Robbers Dissertation

Robbers Dissertation
Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the Ruperto-Carola University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
Put forward by
Diplom-Physiker Georg Robbers
Born in Freiburg im Breisgau, Germany
Oral examination:
03.06.2009
Properties of Quintessence
Referees:
Prof. Dr. Christof Wetterich
Prof. Dr. Matthias Bartelmann
Eigenschaften der Quintessenz
Wir untersuchen kosmologische Modelle mit Quintessenz, einer im gesamten
Universum fast homogen verteilten Energieform, die heute etwa drei Viertel zur
gesamten Energiedichte des Universums beiträgt. Wir verwenden Beobachtungsdaten, um Obergrenzen für die Energiedichte der Quintessenz auch zu früheren
Zeiten zu bestimmen, und analysieren die Rolle, die diese frühe Quintessenz
bei der Interpretation von kosmologischen Beobachtungen spielt. Weiterhin untersuchen wir mögliche Abweichungen von dem Gravitationsgesetz der Allgemeinen Relativitätstheorie auf sehr großen Längenskalen. Schließlich betrachten
wir dann Modelle, in denen die Masse der Dunklen Materie oder der Neutrinos
von dem Quintessenz-Feld abhängt, und stellen Erweiterungen des Programms
Cmbeasy vor, die es erlauben, die kosmologischen Vorhersagen dieser Modelle
zu berechnen.
Properties of Quintessence
We study cosmological models with quintessence, a form of energy that is distributed almost homogeneously in the Universe, and today makes up about
three fourths of its energy density. We use observational data in order to infer
upper bounds for the energy density of quintessence also during early times, and
analyze the impact of this early quintessence on the interpretation of observational data. Furthermore, we investigate deviations from the law of gravity as
described by the theory of General Relativity on very large scales. Finally, we
consider models in which the mass of the dark matter or of neutrinos depends
on the quintessence field, and present extensions of the software Cmbeasy, that
allow to compute the cosmological predictions of these models.
Contents
Contents
i
1 Introduction
1.1 Big-Bang Cosmology . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Properties of Quintessence . . . . . . . . . . . . . . . . . . .
1
1
4
2 The
2.1
2.2
2.3
Homogeneous and Isotropic Universe
The Background Equations . . . . . .
Distance Measures . . . . . . . . . . .
Scalar Field Dark Energy . . . . . . .
2.3.1 Parameterizations . . . . . . .
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7
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17
18
21
26
33
34
34
35
4 Early Dark Energy
4.1 Cosmological Implications . . . . . . . . .
4.2 Constraints on Early Dark Energy . . . .
4.2.1 Models and Parameterizations . . .
4.2.2 Observational Data . . . . . . . . .
4.2.3 Parameter Constraints . . . . . . .
4.2.4 Constraints from Ly-α data . . . .
4.3 Early Dark Energy and the CMB . . . . .
4.3.1 Baryon Acoustic Oscillations . . .
4.3.2 Miscalibrating the Standard Ruler
4.4 Prospects . . . . . . . . . . . . . . . . . .
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37
37
39
39
41
42
45
48
51
52
55
Planck Mass at the Horizon Scale
Effective Planck Mass . . . . . . . . . . . . . . .
The Planck Mass and Early Dark Energy Models
Cuscuton Cosmologies . . . . . . . . . . . . . . .
5.3.1 Cuscuton and k-Essence . . . . . . . . . .
5.4 Running of the Planck Mass? . . . . . . . . . . .
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57
58
59
60
62
63
3 Perturbations
3.1 The Cosmic Microwave Background .
3.2 Metric and Fluid Perturbations . . .
3.3 The Distribution Function . . . . . .
3.4 Initial Conditions . . . . . . . . . . .
3.5 Numerical Tools . . . . . . . . . . .
3.5.1 Cmbeasy . . . . . . . . . . .
3.5.2 AnalyzeThis! . . . . . . . .
5 The
5.1
5.2
5.3
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i
6 Coupled Dark Energy
6.1 The Formalism . . . . . .
6.2 Linear Perturbations . . .
6.3 Matter Fluctuations . . .
6.4 The Growth Rate . . . . .
6.5 The Coupling Strength . .
6.6 Newtonian Approximation
6.7 The Non-Linear Regime .
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67
68
71
72
74
75
77
79
7 Growing Neutrino Quintessence
7.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Linear Perturbations . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Neutrino Clustering . . . . . . . . . . . . . . . . . . . . . . . . .
83
83
86
87
8 Conclusions
93
Bibliography
95
Acknowledgments
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107
1 Introduction
1.1 Big-Bang Cosmology
The Theory of General Relativity (Einstein 1916) describes gravity as a geometrical effect caused by the curvature of spacetime. It can be applied to our
Universe as a whole, and remarkably, even a century after it has been proposed,
General Relativity is still the fundamental framework on which modern cosmology is built. However, when the observations of distant supernovae discovered
that the expansion of our Universe is accelerating (Riess et al. 1998; Perlmutter et al. 1999), cosmologists had to accept that on large, cosmological scales,
gravity might not work as our intuition tells us: all objects should be attracted
to each other. But rather than slowing down, the relative velocities of distant
galaxies are increasing. This either means that gravity behaves differently than
previously thought; or it requires the addition of some mysterious form of energy with exotic gravitational properties, the dark energy, or quintessence, to
the cosmic inventory. The properties of this dark energy are the subject of this
thesis.
But even before the supernovae observations, there had been hints that the
total matter density in the Universe might be well below the ‘critical density’
required for a geometrically flat universe (e.g. Bahcall et al. 1995). When observations of the Cosmic Microwave Background found that the Universe is indeed
geometrically flat (de Bernardis et al. 2000), the combination of these discoveries
made a very strong case for some form of dark energy.
In addition to the dark energy, there also is very strong evidence that ordinary
matter, the baryons (including electrons etc.), makes up only a fraction of the
total matter density. One piece of observational evidence is for example the
shape of the rotation curves of galaxies, which shows that the stars revolve
around the center of galaxies much faster than they would in the gravitational
potential of only the baryonic matter in the galaxy. The rest of the matter
content of the Universe must therefore be in the form of ‘dark’ matter, which
gravitationally behaves like ordinary matter, but does not interact (or, at least,
very weakly) with the baryons and photons in the standard model of particle
physics.
The accelerating Universe requires that the dark energy must be the dominating component in the Universe today; all observations point to a universe with
about 70% of its energy density today accounted for by the dark energy and
roughly 30% in the form matter, where only 5% is in baryonic matter, and the
rest in dark matter. We further know that neutrinos with a certain mass could
be responsible for up to a couple of percent of the present-day energy density;
however, as we have yet to determine the precise value of the neutrino mass,
this number is somewhat model-dependent. The contribution of photons to the
1
CHAPTER 1. INTRODUCTION
total energy density, in contrast, can be determined very accurately, since it is
almost entirely contained in the Cosmic Microwave Background radiation, the
CMB. The spectrum of the CMB is well described by a black-body spectrum,
with a very precisely measured temperature of Tcmb = 2.725 ± .001 K (Fixsen
et al. 2009). The contribution of photons to the total energy density of the
Universe at the present day then is of the order of only 10−5 .
Much of the standard picture of big-bang cosmology can be understood if we
accept the notion - and we are going to put this on a mathematically sound
footing in the next chapter - that we can describe the expanding universe in
a first approximation by a single function of time that parameterizes the expansion of spacetime. In a flat universe, this parameter, the scale factor a, can
conveniently be normalized to a0 = 1 today, and was correspondingly smaller
in the past. A direct consequence of the cosmological expansion is the redshift
z of photons emitted by a distant source and detected by a local observer: the
wavelength of these photons will be stretched by a fractional amount z = 1/a−1
by the expansion. In cosmology, the redshift and the scale factor are both often
used as a proxy for time.
The expansion also affects the photons of the Cosmic Microwave Background.
Therefore, back in time, when the scale factor was smaller, so was the wavelength
of the photons; and since the energy of a photon is given by the inverse of this
wavelength, the photon energy was larger by a factor of 1/a. The energy density
stored in the photons, the number density times the energy per particle, then
scales as a−4 , since number density is inversely proportional to volume, and
therefore scales as a−3 .
Similarly, the energy density of matter, baryonic or dark, should scale as the
number density according to a−3 , if the rest mass of such particles is constant
with time. Going back in time, even if the energy density of photons is miniscule
today, at some point radiation was more important than the matter density, if
only the scale factor was small enough. Hence, the cosmological evolution is
governed by different epochs: today, below a redshift z . 0.5, the dominating
component is the dark energy. Earlier, up to a redshift of roughly z ≈ 3000,
matter dominated the overall energy density; this is the period when the structures in the Universe began to form. And even before this, radiation was the
dominant component.
Big-Bang cosmology extends back even further. A very important cornerstone of the paradigm is inflation, a very early phase (t ∼ 10−35 seconds)
of exponential expansion. Inflation is responsible for the initial generation of
the fluctuations that seeded the present-day structures in the Universe, and
explains why the Microwave Background has the same temperature even in
directions that never were in causal contact before the photons were emitted. Inflation also naturally suggest that our Universe should be close to spatially flat, and we are going to assume that this holds exactly throughout this
work. A full description of Big-Bang cosmology also includes Baryogenesis,
the generation of the matter-antimatter asymmetry, and Nucleosynthesis (at
1 second < t < 3 min, ∼ z ≈ 1010 ), i.e. the production of the first light elements.
In this thesis, however, we will try to trace the evolution of the dark energy
2
CHAPTER 1. INTRODUCTION
only from the epoch of radiation domination until today. As we will see, for
the purposes of this work, the state of the Universe at the end of the very
early history can then be encoded in very few parameters, and we will not
require a detailed knowledge of the physics that governs that earliest stage of
the cosmological evolution.
The simplest form of a dark energy is the cosmological constant, which, when
added to Einstein’s Equation, can still describe this accelerating Universe in very
good agreement with the observations. But in order to explain cosmic acceleration, the cosmological constant has to have a particular numerical value, which
is of order 10−120 (in units where the Newton constant as well as the speed of
light and the reduced Planck constant are all set to unity, G = c = ~ = 1). This
is contrary to the usual quantum-theory folklore that the cosmological constant
should be of order unity, or - by some symmetry or dynamical mechanism possibly zero.
Furthermore, if the dark energy is indeed a cosmological constant, then there
seems to be no fundamental reason for why the crossover to a dark energy
dominated universe happened only recently (in cosmological timescales). From
the different scaling behavior of matter and radiation that we have seen above,
such a crossover from radiation to matter domination is simply a consequence
of the expansion and the age of the universe, but for the cosmological constant,
for which the energy density does not change at all in time, there is no such
compelling answer to the ‘why now?’ question, and one is led to accept it as a
mere coincidence.
A much more attractive setting than a pure cosmological constant in this
regard are cosmological models with light scalar fields (Wetterich 1988b; Ratra
& Peebles 1988; Wetterich 1988a; Peebles & Ratra 1988; Frieman et al. 1995).
Such a ‘quintessence’ (Caldwell et al. 1998) or ‘cosmon’ (Peccei et al. 1987)
component, has dynamics, and therefore allows for scenarios in which the dark
energy density is evolving with time. Indeed, many quintessence models posses
attractor solutions (Wetterich 1995; Zlatev et al. 1999; Copeland et al. 1998), so
that over a wide range of initial conditions, the scalar field is driven towards a
solution where it tracks the dominant component, and hence its energy density is
subdominant compared to radiation or matter, but of a comparable magnitude,
and no ‘finetuning’ of the order of 10−120 is needed for the initial conditions in
the theory.
Pure tracker models do not answer the ‘why now’ question, as they need a
mechanism that triggers the change from the tracking behavior to the accelerating regime with dark energy domination; such trigger events can be parameterized phenomenologically through specific features in the potential (or the
kinetic term) of the scalar field, thereby dodging the question, but also dynamic solutions have been explored. One particular route in this regard are
the additional dynamics due to a coupling between the scalar field and other
components, which is one specific quintessence property that we will discuss in
this work.
3
CHAPTER 1. INTRODUCTION
1.2 The Properties of Quintessence
More generally, the topics of this thesis are the properties of scalar field dark
energy models; we study their cosmological implications, and - where possible compare their predictions with observational data. As an introduction, Chapter 2 is dedicated to a very brief survey of the homogeneous and isotropic universe, and the mathematical formulation in terms of the Friedmann-RobertsonWalker metric in General Relativity. Chapter 3 then sets the stage for linear
perturbations around the homogeneous background, introducing the very basics
of cosmological perturbation theory. We also introduce Cmbeasy, the computer
code that we use throughout this work to numerically compute the predictions
of cosmological models.
In Chapter 4, we begin our investigation of quintessence properties by studying its time evolution. Specifically, we consider cosmological models with ‘early
dark energy’. In these scenarios, the dark energy contributes a substantial
fraction to the total energy density even at earlier times, unlike for example
a cosmological constant. By comparing the predictions of several early dark
energy models with a comprehensive set of observational data, we find upper
bounds for the abundance of the dark energy during the time before last scatter
and during structure formation (Doran, Robbers, & Wetterich 2007a). At the
same time, we compare the values for the standard cosmological parameters in
these models to the values that are commonly inferred in the context of the
ΛCDM model, and in ‘standard’ quintessence models, and search for possible
differences. In the second part of this chapter (Linder & Robbers 2008), we
investigate one particular question in more detail, namely whether early dark
energy models have distinct signatures in the power spectra of the Cosmic Microwave Background, i.e. whether a standard quintessence model and one with
a certain fraction of early dark energy can have indistinguishable power spectra. We are particularly interested in this as the CMB is used to ‘calibrate the
standard ruler’, that is, to measure the size of the sound-horizon, at the time
of last scatter; this same scale can also be observed at a time much closer to
today, through measurement of the ‘Baryon Acoustic Oscillations’ (BAO) in the
distribution of matter. These two measurements of the same length scale at two
very different times of the cosmological evolution provides very good leverage to
constrain the background evolution of the Universe. In order to interpret BAO
observations correctly, however, one needs to properly account for the possibility
of early dark energy, as we will show.
The boundary region between scalar field dark energy models and modifications to General Relativity is explored in Chapter 5 (Robbers, Afshordi, &
Doran 2008). We focus on the Planck Mass, which in General Relativity is
the constant that quantifies the strength of the coupling between the energy
momentum and the space time metric, and consider the possibility that the
value of the Planck Mass might change at the scale of the cosmological horizon.
Such a change is a generic feature of certain kinds of early dark energy models, and shows how intimately related the law of gravity is to the properties of
quintessence. This connection is even more evident in the ‘Cuscuton’ models,
which can be formulated as a specific limit of scalar field dark energy models of
4
CHAPTER 1. INTRODUCTION
the k-essence type. In this kind of model, the scalar degree of freedom drops out
of the equations, reducing the theory to a constraint system for the evolution
of the remaining matter content. The corresponding change of the Planck Mass
at the horizon scale can be constrained by observations, and this is the aim of
Chapter 5.
Another rather fundamental property of quintessence is studied in Chapter 6,
namely a possible coupling between dark matter and dark energy. The formalism for a coupled dark matter - dark energy fluid can be written in a rather
general way, and we retain this generality in our numerical implementation in
Cmbeasy. We then consider a specific example of a coupled dark energy model,
and investigate its cosmological properties. The output of our numerical implementation for these specific examples has also been used to perform N-body
simulations (Baldi 2009), and we present the combined results of the complete
numerical treatment of these models from the background level to the non-linear
regime (Baldi, Pettorino, Robbers, & Springel 2008).
Chapter 7 then extends the treatment of quintessence properties to couplings
to neutrinos. The ‘growing neutrino quintessence’ scenario that we investigate
provides a possible answer to the ‘why now?’ question, and here we show that its
properties on the level of linear perturbations are quite novel (Mota, Pettorino,
Robbers, & Wetterich 2008; Pettorino, Mota, Robbers, & Wetterich 2009).
Finally, Chapter 8 gives a short summary of the results of this work.
5
CHAPTER 1. INTRODUCTION
6
2 The Homogeneous and Isotropic
Universe
A fundamental assumption in modern cosmology is that our Universe is homogeneous and isotropic, if just the scales under consideration are large enough. Built
on this hypothesis is an elaborate theoretical apparatus that is very successful
in explaining observational data, and in this chapter we will briefly review the
basics of this framework. Although our Universe certainly is not homogeneous
on scales of galaxies or even clusters of galaxies, an isotropic and homogeneous
description of our Universe is nevertheless immensely useful. Many cosmological observations can be either understood directly in these terms, e.g. distance
measurement through the observation of SnIa, or by considering small perturbations around this homogeneous and isotropic background in linear theory, e.g.
the Cosmic Microwave Background. In fact, in this thesis, we will only once (in
Chapter 6) leave the grounds of linear perturbation theory.
2.1 The Background Equations
General Relativity describes the four-dimensional spacetime of our Universe by
a pseudo-Riemannian manifold together with a metric g. Since we assume that
our Universe is homogeneous and isotropic on sufficiently large scales, the spacetime therefore must admit a slicing in homogeneous and isotropic 3-spaces on
constant time hypersurfaces. Then, there is a preferred geodesic time coordinate
t, ‘cosmic time’, such that the 3-spaces of constant cosmic time are maximally
symmetric. In principle, these constant-time hypersurfaces can have either positive, negative or zero curvature. In this thesis, we assume a flat universe, i.e.
that the curvature vanishes. Hence, the metric has the Friedmann-RobertsonWalker form,
ds2 ≡ gµν dxµ dxν = −dt2 + a(t)2 δij dxi dxj ,
(2.1)
with scale factor a(t), Greek indices running from 0 through 3, and Latin indices
from 1 through 3. The scale factor can conveniently be normalized to be unity
today, so that we can write a0 = a(t0 ) = 1, denoting present-day values by a
subscript 0. It is useful to define the conformal time τ by dτ ≡ a−1 dt, so that
the metric can be written as
ds2 = a(τ )2 −dτ 2 + δij dxi dxj .
(2.2)
The Hubble parameter H is related to the scale factor via
H ≡ a−1
ȧ
da
≡ ,
dt
a
(2.3)
7
CHAPTER 2. THE HOMOGENEOUS AND ISOTROPIC UNIVERSE
where our convention throughout is that a dot denotes derivatives with respect
to cosmic time, whereas a prime represents derivatives with respect to conformal
time - the comoving Hubble parameter H then is
H≡
a′
= aH.
a
(2.4)
The value of the Hubble parameter today is conventionally expressed as H0 =
100 h km/sec/Mpc, where h ≈ 0.7 is the Hubble parameter. The energy momentum tensor in a homogeneous and isotropic universe is
−ρ g00
0
(Tµν ) =
,
(2.5)
0
p gij
where ρ(t) and p(t) are the total energy density and pressure, respectively. The
present-day energy density of the flat Universe is called the ‘critical density’,
ρcrit = 1.88 × 10−29 h2 gm cm−3 = 8.10 × 10−47 h2 GeV4 ; the energy density
ρi of a species i is often expressed as the fraction of the total energy density,
Ωi = ρi /ρtot . The relation between energy density and pressure is parameterized
by the equation of state w(t),
p(t) = w(t)ρ(t).
(2.6)
One can also write this equation for the individual contributions to the total
energy density and pressure, where e.g. for non-relativistic matter or dust, the
pressure vanishes, wm = 0, while for photons and massless neutrinos wr = 1/3 =
const, and a cosmological constant has wΛ = −1 = const.
From the metric, one computes the Ricci Tensor Rµν and the Ricci scalar R
via the Christoffel symbols to get the left hand side of Einstein’s Equations,
relating the geometry of the Universe to its energy content:
1
Gµν ≡ Rµν − gµν R = 8πG Tµν ,
2
(2.7)
where G is Newton’s constant, and a possible cosmological constant is taken
to be part of the energy-momentum tensor. Defining the reduced Planck Mass
MP by MP ≡ (8πG)−1/2 , the 0 − 0 part of this equation yields the Friedmann
Equation,
3MP2 H 2 = ρ(τ ),
(2.8)
and the space-space components give
1
ä 1 ȧ 2
p.
+
=−
a 2 a
2MP2
(2.9)
These equations can be combined to write
ρ̇ = −3(ρ + p)
ȧ
,
a
(2.10)
which can also be obtained as a consequence of energy momentum conservation,
T µν;µ = 0.
8
CHAPTER 2. THE HOMOGENEOUS AND ISOTROPIC UNIVERSE
If w = p/ρ is constant, Eq. (2.10) is solved by
a 3(1+w)
0
,
ρ = ρ0
a
(2.11)
so that it is easy to see how the energy density of any component with a constant
equation of state changes with the scale factor. For example, pressureless matter
scales as ρm ∝ a−3 , while radiation with w = 1/3 scales as ρr ∝ a−4 . One can
also insert this relation in the Friedmann Equation, Eq. (2.8), to see how the
scale factor evolves if the energy is dominated by one particular component with
a constant equation of state. One finds that the scale factor evolves as
a ∝ t2/3(1+w)
∝ τ 2/(1+3w) ,
w = const (but 6= −1),
a ∝ t1/2
∝ τ,
w = 1/3
a∝t
2/3
2
∝τ ,
(2.12)
w = 0 (dust matter),
(2.13)
(radiation).
(2.14)
If the equation of state of a particular component varies with time, the energy
density involves an integral over its time evolution,
Z a ′
da ′
1 + w(a ) .
(2.15)
ρx ∝ exp −3
a′
It is often very useful to consider energy density and pressure of certain species
in terms of their distribution function. For example, the description of perturbations around the homogeneous background in the tightly coupled photon-baryon
fluid, responsible for the emergence of the anisotropies in the CMB, is described
by following the evolution of the distribution function while keeping track of
the Thomson scattering via the collision term in the Boltzmann Equation. Of
particular interest to us is the cosmological evolution of neutrinos, since we
will study the effects of a coupling of neutrinos to dark energy in Chapter 7.
Neutrinos have a Fermi-Dirac distribution,
f0 (ǫ) =
1
gs
,
3
ǫ/k
T
0 + 1
b
hp e
(2.16)
where as usual gs is the number of spin degrees of freedom and hp is Planck’s
constant, kB and Boltzmann’s constant (which we set to 1 in our units). With
q denoting the comoving momentum, ǫ is given by
ǫ2 = q 2 + m2ν a2 .
(2.17)
The energy density and pressure in terms of the distribution function are then
determined by the phase space integral,
Z
1
q 2 dqdΩ ǫf0(q), and
(2.18)
ρν =
a4
Z
1
q2
2
pν =
.
(2.19)
q
dqdΩf
(q)
0
3a4
ǫ
We have left mν as a constant here, but note that in Chapter 7 we will have
mν = mν (φ), which does introduce certain modifications to the equations.
9
CHAPTER 2. THE HOMOGENEOUS AND ISOTROPIC UNIVERSE
2.2 Distance Measures
In order to infer the geometrical properties of the Universe from observations
we need to measure distances in the Universe. We can use the photons emitted by astronomical sources to determine their redshift, assuming we know the
wavelength with which they were emitted, for example by considering a specific
spectral line. If a photon with wavelength λ is emitted at a time τ , it will reach
us today with a wavelength
λobs =
λ
= (1 + z) λ.
a(τ )
(2.20)
This defines the redshift z,
1
.
(2.21)
a
The comoving distance between us and a distant source at a redshift z, or
equivalently, a scale factor a is given by
Z 1
Z t0
dt′
da′
=
.
(2.22)
χ(a) =
′
′2
′
t(a) a(t )
a a H(a )
z+1=
Another possible measure of distance is the angular diameter distance DA to
an object with a given physical size l, seen at redshift z1 . The angular diameter
distance, as the name suggests, is defined via the angle ϑ subtended by the
object,
DA = l/ϑ .
(2.23)
The comoving size of the object is l/a, and the comoving distance is given by
Eq. (2.22), so that
χ
.
(2.24)
DA = a χ =
1+z
Finally, distances can also be inferred from the flux F (i.e. the energy received
per unit time per unit area) from a source at redshift zS with a known luminosity
L (i.e. the energy emitted per unit time); the luminosity distance is defined as
DL (zS ) ≡
L
4πF
1/2
.
(2.25)
The relation of this definition the angular diameter distance can be deduced by
considering the energy emitted in a proper time interval of the source, dtS =
a(τS )dτ , which is La(τS )dτ . This energy is then redshifted, by a factor of
1/(1 + zS ) = a(τS ). It is also distributed over a spherical shell with the radius
χ(zS ), so that flux per proper time of the observer at aobs = 1 is
F =
La2 (zS )
.
4πχ2 (zS )
(2.26)
Replacing F in the definition Eq. (2.25) yields the relation between the different
distance measures in the flat Universe,
DL = (1 + z)χ = (1 + z)2 DA .
10
(2.27)
CHAPTER 2. THE HOMOGENEOUS AND ISOTROPIC UNIVERSE
Using the Friedmann Equation to replace H(z) in the definition of the comoving
distance, Eq. (2.22), we can now relate the luminosity distance to the energy
content of the Universe, and write relations like
Z
−1/2
1+z z ′ 0
dz Ωγ (1 + z ′ )4 + Ω0m (1 + z ′ )3 + Ωd (z ′ )
,
(2.28)
dL (z) =
H0 0
where Ω0r and Ω0m are the fractions of the total energy density today of radiation
and matter with the standard scaling behavior, and we anticipate the existence
of a dark energy component with an - at least a priori - arbitrary time evolution
of the energy density Ωd (z).
