E
XPLORING THE
C
OSMOLOGY OF
M
ODIFIED
G
RAVITY
:
THEORY AND PHENOMENOLOGY OF MODELS
EXHIBITING WEAKLY BROKEN GALILEON
INVARIANCE
THESIS
submitted in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
in PHYSICS
Author : Carlos Mateos Hidalgo Student ID : s2076861 Supervisor : Dr. A. Silvestri J . Papadomanolakis S. Peirone 2ndcorrector : Prof. Dr. A. Achucarro
E
XPLORING THE
C
OSMOLOGY OF
M
ODIFIED
G
RAVITY
:
THEORY AND PHENOMENOLOGY OF MODELS
EXHIBITING WEAKLY BROKEN GALILEON
INVARIANCE
Carlos Mateos Hidalgo
Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands
July 16, 2019
Abstract
The problem of the cosmological constant together with the tension in the observations of the present value of the Hubble parameter has brought about the search of alternative theories to the Standard Model of Cosmol-ogy. One of the most promising ones is Modified Gravity. In this thesis, we explore scalar-tensor theories that are invariant under weakly broken galileon (WBG) transformations. We have derived the background cos-mology and found attractor solutions that track a De Sitter Universe at late times, solving the coincidence problem and preventing from fine tun-ing issues. We have implemented the model into EFTCAMB, an
Einstein-Boltzmann solver that employs the effective field theory of dark energy, and computed the power spectrum of temperature-temperature CMB ani-sotropies and the matter power spectrum. Our results, based on the the-oretical predictions, are promising. While being compatible with LCDM at small scales, WBG can lower the ISW tail of the TT models on the CMB power spectrum, making it potentially favoured by observations.
Contents
1 Preface 6
2 Introduction 8
2.1 A homogeneous and isotropic universe . . . 8
2.1.1 Dynamics of the Universe . . . 9
2.2 The cosmological constant . . . 12
2.2.1 The problem of the cosmological constant . . . 13
2.3 The Standard Model of Cosmology . . . 14
2.4 Beyond the Standard Model of Cosmology . . . 16
2.4.1 Dark energy as a modified form of matter . . . 17
2.4.2 Dark energy as a modification of gravity . . . 18
3 The effective field theory of dark energy 20 3.1 Cosmological perturbations . . . 21
3.2 A unifying language . . . 22
3.2.1 Mapping to EFT: two examples . . . 23
3.3 EFT mapping . . . 24
3.3.1 A simple example . . . 26
3.4 Stuckelberg mechanism . . . 26
3.5 Viability conditions . . . 27
4 Numerical tools for the study of DE 30 4.1 The Einstein-Boltzmann equations . . . 30
4.1.1 The Boltzmann equations . . . 31
4.1.2 The Einstein equations . . . 32
4.2 Cosmological observables . . . 33
4.3 Implementation of an Einstein-Boltzmann solver . . . 35
4.3.1 Structure of eftcamb . . . 35
4.3.2 Viability conditions . . . 36
5 Weakly broken galileons 38 5.1 Galileons in flat space . . . 38
5.2 Galileon model with gravity . . . 39
5.2.1 Weakly broken symmetry . . . 40
5.3 Most general theory with WBG invariance . . . 43
CONTENTS 3
6 Results I: background evolution 46 6.1 Dynamical equations . . . 46 6.2 Tracker solution . . . 48 6.3 Expansion history along the tracker . . . 54
7 Results II: numerical analysis 56 7.1 Theoretical set up . . . 56 7.2 Numerical results of WBG . . . 58
8 Conclusions and outlook 70
A Full EFT mapping for Galileons and GLPV theories 74 A.1 Generalized galileon mapping . . . 74 A.2 GLPV mapping . . . 75
List of Figures
2.1 Distribution of galaxies as measured by the 2dF Galaxy Redshift Survey. 9
2.2 CMB map reconstructed from WMAP observations. . . 14
2.3 Linear matter power spectrum measured by several astronomical sur-veys. . . 15
2.4 Hubble diagram for the low-redshift and high-redshift SNe Ia from HSST collaboration. . . 16
4.1 eftcamb flag structure . . . 36
5.1 Conventions for the external legs in the Feynmann diagram. . . 41
5.2 1PI vertices with one graviton line and two or three scalar fields. . . . 41
7.1 Background expansion history of WBG along the tracker solution . . 59
7.2 χ evolution for different choices of s and p. . . 59
7.3 Stability map for WBG with c5 = 0 . . . 60
7.4 Background EFT functions for WBG: Horndeski sector . . . 61
7.5 Second order EFT functions for WBG: Horndeski sector . . . 62
7.6 Power spectra for WBG, case I: sampling s. . . 63
7.7 Power spectra for WBG, case I: sampling p. . . 63
7.8 Power spectra for WBG, case I: sampling c4. . . 64
7.9 Power spectra for WBG, case I: cT = 1. . . 64
7.10 Stability map for WBG with c5 6= 0, s = 2 . . . 65
7.11 Stability map for WBG with c5 6= 0, s 6= 2 . . . 66
7.12 Background EFT functions for WBG: beyond Horndeski sector. . . . 66
7.13 Second order EFT functions for WBG: beyond Horndeski sector. . . . 67
7.14 Power spectra for WBG: case II. . . 67
7.15 Summary of WBG models against Planck ’s TT modes of the CMB anisotropies. . . 68
A mis padres.
