
pβ 2 1q
1
2
1
k pρΛ {ρΛ0 q 2
β
β
k pρΛ {ρΛ0 q 2
β
1
1
,
(2.2.12)
35
103
300
10
0.1
ΩΛ >ΩΛ
2
10
0
∞
250
10
k>0
101
0.2
200
0.3
0
10
ρΛ /ρΛ
ΩΛ <Ω
Λ
0
−1
10
i
Q/G1/2
10
0
∞
0.4
150
10
0.5
k<0
100
10
10−2
0.6
0.7
50
10
−3
10
10−4
0.8
0.9
0
10
0
0.2
0.4
ΩΛ
0.6
0.8
1
0
−50
10
10
0
−100
∞
Figure 2.1
−150
10
ΩΛ −ΩΛ
10
Figure 2.2
Figure 2.1: The curve ΩΛ0 ΩΛ8 shown in the plane ΩΛ0 Q (model with n 3{2).
1
The parameter Q is expressed here in term of the dimensionless quantity Q{G 2 , where
G is the Newton’s constant. The physical solutions (k 0) are situated below the
curve ΩΛ0 ΩΛ8 .
Figure 2.2: The ratio ρΛi {ρΛ0 obtained from eq. (2.2.19) and presented for different
values of ΩΛ0 (indicated on the figure) as a function of the difference between ΩΛ8
and ΩΛ0 (model with n 3{2). The dashed line represents the ratio between the
theoretical value and the observed value of dark energy density obtained in the context
of the usual ΛCDM model ( 10120 ).
where k is a dimensionless integration constant, SQ
sign Q and β SQ
a
1
96πG{Q2 .
The scale factor a can be obtained by integrating H,
a ã0 exp
»
Hdt
ã0
1
ρΛ
β
1
6
β
k pρΛ {ρΛ0 q 2
2{3 SQ
.
(2.2.13)
In order to have a 1 at t t0 , the value of the constant ã0 is set to
ã0
1
β
ρΛ06
p1
kq {
2 3
SQ
.
(2.2.14)
The solutions obtained can be divided into two classes depending on the sign of
the constant k. If we only consider dark energy and dark matter, it can be shown that
the solutions with k
¡ 0 are characterized by the absence of an initial singularity and
by the absence of an early phase dominated by matter (which is needed for structures
formation). The converse is true for k
0, i.e. the solutions are characterized by the
−200
10
0
36
(a)
(b)
(c)
Λ→ DM
k>0
Λ→ DM
k<0
DM→Λ
k<0
ΩΛ, Ωm
1
0
2
|a|
10
0
10
−2
10
tn
ton
t0
tn
ton ti
t0
ti
t0 toff
tn
Time
Figure 2.3: Examples of the evolution (for the model with n 3{2) of the dark energy
density parameter ΩΛ (thick line), the matter density parameter Ωm (thin line) and
the modulus of the scale factor (|a| – for t ti , a P I) in absence of radiation and
baryons for an expanding Universe (H0 ¡ 0) with (a) ΩΛ0 ¡ ΩΛ8 , (b) ΩΛ0 ΩΛ8 1
and (c) ΩΛ8 ¡ 1. The dashed lines correspond to the solutions that we would obtain
by letting ρm become negative and the shaded areas to the mathematical solutions
for times prior to the singularity. We have considered that the interaction starts at
ton tw for (a) and stops at toff tw for (c). For (a), we can notice the absence of
an initial singularity and of an early phase dominated by matter
presence of an initial singularity and by the presence of an early phase dominated by
matter.
To find an expression for k, the initial conditions rρΛ0 , Q, H0 s could be used, how-
ever, it will be more convenient to replace the first two parameters by the current and
the late-time value (supposed to be constant for n 3{2) of the dark energy density
parameter ΩΛ . We can rearrange eq. (2.2.12) in order to have
ΩΛ ptq In the limit where t
value, ΩΛ
Ñ
S
ΩΛQ8 .
ρΛ
ρtot
ρΛ
3H 2
8πG
β2 1
k pρΛ {ρΛ0 q 2
β
1
β
k pρΛ {ρΛ0 q 2
β
1
2 .
(2.2.15)
1
Ñ 8, ρΛ becomes 0 as expected and ΩΛ approaches a constant
The parameter ΩΛ8 depends only on the value of Q and is defined
37
as
ΩΛ8
ββ 11 .
(2.2.16)
We have to keep in mind that ΩΛQ8 truly represents the late time value of ΩΛ ptq only
for the decay of dark energy. For the decay of dark matter, we have shown that the
S
model breaks down in the future at t
tn, hence before reaching the “late time”
(actually, as we will see, the model breaks down even before tn ). In this case, ΩΛ8
has to be considered only as a parameter related to the strength of the interaction.
Setting t t0 in eq. (2.2.15) and solving for k yields
1
k rΩΛ0 , ΩΛ8 , H0 s where SH0
sign H0.
SH0 ΩΛ8 ΩΛ2 0
1
ΩΛ2 8
(2.2.17)
,
ΩΛ8
For an expanding Universe (H0 ¡ 1), the sign of k depends
1
SH0 ΩΛ2 0
1
2
S
only on whether ΩΛ8 is greater or smaller than ΩΛ0 . Since ΩΛ0 and ΩΛQ8 are both
included in the interval r0, 1s, k is necessarily negative for the decay of dark matter
¡ ΩΛ8 ) or negative (if ΩΛ ΩΛ8 ) for the decay
The solutions with k 0 correspond to a situation where the ratio
but could be either positive (if ΩΛ0
of dark energy.
0
ρm {ρΛ is constant for all time (at least as long as radiation is neglected) and which is a
peculiar case of the solutions found in refs. [146–150]. The curve ΩΛ0
ΩΛ8 is shown
in the plane ΩΛ0 Q in figure 2.1. For the decay of dark energy, the relevant range of
values for Q is roughly situated between 102G 2 and 102 G 2 . Below the lower bound,
1
1
the interaction becomes insignificant and we recover approximately the ΛCDM model
1), whereas above the upper bound, ΩΛ remains close to zero for all time.
From eq. (2.2.13) it should be obvious that only the solutions with k 0 feature
(ΩΛ8
a singularity and that it occurs at the time
ti
1
k1
1
β
tn .
(2.2.18)
¡ 0, this singularity occurs in the past (ti t0 ) and could be considered as
the initial time of the Universe. Evaluating eq. (2.2.5) at t ti , we find an expression
For H0
38
for the ratio of the initial and the current value of the dark energy density
ρΛi
ρΛ0
pkq ΩΛ8 ΩΛ ΩΛ8 ΩΛ8 ΩΛ
1
2
2
β
1
2
2
β
0
1
2
1
2
(2.2.19)
.
0
The case where ΩΛ0
ΩΛ8
could in principle provide an explanation for both
the coincidence and the cosmological constant problems. Indeed, when radiation is
neglected, we have ρm {ρΛ
1{ΩΛ 1, hence the condition ΩΛ ΩΛ8 implies that
the current value of the ratio ρm {ρΛ is now typical of a large period of time and there
is no more “coincidence” problem. Moreover, if ΩΛ ΩΛ8 , the ratio ρΛ {ρΛ can
0
0
i
0
be arbitrarily large. However, if the difference between the theoretical value of ρΛi
and the value of ρΛ0 obtained for the ΛptqCDM model remains of several orders of
magnitude, we see from figure 2.2 that a fine tuning of the quantity ΩΛ8 ΩΛ0 would
be needed to explain it.
To complete the analysis of the MΛD-era, we must come back to the problem
encountered in the future for the decay of dark matter. It easy to verify that at tn ,
the density parameter of dark energy is greater than 1, which constitutes a violation
of the null energy condition (w
1q. Setting ΩΛ 1 in eq. (2.2.15) and solving for
t, we actually find that the null energy condition starts to be violated at
tw
1
kkw
where
1
β
(2.2.20)
tn ,
1
kw rΩΛ0 , ΩΛ8 , H0 s Sk SQ ΩΛ2 8
1
2
Sk SQ ΩΛ8
ΩΛ8
(2.2.21)
1
and Sk is defined as the sign of the constant k. The density parameter ΩΛ could
become greater than 1 only if we let ρm become negative after tw . To prevent this
problem, when the dark matter has completely decayed in dark energy (ρc
0), we
should consider that there is no more interaction and the dark energy density becomes
a constant. If we neglect the baryon density, the interaction stops at toff
tw and the
39
late time(t ¡ toff ) value of ρΛ is then given by
ρΛw
kk
w
2
β
(2.2.22)
ρΛ0 .
1 ΩΛ and |a| are shown in figure 2.3 for ΩΛ ¡ ΩΛ8 ,
¡ 1. As we can see, the null energy condition is also
The solutions for ΩΛ , Ωm
1 and ΩΛ8
violated for the decay of dark energy, but this time the violation occurs in the past.
ΩΛ0
ΩΛ8
0
¡ tw ) for k 0 but not for
We could simply consider that ρΛ is constant before ton tw . However, the
The presence of the singularity prevents this problem (ti
k
¡ 0.
reason why the dark energy should start to decay at tw would be unclear. It would
also be possible to consider that decay starts after tw (when ρm is non-zero), which
would lead to a solution with an initial singularity and an early phase dominated by
matter. That could be the case for instance if, for some reason, the dark energy decay
starts only when the dark matter becomes sufficiently diluted. However, we do not
have considered any threshold of this kind for the case k
0. In order to be consistent
and to keep our model as simple as possible, we will not introduce any threshold, but
rather consider that the model is not valid for k
2.2.2
¡ 0.
Radiation dominated era (RD-era): n 3{2
Until now, the radiation has been neglected. This approximation clearly does not
hold at early times. Solving eq. (2.2.1) leads to the usual result (which holds for all
time)
ρr
ρr a4 .
(2.2.23)
0
During the RD-era, we can neglect the dark energy and the matter densities (ρr
"
ρm , ρΛ ). Using the Friedmann equation, we can easily find the Hubble parameter
H
where Hr0
a
p8πG{3qρr
0
2Hr a2 .
(2.2.24)
0
and integrate it to obtain an expression for the scale factor
a 2Hr0 pt teq q
a2eq
1
2
.
(2.2.25)
40
Λ→DM
teq=2506 kyrs
1
0.8
DM→Λ
teq=64 kyrs
t0=18.9 Gyrs
ΩΛ =0.75
t0=13.6 Gyrs
ΩΛ =1.05
∞
∞
0.6
0.4
0.2
0
teq=191 kyrs
1
ΩΛ, Ωm, Ωr
0.8
t0=15.3 Gyrs
ΩΛ =0.85
teq=49 kyrs
t0=13.2 Gyrs
teq=41 kyrs
t0=12.9 Gyrs
ΩΛ =1.15
∞
∞
0.6
0.4
0.2
0
teq=93 kyrs
1
0.8
t0=14.2 Gyrs
ΩΛ =0.95
ΩΛ =1.25
∞
∞
0.6
0.4
0.2
0
−8
10
−6
10
−4
−2
10
10
Time [Gyrs]
0
2
10
10
−8
−6
10
10
−4
−2
10
10
Time [Gyrs]
0
10
2
10
Figure 2.4: Evolution (for the model with n 3{2) of dark energy (ΩΛ , thick line),
matter (Ωm , thine line) and radiation (Ωr , dashed line) density parameter for different
values of ΩΛ8 and with ΩΛ0 0.73 and H0 70 km s1 Mpc1 . The values of Ωb0 and
Ωr0 are set using eq. (2.3.12) and eq. (2.3.13). The time has been redefined setting
ti 0, hence the age of the Universe is given by t0 .
The value of the integration constant has been set in order to have aeq
aRD pteq q, where teq is defined as the time where ρr
now occurs at
ρΛ
aM ΛD pteq q ρm . The initial singularity
2
ti
aeq
teq 2H
(2.2.26)
.
r0
We cannot find an analytical expression for teq ; instead, we have to make the approximation that the solutions found for a and H during the MΛD-era hold up to teq and
solve numerically the equation
ρrM ΛD pteq q ρmM ΛD pteq q
4
ρr0 a
M ΛD pteq q 2
3HM
ΛD pteq q
.
8πG
ρΛM ΛD pteq q,
(2.2.27)
To find an expression for the matter density ρm during the RD-era, we can use the
expressions found for ρΛ and for H (eqs. (2.2.5) and (2.2.24)) to solve the matter
41
conservation equation (eq. (2.2.2)). Setting the value of the integration constant in
order to have ρmeq
ρm pteq q ρm
M ΛD
RD
ρm
ρm
eq
a
aeq
pteq q, we get
3
F2 ptqs a3 ,
rF1 ptq
(2.2.28)
where F1 is given by
F1
ρΛeq a3eq ρΛ a3
3Hr0 |tn |ρΛ2 0 ρΛ2 eq aeq ρΛ2 a
1
1
1
(2.2.29)
and F2 by
F2
6Hr20 t2n
α
arctan
a
α
arctan
a eq
α
ρΛ0 , with α b
2Hr teq
0
p tn q a2eq .
(2.2.30)
In the limit where Q approaches 0, F1 and F2 are reduced to F1
F2 3Hr |tn|paeq 0
aqρΛ0 , so the second term in eq. (2.2.28) vanishes and we recover the expected expression for the ΛCDM model.
As a consequence of the approximations made for the MΛD-era and for the RD-era,
ρtot
3H 2{p8πGq in the vicinity of teq .
To evaluate ρtot (and Ωi ) we should instead
explicitly add ρΛ , ρm and ρr together. The evolution of the density parameters are
shown in figure 2.4 for different values of ΩΛ8 . For smaller values of ΩΛ8 , the age of the
Universe is larger. There is also an inverse relationship relating ΩΛ8 and the duration
of the RD-era and of the ΛD-era, whereas the duration of the MD-era decreases as
ΩΛ8 becomes smaller. The variation of these time intervals becomes increasingly
important as ΩΛ8 approaches ΩΛ0 , leading to solutions significantly different from
what we would have obtained in the ΛCDM model, even for t
t0 .
We can then
expect that these solutions will be disfavoured by the observational constraints.
2.3
Observational Constraints
In this section, we explain our methodology for using the currently available data
to constrain the three free parameters (ΩΛ0 , ΩΛ8 , H0 ) of the model. Following what
was done in refs. [151–154] to constrain cosmological models with interacting dark
sectors, we have considered the tests described below involving the distance modulus
42
µ of type Ia supernova (SNeIa) and gamma-ray bursts (GRB), the baryon acoustic
oscillation (BAO), the cosmic microwave background (CMB) and the observational
Hubble rate (OHD). In refs. [151–154], the authors also use the gas mass fractions in
galaxy clusters as inferred from x-ray data to constrain their models, however since
this method involves many uncertain parameters, we do not consider it here.
The numerical analysis has been performed on a three dimensional grid. Each axis
corresponds to one of the free parameters (ΩΛ0 , ΩΛ8 , H0 ) and the distance between
0.001, ∆ΩΛ8 0.001 and ∆H0 0.01
the points is given on each axis by ∆ΩΛ0
km s1 Mpc1 . For each point, we have computed the sum of the χ2 of each data set
χ2tot
χ2µ
χ2OHD
χ2BAO
χ2CM B .
(2.3.1)
The best fit point corresponds to the point which minimizes the value of χ2tot
χ2min . For the ΛptqCDM model (3 free parameters), the 1-σ confidence region (68.3%)
¤ χ2min
χ2tot ¤ χ2min
corresponds to the points for which χ2tot
(95.45%), to the points for which
3.53 and the 2-σ confidence region
8.02. For the ΛCDM model (2
free parameters), these two regions correspond respectively to the points for which
χ2tot
¤ χ2min
2.3.1
2.30 and χ2tot
¤ χ2min
6.18.
Distance modulus µ of SNeIa and GRB
The distance modulus is the difference between the apparent magnitude m and
the absolute magnitude M of an astronomical object. Its theoretical value for a flat
Universe is given by
where h
µth pz q 5 log10
DL pz q
h
42.38,
(2.3.2)
H0 /(100 km s1 Mpc1 ) and the Hubble free luminosity distance DL
defined as
DL
H0p1
zq
»
z
0
dz
.
H
is
(2.3.3)
The best fit is obtained by minimizing the χ2 function
χ2µ rΩΛ0 , ΩΛ8 , H0s ¸ rµ
i
obs
pziq µthpzi qs2 ,
σi2
(2.3.4)
43
z
H pz q
1-σ uncertainty
z
H pz q
1-σ uncertainty
0 [164]
74.2
3.6
0.48
97
62
0.1
69
12
0.88
90
40
0.17
83
8
0.9
117
23
0.24 [165]
79.69
3.61
1.3
168
17
0.27
77
14
1.43
177
18
0.34 [165]
83.80
4.55
1.53
140
14
0.4
95
17
1.75
202
40
0.43 [165]
86.45
4.96
Table 2.1: Observational values of H pz q (in km s1 Mpc1 ) used to constrain the
model. Unless indicated otherwise, the values are taken from table 2 in [163].
where σi is the 1-σ uncertainty associated with ith point. The observational data
used are the 557 distance modulii of SNeIa assembled in the Union2 compilation [24]
(0.015 z
1.40) and the 59 distance modulii of GRB from [25] (1.44 z 8.10).
The combination of these two types of data covers a wide range of redshift providing a
more complete description of the cosmic evolution than the SNeIa data by themselves.
2.3.2
Observational H pz q data (OHD)
As a function of the redshift, the Hubble parameter is given by
H pz q 1
1
dz
.
z dt
(2.3.5)
Therefore it is possible to measure H pz q through a determination of dz {dt. In [161],
Jimenez et al. demonstrated the feasibility of the method by applying it to a z
0
sample. Simon et al. [162], completed by Stern et al. [163] have applied this method
using the differential ages of passively evolving galaxies to derive a set of 11 observational values for H pz q in the range 0.1 z
1.8. In addition to these values, we also
constrain the model using the value of H at z 0 obtained by Riess et al. [164] from
the observations of 240 Cepheids, as well those obtained by Gazstañaga et al. [165]
at z
0.24, z 0.34 and z 0.43 using the radial BAO peak scale as a standard
ruler (see ref. [165] for more details). The best fit is obtained by minimizing the χ2
function
χ2OHD
rΩΛ , ΩΛ8
0
¸ rH
,H s 0
i
The data used are summarized in table 2.1.
obs
pziq Hthpzi qs2 .