Supernovae of Type Ia
The relation Eq. (2.28) is a powerful tool to infer the composition of our Universe, if we can determine the left-hand side from observations, and historically,
it were exactly such measurements (Perlmutter et al. 1999; Riess et al. 1998)
that made the need for Ωd 6= 0 apparent, namely the observation of supernovae
of type Ia. Supernovae can be classified by their spectra, where type Ia supernovae are those that do not have any hydrogen or helium lines, together with
an absorption structure due to Si II lines at wavelengths of about λ ≈ 610 nm.
They are the result of the explosion of white dwarf stars when their mass grows
to near the Chandrasekhar limit in a binary system. The measurements are
usually discussed in terms of their absolute magnitudes M compared to the apparent magnitude m, which are related to fluxes by m = −(5/2)log(F ) + const.
As the flux scales like dL−2 , the apparent magnitude is m = M +5log(dL )+const,
and one arrives at the distance modulus
dL
,
(2.29)
m − M = 5log
10 pc
which is the main quantity of cosmological interest determined from supernovae
surveys. The usefulness of SnIa comes from the fact that they are ‘standard
candles’, a fancy term to indicate that they have the same absolute magnitudes,
or are at least ‘standardizable candles’, i.e. the intrinsic differences can be
corrected for. While the original samples contained about ∼ 50 supernovae
covering a redshift range up to z . 1, current compilations cover between 300
and 400 SnIa (Kowalski et al. 2008), and the outlook is that future probes will
increase the number to tens of thousands of supernovae discoveries (for a recent
overview of future observational prospects, see e.g. Howell et al. 2009), so that
the statistical errors are bound to decrease significantly, and understanding the
systematic errors in SnIa data becomes increasingly important.
The crucial piece of evidence for dark energy from the first SnIa observations then was the discovery that at recent cosmological times, the deceleration
parameter q ≡ ä/aH 2 < 0. From Eq. (2.9) we see that during matter and
radiation domination, q > 0 and ä < 0, i.e. gravity slows down the expansion.
Only a component that has p < −ρ/3, and therefore w < −1/3, can cause the
expansion to accelerate, ä > 0. This is the defining property of the dark energy.
11
CHAPTER 2. THE HOMOGENEOUS AND ISOTROPIC UNIVERSE
Baryon Acoustic Oscillations
Another way of measuring distances are the baryon acoustic oscillation, which
we mention here as one example of the many other cosmological probes that
have confirmed the picture of the accelerating universe.
When measuring the large-scale correlation function of galaxies, one finds a
peak at a separation of about 100 h−1 Mpc. This peak is a remnant of the
sound waves in the photon-baryon fluid in the early universe, the acoustic oscillations. In the coupled photon-baryon fluid, these waves can propagate, until at
recombination, the wave propagation effectively stops. The comoving distance
that a sound wave can travel from the initial seed of a perturbation to the time
of recombination is a characteristic length scale, known as the ‘sound horizon’,
denoted by s. It is this scale that is responsible for the observed peak. Its size
depends both on the expansion history, and the properties of the photon-baryon
plasma, encoded in its speed of sound cs , and is given by
Z ∞
cs
,
(2.30)
dz
s=
H(z)
zdec
where zdec ∼ 1100 is the redshift of decoupling between the photons and the
baryons. Measuring this scale in the in clustering of galaxies in the transverse
and line-of-sight directions yields a determination of χ(z)/s and of s H(z), respectively. The power of measurements of the baryon acoustic oscillations now
lies in the fact that we can observe the imprint of the acoustic oscillations in the
CMB, thereby calibrating the sound horizon at a very high redshift, and then
measure this length scale at different redshifts in the large-scale baryon power
spectrum. The ratios of these distances then determine the angular-diameterdistance to redshift relation, yielding a powerful geometric measurement of the
expansion history of the Universe. The main observational obstacle is the weakness of the acoustic feature in the correlation function (≈ 10% contrast), and
so the first detection (at z ≈ 0.35) was only reported a few years ago, by Eisenstein et al. (2005). A crucial ingredient for the correct interpretation of these
distance ratios is obviously the precise determination of the size of the sound
horizon from the CMB, and we will come back to this issue in Chapter 4 in the
context of early dark energy models.
The detection of the baryon acoustic peak, however, is not only significant
due to its potential for accurate geometric measurements, but also because it
provides a confirmation of the paradigm that large scale inhomogeneities grow
from small perturbations at z ∼ 1100 by gravitational clustering, and because
the observed size of the feature cannot be reproduced without dark matter, and
therefore confirms its existence.
2.3 Scalar Field Dark Energy
If we assume that the measured acceleration is not due to a cosmological constant, but caused by a dynamical dark energy component, then the most natural
candidate is a scalar field. We can write down the action of such a scalar field
12
CHAPTER 2. THE HOMOGENEOUS AND ISOTROPIC UNIVERSE
φ with potential V , as
S =−
Z
√
1
µ
d x −g ∂µ φ∂ φ + V (φ) .
2
4
(2.31)
Varying the action with regard to the metric yields the energy momentum tensor
Tµν
√
−2 ∂
√
−gL
µν
−g ∂g
1
α
= ∂µ φ ∂ν φ − ∂α φ∂ φ + V gµν ,
2
=
(2.32)
(2.33)
and, in the homogeneous background case where spatial derivatives vanish, to
an energy density
1
ρφ = −T0 0 = 2 φ′2 + V (φ),
(2.34)
2a
and pressure
1
1
(2.35)
pφ = Ti i = 2 φ′2 − V (φ).
3
2a
Hence, defining the kinetic energy T ≡ φ̇2 /2, the equation of state becomes
wφ ≡
T −V
p
=
.
ρ
T +V
(2.36)
The equation of motion for the field becomes
φ′′ + 2
a′ ′
∂V
φ + a2
= 0.
a
∂φ
(2.37)
Furthermore, by inserting the expression for the pressure in Eq. (2.9), we can
see that in order to get wφ < −1/3, the condition
φ̇2 < V (φ)
(2.38)
needs to be fullfilled, which in standard quintessence models is reached by very
flat potentials, but can theoretically also be obtained by any other mechanism
that makes φ̇ small, i.e. slows or completely stops the evolution of the field.
One of the prototypical quintessence potentials is the exponential potential
(Wetterich 1988b, 2008),
V (φ) = MP4 exp(−α φ/MP ),
(2.39)
which admits scaling solutions for the evolution of the field, where the equation
of state follows the dominating component, i.e. radiation or matter. In the
scaling regime, the constant α determines the fraction that the dark energy
contributes to the total energy density via
Ωφ =
n
,
α2
(2.40)
where n = 4 (3) during radiation (respectively matter) domination. However,
the scaling solutions do not simply convert to the accelerated solutions where
13
CHAPTER 2. THE HOMOGENEOUS AND ISOTROPIC UNIVERSE
dark energy dominates today. Modifications of a pure exponential potential,
such as a polynomial prefactor (Albrecht & Skordis 2000),
V (φ) = VP (φ) exp(−α φ/MP ),
(2.41)
where the prefactor is for example of the form
VP (φ) = (φ − B)β + A
(2.42)
with constants B, β and A. Such modifications can lead to a local minimum in
the potential, in which the field can be trapped and cause the onset of acceleration. In Chapter 7, we will encounter a dynamical mechanism that combines
the pure exponential potential with a possibility to obtain late-time acceleration,
thereby making any additional modifications of the exponential redundant.
A second prototypical potential is the inverse power law (Ratra & Peebles
1988)
φ −α
,
(2.43)
V (φ) = A
MP
which has been widely studied in the literature (e.g. Caldwell, Dave, & Steinhardt (1998); Zlatev, Wang, & Steinhardt (1999); Binetruy (1999); see also
Copeland et al. (2006) and references therein). The dynamics in these models
can lead to a tracking regime where the equation of state is given by
wφ ≈
α wBgrnd − 2
,
α+2
(2.44)
where wBgrnd is the equation of state of the dominant (background) component.
The observed quintessence equation of state today restricts the combination of
A and α in these models, requiring 0 < α . 1. The scale A is then set by the
fraction of quintessence observed today - for smaller and smaller α this is akin
to finetuning the value of the cosmological constant. Owing to the popularity
of this class of models, we will use this potential as a starting point for a set of
example models with a dark matter dark energy interaction in Chapter 6.
One can also consider modifying the kinetic term of scalar field dark energy
models, and write more general scalar field actions formulated as functions of the
field φ and of X ≡ −(1/2)∂µ φ∂ µ φ. In these ‘k-essence’ theories (ArmendarizPicon et al. 2000, 2001), we have
Z
√
(2.45)
S = d4 x −g p(φ, X),
where the cosmic pressure is given by the lagrangian density, p. The corresponding energy density is
∂p
ρ = 2X
− p.
(2.46)
∂X
Certain choices of p, for which the field tracks the radiation component with
wφ = 1/3 early on, but transitions to wφ at the onset of the matter era, have
been proposed as solutions to the coincidence problem. An often-employed
simplification in the choice of p(φ, X) is to only consider
p(φ, X) = f (φ)p̃(X),
14
(2.47)
CHAPTER 2. THE HOMOGENEOUS AND ISOTROPIC UNIVERSE
and tracker models similar to the standard quintessence case can then also be
found for example (Chiba et al. 2000) by choosing f (φ) ∝ φ−α and p̃(X) =
(−X + X 2 ). For 0 < α < 2, the k-essence is characterized by a constant
equation of state, wφ = −1 + (α/2)(1 + wBgrnd ), until it starts to dominate
the energy density budget, at which point wφ → −1. In this work, we will not
consider the general case of k-essence cosmology, but study a particular example
in Chapter 5.
2.3.1 Parameterizations
Since a clear front runner for a complete particle physics model of dark energy
has not yet emerged, it is often times convenient to consider the dark energy
not directly in terms of a specific potential for the scalar field, but instead
starting from a parameterization of the time evolution of its energy density
Ωd (a), or of its equation of state w(a). In this way, one can hope to capture
the important physical properties and study their consequences, and employ
observations to search for hints towards the underlying fundamental theory; we
will make extensive use of this approach in Chapter 4.
Converting from the equation of state to the energy density and vice versa can
easily be done analytically, using Eq. (2.15). For a universe with certain fractions
of the total energy density stored in dark energy, matter, and relativistic species,
we have Ωtot = Ωd + Ωm + Ωrel , and find the derivative of Ωd with respect to
the scale factor,
dΩd
da
=
=
d ρd
(2.48)
da ρcrit
ρd
1
3ρd
(1 + w)ρcrit −
(−3ρcrit − ρrel − 3w ρd ) ,
−
2
a
a
ρcrit
which simplifies to
dΩd /da
= 3(1 − Ωd )a−1 w + Ωrel a−1 .
Ωd
(2.49)
Dividing both sides by (1 − Ωd), using Ωd = 1 − Ωm − Ωrel on the right hand side
and re-expressing Ωrel via the scale factor of equality aeq as Ωrel = (aeq /a)Ω0m ,
one finds that the equation of state is related to the energy density via
wd (a) =
−dΩd /d ln a 1 1
+
.
3(1 − Ωd )Ωd 3 1 + aaeq
(2.50)
When neglecting radiation, which is a good approximation in the late universe,
this is equivalent to
−dΩd /d ln a
wd (a) =
.
(2.51)
3(1 − Ωd )Ωd
In the same vein, the potential V (a) of the scalar field φ with a canonical
kinetic term can be reconstructed from a given functional form of the equation
15
CHAPTER 2. THE HOMOGENEOUS AND ISOTROPIC UNIVERSE
of state via
1
[1 − w(a)] ρ(a),
2
p
Z
1 + w(a) p
φ(a) =
da
ρ(a),
aH(a)
V (a) =
which holds as long as φ̇ 6= 0.
16
(2.52)
(2.53)
3 Perturbations
This chapter gives a short summary of the theory of linear perturbations around
the homogeneous background described in Chapter 2. As we have stated from
the beginning, we expect this treatment to yield useful results, simply because
the Universe looks more and more homogeneous the larger the scales become.
Even more importantly, gravity should usually cause inhomogeneities to grow,
so that the structures that we do observe have evolved out of very small initial fluctuations. That in turn means that linear perturbation theory should
be a rather good description especially of the early times of the cosmological
evolution. The theory that we - very briefly - review in this chapter indeed
yields a remarkably precise description of many cosmological observables, and
in particular of the fluctuations visible in the CMB, which are still very small
even at the present day, and can to a very good accuracy be calculated entirely
in linear theory.
The results of these discussions will not be a prediction for e. g. the exact
pattern of the temperature anisotropy that we observe on the sky, nor for the
particular matter distribution in the Universe; this would be pointless, as we
assume that the initial fluctuations present at the end of inflation are a particular realization of random variables set by a stochastic process. Instead, the
predictions will be about statistical properties of these quantities, such as their
correlation functions. If the perturbations of the model under consideration are
Gaussian, which we are going to assume, then the full information is contained
in the Fourier transform of the two-point correlation function, the power spectrum. For a scalar quantity X(x), the power spectrum PX is defined in terms
of its Fourier transform X(k) as
X(k) X ∗ (k′ ) = (2π)3 δD (k − k′ )PX (k),
(3.1)
where letters in bold face denote 3-vectors. We assume that inflation generates
the initial perturbations, and we describe this by the initial perturbation for the
metric perturbation Ψ1 , with a power spectrum parameterized as
ns −1
k
2π 2
.
(3.2)
PΨ (k) = 3 As
k
k⋆
The parameters are the initial scalar amplitude As , given at a pivot scale k⋆
(our choice is usually k⋆ = 0.05 Mpc−1 , but k⋆ = 0.002 Mpc−1 is also popular),
and the scalar spectral index ns . We will restrict the discussion here to scalar
perturbations, and therefore do not consider e.g. the initial amplitude of tensor
fluctuations AT . This shape of the initial power spectrum is a fundamental
input, justified a posteriori by its ability to fit observations; in our parameter
1
The precise definition of the perturbation variables will be given below.
17
CHAPTER 3. PERTURBATIONS
estimation analysis in Chapter 4, both ns and As will be free parameters, to
be determined from observational data. For the analysis of the evolution of
perturbations, we only assume that the initial values of all perturbation variables
can be set from this input; the method to obtain these initial conditions is then
discussed at the end of this Chapter, in Section 3.4.
To motivate the following mathematical considerations, we start by introducing the CMB and the power spectrum of the temperature and polarization
anisotropy; together with the matter power spectrum, these are the observable
quantities that we want to compute. The purpose of the theoretical machinery
in this chapter therefore is to relate the initial power spectrum PΨ to these observables. This overview of the theoretical framework is rather brief, since we
focus only on the modeling of the dark energy fluctuations together with dark
matter and neutrinos, as this is the foundation for the treatment of coupled
dark energy models in Chapters 6 & 7. At the end of this chapter, we introduce Cmbeasy, the numerical tool we will use in the subsequent chapters to
actually compute predictions for dark energy models and compare them with
observations, using the framework outlined here.
3.1 The Cosmic Microwave Background
At early times, when the cosmic fluid is dominated by radiation, reaction rates
are much faster than the expansion rate, so that the fluid is in thermal equilibrium. The neutrinos are the first to decouple, since they are kept in equilibrium
via their weak interaction with electrons. For the photons, the scattering off of
free electrons via Thomson scattering is the dominant process. The scattering
rate of Thomson scattering is
Γ T = σT n e ,
(3.3)
where σT = 6.6524 × 10−25 cm2 is the Thomson scattering cross section. The
scattering rate depends also on the number density of free electrons, ne . As long
as the recombination process,
p + e− ↔ H + γ
(3.4)
is rapid compared to the expansion, the ionization fraction xe ≡ ne /(ne + nH ),
i.e. the number density of free electrons divided by the total number density
of neutral hydrogen and electrons, will be given by an equilibrium distribution
(well described by the Saha equation). The mean free path of a photon λC is
given by
′
(3.5)
λ−1
C ≡ τ ≡ ne σT a,
which also defines an absorption coefficient τ ′ ; the integral over conformal time
of this coefficient is the optical depth, τ . As long as the ionization fraction is
close to unity, xe ≈ 1, the mean free path of a photon is λC ∼ 2.5 Mpc, which is
about two orders of magnitude smaller than the horizon at recombination. For
scales larger than this, λ ≫ λC , the photons are tightly coupled to the electrons
18
CHAPTER 3. PERTURBATIONS
by Thomson scattering (and the electrons are in turn tightly coupled to the
protons by Coulomb interactions).
As the Universe expands there will be a certain point in time, when there
are not enough photons with energies above the binding energy to ionize the
hydrogen. At a temperature of about T ∼ 4000K ∼ 0.4 eV, zrec ∼ 1400, the
protons begin to recombine to neutral hydrogen, the free electron fraction drops,
and the mean free path of a photon becomes larger than the Hubble scale. The
exact evolution of ne (τ ), during this time depends on the details of the reactions
by which the electrons and protons recombine to neutral Hydrogen and Helium,
and can only be solved numerically. Computer codes like Cmbeasy usually
employ numerical approximations calibrated by more detailed simulations of the
recombination processes, such as Recfast (Seager et al. 2000; Wong et al. 2007),
in order to obtain the ionization history for a given background cosmology.
Eventually, the photons decouple from the electrons, and the Universe becomes
transparent (at T ≈ 0.26 eV or redshift z ≈ 1100). Hence, the fluctuations
of the energy density in the photon/baryon plasma before recombination are
reflected in the temperature anisotropies that we measure today.
However, there is one important caveat: observations find very little neutral
hydrogen in the intergalactic medium (Fan et al. 2006), apart from so-called
Ly-α-clouds, i.e. clouds of neutral hydrogen that lead to absorption features in
the spectra of distant quasars (see Sec. 4.2.4). One is therefore led to conclude
that the Universe became reionized at low redshifts. In practice, we account for
this by introducing the optical depth to the last scattering surface,
Z t0
τ = σT
a(t)ne (t)dt,
(3.6)
tlss
as an additional cosmological parameter. So far, we have only mentioned the
temperature anisotropies. Yet, the CMB contains much more information. Since
the Thomson scattering cross section depends on the polarization of the outgoing photon in the scattering process, information is not only contained in the
temperature anisotropies, but also in the polarization of the CMB photons.
Polarization
In Thomson scattering, for an outgoing photon that has a polarization vector
that lies in the scattering plane, the cross section is reduced by a factor cos2 β,
where β is the scattering angle. There is no such reduction for a polarization
normal to the scattering plane. Hence, if the incoming photons are isotropically
distributed, the polarization will still average out, but if we are in a situation
where there is a higher intensity of photons coming in from one direction than
from a direction perpendicular to it, a net polarization will build up; therefore, a quadrupole anisotropy (in the electron reference frame) will produce a
polarization of the outgoing photon beam.
The polarization that we observe on the microwave sky is described by the
Stokes parameters (see e.g. Dodelson 2003), Q, U , and V . In a basis ǫ1 , ǫ2 that
forms a right-handed orthonormal system with the incoming beam, Q and U
measure linear polarization. Q is the difference of linear polarization along ǫ1
19
CHAPTER 3. PERTURBATIONS
and ǫ2 , U is the same, but along vectors ǫ′1 and ǫ′2 rotated by 45◦ each. The
fourth Stokes parameter known from electrodynamics, V , which describes circular polarization, is zero for the CMB - Thompson scattering does not produce
circular polarization.
The Power Spectra
The temperature anisotropies ∆(n) of the microwave sky, observed today in a
direction n, can be expanded in spherical harmonics,
X
∆T
(n) =
alm Ylm (n).
T
∆(n) ≡
(3.7)
l,m
The two-point correlation then is
∆(n)∆(n′ ) =
X
l,l′ ,m,m′
′
∗
′
hal,m a∗l′ ,m′ iYlm (n) Ylm
.
′ (n )
(3.8)
For a distribution of ∆T /T that is independent of n, we have
halm a∗l′ m′ i = δll′ δmm′ Cl ,
(3.9)
which defines the coefficients Cl . For the temperature anisotropy, we have
l
′
∗
X
X
′
∆(n)∆(n′ ) =
Cl
Ylm (n) Ylm
′ (n )
l
m=−l
1 X
(2l + 1)ClT T Pl (n · n′ )
4π
=
(3.10)
(3.11)
l
For the polarization, we expand the combination Q ± iU in terms of spinweighted spherical harmonics, s Ylm of weight s = ±2, which defines the coefficients elm and blm ,
(Q ± iU )(n) =
l
X X
l=2 m=−l
(elm ± iblm )±2 Ylm (n).
(3.12)
and corresponding power spectra
ClE = h|elm |2 i,
(3.13)
ha∗lm elm i
(3.15)
ClB
ClT E
2
= h|blm | i,
=
(3.14)
The next step now is to use perturbation theory to connect the initial fluctuations to the power spectra of the CMB.
20
CHAPTER 3. PERTURBATIONS
3.2 Metric and Fluid Perturbations
The mathematical framework of linear perturbations theory in cosmology is
well-known. The use of gauge-invariant variables in cosmology is for example
reviewed in a seminal article by Bardeen (1980). The formalism that we use here
(and in the numerical implementation in Cmbeasy) follows Kodama & Sasaki
(1984). Our notation is identical to Doran (2005); a very detailed and pedagogical review of perturbation theory (and the CMB) is given by Durrer (2008),
where the only notational difference is the opposite sign of the gravitational
potential Φ.
The general strategy will be to compute the perturbed versions of the metric
as well as of the energy momentum tensor, so that one can then use Einstein’s
equation and the conservation equations to relate the metric to the energymomentum perturbations. This yields evolution equations for the perturbation
variables, which can for special cases be solved analytically, but in general need
to be computed numerically - in our case by Cmbeasy.
Decomposition of Perturbation Variables
The metric and energy-momentum perturbations on constant-time hypersurfaces can decomposed into Eigenfunctions of the 3-dimensional Laplacian,
∆Qk = −k2 Qk .
(3.16)
separating the perturbations into scalar, vector and tensor parts, which evolve
independently. In a flat universe, we have
Qk = exp(i k x),
(3.17)
so that the decomposition is nothing else than a Fourier transform, and therefore
e.g. the energy density is given by
Z
1
d3 k δρ(τ, k) Qk (x)
(3.18)
ρ(τ, x) = ρ +
(2π)3
This decomposition also works for vectors V, since they can be written as the
sum of a gradient and a rotation part,
V = gradϕ + rotB,
(3.19)
where ϕ is a scalar function. Together with the definitions of
Qi ≡ −k−1 Q,i
Qij ≡ k−2 Q,ij
and
1
+ δij Q,
3
the contribution of a scalar perturbation B(τ, k) to a vector field Bi can be
expressed as
Bi (τ, x) = B(τ, k)Q(k, x).
(3.20)
21
CHAPTER 3. PERTURBATIONS
Correspondingly, the scalar contributions to a tensor field are given by
Hij (τ, x) = HL (τ )Qδij + HT (τ )Qij .
(3.21)
This only gives the basis functions for scalar type perturbations, but basis functions for vector and tensor perturbations can be derived accordingly. Since we
will focus on scalar perturbations in this work, we do not repeat their definition
here, and specialize our treatment to the scalar case - the general case can be
found in any of the reviews referenced above.
Gauge Transformations
In the context of linear perturbation theory around a Friedmann-RobertsonWalker background, the application of general coordinate transformations (diffeomorphisms) becomes problematic. One could still make use of the usual
transformation laws of general relativity, but a given coordinate transformation
will, in general, not keep the FRW background invariant, which, in our context,
is too high a price to pay. If we consider small perturbations hµν around the
FRW background metric ḡ,
gµν = ḡµν + a2 hµν ,
(3.22)
we only allow transformations that leave ḡ invariant, and therefore deviate only
in first order from the identity. One can show (see Doran 2005, for a step-by-step
calculation) that such an infinitesimal coordinate transformation,
xµ → x̃µ = xµ + ǫ(τ, x),
(3.23)
can be written using the Lie derivative Lǫ , so that for example the metric perturbation h transforms as
h → h + a−2 Lǫ ḡ.
(3.24)
In cosmology, these infinitesimal coordinate transformations are called ‘gauge
transformations’. Conventionally, the transformation vector ǫ is parameterized
by its scalar and vector parts, T and L,
τ̃
x̃
i
= τ + T (τ )Q(x)
i
(3.25)
i
= x + L(τ )Q (x.
The usefulness of these transformations will become obvious in a moment, since
we now define the perturbations of the metric and the energy-momentum tensor;
one can then choose the functions T and L such that the general forms of the
perturbed metric and energy-momentum tensor have the most convenient form.
Metric Perturbations
For the case of scalar perturbations, Eq. (3.20) and Eq. (3.21) are the definitions
that can be inserted into the general form of the line element in a perturbed
FRW-metric, which is given by (Bardeen 1980)
ds2 = a(τ )2 −(1 + 2AQ) dτ 2 − 2Bi dτ dxi + (δij + 2Hij ) dxi dxj .