1
Preface
Cosmology is an observational science. As much as we would like to we cannot ex-periment with the Universe, play with gravity or change its energy content. Thus, we rely on observations to learn from the Cosmos and describe it. As plenty of evidence supports, our Universe is currently undergoing an epoch of accelerated expansion, what we call cosmic acceleration. Cosmologists rely on dark energy to explain this fact, for which the simplest candidate is the cosmological constant (CC): a constant energy density across the Universe that is driving the acceleration. The Standard Model of Cosmology (SMC) states that we live in a flat Universe in which about 70% of its energy content is in the form of this CC. Despite its simplicity, the CC is still a puzzle for physicists. The lack of theoretical understanding worrisome and the tensions in the SMC, has brought cosmologists to search for different approaches to dark energy.
In this thesis we explore the cosmology of modified gravity (MG), one of the most promising alternatives to the cosmological constant. The idea behind MG is simple: introduce modifications to general relativity that can generate a self accelerating universe at cosmological scales without altering the cosmology at small scales. The number of MG models that has been proposed is overwhelming. In order to deal with such a plethora of theories the Effective Field Theory (EFT) of dark energy was proposed. This powerful approach offers a universal language to map every particular model to the EFT framework. Our project studies the cosmology of weakly broken galileons (WBG) and implemented it into an Einstein-Boltzmann (EB) solver to obtain predictions from it.
The present thesis is structured as follows. We begin with an introduction to the Standard Model of Cosmology in chapter2where we offer a self contained description of the topic, from the cosmological principle to theories beyond the Standard Model of Cosmology. Chapter 3 is devoted to the EFT of dark energy, its language and the theory behind it. In chapter ?? we describe how the Einstein-Boltzmann (EB) equations are implemented into cosmological codes that can compute observables. In particular, we will focus on eftcamb, an extension of the EB solver camb that implements the EFT framework. Once all the machinery is ready and understood, we build the MG model we will study, WBG, that will be discussed in chapter 5. The theoretical results are worked out in chapter6, where we study the background
7
dynamics. The results coming from the implementation of the model into eftcamb are discussed in chapter7. We end up with the conclusions of the project and future prospect in chapter 8.
2
Introduction
In this chapter, we present a self contained introduction to Cosmology. We will start by the foundation of the standard model of cosmology (SMC), the cosmological principle, in section 2.1. That will takes us to the mathematical description of a universe that satisfies this principle and the study of its dynamics, given by the Friedmann equations. The expansion of the universe is explained by the inclusion of the cosmological constant as dark energy. We will devote section2.2 to the study of the cosmological constant and its problems. The SMC is properly described in section 2.3. We finish this chapter with section 2.4, where we review the main alternatives beyond the SMC.
2.1
A homogeneous and isotropic universe
The cosmological principle is at the heart of the standard model of cosmology and states that the Universe is homogeneous and isotropic at large enough scales. Ho-mogeneity refers to the fact that the Universe looks the same from any point in space, while isotropy assumes that the universe is the same in every direction.