σi2
(2.3.6)
44
2.3.3
Baryon acoustic oscillation (BAO)
The use of BAO to test dark energy models is usually made by means of the
distance parameter A. However, as pointed out in ref. [155], A is not appropriate
to test a model with matter production associated with the decays of dark energy.
Instead, we can use the dilation scale
z
DV pz q c
H pz q
» z
0
dz
H pz q
2 1{3
(2.3.7)
.
The ratio rs pzd q{DV pz q, where rs pzd q is the comoving sound horizon size at the drag
0.35 by SDSS [32] and at z 0.20 by 2dFGRS
epoch, has been observed at z
[157]. It would be possible to proceed as in refs. [152, 154] and use a fitting formula
(developed for the ΛCDM model) to find rs pzd q. However we can avoid this if we
follow refs. [151, 153] and minimize the χ2 of the ratio DV0.35 {DV0.20 given by
χ2BAO rΩΛ0 , ΩΛ8 s rpDV {DV qth pDV {DV qobss2 .
0.35
0.20
0.35
0.20
2
σ0.35
{0.20
(2.3.8)
In addition to being independent of rs , this ratio is also independent of H0 . The
observed value for DV0.35 {DV0.20 is 1.736 0.065 [157].
2.3.4
Cosmic Microwave Background (CMB)
The values extracted from the 7-year WMAP data for the acoustic scale (lA pz q),
for the CMB shift parameter (R), and for the redshift at the decoupling epoch (z )
can be used to constrain the model parameters. The CMB shift parameter and the
acoustic scale are respectively defined as
R
b
and
lA
Ωm0 H02
³z
π 0
³8
z
» z dz
H
0
dz
H
cs dz
c H
.
(2.3.9)
(2.3.10)
45
Since the sound velocity cs is given by
cs
c
Ωb0
9
4 Ωγ0 p1 z q
3
1{2
,
(2.3.11)
two additional free parameters are needed to determine the acoustic scale, namely the
current value of the density parameter of baryons (Ωb0 ) and of radiation (Ωγ0 ). Constraining the model with these two additional parameters will require in an increased
computational cost. However as suggested in ref. [155], we can use the values obtained
in the context of the ΛCDM cosmology. This is motivated since the radiation and
the baryons are separately conserved, and because we want to preserve the spectrum
profile as well the nucleosynthesis constraints. The observational results from 7-year
WMAP data [30] are
Ωb0
2.25 102h2
and
Ωγ0
2.469 105h2.
(2.3.12)
Considering the high redshift values involved here, the dynamical effects of radiation cannot be neglected. We therefore must include the density parameter of
radiation, Ωr0 , as an extra initial parameter. However, this quantity is related to the
density parameter of photons through
Ωr0
7
8
1
4
11
4
3
Neff
Ωγ0 .
(2.3.13)
Hence, providing that the effective number of neutrino species, Neff , is determined,
we still have only three parameters to constrain. The departure of Neff from 3 is
due to neutrino heating by e annihilations in early Universe. The value inferred
from observations, usually close to 3 (3.04 [158], 3.14 [159]), was recently updated to
4.34 [30].
The decoupling epoch occurs when the expansion time becomes less than the
Thomson scattering time. From a practical point of view, z may be computed by
finding the redshift value corresponding to an optical depth of unity
τ pz q » z
0
ne σT c
p1 zqH dz
1,
(2.3.14)
46
where σT is the Thompson cross-section. To evaluate this integral, we need to know
the evolution of the electron density ne as a function of the redshift, which is not
simple because of its dependence on the recombination process. To avoid to do this
complex computation each time that one need to know the value of z , a fitting
formula has been developed for the ΛCDM model [160]
z
1048r1
0.00124pΩb0 h2 q0.738 sr1
g1 pΩm0 h2 qg2 s,
(2.3.15)
where
0.0783pΩb h2q0.238 p1 39.5pΩb h2 q0.763 q1,
g2 0.560p1 21.1pΩb h2 q1.81 q1 .
g1
0
0
0
(2.3.16)
(2.3.17)
Following refs. [152, 154], we approximate the value of z in the ΛptqCDM model
using eq. (2.3.15). However, from eq. 2.3.14, it is obvious that even if we set all the
parameters, except Q, to the same values, the decoupling redshift for the interacting
and the non-interacting cases will be different as a consequence of a different redshiftdependence for the Hubble parameter. Hence, the validity of this approximation may
be questioned; we will come back to this issue in the results section.
Defining v
plA lAobs, R Robs, z zobsq, the best fit is obtained by minimizing
the χ2 function
χ2CM B rΩΛ0 , ΩΛ8 , H0 s vMv t ,
(2.3.18)
where M is the inverse variance-covariance matrix from the 7-year WMAP data
2.305
M 29.698
6825.270 113.180 Æ
29.698
1.333 113.180
1.333 Æ
(2.3.19)
3.414
and the observed value are also taken from the 7-year WMAP data, lA pz q 302.09 0.76, Rpz q 1.725 0.018 and z
1091.3 0.91.
47
2.4
Results and discussion
2.4.1
Case n 3{2
The best fit values obtained for the parameters of the ΛCDM model and the
corresponding χ2min are shown in table 2.2 for three different values of Neff : 3.04, 3.14,
and 4.34. Comparing these values to those obtained from the 7-year WMAP data [30]
(ΩΛ0
0.725 0.016, H0 70.2 1.4 km s1Mpc1), we find a better agreement
when we use Neff =3.04. Moreover, the χ2min is minimized for this value. For these
reasons, we will use Neff
3.04 to constrain the interacting model.
For the ΛptqCDM model, we present the constraints obtained from each ob-
servational data set considered separately (distance modulus µ, OHD, BAO, and
CMB) as well from two different combinations of them: C3 (µ+OHD+BAO) and C4
(µ+OHD+BAO+CMB). The best fit values and the corresponding χ2min are shown in
in table 3.1 and the projections of the 1-σ and 2-σ confidence regions in the planes
ΩΛ0
ΩΛ8 , ΩΛ H0 and ΩΛ8 H0, are shown in figure 2.5.
0
ΩΛ8
ΩΛ0
Neff
0.035
0.051
H0 (km s1 Mpc1 )
0.61
χ2min
1.00
1
69.82 0.50 0.91
576.178
0.033 0.051
0.014
0.76
1
69.85 0.49
576.638
C4 3.04 0.718 0.009
0.012 0.019
0.55 0.82
0.009 0.014
0.48 0.75
C4 3.14 0.712 0.012 0.017
1
69.75 0.57 0.83
577.083
0.018
0.78
C4 4.34 0.635 0.012
1
69.02 0.51
613.694
0.014 0.021
0.55 0.83
Table 2.2: Best fit values for the ΛCDM model parameters and the corresponding χ2min
for three different values of Neff inferred from observations 3.04 [158], 3.14 [159], and
4.34 [30]. We have considered two different combinations of observational constraint,
C3 (µ+OHD+BAO) and C4 (µ+OHD+BAO+CMB). The results obtained from C3
are weakly sensitive to the value of Neff . The limits are the 1-σ and the 2-σ extremal
values.
C3
3.04
0.723
We consider the combination of constraints C3 , which excludes the CMB constraints, because there is a possible circularity problem in our analysis of the CMB
data. Indeed, we have used the values of Ωb0 , Ωγ0 and Ωr0 obtained in the context of
the ΛCDM model and we have approximated the value of z using a fitting formula
developed for the same model. In the latter case, the difficulty to obtain the exact
48
µ
OHD
BAO
CMB
C3
C4
0.122
0.804 0.196
0.570
0.812 0.188
0.775
0.710 0.115
0.189
0.753 0.142
0.106
0.720 0.013
0.013
0.746
0.178
H0 (km s1 Mpc1 )
ΩΛ8
ΩΛ0
0.172
0.196
0.690
0.188
0.812
0.161
0.315
0.229
0.153
0.020
0.021
0.254
0.355
1.174 0.970
0.970
0.681 2.743
0.675
1.011 0.254
0.038
1.099 0.549
0.301
1.006 0.020
0.019
1.061
0.770
00.461
01.660
01.090
40.695
00.680
01.116
00.046
00.975
00.408
00.030
00.029
01.383
00.87 01.30
10.50
72.68 07.99
11.67 16.06
00.97
69.94
01.49
-
69.77 68.89
69.97
14.08
11.42
00.91
00.84
00.52
00.52
23.80
16.11
01.38
01.25
00.79
00.79
χ2min
χ23
χ24
565.834
565.878
566.182
7.970
8.824
8.790
0.000
1.185
1.240
0.015
-
0.054
575.888
-
-
576.267
-
-
Table 2.3: Best fit values for the interacting model (n 3{2) parameters and the
corresponding χ2min associated with each data set and two combinations of them:
C3 (µ+OHD+BAO) and C4 (µ+OHD+BAO+CMB). The two last columns, χ23 and
χ24 , represent the partial contribution of each data set to the value of C3 and C4 ,
respectively. For the sake of comparison, the values obtained for the ΛCDM model
using the same constraints are also shown. The limits are the 1-σ and the 2-σ extremal
values excluding the non-physical region ΩΛ0 ¡ 1.
value of z comes from the presence of the electron density ne , whose value at different redshifts depends on the recombination history, in the computation of the optical
depth (eq. (2.3.14)). Hence, in order to have a rough idea of the error induced by
this approximation, we have simply removed the electron density from eq. (2.3.14) to
get the integral
³ z
0
dz {pH pz qp1
z qq (where z is given by eq. (2.3.15)) and compared
the values obtained for the interacting and the non-interacting cases. If the difference
is not too large, that means that up to z , the evolution of the Hubble term and
thus of the scale factor, are similar in both cases; this in its turn implies that the
recombination history has also to be similar since for the comparison, all the initial
parameters are fixed to the same value (except Q). In figure 2.6, we see that for the
CMB constraints considered alone, the percentage of difference reaches up to
in the 2-σ confidence region and
5% in the 1-σ confidence region.
7.5%
When all the
data are considered, the percentage of difference in the 2-σ and in the 1-σ confidence
region falls respectively below 2.5% and 2%. Hence, we can expect that eq. (2.3.15)
provides a reasonable approximation for z , at least in the confidence regions obtained
from the combination of constraints C4 .
49
(a)
(b)
BAO
90
ΩΛ
∞
H0 [(km/s)/Mpc]
OHD
4
3
µ
CMB
2
90
OHD
OHD
H0 [(km/s)/Mpc]
5
(c)
80
µ
70
60
1
80
µ
70
60
CMB
CMB
0
0
0.5
1
ΩΛ
1.5
2
50
2.5
0
0.5
0
1
ΩΛ
1.5
2
50
2.5
0
1
2
0
ΩΛ
3
4
5
∞
74
74
1.3
1.1
CMB
∞
ΩΛ
H0 [(km/s)/Mpc]
µ
1
0.9
0.8
0.7
0.6
0.5
OHD
µ
70
68
BAO
0.6
0.7
ΩΛ
0.8
0.9
OHD
72
H0 [(km/s)/Mpc]
1.2
OHD
66
0.5
0.6
CMB
0.7
ΩΛ
72
µ
70
68
CMB
0.8
0.9
66
0.6
0.8
ΩΛ
1
1.2
Figure 2.5: Projection (a) in the plane ΩΛ0 ΩΛ8 , (b) in the plane ΩΛ0 H0 and (c) in
the plane ΩΛ8 H0 of the 1-σ (darker colour) and the 2-σ (lighter colour) confidence
region (for the model with n 3{2) obtained from four different types of observational
data, the distance modulus µ (blue), BAO (yellow), CMB (pink), OHD (green) and the
from combination of all of them (C4 , µ+OHD+BAO+CMB) (gray). The confidence
regions obtained from combination of observational constraints C3 (µ+OHD+BAO)
are shown in the lower panels (dotted contours). The region situated at the right
of the dashed line (ΩΛ0 1), involved negative matter energy density and must be
considered as non-physical. The solid line (ΩΛ8 1) sets the separation where dark
energy is decaying in dark matter (below the line) and where dark matter is decaying
in dark energy (below the line). Below the dot-dashed line (ΩΛ0 ΩΛ8 ), k ¡ 0 and
above, k 0. The best fit parameters are also indicated by a black dot in each panel.
We can notice that the confidence regions obtained from BAO (1-σ) , from OHD
(1-σ) and from the distance modulus µ (2-σ) extend partially beyond ΩΛ0
1. How-
ever, a value of ΩΛ0 greater than 1 would imply a negative value for the current
energy density of matter (ρm0
0) and therefore must be considered as non-physical.
Similarly, we can also notice that the 1-σ confidence regions obtained from BAO and
OHD extend (marginally in the case of OHD) into region of the parameter space where
¡ 0 (below the dot-dashed line on figure 2.5). We have shown in section 2.2.1 that
the solutions with k ¡ 0 necessarily reach a point where ρm becomes negative if we
k
50
>25
1.5
20
15
1.4
10
1.3
ΩΛ
∞
5
0
1.2
−5
1.1
−10
−15
1
−20
0.9
0.35
0.45
0.55
0.65
ΩΛ
0.75
0.85
<−25
0
³z
Figure 2.6: Percentage of difference on the value of the integral 0 dz {pH pz qp1 z qq
evaluated for the ΛCDM model and for the ΛptqCDM model for a corresponding
value of ΩΛ0 (z is given by eq. (2.3.15)). A positive difference means that the ΛCDM
value is larger than the ΛptqCDM one and a negative difference, the converse. The
dashed lines and the solid lines represent respectively, the 1-σ and the 2-σ confidence contours obtained from the CMB data, and from the combination of data C4
(µ+OHD+BAO+CMB).
are looking sufficiently far in the past. However, for the redshifts involved in the
computation of the BAO and the OHD constraints, this point is never reached and
ρm remains always positive. Hence, in this case, there is no reason to consider these
constraints as non-physical. In any case, these two regions are excluded by at least
2-σ from the combinations of constraints C3 and C4 .
Regarding the best fit values obtained, the processes of decay of dark matter into
dark energy (ΩΛ8 ¡ 1) is favoured by only one constraint (BAO), while the decay
of dark energy into dark matter (ΩΛ8 1) is favoured by the three other, as well
as by the combinations C3 and C4 . However, all the 1-σ confidence regions extend
below and above the the plane ΩΛ8 1 (which corresponds to the ΛCDM model)
and therefore both processes still are statistically allowed.
As shown in section 2.2.1, we need to have ΩΛ8 ΩΛ0 in order to relate and
explain the cosmological constant problem and the coincidence problem. However, as
we can see on figure 2.5, these values are excluded by at least 2-σ due to constraints
51
C4 . Actually, the closest point to the line ΩΛ8 ΩΛ0 situated on the 2-σ contour
is ΩΛ0 0.718, ΩΛ8 0.983. For these parameters, the value of ρΛ remains mostly
constant (ρΛi {ρΛ0
1.04), and a coincidence problem is still present since the current
ratio of matter and dark energy density (ρm {ρΛ 0.393) lies in the transition zone
0
0
between the point where dark energy starts to dominate and the late-time phase where
the ratio ρm {ρΛ becomes approximately constant (ρm8 {ρΛ8 0.017).
The best fit values obtained for the ΛCDM model from the constraints C4 (table 2.2) are included in the 1-σ confidence region of the interacting model. Moreover,
the χ2min is only slightly smaller for interacting model (576.267 vs. 576.638). Hence,
we may wonder if the introduction of an extra-parameter (ΩΛ8 ) is really justified.
The improvement of the χ2min value may be assessed by the mean of the Bayesian
information criterion [166], defined as
BIC 2 ln Lmax
K ln N
pχ2min
Cq
K ln N,
(2.4.1)
where Lmax is the maximum likelihood, K is the number of parameters for the model
(2, for the ΛCDM model, 3 for the interacting model), N the number of data points
used in the fit (N
635) and C a constant independent of the model used. Following
ref. [167], we will regard a difference of 2 for the BIC as a non significant, and of 6 or
more as very non-significant improvement of the χ2min value. Since we get ∆BIC
6.08,
we can conclude that the addition of an extra parameter is not warranted by the
marginal decrease in the value of χ2min .
If we consider now the combination of observational constraints C3 (which excludes
the CMB data), the improvement of the χ2min value remains insignificant (∆BIC =6.15)
and the closest point to the line ΩΛ8 ΩΛ0 on the 2-σ contour is now ΩΛ0 0.615 ,
ΩΛ8 0.693, which leads to the ratio ρΛi {ρΛ0 3.37, still far to being able to explain
the discrepancy with the theoretical value. The current value of the ratio ρm {ρΛ is
now closer to the late-time value (0.443 vs. 0.626), but not sufficiently to explain the
coincidence problem.
52
2.4.2
General case
The motivation to study the model with n
3{2 was purely phenomenological
and did not rely upon any deep principles. Then, it will be interesting to extend our
analysis to general values of n. However, we do not perform an analysis as detailed
as for n 3{2 and restrict it to constrain the model parameters in the range n 0 to
n
10.
The results are presented in figure 2.7 for the combination of observational
constraints C3 and C4 .
We can see on figure 2.7 that the best fit values for ΩΛ0 and H0 are nearly constant
for both C3 and C4 , and for the latter, this is even the case for the confidence intervals.
These results seem a priori very surprising, but could be understand if the constant
Q is sufficiently small (though still non-zero). Indeed, in the limit of a small Q, it is
possible to reproduce exactly the same cosmological evolution, for t
t0 , using two
different values of n.
If we suppose that the term QρnΛ can be neglected in the matter energy conservation equation (eq. (2.2.2)), we find a dependence on n only in the energy density
ρΛ and in the quantities derived from it. We are then interested to determined the
conditions needed to have ρΛn ptq ρΛñ ptq, with n ñ. In the limit where Q is small,
the solutions found for ρΛ (eq. (2.2.3)) become
ρΛn ptq ρΛ0
QρnΛ t.
0
(2.4.2)
Hence the condition to have ρΛn ptq ρΛñ ptq is given by
n
Q Q̃ρΛñ
,
0
(2.4.3)
which provides an explanation to the exponential relationship observed on the lower
panels of figure 2.7 between the best fit value of Q and n.