(3.26)
22
CHAPTER 3. PERTURBATIONS
We can now apply the general transformation Eq. (3.25) to obtain the behavior
of A, B i and Hi,j under gauge transformations. From the explicit calculation,
one finds that T and L can be chosen in such a way that two of the perturbation
variables vanish. Common choices are the synchronous gauge, defined by setting
A = B = 0, and the longitudinal, or conformal Newtonian gauge, which is
defined by the choice HT = B = 0. In this thesis, we always use quantities
in longitudinal gauge, or gauge-invariant quantities, which can be constructed
by combining the gauge-dependent quantities in a suitable way. Such gauge
invariant quantities were constructed by Bardeen (1980), namely
a′ −1
k σ − k−1 σ ′ , and
(3.27)
a
1
a′
Φ ≡ HL + HT − k−1 σ,
(3.28)
3
a
which are called the Bardeen potentials. In longitudinal gauge, we have
Ψ≡A−
σ ≡ k−1 H ′ T − B = 0.
(3.29)
so that the line element corresponding to the perturbed metric becomes
ds2 = a(τ )2 −(1 + 2Ψ)dτ 2 + (1 + 2Φ)δij dxi dxj ,
(3.30)
where we do not write out the basis functions Q and its counterparts explicitely
anymore.
Starting from the perturbed metric, it is now an algebraic exercise to calculate
the perturbed version of the left hand side of the Einstein Equation, Eq. (2.7).
The next step then is to compute the perturbations of the energy momentum
tensor.
The Perturbed Energy-Momentum Tensor
For the energy-momentum tensor, we consider a fluid, for which
Tνµ = p δνµ + (p + ρ)uµ uν + πνµ .
(3.31)
The four-velocity u here is the velocity of the fluid restframe with respect to
the coordinate frame. From the condition uµ uν gµν = −1, the components of
u can be shown to be
u0 = −a(τ )(1 + A(τ )),
(3.32)
ui = a (v(τ ) − B(τ )) Qi .
The spatial part contains the definition of the velocity perturbation v. Similarly,
the density perturbation δ to the background value ρ̄ is defined by
ρ ≡ ρ̄ (1 + δ Q) ,
(3.33)
but we ususally drop the overbar for background quantities, since there is little
potential for confusion. The spatial trace and the traceless part, in turn, define
the perturbations πL and Π via
pδji
πji
≡ p̄ [1 + πL Q] δi j ,
i
≡ p̄ Π Q j .
23
(3.34)
CHAPTER 3. PERTURBATIONS
After inserting these definitions of the perturbations δ, v, πL and Π in Eq. (3.31),
we can - just like we did for the perturbed metric - apply the general transformation Eq. (3.25) to determine these perturbation variables in the different gauges;
a very useful reference for the comparison between the longitudinal gauge that
we employ here and the also very popular synchronous gauge is the review by
Ma & Bertschinger (1995).
As we will often switch between gauge-invariant combinations V and Dg of
these variables and the longitudinal gauge, the explicit conversion between these
two sets is useful enough to write out explicitely, and is given by (e.g. Kodama
& Sasaki 1984)
V
Dg
1
≡ v − HT′ = v longit
k
1
≡ δ + 3(1 + w) HL + HT
3
= δlongit + 3(1 + w)Φ
1 p̄′
δ.
Γ ≡ πL −
w ρ̄′
(3.35)
(3.36)
(3.37)
(3.38)
Since we either use gauge-invariant quantities, or longitudinal gauge, we drop
the explicit ‘longit’ designation for v and δ from now on.
Evolution of Perturbations
After these preparations, the final steps are to first use Einstein’s Equations
to relate the perturbed metric perturbations to the energy-momentum perturbations, and then, to write down the conservation equations for the energymomentum. This yields the differential equations that govern the evolution of
the perturbations. The perturbed Einstein Equations yield:
a′
a′
2
2
2
′
a ρ δ = 2MP k Φ + 3
(3.39)
Φ − Ψ ,
a
a
′
a
Ψ − Φ′ ,
(3.40)
a2 v (ρ + p) = 2MP2 k
a
a2 p Π = −MP2 k2 (Φ + Ψ) .
(3.41)
Converting the longitudinal variables to their gauge-invariant counterparts, one
finds the perturbed Einstein Equations in gauge-invariant variables:
a2 ρ D = 2MP2 k2 Φ
′
a
′
2
2
Ψ−Φ
a (ρ + p)V = 2MP k
a
a2 p Π = −MP2 k2 (Φ + Ψ) .
(3.42)
(3.43)
(3.44)
We can combine these equations to obtain expressions for Φ and Ψ,
Ψ = −Φ − MP−2 k−2 a2 p Π
P 2
α
α a ρα Dg
P
Φ =
.
2MP2 k2 + α 3ρα a2 (1 + wα )
24
(3.45)
(3.46)
CHAPTER 3. PERTURBATIONS
This formulation is useful for the numerical integration, as the right hand side
only contains known variables at each timestep, and hence one can also compute
Φ′ and Ψ′ from these expression already during the integration.
Finally, T µν;µ = 0 leads to
d
a′
(ρ δ) = 0,
(ρ + p) 3Φ′ + k v + 3 ρ(δ + w πL ) +
a
dτ
together with
′
a v v′
v
2
+
+ (ρ′ + p′ ) − (ρ + p)Ψ = 0.
p Π − πL + (ρ + p) 4
3
ak
k
k
(3.47)
(3.48)
We can define the adiabatic sound speed as c2s = ρ̇/ṗ and isolate δ′ and v ′ , so
that we find an expression suitable for numerical integration:
a′
a′
δ′ = −(1 + w) kv + 3Φ′ − 3 wΓ + 3 δ (w − c2s ),
(3.49)
a
a
a′
w
2
c2s
v′ =
kδ +
k Γ − Π . (3.50)
3c2s − 1 v + k Ψ +
a
1+w
1+w
3
These expressions can also be converted to the gauge invariant variables defined
in Eq. (3.35), where we find
a′
a′
Dg − kV (1 + w) − 3 wΓ
(3.51)
a
a
a′
c2 k
wk
2
Γ− Π .
= k Ψ − 3c2s Φ + (3c2s − 1)V + s Dg +
a
1+w
1+w
3
Dg′ = −3(c2s − w)
V′
As these equations in the fluid-language are derived from the energy-momentum
tensor Eq. (3.31), they are valid for any component individually, provided its
energy-momentum is conserved, since it was the conservation of energy momentum that yields the evolution equations Eq. (3.47) and Eq. (3.48). A particularly
simple case, for example, are perturbations of cold dark matter, for which we
have w = c2s = 0, so that Eq. (3.51) reduces to:
′
Dgc = −kVc
Vc′ = −
a′
a
Vc + k Ψ.
(3.52)
(3.53)
To model the dark energy, it is useful to formulate the system in terms of an
arbitrary speed of sound (Bean & Dore 2004; Caldwell & Doran 2005; Erickson
et al. 2002) . In the rest frame of the fluid, we define
µ2 ≡
δp̃
,
δρ̃
(3.54)
which describes the actual speed with which perturbations propagate, which in
general is different from the adiabatic sound speed c2s . In this expressions, p̃ and
ρ̃ denote quantities in the rest frame of the fluid. From there, we can transform
to a random frame (Kodama & Sasaki 1984),
v
δ̃ = δ + 3H(1 + w) ,
k
25
(3.55)
CHAPTER 3. PERTURBATIONS
and find that the pressure perturbation δp is given by
V
w′
2
2
δp = µ δρ + ρ
4H(µ − w) +
.
k
1+w
(3.56)
A canonical scalar field has µ2 = 1, whereas in more general dark energy theories, like k-essence, the speed of sound can be time and scale dependent. Together, the equations combine to give the evolution equations for the perturbations of the dark energy, namely
δpφ
− wδφ ),
δφ′ = −3(1 + wφ )Φ′ − k(1 + wφ )Vφ − 3H(
ρφ
δpφ
′
vφ = (3wφ − 1)H(1 + wφ )vφ + k (1 + wφ )Ψ +
.
ρφ
(3.57)
(3.58)
When dark matter is coupled to the dark energy, the case we consider in
Chapter 6, only the combined energy momentum tensor is conserved, but the
individual conservation equations will obtain a source term that accounts for
the coupling - the changes to the formalism presented here will be outlined in
Section 6.1.
3.3 The Distribution Function
The fluid formulation is only appropriate for the dark energy and matter perturbations; neutrinos and photons can only be described by following their full
distribution function. As an example of this treatment, we review the case of
massive neutrinos here, in anticipation of the study of a dark energy - neutrino
coupling in Chapter 7, which will be based on this formalism.
When writing down the perturbed distribution function, one usually chooses
the comoving momentum qi = api as a coordinate in phase space, instead of the
proper momentum pi measured by an observer at a fixed spatial coordinate xi ,
and splits qi into magnitude q and direction n, qj = qnj , with ni ni = δij ni nj =
1. We follow Ma & Bertschinger (1995), and denote the perturbation to the
zeroth-order f0 by Ψps and write
f (xi , τ, q, nj ) = f0 (q) 1 + Ψps (xi , τ, q, nj ) .
We can recover the energy momentum tensor via
Z
1 Pµ Pν
f (xi , P, τ ),
Tµν = d3 P √
−g P 0
(3.59)
(3.60)
noting that in longitudinal gauge the conjugate momenta Pi of the coordinates
xi are related by
Pi = a(1 + Φ)pi .
(3.61)
For example, this relation leads to Eq. (2.18) in the unperturbed case for neutrinos; here, we will use this relation to recover the perturbations to the density
26
CHAPTER 3. PERTURBATIONS
and pressure from the perturbation of the distribution function, Ψps . The time
evolution of the distribution function is described schematically by
df
= C[f ].
dτ
(3.62)
This is the Boltzmann Equation, where the right hand side describes changes in
the distribution function due to collision between species. The detailed computation of this term is crucial when considering the change in f for the coupled
photon-baryon fluid before decoupling. Luckily, since we concentrate on the
neutrinos, we do not need to worry about this term, and only need to compute
the collisionless Boltzmann equation:
∂f
dxi ∂f
dq ∂f
dni ∂f
df
=
+
+
+
= 0.
i
dτ
∂τ
dτ ∂x
dτ ∂q
dτ ∂ni
(3.63)
The last term is a second order quantity, and is therefore neglected in the linear
calculation (without perturbations, the distribution function does not depend
on direction, and the momenta - and their direction - can only change in the
presence of a potential, which is also a first order quantity). The dxi /dτ can be
computed using P µ ≡ dxµ /dλ. The geodesic equation
P0
dP µ
+ Γµαβ P α P β = 0,
dτ
(3.64)
can be used to obtain the momentum derivative
dq
= −Φ′ − ǫ(q, τ )ni ∂i Ψ,
dτ
(3.65)
and we recall the definition of ǫ2 = q 2 + m2ν a2 from Chapter 2. We have spelled
out the momentum derivative explicitely, since we already have the case of
neutrinos with a varying mass in mind, where mν = mν (φ), so that this term in
the Boltzmann Equation will acquire extra terms in Chapter 7. Combining the
results for the individual terms, the Boltzmann Equation then can be expressed
as
∂Ψps
q
d ln f0
ǫ
′
+ i (k · n)Ψps +
(3.66)
−Φ − i (k · n)Ψ = 0.
∂τ
ǫ
d ln q
q
The only directional dependence of this equation comes via (k · n), which motivates the expansion using Legendre polynomials. Defining
ξ ≡ k−1 k · n,
(3.67)
To solve the Boltzmann Equation for massive neutrinos, Ψps is now expanded
in a Legendre series,
Ψps
∞
X
=
(−i)l (2l + 1)Ψps,l Pl (ξ)
(3.68)
l=0
with Legendre Polynomials Pl . The relation between the perturbation variables
in k-space and the expansion of the perturbed distribution function is then given
27
CHAPTER 3. PERTURBATIONS
by:
δρν
−4
= 4πa
Z
q 2 dq f0 (q)ǫΨps,0
Z
q2
4π −4
a
q 2 dq f0 (q)Ψps,0
3
ǫ
Z
(ρν + pν )v = 4πa−4 q 2 dqqf0 (q)Ψps,1
Z
q2
8π −4
a
q 2 dq f0 (q)Ψps,2
(ρν + pν )σν =
3
ǫ
δpν
=
(3.69)
(3.70)
(3.71)
(3.72)
However, we still need to solve the Boltzmann Equation, and express the solution
in a form that can be numerically evaluated. The well-known procedure to do
exactly that operates along the following lines:
• First multiply the k-space version of Eq. (3.66) by Pl (ξ),
• insert the expansion of Ψps ,
• integrate both sides of the equation over ξ:
R1
−1 dξ,
• and then consider the orthonormality relations of the Legendre polynomials, and their recurrence relation (l + 1)Pl+1 (ξ) = (2l + 1)ξPl (ξ).
The end result of this calculation is a hierarchy of expressions for the conformal
time derivatives of the moments Ψps,l , namely:
d ln f0
qk
Ψps,1 + Φ′
,
ǫ
d ln q
ǫk d ln f0
qk
(Ψps,0 − 2Ψps,2 ) − Ψ
,
3ǫ
3q d ln q
qk
[lΨps,l−1 − (l + 1)Ψps,l+1 ] , l ≥ 2.
(2l + 1)ǫ
Ψ′ps,0 = −
Ψ′ps,1 =
Ψ′ps,l =
(3.73)
This is an infinite set of equations, one for each moment l = 0..∞. Fortunately,
higher multipole moments decay rapidly, and in practice only the first few multipoles need to be evolved numerically (∼ 30 already yields an accuracy that
is sufficient for most purposes). Instead of setting the last multipole to zero,
approximating the behavior of the first non-evolved one helps to improve the
accuracy; there are a few approximation schemes that have been proposed in
the literature, but the original one by Ma & Bertschinger (1995) works quite
well, setting
Ψps,lmax+1 ≈
(2lmax + 1)ǫ
Ψps,lmax − Ψps,lmax−1 .
qkτ
(3.74)
This completes the set of equations necessary to numerically compute the evolution of the Boltzmann hierarchy for massive neutrinos from some suitable initial
conditions.
28
CHAPTER 3. PERTURBATIONS
Photons, Baryons, and Electrons
What remains is the derivation of the evolution of the perturbations for the
photons, and the baryons. Conceptually, the derivation of the photon evolution
follows the same steps as for the massive neutrinos (indeed, for massless particles
the phase space integrals in the equations can be simplified considerably). What
makes photons more interesting, however, are their interactions with the baryons
and electrons.
The mathematical description of the CMB polarization is one of the more
intricate parts of the theory of the Cosmic Microwave Background. However,
none of our considerations in the following introduce any changes to the standard
treatment (literature that uses the same notation as the numerical implementation in Cmbeasy includes Durrer 2008; Doran 2005). When considering a dark
energy component coupled either to dark matter (Chapter 6) or to neutrinos
(Chapter 7), only the perturbation equations that we have outlined above will
be modified. Nevertheless, since the physical observables are the photons of the
CMB, let us briefly sketch the remaining steps towards the description of the
anisotropies that we see today.
Boltzmann Equations
Instead of using the photon distribution function directly, we use the temperatrue perturbation ∆ as a more intuitive variable. This is, of course, directly
connected to the distribution function of the photons, so that we can relate the
first few moments of the temperature perturbation ∆ to the fluid perturbation
variables in a given gauge in a way similar to the neutrino case, namely via
δγ
= 4Φ + ∆0 ,
(3.75)
Vγ = ∆ 1 ,
(3.76)
12
∆2 .
(3.77)
Πγ =
5
Instead of ∆, which is again a gauge-dependent variable, we can also define a
gauge-invariant quantity,
1
(3.78)
M ≡ ∆ + 2 HL + HT + iξk−1 HT′ − ξ 2 HT ,
3
which, if ∆ is in longitudinal gauge, reduces to
M = ∆ + 2Φ,
and can be expanded in terms of Legendre polynomials 2
X
M(τ, x, n) =
(−i)l Ml (k, τ )Pl (n)Q.
(3.79)
(3.80)
l
The polarization is expanded via spin-weighted spherical harmonics,
r
X
4π
Y 0 (n)Q.
(Q ± iU )(τ, x, n) =
(−i)(El ± iBl ) ×
2l + 1±2 l
(3.81)
l=2
2
Note that Q is different from the eigenfunctions of the 3-d Laplacian Q; see Eq. (3.17).
29
CHAPTER 3. PERTURBATIONS
For scalar perturbations, Bl = 03 . The Boltzmann equation for M is given by
M′ + ikξM + κ′ M
′
= iξk (Φ − Ψ) + κ
i
h√
1
1 γ
∆ − iξVb + P2 (ξ) 6E2 − M2 ,
4 g
10
(3.82)
whereas for the polarization one finds
i
h√
κ′
(3.83)
(P2 (ξ) − 1) 6E2 − M2 .
10
These two equations need to be solved in addition to the evolution equations for
neutrinos, baryons and dark matter and the dark energy, which is again done
by inserting the multipole expansions Eq. (3.80) and Eq. (3.81), and then of
projecting out multipoles, resulting in a hierarchy of equations for Ml and El .
Q′ + ikξQ + κ′ Q =
Mapping Anisotropies onto the Sky
The actual observed quantity, of course, is the temperature (and polarization)
anisotropy today, so we want to know M(τ0 , ξ); the clever and now standard
way of how to compute this from the hierarchies for Ml and El is due to Seljak
& Zaldarriaga (1996), who showed how it is possible to cleanly separate the
integration into a geometrical term and a source term, where only the source
term depends on the El and Ml (and only on the first few of them). This
line of sight approach allows to calculate the source terms, which depend on
the system of evolution equations for all the perturbation variables we have
considered so far, for only relatively few k-modes, as they only vary slowly with
the wavenumber. The integration over the geometrical part can then be done
afterwards; this approach saves a large amount of computational time - about
two orders of magnitude on then-standard CPUs.
The first step in this approach is to define a function L,
L ≡ exp(iξkτ ) exp(κ(τ )M
(3.84)
and then note that in terms of this function, the Boltzmann equation Eq. (3.80)
is given by:
1 2
′
iξkτ κ(τ )
′ 1 γ
D − iξVb − (3ξ − 1)C ,
(3.85)
L =e e
iξk(Φ − Φ) + κ
4 g
2
√
where C = (M2 − 6E2 )/10. The temperature anisotropy today then is
M(ξ, τ0 ) = eiξkτ0 e−κ(τ0 ) L(τ0 ),
(3.86)
so L′ needs to be integrated over conformal time; the integration can be rewritten
(i.e. integrated by parts) to eliminate any dependence on ξ that appeared on the
right hand side of Eq. (3.80). It is also useful to define the visibility function,
g ≡ κ′ exp [κ(τ ) − κ(τ0 )] .
3
(3.87)
Weak lensing of the CMB introduces a B-mode component, which we do not discuss here;
see Lewis & Challinor (2006) for a review of CMB lensing, and Robbers (2006) for a
discussion of the numerical implementation in Cmbeasy. An example B-mode power
spectrum is shown in Fig. 4.8.
30
CHAPTER 3. PERTURBATIONS
This function is sharply peaked at the time of recombination, and will encode the
fact that we see the CMB (almost) as it was at that time. These manipulations
result in the final expression of the temperature anisotropy,
Z τ0
exp [iξk(τ − τ0 )] ST (k, τ )dτ,
(3.88)
M(ξ, τ0 ) =
0
with the source term
ST
3 ′
3
Vb
+ 2 C + g′′ 2 C
= −e
Φ −Ψ +g
k
k
2k
′
V
1
C
3
+ g Dgγ + b − (Φ − Ψ) + + 2 C ′′
4
k
2 2k
k(τ )−k(τ0 )
′
′
′
(3.89)
(3.90)
In this expression, the most important features the temperature anisotropies
that we observe today are already apparent: the last term has a prefactor of g,
therefore the anisotropies get their largest contribution from the density contrast
in the photon gas from that time - in fact, the behavior of Dgγ is the main
contribution towards the oscillations in the Cl spectrum. The Vb term (Doppler
term) describes the relative motion of emitter and observer. The term with
[Φ′ − Ψ′ ] is the integrated Sachs-Wolfe effect, describing the red- or blue-shift
of a photon that travels through an evolving gravitational potential; finally
the [Φ − Ψ] term is responsible for the ordinary Sachs-Wolfe effect, the main
contribution to the power spectrum on very large scales.
The Multipole Spectrum
This brings us full circle, as we can now express the Cl s that we had introduced
in Section 3.1 (p. 20) in terms of the temperature anisotropy. First, we note that
the evolution of the perturbation quantities does not depend on the direction of
k, so that for a perturbation A(k, τ0 , n) we can write
A(k, τ0 , n) = ψ ini (k)A(k, τ0 , n),
(3.91)
where the power spectrum of ψ ini (k),
hψ ini (k) ψ ini (k′ )∗ i =
2π 3
PΨ (k)δD (k − k′ ),
k3
(3.92)
is the input spectrum of Eq. (3.2) and encodes the initial correlation. To obtain
the Cl , we first project out the individual alm , and then insert the Fourier
decomposition,
Z
alm =
dn [Ylm (n)]∗ A(x0 , τ0 , n)
(3.93)
Z
Z
d3 k
∗
m
=
dn [Yl (n)]
A(k, τ0 , n) exp(ikx).
(3.94)
(2π)3
We can find the expression for the temperature anisotropy, M, by substituting
A by the integral expression we found for M in Eq. (3.88), and use the definition
31
CHAPTER 3. PERTURBATIONS
of the Cl in terms of the alm , so that we have finally have an expression
temperature power spectrum:
1 X ∗
ha alm i
ClT T =
2l + 1 m lm
Z
d3 k
1
PΨ (k)
=
2l + 1
(2π)3
2
X Z
m
×
dτ dn Yl (n) exp(ikξ[τ − τ0 ]) ST (k, τ ) .
for the
(3.95)
(3.96)
(3.97)
m
The last step then is to use the following relation for the spherical Bessel functions jl :
Z
p
dn Ylm (n) exp(iξx) = 4π(2l + 1)il jl (x)δ0m .
(3.98)
The integral over n in Eq. (3.95) will therefore give a spherical Bessel function
in the final result. If we define
Z
(3.99)
MTl = (2l + 1) dτ jl (k[τ0 − τ ]) ST (k, τ ),
the ClT T can be written in the following way:
Z
d3 k
4π
TT
PΨ (k) [Ml (k)]2 .
Cl =
(2l + 1)2
(2π)3
(3.100)
From this expression, the recipe for a numerical computation of the Cl is obvious:
• evolve the perturbation equations for a suitably chosen set of Fourier
modes k from their initial conditions to today and save the source term
ST at a sufficient number of timesteps τi
• compute the result of folding the source terms with the spherical Bessel
functions
• square the result, and obtain the final ClT T by integrating over Fourier
modes k times the initial power spectrum PΨ (k).
As to polarization, it is remarkable that the final result can be expressed just as
concisely as for the temperature, even though the complete derivation is more
involved than for the temperature (among other things, we have to deal with
spin-weighted spherical harmonics, which also depend on the chosen polarization
basis orthogonal to n). The source term for the polarization can be written as
SE = 3g
√
C
,
4 [k(τ − τ0 )]2
(3.101)
6E2 )/10 as above, and together with the definition for ∆E
l ,
s
Z
(l − 2)!
E
∆l = (2l + 1)
dτ jl (k[τ0 − τ ]) SE (k, τ ),
(3.102)
(l + 2)!
where C = (M2 −
32
CHAPTER 3. PERTURBATIONS
the polarization power spectrum and the cross correlation with temperature are
given by
ClEE
=
ClT E =
Z
2
d3 k
4π
PΨ (k) ∆E
l (k)
2
3
(2l + 1)
(2π)
Z
d3 k
4π
PΨ (k)∆E
l (k)Ml (k).
(2l + 1)2
(2π)3
(3.103)
(3.104)
The only missing piece now is how to set the initial values of the perturbation
variables, i.e. relating the initial fluctuations in the densities, velocities and so
forth at early times to the initial curvature perturbation and its power spectrum,
Eq. (3.2).
3.4 Initial Conditions
The strategy for finding the initial conditions is to consider the evolution equations for each k−mode at early times, during radiation domination, in the limit
where x ≡ kτ ≪ 1 when the mode is still far outside the horizon. One can then
expand the evolution equations for the perturbation variables in powers of x, and
write down the system with the perturbed Einstein Equations. The solutions
to these equations are classified using the gauge-invariant entropy perturbation
Sα,β between two species α and β (see e. g. Kodama & Sasaki 1984)
Sα,β ≡
Dgα
Dgβ
−
.
1 + wα 1 + wβ
(3.105)
Adiabatic initial conditions correspond to the case when all Sα,β vanish. To
classify the remaining modes, one introduces the curvature perturbation4 on
hyper-surfaces of uniform energy density of the component α,
1
δρα
,
(3.106)
ζ = HL + HT +
3
3(1 + wα )ρα
where the total curvature perturbation on slices of uniform total energy density
is given by
P
1
α δρα
,
(3.107)
ζtot = HL + HT + P
3
α 3(1 + wα )ρα
and for a constant w, we have ζtot ∝ Ψ. Besides the adiabatic initial conditions,
one defines the ‘isocurvature’ modes by requiring ζtot = 0, while one species
σ has Sα,β 6= 0, and for all other species we still require Sα,β = 0. In the
remainder of this work, we only consider adiabatic initial conditions, for which
explicit expressions in a universe with quintessence are given by Doran et al.