In terms of the language we use to describe space-time, differential geometry, the cosmological principle translates into the so-called Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, that describes a homogeneous, isotropic and expanding universe:
ds2 = gµνdxµdxν = −dt2+ a(t)2γijdxidxj, (2.1)
where ds2 is the line-element, gµν is the metric, the indices denotes with Greek
characters are space-time indices, µ, ν = {0, 1, 2, 3}, while the latin ones are spatial only indices, i, j = {1, 2, 3}. The spatial part of the metric, γij, corresponds to a
three dimensional space with a constant curvature, K. In spherical coordinates:
γijdxidxj =
dr2
1 − Kr2 + r
2 dθ2+ sin2θφ2 , (2.2)
where a(t) is a time dependent function called scale factor and r is called the co-moving coordinate. The curvature of the Universe is given by K so that K = 0 corresponds to a flat universe; K = +1, a closed universe and K = −1 is an open
2.1. A HOMOGENEOUS AND ISOTROPIC UNIVERSE 9
Figure 2.1: Distribution of galaxies as measured by the 2dF Galaxy Redshift Survey. In this image, taken from (1), it is manifest that the universe is isotropic and homogeneous at large scales.
universe. The FLRW metric defines a maximally symmetric universe consequence of assuming homogeneity and isotropy. We say a spacetime is maximally symmetric when it has the maximum number of killing vectors: in this case homogeneity can be thought as invariance under space time translation while the latter invariance under rotations.
It is usually useful to do the following conformal parametrization of these coor-dinates
ds2 = a(t)2 −dτ2+ γ
ijdxidxj , (2.3)
where τ is the conformal time and it is given by:
dτ = dt
a (2.4)
2.1.1
Dynamics of the Universe
In order to study the evolution of the universe we need to look at gravity, the force governing the cosmological dynamics. Even though is is the weakest of the four fundamental forces, gravity is the one with largest range. Einstein’s theory of general relativity (GR) describes the gravitational interactions as the consequence of the curvature of spacetime. The acceleration is thus caused by the geodesic motion in a curved spacetime. GR has been successfully tested in solar system scales as well as in weak gravity regimes. The Einstein equations can be written as follows,
Gµν = 8πGTµν. (2.5)
where G is the Newton gravitational constant, Tµν is the energy-stress tensor and
Gµν is the Einstein tensor given by
Gµν = Rµν−
1
10 2. INTRODUCTION
where Rµν is the Ricci tensor and R = Rµµ is the Ricci scalar. From this equation it
follows that the energy momentum tensor is conserved,
∇µTνµ= 0 , (2.7)
since ∇µ(Rµν− 12Rgµν) = 0 due to the Bianchi identity.
The right hand side of Einstein equations describes the energy content of the universe while the left hand side comprises the information about the geometry of spacetime. Therefore, given the FLRW metric and the energy content of the universe, the Einstein equations yield the dynamics of our universe. Let us look at each side individually.
Geometry
It is given by the left hand side of equation (2.5). We can calculate the Einstein tensor from the definition of the Ricci tensor in terms of the Christoffel symbols that are computed from the metric. The definitions of these geometrical quantities can be found in the literature, see reference (2). The components of Gµν are:
G00 = −3 H2+ K/a(t)2 , Gi0 = G0i = 0 , (2.8) Gji = −3 3H2+ 2 ˙H + K/a(t)2δij .
where H = ˙a/a is called the Hubble parameter and describes the expansion rate of the universe. A dot represents the derivative with respect to cosmic time.
Energy content
It is included in the right hand side of the Einstein equation. In cosmology, we assume that all the energy content of the universe behaves as a perfect fluid described by the energy-momentum tensor,
Tνµ= (ρ + P )uµuν + P δνµ, (2.9)
where uµ = (−1, 0, 0, 0) is the 4-velocity of the fluid in comoving coordinates, ρ is the desnsity and P the pressure. For the way this tensor is constructed we can see that the (00) component corresponds to the density,
T00 = −ρ , (2.10)
while the spatial part (ij) corresponds to the pressure of the fluid,
2.1. A HOMOGENEOUS AND ISOTROPIC UNIVERSE 11
Friedmann equations
Putting this information all together on the Einstein equation yields:
(00) → H2 = 8πG 3 ρ − K a2 , (2.12) (ii) → 3H2+ 2 ˙H = −8πGP − K a2 . (2.13)
The equation obtained from the (00) component is called the first Friedmann equa-tion. Regarding the (ii) component, it can be reshaped using the first Friedmann equation to give: ¨ a a = − 4πG 3 (ρ + 3P ) , (2.14) known as the second Friedmann equation or acceleration equation. Differentiating equation (2.25) and employing (2.14) yield to the continuity equation,
˙
ρ + 3H(P + ρ) = 0 . (2.15) It is useful to define the dimensionless energy density parameter,
Ωi =
ρi
ρcr
, (2.16)
where ρcr is the critical density of the universe and it is given by
ρcr=
3H02
8πG, (2.17)
where H0 is the value of the Hubble parameter at present. With this notation, the
first Friedmann equation can be written as
Ωm+ Ωr+ Ωk = 1 . (2.18)
Interpreting curvature as other energy fluid with Ωk = HK2a2.