For the best-fit values, the same cosmological evolution is closely reproduced independently of the value of n, hence we find the same value for χ2min . Concerning the
1-σ and 2-σ confidence regions, the results are weakly sensitive to n for the constraints
C4 , (for the higher values of n, Q becomes sufficiently large to start to observe a weak
53
(a)
(b)
1
0.8
Ω
Λ
0
0.9
0.7
0.6
−1
Mpc ]
0.5
72
−1
71
H [km s
70
0
69
68
10
Q/Q
BFV
5
0
|Q| [(kg m )
−3 1−n −1
s ]
−5
200
10
100
10
0
10
1
3
5
n
7
9
1
3
5
n
7
9
Figure 2.7: The panels in the first three rows show the best fit values (solid line),
the 1-σ (dark shade) and the 2-σ (light shade) confidence intervals for ΩΛ0 , H0 and
Q{QBF V at different value of n. In the panels of the last row, the dots represents
the best fit values QBF V and the solid line is given by eq. (2.4.3) where we have used
the best fit value found in section 2.4.1 (ñ 3{2q for Q̃ and ρΛ0 . For Q{QBF V , since
QBF V turns out to be always negative, Q{QBF V ¡ 0 corresponds to dark matter
decay and Q{QBF V 0 to dark energy decay. These results were obtained using
the combination of constraints C3 (µ+OHD+BAO) in (a), and the combination of
constraints C4 (µ+OHD+BAO+CMB) in (b).
deviation from the exponential relationship) but are more much important for the
54
constraints C3 . However, even in this case, the constraints obtained do not allow ρΛ
to decay from an initially large value to that observed today.
2.5
Conclusion
In this paper, motivated by the hope to shed some light on the coincidence and
the cosmological constant problems, we have applied the phenomenological decay law
ρ9 Λ
QρnΛ , originally proposed in ref. [4] to described the decay of dark energy into
radiation, to study the interaction between the dark sectors. In order to ameliorate
these two problems, we were primarily interested in the decay of dark energy into
dark matter (Q ¡ 0), but we have also considered the decay of dark matter into dark
energy (Q 0).
From dimensional analysis, we have shown that the model with n 3{2 (and only
this model) leads to solution where the ratio of energy densities ρm {ρΛ is constant at
late time. An important feature of this model is the possibility to have, in addition
to this late phase (which could explain the coincidence problem), an early phase
qualitatively similar to what is obtained in the ΛCDM model (which already provides
a good fit to the observational constraints).
To constrain the three free parameters of the model with n 3{2, i.e. the current
density parameter of dark energy (ΩΛ0 ), the late-time density parameter of dark en-
ergy (ΩΛ8 related to Q) and the current Hubble parameter (H0 ), we have mainly
followed the procedure of refs. [151–154] and considered the observational constraints
involving the distance modulus µ of type Ia supernova (SNeIa) and gamma-ray bursts
(GRB), the baryon acoustic oscillation (BAO), the cosmic microwave background
(CMB) and the observational Hubble rate (OHD). The constraints obtained from the
CMB data could be affected by a circularity problem. To constrain the ΛptqCDM
model with the CMB data, we have used some parameters obtained from the usual
ΛCDM model (Ωb0 , Ωγ0 , Ωr0 , z ). Although we have argued that it was legitimate
to do so, that could possibly lead to biased results. For this reason, we have presented our result considering two different combinations of observational constraints,
C3 (µ+OHD+BAO) and C4 (µ+OHD+BAO+CMB).
55
0.720, ΩΛ8 ¡ 1 corresponds to Q 0 and
For the constraints C4 , the best fit parameters are given by ΩΛ0
69.77 km s1Mpc1. Since ΩΛ8
ΩΛ8 1 to Q ¡ 0, the process involved at the best fit is the decay of dark matter into
1.006 and H0
dark energy and in fact exacerbates the coincidence and the cosmological constant
problems. We have shown that in order to explain both the coincidence and the
ΩΛ8 . The point lying in the
2-σ confidence region where these parameters are closest is ΩΛ 0.718, ΩΛ8 0.983.
For these values, ρΛ remains nearly constant (ρΛ {ρΛ 1.04) and the current value
of the ratio ρm {ρΛ is far from the late-time value (0.393 vs. 0.017).
For the constraints C3 , the best fit parameters (ΩΛ 0.753, ΩΛ8 1.099 and
H0 69.97 km s1 Mpc1 ) also correspond to the decay of dark matter into dark
energy. The closest point (in the 2-σ confidence region) to the line ΩΛ ΩΛ8 is
now ΩΛ 0.615, ΩΛ8 0.693. At this point, the value of ρΛ {ρΛ remains far
away from the desired value ( 3.37), but the current and late-time value of the
ratio ρm {ρΛ are now of the same order of magnitude (0.626 and 0.443, respectively).
cosmological constant problems, we need to have ΩΛ0
0
i
0
0
0
0
i
0
However, to provide a convincing explanation to the coincidence problem, we should
have ρm0 {ρΛ0
ρm8 {ρΛ8 , which is not the case here.
From a statistical point of view, we do not find that the addition of an extra
parameter was justified. The χ2min values obtained for the ΛptqCDM model were only
slightly smaller than the value obtained for the ΛCDM, 575.888 vs. 576.178 for C3
and 576.267 vs. 576.638 for C4 . In both cases, according to the Bayesian information
criterion (BIC), the improvement of the χ2min is considered to be insignificant (∆BIC
6.15 for C3 and ∆BIC
6.08 for C4).
Finally, we have extended our results to generic values of n ranging form 0 to
10. The values of χ2min and the corresponding values for ΩΛ0 and for H0 are nearly
constant, whereas the value of Q have a exponential dependence on n. We have shown
that these results may be understood in the limit of a small Q. Since the same values
of χ2min are found, the conclusion that we drew for the BIC holds for any value of n,
i.e. the improvement of the χ2min is insignificant. Moreover, even if we consider the
56
2σ-confidence region, the constraints obtained do not offer any explanation for the
cosmological constant problem.
Since when we consider the exclusion limits obtained from observational data, the
model studied in this article fails to provide an explanation to the coincidence and the
cosmological constant problems, and since this model is also disfavoured by the BIC,
we can conclude that the usual ΛCDM model remains the most reasonable description
of the Universe.
Acknowledgements
We would like to thank James Cline for helpful comments on the preliminary
version of this article. This work has been supported by the Fonds de recherche du
Québec - Nature et technologies (FQRNT) through its doctoral research scholarships
programme.
Chapter 3
Can ΩΛ remain constant to late
times?
The work presented in this chapter has been accepeted for publication in General
Relativity and Gravition [2]. We extend the work presented in the previous chapter to
the case to of spacetime with spatial curvature (κ
0).
The difference from the flat
case is not only quantitative since two different late-time solutions (instead of only
one) are now allowed.
Abstract
With the objective to find a solution to the cosmological constant and the coincidence
problems, we consider a cosmological model where the dark sectors are interacting
together through a phenomenological decay law ρ9 Λ
QρnΛ in a FRW spacetime with
spatial curvature. We show that the only value of n for which the late-time matter
ρm{ρΛ ) is constant (which could
provide an explanation to the coincidence problem) is n 3{2. For each value of Q,
energy density to dark energy density ratio (r m
there are two distinct solutions. One of them involves a spatial curvature approaching
0) and is stable when the interaction is weaker than a critical
?
value Q0 32πG. The other one allows for a non-negligible spatial curvature
(ρk 0) at late times and is stable when the interaction is stronger than Q0 . We
zero at late times (ρk
constrain the model parameters using various observational data (SNeIa, GRB, CMB,
BAO, OHD). The limits obtained on the parameters exclude the regions where the
57
58
cosmological constant problem is significantly ameliorated and do not allow for a
completely satisfying explanation for the coincidence problem.
3.1
Introduction
It is now more than a decade since the first observations of type Ia supernovae
suggesting that the Universe is currently experiencing a phase of accelerated expansion
were done [19–21]. Since then, improved measurement of supernovae distance [22,
24] and additional evidence based, for instance, on the measurement of the cosmic
microwave background [27–30, 39] or on the apparent size of the baryons acoustic
oscillations [126, 127] have led to the same conclusion. The ΛCDM model is currently
considered to be the most successful cosmological model by reason of its simplicity
and of the quality of the fit to the data that it provides. In this model, the Universe
is composed of, in addition to ordinary matter (radiation, baryon), a pressureless cold
dark matter fluid and a cosmological constant Λ, the simplest form of dark energy.
However, despite the excellent agreement with the observational data, the ΛCDM
model is facing two theoretical difficulties, namely the cosmological constant problem
[105] and the coincidence problem [168]. Regarding the first one, there is a discrepancy
123 orders of magnitude between the value of the energy density expected from
theoretic computation and the value inferred from observations (ρΛ {ρΛ 10123 ).
of
obs
th
As for the second one, according to the observations, the current values of the energy
densities of matter and of dark energy are of the same order of magnitude(ρm0 {ρΛ0
Op1q). This is not strictly incompatible with the model, but however, requires a fine
tuning of the initial conditions of the model.
A possible way to avoid these problems would be to replace the cosmological
constant Λ by a cosmological term, Λptq, which is allowed to vary in time (see for
instance [151–153, 155, 169–177] for some recent examples and [104, 123, 124] for a
review). Hence, it would be possible for the dark energy density to decrease from
an initial large value, consistent with the theoretic expectation, to a smaller one,
consistent with the current value inferred from the observations. In [1], we proposed
a phenomenological model, referred to the ΛptqCDM model, where the dark fluids
59
are interacting together in a flat spacetime with an energy transfer rate of the form
QΛ 9ρnΛ . We mainly focused on the case where n
3{2 since,
as we showed, it is
the only one for which the ratio of matter to dark energy densities remains constant
at late times (rm
ρm {ρΛ const).
This could have provided an explanation for
the coincidence problem since the current value of rm could thus become typical of
late times; however it turned out that the region of the parameter space where the
coincidence and the cosmological constant problems are solved (or at least significantly
alleviated) are excluded by the observational constraints.
From a more fundamental point of view, as suggested by Polyakov [140, 141, 180,
181], this interaction in the dark sector could find its origin in an instability of de
Sitter spacetime which would lead to a process where a fraction of the dark energy
is converted into particles. In [182], a concrete cosmological model based on the
Polyakov suggestion was built and it was argued that the interaction term should
have precisely the form considered in [1] (Q
Q0 ρ3Λ 2).
{
For this reason, we think it
is worth to complete the analysis presented in our previous work for a flat spacetime
by considering one with spatial curvature. As it will be shown, the difference from
the flat case is not only quantitative since two different late-time solutions (instead of
only one) are now allowed. Moreover, we had previously set the value of the energy
density of radiation to that obtained in the context of the ΛCDM model in order to
reduce the number of parameters to constrain. Here, we will consider this quantity
as a free parameter.
3.2
3.2.1
Models
Basic equations
In a Friedmann-Robertson-Walker (FRW) spacetime, if the Universe content is
modeled by perfect fluids, its continuity equation is given by
ρ9 3H pρ
pq,
(3.2.1)
a{a (a is the scale factor) stands for the Hubble term, ρ is the sum of the
°
energy density of each fluid (ρ i ρi ) and similarly, p is the sum of the pressure of
where H
9
each fluid (p ° p ). Defining Q ρ
60
3H pρi
pi q, we obtain a continuity equation
° Q 0. These equations can be more
for each fluid, subject to the the condition
i
i
i
9i
i
conveniently written as
ρ9 i
3H p1
i
wi qρi
(3.2.2)
Qi ,
pi{ρi ) depends on
the nature of the fluid (wm 0 for cold matter (dark and baryonic), wr 1{3 for
radiation and wΛ 1 for dark energy). As we can see from the previous equation,
where the value of the equation of state (EoS) parameter (wi
the variation of the energy density could be the result of two different mechanisms.
° Q 0, it must be interpreted as
The first term on the RHS represents the usual energy density dilution caused by the
cosmic expansion. As for the other term, since
i
i
a possible energy transfer between the fluids. A positive value (Qi
gain of energy for the fluid (source term), and negative value (Qi
¡ 0) constitutes a
0), a loss of energy
(sink term). In the ΛCDM model, the energy of each fluid is conserved separately,
i.e. Qi
0 for all of them.
In addition to eq. (3.2.2), to completely specify the time evolution we also need
the Friedmann equation, which takes its usual form1
H2
8πG
pρ
3
ρk q.
(3.2.3)
Now it remains only to specify the interaction terms Qi for the ΛptqCDM model.
Following our previous work [1], we will chose the interaction term between dark
energy and dark matter (Qdm
QΛ ) to be QΛ QρnΛ, where Q is a parameter to
constrain. In order to find an explanation to the coincidence problem, we will try to
find under which conditions, if any, it is possible to obtain a phase during which the
1
In order to have a more compact notation, we treat here the contribution of spatial
curvature as a fictitious fluid whose energy density is defined as ρk 3κ{8πGa2 .
The curvature parameter κ, whose dimensions are (length)2 , is negative for an open
Universe and positive for a closed one. In eq. (3.2.2), wk 1{3 and Qk 0. A
non-zero value for Qk would be inconsistent with the FRW metric.
61
ratio of the dark matter to the dark energy densities (rm
ρm{ρΛ ) remains constant
0). To simplify our analysis, we will first consider the case of an era dominated
by dark energy and matter (Λm-dominated era, ρΛ , ρm " ρk , ρr ), and subsequently,
(r9m
that of an era dominated by dark energy, matter and curvature (Λmk-dominated era,
ρΛ , ρm , ρk
" ρr).
3.2.2
r9m
0 during a Λm-dominated era
In [1], we have already shown that is impossible to have a constant ratio rm
during Λm-dominated era unless that n 3{2. Indeed, using the continuity equations
(eq. (3.2.2)) for dark energy and matter, we can show that
r9m
rm qρΛn1 .
3Hrm Qp1
If we suppose that at some point, rm reaches a constant value r̃m , hence r9 m
(3.2.4)
0 at
this point, and the parameter Q will be related to this value through
Q 3
r̃m
1 r̃m
HρΛ1n .
(3.2.5)
Since Q is a constant, the product Hρ1Λn must also be a constant. During a Λm-era,
the Hubble term may be written as
H
rm q
8πG
p1
3
1
2
1
ρΛ2 .
(3.2.6)
The plus sign stands for an expanding Universe and the minus sign for a contracting
one. We will only consider the former case (H
¡ 0).
Inserting this expression into
eq. (3.2.5), we see that the only consistent value for n is 3{2, which leads to
?
Q 24πG
p1
r̃m
r̃m q 2
1
.
(3.2.7)
The negative value for the interaction term implies that the energy transfer must
occur from dark energy to dark matter in order to reach a phase with a constant ratio
r̃m in an expanding Universe. If we invert this equation, we finally get an expression
62
for r̃m
r̃m
Q2
48πG
d
1
1
which is shown in figure 3.1.
In [1], we have shown that, provided that Q
experience a late phase with r9m
0.
96πG
Q2
(3.2.8)
,
0, a flat Universe will necessarily
For a non-flat Universe, this result does not
necessarily hold since the energy associated with the spatial curvature could possibly
become non-negligible before that this phase has been reached. Thus, we have to
determine if ρk decreases slower or faster than ρΛ and ρm . In the ΛCDM model, to
answer this question, we simply have to compare the EoS parameter of the fluids; a
smaller value implies that the energy density will decrease slower. In the ΛptqCDM
model, to take in account the effect of the energy transfer between the fluids, we need
to look at the effective EoS parameter, defined as wieff
Q
.
wi 3Hρ
The continuity
i
i
equation now takes the same form as in the ΛCDM model
ρ9 i
Since Qk
3H p1
wieff qρi .
0, wkeff 1{3 for the curvature.
(3.2.9)
For dark energy and matter, in the
case where rm is approaching r̃m , the effective EoS parameters become wΛeff
1{p1
wmeff r̃m q. Therefore, the energy density ρk will decrease faster than the two
other (wkeff
¡ wΛeff wmeff ) if r̃m 2 (or equivalently, if Ω̃Λ ¡ 1{3).
In this case the
approximation of a Λm-dominated era will remain accurate for ever. Conversely, ρk
¡ 2 (Ω̃Λ 1{3).
will decrease slower if r̃m
In this case, it would be possible to find
solutions where rm approaches r̃m for a certain time, but eventually the assumption of
a Λm-dominated era will become invalid. We then need to look what happen during
a Λmk-dominated era.
3.2.3
r9m
0 and rk 0 during a Λmk-dominated era
9
During a Λmk-dominated era, the Hubble term may be written as
H
8πG
3
p1
rm
rk q
1
2
1
ρΛ2 .
(3.2.10)
63
As in the previous section, we will consider only the case of an expanding Universe
¡ 0). Moreover, the product Hρ1Λn must still be a constant in order to have
rm 0 (c.f. eq. (3.2.5)). But now, this product involves the curvature to dark energy
?
3{2n
. For n 3{2,
density ratio (rk ρk {ρΛ ) and is proportional to 1 rm rk ρΛ
(H
9
this quantity will be a constant only if, in addition to the ratio rm , the ratio rk is also
a constant. In this case, it is useful to derive from eq. (3.2.2) an equation for the time
derivative of rk
r9k
Setting r9k
2Hrk QrkρΛn1 .
(3.2.11)
0 (rk r̃k), we obtain an equation analogous to eq. (3.2.5)
Q 2HρΛ1n .
(3.2.12)
These two expressions for the parameter Q must be equivalent and that will be the
case only if rm
2, which leads to
Q
2?
24πGpr̃k
3
3q 2 .
1
(3.2.13)
As in eq. (3.2.7), the interaction term is negative. Inverting this equation, we get
r̃k
3
Q2
32πG
1
(3.2.14)
.
This function is shown in figure 3.1. In this figure, one sees that for each value of
Q, there are actually two solutions for which r9m
0 (corresponding to r̃m and r̃k).
Hence, we have to find under which conditions one or the other solution (if any) will
be relevant. First of all, we can notice that the two solutions are equivalent when
prm, rk q pr̃m, r̃kq p2, 0q (which corresponds to Ω̃Λ 1{3). In this case, the interac?
tion parameter is given by Q0 32πG. It turns out that the cosmic evolution will
be qualitatively different depending on whether the strength of interaction is weaker
(|Q| |Q0 |) or stronger (|Q| ¡ |Q0 |) than this critical value.