(2003)
4
4
4
Dgν = Dgγ = Dgb = Dgc = (1 + wφ )−1 Dgφ ,
(3.108)
3
3
3
4
There are slightly different definitions of this quantity in the literature; we follow Doran
et al. (2003).
33
CHAPTER 3. PERTURBATIONS
and
5
Vc = Vγ = Vν = Vφ = − P,
4
(3.109)
where P = (15 + 4Ων )−1 . The overall amplitude of the perturbations for each
k−mode is then set by calculating Ψ via Eq. (3.45), and normalizing according
to the amplitude of the initial power spectrum of Eq. (3.2), determined by the
specific values of As and ns .
3.5 Numerical Tools
3.5.1 Cmbeasy
We have now all the ingredients to numerically evaluate the background history,
the evolution of perturbations, and the CMB power spectra for cosmologies with
a dark energy component, and this will be the topic of the next chapters. The
results that we discuss there have been computed using Cmbeasy (Doran 2003),
updated and extended by implementing the respective models, e.g. coupled
dark energy cosmologies (Chapter 6) and growing neutrino quintessence models
(Chapter 7).
Cmbeasy is implemented in the C++ programming language, and is based
on Cmbfast (Seljak & Zaldarriaga 1996), but almost all of its code has been
re-implemented; it is, however, not a completely independent implementation.
From the discussion of the theory of the homogeneous background in Chapter 2 and of the perturbations around this background in this chapter, it is clear
that the computation of model predictions can be split in separate parts:
• computing the expansion history for a given model,
• calculating the thermal history; as discussed on p. 19, we rely on a version
of Recfast (Seager et al. 2000; Wong et al. 2007) ported to C++,
• evolving the perturbations from their initial conditions, and compute the
‘observables’, such as the CMB spectra following the recipe outlined on
p. 32, and e.g. the matter power spectra.
The output generated by this procedure can then be compared to observational
data, in order to asses the viability of a given model. The predictions of a particular model depend on the input parameters, of which the ΛCDM-model has
at least six: the baryon and cold dark matter densities today, the value of the
Hubble paramater, the slope and amplitude of the initial fluctuations, and the
optical depth to the last scattering surface. Then there are the neutrino masses,
possibly a running of the spectral index, the amplitude of tensor fluctuations
and so forth, so that the dimensionality of the parameter space quickly becomes
rather high, especially when considering dark energy models that are characterized by additional parameters; we therefore need methods to reduce the amount
of computational resources needed to explore the parameter space.
34
CHAPTER 3. PERTURBATIONS
3.5.2 AnalyzeThis!
Specifically, when we study cosmological models in the following, we want to
quantify how likely it is that a particular model, with a specific set of values for
its parameters, denoted m, is the true description of our Universe, given a set
of observational data di with associated errors σi . That is, we want to find the
propability π(m|{di , σi }). However, this is not exactly the quantity that we can
compute directly. Rather, the results of our computation are either quantities
like the luminosity distances, or expectation values and variance for e.g. the
temperature of the CMB in different directions for a given model, that is to say
we compute π({di , σi }|m). The connection between these two quantities is of
course Bayes’ theorem, which states
π(m|{di , σi }) =
π({di , σi }|m) π(m)
,
π({di , σi })
(3.110)
where π(m) is the prior, encoding our prior knowledge of the parameter space,
e. g. the bounds on the values that the model parameters can take, and
π({di , σi }) is the evidence. In the following, we will estimate the confidence
intervals for parameters within the context of one cosmological theory (e.g.
ΛCDM) at a time, so that the total probability should be normalized to unity,
and the evidence is only an overall normalization factor and hence unimportant.
It is, however, the crucial quantity in the context of Bayesian model comparison
techniques (see e.g. Trotta 2008).
The most obvious way to compute the posterior distribution would be to
interpolate it from a sufficient number of likelihood evaluations on a grid.
This approach is however infeasible for a high dimensionality of the parameter space. There are several techniques that solve this problem, and the method
of choice for cosmological parameter estimation is the Markov Chain Monte
Carlo (MCMC) approach, made popular in cosmology by the implementation
in CosmoMC (Lewis & Bridle 2002), which employs Camb (Lewis et al. 2000)
as its underlying Boltzmann code. In the following, we use the MCMC implementation of Cmbeasy, christened ‘AnalyzeThis!’, as originally presented in
Müller (2005) as well as Doran & Müller (2004), with several extensions and
modifications.
35
CHAPTER 3. PERTURBATIONS
36
4 Early Dark Energy
In the previous chapters, we have covered the foundations necessary to study
specific dark energy models, to compute their predictions, and to compare those
predictions with observational data. In this chapter, we will apply this machinery to the investigation of ‘early dark energy’ models, i.e. cosmological models
in which the dark energy contributes a non-negligible fraction to the total energy budget even at high redshifts. In this regard, this class of early dark energy
models is distinct not only from ΛCDM cosmologies, but also from many standard scalar field models of dark energy, where the dark energy component is
assumed to be of no importance at all for redshifts higher than a few. We will
now briefly review the effect of early dark energy on the cosmological evolution.
We then compare a sample of early dark energy models to observational data
and compute upper bounds for the amount of early dark energy. At the same
time, we compare these early dark energy models to the standard ΛCDM case
and other scalar field dark energy models.
In the second part of this chapter, we ask the question whether it is possible
to detect early dark energy in the CMB power spectra. We show quantitatively
how assuming a negligible fraction of early dark energy for the analysis of observations of the baryon acoustic oscillations leads to a substantially biased (i.e.
wrong) interpretation of the data, and to the incorrect inference of cosmological
parameters.
4.1 Cosmological Implications
The presence of early dark energy (EDE) can potentially have an impact on all
three levels of cosmological evolution, the background, the linear perturbations,
and possibly even on the non-linear formation of structure. These effects have
been studied in the literature, for example the structure formation in the linear
regime by Ferreira & Joyce (1997); Doran et al. (2001). The effect on the
CMB power spectrum was investigated by Doran & Lilley (2002), and then by
Caldwell et al. (2003), who considered the compatibility of the CMB spectra
of these models with the results from the WMAP satellite obtained through
one year of observations. For a more recent review of dark energy models that
includes a discussion of early dark energy, see e.g. Linder (2008) and references
therein.
On the background level, the presence of early dark energy changes the evolution of the Hubble parameter. This is easy to see if we assume that the dark
energy scales as a constant fraction of the energy density, i.e. contributes a
constant fraction of ΩEDE to the total energy budget. The Hubble parameter
37
CHAPTER 4. EARLY DARK ENERGY
then depends on the energy density as
1
[ρmatter,radiation (z) + ρEDE (z)]
H 2 (z) =
3MP2
1
=
ρmatter,radiation (z) + H 2 (z)ΩEDE (z)
2
3MP
1 ρmatter,radiation (z)
=
3MP2 1 − ΩEDE (z)
(4.1)
At high redshifts the Hubble parameter therefore scales as H(z) ∼ (1−ΩEDE )−1/2 .
This also means that cosmological distance scales in an early dark energy universe are changed compared to e.g. a ΛCDM-universe - this will, for example,
change the angle under which we see the acoustic peaks in the CMB.
The change in the expansion rate induced by the presence of early dark energy
during the radiation era (at T ∼ 1 MeV) affects Big Bang Nucleosynthesis, and
hence the light element abundance inferred from observations, can constrain
the amount of early dark energy. Current limits are ΩEDE (T ∼ 1MeV) . 0.2
(Cyburt et al. 2005), considerably weaker than the bounds we will find from our
analysis.
Early dark energy also affects structure formation - the growth of linear perturbations is suppressed (Ferreira & Joyce 1998). Hence, in a universe in which
dark energy is important when structures form, these structures grow slower.
To quantify the effect, we consider an average of the dark energy contribution
towards the total energy density during structure formation Doran et al. (2001),
Z ln atr
−1
d
Ωd (a) d ln a,
(4.2)
Ω̄sf ≡ [ln atr − ln aeq ]
ln aeq
where the scale factor atr is chosen as atr = 1/3. The suppression of growth can
be understood by considering the sub-horizon limit of the equation of motion
for the dark matter perturbations (Ferreira & Joyce 1998),
3
′′
′
(4.3)
δm
+ Hδm
− H2 Ωm δm = 0.
2
If the dark energy is negligible during structure formation, we have Ωm ≈ 1,
and δm ∝ a. In an early dark energy cosmology Ωm < 1, and the solution of
Eq. (4.3) is
i
h√
δm ∝ a
25−24Ω̄dsf −1 /4
d
≈ a1−3Ω̄sf /5 .
(4.4)
Hence, all k−modes inside the horizon at a given time will suffer this suppresion.
Modes that had already been inside the horizon around matter-radiation equality will all suffer the same suppression. As an approximation to the effect on
σ8 , the rms density fluctuations averaged over 8 h−1 Mpc spheres, Doran et al.
(2001) suggested
s
τ0 (EDE)
σ8 (EDE)
−1
d
−(1+w̄ )/5
≈ (aeq )3Ω̄sf /5 (1 − ΩΛ
,
(4.5)
0)
σ8 (Λ)
τ0 (Λ)
where w̄ is a suitable average of the dark energy equation of state from matter
radiation equality until today; this estimate roughly gives a decrease of σ8 by
about 50%, if Ω̄dsf is increased by 10%.
38
CHAPTER 4. EARLY DARK ENERGY
4.2 Constraints on Early Dark Energy
4.2.1 Models and Parameterizations
The aim of our study is to determine the amount of early dark energy that
is compatible with observations. In order to obtain robust results, we do not
choose one specific dark energy model, but consider a broad sample of models
and parameterizations. We also include models that by construction do not have
an early dark energy component, in order to assess potential differences in the
inferred parameter bounds on the standard cosmological parameters. Naturally,
we include the ΛCDM models for comparison purposes.
Leaping Kinetic Term Quintessence
The first model we consider is a quintessence model with a ‘Leaping Kinetic
Term’ (the ‘LKT’ model). The LKT model was introduced by Hebecker & Wetterich (2001), who discussed the naturalness of different quintessence scenarios
from a field theoretical point of view by writing the scalar field Lagrangian as
1
L(φ) = (∂φ)2 k2 (φ) + MP4 exp(−φ/MP ),
2
(4.6)
and considering specific forms of the φ-dependent kinetic coefficient k(φ) > 0.
As one particular case for our comparison of different dark energy models we
choose the example k(φ) to be given by
k(φ) = kmin + tanh(φ − φ1 ) + 1.
(4.7)
Reasonable values for the constants kmin and φ1 that lead to a typical quintessence
evolution of the scalar field are e.g. kmin ≈ 0.1 and φ1 ≈ 276. The scalar field
will then first track the component that dominates the background (first radiation, then matter), and finally, at a time close to today, the ‘leap’ in the kinetic
coefficient will cause the quintessence component to become dominant and lead
to late-time acceleration. It is noteworthy that one can rewrite Eq. (4.6) as a
completely equivalent Lagrangian, but one that has a standard kinetic term by
defining a function K(φ) such that
k(φ) =
∂K(φ)
,
∂φ
(4.8)
and introducing a new field variable χ = K(φ), so that Eq. (4.6) becomes
1
L(φ) = (∂φ)2 + MP4 exp −K −1 (χ/MP ) .
2
(4.9)
In the numerical implementation in Cmbeasy we use this rewritten form of the
potential.
39
CHAPTER 4. EARLY DARK ENERGY
Parameterizing the Energy Density I
As the first of two examples of parameterizations of the energy density of the
quintessence component, we consider the form proposed by Wetterich (2004),
with Ωd (a) given as
eR
,
(4.10)
Ωd (a) =
1 + eR
where R(a) is defined by
R(a) ≡ ln
Ωd (a)
,
1 − Ωd (a)
(4.11)
and can be written in terms of a parameter b and a constant R0 ,
R(a) = R0 −
3w0 ln(a)
.
1 − b ln(a)
(4.12)
The introduction of the parameter b is advantageous because it is directly related
to the asymptotic early-time value of the energy density, Ω⋆ , by the relation
−1
Ω0d
1 − Ω⋆
+ ln
.
(4.13)
b = −3w0 ln
Ω⋆
1 − Ω0d
Additionally, this also allows to give a simple form of the equation of state w(a),
w0
,
(4.14)
w(a) =
[1 − b ln(a)]2
so that this parameterization can also be considered a particularly simple parameterization of the equation of state, which allows for the possibility of early
dark energy, completely specified by the parameter set of (Ω0d , w0 , Ω⋆ ). In this
class of models, the energy density Ωd rather slowly relaxes from today’s Ω0d
to its asymptotic value of Ω⋆ at early times. Observational constraints on the
parameters of this model, including Ω̄dsf , were computed by (Doran et al. 2005);
our investigation here extends this study by the inclusion of more (and updated)
data.
Parameterizing the Energy Density II
As a second way to parameterize the evolution of the fraction of dark energy,
we include in our analysis the class of models proposed in Doran & Robbers
(2006), which are discussed at length in Robbers (2006). Here, Ωd (a) is given
by
Ω0 − (1 − a−3w0 )Ωed
+ Ωed (1 − a−3w0 ).
(4.15)
Ωd (a) = d0
Ωd + (1 − Ω0d )a3w0
This specific form of Ωd (a) (which we will call the ‘Ω(w0 , Ωed )’ model) is motivated by two considerations: firstly, that the late-time behavior should be as
close as possible to a cosmological constant, and secondly, that the fraction
of dark energy at early times should be given by a single number, Ωed (and is
therefore necessarily constant at early times, before changing to a cosmologicalconstant-like behavior for redshifts z . 2). The amount of dark energy therefore
is never smaller than a given Ωed , which is one of the three input parameters
Ω0d , w0 , Ωed for this parameterization.
40
CHAPTER 4. EARLY DARK ENERGY
Interpolating model
Allowing more freedom in the time-evolution of the energy density, we also
consider a case where Ωd is linearly (in ln a) interpolated between values at
different redshifts zi ; we choose the zi as zi = 1, 3, 10, 100, 1100. In particular,
this includes the possibility of a non-monotonic evolution.
Parameterizing the Equation of State I
Instead of parameterizing the energy density, we can also choose a specific form
for the equation of state, as we have seen in Section 2.3. A very popular choice
to parameterize the equation of state is
w(a) = w0 + w1 (1 − a) = w0 + w1
z
,
1+z
(4.16)
which has originally been discussed in the context of cosmology by Chevallier
& Polarski (2001) and Linder (2003). This particular w(a) is characterized by a
a linear redshift behavior at low redshifts while at high redshifts w → w0 + w1 ;
early dark energy models cannot be described by this parameterization. It is,
however, a very useful tool for the description of models with a dynamical dark
energy component, but no early dark energy.
Parameterizing the Equation of State II
Another widely used version of a parameterization for the equation of state of
the dark energy (the ‘C&C’ parameterization) has been introduced by Bassett
et al. (2002), and later been generalized by Corasaniti & Copeland (2003). The
evolution of w(a) is in this case being written as
am
c
w(a) = w0 − w0 ×
1 + e ∆m
1 + e−
a−am
c
∆m
a−1
×
1 + e− ∆m
1
,
(4.17)
1 − e ∆m
where we have set an additional parameter wm , which in the original version
of Eq. (4.17) describes the value of w during matter domination, to zero. An
equation of state described by this formula features two distinct phases; it first
stays close to its value today w0 , and for small scale factors it is essentially zero.
This evolution is described by two parameters apart from w0 , the equation of
state today: A scale factor am
c , which signifies the scale factor where w(a) turns
from w0 to zero, and a width, ∆m , which is a measure for the width of this
transition.
4.2.2 Observational Data
The cosmological predictions for all of these models are computed using Cmbeasy,
and then compared with a variety of observational data. We first compare only
with the CMB data from WMAP (the release after three years of data). In a
second step, we additionally include more sets of CMB data (which extend to
higher Cl in the TT-spectra), data from SnIa surveys, as well as from large scale
41
CHAPTER 4. EARLY DARK ENERGY
structure. Specifically, we use the data from the extended Very Small Array, a
radio telescope located at the Teide Observatory site in Tenerife, which provides
data points for the TT spectrum out to l ∼ 1500 (Dickinson et al. 2004), as well
as data from the Cosmic Microwave Background Imager (CBI) extending out
to even slightly higher multipoles closer to l ∼ 2000, where we use data released
in 2004 (Readhead et al. 2004). Our set also includes data from ACBAR (the
Arcminute Cosmology Bolometer Array Receiver, observed from the Viper telescope at the South Pole (Kuo et al. 2004), and finally from the 2003 flight of
the balloon-borne Boomerang probe (MacTavish et al. 2005). While ACBAR’s
highest-l binned data point is at l = 2500, the Boomerang data is the most
accurate of our set in the range 600 < l < 1200 for the TT-spectrum.
The distribution of matter in our models is probed by comparing the linear
matter power spectra with data from the Sloan Digital Sky Survey (Tegmark
et al. 2004b) and the 2dF Galaxy Redshift Survey (Percival et al. 2001). For
this analysis, we marginalize over the bias between the galaxy power spectrum
observed by these surveys and the underlying distribution of cold dark matter,
i.e. we do not use any information on the normalization of the measured power
spectra. As we will see below, the inclusion of measurements of the overall
normalization can provide quite powerful additional constraints on the allowed
fraction of early dark energy, which is to be expected from its influence on the
growth of perturbations as we have discussed at the beginning of this chapter.
Finally, we also make use of the results of SnIa surveys, where we include the
first-year data from the Supernovae Legacy Survey (Astier et al. 2005) and the
Gold Set of Riess et al. (2004).
4.2.3 Parameter Constraints
After these preparations, we use Cmbeasy and its MCMC implementation AnalyzeThis! to finally compute the parameter constraints on the standard cosmological parameters, including the quantity that we are most interested in,
the fraction of dark energy during structure formation, Ω̄dsf . Looking first at
our interpolated model, we can see in Fig. 4.1 that already from the WMAP
data alone the time evolution of dark energy is quite well constrained. It is also
apparent that also for the combined set of data, the values typical for a ΛCDM
cosmology are always inside the one-sigma bounds from the MCMC analysis.
However, at redshifts z = 3 and higher, the amount of dark energy is allowed
to be considerably larger.
As a general result from our Monte-Carlo analysis, we find that cosmological
models with a dynamical dark energy component fit the observational data as
well as ΛCDM, but not better - the addition of more degrees of freedom in the
form of additional parameters for the evolution of the dark energy component
does not lead to correspondingly better likelihood values. Our complete results are shown in Fig. 4.2, which includes both the constraints on the standard
cosmological parameters, as well as their scatter between the different models. In our analysis, we compute the predictions of our sample of cosmological
models depending on the standard cosmological parameters: the present matter energy fraction Ω0m and baryon energy fraction Ω0b , the Hubble parameter
42
CHAPTER 4. EARLY DARK ENERGY
0.8
0.7
0.6
Ωd
0.5
0.4
0.3
0.2
0.1
0
0.1
1
10
redshift
100
1000
Figure 4.1: The fraction of dark energy compatible with observations in the interpolated model. Shown are the one-sigma confidence intervals at
the different redshifts. Red bars correspond to the result from the
WMAP data alone, whereas the blue bars correspond to the combined set described in the text. The black line shows the evolution
of ΩΛ in a ΛCDM-Universe with the WMAP 3-year best fit values
for the standard cosmological parameters.
h, optical depth τ , scalar spectral index ns and the initial scalar amplitude
As , which we took into account using the observationally relevant combination
ln(1010 As ) − 2τ . Fig. 4.2 summarizes the results for these parameters for our
choice of flat priors on all parameters. Fig. 4.2 also includes the results for
Ω̄dsf and σ8 . Naturally, all models additionally depend on their respective dark
energy parameters.
The results for the standard cosmological parameters show little deviation
from their values in ΛCDM-cosmologies - the choice of a particular dark energy
model does not lead to any significant shift of the confidence intervals. The trend
for models that allow for comparatively large fractions of early dark energy to
have correspondingly lower σ8 is clearly visible. Most surprising though are
the confidence intervals for Ω̄dsf for the interpolated model. While we have seen
in Fig. 4.1 that the preferred amount of dark energy at the redshift points zi
always includes the ΛCDM-value, this model allows for considerably larger dark
energy fractions at higher redshifts than the other models, and in particular
much higher values than a ΛCDM universe. This is mainly due to the fact that
the evolution of the dark energy density in this model is not required to be
monotonic, which can lead to substantial contributions to the integrated SachsWolfe effect. In fact, the free Ωd at the different redshift points zi can in a sense
‘manufacture’ the shape of the large-scale TT-spectrum to be compatible with
the WMAP data even for rather large values of Ω̄dsf . The relative independence
of Ωd in a given redshift bin from its neighboring values in effect allows for a
43
CHAPTER 4. EARLY DARK ENERGY
Figure 4.2: Monte-Carlo results for the cosmological parameters. The results
shown on the left-hand side are for the comparison to WMAP 3year data alone, the right-hand side corresponds to the results for
the combined set described in the text. The errorbars depict the
one-sigma bounds for the different models; the shaded regions show
the corresponding bounds for ΛCDM-cosmologies. The crosses mark
the parameter values for the best-fit models in the respective MonteCarlo chains.
44
CHAPTER 4. EARLY DARK ENERGY
68% confidence limit
(−)0.004
95% confidence limit
(−)0.008
best fit
Ωm h2
0.144(+)0.004
Ωb h2
0.0227(+)0.0006
h
0.66(+)0.02
(−)0.02
0.66(+)0.03
(−)0.03
0.67
τ
0.11(+)0.03
(−)0.03
0.11(+)0.05
(−)0.05
0.09
ns
0.97(+)0.02
(−)0.01
0.97(+)0.03
(−)0.03
0.96
ln(1010 As ) − 2τ
2.91(+)0.02
(−)0.02
2.91(+)0.03
(−)0.03
2.92
Ωe
< 0.008
< 0.017
0.0006
w0
< −0.92
< −0.84
−1
0.144(+)0.009
(−)0.0006
(−)0.0012
0.0227(+)0.0013
0.144
0.0224
Table 4.1: Marginalized one-dimensional likelihood constraints for the cosmological parameters for our complete data set described in the text,
including SnIa observations, matter power spectra, and CMB observations, together with Ly-α data from Seljak et al. (2006); McDonald
et al. (2005). The corresponding 2d-plots for the early dark energy
parameter Ωe are shown in Fig. 4.3.
relatively large number of models in the Monte-Carlo chains with these high
Ω̄dsf values. The models with a high Ω̄dsf then have a substantially lower σ8 , and
the combination of the CMB data with the other observations in our combined
data set then leads to a clear decrease in the allowed range for Ω̄dsf also for the
interpolated model, but the upper 2 − σ limit is still only Ω̄dsf . 6.5% for our
combined set.
Summarizing the results for the other models, we find the expected very
small upper (2σ) bound of ∼ 0.5% for the amount of early dark energy for
the w(a) = w0 + w1 (1 − a) parameterization, which is ‘by design’ limited to
have dark energy fractions close to the ΛCDM values. For all other models, the
combined sets a constraint of less than about three percent, and the Ω(w0 , Ωed )
model yields Ω̄dsf . 4%.
4.2.4 Constraints from Ly-α data
We now consider the constraints on the abundance of early dark energy that
can be inferred from observations of the Lyman-α forest, using the data set of
Seljak et al. (2006), which is obtained from the absorption of the Ly-α line in
quasar spectra by the Sloan Digital Sky Survey (McDonald et al. 2005).
These observations provide a constraint on the linear matter power spectrum,
and given the effect of early dark energy on the perturbation growth, one would
expect that Ly-α data can improve the constraints on Ω̄dsf that we have obtained
in the last section; that this is indeed the case was first pointed out in Afshordi
45
CHAPTER 4. EARLY DARK ENERGY
et al. (2007a).
The dataset essentially provides a measurement of the amplitude and the
slope of the linear power spectrum at a scale of ∼ 1 Mpc−1 and at a redshift
of z ≃ 3. The measurement is derived from the observation of the absorption
of the Ly-α lines of ∼ 3000 spectra of quasars by the neutral hydrogen in the
intergalactic medium.
The conversion from the observed spectra to a ‘measured’ linear power spectrum involves the use of numerical simulations that assume that the underlying
cosmological model is a ΛCDM model. Hence, there is a potential that the application to early dark energy models could introduce additional errors, especially
if the non-linear growth of density perturbations behaved substantially different
in early dark energy models (as suggested by analytic estimates in Bartelmann
et al. (2006)). On the other hand, the constraints on the early dark energy
abundance that we have found above are already relatively substantial (so that
an eventual error would expected to be small compared to the present observational uncertainties), and at the scale of consideration, linear effects should
dominate. Additionally, results from N-body simulations (Francis et al. 2008a,b;
Grossi & Springel 2008) suggest that correctly taking into account the change
in the linear growth factor is sufficient to capture the non-linear effects. We
therefore proceed with the analysis, keeping this potential caveat in mind.