The first Friedmann equation (2.25) and the continuity equation (2.15), describe the dynamics of the universe and lie at the base Cosmology is built on. In particular, they tell us the rate at which the Universe is expanding and the evolution of its components. Regarding the latter, it is usual to consider the equation of state of a perfect fluid, given by the dimensionless ratio of pressure and density:
w = P
ρ . (2.19)
With this definition, The Friedmann equations allow us to find the evolution of the scale factor,
a ∼ (t − ti)2/(3(1+w)), (2.20)
and the evolution the energy density of one particle,
ρ ∼ a(t)−3(1+w). (2.21) Let us consider how this affects each of the components we observe in the uni-verse:
12 2. INTRODUCTION
• Non-relativistic matter: such as baryonic or cold dark matter. This kind of matter is pressureless and thus w = 0. Therefore,
ρm ∼ a(t)−3. (2.22)
This result is very intuitive since matter will dilute because of the volume of the universe increases with the expansion.
• Relativistic matter: radiation as photons and neutrinos. Radiation is charac-terized by w = 1/3, therefore
ρr ∼ a(t)−4. (2.23)
This indicates that radiation dilutes faster than non relativistic matter. This is due to the fact that not only is radiation diluted as the volume of the universe expands, but also it gets redshifted. This effect accounts for the decrease of the frequency of the radiation when space expands.
As we can see form the previous equations, the expansion of the universe quickly dilutes all energy constituents. Non relativistic matter will be the first species to become subdominant followed by matter. As we will see below, the late time universe is dominated by dark energy driving the cosmic acceleration.
2.2
The cosmological constant
As we saw in the previous section, the energy-momentum tensor is a conserved quantity. Since the metric satisfies ∇µgµν = 0, Einstein equations admit the addition
of the term Λgµν sice ∇µgνµ= 0, where Λ is the so called cosmological constant. The
Einstein equations with this new term read
Rµν −
1
2Rgµν+ Λgµν = 8πGTµν. (2.24) The cosmological constant was added by A. Einstein to his equation based on the idea that the Universe should be static. As we saw in expression (2.20), the Friedmann equations predict an expanding, dynamical universe. To fix this problem Einstein added the cosmological constant to the equations that result in the following Friedmann equations: H2 = 8πG 3 ρ − K a2 + Λ 3 , (2.25) 3H2+ 2 ˙H = −8πGP − K a2 + Λ 3 . (2.26) These equations provide a static universe ( ˙a = ¨a = 0) given the domination of a pressureless fluid of constant density. As shown by Lemaitre, such a solution is unstable and thus unfeasible. It was some years after that the expansion of the uni-verse was evidenced by the observation of the redshift of neighbour galaxies when Einstein named the cosmological constant his ”biggest blunder”. However, the ad-dition of the cosmological constant was mathematically permitted and proved to be
2.2. THE COSMOLOGICAL CONSTANT 13
very useful to describe the late time universe.
The cosmological constant can be interpreted as a negative pressure fluid with equation of state w = −1. Upon its inclusion, and considering equations (2.22) and (2.23), the first Friedmann equation for a universe with matter and radiation can be written as H2 = H02 Ωm,0a(t)−3+ Ωr,0a(t)−4− K H2 0a2 + Λ 3H2 0 . (2.27) De Sitter universe
W. De Sitter found in 1917 a solution to the Einstein equations that represented the first model of an accelerated Universe. The De sitter universe is given by the following metric,
ds2 = dt2+ eHtdxidxi, (2.28)
for a empty universe. This metric gives a universe that grows exponentially with a(t) = eHt yielding a expansion rate:
H(t) = ˙a a =
r Λ
3 . (2.29)
The De Sitter universe describes very well the behaviour of the late time universe, when the cosmological constant is the dominant energy density and matter and radiation have been diluted by the expansion of the Universe.