In figure 3.2, two examples of trajectories in the plane rm
rk
are shown, one
where the interaction is weaker than Q0 and the other where it is stronger. In each
case, the plane is divided into two regions. In the figure, the boundary between
64
them is represented by a dashed line. For the region situated above this line, all the
trajectories end at the same point. For the weak case, this point corresponds to the
flat solution found in the previous section (prm , rk q pr̃m , 0q) and for the strong case,
to the non-flat solution found in the current section (prm , rk q p2, r̃kq). In the region
situated below the dashed line, the fate of the Universe will be the same no matter the
value of Q; it will eventually reach a point on the line rk
prm
1q (where H
0)
and then recollapse. As for the trajectories starting exactly on the boundary between
these two regions, they will also end in a point determined only by the value of Q, but
conversely to the upper region, this point corresponds now to the non-flat solution for
the weak case and to the flat solution for strong case. However, as we can see from
the figure, these solutions are unstable since any small perturbation which takes the
trajectory slightly away from the dashed line will make it diverge toward the stable
solutions in the upper region or toward a recollapsing point in the lower region. For
the strong case, the boundary between the two regions is set by the line rk
0, while
for the weak case, the boundary is entirely situated below this line (in this case we
cannot obtain an analytic expression to describe it). Thus, an open Universe (rk
will always evolve up to reach a stable point, while for a closed Universe (rk
¡ 0)
0), that
will be possible for a given initial point (rm0 , rk0 ) only if the interaction is sufficiently
weak (at least |Q| Q0 ).
Here we have to keep in mind that in order to be able to explain the coincidence
problem, it is not sufficient to find a solution for which the ratio rm becomes constant;
the current value of the ratio (rm0 ) must also be close to this constant value . For the
non-flat solution, this means that this value should be close to two (rm0
2).
This
value is so different from that obtained for the ΛCDM model (which already provides
a good fit to data) that it is reasonable to expect that this solution (and the larger
values of |Q| associated with it) will be excluded by the observational constraints. If
the larger values of |Q| are excluded, that will also affect the ability of the ΛptqCDM
model to solve the cosmological constant problem. Indeed, if the interaction is too
weak, that will not be possible for the dark energy density to decay from an initial
large value to the small one observed today. Actually, the condition rm
2 for the
65
non-flat solution results from the fact that, for n
3{2, in order to have a constant
value of rm , rk must also be a constant (cf. eq. (3.2.12)). It would then be interesting
to verify whether it is possible to obtain a solution with r9m
0 and rk 0 during a
9
Λmk-dominated era if we consider a different value of n. In other words, we would like
a
to check if it is possible to find a solution to Hρ1Λn 9 1
for n 3{2.
3.2.4
r9m
{ ptq const
rk ptqρΛ
3 2 n
r̃m
0 and rk 0 during a Λmk-dominated era?
9
To derive eq. (3.2.5), which implicitly implies that r9m
0, we have used the
continuity equations of dark energy and of matter. During a Λmk-dominated era,
the description of the Universe also involved the continuity equation of curvature and
the Friedmann equation. We can use these two equations to check whether there are
3{2 that are consistent with the condition rm 0.
other values than n
9
From the
Friedmann equation (eq. (3.2.10)), we get
ρk
In order to have r9m
3
H 2 p1
8πG
rm qρΛ .
(3.2.15)
0, the Hubble term must be given by eq. (3.2.5), i.e.
p1 r̃mqQ ρn1 .
H
Λ
3r̃m
(3.2.16)
Hence, differentiating eq. (3.2.15) and replacing rm by r̃m yields
ρ9 k
1qc
pn12πG
2
1
r̃m
r̃m
2 Q3 ρΛ3n3 rp1
r̃m qQs ρnΛ .
(3.2.17)
This expression has to be compared to the continuity equation of curvature (eq. (3.2.2)),
which becomes, using the expression of ρk and H given by respectively by eqs. (3.2.15)
and (3.2.16)
ρ9 k
1
36πG
1
r̃m
r̃m
3 3
Q
ρΛ3n 3
2
3
1
r̃m
r̃m
p1
r̃m qQ ρnΛ .
These two expressions for ρ9 k are equivalent in two cases: (r̃m
(r̃m
1,
(3.2.18)
2, n 3{2) and
n unfixed). We have already considered the first one in the previous
66
section. The second one must be rejected, since, in addition to involve a violation
of the weak energy condition (ρ
¥ 0) either for ρΛ or for ρm, this solution has been
obtained from a division by zero in eq. (3.2.5). Hence, we conclude that the only
0 during a Λmk-dominated era is n 3{2.
impossible to have simultaneously rm 0 and rk 0.
value leading to r9m
9
Consequently, it is
9
8
r̃m , r̃k
6
4
2
0
−2
1
Ω̃Λ
0.8
0.6
0.4
Ω̃ Λ =1/3
0.2
0
−2
10
−1
10
0
1
10
10
2
10
3
10
−q
Figure 3.1: In the upper panel, the functions r̃m (solid line) and r̃k (dashed line) are
shown as a function of the dimensionless parameter q Qp1{G1{2 q. In the lower panel,
the density parameter of dark energy ΩΛ p1 rm rk q1 is shown for the solution
found in section 3.2.2, prm , rk q pr̃m , 0q Ñ Ω̃Λ p1 r̃m q1 , (full line) and for the
solution found in section 3.2.3, prm , rk q p2, r̃k q Ñ Ω̃Λ p3 ?
r̃k q1 , (dashed line).
? In
both panels, the vertical line corresponds to the point Q0 32πG (q0 32π 10) where the two solutions are equivalent prm, rkq pr̃m, r̃kq p2, 0q.
67
q =0. 75q0
<
4
<
3
>
2
>
2
<
−3
0
1
2
3
4
5
rm
<
6
7
8
rk
<
<
<
<
<
<
<
<
10
−5
<
<
>
−4
9
<
>
−3
<
<
<
>
<
−4
<
−2
<
<
<
>
0
−1
<
>
−2
1
<
<
<
<
>
>
>
>
>
>
>
>
>>
>
>
>
3
<
>
1 >
>
0 >>
>>
>>
−1
−5
>
<
<
<
q =1. 25q0
5
<
>
4
rk
<
>
5
<
<
0
1
2
3
4
5
6
rm
7
8
9
10
Figure 3.2: Examples of trajectories in the plane rm rk for an expanding Universe
(H ¡ 0) where the radiation is neglected (ρr 0). In the left panel, the interaction
parameter is smaller (in magnitude) than the critical value q0 (q 0.75q0 ), and in
the right panel, it is larger (in magnitude) than q0 (q 1.25q0 ). The solution found
in section 3.2.2 for which r9m 0 (r̃k 0, r̃m given by eq. (3.2.7) is represented by
a square mark, and that found in section 3.2.3 (r̃m 2, r̃k given by eq. (3.2.13), by
a circular mark. The positive values of rk correspond to a negatively curved space
(open Universe) and the negative ones to a positively curved space (closed Universe).
The area under the line rk prm 1q corresponds to a non-physical region where
H 2 0.
3.3
Results and discussion
To assess the validity of the ΛptqCDM model, we have constrained the model
parameters using the methodology described in appendix A. For the ΛCDM model,
the continuity equations eq. (3.2.2) of the four fluids (dark energy, matter, radiation
and curvature) involve five parameters. They can be chosen as ρΛ0 , ρm0 , ρr0 , ρk0 and
H0 , where as usual, the subscript zero refers to the current value of these quantities.
However, due to the Friedmann equation, only four of them are independent. For the
ΛptqCDM model we have in addition to consider the interaction parameter Q. It will
be more convenient, but completely equivalent, to express our results in terms of the
following dimensionless parameters
Ωi
ρ ρiρ
k
,
h
H
,
100 km s1 Mpc1
q
1
1
G2
Q.
(3.3.1)
68
ΛCDM
ΛptqCDM
0.729
0.730
0.024
0.034
0.027 0.040
0.029 0.039
0.031 0.043
h0
ΛCDM
ΛptqCDM
0.698
0.698
Ωr0 (105 )
Ωm 0
ΩΛ0
0.006
0.009
0.006 0.009
0.007 0.010
0.007 0.010
0.274
0.283
0.027
0.039
0.020 0.028
0.053 0.075
0.031 0.039
8.55
9.07
0.87
q
χ2min
0
584.308
584.038
0.686
0.198 0.528
0.894 1.235
1.27
0.19 0.26
2.78 3.93
0.73 0.79
Ωk0 (102 )
00.912
0.281 00.637
01.009 01.457
01.975 02.395
1.250 04.246 05.714
Table 3.1: Best-fit values for the model parameters (ΩΛ0 , Ωm0 , Ωr0 , h0 and q) and the
corresponding χ2min for the ΛCDM and the ΛptqCDM models. The value for Ωk0 has
been computed using the relation Ωk 1 Ω. The limits are the extremal values of
the 1-σ and the 2-σ confidence regions (shown in figure 3.3 for the ΛptqCDM model).
Concretely, we have constrained the following five parameters: ΩΛ0 , Ωm0 , Ωr0 , h0 and
1, where Ω °ik Ωi.
q. The current value of the density parameter of curvature, Ωk0 , may be obtained
from the relation Ω
Ωk
The value of each parameter at the best-fit point and the corresponding χ2min are
shown for both models in table 3.1. The limits are the extremal values of the 1-σ
and the 2-σ confidence regions and they are shown in figure 3.3 for the ΛptqCDM
model. At the best-fit point, the results that we obtained for the ΛptqCDM model
are not too much different from those of the ΛCDM model. Indeed, the values of the
best-fit parameters of the ΛCDM model are all included in the 1-σ confidence region
of the ΛptqCDM and, except for the interaction parameter q, the converse is also true.
Moreover, in figure 3.4, we can see that the evolution history of each fluid is relatively
similar for each model.
For the ΛptqCDM model, the best-fit value of the interaction parameter indicates
that the decay of dark energy into dark matter (q
0) is favoured over the inverse
process. This result contrasts with that found in [1] where the best-fit point (for a
flat spacetime) was situated in the positive values of q. For q
¡ 0, the cosmological
constant problem becomes actually worse, since the dark energy density is increasing with time and concerning the coincidence problem, as it is clearly shown from
eq. (3.2.7) and eq. (3.2.13), the interaction parameter must be negative in order to
69
Ωr0 (×10− 4 )
Ωm 0
0.35
0.25
1.20
1.00
0.80
Ωk0
0.00
−0.05
h0
0.70
0.69
q
0.50
−0.50
−1.50
0.70
0.75
ΩΛ0
0.25
Ωm 0
0.35
0.80 1.00 1.20
−4
Ωr0 (×10
)
−0.05
Ωk0
0.00
0.69
0.70
h0
Figure 3.3: Projections of the 1-σ and 2-σ confidence regions obtained from observational constraints (c.f. appendix A) in the planes formed by the combination of two
of the following parameters: ΩΛ0 , Ωm0 , Ωr0 , Ωk0 , h0 and q. The best-fit values are
also indicated by a dot in each panel. The energy density parameter of curvature was
computed using the relation Ωk 1 Ω. A negative value (Ωk0 0) corresponds to a
positive spatial curvature (closed Universe) and a positive value (Ωk0 ¡ 0), to a negative space curvature (open Universe). In the panels where the interaction parameter
q is involved, the negative values (q 0) correspond to the decay of dark energy into
dark matter and the positive values (q ¡ 0), to the inverse process.
possibly have a solution with a constant matter to dark energy density ratio at late
times. In figure 3.3, we see that the presence of spatial curvature leads to an extension
for the interval of confidence of each parameters relative to the flat case represented
by the line Ωk0
0.
In the case of the matter energy density Ωm0 and the radiation
energy density Ωr0 , this extension is clearly larger in the positive direction, while for
the interaction term q, it is in the negative direction. This latter is important since it
allows for a large and negative value of q, which is more likely to solve the cosmological
and the coincidence problems.
70
Ωi
ΛCDM (BFV)
−8
−6
10
10
−4
−2
0
2
10
10
10
Time [Gyrs]
1
¡
¢
−1
mi n rm , rm
Λ(t)CDM (BFV)
Λ(t)CDM (2−σ)
1
0.8
0.6
0.4
0.2
0
10
−8
−6
10
10
−4
−2
t0=13.86 Gyrs
0.8
0
2
10
10
10
Time [Gyrs]
10
−8
−6
10
10
−4
−2
0
2
10
10
10
Time [Gyrs]
t0=13.92 Gyrs
10
t0=14.35 Gyrs
0.6
0.4
0.2
0
−1
10
1
10
Time [Gyrs]
3
10
−1
10
1
10
Time [Gyrs]
3
10
−1
10
1
10
Time [Gyrs]
3
10
Figure 3.4: Upper row: evolution of the energy density parameter of dark energy (solid
thin, blue), matter (dashed, red), radiation (dot-dashed, green) and curvature (solid
1q as a function of the
thick, black) as a function of the time. Lower row: min prm , rm
time. In each figure, the vertical line represents the current time t0 and its numerical
value is indicated in the lower row. The first two columns correspond respectively to
the best-fit values of the ΛCDM model and of the ΛptqCDM (c.f. table 3.1) and the
last row to the value of the parameters minimizing the quantity ∆rm {rm0 in the 2-σ
confidence region (c.f. figure 3.5).
In terms of the dimensionless parameter q, the critical value Q0 , which set the limit
between the two classes of solutions (flat and non-flat) becomes q0
?
32π 10.
The largest (in terms of magnitude) negative value which lies the 2-σ confidence region
is q
1.433;
thus only the flat solutions are relevant here. We can then obtain
the late-time value of rm from the value of the interaction parameter by inverting
eq. (3.2.7), as we did in figure 3.5. In this figure, we have also shown the points, in
the 1-σ and 2-σ confidence regions, which minimize the relative variation between the
current and the late-time value of the matter to dark energy density ratio, ∆rm {rm0
prm r̃mq{rm .
0 in order to explain
the coincidence problem (thus the current value of rm would be typical for t ¡ t0 ).
However, we get ∆rm {rm 0.70 in the 1-σ region and ∆rm {rm 0.63 in the 2-σ
region. Since for the ΛCDM model, ∆rm {rm Ñ 1, one can argue that the coincidence
problem is alleviated in the ΛptqCDM model. However, an interesting way to visualize
1 q as a function of
the coincidence problem is to plot the function F min prm , rm
0
0
Ideally, we would like to obtain ∆rm {rm0
0
0
0
71
the time (figure 3.4). For the ΛCDM model (at the best-fit point), this function is
characterized by an early and a late phase where F
median one where F
0 and forms a peak.
0 which are separated by a
The duration of this median phase is
very narrow in comparison to the entire Universe history and the coincidence problem
consists in the fact that we are currently situated in it. If we look now at the plot
for the point minimizing ∆rm {rm0 in 2-σ region in the ΛptqCDM model, we can see
0 and
a median one where F 0 and forms a peak. However, for the late phase, F 0
and becomes approximately constant (F Ñ r̃m 0.17). Since rm 0.47, the order
of magnitude of the current value of rm is now typical for t ¡ t0 . In this sense the
that the function F is characterized, as before, by an early phase where F
0
coincidence problem is alleviated. However, as was the case for the ΛCDM model, we
are currently situated in the median phase, which remains narrow compared to the
whole Universe history. Moreover, from the fluid evolution shown in figure 3.4, we
can see that at t0 , it is not only the energy densities of matter and of dark energy
that are of the same order of magnitude, but also that of curvature, which is actually
a triple coincidence problem. Hence, we can conclude that for the parameters range
which are consistent with the observations, the ΛptqCDM model fails to provide a
completely satisfying explanation to the coincidence problem.
For both models, the discrepancy between the observed and the predicted values
of the dark energy density remains roughly the same every everywhere in the 1 and
the 2-σ confidence regions (ρΛ0 {ρΛth
10123).
For the ΛptqCDM model, to explain
this result, we have made the hypothesis that the dark energy density could decrease
from an initial large value (ρΛi , evaluated at a 0), to the current observed one (ρΛ0 ).
Inside of the 1-σ confidence region, the largest value that we get for the ratio ρΛi {ρΛ0
is 1.4, and inside of the 2-σ region, ρΛi {ρΛ0
ρΛi {ρΛ0
10
123
1.6. These are very far from the ratio
needed to solve the cosmological constant problem. Hence, we can
conclude that for the values of parameters that are consistent with the observations,
ΛptqCDM is unable to provide an explanation to this problem.
Even if ΛptqCDM fails to provide an explanation to both of the cosmological
problems, we notice that the χ2min values obtained for the interacting model (584.038)
72
0.25
0.2
∆rm =0. 29
0.15
r̃m
∆rm =0. 31
0.1
0.05
0
0
0.1
0.2
0.3
rm0
0.4
0.5
0.6
0.7
Figure 3.5: Projections of the 1-σ and the 2-σ confidence regions obtained from
observational constraints in the plane rm0 r̃m . The parameter r̃m is obtained from
eq. (3.2.7) and the parameter rm0 is the current value of the matter to dark energy
density ratio (rm0 Ωm0 {ΩΛ0 ). The points minimizing the quantity ∆rm {rm0 in the
1-σ and the 2-σ confidence regions are indicated by the circular marks. The negative
values of r̃m which, inserted in eq. (3.2.7), leads to a positive values of Qr̃m are not
represented. Indeed, in this case r̃m can be used as a parameter to characterize the
decay of dark matter to dark energy but do not represent the physical ratio ρm {ρΛ
since we suppose that the interaction stops when all the dark matter has decayed
(rm 0).
is slightly better than that obtained for the ΛCDM model (584.308). However, the
interacting model involves an additional parameter to constrain (q), hence it is not
surprising that it provides a better fit to data. To take into account the different
number of parameters, the significance of the improvement of the χ2min value may be
assessed by the mean of the Bayesian information criterion [166], defined as
BIC 2 ln Lmax
K ln N
pχ2min
Cq
K ln N,
(3.3.2)
where Lmax is the maximum likelihood, K is the number of parameters for the model
(4, for the ΛCDM model, 5 for the ΛptqCDM), N the number of data points used in
the fit (N
646) and C a constant independent of the model used. Following [167], we
will regard a difference of 2 for the BIC as a non-significant, and of 6 or more as very
non-significant improvement of the χ2min value. Using the ΛCDM model as reference,
6.2. Hence the addition of an extra parameter is not warranted by the
marginal decrease in the value of χ2min . Since the ΛptqCDM model is not able either to
we get ∆BIC
provide a satisfying explanation to the cosmological and to the coincidence problems
we must conclude that the ΛCDM model remains the most satisfying one.
73
3.4
Conclusion
Despite of successes (simplicity, good fit to data), the ΛCDM model is not completely satisfying because of the existence of the coincidence problem, and more importantly, of the cosmological constant problem. These two problems have been actively
studied since the discovery of the accelerated expansion of the Universe, but none
of the proposed solutions has been able to convince unanimously the cosmological
community. In [1] a cosmological model where dark energy and dark matter interacts
through a term QΛ
for n
QρnΛ in flat spacetime was considered.