We use the Ω(w0 , Ωed ) model as the model for an early dark energy cosmology,
and add the Ly-α data to the full set of observational data that we used in the
last section; the resulting confidence intervals for the cosmological parameters
are shown in Tab. 4.1, and the two dimensional contours for the early dark
energy parameter Ωed in Fig. 4.3. The decrease in the upper bound for the early
dark energy abundance is quite significant, as we find Ωed . 1.7% for the 95%
confidence limit, compared to about 4% without the Ly-α data. The addition
of a constraint on the linear overdensity provides the strongest of the limits for
early dark energy.
Other Constraints
The power of measurements of the amplitude of the (linear) growth of perturbations was also illustrated by Xia & Viel (2009). They parameterized the dark
energy equation of state as
−1
w0
C
,
(4.18)
a
w(a) = −1 + 1 −
1 + w0
where w0 is the value of the equation of state today, and the parameter C
determines the change in the equation of state from its present day value to
its asymptotic value of zero in the past. This is a particular version of the
parameterization discussed by Linder & Huterer (2005); that version also allows
to specify an arbitrary asymptotic value for the equation of state in the past,
and added a parameter to independently specify the redshift of the transition
and its rapidity, while still providing an analytic expression for H(z).
Using a combination of CMB and BAO data, together with SnIa, Xia & Viel
quote 95% confidence upper limits on the amount of dark energy at the last
46
CHAPTER 4. EARLY DARK ENERGY
0.160
0.025
0.155
0.024
Ωbh2
Ωmh2
0.150
0.145
0.023
0.140
0.022
0.135
0.021
0.130
0
0.005
0.010
0.015
0.020
0
0.025
0.005
0.010
Ωe
0.015
0.020
0.025
0.015
0.020
0.025
0.015
0.020
0.025
Ωe
0.74
0.20
0.72
0.15
0.70
τ
h
0.68
0.10
0.66
0.64
0.05
0.62
0.60
0
0.005
0.010
0.015
0.020
0
0.025
0.005
0.010
Ωe
Ωe
1.02
-0.80
1.00
-0.85
ns
w0
0.98
-0.90
0.96
-0.95
0.94
0.92
0
0.005
0.010
0.015
0.020
0.025
Ωe
-1
0
0.005
0.010
Ωe
Figure 4.3: Likelihood constraints for the 1σ− (blue shades delimited by the
white line) and 2σ (green shades delimited by a black line) confidence
levels for the cosmological parameters in the Ωed − x plane, where
x ǫ {Ωm h2 , Ωb h2 , h, τ, ns , w0 }, for our complete data set described
in the text, including SnIa observations, matter power spectra, and
CMB observations, together with the Ly-α data set (Seljak et al.
2006; McDonald et al. 2005).
47
CHAPTER 4. EARLY DARK ENERGY
scattering surface of about 2%, and of about 6% during structure formation,
with a somewhat different definition of the average amount of dark energy during structure formation, so that these results are compatible with what we had
found above. They also find that adding information on the power spectra from
Ly-α observations improves their constraint on the amount of dark energy at
last scattering by a factor of at least five, and when using measurements of the
growth factor (and not only the power spectra), by a factor of fifteen, showing
how crucial the inclusion of high-redshift probes of the growth of perturbations
is. Apart from the different dark energy parameterization, the higher improvement on the constraint compared to what we have found above can be attributed
to the choice of Ly-α data sets, since they also included the dark matter power
spectrum information inferred from Ly-α by Viel et al. (2004) in addition to the
SDSS data in our analysis. Furthermore, they include the values of the growth
factor from these data, so that the smaller upper bound is not surprising. However, the caveat still applies that the conversion from the observed Ly-α flux
power spectrum to the ‘measured’ linear dark matter power spectrum involves
numerical simulations of the cosmological and astrophysical processes is based
on a ΛCDM cosmology.
The analysis by Xia & Viel also uses luminosity distances inferred from 69
Gamma-Ray Bursts (GRBs) reaching up to redshift of z > 6, (Schaefer 2007),
but notes that the use of GRBs as ‘standard candles’ is still in its infancy,
and the procedures currently in use to derive luminosity distances do involve
assumptions of the input cosmology, i.e. assume a ΛCDM model.
While the upper bounds that we have discussed in this section might seem
small, especially when compared to today’s value of Ωd , one should keep in mind
that in the concordance ΛCDM model, the cosmological constant contributes
less than half a percent to the energy density during structure formation (i.e.
to the average Ω̄dsf ), while its importance before the time of last scatter is of the
order of 10−9 . In Section 4.4, we will discuss the prospects of how constraints
on early dark energy are expected to improve in the future.
4.3 Early Dark Energy and the CMB
Having computed bounds on early dark energy from a compilation of cosmological observations, we now return to considering only the CMB without adding
any other data. The CMB alone yields strong constraints on the fraction of
early dark energy, as we have just seen from the WMAP-only constraints. But
is CMB data enough to unambiguously detect the presence of an early dark energy? Since we can easily decrease the amount of early dark energy in a suitable
parameterization of Ωd (a) down to the limiting case of a cosmological constant,
we are actually interested in finding out whether an early dark energy cosmology
with a physically significant amount of early dark energy is detectable by CMB
observations alone. The results obtained in the last section clearly demonstrate
that for current CMB data, this is not the case - an early dark energy fraction
on the percent-level is certainly compatible with the data, as is a pure ΛCDM
model, so that we cannot cleanly distinguish between these cases relying on the
48
∆Cl
20
10
100
1000
10
0
∆Cl
2
500
1000
1500
100
0
-100
-200
-300
2000
20
1
0
0
-1
2
2
-10
0
2
2000
-500
l(l+1) Cl/2π [µK ]
∆Cl
4000
0
l(l+1)Cl/2π [µK ]
6000
500
l(l+1)Cl/2π [µK ]
CHAPTER 4. EARLY DARK ENERGY
-20
500
1000
1500
-40
2000
l
Figure 4.4: CMB power spectra for TT (top panel), TE (middle), and EE (bottom) modes of different Ωed = 0.03 EDE models (listed in Table 4.2)
are plotted using the right axis scale. Their differences relative to
the fiducial (w0 = −0.95, wa = −0.1) model are shown using the
left axis scale, and lie within the grey shading surrounding the zero
difference level that shows the cosmic variance limit.
data that is available today.
However, this might soon change, as the precision of CMB observations approaches the fundamental limit dictated by cosmic variance. The question then
is whether this increased precision will allow for a clear distinction between
early dark energy models and theories with a ‘fading’ dark energy, i.e. theories that predict a negligible dark energy component at early times, or even a
cosmological constant.
To answer this question, we will now investigate whether the CMB power
spectra in two dark energy models, one with and the other without early dark
energy, can be similar enough to be indistinguishable, even if we already had
CMB measurements only limited by cosmic variance up to Cl = 2000.
The two example models we chose are the (Ω0d , w0 , Ωed ) parameterization of
Eq. (4.15) introduced in 4.2.1 as an example of an early dark energy (EDE)
model, and the (w0 , w1 ) described in 4.2.1 as a representative for the family of
fading dark energy models.
The result is shown in Fig. 4.4, demonstrating that EDE models can indeed reproduce the power spectra of a reference model with a fading dark energy component. In particular, each panel in Fig. 4.4 shows the respective
power spectrum (TT, TE and EE) of a fiducial fading dark energy model with
(w0 , w1 ) = −0.95, w1 = −0.1), and on top the corresponding spectra of EDE
models with Ωed = 0.03 and w0 = −0.95 and their difference to the fiducial
spectra. The differences are always well within cosmic variance limits.
49
CHAPTER 4. EARLY DARK ENERGY
fiducial
A
B
C
D
——
––
···
·–·
– ··
w0 = −0.95, wa = −0.1, no
EDE
matching 2 < CℓT T < 2000
Ω0m = 0.28, matching 2 <
CℓT T < 2000
matching 2 < CℓT T,EE < 2000
Ω0m = 0.28, matching 2 <
CℓT T,EE < 2000
Table 4.2: Fitting requirements for the models shown in Fig. 4.4.
The matching spectra were found by running MCMC chains with Cmbeasy,
using the power spectra of the fading model as the ‘observed’ data, and then
varying the other cosmological parameters of the EDE model. For the chains
used to obtain the spectra in Fig. 4.4, we allowed the baryon density Ω0b h2 to
vary, as well as the overall matter density Ω0m h2 , the Hubble parameter h, optical
depth τ and the scalar spectral index ns . The amplitude of the fluctuation was
also allowed to vary, by including the conventional combination of As exp−2τ as
a Monte-Carlo parameter for the chains. In particular, we have not included
a potential running of the spectral index, do not allow w0 to vary, and have
assumed neutrinos to be massless, so there exist considerably more degeneracies
that make it even more difficult to look for a clear signal of early dark energy
in the CMB.
The matching EDE spectra shown are examples from Monte-Carlo chains
with four different matching requirements summarized in Tab. 4.2 - matching
either the temperature-temperature spectrum only, or both the TT and the EEpolarization spectra, as well as additionally requiring the matter density Ω0m to
be in exact agreement with the value in the fiducial fading model (Ω0m = 0.28).
The precision of the hypothetical future CMB experiment assumed to have
‘measured’ the fiducial spectrum was set to be cosmic-variance limited up to
l = 2000 for a observed sky-fraction of 81%.
These matching spectra then are not cleanly distinguishable: e.g. for model
A, the overall χ2 relative to the fiducial fading model for all Cl < 2000 and
six cosmological parameters is only 14 - the corresponding numerical value for
unbinned multipoles e.g. for the five-year data from WMAP is a χ2 ∼ 1000
comparing to their best-fit ΛCDM-model over the range 33 ≤ ℓ ≤ 1000 (Nolta
et al. 2009). When looking at binned multipoles, the WMAP team obtains a
reduced χ2 = 1.04, compared to a χ2 = 0.3 for the EDE model relative to the
fiducial fading model, when using the same binning.
We can therefore conclude that even a CMB observation able to deliver accurate Cl -measurements up to ℓ = 2000 does not enable us to detect a clear signal
of early dark energy in the CMB power spectra. In 4.4 we will show an example
B-mode polarization spectrum, which would discriminate between our example
models, but might be difficult to measure due to its much lower amplitude. We
will now study the consequences of the results we have obtained so far for the
interpretation of data from measurements of the baryon acoustic oscillations.
50
CHAPTER 4. EARLY DARK ENERGY
14500
dlss [Mpc]
14450
14400
14350
14300
14250
148
148.5
149
149.5
150
150.5
s [Mpc]
Figure 4.5: The distance to last scattering, dlss and the sound horizon scale s
as determined by the CMB power spectrum. Their ratio, i.e. the
slope of the contours is particularly well constrained. The white
and black lines give the 68.3% and 95.4% confidence limits for a
CMB experiment limited only by cosmic variance up to l = 2000.
The outer contours light green to dark blue) are the result when
using only the temperature power spectrum and the dashed lines
correspond to using only multipoles of l ≥ 40. The smaller contours
from dark red to light gold include the polarization E-mode spectra
and the TE cross-correlation spectra for a cosmic variance limited
experiment. The star marks the input values used for generating the
“observed” spectrum
4.3.1 Baryon Acoustic Oscillations
As we have seen in the introduction in Chapter 2, the information one can
deduce from the BAO measurements relies on an accurate determination of the
standard ruler. On the other hand, since early dark energy changes the distance
scales according to Eq. (4.1), both the distance to the last scattering surface,
dlss , and the sound horizon s will be modified in an universe with an early dark
energy component. The size of the sound horizon mainly depends on the sound
speed cs of the coupled photon-baryon fluid, and therefore on baryon and photon
densities, and the redshift zdec when the photons and baryons decouple. The
redshift of decoupling is the result of atomic physics and remains unaffected
by an early dark energy component. Hence we can directly deduce from the
expression for the sound horizon s,
s=
Z
∞
dz
zdec
cs
,
H(z)
(4.19)
that the influence of early dark energy on the scale of the sound horizon is
determined by the change in H(z). Indeed, we see that Eq. (4.1) yields
sEDE
∼ (1 − ΩE DE)1/2 .
snoEDE
51
(4.20)
CHAPTER 4. EARLY DARK ENERGY
for the shift in the sound horizon caused by the presence of a non-negligible
early dark energy component (for a detailed discussion see Doran et al. 2007b).
The conformal distance also changes, of course, as
d(z) =
Z
z
0
dz ′
,
H(z ′ )
(4.21)
which directly gives the change of the distance to the last scattering surface,
dlss . That we could accurately match the CMB spectra of cosmologies with and
without EDE in Section 4.3 is simply a consequence of the fact that the CMB
power spectra measure the ratio, dlss /s, much more precisely than the individual
distances themselves, as seen in Fig. 4.5, which depicts the likelihood contours
in the dlss -s plane for a cosmic variance limited experiment (here for a ΛCDM
universe). Since the contribution of the dark energy to H(z) is time-dependent,
the change in the distances and scales also depends on time, so that an early
dark energy cosmology can match the angular scale given by dlss /s at the time
of last scattering of a model with fading dark energy, and produce matching
CMB spectra, while the ratio will be distinctly different at other times.
The same experiment can be done with our two example cosmologies with
matching CMB spectra from Section 4.3. As we have shown there, both the
EDE and the non-EDE model can produce matching spectra, and therefore the
same dlss /s. However, we cannot recover the distances individually from the
CMB spectra alone, as we would like for the BAO measurements. Since we do
not know a priori what the true cosmology is, and specifically, whether it has
an EDE component or not, this has to be taken into account when determining
the scale of the standard ruler in BAO measurements. For example, assuming
that the our real Universe has only fading dark energy and is described by
the fiducial (w0 = −0.95, w1 = −0.1) model from Section 4.3, then trying to
determine cosmological parameters by fitting an EDE model will lead to biased
results, as can be seen in Fig. 4.6. In particular, neglecting the possibility of
early dark energy when calibrating the standard ruler for BAO studies will bias
the values of the cosmological parameters obtained from such observations.
We can now calculate the propagation of the error introduced by assuming a
wrong cosmology and subsequently miscalibrating the standard ruler. This error
propagates into the cosmological parameters as determined in BAO measurements - and this calculation then shows that the effects of such a miscalibration
is indeed important for the interpretation of future cosmological observations,
as we will now show in the next section.
4.3.2 Miscalibrating the Standard Ruler
To asses the magnitude of the error introduced by a miscalibrated standard ruler
due to ignoring the effect of early dark energy, we use the Fisher information
matrix for a data set consisting of a vector O = {O1 , ..., Ok } of observed quantities. The probability distribution f of these quantities then depends on the
cosmological parameters p = {p1 , ..., pn } that we want to estimate. The Fisher
52
CHAPTER 4. EARLY DARK ENERGY
13400
dlss [Mpc]
13300
13200
13100
13000
12900
138
140
139
141
142
143
s [Mpc]
Figure 4.6: Likelihood contour at 95.4% cl in the s−dlss plane for EDE cosmologies with Ωed = 0.03 that match the CMB temperature power spectrum of the fiducial (w0 = −0.95, wa = −0.1) model without early
dark energy. Note that the matching almost automatically preserves
the geometric shift factor, i.e. the slope dlss /s, but not the individual
distance scales. The red star marks s and dlss of the fiducial model;
the shift in the standard ruler amounts to 2.5%.
matrix Fij is defined by
Fij ≡ −
∂ 2 ln f (O; p)
∂pi ∂pj
,
(4.22)
and its inverse Fij−1 gives the best possible covariance matrix for the measurement errors on these parameters. We can then quantify how a miscalibration in
the Ok propagates to the inferred values for the cosmological parameter pi via
δpi = (F −1 )ij
X
∆Ok
k
1
∂Ok
.
2
∂pj σ (Ok )
(4.23)
In our case, the quantities observed by the radial and transverse BAO modes,
Ok , are the ratio d˜ = d/s of the angular diameter distance at redshifts zk to
the sound horizon, as well the ratio of the proper distance interval to the sound
horizon, H̃ −1 = H −1 /s. We will assume a future BAO experiment that will
be able to measure these quantities to 1% precision at four different redshifts,
zk = 0.4, 0.6, 0.8, 1, a precision that will require surveys beyond the planned
WFMOS survey (Bassett et al. 2005), or e.g. the BOSS survey (The SDSSIII Collaboration 2008). For the accuracy of the determination of the sound
horizon at high redshift, we use the precision of the Planck satellite (Tauber
2000). If our real Universe has an EDE component, and if we were then to
deduce the value of cosmological parameters of this EDE-universe (we assume
Ωed = 0.03, w0 = 0.95), but incorrectly assumed that the dark energy evolution
could be fit using the w(a) = w0 + w1 (1 − a) parameterization, the result would
53
CHAPTER 4. EARLY DARK ENERGY
0.8
0.6
0.4
w1
0.2
x
0
-0.2
-0.4
SN + σΩ =0.01
-0.6
-0.8
BAO+CMB
shifted expansion history
-1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5
w0
Figure 4.7: Early dark energy density Ωed can cause miscalibration of the standard ruler used by the baryon acoustic oscillation probe. Assumed
here is a true model with Ωed = 0.03 - this shifts the dark energy properties derived with BAO+CMB data from their true values (black x)
to those marked by the red star, with the dark red confidence contour. The contours indicate the 68% confidence limit. Distance
probes such as supernovae that do not rely on the standard ruler
recover the true expansion history (here shown with a prior on Ωm ).
be that of Fig. 4.7. We would obtain a value of w1 ≡ wa = −0.66 instead of the
true −0.95, and even the 68% confidence limits overlap only slightly.
In contrast, supernovae observations would be able to recover the true value
w0 today, as would other observations that do not rely on the calibration of a
high-z quantity influenced by the dark energy - shown in the plot are results
that could be obtained by a survey like SNAP (Aldering et al. 2002), that would
measure the luminosity distance at low redshift ( 0 < z . 1.7 ) through SnIa, and
yield a constraint σΩ on the matter density through weak lensing measurements.
This is the direct result of the fact that at low redshifts, the luminosity distances
expansion histories of both models are the same to a precision of 0.02%, as the
best-fit (w0 , w1 ) model can replicate the expansion history of the EDE model
very well at low redshifts (for about z . 2 in our case), and it is only at high
redshifts where the difference becomes substantial.
To avoid misleading conclusions from BAO observations, one has has to account for this uncertainty. This will effectively increase the confidence limits on
the derived cosmological parameters, but allow to recover the true cosmology.
In the context of BAO observations, this is easily done by also fitting for the
absolute ruler scale, i.e. including the uncertainty δs in the sound horizon as an
54
CHAPTER 4. EARLY DARK ENERGY
additional parameter in the analysis (see also Linder 2007).
For the model example models considered here, accounting for the possibility
of early dark energy in this way increases the area of the error contours in Fig. 4.7
by a factor of 2.3. This shows that the potential presence of an early dark energy
component considerably weakens the potential power of BAO constraints.
4.4 Prospects
We have restricted our discussion of early dark energy almost entirely to the
regime accessible by linear perturbation theory, and not investigated the effects
of an early dark energy component on the non-linear regime of structure formation at all. Semi-analytical calculations within the spherical collapse model
(Bartelmann et al. 2006) find that in early dark energy models, the linear collapse parameter δc is lowered, leading e.g. to a significant increase in clusters
at redshifts above unity. The effect on the statistics of gravitational arcs have
been studied by Fedeli & Bartelmann (2007), and the impact on the cluster
sample that forthcoming surveys will observe has been discussed by Fedeli et al.
(2008); Waizmann & Bartelmann (2008) specifically study the cluster sample
that the Planck satellite will measure, and find that a fraction Ω̄dsf & 4% would
be necessary for a clear detection, but also that EDE cosmologies lead to cluster samples that are considerably less contaminated by false detections than
in ΛCDM. In contrast to the analytical estimate, numerical studies performing
N-body simulations (Francis et al. 2008a; Grossi & Springel 2008; Francis et al.
2008b) conclude that standard estimates for the abundance of dark matter halos
continue to work in EDE cosmologies, provided the change in linear growth is
properly taken into account. This would suggest that finding a clear signal of
EDE might be harder than originally thought.
Independently of this debate, probes sensitive to the linear structure growth
remain a viable observational path. For example, when discussing the degeneracy of the CMB spectra between EDE and fading dark energy models in
Section 4.3, we have skipped the B-mode polarization, which is one such possibility.
The B-mode polarization is not intrinsically present in the CMB, but is generated by the gravitational deflection of the photons by the large scale structure
between the last scattering surface and the observer (for a review, see Lewis &
Challinor 2006). The effect of the lensing of the CMB can roughly be described
as smearing out the acoustic peaks, and has an effect of a few percent at high
multipoles (l ∼ 2000) on the TT spectra. The total angle by which the photons
are deflected is on the order of a few arcminutes, but since the deflection angles
are correlated over scales of typically about two degrees, the acoustic peaks are
broadened by a few percent. The largest contribution to the lensing potential
comes from redshifts of a few (see e.g. Acquaviva & Baccigalupi 2006), and
therefore provides a probe of the changed growth in early dark energy models.
A method combining gravitational lensing of the CMB with cosmic shear surveys in a way that yields a measurement of the ratio of comoving distances to different redshifts was proposed by Das & Spergel (2008). The authors concluded
55
CHAPTER 4. EARLY DARK ENERGY
e
Ωd = 0.001
0.08
e
Ωd = 0.01
e
0.06
0.04
Cl
BB
2
[µK ]
Ωd = 0.05
0.02
0
500
1000
Multipole l
1500
2000
Figure 4.8: Power spectra of the B-mode polarization for three Ω(w0 , Ωed )-models
with the same initial fluctuation amplitude and different values
of Ωed , while all other parameters are kept fixed, as computed by
Cmbeasy. Normalizing all models to the same σ8 today would
show the inverse effect, i.e. an enhancement of the B-mode signal
by early dark energy.
that their method is unfortunately rather insensitive to the early dark energy
contribution, particularly to the parameterization in terms of the Ω(w0 , Ωed )model. Nevertheless, it is very useful for determining the late-time behavior of
dark energy, and could potentially yield tighter constraints when the early dark
energy behaves differently than the parameterization of the Ω(w0 , Ωed ) model.
As for the B-mode spectra, the biggest observational difficulty is their low
overall amplitude, which can be seen from the example spectra in Fig. 4.8;
this is well beyond the reach of current experiments. That might change with
future probes, though. Experiments that are contemplated include for example
CMBpol (Zaldarriaga et al. 2008), which is designed to measure the polarization
spectra, including the B-mode, with ‘ultra-high’ precision. Two groups, de
Putter et al. (2009) as well as Hollenstein et al. (2009), have computed forecasts
for the expected constraints on Ωed that could be achieved, finding a sensitivity
down to Ωed . 0.002 for CMBpol; a combination with weak lensing data from
future weak lensing surveys (Amara & Refregier 2007) is projected to be able
to constrain Ωed to ∼ 10−4 , so that the outlook of possibly detecting an early
dark energy component in the future seems promising.
56
5 The Planck Mass at the
Cosmological Horizon Scale
The last chapter presented an overview of different models of early dark energy,
and their general features in comparison with ‘fading’ dark energy and the concordance ΛCDM - model. In this chapter, we will explore the ‘boundary region’
between these scalar field dark energy models and modifications to the law of
gravitation as formulated by the Theory of General Relativity. The specific
question we are asking here is: Does the Planck Mass change at the cosmological horizon scale? This question immediately requires that the Planck Mass,
which in General Relativity quantifies the strength of the coupling between the
space-time metric and the matter and energy content of the Universe, now becomes a scale-dependent quantity, an ‘effective’ Planck Mass. In the limit of
a vanishing scale dependence, we then recover standard General Relativity. In
Section 5.2 we will see that we can reinterpret the tracker models of the dark
energy that we have studied in the last chapter as causing such a change in the
value of the effective Planck Mass, leading to a ‘glitch’ in the value between
sub-horizon and super-horizon scales; it is this direct connection that motivates
the investigations in this chapter.
In early dark energy cosmologies, however, this feature, while interesting
enough to merit closer study, is more of a by-product of the presence and the
dynamics of the scalar field; there exists however, a class of theories, called ‘Cuscuton’ models (Afshordi et al. 2007b), where this mismatch between small-scale
and large-scale values of the Planck Mass emerges as a central property of a theory that does not contain a new scalar field dark energy component, but instead
describes a modification of the law of gravity that leads to this mismatch, and
therefore is a much more natural setting for this discussion. Additionally, as we
will see below, one can still describe ‘Cuscuton’ models in a language that is
indeed very similar to the scalar field dark energy we have used so far, showing
the direct relation to the models we have discussed in Chapter 4, and pointing
out yet another aspect of the close connection of (scalar field) dark energy and
modified gravity models.