2.2.1
The problem of the cosmological constant
Despite the simplicity of the cosmological constant, it comes with several, difficult problems that make cosmologists discontent. We interpret Λ as the energy density of vacuum and its value can be predicted from Quantum Field Theory. An estima-tion can be made by considering the contribuestima-tion of all the known particles in the Standard Model to the cosmological constant. To do so, we consider these particles as fields that can be describe as a series of harmonic oscillator. The vacuum energy would be given by hρi ∼ Z ΛU V O d3k (2π)3 1 2~Ek ∼ Z ΛU V 0 dk k2√k2+ m2 ∼ Λ4 U V , (2.30)
where ΛU V is the cutoff of the theory, namely the highest energy at which we can
trust predictions. This is given by the weak interaction having ΛU V ∼ T eV .
There-fore,
Λtheory∼ 10−60MPl. (2.31)
On the other hande the observed value of the cosmological constant is
ΛObservations ∼ 10−120MPl. (2.32)
This 60 orders of magnitude of difference between theory and observations is a worrisome problem since the value of λ seems to be fine tuned.
14 2. INTRODUCTION
2.3
The Standard Model of Cosmology
The Standard Model of Cosmology (SMC) is the current and simplest model that describes the cosmology of our universe tested against the following cosmological observations.
Cosmic Microwave Background
The Cosmic Microwave Background (CMB) is a black body radiation with temper-ature 2.7K that is observed from every direction in the sky. This almost perfectly isotropic radiation comes from the time when the universe became transparent. Af-ter the Big Bang, photons and matAf-ter particles were coupled, evolving together and forming what we know as the primordial plasma. When the Universe was 380 000 years old it was cool enough for photon to decouple from the primordial plasma and travel freely throughout space. Today, we receive this redshifted radiation as the CMB.
Figure 2.2: CMB map reconstructed from WMAP observations. Image credit: (3)
.
The study of the angular size of the temperature anisotropies in the CMB has determined that the universe is flat, namely K = 0.
Abundances of light elements
The Big Bang Nucleosynthesis (BBN) is a process that formed the light elements that we observe today, such as deuterium and helium, when the primordial was cool enough. Not only do the predictions of BBN on light element abundances agree with observations, but also place strong constraints on the energy density parameter of dark energy. An upper bound can be found form the fact that the expansion rate of the universe modify the proton to neutron ratio at freeze out inevitably changing the primordial abundances of light elements.
2.3. THE STANDARD MODEL OF COSMOLOGY 15
Large Scale Structure
Large scale structure (LSS) is found in galaxy cluster observations and it is a strong evidence in favour of dark energy and the cosmological constant. As shown in (4), the measured power spectrum of galaxy distribution depends on the matter density parameter which radically changes with the presence of dark energy.
Figure 2.3: Linear matter power spectrum measured by several astronomical surveys. Im-age credit: (5)
.
Supernova observations
The first strong piece of evidence of the expansion of the universe was given by the observation of supernovae of supernovae by the High Redshift Supernovae Search Team (HSST) (6). The survey determined that the stellar objects were receding from us and placed constraints on the value of H0, and the energy density parameter of
matter and dark energy.
In Figure (2.4) we can see the Hubble diagram created form the SN observations at different redshifts. The quantity m − M is directly related to the luminosity distance.
These observations support the SMC that describes a flat universe with the following energy content distribution: 4% baryonic matter, 23% cold dark matter and around 73% of dark energy understood as the cosmological constant. Precisely due to the two main energy components, the SMC is usually called LCDM standing up for the cosmological constant (L) and cold dark matter (CDM). The model is fully specified by six parameters that are shown in the table below:
16 2. INTRODUCTION
Figure 2.4: Hubble diagram for the low-redshift and high-redshift SNe Ia from HSST col-laboration. Image credit: (6)
.
Parameter Value Description Ωb 0.02233 ± 0.00015 Baryonic energy density
Ωc 0.1198 ± 0.0012 CDM energy density H0 67.37± 0.54 Hubble parameter τ 0.0540 ± 0.0074 Optical depth ln(1010A s) 3.043 ± 0.014 Scalar amplitude ns 0.9652 ± 0.0042 Spectral index
Table 2.1: Parameters of the SMC, the values correspond to the Planck TT,TE,EE+lowE+lensing 2018 results (7).
2.4
Beyond the Standard Model of Cosmology
The lack of theoretical understanding behind the cosmological constant is a wor-risome problem. Furthermore, there are some tensions in the LCDM model. In particular, the value of the Hubble parameter evaluated today observed by Planck is in tension with its local measurements (8). The amplitude of the linear power spectrum on the scales of 8h−1Mpc is also in disagreement with weak lensing obser-vations such us KiDS mission (9). As an alternative to LCDM many other cosmo-logical models have arisen to explain the cosmic acceleration without relying on the
2.4. BEYOND THE STANDARD MODEL OF COSMOLOGY 17
cosmological constant.