It has been shown that
3{2, this model could have provided an elegant solution to both problems,
but the values of the parameters required were excluded by observational constraints.
Given the importance of the cosmological constant and the coincidence problem,
we were motivated to complete the analysis of the model by checking whether this
result holds also in the presence of spatial curvature. We have shown that even in
that case, n
3{2 remains the only value for which it is possible to find a late-time
cosmology with a constant ratio of dark matter to dark energy density. Depending on
the strength of the interaction, the Universe will be nearly flat (ρk
or will admit spatial curvature (ρk
0) for |Q| |Q0|
0) for |Q| ¡ |Q0| at late times.
By constraining the model using observational data, we have found that within
the 2-σ confidence region, the cosmological constant problem remains as severe as in
the ΛCDM model. For the coincidence problem, the situation is different. It is now
possible to find points in the 2-σ confidence region for which the current value and
the late-time value of the ratio of dark matter to dark energy density are of the same
order of magnitude, mathcalO prm0 {r̃m q
1.
However, for these points, the current
0 and rm 0. Hence, we
cannot conclude that, in presence of spatial curvature, the ΛptqCDM model provides
time t0 is situated in the short time interval for which rm
a completely satisfying solution to the coincidence problem.
9
74
Acknowledgements
The author wants to thank James Cline for his comments on this manuscript.
This work was partly supported by the Fonds de recherche du Québec - Nature et
technologies (FQRNT) through the doctoral research scholarships programme.
Chapter 4
Dark energy in thermal equilibrium
with the cosmological horizon?
The work presented in this chapter was originally published in Physical Review D
[3]. Unlike to the two previous chapters, we do not consider a interaction term Q which
leads to a constant ratio of the energy density of dark energy and matter (ρm {ρΛ
const) at late time, but rather, which involves a thermal equilibrium between dark
energy and a cosmological horizon (Tde
Th).
Abstract
According to a generalization of black hole thermodynamics to a cosmological framework, it is possible to define a temperature for the cosmological horizon. The hypothesis of thermal equilibrium between the dark energy and the horizon has been
considered by many authors. We find the restrictions imposed by this hypothesis on
the energy transfer rate (Qi ) between the cosmological fluids, assuming that the temperature of the horizon has the form T
b{2πR, where R is the radius of the horizon.
We more specifically consider two types of dark energy: Chaplygin gas (CG) and dark
energy with a constant EoS parameter (wDE). In each case, we show that for a given
radius R, there is a unique term Qde that is consistent with thermal equilibrium. We
also consider the situation where, in addition to dark energy, other fluids (cold matter,
° Q 0).
radiation) are in thermal equilibrium with the horizon. We find that the interaction
terms required for this will generally violate energy conservation (
75
i
i
76
4.1
Introduction
In the late 1990s, observations of supernovae [19–21] have suggested that the Universe is undergoing a state of accelerated expansion. Since then, additional evidence
leading to the same conclusion have been found [22, 24, 27–30, 39, 126, 127]. In the
context of general relativity (GR), the equation of state (EoS) parameter of a fluid,
defined as the ratio of its pressure over its energy density (w
p{ρ), must be smaller
1{3 in order to be able to drive the accelerated expansion of the Universe.
Since normal matter satisfies the strong energy condition (w ¥ 0), this condition is
than
not fulfilled. Hence, two main approaches have been proposed to explain the acceleration: one of them consists to replace the GR by a modified gravity theory (see
e.g. Refs. [128–130]) and the other, to keep GR while introducing a new cosmic fluid,
known as dark energy (see Ref. [104] and references therein), endowed with a sufficiently large and negative EoS parameter (wde
1{3).
Alternatively, it has also
been proposed that the (apparent) acceleration could be only an artifact caused by
the spatial inhomogeneity of the Universe (see e.g. Refs. [80, 131]).
The ΛCDM model is the simplest cosmological model which provides a reasonably
good fit to the observational data. In this model, the two main components of the
Universe are currently a form of dark energy provided by a cosmological constant (Λ)
and a pressureless fluid known as cold dark matter (CDM). In addition to these two
fluids, the Universe is also composed of ordinary matter (radiation, baryons). However, despite the excellent agreement with the observational data, the ΛCDM model is
facing two theoretical difficulties. The most serious one concerns the value of the dark
energy density and is known as the cosmological constant problem [105]. Indeed, there
123 orders of magnitude between the value expected from theoretical computations and the value inferred from observations (ρde {ρde 10123 ).
is a discrepancy of
obs
th
The other one, dubbed coincidence problem [168], relies on the observation that the
values of the matter energy density and of the dark energy density are currently of the
same order of magnitude. Unlike to the previous problem, this is not incompatible
with the theory. However, since the matter energy density is diluted proportionally
77
to the volume of the Universe as it is expanding (ρm 9a3 ) while the dark energy density remains constant (ρde
const), the period of time during which ρm {ρde Op1q
corresponds to a very narrow window in the Universe history. To currently lie in
this window, a fine tuning of the initial conditions of the model is needed. However,
it is worth mentioning that about a decade before the discovery of the accelerated
expansion of the Universe, anthropic arguments were already addressing both problems [100, 105].
A possible way to circumvent these problems, without the recourse to anthropic
arguments, would be to allow the dark energy density to vary in time (ρde
const).
It would then be possible for the dark energy density to decrease from an initial
large value, consistent with the theoretical computation, to a smaller one, consistent
with the current value inferred from the observations. Moreover, that could also
extend duration of the period during which ρm {ρde
Op1q.
A variable dark energy
density could be obtained or by considering an EoS parameter wde different from 1,
either by allowing an energy transfer between the dark energy and another fluid (or
by considering these two ways together). Several forms of dark energy models have
been proposed, including quintessence [114], phantom fields [14], tachyon fields [190],
Chaplygin gas [13], agegraphic dark energy [191] and holographic dark energy [192],
to name few.
To study the thermodynamical implications of these models, the determination
of the dark energy temperature is a question that must inevitably be addressed. A
hypothesis often used [193, 194] is that the dark energy temperature is proportional
to that of the cosmological horizon (Tde 9Th ). Indeed, according to a generalization
of black-hole thermodynamics to a cosmological framework, it is possible to define a
temperature for the horizon which is related to its surface gravity (see section 4.3.1
for more details). A stronger hypothesis [5, 195–206], albeit more motivated, consists
in considering that the dark energy fluid and the horizon are in thermal equilibrium
(Tde
Th).
An argument presented in Ref. [5] and reused in Refs. [195–202] states
that if this were not the case, then the “energy would spontaneously flow between the
horizon and the fluid (or vice versa), something at variance with the FRW geometry”.
78
Following this argument, some authors [199–202] have even extended this hypothesis
to the other fluids, assuming that the thermal equilibrium between the horizon and a
given fluid must hold at least for late time.
Although this assumption may be questionable (especially in regard to its extension to other fluids), the objective of this paper is not to directly discuss of its
validity. Instead, we will demonstrate that in order to maintain thermal equilibrium
between a given fluid and the horizon, a specific energy transfer rate is required, which
constitutes a restrictive condition for its application.
4.2
4.2.1
Dynamics
Interacting fluids
In a Friedmann-Robertson-Walker (FRW) spacetime, the continuity equations for
a model allowing interactions between the different cosmic fluids (dark energy, dark
matter, baryonic matter and radiation) are given by
ρ9 i
3H p1
wi qρi
Qi .
(4.2.1)
If we treat the curvature as fictitious fluid, this equation can also be used to describe
the evolution of its energy density ρk
defined as H
3k{8πGa2
1
. Since the Hubble term is
a{a, where a is the scale factor, in absence of interaction (Qi 0),
9
the solution to this equation is
ρi
Here, we have set a0
ρi a3p1
wi
0
q.
(4.2.2)
1 (in this paper, the subscript 0 refers to the current value of
a variable). The Friedmann equations can be written as
H2
2
M3p
¸ρ ,
i
(4.2.3)
i
The curvature parameter k, whose dimensions are (length)2 , is negative for an
open Universe and positive for a closed one.
1
79
Mp2
H 2
9
¸
p1
wi qρi ,
(4.2.4)
i
p8πGq1{2 is the reduced Planck mass (throughout this chapter, we will
use a unit system where ~ kB c 1). The LHS of Eq. (4.2.1) has the same form as
in the noninteracting case, where H a{a (a is the scale factor) stands for the Hubble
term, ρi , for the energy density of a given fluid and wi pi {ρi (pi is the pressure), for
where Mp
9
the equation of state (EoS) parameter of this fluid. The values of these parameters
1{3), for curvature (wk 1{3) and, in absence
of interaction (see section 4.3.2), for dark and baryonic matter (wdm wb 0). For
dark energy, wde is not necessarily fixed to 1 as in the ΛCDM model and could even
are the usual ones for radiation (wr
be variable. The RHS of the equation represents the possible interactions between
the fluids. A positive value (Qi
¡ 0) represents a gain of energy for the fluid, and
0), a loss. The ensemble of these terms is subject to the energy
°
conservation condition i Qi 0. It is to be noticed that the interaction is allowed
negative value (Qi
only between the real fluids. For the curvature, the interaction term Qk must be zero,
otherwise it would imply that the curvature parameter k is variable, which would be
inconsistent with the FRW metric.
4.2.2
Types of dark energy
The exact nature of dark energy is not known and several models have been
proposed. In section 4.3.2, we will obtain an expression for the form of the interaction
term which is required to have a thermal equilibrium between a generic type of dark
energy and the cosmological horizon. To provide a specific example, we will consider
the case of the Chaplygin gas (CG)2 which was the first form of dark energy for which
2
We had previously also considered the example of holographic dark energy (HDE),
but the hypothesis of thermal equilibrium with the cosmological horizon is actually
not consistent for this form of dark energy. The entropy of the HDE is related to the
3{4
Bekenstein-Hawking entropy associated with the horizon through Shde SBH 9R3{2
[207, 208], where R is the horizon radius. We also know that in a volume V R3 , the
3
thermal entropy of an effective quantum field theory is given by Shde 9R3 Thde
[207].
80
the hypothesis of the thermal equilibrium with the horizon was considered [5]. A
Chaplygin gas [13] is a fluid for which its pressure and energy density are related
through
ρ2cg8
,
pcg ρcg
(4.2.5)
where the constant ρcg8 is the late-time value of ρcg . In absence of interaction (Qcg
0), the solution to the continuity equation is
ρcg
Equivalently, we can write
ρ2cg8
pρ2cg ρ2cg8 qa6,
0
(4.2.6)
2
(4.2.7)
1 pρ2 {ρ12 1qa6 .
cg
cg8
At early times (a ! 1), the CG behaves like cold matter (wcg 0) and at late times
(a " 1) like a cosmological constant (wcg 1) providing a unified form of dark
wcg
b
ρcg8
ρcg
0
matter and dark energy.
Since Eq. (4.2.7) is a function of the scale factor, as a complement to CG, we will
consider a second type of dark energy for which the EoS parameter has a fixed value
(wDE).
4.3
4.3.1
Thermodynamics
Cosmological horizon temperature
Since the seminal works of Hawking [209] and Bekenstein [210] in the seventies, the
thermodynamical properties of black holes have been widely studied. One of the most
well known feature is that, as consequence of the existence of an event horizon, the
stationary (or quasistationary) black holes behave like black bodies emitting thermal
radiation with a temperature proportional to the value of the surface gravity evaluated
Thus, the temperature of the HDE should scale as Thde 9R1{2 which is not consistent
with the hypothesis of thermal equilibrium since Thde Th 9R1 (see Eq. 4.3.9).
81
on the horizon
Th
κ
2π
.
(4.3.1)
A first extension of black hole thermodynamics to a cosmological framework was done
by Gibbons and Hawking in Ref. [211] by considering de Sitter space. In this case,
³
a3{Λ, thus the temperature is given by
the surface gravity on the event horizon is given by the inverse of the horizon radius
(RE
a
tend dt 2
),
t
a
κ 1{RE
Th
1
2πR
(4.3.2)
.
E
Unlike to de Sitter space, the event horizon is not always well defined for FRW space-
b
time. However, it has been argued [212, 213] that it is actually the apparent horizon
(RA
1{
H 2 13 Mp2 ρk ), and not the event horizon, that is responsible for Hawking
radiation (in the case of de Sitter space, the two horizons coincide). It worth mentioning that for de Sitter space, the event horizon radius has a constant value, while
for a FRW spacetime, the value of the apparent horizon radius varies. To compute
the surface gravity, this could be problematic. Indeed, this quantity is usually defined
in terms of Killing horizons, which work well in stationary (or quasistationary) situations. For the dynamical situations where no such horizons exist, several definitions
have been proposed (see [214, 215] for a review). If we consider a generic spherically
symmetric spacetime, the line element is given by
ds2
hab dxadxb
r̃ 2 dΩ2 ,
(4.3.3)
t, x1 r, r̃ aptqr and dΩ2 dθ2 sin2 θdφ2. For the FRW spacetime,
the 2-dimensional metric hab is given by diagp1, a2 {p1 kr 2 qq. A frequently used
where x0
The upper integration limit is given by tend 8 in an eternally expanding model
and by the time of the big crunch in a recollapsing model. This expression may also
a
be computed as RE a aend Hda2 a , where aend aptend q.
3
³
82
definition of the surface gravity has been proposed by Hayward in Ref. [216]:
1
κ ∇ ∇r̃
2
2?1h Ba
?
hhab Bbr̃
.
(4.3.4)
Here, the divergence and gradient refer to the two-dimensional space normal to the
spheres of symmetry. An evaluation of this expression at r̃
RA gives κ p1 ǫq{RA,
where ǫ R9 A {p2HRA q. Thus the horizon temperature is given by
Th
1ǫ
.
2πR
(4.3.5)
A
An alternative definition [194] for the dynamical surface gravity is
κ
where χ
hab Bar̃Bbr̃ 3 .
At r̃
1
r̃
,
B
r̃ χ 2
2
RA
(4.3.6)
RA, the surface gravity is then given by κ 1{RA,
and the horizon temperature by
Th
1
2πR
.
(4.3.7)
A
Among the papers where a thermal equilibrium between the horizon and the dark
energy is considered, both Eq. (4.3.5) [201–204] and Eq. (4.3.7) [196–199, 205] are
commonly used as a definitions of the horizon temperature. Although it has been argued [217] that the ǫ term can be neglected in certain situations, these two expressions
are generally different and one can wonder whether one definition is better motivated
than the other. In favor of Eq. (4.3.7), it was shown in Ref. [218], using the tunneling
approach, that an observer inside the apparent horizon of a FRW Universe will see a
thermal spectrum with a temperature given by Th
1{p2πRAq, without the extra ǫ
term. It is also interesting to notice that using this expression for the temperature, it
4
It is to be noticed that the radius of the apparent horizon, RA , is defined as the
value of r̃ for which the scalar χ vanishes (which implies that the vector ∇r̃ is null on
the apparent horizon surface).
83
is possible to recover the second Friedmann equation (Eq. (4.2.4)) from the first law of
thermodynamics [217]. Some authors still consider the event horizon as the relevant
one and use Eq. (4.3.2) to define the horizon temperature [5, 195, 200] (see however
Ref. [206] where Eq. (4.3.6) is evaluated at r̃
RE , which leads to Th RE {p2πRA2 q).
In Refs. [194, 198, 205], the horizon temperature is assumed to be proportional to its
de Sitter value, i.e.
Th
b
2πR
(4.3.8)
,
H
where b is a constant parameter and RH the Hubble radius (RH
1{|H |).
It would
be interesting to consider all these different definitions, but for the sake of conciseness
we will restrict our attention (while keeping in mind that there is no clear consensus
on how the horizon temperature should be defined and which horizon should be considered) to the case where the temperature has the dependence on the horizon radius
given by
b
.
2πR
Th
Here R could stand for, with b
(4.3.9)
1, the event horizon radius (Eq. (4.3.2)) and the
apparent horizon radius (Eq. (4.3.7)), as well for the Hubble radius (Eq. (4.3.8)).
4.3.2
Conditions for thermal equilibrium
To find the form of the energy transfer rate Qi required to maintain thermal
equilibrium between a fluid, whose the continuity equation is given by Eq. (4.2.1),
and the cosmological horizon, we will first derive an equation for the temperature
evolution for this fluid. Our derivation is similar to that presented in Ref. [219]. The
starting point is the Gibbs equation, Ti dSi
dEi
pi dV . For simplicity we will
consider a comoving volume of unit coordinate volume and hence a physical volume
of V
a3.
Since the energy of the fluid is given by Ei
Gibbs equation as
dSi
ρi T
i
pi
dV
V
dρi .
Ti
ρiV , we can rearrange the
(4.3.10)
84
From this expression for the entropy, we can show that the integrability condition
B B Si B B Si
BV BTi N ,V N ,T
BTi BV N ,T N ,V
i
implies that
Ti
i
i
i
Bpi pρ
i
BTi N ,V
pi q
V
i
i
i
Bρi BV N ,T
i
(4.3.11)
(4.3.12)
.
i
Except for the cases where the derivatives vanish or are ill defined (e.g. for the DE
in ΛCDM model) this equation is equivalent to
Ti
Bpi pρ
i
Bρi N ,V
i
B
Ti
BTi
pi q
V
Bρi N ,V
BV N ,ρ
i
i
(4.3.13)
.
i
Since we can express the temperature as a function of the volume and the energy den-
Tipρi , V q), its time derivative may be expressed as Ti pBTi{Bρiqρi pBTi{BV qV .
The time derivative of the physical volume V a3 is V 3HV ; then using also
9
sity (Ti
9
9
9
Eq. (4.2.1) to replace ρ9i , we get
Ti 3H pρi
9
Ti
BTi
B
pi q
V
Bρi N ,V
BV N ,ρ
i
i
Qi
i
BTi Bρi N ,V
(4.3.14)
i
The expression in the square brackets is identical to the RHS of Eq. (4.3.13); hence,
we can write the temperature evolution equation as
T9i
Ti
3H
B pi Bρi N ,V
Qi
Ti
i
BTi .
Bρi N ,V
(4.3.15)
i
Now to find the form of the energy transfer rate required to have thermal equilibrium (Q̃i ) between the cosmic fluid and the cosmological horizon (Ti
Th T q, we
must simply solve the preceding equation for Qi and replace the temperature by the
expression given by Eq. (4.3.9),
Q̃i
b
2π
Bρi BT N ,V
i
3HRpi R9
,
R2
1
(4.3.16)
85
where p1i
pBpi{BρiqN ,V .
i
In the peculiar case where the energy density of the
fluid depends only on the temperature, we can replace the first partial derivative
in Eq. (4.3.16) by a total derivative and write dρi {dT
simple manipulations, to
ρi{T . This leads, after some
9
9
R
.