Since MP = (8πG)−1/2 ≃ 2.44 × 1018 GeV, a change in the Planck Mass
can equivalently be described in terms of Newton’s constant G as a fifth-force
modification to the inverse square law,
m1 m2 (5.1)
1 + αe−r/λ ,
V (r) = −G
r
where m1 and m2 are the masses of (point-like) gravitating objects, while α and
λ quantify the strength and the scale of the new interaction. In this formulation,
the effective Newton’s constant smoothly goes from G on large scales (r ≫ λ)
57
CHAPTER 5. THE PLANCK MASS AT THE HORIZON SCALE
to G(1 + α) on small scales (r ≪ λ). Current constraints from torsion-balance
experiments (e.g. Kapner et al. 2007) to solar-system/planetary dynamics and
lunar laser-ranging (e. g. Williams et al. 2004) severely limit α in the range
10−2 cm < λ < 1016 cm, (see Adelberger et al. 2003, for an overview), and largescale structure in the Universe has been used to obtain constraints on α even
on scales of λ ∼ 1025 cm (Sealfon et al. 2005).
With a numerical implementation of the Cuscuton models in Cmbeasy, we
will then be able to compute observational constraints on the possible running of
the Planck mass, which, together with the constraints on the abundance of early
dark energy from Chapter 4, show that current observational data constrain
deviations from Einstein gravity even on the scale of the cosmological horizon,
or the Hubble radius (λ ∼ c/H ∼ 1028 cm).
5.1 Effective Planck Mass
In order to quantify the size of the possible change of the Planck Mass as we approach and cross the cosmological horizon, General Relativity’s constant Planck
Mass MP in Einstein’s Equation,
Gµν = Mp−2 Tνµ ,
(5.2)
should be replaced by a scale dependent function. The effective Planck Mass
Mp,eff (|k|), which we define in Fourier space as a function of the vector k as
−2
Mp,eff
(|k|) ≡
0∗ i
hδG00,k δT0,k
0 δT 0∗ i
hδT0,k
0,k
,
(5.3)
0
0
0 and δG0
where δT0,k
0,k are the spatial Fourier transforms of δT0 and δG0 on
a given spatial hypersurface, and ‘h i’ represent ensemble averages. Another
possible definition would have been
−2
≡
Mp,eff
δG00
,
δT00
(5.4)
−2
as a direct measure of the response of G00 to perturbations of
taking Mp,eff
the energy density. But this simple definition becomes ill-defined when the
denominator crosses zero; additionally, our choice of definition has the advantage
of making the scale-dependence explicit, being a function of a single physical
scale |k|.
Obviously, the definition of Eq. (5.3) depends on the gauge we choose to compute the perturbations. However, on small scales we recover Newtonian gravity,
and the effective Planck Mass will therefore be gauge independent. On superhorizon scales (k ≪ H), gauge transformations can change the effective Planck
mass only if the Planck mass associated with the background expansion is different from the Planck mass associated with the perturbations; for our analysis,
we assume that this is not the case, and the large-scale value of the Planck
Mass will be identical to the Planck Mass value in the Friedmann Equation,
58
CHAPTER 5. THE PLANCK MASS AT THE HORIZON SCALE
and therefore gauge independent. Only the details of the transition will depend
on the choice of gauge.
Referring to the cosmological sub-horizon (k ≫ H) and super-horizon (k ≪
H) scales as the UV and IR scales, respectively, we denote the corresponding
values of the effective Planck mass by Mp,UV and Mp,IR . In this notation, the
UV-IR mismatch can then be expressed by the dimensionless parameter α:
2
2
Mp,IR
= Mp,UV
(1 + α).
(5.5)
One can, of course, imagine many other forms of variations of the Planck
Mass and equivalently, Newton’s constant. When we consider possible couplings
between the scalar field dark energy and matter, as we will do in the next
chapter, one also finds an effective Newton’s Constant, modified by the strength
of this coupling. Theories of gravity other than General Relativity in general
predict a time- and/or space-dependent G, as do e.g. scalar-tensor theories
(Linde 1990; Garcia-Bellido et al. 1994; Clifton et al. 2005; Nagata et al. 2004;
Pettorino et al. 2005). In other words, possible deviations from Einstein gravity,
or alternatively, other energy components that are not accounted for in the total
energy momentum tensor, Tµν , may lead to an effective (or dressed) Planck mass
that could run with time and/or the energy/length scale of the interactions. In
fact, the isolated consideration of the gravity theory under the assumption that
the other ‘constants’ of nature are indeed constants, is already a very strong
simplification (see e.g. Stern 2008; Dent et al. 2009, and references therein).
In our investigation here, we concentrate on cases where α does not depend on
redshift, and where the scale at which this transition occurs is the only natural
macroscopic scale of the problem, namely the cosmological horizon size; the only
time dependence of the transition scale therefore is through the evolution of the
cosmological horizon.
A scale dependence of the Planck Mass that can be written in the form of
Eq. (5.5) can also be found in “Einstein-Aether” theory, where the “aether” is a
Lorentz-violating, fixed-norm and time-like vector field (Jacobson & Mattingly
2001; Foster & Jacobson 2006; Carroll & Lim 2004). In contrast to both the
tracking dark energy models and the Cuscuton that we consider below, the
coupling of the vector field to the metric renormalizes both the IR and UV
values of the effective Planck mass, so that the ‘bare’ value is seen neither on
small nor on super-horizon scales.
5.2 The Planck Mass and Early Dark Energy Models
The connection between early dark energy models and a rescaled Planck Mass
is easy to see on the background level. Assuming an exponential potential for
the scalar field,
(5.6)
V (φ) = Mp4 e−λφ/Mp ,
the constant fraction of early dark energy Ωφ is during matter domination given
by
3
(5.7)
Ωφ = 2 ,
λ
59
CHAPTER 5. THE PLANCK MASS AT THE HORIZON SCALE
as we had seen in Section 2.3. As long as Ωφ is a constant, we see that with the
definition
3
2
2
(5.8)
Mp,IR = Mp,U V 1 − 2 ,
λ
the Friedmann Equation can be written as
ρm
,
H2 =
2
3Mp,IR
(5.9)
interpreting the early dark energy as a simple rescaling of the Planck Mass. This
interpretation is consistent with our definition of the effective Planck Mass by
Eq. (5.3), which gives
δρφ + δρm
−2
,
(5.10)
Mp,eff
(k) =
δρm
if we assume adiabatic initial conditions, as given in Section 3.4; the choice of
adiabatic initial conditions is a precondition for the validity of this analysis.
5.3 Cuscuton Cosmologies
Leaving the early dark energy models aside for a moment, we now turn to the
‘Cuscuton’ models, in order to see how a UV-IR mismatch in the values of the
Planck Mass appears in this framework. Introduced by Afshordi et al. (2007b),
this theory is a scalar field dark energy model with a particular action for the
field ϕ that results in several intriguing characteristics. Most importantly, even
though the action is formulated as the action of a scalar field, the form of
the action dictates that the field degree of freedom is merely a non-dynamical
auxiliary field. The equation of motion does not have any second-order time
derivatives, and the behavior of ϕ is completely determined by the fields it
couples to. In other words, ϕ does not introduce an additional degree of freedom,
like conventional dark energy models do. Instead, it merely acts as a constraint,
changing the dynamics of the theory. This explains the name ‘Cuscuton’, which
is the Latin name for dodder, a genus of parasitic plants. Then, as there is no
additional degree of freedom in the theory, it is indeed more a theory of modified
gravity, and this makes it a suitable framework to explore a possible mismatch
of the Planck Mass between the UV and IR regimes.
Formally, the Cuscuton action is given by
q
Z
2
4 √
µ
(5.11)
SQ = d x −g κ |∂ ϕ∂µ ϕ| − V (ϕ) ,
where ϕ is a scalar field, and κ and m are constants of theory with the dimensions
of energy. The general cosmological aspects of the Cuscuton have been outlined
by Afshordi et al. (2007a). The first main property of interested to us here is the
behavior in the homogeneous limit of the field, i.e. the cosmological background
evolution. In this limit, the definition of the Cuscuton action Eq. (5.11) implies
that the kinetic term of the field is a total derivative, so that the action for ϕ is
simply
Z
Sφhomog. = −
60
d4 x V (ϕ),
(5.12)
CHAPTER 5. THE PLANCK MASS AT THE HORIZON SCALE
as long as the field does not couple to any other fields, which we assume here.
Then, the field equation can be written in the form
3κ2 Hsgn(ϕ̇) + V,ϕ (ϕ) = 0.
Together with the Friedmann Equation, this relation leads to
2
MP
2
V,ϕ
(ϕ) − V (ϕ) = ρbkg ,
3κ4
(5.13)
(5.14)
where ρbkg is the total energy density without the contribution from the Cuscuton. This relation shows how, once V (ϕ) is chosen, the field does not have
its own dynamics, but has its value determined by the background density. We
can also use this equation to rewrite the Friedmann Equation (for ϕ̇ < 0) as
H2 =
1 ρbkg + V V,ϕ−1 (3κ2 H) ,
2
3MP
(5.15)
−1 denoting the inverse function of V . The nature of the Cuscuton as
with V,ϕ
,ϕ
a modification of gravity is evident here. Even though we use the same fieldtheoretical language as for our scalar field models, writing down the action for a
Cuscuton ‘field’ ϕ, this variable can be eliminated from the resulting equations,
and the cosmology is described by a modified Friedmann equation which is fixed
once we specify the potential V (ϕ).
For our investigation, we set
1
V (ϕ) = V0 + m2 ϕ2 ,
2
(5.16)
i.e. a simple quadratic potential. Other possibilities are discussed in Afshordi
et al. (2007a). With this choice, V0 will simply contribute to the cosmological
constant. The quadratic part, instead, maintains a constant fraction ΩQ of the
total energy density, which can easily be seen by taking the square of Eq. (5.13),
which together with the Friedmann Equation shows that this fraction is
ΩQ =
1 2 2
2m ϕ
ρtot
=
3κ4
.
2MP2 m2
(5.17)
Inserting this in Eq. (5.15) then shows that the Friedmann Equation, modified
by the Cuscuton constraint, can be written in its usual, un-modified way, but
with a rescaled Planck Mass,
2
2
Mp,IR
= Mp,UV
−
3κ4
.
2m2
(5.18)
As the Cuscuton does not represent a real degree of freedom, it is not surprising
that the Cuscuton perturbation is a direct function of the metric perturbations
(assuming Φ = Ψ), and can be written as (Afshordi et al. 2007a):
δϕ =
3ϕ̇(Φ̇ + HΦ)
,
k2 /a2 − 3Ḣ
61
(5.19)
CHAPTER 5. THE PLANCK MASS AT THE HORIZON SCALE
which makes it possible to eliminate the Cuscuton perturbation from the perturbed Einstein Equations, so that e.g. in a matter dominated universe,


2
2
k
Ḣ + 3H Ωm )  3H + 9H(2
Φ̇
+
HΦ
+ (2MP2 )−1 δρm = 0. (5.20)
Φ
+
a2
2
2
2 k /a − 3Ḣ
The quantity of interest for us, however, is the effective Planck Mass, and since
we take the Cuscuton as not contributing to the energy-momentum tensor but
as part of the gravitational action, we can use these expressions and Eq. (5.10)
to find that
−1
−1 3κ4
k2
k2
2
2
,
(5.21)
Mp,eff ≃ Mp,UV −
1+
1−
2m2
3H 2
3Ḣ
to the lowest order in κ, in a flat matter-dominated Universe, which again shows
that the IR limit of our definition of the effective Planck Mass is consistent with
the background result above, and that in our numerical calculations below, only
the exact transition between the IR and UV regimes will depend on our choice
of gauge.
5.3.1 Cuscuton and k-Essence
For completeness, let us very briefly mention the connection between the kessence models introduced in Section 2.3 and the Cuscuton. If we consider
k-essence actions written as
Z
1
4 √
(5.22)
Sϕ = d x −g F (X, ϕ) − V (ϕ) ,
2
with X ≡ ∂µ ϕ∂ µ ϕ, we see that the Cuscuton action corresponds to a particular
choice for F. Important in our context is the relation for the speed of sound µ
(as defined in Section 3.2), which can in k-essence theories be computed by
µ2 =
1
,
1 + 2XF ′′ /F ′
(5.23)
where only for this subsection we use ′ to denote the derivative with respect to
X. Inserting the definition of the Cuscuton action in this equation, one finds that
µ = ∞. Cuscuton models therefore are a very specific limiting case of k-essence
theories, namely the limit of an infinite speed of sound. On one hand, this is
intimately connected to the fact that the scalar degree of freedom can be eliminated from the dynamical equations, on the other hand, it immediately raises
questions about the causality of the theory (see Bonvin et al. 2006; Babichev
et al. 2008; Adams et al. 2006; Bruneton 2007; Vikman 2007, for the debate of
whether or not µ > 1 is a fundamental problem in k-essence theories in general).
We will not repeat the very detailed discussion of this question carried out in the
original article by Afshordi et al. (2007b), but merely state the result, namely
that perturbations of the Cuscuton cannot carry any microscopic information,
and therefore do not violate causality. This again expresses the fact that δϕ is
62
CHAPTER 5. THE PLANCK MASS AT THE HORIZON SCALE
2
Mp, eff
1.00
0.95
1e-04
1e-03
k / h [Mpc-1]
1e-02
Figure 5.1: Transition between the IR and UV regimes for the effective Planck
mass (in units of Mp,UV ) for the quadratic Cuscuton with α = −0.05
(red, straight line). The transition for a canonical scalar field model
(µ2 = 1) is depicted in green (dashed line). For µ2 & 10, the transitions virtually coincide with the quadratic Cuscuton (for which
µ2 = ∞).
given by a constraint equation, and cannot individually carry any information.
It could, in principle, change causal properties of other fields that it couples to,
but again, Afshordi et al. (2007a) show explicitely that for metric perturbations
coupled to the Cuscuton, which is the case we are interested in here, the infinite
speed of sound is not equivalent to an instantaneous interaction which would
directly violate causality, and that there is no (at least no obvious) violation of
causality in Cuscuton cosmologies.
5.4 Running of the Planck Mass?
In order to compute the observational constraints on the possible UV-IR mismatch of the Planck Mass, parameterized by α in Eq. (5.5)f, we have implemented the Cuscuton models in Cmbeasy. An example of the transition is
shown in Fig. 5.1, where we can see that for a canonical scalar field model with
µ2 = 1, the transition shifts to slightly smaller scales, by about 30%. As we will
see belowe, this slight shift has only a marginal effect on the allowed size of α.
The phenomenology of the quadratic Cuscuton model is very similar to the
early dark energy models of Chapter 4, as the infinite speed of sound introduces
no significant changes (but see Afshordi et al. 2007a, for the explicit calculations
for the Cuscuton models). In particular, with α ≡ −ΩQ ≪ 1, the dependence
of the growing mode of the matter density perturbations and the gravitational
potential during matter domination is identical to the one in an early dark
energy model,
2
2
2
(5.24)
δ ∝ t 3 + 5 α ⇒ Φ ∝ t 5 α.
The notable distinction though, is the fact that unlike the fractional energy
density in a tracking dark energy model, α is not a priory required to be negative,
so instead of the general suppression of growth that perturbation modes in early
dark energy models would experience, we now have either a suppression or an
enhancement. This directly translates to the matter power spectra, and also to
the CMB, where for example the contribution from the Integrated Sachs-Wolfe
63
rel. likelihood
CHAPTER 5. THE PLANCK MASS AT THE HORIZON SCALE
1
0.5
0
-0.05
0
α
-0.025
0.025
0.05
Figure 5.2: Observational constraints on the UV/IR Planck mass mismatch
parameter, α, from 3 years of WMAP data alone Hinshaw et al.
(2007)(red, dashed line), and our compilation (see the text) of cosmological observations (black straight line).
effect can now either lead to less power on the large angular scales (for α > 0)
or to an increase of the lower CMB multipoles (for α < 0).
We compute the constraints that these effects impose on the allowed range
for α using our modified version of Cmbeasy, with the result shown in Fig. 5.2.
The 3-year CMB power spectrum of WMAP (Hinshaw et al. 2007) constrains α
to −0.005 ± 0.040 at the 95% confidence level, which is in line with the allowed
size of α we would expect from our constraints on the abundance of early dark
energy in Chapter 4.
We then also use data from the distribution of luminous red galaxies from the
Sloan Digital Sky Survey (SDSS, Tegmark et al. 2006) to probe the effects of
large scale structure data on our constraints. For the SDSS data, we marginalize over the bias. We also include constraints on the cold dark matter power
spectrum from observations of the Lyman-α forest (Seljak et al. 2006), as we
have done for the early dark energy models in Section 4.2.4. Adding the results
from Supernovae Ia observations (Riess et al. 2007) as well as the observation
of the baryon acoustic oscillations (BAO) (Eisenstein et al. 2005) to this large
scale structure data (in order to decrease degeneracy with other cosmological
parameters), together with the WMAP data. As our central result, we find the
UV/IR mismatch parameter, α, to be tightly constrained to −0.004 ± 0.021
(95% confidence limit) by our complete set of observational data.
The effect of the sound speed of the scalar field on the bounds on a UV/IR
glitch is marginal. If we recompute the constraints for a scalar field model
with µ2 = 1, instead of the Cuscuton case of µ2 = ∞, the bounds on α are only
slightly relaxed, namely to −0.004±0.024 at the 95% confidence level. Together
with the results for α from the early dark energy models, we therefore conclude
that the constraints on α are insensitive to the details of UV/IR transition, since
most of the observable consequences of the mismatch occur on small sub-horizon
scales.
We can also compare our constraints to the Einstein-Aether theories that we
had very briefly mentioned above; there, the glitch in the Planck Mass is a
function of three constants of the theory c1,2,3 , i.e.
2
2
Mp,IR
= Mp,UV
− (2c1 + 3c2 + c3 ),
64
(5.25)
CHAPTER 5. THE PLANCK MASS AT THE HORIZON SCALE
2
2
but in contrast to our investigation, both Mp,IR
and Mp,UV
are ‘screened’
values of the bare Planck Mass in the theory. Recently, Zuntz et al. (2008)
used a modified version of Cmbeasy to obtain cosmological constraints from
a combination of CMB and large scale structure data, finding 1σ bounds of
(c1 = −0.26 ± 0.12, c2 = 0.20 ± 0.09, c3 = − 0.12 ± 0.05), so that an addition
of the mean parameter values would yield α ∼ −0.04, broadly compatible with
the constraints we found here.
As the precision of cosmological observations is continuously increasing, the
bounds on any mismatch between UV and IR Planck Masses will improve accordingly, but already with current data, we have shown in this chapter that
such a mismatch is constrained to less than 1.2% for the Planck Mass at the 95%
confidence level, severely restricting any such deviation from Einstein Gravity
at the largest cosmological scales.
65
CHAPTER 5. THE PLANCK MASS AT THE HORIZON SCALE
66
6 Coupled Dark Energy
When considering scalar field dark energy in the previous chapters, we have always assumed that gravity is the only way of interaction between the scalar field
and the other components in the Universe. In principle, there is no fundamental
reason why we should postulate the absence of other interactions, and in fact, it
is well known that such a possible interaction between the scalar field and other
matter species could be an important part of any complete description of cosmology (Wetterich 1995; Amendola 2000). It is therefore not surprising that the
literature on this ’Coupled Quintessence’ is extensive, ranging from studies of
Coupled Quintessence as a solution to the coincidence problem and similar questions (Quartin et al. 2008; Brookfield et al. 2008; Gromov et al. 2004; Mangano
et al. 2003; Anderson & Carroll 1997), to its observational characteristics and
implications (see e.g. Bertolami et al. 2007; Wang et al. 2007; Guo et al. 2007;
Mainini & Bonometto 2007; Lee et al. 2006; Farrar & Peebles 2004; Gubser &
Peebles 2004; Farrar & Rosen 2007, and references therein).
As one can a priori imagine many different forms of a dark energy coupling to
other species, or e.g. an arbitrary time dependence, there is a vast theory space
to explore, as one hopes to gain physical insight, and in the age of ‘precision
cosmology’, quantitative numerical studies and comparison of their result with
observations are indispensable; the number of such studies in the literature,
however, is comparatively small (examples include Bean et al. 2008; Kesden &
Kamionkowski 2006; Amendola & Quercellini 2004; Amendola 2001).
This chapter extends our previous consideration of scalar field dark energy
models to cases with a coupling between cold dark matter and the scalar field.
The formalism presented here has then been implemented in Cmbeasy, and we
will use this code to show the phenomenology of one specific example of coupled
dark energy, i.e. for a specific choice of the scalar field potential and the dark
matter - dark energy coupling. The numerical results for these specific models
have then been used by Baldi (2009) as input for a modified version of the
N-body code Gadget-2 (Springel 2005), in order to investigate the non-linear
regime. Together with the extensions and modifications of Cmbeasy described
here, these two implementations form a complete numerical pipeline for the
study of general coupled dark energy models, as presented in Baldi, Pettorino,
Robbers, & Springel (2008).
Of course, dark energy could also interact with other particle species, not only
the dark matter. The coupling to baryons is tightly contrained by experimental
tests of gravity on scales of the solar system and below (see e.g. Carroll et al.
2009; Damour et al. 1990), but a dependence of the neutrino mass on the value
of the scalar field is much less restricted by current observations. In fact, a
neutrino with a time varying mass, depending on the scalar field, can have a
very rich phenomenology, and suggest for example a very intriguing solution to
67
CHAPTER 6. COUPLED DARK ENERGY
the ‘why now?’ question, together with substantial implications for structure
formation. We will postpone the study of this particular kind of coupling to the
next chapter, and focus on the dark matter - dark energy coupling here.
6.1 The Formalism
The action of the system of a dark energy scalar field coupled to matter fields
ψ can be written as
Z
1 µ
4 √
(6.1)
Scoupled = d x −g − ∂ φ∂µ φ − U (φ) − m(φ)ψ̄ψ + Lkin [ψ]
2
The specific form of the interaction between dark matter and dark energy is
encoded in the dependence of the mass of the matter fields on the value of the
scalar field φ, i.e. the form of the coupling is specified by choosing a particular
m(φ).
The coupling then naturally changes the evolution of the scalar field, which
is now governed by a modified Klein-Gordon equation,
φ′′ + 2Hφ′ + a2 U,φ (φ) ≡ −a2 S,
(6.2)
which is the result of varying Eq. (6.1) with respect to φ, and where H = a′ /a
is the conformal Hubble parameter and U,φ is the derivative of the potential
of the scalar field with respect to φ. The coupling changes the ordinary field
equation through the appearance of an external source term S, which contains
the φ-dependent mass term,
S≡
∂ ln m(φ)
∂ ln m(φ)
hm(φ)ψ̄ψi =
(ρ − 3p).
∂φ
∂φ
(6.3)
The energy density and pressure on the right hand side are the ones of the
species that the dark energy couples to, so in the case considered here, they
reduce to the energy density of cold dark matter, ρc . The energy-momentum
tensor of the combined dark matter - dark energy fluid is conserved,
X
∇ν T(α) νµ = 0,
(6.4)
α
but the divergences of the contributions of the individual species α (in this case
dark matter and dark energy), do not vanish as they normally would, but are
given by a source term Q(α)µ .
∇ν T(α) νµ = Q(α)µ .
(6.5)
As the combined energy-momentum of this multifluid system is conserved, the
sum of the source terms must vanish,
X
Q(α)µ = 0,
(6.6)
α
68
CHAPTER 6. COUPLED DARK ENERGY
model name
RP1
RP2
RP3
RP4
RP5
β
0.04
0.08
0.12
0.16
0.2
Table 6.1: Value of the coupling parameter β for our example models with an
inverse power law potential.
which in the case considered here is equivalent to
Qc = −Qφ .
(6.7)
We can combine this with the usual expression for the energy-momentum tensor
of the scalar field and Eq. (6.2) to obtain the relation between the Qα and S,
which yields
Q(φ)µ = S∇µ φ.
(6.8)
The evolution of the background densities in a coupled cosmology is determined
by the conservation equation Eq. (6.4). For notational convenience, we define
1
Qα ≡ − Q(α)0
a
(6.9)
hα ≡ ρα + pα = (1 + wα )ρα ,
(6.10)
as well as
so that the background densities then evolve according to
ρ′α = −3Hρα (1 + wα ) + aQα .
(6.11)
As we restrict our consideration here to a coupling between cold dark matter
and φ, this is explicitely:
ρ′φ = −3Hhφ + aQφ
ρ′c
(6.12)
= −3Hhc + aQc .
As examples for a particular coupled dark energy cosmology, we take a standard scalar field with an inverse power law potential (Ratra & Peebles 1988)
U (φ) = U0 φ−α ,
(6.13)
m(φ) = m0 e−βφ/MP .
(6.14)
and choose the coupling as
The constant U0 in the potential is then set such that we obtain a given presentday energy density of the dark energy, and the cold dark matter mass m0 is
determined by the dark matter energy density today. The slope of the potential
and the coupling strength are chosen in as in Maccio et al. (2004), who carried
69
CHAPTER 6. COUPLED DARK ENERGY
1
Ωcdm-ΛCDM
Ωr -ΛCDM
ΩΛ −ΛCDM
Ωcdm-RP5
Ωr -RP5
ΩQ -RP5
0.9
0.8
0.7
Ωi
0.6
0.5
0.4
0.3
0.2
0.1
0
0.0001
0.001
0.01
scale factor a
0.1
1
Figure 6.1: The evolution of energy densities in the coupled model RP5 with
a coupling strength of β = 0.2 compared to a concordance ΛCDM
cosmology. Both models have the same present-day values of the
background densities (as determined from WMAP 5-year data for
ΛCDM), but differ considerably in the past. The shift of matterradiation equality is evident, as is the φMDE phase of the RP5
model. In the language of Chapter 4, this leads to an average fraction
of dark energy during structure formation of Ω̄dsf ∼ 2%.
out a first N-body simulation of these particular models. The example models
are labeld RP1 through RP5, with α = 0.143, and coupling strengths β = const
as shown in Table 6.1. Note that the numerical values of β differ from the ones
in Maccio et al. (2004) due to a different convention in the definition of the
coupling in Eq. (6.14).
This parameter choice then allows us to compare our numerical implementations to this previous work; the formalism here and the extension of Cmbeasy
and Gadget-2, however, are written in terms of a general coupling source S
with the corresponding Qα , as defined above.
For our example models with the coupling of Eq. (6.14), S is simply given by
S=−
β(φ)
ρc .
MP
(6.15)
The resulting evolution of the background densities is illustrated by Fig. 6.1,
which compares the example model RP5 to a concordance ΛCDM universe. A
typical feature of this type of coupled dark energy cosmologies is the appearance
of a phase of constant, but finite, Ωφ before the scalar field begins to dominate
- a particular realization of early dark energy. Assuming a vanishing energy
density for all components other than cold dark matter and the scalar field,
the dark energy density in this phase for a constant β is Ωφ = 23 β 2 . The
phenomenology of this epoch with an early dark energy component, however,
is substantially different from the uncoupled scalar field dark energy models
70
CHAPTER 6. COUPLED DARK ENERGY
discussed in Chapter 4, as we will see below. In the nomenclature of the phase
space analysis of coupled quintessence by Amendola (2000), this phase is called
the φM DE, as this is not the pure matter dominated epoch (MDE) found in
an ordinary ΛCDM, but contains also a fraction of dark energy that depends
on the strength of the coupling. Also apparent in Fig. 6.1 is a shift of the time
of matter-radiation equality to earlier times, which for the very high coupling
strength of model RP5 is substantial.
6.2 Linear Perturbations
With the general formalism of the previous section at hand, one can now compute the linear perturbations around the homegeneous background, exactly as
described in Chapter 3, but this time keeping the additional terms that result from the dark matter - dark energy coupling. The perturbed versions of
energy-momentum conservation, Eq. (6.4), now are, in the longitudinal gauge,
(ρα + pα ) 3Φ′ + kvα + 3Hρα (δα + wα πLα ) + (ρα + δα )′ = −δQ(α)0 (6.16)
for the energy density perturbation of the species α, as well as
2
Πα − πLα k + 4Hhα + (hα vα )′ − hα Ψ = δQ(α)i ,
pα
3
(6.17)
for the velocity part. These equations are identical to the ones obtained in
the uncoupled case, Eq. (3.47) and Eq. (3.48), but for the perturbations of the
source terms on the right hand side, expressing the fact that instead of two independent sets of equations for cold dark matter and the dark energy fluid, we
are dealing with one multicomponent fluid. In the evolution equations below,
we substitute expressions that contain terms containing δQ(α) by the corresponding expressions in terms of the gauge-invariant perturbation of the source
S, denoted by DS. We now denote the adiabatic sound speed of species i by
c2i = ρ̇/ṗ. Following Kodama & Sasaki (1984), and again using the same gaugeinvariant variables and notation as Doran (2005) and Durrer (2008), Eq. (6.16)
and Eq. (6.17) can be rewritten as evolution equations for the gauge-invariant
density perturbations Dgα (which, as a reminder, are related to the longitudinal
δ via Dgα = δα + 3(1 + w)Φ). A tedious calculation leads to the following set of
evolution equations:
aQφ
Dgφ
(6.18)
Dg′φ = − 3H(1 − wφ ) +
ρφ
aQφ
2
2
− k (1 + wφ ) + 3H(1 − cφ ) 3H(1 + wφ ) −
Vφ /k
ρφ
aQφ
+9H(1 + wφ )(1 − c2φ )(1 −
)Φ
3Hhφ
"
#
′
′
′
Q
φ′
H
1
Φ
a
φ
− DS − SX ′ + Qφ
− 1 Φ − Qφ −
Φ
+
ρφ
a
a
H2
H
H
aQφ
aQφ
kDgφ
′
+ 2H −
Vφ =
Vφ − 3 1 −
kΦ + kΨ
(6.19)
1 + wφ
hφ
3Hhφ
71
CHAPTER 6. COUPLED DARK ENERGY
The evolution of cold dark matter perturbations is governed by the two equations:
a
aQc
a Q′c
φ′
SX ′
aQc c
Φ + Qφ Ψ + DS +
Dg − kVc +
A−
,
ρ
ρc
ρc H
ρc
ρc
ρc
c
Sφ′
Sφ′
= − H+
Vφ + kΨ,
(6.20)
Vc +
ρc
ρc
Dg′c = −
Vc′
where A denotes a combination of the metric perturbations,
H′
1
A = Ψ − 1 − 2 Φ − Φ′ ,
H
H
(6.21)
and X is the gauge-invariant perturbation of the scalar field, φ, for which we
have
Vq
(6.22)
X = φ′ .
k
In the equations above, only its derivative appears, which can be substituted by
X ′ = (−2Hφ′ − a2 U,φ − a2 S) Vφ /k + φ′ Vφ /k,
(6.23)
completing the set of evolution equations for the coupled dark matter - dark
energy fluid. The other perturbation equations of Chapter 3 remain untouched,
in particular the equations for the gravitational potentials Φ and Ψ.
These equations have been implemented in Cmbeasy, allowing to compute
CMB and matter power spectra for any coupling specified by an expression for
S, and the corresponding perturbation, DS. For our example models, DS is
given by
β,φ
β(φ)
β(φ)φ′
c
ρc Dg − 3 1 +
ρc X.
(6.24)
DS = −
Φ −
MP
3MP H
MP
The temperature-temperature CMB power spectrum of the set of example
models is shown in Fig. 6.2. Even more pronounced than the shift of the peaks
is the strong suppression of power with increasing coupling. A considerable part
of this suppression is caused by the fact that a higher coupling also entails a
higher initial energy density of cold dark matter, in order for all example models
to reach identical fractional energy densities today. Already from this plot, it
appears that it is very unlikely that a coupling strength as high as the one of
RP5 could be compatible with present day observations, even if changes in other
cosmological parameters could partly compensate for the change in the power
spectra.
6.3 Matter Fluctuations
The effects of a coupling can also be very substantial for matter perturbations,
as can be seen from the cdm-power spectrum of the example models in Fig. 6.3.
At first, it seems surprising that the amplitude increases with higher coupling.
After all, in Chapter 4 we had seen that the presence of an early dark energy
72
CHAPTER 6. COUPLED DARK ENERGY
6000
ΛCDM
RP1
RP2
RP3
RP4
RP5
2
l(l+1) C l / 2 π [µ K ]
5000
4000
3000
2000
1000
0
0
1000
500
1500
2000
l
Figure 6.2: TT power spectra for models RP1 to RP5. Shown for comparison
is a ΛCDM model with best fit parameter values determined from
WMAP 5-year data.
ΛCDM
RP1
RP2
RP3
RP4
RP5
1e+06
P(k) [ h
-3
3
Mpc ]
1e+08
10000
100
1
0.0001
0.01
k/h [Mpc
-1
1
]
Figure 6.3: The power spectrum of linear cold dark matter perturbations for the
example models RP1 to RP5. Shown in black is a fiducial ΛCDM
model with best-fit parameter values from the 5-years of WMAP
data. The enhanced structure growth in the linear regime is evident,
in particular for model RP5 with the highest value of the coupling,
β = 0.2.
73
CHAPTER 6. COUPLED DARK ENERGY
component during the period of structure formation suppresses the growth of
matter fluctuations, and from Section 6.1 we know that in the φMDE, we expect
Ωφ ≈ 23 β 2 . Of course, the higher matter fraction at early times in the example
models could be partially responsible for the effect seen here, but the enhanced
growth is actually a quite generic feature of our class of models, and can even
be understood analytically, e.g. from the explicit solutions for the evolution of
matter perturbations during the φMDE in the small-scale limit presented by
Amendola & Tocchini-Valentini (2002). Deriving Eq. (6.20), and taking the
limit k → ∞, one arrives after some algebraic manipulations at the expression
for the matter perturbation growth in terms of the time variable α = log a,
!
3
Ḣ
− β φ̇ Ḋgc − (1 + 2β 2 )Dgc Ωc = 0,
(6.25)
D̈gc + 1 +
H
2
where - only in this equation - overdots are derivatives with respect to α, and
Ωb = 0 is assumed. This can be solved analytically (see also the discussion in
Di Porto & Amendola 2008), and one finds
2
Dgc ∼ a1+2β .
(6.26)
This behavior is illustrated in Fig. 6.4, which shows the numerical results of the
growth factor Dgc /a. In this figure, the growth factors are normalized to unity
today, and it is apparent that the coupled models reach the same amplitude of
the density contrast as the ΛCDM model starting from much lower values in
the past.
6.4 The Growth Rate
We can also fit the growth of matter perturbations in our models by an analytic
expression; such a phenomenological fit is especially useful when compared to the
standard expressions commonly used for ΛCDM. Conventionally, one presents
this kind of fit as an approximation of the the growth rate f (a), which is defined
as
d ln Dgc
.
(6.27)
f (a) ≡
d ln a
It is well known that for ΛCDM cosmologies the total growth rate is well approximated by a power of the total matter density
f ≈ ΩγM ,
(6.28)
with γ = 0.55, roughly independent of the cosmological constant density (Peebles 1980).
Amendola & Quercellini (2004) found that this fit can be extended to coupled
dark energy models at low redshifts by setting γ = 0.56(1 − β 2 ). A more general
fit, extending the validity to higher redshifts and to models with a growth faster
than the standard ΛCDM case, was proposed by Di Porto & Amendola (2008).
However, it is only valid for models with no admixture of uncoupled matter,
whereas in our case we also have a baryonic component. We suggest here a fit
74
CHAPTER 6. COUPLED DARK ENERGY
1.6
ΛCDM
RP1
RP2
RP3
RP4
RP5
D+ / a
1.4
1.2
1
1
10
z
100
Figure 6.4: Growth factors for the example models compared to the reference
ΛCDM model, for which the growth factor is approximately constant during matter domination. Plotted is the gauge-invariant Dgc
(labeled D+ in the plot), numerically evaluated using Cmbeasy,
divided by the scale factor a.
for the combined growth rate, i.e. the one computed by weighting the matter
fluctuations at any given redshift with the corresponding Ω(cdm, b) . For the class
of our example models with α = 0.143, we find that a phenomenological fit of
the form
ΩCDM
ǫc β 2 ) ,
(6.29)
f (a) ∼ ΩγM (1 + γ
ΩM
with γ = 0.56 and ǫc = 2.4 reproduces the growth rate with a maximum error of
∼ 2% over a range of coupling values between 0 and 0.2 and for a cosmic baryon
fraction Ωb at z = 0 in the interval 0.0− 0.1. For models with a different slope of
the potential the error is not much bigger, e.g. for α = 2.0 the maximum error
increases to ∼ 4% in the same range of coupling and baryon fraction. Fig. 6.5
compares the result of this fitting formula to the numerical result, showing the
good agreement, and compares this to the fit one would obtain from the fitting
function Eq. (6.28), which is a good fits only for the ΛCDM case.
6.5 The Coupling Strength
Before we move on to the non-linear regime and the implications of a darkmatter dark energy coupling on smaller scales, we can ask the question of how
realistic the size of β in our example models really is - and indeed the power
spectra in Fig. 6.2 and Fig. 6.3 clearly suggested that the higher end of the
range for β chosen for the examples can serve only as an illustration of the
75
CHAPTER 6. COUPLED DARK ENERGY
1.2
growth index f
1
0.8
RP5 - numerical
RP5 - new fit
0.55
ΩM
0.6
0.4
1
10
z+1
100
Figure 6.5: The growth rate f (a), evaluated numerically for model RP5, and
the phenomenological fit of Eq. (6.29), which at this scale is indistinguishable from the numerical computation. Also shown is the ‘fit’
f = Ω0.55
M , which is a very good approximation in ΛCDM cosmologies, but less so for coupled dark energy models.
1.0
0.9
rel. likelihood
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
β
Figure 6.6: Observational constraints on the strength β of the dark matter dark energy coupling for quintessence models with an inverse power
law potential and α = 0.143. The blue and light green bars show
the 68% and 95% confidence limits. As expected from the example
CMB and matter power spectra for the models RP1 through RP5,
only the smaller coupling strengths of RP1 and RP2 are in the region
that can give reasonable fits to our set of observational data.
76
CHAPTER 6. COUPLED DARK ENERGY
68% confidence limit
(−)0.003
95% confidence limit
(−)0.006
Ωm h2
0.136(+)0.003
Ωb h2
0.0222(+)0.0006
h
0.69(+)0.02
0.69(+)0.03
β
< 0.04
< 0.08
τ
0.08(+)0.02
(−)0.02
0.08(+)0.03
ns
0.96(+)0.01
(−)0.01
0.96(+)0.03
ln(1010 As ) − 2τ
2.92(+)0.02
(−)0.02
2.92(+)0.04
σ8
0.81(+)0.03
(−)0.03
0.81(+)0.06
(−)0.0005
(−)0.01
0.136(+)0.006
(−)0.0010
0.0222(+)0.0011
(−)0.03
(−)0.03
(−)0.02
(−)0.04
(−)0.05
Table 6.2: Observational constraints on the cosmological parameters for coupled
dark energy models of the type of our set of examples and the set of
cosmological probes described in the text.
effects induced by the coupling, and that e.g. RP5 does not represent a realistic
description of the Universe we observe.
Observational constrains on β are shown in Fig. 6.6, from comparing the predictions of the class of our example models (i.e. an inverse power law potential
with α = 0.143 and a constant, positive coupling as defined by Eq. (6.14)), to
the CMB data from five years of observations from WMAP (Nolta et al. 2009),
the distribution of matter in the Universe as reported by the Sloan Digital Sky
Survey (Tegmark et al. 2006, 2004a) (we marginalize over the bias), the Baryon
Acoustic Peak (Eisenstein et al. 2005), and a compilation of supernovae data
(Riess et al. 2007). While the usual cosmological parameters are allowed to
vary, i.e. the Hubble parameter h, the matter densities today Ω0m h2 and Ω0b h2 ,
the optical depth τ , scalar spectral index ns and the initial amplitude of fluctuations As via the combination ln(1010 As ) − 2τ , the initial energy density of
cold dark matter and the constant prefactor of the quintessence potential are
adjusted such that the given energy densities today are reached. The complete
set of confidence intervals for the parameters is summarized in Table 6.2.
From this set of observational data, the size of the coupling is constrained to
β < 0.08 at a confidence limit of 95%. This result is in very good agreement
with the findings of Bean et al. (2008), who report a constraint for the coupling
strength of β < 0.07 for a cold dark matter - dark energy coupling for dark
energy models with inverse power law and exponential potentials.
6.6 Newtonian Approximation
A coupling between dark matter and dark energy will also influence the physics
on scales smaller than the cosmological scales considered in the previous section.
77
CHAPTER 6. COUPLED DARK ENERGY
For a preliminary understanding of the physical effects that we expect on these
smaller scales, we consider in this section the Newtonian limit, i.e. the limit in
which λ ≡ H/k ≪ 1. Following e.g. Amendola (2004), (see also the discussion
in Maccio et al. 2004) one can show that in this limit the perturbation of the
scalar field can be expressed as
δφ ∼ 3λ2 Ωc β(φ)δc ,
(6.30)
Using this expression then allows to derive the approximate formula for the
gravitational potential, namely
Φ∼
3 λ2 X
Ωα δα .
2 M2
(6.31)
α6=φ
This can be substituted in Eq. (6.20), and then yields an expression for the
effective gravitational potential Φc felt by the dark matter in the Newtonian
limit,
β(φ)
δφ,
(6.32)
Φc ≡ Φ +
M
where we can replace Φ and δφ using Eq. (6.31) and Eq. (6.30). With the N-body
simulations in mind that will eventually study the small-scale behavior of our
coupled dark energy models in detail, we can convert the resulting expression
to real space, and find
∇2 Φ c = −
a2 X
a2
ρc δc 1 + 2β 2 (φ) −
ρα δα ,
2
2
(6.33)
α6=φ,c
where the last term is the contribution of the uncoupled components in the
universe. We can absorb the coupling factor in this equation into an effective
gravitational constant, and define
G̃c = GN [1 + 2β 2 (φ)],
(6.34)
where GN is the usual value of Newton’s constant. It immediately follows that
the coupling not only changes the value of the gravitational constant, but also
that the effective gravitational interaction becomes time dependent as soon as
β is a function of the scalar field φ. Of particular interest are the modifications
to the Euler equation, which can be derived starting from Eq. (6.20), taking
the Newtonian limit and inserting the expression for the effective gravitational
potential Φc . The final result can be written as
β(φ) ′ ~
′
~
φ ∇~vc +
∇~vc + H −
M


X
3 2
H Ωc δc + 2Ωc δc β 2 (φ) +
Ωα δα  = 0 .
(6.35)
2
α6=φ,c
This formula is useful because it can be used to derive the acceleration of a
single cold dark matter test particle in empty space, at a distance r from the
78
CHAPTER 6. COUPLED DARK ENERGY
origin, caused by a mass Mc at the origin. Mc will change, depending on the
coupling as
R
dφ
(6.36)
M̃c ≡ Mc e− β(φ) da da .
Then, assuming that the density of this mass is much larger than the background
density, one can write the density contrast due to Mc as
Ωc δc =
8πGM̃c δ(0)
,
3H2 a
(6.37)
where δ(~r) is the Dirac distribution.
Inserting this in Eq. (6.35) and again converting to real space, yields the
acceleration equation for the particle at position ~r:
~ G̃c M̃c .
~v˙ c = −H̃~vc − ∇
r
(6.38)
Here, we have defined H̃ by
H̃ ≡ H
β(φ) φ̇
1−
M H
!
.
(6.39)
In standard cosmologies, for β = 0, this term is the well-known Hubble-drag, or
friction term. In coupled dark energy models, we see that the coupling induces
an extra coupling-dependent contribution to this term, effectively leading to an
’anti-friction’ (in the case of a positive coupling) for the acceleration of dark
matter particles. This effect is in addition to the modification of the gravitational constant that the dark matter particles feel, G̃ as defined in Eq. (6.34),
and the varying mass of the dark matter particles.
This also summarizes the changes that need to be incorporated into any Nbody code in order to study the non-linear regime of structure formation.
6.7 The Non-Linear Regime
A complete N-body code package that can simulate the non-linear structure
formation in coupled dark energy models has been developed by Baldi (2009),
building on the Gadget-2 code (Springel 2005). This code takes the output of
the modified Cmbeasy version we have described here, i.e. the evolution of the
background quantities like the energy densities including the scalar field and the
Hubble rate, the mass variation of cold dark matter particles, the linear growth
factor and matter power spectra as computed by Cmbeasy as input and then
evolves a system of particles with those linear properties from a given redshift
on (in the simulations considered here, from zi = 60), making also the nonlinear properties of coupled dark energy cosmologies accessible to quantitative
study. In this section, we briefly report the results for our example models RP1
through RP5 (for the complete and detailed discussion, see Baldi 2009).
For the set of example models RP1 through RP5 (as well the reference ΛCDM
model), two sets of simulation runs were carried out. The first set, a lowresolution one with a large box size of 320 h−1 Mpc and 2 × 1283 particles (mass
79
CHAPTER 6. COUPLED DARK ENERGY
resolutions were Mb = 1.9 × 1011 h−1 M⊙ and Mcdm = 9.2 × 1011 h−1 M⊙ with a
smoothing length es = 50 kpc). The second set, with a much higher resolution,
used a smaller box size of 80 h−1 Mpc with considerably more particles, 2 × 5123
(here the mass resolutions were Mb = 4.7 × 107 h−1 M⊙ and Mcdm = 2.3 ×
108 h−1 M⊙ with a smoothing length es = 3.5kpc).
A first result of the simulations is that the mass function, i.e. the number
of dark matter halos of a certain mass that form in our cosmologies, is well
described by the fitting function given in Jenkins et al. (2001) and also the
one by Sheth & Tormen (1999), which yields a a marginally better fit. Both
these fitting functions use the linear power spectra and growth factor computed
numerically using Cmbeasy to predict the mass function. The results then are
good fits for all values of the coupling and over the entire redshift range of the
simulations.
A second result is the evolution of the bias between the amplitude of the
density fluctuations of baryons and cold dark matter. As we have seen in the
linear analysis, baryons and cold dark matter evolve differently, since only the
dark matter is subjected to the additional forces due to the coupling (Mainini
2005; Mainini & Bonometto 2006). This linear bias is present on all scales, and
in our coupled models substantially different from the ΛCDM prediction. In the
non-linear simulations, an additional bias will develop due to the hydrodynamic
pressure forces acting on the baryons, but not on the collisionless dark matter
particles. The interesting result of the N-body runs now is that the effect of the
dark matter - dark energy coupling is also important at smaller scales, in the
mildly non-linear regime, and enhances the different clustering rates of baryons
and cold dark matter. Even when artificially switching of hydrodynamic forces,
and treating both baryons and dark matter as collisionless, a stronger coupling
still leads to visibly higher bias due to the different effective gravitational interactions for both species.
The simulations also clearly show this effect at the center of massive collapsed structures. Indeed, halos in simulations with higher coupling contain
fewer baryons than expected based on their mass and baryon fraction of the
background cosmology. This difference is in principle observable, as e.g. X-ray
measurements in clusters would not yield the cosmological value. The baryon
fraction within the virial radius r200 , i.e. within the radius enclosing a mean
overdensity 200 times the critical density, can be defined as
fb ≡
Mb (< r200 )
Mtot (< r200 )
(6.40)
The relative baryon fraction then has the expression
Yb ≡
fb
,
Ωb /Ωm
(6.41)
for which the reference ΛCDM-simulations yield values consistent with Yb ∼ 0.92
found by the Santa Barbara Cluster Comparison Project (Frenk et al. 1999),
and with the more recent results of Ettori et al. (2006) and Gottloeber & Yepes
(2007). For the coupled models, however, the simulations show that the relative
80
CHAPTER 6. COUPLED DARK ENERGY
Halo Density profiles for CDM and baryons for Group nr. 29
ΛCDM
RP1
RP2
RP5
ρ / ρcrit
10000
1000
100
10
10
M200(ΛCDM) = 2.79752e+13 h-1 MO•
100
R (h-1 kpc)
1000
Figure 6.7: Density profile at z = 0 of one halo in the different example models
(Baldi 2009). Solid lines show cold dark matter, dot-dashed lines
baryons. The vertical line indicates the virial radius for the ΛCDM
halo. Models with a higher coupling show a lower inner overdensity
compared to ΛCDM.
baryon fraction decreases with increasing coupling, down to a value of Yb ∼
0.86−0.87 for the RP5 case. In all cases, the fractions in clusters are smaller than
the cosmological value; this could potentially alleviate tensions between the high
baryon abundance estimated from CMB observations, and the somewhat lower
values inferred from detailed X-ray observations of galaxy clusters (Vikhlinin
et al. 2006; McCarthy et al. 2007; LaRoque et al. 2006; Afshordi et al. 2007c).
A third result of the N-body simulations is that the density profile of the
halos can, even in the coupled cases, be described by an NFW (Navarro et al.
1997) profile, i.e. as
δ∗
ρ(r)
=
.
(6.42)
ρcrit
(r/rs )(1 + r/rs )2
The parameters in this fit are δ∗ , which sets the characteristic halo density
contrast relative to the critical density, and the scale radius rs . To compare
the characteristic density profiles between different simulations, one can compare corresponding structures in the different simulations. Due to the different
physics involved, there obviously is no one-to-one correspondence, but since all
simulations use the same random seed to produce realizations of the respective
initial power spectra, it is nevertheless possible to identify objects as being the
‘same’ e.g. by demanding that the most bound particle lie within the virial radius of the corresponding structure in the ΛCDM simulation. The simulations
now show that the scale radius for corresponding halos in the coupled models
consistently increases with increasing coupling, typically by about 10% − 35%
81
CHAPTER 6. COUPLED DARK ENERGY
from ΛCDM to RP5, i.e. halos become less concentrated compared to ΛCDM.
The density profile of on example halo chosen from the simulations is shown in
Fig. 6.7, and is essentially the opposite to the results obtained by Maccio et al.
(2004).
For our set of simulations, the main contribution to the lower concentration
can be attributed to the ‘anti-friction’ term of Eq. (6.39). This term induces
an extra acceleration on coupled particles in the direction of their velocity, and
the system responds to this increase in kinetic energy by a small expansion and
corresponding lowering of the concentration in order to maintain an equilibrium
configuration. The decrease of the mass of the cold dark matter particles with
time similarly causes the potential energy of the halo to decrease, also moving the system to a configuration with excess kinetic energy. However, it is
indeed the ‘anti-friction’ term which is the dominant effect, as can be numerically verified be individually switching off either contribution, and comparing
the resulting density contrasts.