Q̃i ρ9 i 1 3p1i H
(4.3.17)
R9
Using the continuity equation (Eq. (4.2.1)) to replace Q̃i , we obtain the following
differential equation
ρ9 i
ρi
1
p1i
wi
R9
.
R
(4.3.18)
which relates the evolution of the energy density to that of the horizon radius. In the
following sections, we will evaluate Q̃i for the two different types of dark energy (CG
and wDE), as well for relativistic and nonrelativistic matter. The energy density of
the relativistic matter depend only on the temperature. We will also assume that is
the case for the Chaplygin gas and the wDE. The only fluid for which we will not use
Eqs. (4.3.17) and (4.3.18) is the nonrelativistic matter.
Chaplygin gas
For the Chaplygin gas, the EoS parameter is given by wcg
derivative of the pressure by p1cg
ρ2cg8 {ρ2cg and the
wcg . Inserting these expressions in Eqs. (4.3.18)
and (4.3.17), we find that the energy density is given by
ρcg
ρcg8
and the interaction term by
Q̃cg
3p1
1
1
1
2
pρ2cg8 {ρ2cg 1qpR0{Rq2
(4.3.19)
0
wcg qHR
R
p1
1 qR9
wcg
ρcg .
(4.3.20)
The expression that we got for the energy density is different from that obtained
in absence of interaction (Eq. 4.2.7), but still consistent with an unified form of
dark energy and dark matter. Indeed, at late times (when R is large), the value
of the energy density approaches ρcg8 , and thus the Chaplygin gas behaves like a
cosmological constant (wcg Ñ 1, Q̃cg Ñ 0) and could drive the accelerated expansion
86
of the Universe. For ρcg8 ¡ ρcg0 , the value of the energy density is increasing in time
(as R is increasing) and ρcg8 represents the maximum value that can be reached.
In this case, the Chaplygin gas cannot play the role of dark matter (wcg 0) since
However, for ρcg8 ρcg0 , the energy density decreases in
time and ρcg8 represents the minimum value that can be reached, which means that
a
wcg ¥ 1 for all time. Actually, for radii smaller than Rptmin q R0 1 pρcg8 {ρcg0 q2 ,
wcg
¤ 1 for all time.
the energy density becomes imaginary. Hence, we must conclude that a thermal
equilibrium between the CG and the cosmological horizon is impossible at early time
(t tmin ). At Rmin , the value of the EoS parameter is wcg
0. Hence, providing that
thermal equilibrium is established after tmin , the solution that we found is consistent
with an unified form of dark energy and dark matter and the thermal equilibrium
hypothesis.
Dark energy with a constant EoS parameter
For a form of dark energy with a constant EoS parameter, p1wde
wwde . The
expressions for the energy density and the interaction term follow directly from
Eqs. (4.3.18) and (4.3.17)
ρwde
Q̃wde
3p1
ρde
0
R
R0
1 wwde
wwde
wwde qHR p1
R
(4.3.21)
,
1 qR9
wwde
ρwde .
(4.3.22)
1. We recover
the dark energy of the ΛCDM model for this value (ρwde const and Qwde 0), but
This expression is valid for any constant EoS parameter except wwde
we cannot conclude that thermal equilibrium with the horizon is possible for this type
of dark energy since, as was pointed after Eq. (4.3.12) our derivation is not valid for
a fluid whose energy density and pressure are intrinsically constant (in this case, we
can even ask whether a temperature can be meaningfully defined).
Other fluids
As mentioned above, some authors [199–205] considered the possibility that, in
addition to dark energy, other fluids could also be in thermal equilibrium with the
87
horizon. We will now consider the implications of this hypothesis. For an ultrarelativistic fluid (photons, neutrinos) the energy density and the pressure are given
by
4σTr4,
ρr
pr ,
3
(4.3.23)
ρr
(4.3.24)
where σ is the Stefan-Boltzmann constant. From Eq. (4.3.16), the interaction term
needed to maintain thermal equilibrium follows immediately:
Q̃r
4HR 4R9
ρr .
R
(4.3.25)
We note that by replacing the variables associated with dark energy in Eq. (4.3.22)
by those associated with radiation, we get the same expression. This is not surprising
since to obtain Eq. (4.3.22), we considered a fluid with a constant EoS parameter and
whose energy density depends only on the temperature, as is the case for radiation
(ρr
4σTr4, wr 1{3).
More generally, all the results of section 4.3.2 hold for any
fluid fulfilling these two conditions, which excludes however nonrelativistic matter. In
particular, Eq. (4.3.21) becomes for radiation
ρr
ρr
4
0
R
R0
.
(4.3.26)
For a nonrelativistic fluid, such as dark matter or baryonic matter, the energy
density and the pressure are given by
nm m
pm nm Tm ,
ρm
where nm
Nm {V
3
nm Tm ,
2
(4.3.27)
(4.3.28)
is the particle number density. Here we consider a single particle
species of mass m, but the generalization to many species is straightforward. Inserting
88
Eqs. (4.3.27) and (4.3.28) into Eq. (4.3.16) leads to
Q̃m
3wm HR 32 wm R9
ρm ,
R
(4.3.29)
where the EoS parameter is given by
wm
pρm m Tm3 T
m
2
(4.3.30)
.
m
Assuming that the rest-energy of the fluid is much larger than its kinetic energy
(m " Tm ), the EoS parameter may be approximated by wm
cold matter is usually considered to be pressureless (wm
Tm{m.
0).
Since wm
! 1,
However, we cannot
use this approximation here since that would imply, according to Eq. (4.3.29), that
Q̃m
0.
Using Eq. (4.3.9), the EoS parameter may be written more conveniently as
a function of the horizon radius
wm
where wm0
b{p2πmR0 q.
wm
0
R0
,
R
(4.3.31)
Inserting the interaction term Q̃m into the continuity
equation (4.2.1) and solving it yields
3
R0 R
ρm ρm0 a3 exp wm0
2
.
R
(4.3.32)
Now we must check whether the interaction terms found are consistent with the
energy conservation condition
°
Qi
° Q̃i
°
eq
Qineq
0.
The summation indices
ieq and ineq refer respectively to the fluids that are in thermal equilibrium with the
horizon, and to those that are not. In the case where at least one of the interacting
fluid is not in equilibrium, we can set
°
Qineq
° Q̃i
eq
in order to fulfill the energy
conservation condition. However, when all the interacting fluids are assumed to be
in thermal equilibrium we must have
expression for the Hubble rate
H
°
3 p1
°
°
Q̃ieq
0, from which we get the following
βieq ρieq
wieq δimeq qρieq
R9
,
R
(4.3.33)
89
where βi
p1
wcg q1 , p1 wwde q1 , 4 and 32 wm respectively for CG, wDE, radiation
and cold matter. The value of δim is 1 when i m and 0 otherwise. The energy den-
sity of the fluids in thermal equilibrium (Eqs. (4.3.19), (4.3.21), (4.3.26) and (4.3.32))
depends only on the horizon radius R and on the scale factor a (for cold matter);
hence, Eq. (4.3.33) can be integrated (at least numerically) in order to find the relationship between these two variables. However, the function Rpaq thus obtained
does not necessarily coincide with one of the three radii (RH , RA , RE ) considered in
section 4.3.1.
To illustrate the previous statement, we will consider the case where wDE and
radiation are in thermal equilibrium and are the only two interacting fluids. This
example is among the simpler to consider because Eq. (4.3.33), which becomes
H
p1
3p1
4ρr
4ρr
1 qρ
wwde
R9
wde
,
wwde qρwde R
(4.3.34)
can be integrated analytically. Inserting the expressions found for ρr and ρwde (Eqs. (4.3.21)
and (4.3.26)) gives
H
4rr0
4rr0
p1
3 p1
1
1 qR̃3wwde
wwde
1
wwde qR̃3wwde
Here, we have introduced the dimensionless radius R̃
R9̃
.
R̃
(4.3.35)
R{R0 and the radiation to
ρr {ρwde ) . Integration of Eq. (4.3.35) yields
4rr
3p1 wwde q
R̃.
(4.3.36)
a
4rr
3p1 wwde qR̃3w
dark energy density ratio at t0 (rr0
0
0
1
3
0
1
wde
0
By differentiating this equation, we find that the scale factor reaches a maximum
value amax at
R̃amax
14rr0 wwwde
wde
1
3 w 1
wde
.
(4.3.37)
Consistently, the expression for the Hubble rate given by Eq. (4.3.34) is zero at
R̃ R̃amax . The expression for the Hubble rate given by the first Friedmann equation
(Eq. (4.2.3)) must also be zero at this point. This condition reduces by one the number
of free parameters in the model. For instance, we can express the value of the energy
90
density of the spatial curvature as
ρk0
¸
i k
2
ρi a
(4.3.38)
,
R̃ R̃amax
where the energy density of the noninteracting fluids (i wde, r) is given by Eq. (4.2.2).
Not surprisingly for a cosmic scenario involving recollapse, we find that the spatial curvature is positive (ρk0
be chosen freely (provided that
0). The value of the remaining parameters can
°
ρi0
i k
ρk0
¥ 0, in order to have H0 P R) and
leads to a self-consistent cosmology where the radiation and wDE and are in thermal equilibrium with a cosmological horizon whose radius is implicitly defined in
Eq. (4.3.36). Now, we want to verify whether this radius coincides either with the
Hubble radius, the apparent radius or the event horizon radius. By solving the equation R̃ R̃H pR̃q 1{|H pR̃q| for the constant ρr0 , we get
ρr0
¸
ρi0 apR̃q3p1
wi
q
ρwde0 R̃p1
1
wwde
q 3M 2 R̃2
p
R̃4 .
(4.3.39)
i r,wde
Solving R̃ R̃A pR̃q for ρr0 leads to the same expression, except that now, the spatial
curvature is excluded from the summation (i k, r, wde). In both cases, we obtain an
expression for the constant ρr0 which is actually a function of R̃. This inconsistency
shows that R̃
R̃H
and R̃
R̃A.
For the event horizon radius, we cannot directly
compare R̃E to R̃ by reason of the integral involved in the definition of this radius.
However, we can compare its time derivative, which is
R9̃ E
RRE H R̃E RE1,
9
E0
0
(4.3.40)
91
to the expression for R9̃ obtained from Eq. (4.3.35). Solving R9̃
replacing R̃E by R̃ yields
RE0
?
3Mp
2
1 2wwde 3wwde
RE
9̃
for RE0 and
1
4r0 wwde R̃wwde 4 p1 wwde qR̃1
°
1 q .
p
1 wwde
3
p
1
w
q
4
i
ρi0 apR̃q
ρr0 R̃
ρwde0 R̃
ir,wde
(4.3.41)
Once again, we obtain an inconsistent equation where a constant is equal to a function
of R̃, showing that R̃
R̃E . Here we have shown that none of the three radius
definitions considered in section 4.3.1 could lead to thermal equilibrium between the
cosmological horizon, radiation and wDE if the other fluids are not interacting. More
generally, when a different combination of fluids is considered, we should proceed
similarly to this example and verify whether the radius obtained from Eq. (4.3.33) is
meaningful or not.
4.4
Summary
When the thermodynamical properties of dark energy are studied, the hypothesis
of (late time) thermal equilibrium between the cosmological horizon and the dark
energy fluid is frequently assumed [5, 195–206] and, in some cases, even extended to
other cosmological fluids [199–205]. The aim of this paper was to find the restriction
imposed by this hypothesis on the energy transfer rate (Qi ) between the fluids.
A first difficulty occurs in defining the temperature of the horizon. In a dynamical
spacetime, such as the FRW spacetime, there is no consensus for which horizon (if
any) should emit Hawking radiation and, for a given choice, what should be the
temperature associated with this radiation. In order to recover different expressions
used in the literature, we have considered a temperature of the form Th
b{2πR,
where R could stand for the Hubble radius (RH ) [194, 198, 205], for the apparent
radius (RA ) [196–199, 205] or for the event horizon radius pRE q [5, 195, 200].
A second difficulty is the unknown nature of dark energy.
We considered a
generic fluid to find the interaction term required to maintain thermal equilibrium
92
(Eq. (4.3.16)), but to go further in our analysis, we specialized to two specific types
of dark energy, namely Chapligyn gas (CG) and dark energy with a constant EoS
parameter (wDE). In both cases, we assumed that the energy density depends only
on the temperature. This leads to interaction terms given by Eq. (4.3.20) for CG and
Eq. (4.3.22) for wDE. These results illustrate that if we assume thermal equilibrium
between the dark energy and a horizon of radius R, we cannot choose the interaction
term Qi freely (if an other type of dark energy is considered, its interaction term
can be derived from Eq. (4.3.16), just as we did for CG and wDE). Conversely, if
we impose a specific choice for the interaction term, the radius R will be determined
by inverting these equations, which will not necessarily correspond to a physically
meaningful horizon.
Finally, we found the interaction terms for which radiation (Eq. (4.3.25)) and
° Q 0, it is nontrivial to propose
cold matter (Eq. (4.3.29)) are in thermal equilibrium with the horizon. Since the
ensemble of the interaction terms must satisfy
i
i
a cosmological model for which all the interacting fluids are in thermal equilibrium
with the horizon. Indeed, in this case, the horizon radius will be determined by
Eq. (4.3.33) and will not necessarily be physically meaningful. With this regard, the
hypothesis where the dark energy is the only fluid in thermal equilibrium with the
horizon is better motivated. Moreover, since the baryons and the photons densities
are tightly bound by the big bang nucleosynthesis (BBN) constraints and by the CMB
constraints, an interaction between dark energy and dark matter is more likely to be
consistent with the observational data. In this case, the interaction terms will be
given by Qde
Q̃de and Qdm Q̃de , where Q̃de is given by Eq. (4.3.20) for CG, by
Eq. (4.3.22) for wDE or by an analogous expression derived from Eq. (4.3.16) if an
other type of dark energy is considered.
To conclude, we can remind to the reader that it is possible to obtain dynamical dark energy without the recourse of an interaction with an other fluid if its
EoS parameter is different from -1. In particular, certain noninteracting models
involve a variable EoS parameter wde ptq (see [39, 120, 121, 220–222] and references
therein). In this case, the temperature evolution equation (Eq. 4.3.15) becomes
93
T9de {Tde
3H pwde
ρde B wde {B ρde q. As a future perspective, it would be interest-
ing to find under which conditions a thermal equilibrium with the horizon (Tde
Th)
is possible for these kind of models.
Acknowledgements
The author want to thank James Cline for his comments on this manuscript.
This work was partly supported by the Fonds de recherche du Québec - Nature et
technologies (FQRNT) through the doctoral research scholarships programme.
Chapter 5
Particle production from vacuum
energy in the causal diamond
In the previous three chapters, we have treated the decay of vacuum energy from
a phenomenological perspective. In the current one, we considered this problem from
a more fundamental point of view. We review the work presented in [7] where the
phenomenon of particle production from the vacuum energy is described by solving a
generalization of the Klein-Gordon equation in curved spacetime. We use this result
to check if the entropy released by the vacuum decay could affect the explanation
provided by the causal entropic principle for the observed small value of the current
dark energy density.
Abstract
The causal entropic principle (CEP) has been developed as a refined version of the
conventional anthropic principle to understand the observed value of the cosmological
constant. In this approach, the probability to observed the value of the cosmological
constant in the different vacua is assumed to be proportional to the total amount of
entropy produced in a causally connected region known as the causal diamond. Despite that less restrictive assumptions are considered than in the more conventional
anthropic approach, the observed value of the cosmological constant is in good agreement with the probability distribution obtained. Recently, increasing attention has
been dedicated to the study of particle creation from the vacuum energy. The aim
of this paper is to verify under which conditions this production of particle could
94
95
affect the results previously obtained. For the model considered, we show that to be
consistent with CEP, the duration of the particles production should not exceed an
upper bound determined by the mass of the particle created.
5.1
Introduction
Since the late 90s, an increasing amount of cosmological evidence [19–22, 24, 30,
39, 127] indicates that our universe is currently experiencing a phase of accelerated
expansion. The simplest explanation for this acceleration is provided by Einstein’s
cosmological constant Λ, to which correspond an energy density ρΛ
a negative pressure pΛ
ρΛ.
Λ{p8πq and
Assuming that the cosmological constant originates
from vacuum fluctuations, the theoretical expectation for the value of ρΛ is evaluated
by the sum of zero-point energy of quantum fields up to UV cut-off situated at the
Mpl ), which yields ρΛ Mpl4 1 (when no explicit units are
given, we set ~ G c kB 1). This is many orders of magnitude larger than
the value inferred from cosmological observations, ρΛ 1.25 10123 [223].
Planck scale (kmax
obs
This severe discrepancy, known as the cosmological constant problem, strengthens
the hypothesis that the energy density of the vacuum might not be uniquely determined from a fundamental theory, but is rather an environmental variable which takes
different values in different regions of the Universe. For instance, the string theory
landscape admits as many as 10500 vacua with different values of ρΛ [98, 99]. According to the anthropic principle (which is based on the tautological idea that to
measure any physical parameter, there must be observers to make the measurement),
the probability to observe one of these values depends up to which point the latter is
consistent with the existence of observers (see e.g. [224]).
More precisely, for a given distribution of ρΛ among the different vacua, our expectation for the value of the vacuum energy depends on two questions [225]: (1)
how the anthropic constraints affect this distribution and (2) how do we compare the
distinct observations of a diverging number of observers. To answer to the first question, we must first determine what are the constraints to consider, i.e. what are the
requirements for the existence of observers. In his seminal paper on the subject [100],
96
Weinberg only made the assumption that galaxies are a prerequisite for the existence
of observers and then concluded that the observed value of ρΛ must lie in a range
compatible with galaxy formation, estimated to be
ρΛ ρΛ 100ρΛ
obs
obs
time. However, following the discovery of dwarf galaxies at high redshift (z
at the
10),
this range has been modified, which has weakened the anthropic explanation for the
value of ρΛ . In reaction, to narrow the range of possible values, increasingly specific
conditions for life have been posited. However, since the nature of possible observers
is ambiguously defined, these assumptions may appear somewhat arbitrary.
Concerning the second question, it is reasonable to assume that the likelihood
to observe a given value of ρΛ does not only depend on the presence or the absence
of observers in the different vacua, but also on their number. The problem with
this approach is that the spatial volume of the vacua with non-compact geometry
(corresponding to a flat or to an open universe) is infinite, thus if these vacua admit
observers, their number will be infinite. It has then been proposed to weight the vacua
by the number of observers per baryon, per photon or per unit of mass [101, 226, 227].
However, the motivations behind these regularization schemes are questionable. For
instance, as pointed out in [6], if there is an infinite number of baryons, it is not clear
why it should matter how efficiently they are converted to observers. Moreover, in
vacua with different baryon-to-photon ratios, these weighting methods yield different
results.
The causal entropic principle (CEP) has been proposed in [6] to address in a more
satisfactory manner the two above questions and is based on two independent ideas.
(1) Since any act of observation requires free energy to be done, which leads to an
increase in the entropy, it is assumed that the latter is a good proxy for observers.
Thus, instead of counting the number of observers based on questionable conditions for
the emergence of life, the different vacua can be weighted according to the amount of
entropy produced in each of them. (2) To weight each vacuum, we should only consider
the number of observers or the entropy produced within a region causally accessible to
a single observer. This prescription is motivated by the physics of black holes where
such a local point of view should be adopted to preserve the linearity of quantum
97
mechanics [228–230]. More specifically, the region considered in [6], known as causal
diamond [231], is defined as the intersection of the past lightcone of a future endpoint
of a geodesic (at t
Ñ 8), with the future light-cone of a point corresponding to the
earliest time (ti ) where entropy is produced in the matter sector1 (this represents the
largest portion of spacetime that can be probed and across which matter can interact).
Other definitions of causal regions have been considered, for instance the causal patch
(defined by the past lightcone of a future endpoint of a geodesic) in [232], but the
results obtained are less consistent. Assuming that before the anthropic selection, all
the vacua are equally likely, the CEP simply asserts that the probability distribution
of ρΛ is given by the total amount of entropy produced in the causal diamond
»8
ds
dP
(5.1.1)
9
∆S dt Vc ptq .
dρΛ
dt
ti
Restricting to the vacua with ρΛ
¡ 0, the range of likely values of ρΛ could be more
easily visualized if we consider instead the probability distribution in log ρΛ
»8
ds
dP
9
ρΛ
dt Vc ptq .
d log ρΛ
dt
ti
(5.1.2)
The expression for comoving volume of the causal diamond, Vc , is given in eq. (5.2.5)
and ds{dt represents the rate of entropy production per comoving volume. After
having examined different sources of entropy production, it was found in [6] that the
most important one is the radiating dust heated by starlight. As shown in figure 5.3,
the resulting probability distribution turn out to be in excellent agreement with the
observed value of ρΛ . This is commented with more details in section 5.3. In [6],
the analysis has been restricted to the sublandscape of positive values of ρΛ with
1
The entropy produced in the matter sector (massive particles, radiation) has to
be distinguish of the Bekenstein-Hawking entropy associated with black holes and
cosmological horizons. The latter could be actually quite large, but does not seem
relevant to the existence of observers. Moreover, by definition, these horizons are not
located inside the causal diamond, thus their entropy should not be taken in account
to weight the vacua.
98
all the other parameters fixed to the values measured in our own universe, but has
since then been extended to include, for instance, the distribution probability over
the density contrast, the spatial curvature, and to negative values of ρΛ [232–235].
The effect of inhomogeneities are discussed in [236]. In [235–237], the annihilation or
the decay of dark matter has been considered as a possible competitor to radiating
dust as dominant source of entropy production. A critical point of view on the CEP
may be found in [238, 239].
Alternatively to the environmental approach, it has also been hypothesized that
the energy density of the vacuum could be a dynamical term evolving in time to
explain the cosmological constant problem. It is known since the pioneering works of
Parker [240–243] (see [244] for a review of the early work on the subject) that quantum
effects in curved spacetime could lead to the spontaneous creation of particles from the
vacuum energy. There is a hope that this effect could lead to a dynamical relaxation of
the value of ρΛ , from an initial large value, consistent with the theoretical expectation,
to a smaller one, consistent with the current value inferred from the observations. The
development of a realistic cosmological model based on this mechanism is not a simple
task. As a first step in this direction, the simpler case of particles creation in a de
Sitter spacetime has been studied in [7, 112, 113, 137, 140, 141, 180, 245–247].
The environmental approach to explain the observed value of ρΛ and the mechanism where particles are created from the vacuum energy are not mutually exclusive.
However, such a process could release a large amount of entropy and thus possibly
modify the probability distribution for ρΛ obtained from the CEP. We will check
whether it is the case for the model presented in section VII of [7] (free, massive, and
conformally couple scalar field). For this, we will firstly use the results found in [7] to
obtain an expression for the entropy production rate. Although this expression has
been obtained from computations involving a de Sitter spacetime (involving only vacuum energy), we will directly transpose it in the more realistic cosmology described
in section 2 (involving vacuum energy and matter). This is of course a rough approximation, but in the absence of a more realistic model, this will provide a first insight
in how the particle production from the vacuum energy could affect the CEP.
99
This paper is divided as follows. In section 2, we found an expression for the
comoving volume of the causal diamond for an universe containing vacuum energy
and matter. In section 3, we briefly reviewed the results obtained from the CEP
with the radiating dust by starlight as dominant source of entropy. In section 4, we
compute the entropy produced from the particles creation from the vacuum energy
and compare the results to those obtained for the radiating dust.
5.2
Causal diamond
The line element of the FRW spacetime is given by
ds2
where
d2 x dt2 aptq2 d2x,
d2 r
1 κr 2
r 2 pdθ2
The curvature parameter κ can take the values
(5.2.1)
sin2 θdφ2 q.
(5.2.2)
1, 0 or 1, respectively for an open,
a flat or a closed universe. Since the causal diamond is defined as the intersection of
³
two lightcones, it will be convenient to introduce the conformal time τ ptq dt{aptq
and write the line element as
ds2
apτ q2 d2τ
d2 x .
In this coordinate system, the radial light-rays obey dτ
(5.2.3)
d|x|, hence the comoving
radius of the causal diamond is simply given by
r ptq where ∆τ
∆τ
2
∆τ
2
τ ,
τ p8q τ p0q is the total conformal lifetime of the universe.
(5.2.4)
The volume
of the causal diamond is given by
Vc ptq The maximum value of Vc is reached at τmax
4π 3
r .
3
(5.2.5)
∆τ {2. To compute the conformal time
τ , we must determine what is the time dependence of the scale factor. In a FRW
100
spacetime, its evolution and that of the energy density of the different cosmic fluids,
ρi , are described by the coupled equations
H ptq2
8π3
¸
ρi ρ9 i ptq 3H p1
where H
a{a is the Hubble
9
κ
,
a2
(5.2.6)
wi qρi ,
(5.2.7)
parameter and wΛ
1,
wm
0 and wr
1{3,
respectively for vacuum energy, cold matter and radiation. In [6], a flat geometry
was assumed (κ 0). Moreover, it was also shown that the radiation can be neglected (ρr
0) without affecting significantly the results obtained. This leads to the
following solution for the scale factor and the total energy density
aptq a0 sinh {
2 3
where tΛ
ρtot ptq ρΛ
a
3t
2tΛ
(5.2.8)
,
ρm ptq ρΛ coth
2
3t
2tΛ
(5.2.9)
,
{
3{p8πρΛ q. In [6], a0 was set to tΛ , but the physics does not depends on
2 3
the value of this parameter. For the scale factor given by eq. (5.2.8), the conformal
time is given by
τ ptq { cosh2{3
1 3
tΛ
3t
2tΛ
2 F1
5 1 4
, ; ; cosh2
6 3 3
3t
2tΛ
,
(5.2.10)
where 2 F1 is the Gaussian hypergeometric function. The evolution in time of Vc is
shown in figure 5.1 for three different values of ρΛ . In each case, the volume reaches
a maximum Vmax
11.6tΛ at tmax 0.23tΛ.
conformal lifetime of the universe is finite, ∆τ
It is straightforward to verify that the
N∆tΛ1{3 , with N∆ 2.804.
101
−110
x 10
5
0. 1ρΛo bs
Vc [ Mpc3 ]
4
3
2
ρΛo bs
1
10ρΛo bs
0
0
5
10
15
20
25
t [ Gyr]
Figure 5.1: Comoving 3-volume Vc of the causal diamond (with a0
of time for ρΛ p0.1, 1, 10q times ρΛobs .
5.3
tΛ2{3) as a function
Entropy produced from radiating dust
The rate of entropy production resulting from the scattering of starlight by dust
can be computed as
ds
dt
»t
0
dt1 Γpt t1 qρ9 Æ pt1 q.
(5.3.1)
The function Γpt t1 q represents the rate of entropy production per stellar mass of a
stellar population born at a time t1
t (the details of the computation may be found
in [6, 232, 235]) and ρ9 Æ is the stellar formation rate (SFR) which is defined as the
rate of stellar mass production per comoving volume. In [6], two different models of
SFR [248, 249] were considered to illustrate the dependence of the density probability
on the time dependence of star formation. In [235], a third model [250] is considered
to vary the SFR in response to changes in the vacuum energy density, and also to
changes in the amplitude of the cosmological perturbation. Finally, in [232] a model of
SFR specifically developed for multiverses [251] was used. When only positive values
of ρΛ are considered and all the other parameters are fixed to the values measured in
our own universe, the densities probability obtained from these different SFR models
102
are, on a qualitative level, relatively similar. Thus, for simplicity, we will only show
the results obtained from the SFR model developed in [248], which is the simplest to
compute. The rate of entropy production in our own universe is shown in figure 5.2
and is peaked at t 2.3 Gyr. This corresponds to the time where the birth and the
death rates of stars are equal (the peak of the SFR occurs earlier, at t 1.7 Gyr). The
tail of the function vanishes slowly due the presence of low mass stars which maintain
the entropy production for a long time. In other vacua, the different values of ρΛ affect
the fraction of baryons that are incorporated in the halos. However, in the approach
adopted in [6], it was shown that the SFR is mildly affected by this effect (the position
of the peaks remains roughly the same). The corresponding probability distribution
is shown in figure 5.3. The CEP successfully explains the value measured in our own
1.25 10123) which lies within the 1-σ error band located around
the median value of the distribution (ρΛ 5.6 10123 ). The preferred values for ρΛ
universe (ρΛobs
are those for which tmax (the time where the Vc is maximum) is close to the peak of
entropy production (t 2.3 Gyr) since it is for those cases that the causal diamonds
capture more entropy in proportion to their sizes. In [232,235], the models considered
allow a more accurate treatment of the dependence of the SFR on the value of ρΛ ,
but it remains true that the preferred values of ρΛ are those for which tmax and the
peak of the SFR roughly coincide.
103
193
6
x 10
ds/dt [ yr−1 Mpc−3 ]
5
4
3
2
1
0
0
5
10
15
20
25
t [ Gyr]
Figure 5.2: Entropy production rate per unit of comoving volume (with a0 tΛ ) in
our universe as a function of time, from dust heated by starlight, based on the SFR
model developed in [248].
2{3
104
0.4
0.35
0.3
dP/dl og ρΛ
0.25
0.2
0.15
0.1
0.05
0
−126
−125
−124
−123
−122
−121
−120
l og ρΛ
Figure 5.3: Probability distribution over log ρΛ computed from the CEP considering only the dust heated by starlight as source of entropy. The computation was
based on the SFR model developed in [248]. The median value of the distribution (ρΛ 5.6 10123 ) is indicated by the dotted line with the 1-σ error band
(shaded area). The dashed line corresponds to the value observed in our own universe
(ρΛobs 1.25 10123 ).
105
5.4
Entropy produced by particle production from
vacuum energy
Our computation of the entropy released from the production of massive particles
from the vacuum energy is based on the expression found in [7] for the number density
of particles produced (eq. (5.4.18) of the current article). We will start by reviewing
how this result was obtained and then find an explicit expression for ds{dt to make a
comparison with the entropy released by the radiating dust.
The Lagrangian density of a neutral scalar field Φpt, xq with a mass m and a
curvature coupling ξ in an arbitrary spacetime is given by (see e.g. [252])
L
1a
|
g | g µν Bµ ΦBν Φ m2 Φ2 ξRΦ2 ,
2
(5.4.1)
where R is the Ricci curvature scalar. Applying the Euler-Lagrange equation leads
to the generalization of the Klein-Gordon equation
pl m2 ξRqΦ 0,
(5.4.2)
with the d’Alembertian operator given by
l a1 Bµ
|g|
a
|g|gµν Bν
(5.4.3)
.
For a FRW spacetime (eq. (5.2.1)), eq. (5.4.2) becomes
B2
Bt2
B ∇2
3H 2
Bt a
2
m
ξR Φ 0.
(5.4.4)
The operator ∇2 is the spatial part of the Dalembertian operator and is equivalent
to the usual Laplacian operator (∇2
BiBi) in a flat spacetime.
To solve eq. (5.4.4)
it is convenient to decompose the scalar field Φ in terms of creation and annihilation
operators
Φ
»
rdks
ak φk ptqYk pxq
a:k φk ptqYk pxq .
The functions Yk are the solutions of the equation ∇2 Yk
(5.4.5)
pκ k2 qYk where k |k|.
The expressions for the integration measure rdks depend on the value of the curvature
106
parameter κ and may be found in [252,253] (for a flat spacetime, it is simply given by
rdks d3k). Inserting the decomposed expression for Φ into eq. (5.4.4) and defining
fk a3{2 φk , we obtain
fk ωk2ptqfk 0,
(5.4.6)
:
which has the form of a harmonic oscillator with a time dependent angular frequency
ωk2
ptq k2 κ
a2
2
m
ξ
1
6
R
1
2
9
H
H2
2
In [7], the scale factor and the curvature coupling were taken to be ξ
coupling) and a
eHt et{t
.
(5.4.7)
1{6 (conformal
(expanding patch of a de Sitter spacetime). If matter
Λ
is created from the vacuum energy, the expression for the scale factor is expected to
change. However, it was shown that the energy density of the particles created is
redshifted sufficiently fast to set ρm
0, in the case where the back reaction on ρΛ is
neglected. Then, eq. (5.4.7) becomes
ωk2 ptq k2
γ2
,
t2Λ
e2t{tΛ
(5.4.8)
m2 t2Λ 1{4. Following [7], we will only consider the case where the particles
created are “massive”, i.e. γ ¡ 0. It is easy to verify by direct replacement that the
two functions
π
e
e
eπγ Jiγ pz q Jiγ pz q ,
(5.4.9)
fkin ptq H e2πγ 1
2iγ Γp1 iγ q
?
J pz q,
(5.4.10)
f out ptq with γ 2
πγ
2
iπ
4
k
iγ
2Hγ
are solutions of eq. (5.4.6) for the frequency given by eq. (5.4.8). The functions Jiz pz q
are Bessel functions of the first kind and z
pk{H q exppHtq. The labels in and out
refer to the fact that these solutions are the normalized adiabatic vacuum solutions
in the infinite past and in the infinite future. More explicitly
lim fkin ptq Ñ8
1
?2ω
lim fkout ptq ?1
t
Ñ8
t
eiΘk ,
(5.4.11)
eiΘk ,
(5.4.12)
k
2ωk
107
with
Θk ptq »
dt ωk ptq.
(5.4.13)
The in and the out solutions are related through the Bogoliubov transformation
fkin
αfkout
βfkout , where
?2πγ e
e
,
(5.4.14)
α iγ
2πγ
2 Γp1 iγ q e 1
?2πγ e
e
β iγ
.
(5.4.15)
2πγ
2 Γp1 iγ q e 1
As usual, it can be verified that |α|2 |β |2 1. A non-zero value for β implies that
iπ
4
3πγ
2
iπ
4
πγ
2
a process of particle creation happens. In a general situation, the expression for the
mean adiabatic particle number in the mode k is given by
Nk
Nk p1
2Nk q|βk ptq|2 ,
where Nk , is the particle number at a time t0 for which |βk pt0 q|2
(5.4.16)
0.
If we assume
that there are no particles at past infinity, the particle number in the far future is
simply given by Nk
|β |2 pe2πγ 1q1.
This nonzero value was expected, since
eqs. (5.4.6) and (5.4.8) described an over the barrier scattering. Thus an incident
wave interacting with this potential will not be totally transmitted; a part will also
be reflected. The value of Nk ptq for an arbitrary time t may be computed using the
WKB approximation as was done in [7]. However, it was also shown that most of the
particles produced in a given mode are created at a time located around
tk
tΛ ln
tΛ k
γ
,
(5.4.17)
which corresponds to the real part of the complex time t̃k for which ωk pt̃k q
0.
Hence, to a first approximation, we can consider that for each mode, all the particles
are produced by a single event occurring at tk and thus, we can use a step-function
to describe the number density of particles,
Nk
θpt tk q|β |2.
(5.4.18)
108
The corresponding entropy density is given by (see e.g. [254, 255])
s
1
p2πq3
»
d3 k rp1
Nk q lnp1
Nk q Nk ln Nk s .
(5.4.19)
The entropy density per mode is obtained by performing the angular part of the
integral, which yields
ds
dk
2
k
2π
rp1
2
Nk q lnp1
Nk q Nk ln Nk s .
(5.4.20)
Since we are approximating the particle production by a step function, all the entropy
produced in a mode k will enter in the causal diamond at t
tk .
To simplify the
notation, we define the constant sk as
sk
p1 |β |2|q lnp1 |β |2q |β |2 ln |β |2,
and thus we have
ds dk ttk
2πsk2 k2.
(5.4.21)
(5.4.22)
The time at which the particles are created and their wave number are related through
eq. (5.4.17), so eq. (5.4.21) can be equivalently written as
ds
dt
ds dk
dk ttk dtk
3
γ sk 3
2π
a.
2 t4
(5.4.23)
Λ
The total amount of entropy produced in the causal diamond is then given by
∆S
γ 3 sk
2π 2 t4Λ
8
»
0
a3 ptqVc ptqdt
(5.4.24)
The integral in this expression corresponds to the total physical 4-volume of the
causal diamond which diverges. However, in a more realistic model, the particle
creation could not last for ever since the value of ρΛ is expected to decrease as energy
is used to create particles and eventually, the energy available will not be sufficient
to maintain the production. We will then set the upper bound of the integral to toff ,
the time at which the particle production stops. The probability distribution is then
109
given by
3
dP
N∆τ
γ 3 sk
9
I,
d log ρΛ 48π 2 t2Λ
(5.4.25)
where I represents the integral over the dimensionless variable x 3t{2tΛ ,
I
» xoff
0
sinh2 x 1 1
2
N∆τ
2 F1
xq 3
5 1 4
, ; ; cosh 2
6 3 3
2
cosh 3 x
The probability distribution for the limiting cases where toff
dx.