The combination of these results shows that indeed coupled dark energy models are viable cosmological theories. For one specific class of these models, an
inverse power law quintessence with a constant coupling, we have shown here
explicitely that models with β . 0.8 are compatible with current CMB data,
large scale structure observations, and SnIa survey results. Furthermore, the
numerical study of the non-linear regime has shown that the properties of dark
matter halos in these models could actually alleviate present tensions between
observations and the ΛCDM model.
Indeed, the formalism that we have employed and the numerical implementation that we have developed allow to investigate a very large class of coupled
dark energy models, since both the dark energy model and the form of the coupling can be freely specified; this provides the complete numerical framework
necessary to exploit the ever increasing amount of observational data for studies
of the cosmological dark sector.
82
7 Growing Neutrino Quintessence
In the last chapter, we have considered the ramifications of a coupling between
the dark energy component and the dark matter. The coupling strength of the
models we considered was generally weaker than gravity. Now, we turn to a
class of models where this coupling can be considerably stronger, i.e. stronger
than gravity. As we will see, these models, proposed by Amendola et al. (2008)
and Wetterich (2007), then can provide a solution of the ‘why now?’ problem;
this proposed solution is particularly compelling when instead of the mass of
the dark matter it is the mass of the neutrinos that changes as a function of the
scalar field. In this chapter, we will therefore extend the analytical and numerical treatment of dark energy couplings to dark energy - neutrino couplings, and
then, as the main objective of this chapter, explore how linear density perturbations will behave in this kind of scenario.
The idea that cosmic acceleration might be related to neutrinos with a varying mass has been previously investigated in the literature (Fardon et al. 2004;
Peccei 2005) in the ‘MaVaN’ scenarios; here, the slow variation of the dark energy density can be achieved without an extremely flat potential for the scalar
field, but for many choices of scalar-neutrino coupling and scalar field potentials these scenarios suffer from instabilities (Afshordi et al. 2005), where the
attractive force between neutrinos leads to the formation of neutrino ‘nuggets’,
so that the combined fluid behaves like dark matter and is not a viable dark energy candidate any more; there are, however, viable regions of parameter space
(Bjaelde et al. 2008). Also, the cosmology of models with neutrinos coupled
to a light scalar field has been investigated in the literature (Brookfield et al.
2006b,a; Bi et al. 2005), considering scenarios where the neutrinos are heavier
in the past and become lighter during the cosmological evolution, opposite from
the case considered here.
7.1 The Model
The cornerstone the growing neutrino quintessence scenario is a neutrino mass
that depends on the scalar field φ (in units of MP ) responsible for the dark
energy. Here, we consider the specific case of
mν = m̃ν e−β̃(φ)φ ,
(7.1)
with a constant m̃ν . Defining β(φ) ≡ β̃ + ∂ β̃/∂ln φ, the change of mν is given
by
d ln mν (φ)
.
(7.2)
β(φ) = −
dφ
83
CHAPTER 7. GROWING NEUTRINO QUINTESSENCE
If β < 0 and φ, the neutrino mass will increase with time, so that with a neutrino
mass today of mν (t0 ) < 2.3 eV, the mass will have been even smaller in the past.
From the field equation for the field φ and its potential V (φ),
φ′′ + 2Hφ′ = −a2
dU
+ a2 β(φ)(ρν − 3pν ),
dφ
(7.3)
we immediately see that once the neutrinos become non-relativistic, the derivative of the potential can be balanced by the contribution from the β-dependent
term. Hence, depending on the size of the coupling, the evolution of the field φ
will effectively be stopped at a certain point in time; from then on, the scalar
field potential V (φ) will contribute an (almost) constant fraction to the total energy density, just like a cosmological constant, causing the crossover to the dark
energy dominated epoch. The point in time at which this crossover happens
is thus given by the (particle physics) properties of neutrinos and the variation
of their mass, and the crossover is triggered by the neutrinos becoming nonrelativistic; together, this provides a very intriguing explanation of the ‘why
now’ problem.
The ‘force’ that stops the evolution of the cosmon field can, in principle, be
provided something other than neutrinos - any additional dark matter component with a mass increasing with time (with an energy density at some initial
time suitably chosen). This has been discussed by Huey & Wandelt (2006);
the identification of the ‘growing matter’ component with neutrinos, however,
has the distinct advantage of not introducing any new particles, and the existence of a cosmological trigger for the transition, namely the neutrinos becoming
non-relativistic.
As we had seen in Chapter 2, the background energy and pressure of the
neutrinos, which follow a Fermi-Dirac distribution f0 , are given by the phase
space integral, usually expressed in terms of the comoving 3-momentum q =
ap = qn̂ as
Z
−4
ρν = a
q 2 dq dΩ ǫ(φ)f0 (q),
(7.4)
Z
q2
1 −4
a
f0 (q).
(7.5)
q 2 dq dΩ
pν =
3
ǫ(φ)
p
where ǫ = ǫ(φ) =
q 2 + a2 mν (φ)2 , which is a(τ ) times the proper energy
density measured by a comoving observer. In contrast to the uncoupled case,
where mν is a constant, ǫ now depends on the cosmon field value φ due to
the coupling between dark energy and neutrinos. Together with Eq. (7.3), these
equations specify the background evolution for the coupled neutrino-dark energy
system. In terms of energy densities, we can also write
ρ′φ = −3H(1 + wφ )ρφ + β(φ)φ′ (1 − 3wν )ρν ,
ρ′ν
′
= −3H(1 + wν )ρν − β(φ)φ (1 − 3wν )ρν ,
once wν = p/ρ is known from Eq. (7.4).
84
(7.6)
CHAPTER 7. GROWING NEUTRINO QUINTESSENCE
10
0
ρ
10
ρc
ρν
ρr
ρϕ
3
10
10
-6
10
10
-3
-9
-12
10
0
10
1
10
2
10
3
1+z
Figure 7.1: Example of the evolution of the background densities in the growing
neutrino quintessence model as a function of redshift. The dashed
line depicts the evolution of the neutrinos, cold dark matter is shown
as a solid line, dark energy as a dotted and the energy density of
photon as long dashed lines. The coupling has a value of β = −52,
with α = 10 and a neutrino mass of mν = 2.11 eV .
To completely describe the growing neutrino quintessence model, we assume
the potential of the scalar field to be given by an exponential,
V (φ) = MP2 U (φ) = MP4 e−αφ .
(7.7)
If we set the coupling to be constant, β(φ) = const, then the model is described
by only two parameters, the potential constant α, and the coupling strength
β. In particular, at early times, while the neutrinos are still relativistic, the
cosmology will be a typical early dark energy cosmology that we have discussed
in Chapter 4, with the fraction of early dark energy determined by the value of
α.
For concreteness, we choose example values for the constants of the theory,
β = −52, α = 10 and a rather large neutrino mass of mν = 2.11 eV, and compute the evolution of the background densities; the result is shown in Fig. 7.1.
The effect of the neutrino coupling is evident, leading to a sudden transition
from the early dark energy (tracking) behavior of the exponential potential,
when neutrinos are still relativistic and almost massless, to the present phase
with a dominating dark energy component, and the neutrino and dark energy
components having almost constant energy densities. The time after the transition is characterized by oscillations in the coupled dark energy-neutrino fluid
that decay with time. The figure does not extend into the future, which in this
model is governed by an attractor solution where the scalar field and the mas-
85
CHAPTER 7. GROWING NEUTRINO QUINTESSENCE
sive neutrinos dominate; the currently observed values of the matter and dark
energy densities correspond to the transition epoch from matter to scalar field
domination (Amendola et al. 2008).
7.2 Linear Perturbations
Our aim in this chapter is the computation of the evolution of the linear density
perturbations in this class of models. In order to obtain the first-order corrections to Eq. (7.4), one starts from the phase space distribution f , written as the
zeroth-order term f0 and the perturbation Ψps ,
f (xi , τ, q, nj ) = f0 (q) 1 + Ψps (xi , τ, q, nj ) .
(7.8)
As we have done in Chapter 3, we then have to calculate how this distribution
evolves in time, i.e. we have to solve the collisionless Boltzmann equation for the
coupled neutrinos, which gives df (xi , τ, q, nj )/dτ (see e.g. Ma & Bertschinger
1995), but modified to account for the coupling and the changing neutrino mass.
The full derivation of the Boltzmann equations is presented by Ichiki & Keum
(2008) and Brookfield et al. (2006a); note that the equations are written in the
synchronous gauge there, in contrast to the Newtonian gauge employed here.
In the Newtonian gauge, the result is:
q
d ln f0
ǫ
∂Ψps
′
+ i (k · n)Ψps +
−Φ − i (k · n)Ψ
∂τ
ǫ
d ln q
q
2
2
a m ∂ ln mν d ln f0
q
δφ.
(7.9)
= i (k · n)k 2 ν
ǫ
q
∂φ d ln q
As usual, Ψ and Φ are the metric perturbations - all the differences to the
standard case of uncoupled neutrinos, i.e. Eq. (3.66), are contained in the right
hand side of this equation (which for zero coupling vanishes, as we have seen in
Section 3.3).
The perturbation Ψps is now expanded in a Legendre series, exactly as for
uncoupled neutrinos,
Ψps =
∞
X
(−i)l (2l + 1)Ψps,l Pl (k · n)
(7.10)
l=0
with Legendre Polynomials Pl . This then yields, in Newtonian gauge,
d ln f0
qk
Ψps,1 + Φ′
,
ǫ
d ln q
qk
ǫk d ln f0
(Ψps,0 − 2Ψps,2 ) − Ψ
+ κ,
3ǫ
3q d ln q
qk
[lΨps,l−1 − (l + 1)Ψps,l+1 ] , l ≥ 2,
(2l + 1)ǫ
Ψ′ps,0 = −
Ψ′ps,1 =
Ψ′ps,l =
where
κ=−
1 q a2 m2ν ∂ ln mν d ln f0
k 2
δφ.
3ǫ
q
∂φ d ln q
86
(7.11)
(7.12)
CHAPTER 7. GROWING NEUTRINO QUINTESSENCE
This term only appears in the expression for Ψ′ps,1 which is the only one that is
modified by the presence of the coupling, as can be seen from comparing with
Eq. (3.73).
The quantities that we are interested in here, the perturbed energy and pressure as well as the velocity perturbation are also changed, and their expressions
(and the neutrino shear) are then:
Z
∂ǫ
−4
2
δφ dq
δρν = 4πa
q f0 (q) ǫ(φ)Ψps,0 +
∂φ
Z
∂ǫ
4π −4 q 4
a
f0 (q) ǫΨps,0 −
δφ dq
δpν =
3
ǫ2
∂φ
Z
q2
(ρν + pν )θν = 4πka−4 q 2 dq f0 (q)Ψps,1
(7.13)
ǫ
Z
q2
8π −4
a
(7.14)
q 2 dq f0 (q)Ψps,2
(ρν + pν )σν =
3
ǫ
The term ∂ǫ/∂φ is explicitely given by
∂ǫ
a2 m2ν ∂ ln mν
a2 m2ν (φ)
=
= −β(φ)
∂φ
ǫ
∂φ
ǫ(φ)
(7.15)
With all the changes to the neutrino evolution induced by the coupling at hand,
the only missing piece is the perturbed version of the Klein-Gordon equation
Eq. (7.3),
dU
d2 U
Ψ
(7.16)
δφ′′ + 2Hδφ′ + k2 + a2 2 δφ − φ′ (Ψ′ − 3Φ′ ) + 2a2
dφ
dφ
dβ(φ)
2
2
= −a −β(φ)ρν δν (1 − 3cν ) −
δφρν (1 − 3wν ) − 2β(φ)(ρν − 3pν )Ψ ,
dφ
which determines the evolution of the perturbations of the scalar field. This
completes the set of perturbation equations for coupled neutrinos - the perturbations for all other components are only affected through the gravitational
potential, and so their equations (and their respective numerical implementations) remain identical to the uncoupled case.
7.3 Neutrino Clustering
After implementing the equations of the previous section in Cmbeasy, we can
now compute the perturbations of density perturbations; to demonstrate the
behavior in a specific case, we compute the linear perturbations for our example
model of Section 7.1, with β = −52, α = 10 and mν = 2.11 eV. The amplitude of
the initial fluctuations is determined from the CMB anisotropies at zls ∼ 1000.
The resulting density perturbations for two wavemodes are shown in Fig. 7.2
(k = 0.1h/Mpc) and in Fig. 7.3 (k = 1.1h/M pc). As we can see, as soon as the
neutrinos become non-relativistic and their equation of state departs from the
radiation value of wν = 1/3, the neutrino perturbations start to increase, and
87
CHAPTER 7. GROWING NEUTRINO QUINTESSENCE
10
10
12
10
δ
10
10
10
10
10
10
10
8
6
4
2
0
-2
-4
-6
10
10
δc
δν
δϕ
wν
δc (ΛCDM}
10
-8
-10
10
0
10
1
10
2
1+z
Figure 7.2: Evolution of the longitudinal density perturbations for CDM (solid),
neutrinos (dashed) and the scalar field dark energy (dot-dashed)
for k = 0.1h/Mpc. The neutrino equation of state (dotted) is also
shown, as well as the evolution of the cold dark matter perturbations
in the reference ΛCDM model.
eventually even grow larger than the perturbations in cold dark matter. The
dark energy perturbations also grow strongly - and can even overtake the dark
matter perturbations in the linear approximation.
It is, however, clear, that as soon as the size of any of the perturbations
reaches unity, the linear perturbation theory will no longer be able to yield reliable results, and can only give a qualitative picture of the true physics. What
can be learned from the linear analysis, though, is an estimate of the time at
which nonlinearities become important, and how this point in time depends on
the scale. Particularly intriguing is the comparison with the standard ΛCDMpicture in Fig. 7.2. On these large scales, the scales of superclusters, the density
perturbations in a ΛCDM-universe are still in the linear regime today, while in
our example growing neutrino model, the perturbations quickly become nonlinear as soon as the neutrinos are no longer relativistic. Despite the relatively
low fraction of neutrinos, Ων , the strong neutrino clumping quickly becomes the
main source of the gravitational potential, which then drags the cold dark matter fluctuations. At the same time, the large neutrino fluctuations also cause
the inhomogeneities in the cosmon field to grow through their strong coupling.
The corresponding growth of the gravitational potential for different scales is
shown in Fig. 7.5.
In the nonlinear regime, once our linear approximation presented here is no
longer valid, one would expect the neutrino structures to decouple from the
expansion of the background, eventually forming stable ‘neutrino lumps’, like
88
CHAPTER 7. GROWING NEUTRINO QUINTESSENCE
10
10
12
10
10
δ
δν
δϕ
wν
δc (ΛCDM}
10
10
10
10
10
10
10
8
6
4
2
0
-2
-4
-6
10
-8
-10
10 1.0
1.2
1.4
1.6
1.8
2.0
2.2
1+z
Figure 7.3: Same as Fig. 7.2 but for a scale of k = 1.1h/M pc - evolution of the
longitudinal density perturbations.
10
2
1
znl
10
CDM
ν
ϕ
CDM (ΛCDM)
superclusters
10
0
clusters
10
10
-1
-2
10
-3
10
-2
10
-1
10
0
10
1
k/h[M pc−1 ]
Figure 7.4: Redshift of first non-linearities as a function of the wavenumber k
for cold dark matter (solid), neutrinos (dashed) and the scalar field
dark energy (dot-dashed). A reference ΛCDM- model is shown as
the dotted line.
89
CHAPTER 7. GROWING NEUTRINO QUINTESSENCE
10
3
10
2
k2Φ
10
1
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
10
-3
10
-2
10
-1
1
10
10
2
k/h[Mpc−1 ]
Figure 7.5: The gravitational potential in longitudinal gauge as a function of
the wavenumber k for a reference ΛCDM model (dotted), and for
the coupled neutrino model at redshifts z = 0.5 (solid), z = 5 (dotdashed), z = 50 (dashed) (the lines for the latter two redshifts overlap). For high values of Φ, this plot does not show the true physical
gravitational potential, as it is the result of a computation of only
linear perturbations.
the solutions presented by Brouzakis et al. (2008). Inside these neutrino lumps,
the gravitational potential would be much smaller than the huge value of the
linear approximation in our example. A detailed understanding of the nonlinear
regime would also allow to estimate the effect of these structures on the CMB
through the integrated Sachs-Wolfe effect. Since the nonlinear regime of structure formation in this scenario is reached much earlier than in the standard
ΛCDM scenario, and even on very large scales, linear perturbation theory is
not any more sufficient to predict the power spectrum of the Cosmic Microwave
Background, or even the matter power spectra today.
As we have seen, the linear picture is nevertheless useful for a qualitative
understanding of the physical mechanisms. Fig. 7.4 shows, as a function of
wavenumber k, the redshift znl at which the density perturbations of the different components become nonlinear. For comparison, the plot also shows the
behavior of δcdm in a concordance ΛCDM-model, with a corresponding ΩΛ and
massless neutrinos. We define znl as the redshift at which δ(znl ) = 1 for the
respective species. It is therefore a rough measure of the time when the first nonlinearities appear. As we are using linear perturbation theory, only the highest
curve for any given k is reliable, since the lower curves are already affected by
the dragging of the leading component, and in a complete nonlinear treatment,
these dragging effects would be expected to be considerably lower. Fig. 7.4 shows
that at the very largest scales, the universe is still almost homogeneous, and the
90
CHAPTER 7. GROWING NEUTRINO QUINTESSENCE
perturbations remain linear, even in our example growing neutrino model. On
scales from about 4.4 × 103 Mpc down to 14.5 Mpc, however, we find a behavior that is very different from the concordance ΛCDM expectation, where the
neutrino perturbations leave the linear regime, and possibly form large structures whose gravitational potential well will enhance the clustering of the other
cosmological components, including even the scalar field. As we approach the
neutrino free streaming scale, the first component leaving the linear regime is no
longer the neutrinos, but the cold dark matter. It is interesting to remark that
we see less clustering of cold dark matter in this regime in our growing neutrino
model than in the ΛCDM case. This is in fact expected, since as we have seen
in Chapter 4, the presence of early dark energy in the growing neutrino model
will slow the clustering. Additionally, as we have chosen ΩΛ = Ωφ , there is
less matter in our growing neutrino model than in the reference ΛCDM model,
since the massive neutrinos contribute a certain fraction to the total energy density today, unlike the massless neutrinos in the ΛCDM model - for smaller and
smaller neutrino masses, this effect of course becomes correspondingly smaller,
whereas the influence of the early dark energy depends on the choice of α. At
small scales, finally, we find the usual pictures with the cold dark matter already
in the nonlinear regime also for the ΛCDM case, and no clustering of neutrinos,
as they are in the free streaming regime.
To summarize our results, we have shown that in the growing neutrino quintessence scenarios, one typically expects neutrino clustering on supercluster scales;
this is a prediction of a feature that is very distinct from the concordance ΛCDM
paradigm, and the observation of structures on these very large (supercluster)
scales would be a strong hint in favor of the scenario outlined here. When considered together with the results that we had obtained in the last chapter, we
can conclude that quintessence models with a coupling to other species can lead
to vastly different cosmological evolutions, and that further investigation of this
property of the dark energy continues to be a promising route towards revealing
the true nature of quintessence.
91
CHAPTER 7. GROWING NEUTRINO QUINTESSENCE
92
8 Conclusions
In this thesis, we have studied the cosmological implications of a variety of dark
energy models, in search of the fundamental properties of quintessence. Having
surveyed the basics of the framework and the methodology of modern cosmology,
we commenced our investigation in Chapter 4, by considering early dark energy
models. In these class of models, the dark energy can contribute a substantial
fraction to the total energy density even at high redshifts, before last scattering
and during structure formation - unlike in the standard ΛCDM scenario, where
the contribution from the cosmological constant is negligible at early times. We
considered a variety of dark energy models and parameterizations, and compared
the inferred values for the cosmological parameters between early dark energy
models, models with a fading dark energy, and the ΛCDM cosmology.
As the central result, we found that current observational data constrain the
amount of early dark energy to a few percent. We found that the inclusion of
Ly-α data, or more precisely, the linear matter power spectrum inferred from
observations of the Ly-α-forest in the Sloan Digital Sky Survey, yields an upper
bound of Ω̄dsf . 1.7% when the early dark energy is described by the Ω(w0 , Ωed )
parameterization; without the Ly-α-data we found an upper bound of Ω̄dsf . 4%,
from a combination of observational data comprising several sets of CMB data,
large scale structure and SnIa surveys.
We also showed examples of early dark energy models that reproduce the
spectra of fading dark energy models, showing that early dark energy cannot
unambiguously be detected by measurements of the CMB temperature and Emode polarization alone. We then pointed out that future BAO measurements,
when the ‘standard ruler’ is calibrated off the measured CMB, have to properly
account for the possibility of an early dark energy component. This can either be
done by relying on a determination of the early dark energy fraction from some
other, independent, source, or by introducing a calibration factor and thereby
fitting for the absolute ruler scale. This will decrease the power of future BAO
measurements (for a hypothetical future experiment, we found that the area of
the confidence regions increased by a factor of ∼ 2.3). Neglecting the possibility
of early dark energy, however, would lead to a severe misinterpretation of the
data, and the inferred values of the cosmological parameters would be biased.
In Chapter 5, we then considered a scalar field dark energy model with an
exponential potential, which has an early dark energy component, and the Cuscuton model. While Cuscuton models are formulated as scalar field dark energy
models, they actually reduce to a pure constraint system; Cuscuton perturbations do not introduce an additional dynamical degree of freedom, and effectively
amount to a minimal modification of a cosmological constant. This motivated
us to interpret both models as a modification of the law of gravity on large
scales; we expressed this modification in terms of a running of the Planck Mass
93
CHAPTER 8. CONCLUSIONS
at the cosmological horizon scale. Our main conclusion was that cosmological
observations yield an upper bound for this kind of ‘glitch’ in the Planck Mass of
1.2% (at 95% confidence), severely restricting possible deviations from Einstein
gravity at these scales.
We then studied coupled dark energy models. In these models, either the
mass of the dark matter or e.g. of the neutrinos depends on the scalar field.
In Chapter 6 we have presented the general formalism that allows to specify
the form of the coupling as an arbitrary function of the field, independent of
the dark energy model. We have implemented this formalism in Cmbeasy, and
as an example application, explored a set of coupled dark energy models with
an inverse power law potential; for these models, we showed that observational
data constrains the coupling strength to β . 0.08 (at 95% confidence).
We have also modified Cmbeasy to produce input data for N-body simulations for coupled dark energy models. We have reviewed the results from a set
of N-body simulations that were obtained using our computations in the linear
regime as input. These simulations found that a larger coupling strength leads
to a lower average halo concentration. Also, the baryon fraction as well as the
inner overdensity of cold dark matter halos decreased with increasing coupling.
These results show that coupled dark energy models are viable cosmological
models, and might even alleviate present tensions between the ΛCDM model
and observations.
We then extended our investigations of dark energy couplings to the growing
neutrino quintessence scenario. Here, the coupling must be somewhat larger
than gravitational strength, in order to effectively stop the evolution of the
scalar field. We saw that these models lead to a substantial neutrino clustering
on large scales, typically the scale of superclusters, a feature very distinct from
the standard ΛCDM cosmology.
Since non-linear neutrino lumps in this scenario are expected to form at a
redshift of about z ≈ 1, our linear analysis does not apply to the very recent
history of the universe. A full exploration of the non-linear regime is therefore necessary already to consistently compute the power spectra of the Cosmic
Microwave Background; such an investigation could also reveal more intriguing
features of these cosmological scenarios. Certainly, the detection of non-linear
structures at these very large scales would be an important indication for a new
attractive force - mediated by the cosmon.
In summary, all the dark energy scenarios that we have considered are significantly different from the ‘concordance’ model of cosmology, the ΛCDM universe.
We have seen that these different quintessence models, with fundamentally different properties, are in agreement with observations, and that observational
data is not yet precise enough to distinguish between these different alternatives. However, we have also seen that the many quintessence models do make
characteristic predictions, some of which are quite surprising and intriguing; in
combination with the continuing increase in the amount and precision of observational data, this variety in the phenomenology of quintessence is good reason
to remain hopeful and confident that the puzzle of dark energy will be solved.
94
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Acknowledgments
I am very much indebted to my advisor Christof Wetterich, not only for scientific
guidance and support, but also for the opportunity to enjoy science working with
such a great cosmology group at a wonderful place.
I thank Michael Doran for having authored Cmbeasy, showing me its secrets,
and for letting me take care of it.
I thank Matthias Bartelmann for refereeing this thesis, and the entire group at
ITA for many discussions about (early) dark energy and most importantly, real
observations.
I have immensely enjoyed the collaboration both with Eric Linder and Niayesh
Afshordi.
Valeria Pettorino, Iain Brown, David Mota, Juliane Behrend and the entire
cosmology group have made it not only an interesting, but also a fun time.
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