(5.4.26)
" tΛ and toff ! tΛ (in
the vicinity of the peak of the distribution) are shown in figure 5.4. In both cases,
the qualitative features of the distribution are similar. As expected, the probability
density vanishes as ρΛ approaches zero, since the energy available to created particles becomes insufficient (|β |2 Ñ 0). The probability density also vanishes when ρΛ
approaches the value for which γ
0.
In this case, the particle production becomes
very large, but the size of the causal diamond, which is proportional to ρΛ1 , decreases
faster. Between, the distribution reaches a maximum whose position depends on the
mass of the particles produced. The median value of the distribution depends weakly
on the value of toff and is given by ρΛ
0.290m2 for toff ! tΛ and by ρΛ 0.250m2
" tΛ . Thus, the predicted and observed values of ρΛ
m 7 1062 1.2 1034 eV.
for toff
will be comparable if
This mass would be unusually small. For comparison, the masses of the known
particles range from 511 keV for the electron to 173 GeV for the top quark. The
values predicted for ρΛ for more realistic values of mass are many orders of magnitude
larger than ρΛobs . For instance, for m 8.19 1029
distribution is ρΛ
2 1057.
1eV, the median value of the
Thus, it seems unlikely that the CEP and the entropy produced from the particle
creation from the vacuum energy would provide an explanation for the observed value
of ρΛ . Moreover, this could seriously endanger the CEP if the amount of entropy
produced exceeds that released by the radiating dust. To check under which conditions
110
that will be the case, it is convenient to define
pρΛ∆S qppeak
Rpeak pm, toff q pρΛ∆S qdpeak ,
(5.4.27)
which corresponds to the ratio of the peak of ρΛ ∆S for the entropy released by
the creation of particles (p) and by the radiating dust (d). Roughly speaking, if
Rpeak
1, the radiating dust will be the dominant process, and if Rpeak
¡
1, it
will be the particle creation. Thus, the CEP set an upper bound on the duration of
the period during which the particle creation could occur, t̃off , defined as the time
for which Rpeak pm, t̃off q
1 (for a longer duration, more entropy will be produced).
The value of t̃off as a function of the mass is shown in figure 5.5 (right panel) and is
approximately given by
t̃off
10
17
s
1eV
m
3
(5.4.28)
For a light a particle with a mass of 1eV, the probability distribution obtained from
the CEP and the entropy released by the radiating dust will remain unaffected unless
the particle production lasts for more than t̃off
3 Gyr.
For heavier particles, the
limit on the duration of the particles production is far more restrictive. For instance,
for m
100 GeV, the limit is t̃off 1016s.
For such a small time, our computation
of the size of the causal diamond is obviously incorrect. However, on qualitative level,
this result implies that if particles with this mass are created from the vacuum energy,
their production should be confined to the early moment of the universe.
3
3
2.5
2.5
2
2
dP/dl og ρΛ
dP/dl og ρΛ
111
1.5
1
0.5
0
−1.6
1.5
1
0.5
−1.4
−1.2
−1
−0.8
−0.6
0
−1.6
−0.4
−1.4
−1.2
l og (ρΛ /m2 )
−1
−0.8
−0.6
−0.4
l og (ρΛ /m2 )
Figure 5.4: Probability distribution over log pρΛ {m2 q computed from the CEP considering particle production from vacuum energy as the only source of entropy. The
median value of the distribution is indicated by the dotted line with the 1-σ error
band (shaded area). The left and the right panels correspond respectively to the limit
where toff ! tΛ and toff " tΛ in the vicinity of the peak. The median value of the distribution (ρΛ 0.290m2 for toff ! tΛ and ρΛ 0.250m2 for toff " tΛ ) is indicated by
the dotted line with the 1-σ error band (shaded area). The dot-dashed line represents
the value of log ρΛ {m2 for which γ 0.
20
3
1 eV
V
1 Ke
V
1 Me
1 Ge
1 Te
2
10
V
10
V
10
−55
10
10
1
−50
t̃ off [ s]
0
10
0
Log ρΛ
Rpeak
10
10
−45
−1
10
−40
−10
10
−2
10
−35
−20
−3
10
−20
10
−10
10
0
10
t off [ s ]
10
10
20
10
10
0
10
2
10
4
10
6
10
8
10
10
10
12
10
m[ eV]
Figure 5.5: Left panel: Rpeak as a function of toff for different values of mass. Right
panel: t̃off as a function of mass. t̃off is defined as the time for which Rpeak 1, i.e.
the point curve on the left panel intercepts the gray line. The location of the peak
over log ρΛ , corresponding to t̃off and m, is also shown on the right edge of the right
panel.
112
5.5
Summary
According to the causal CEP, the probability to observe a given value for the
vacuum energy density, ρΛ , is proportional to the amount of entropy produced in a
causally connected region know as the causal diamond. It has been found in [6] that
the dominant contribution to entropy production comes from the infrared radiation
of dust heated by starlight. Although less restrictive assumptions are required than in
a more conventional anthropic approach the resulting probability distribution turns
out to be in excellent agreement with observation (see figure 5.3).
Recently, increasing attention has been dedicated to the study of particle creation
from the vacuum energy in a de Sitter spacetime [7, 140, 141, 180, 245–247]. Such a
process could possibly lead to a large production of entropy. The aim of this paper
was to verify under which conditions this could be a competitor to entropy produced
by the radiating dust. In absence of a model describing the decay of vacuum energy
in more realistic cosmology (involving at least a matter dominated era), we have
simply used the expression for the number density of particles produced in a de Sitter
spacetime obtained in [7] and used it to derive an expression for the corresponding
rate of entropy production. This is a rough approximation, but the idea here was to
have a toy model to illustrate how to do the computation and to have a first insight
into the implications of particle production from vacuum energy on the CEP. One
of our main results is that in the case where this new source of entropy would be
the dominant one, then the range of likely values for ρΛ would be many orders of
magnitude larger than the observed value, unless the mass of the particles created is
unusually small (m
1.2 1034 eV). Using the CEP, we have also fixed an upper
bound to the time during which the particle creation could occur (eq. (5.4.28)). Our
computation, was done for a peculiar model [7] and using a rough approximation,
hence further investigation will be required when a more realistic model of vacuum
decay, taking in account the back reaction on the energy density of the vacuum, will
be available.
Chapter 6
Conclusion
Since the introduction of the cosmological constant by Einstein at the beginning
of the 20th century, dark energy has experienced a tumultuous history. The Λ-term
has been disavowed by Einstein itself. It has disappeared and reappeared following
the development of new theories and the improvement of observational measurements,
knowing at least three more noticeable periods of renaissance [256]. Nowadays there
is a large consensus on the fact that the Universe is currently experiencing a phase
of accelerated expansion. The simplest explanation to this acceleration is provided
by the cosmological constant, which has led to the development of the ΛCDM model.
In addition to baryonic matter and radiation, this model involves also dark matter
and provides a good fit to a wide range of observational data (SNeIa, CMB, BAO,
etc.). However this model is facing two important difficulties. One is the severe
discrepancy of about 120 orders of magnitude between the observed value of the dark
energy density and the value expected from quantum field computations (cosmological
constant problem). The other is the puzzling fact that current values of the energy
density of cold matter and dark energy are of the same order of magnitude. Since
their ratio evolves as ρm {ρΛ 9a3 , this should occur only during a very short period of
time (coincidence problem). A possible way to circumvent these two problems is to
consider a cosmological model where the cosmological constant is replaced by a more
generic type of dark energy in which its energy density is allowed to vary in time.
This thesis was dedicated to the study of such models.
113
114
In chapter 2, we have considered a cosmological model, in a flat spacetime, where
dark energy and dark matter are coupled through an interaction term of the form
Q0 ρnΛ . We have primarily focused on the case n 3{2 in which the ratio of matter to
dark energy densities remains constant and thus could possibly provide an explanation
to the coincidence problem. We then proceed to find constraints from the current data
that probes the expansion history. The conclusion is that in the range of parameters
where the two cosmological problems are solved, or at least alleviated, this model
does not provide a good fit to data. While being in the range fitting the data, it is
not as interesting since it does not resolve these problems, nor does it improve the
fit compared to the ΛCDM model. In chapter 3, we have extended the analysis of
this model to case of a spacetime with spatial curvature, obtaining roughly the same
conclusions.
In chapter 4, we have considered the hypothesis proposed in [5] and reused in
[195–202] according to which the dark energy should be in thermal equilibrium with
the cosmological horizon, at least at late time. Assuming that the temperature of the
horizon is inversely proportional to its radius (T 91{R), we have found the generic
form of the interaction term Q required to maintain this equilibrium. The exact
expression depends on the type of dark energy considered. To illustrate our computation,we have considered as examples a Chaplygin gas (which is actually a unified
form of dark energy and dark matter) and a dark energy with a constant EoS parameter. The hypothesis of a thermal equilibrium with the horizon has also been
extended to other fluids by some authors, but we have illustrated that such models
are likely to be inconsistent. In a future work, we should study if the interaction terms
obtained, for different types of dark energy, lead to a cosmological model consistent
with observational data.
In chapter 5, we have reviewed the work presented in [7] in which the particle
production from the vacuum energy is described by the solution in curved spacetime of
the Klein-Gordon equation for a free, massive, and conformally couple scalar field. We
have used the results obtained to check whether the amount of entropy released by the
particle creation could be sufficiently important to affect the probability distribution
115
for the value of the cosmological constant inferred by the causal entropic principle.
This principle asserts that the probability for the parameters of a given vacuum to be
measured by observers is proportional to the amount of entropy produced in causally
connected region. It was found in [6] that the dominant contribution to entropy
production comes from the infrared radiation of dust heated by starlight and the
corresponding probability distribution is good agreement with the observed value of
the dark energy density. We have shown that considered alone, the entropy produced
by the particles created from the vacuum energy favoured values of ρΛ several orders
of magnitude larger than the observed value, unless that mass of the particles created
is usually small (m 1.2 1034 eV). This process could then be a serious threat to
the causal entropic principle. To be preserved, we have found a limit on the duration
of the interaction, which is shorter for more massive particles (t̃off
1017 s
1eV 3
).
m
In
our analysis, we have considered the duration of the interaction as a free parameter.
We are currently trying to find on a way to estimate its value and this should be
included in an improved article.
Appendix A
Curvature tensors
A.1
Definitions
The Riemann tensor is a p1, 3q tensor constructed from the metric gµν and its first
and second derivatives used to express the curvature of a spacetime. It measures to
which extent this spacetime is not locally isometric to a Minkowski spacetime and is
given by
Rρ σµν
BµΓρνσ Bν Γρµσ
Γρµλ Γλνσ Γρνλ Γλµσ ,
(A.1.1)
where the Christoffel symbols are given by
Γλµν
12 gλσ pBµgνσ Bν gσν Bσ gµν q
(A.1.2)
By contracting the first and the third index of the Riemann tensor, we obtain a p0, 2q
tensor know as the Ricci tensor
Rµν
Rλµλν .
(A.1.3)
The Ricci scalar is formed by raising the first index of the Ricci tensor and contracting
it with the second one
R Rµ µ
gµν Rµν
116
(A.1.4)
117
A.2
FRW metric
In section 1.1, we have expressed the FRW metric in the coordinates system
pt, r, θ, φq
1
0
gµν 0
0
0
0
0
a2 t
1 κr 2
0
0
pq
0
a2 ptqr 2
0
0
0
a2 ptqr 2 sin2 θ
ÆÆ
ÆÆ .
Æ
(A.2.1)
The evaluation of the differents objects defined above is straightforward. The non-zero
Christofell symbols are (we do not have repeated the symmetric terms Γαµν =Γανµ )
1aκrH ,
Γ022 a2 Hr 2 ,
Γ122 r p1 κr 2 q,
Γ323 cot θ,
Γ101 Γ202 Γ303 H,
Γ011
where as usual, H
1κrκr ,
Γ033 Γ022 sin2 θ,
Γ133 Γ122 sin2 θ,
Γ233 Γ323 sin2 θ,
Γ212 Γ313 1r ,
2
Γ111
2
2
a{a stands for the Hubble parameter.
9
(A.2.2)
The only non-vanishing
terms in the Ricci tensor are the diagonal ones
3 aa ,
2a 2κ
,
R11 3 aa 1
κr
R22 r 2 paa 2a2 2κq,
R33 R22 sin2 θ.
:
R00
:
9
2
2
:
The Ricci scalar is then given by
R6
:
a
a
(A.2.3)
9
H
2
κ
a2
.
(A.2.4)
Appendix B
Appendix to Chapter 3
B.1
Observational constraints
To constrain the model parameters, we have proceeded similarly to what we did in
[1], i.e. that we have used observational tests involving the distance modulus µ of type
Ia supernova (SNeIa) and gamma-ray bursts (GRB), the baryon acoustic oscillation
(BAO), the cosmic microwave background (CMB) and the observational Hubble rate
(OHD). These data are actually frequently used to constrain the cosmological models
with interacting dark energy [151–153, 155, 169–174]. In our current analysis, there
are two noticeable differences in comparison to our previous study. For the OHD
constraints, we have used an updated dataset [183] and for the CMB constraints, we
have only used the acoustic scale lA since as it was pointed in [174], the CMB shift
parameter R is model dependent and can be used only in the case where the dark
energy density is negligible at the decoupling epoch (which a priori, we do not know).
For each of these dataset, the χ2 function is computed as
χ2
¸ rx
obs
pzi q xth pziqs2 ,
σi2
i
(B.1.1)
where xobs , xth and σi are respectively the observed value, the theoretical value and
the 1-σ uncertainty associated with ith data point of the dataset. The best fit is then
obtained by minimizing the sum of all the χ2
χ2tot
χ2µ
χ2OHD
118
χ2BAO
χ2CMB .
(B.1.2)
119
B.1.1
Distance modulus µ of SNeIa and GRB
The distance modulus is the difference between the apparent magnitude m and
the absolute magnitude M of an astronomical object. Its theoretical value for a flat
Universe is given by
µth pz q 5 log10
DL pz q
h0
42.38,
(B.1.3)
H0 /(100 km s1 Mpc1) and the Hubble free luminosity distance is given
by DL pH0 {cqdL . The luminosity distance, dL , is defined as
» z dz 1 zq
p
dL pz q sinn |Ωk |
.
(B.1.4)
pH0 {cq|Ωk |
0 H pz q{H0
For a closed (Ωk 0), a flat (Ωk 0) and an open Universe (Ωk ¡ 0) the function
where h0
0
0
1
2
0
1
2
0
0
sinn x is respectively equal to sin x, x and sinh x. The observational data used are
the 557 distance modulii of SNeIa assembled in the Union2 compilation [24] (0.015 z
1.40) and the 59 distance modulii of GRB from [25] (1.44 z 8.10).
The
combination of these two types of data covers a wide range of redshift providing a
more complete description of the cosmic evolution than the SNeIa data by themselves.
B.1.2
Observational H pz q data (OHD)
The theoretical values of the Hubble parameter at different redshift is directly
obtained from eq. (3.2.3). The observational data have been taken from ref. [183]
where 28 values measured up to a redshift of z
2.30 have been compiled from
refs. [162, 163, 184–188].
B.1.3
Baryon acoustic oscillation (BAO)
The use of BAO to test a model with interacting dark energy is usually made
[151–153, 155, 169–174] by means of the the dilation scale
DV pz q 1{3
z p1 z q2 2
d pz q
H pz q{c A
.
(B.1.5)
120
Since the (proper) angular diameter distance dA is related to the luminosity distance
dL through dA
dL{p1
z q2 , the dilation scale may also be expressed as
DV pz q z
pH pzq{cqp1
d2
2 L
zq
1{3
pz q .
(B.1.6)
The ratio rs pzd q{DV pz q, where rs pzd q is the comoving sound horizon size at the drag
0.35 by SDSS [32] and at z 0.20 by 2dFGRS [157].
To avoid to have to compute rs pzd q, we will minimize the χ2 of the ratio DV {DV .
The observed value for this ratio is 1.736 0.065 [157].
epoch, has been observed at z
0.35
B.1.4
0.20
Cosmic Microwave Background (CMB)
The values extracted from the 7-year WMAP data for the acoustic scale (lA ) at
the decoupling epoch (z ) can be used to constrain the model parameters (lA pz q
302.09 0.76). Its theoretical value is computed as [29]
lA pz q π p1
z q
dA pz q
rs pz q
(B.1.7)
where rs pz q, the comoving sound horizon at the decoupling epoch, is given by
rs pz q »8
z
cs
dz
.
H
(B.1.8)
Hence, in term of the luminosity distance, the acoustic scale may be expressed as
lA pz q p1
π
dL pz q
³
.
z q 8 dz
z
In these expressions, the sound velocity is given by
cs
c
3
(B.1.9)
cs H
9
Ωb0
4 Ωγ0 p1 z q
1{2
,
(B.1.10)
and following [152, 169–172], we use the following fitting formula [160] to find z
z
1048r1
0.00124pΩb0 h20 q0.738 sr1
g1 pΩm0 h20 qg2 s,
(B.1.11)
121
where
0.0783pΩb h20 q0.238 p1 39.5pΩb h20 q0.763 q1,
g2
0.560p1 21.1pΩb h20q1.81 q1.
g1
0
0
0
(B.1.12)
(B.1.13)
Two additional parameters are needed to determine the acoustic scale, namely the
current value of the density parameter of baryons (Ωb0 ) and of radiation (Ωγ0 ). Constraining the model with these two additional parameters will require in an increased
computational cost. However as suggested in [155], we can use the values obtained
in the context of the ΛCDM cosmology. This is motivated since the radiation and
the baryons are separately conserved, and because we want to preserve the spectrum
profile as well the nucleosynthesis constraints. The observational results from 7-year
WMAP data [30] are
Ωb0
2.25 102h02
and Ωγ0
2.469 105h0 2 .
(B.1.14)
These two quantities are related to two of the constrained parameters since Ωm0
and Ωr0
1
Ωdm
0
7{8p4{11q4{3 Neff Ωγ0 . The combinations of initial parameters Ωm0 , Ωr0
and h0 which lead to a negative energy density for dark matter (Ωdm0
effective number of neutrinos species smaller than three (Neff
from our analysis.
0) or to an
3) have been excluded
Ωb0
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