There are two main approaches to go beyond LCDM. The first one is considering dark energy as a modified form of matter. This approach is based on the idea of introducing dark energy as a dynamical energy fluid in the energy-momentum tensor, Tµν, in the Einstein equations. The second approach consists of modifying general
relativity at large scales to account for the cosmic acceleration, what we call modified gravity. At the end of the day this is just a convenient way of categorizing the theories: the two strategies are completely equivalent and cannot be distinguished. Any change of the energy-momentum tensor can be absorbed by the Einstein tensor and vice versa. In this section we briefly discussed the most important models within each of the approaches.
2.4.1
Dark energy as a modified form of matter
Models of dark energy within this category are characterized for having dynamical equations of motion that changes with time. The main representatives of dynamical dark energy models are quintessence (10) and K-essence (11). These models intro-duce a scalar field with dynamical equation of state that mediates the expansion of the universe. Their solutions have trackers, attractors in which the field falls, that mitigates the coincidence problem. Other models that introduce modified matter are phantom and coupled dark energy. We refer to (4) for a detail description of the aforementioned models.
Quintessence
The action of this model is given the the usual one form general relativity with a canonical scalar field, as below,
S = Z d4x√−g M 2 Pl 2 R + Lφ + SSM, (2.33) where M2 Pl = (8πG) −1 and Lφ= − 1 2g µν∂ µΦ∂νΦ − V (φ) . (2.34)
Since the field is not coupled to the metric this model will not introduce mod-ification to the Friedmann equation. However, it does contribute to the energy-momentum of the matter components. In particular, the density and pressure are given by: ρφ= 1 2 ˙ φ2 + V (φ) , (2.35) Pφ= 1 2 ˙ φ2− V (φ) , (2.36) Therefore the equation of state, defined as the ratio between pressure and density, yields to: wφ= Pφ ρφ = ˙ φ2− 2V (φ) ˙ φ2+ 2V (φ). (2.37)
18 2. INTRODUCTION
Finally, the equation of motion of the field is given by the variation of the action with respect to the field,
¨
φ + 3H ˙φ +∂V (φ)
∂φ = 0. (2.38)
The form of the potential, V (φ), must be fixed so that the energy density of the field becomes dominant at late times. Namely, ρφ ρm at early times. It is in this
sense that we say the field track ρm. Also, in order to have acceleration the equation
of state must satisfy wφ< −1/3, which equivalent to imposing ˙φ2 < V (φ).
K-essence
These models are very similar to the ones above. However, unlike Quintessence, K-essence models include non linear kinetic terms in their lagrangian, as follows,
S = Z d4x√−g M 2 Pl 2 R + P (φ, X) + SSM, (2.39) where X = −1 2g µν∂ µΦ∂νΦ . (2.40)
The tracker behaviour of these models is such that can account for the expansion as the cosmological constant without introducing it. During radiation dominated universe the k-essence will track radiation equation of state. The latter will drop at the start of matter dominated era to track a De Sitter universe. This tracker will remain subdominant until the matter component is diluted and the scalar field can catch up.
2.4.2
Dark energy as a modification of gravity
Models within this category introduce modifications to the geometrical sector of the Einstein equations. The main representatives of modified gravity theories are f(R) (12) and scalar-tensor theories and braneworld scenarios.
f(R) gravity
This modified theory of gravity simply changes the Einstein action of general rela-tivity by a function of the Ricci scalar, R.
S = Z d4x√−gM 2 Pl 2 (R + f (R)) + SSM. (2.41) The modified Einstein equation for f(R) gravities is given by
(1 + fR)Rµν − 1 2gµν(R + f ) + (gµν − ∇µ∇ν)fR= Tµν MPl , (2.42)
where fr = ∂R∂f. Note that these equations are fourth order. The trace equation
yields
(1 − fR)R + 2f − 3fR=
T MPl
2.4. BEYOND THE STANDARD MODEL OF COSMOLOGY 19
From these equations we learn that there is an additional degree of freedom, dubbed the scalaron, given by fR. For each expansion history there will be a family
of f (R) models labelled by a parameter that is related to the length scale of the scalaron. The construction of f(R) must account for the following conditions of viability (13):
• fRR > 0,
• 1 + fR> 0,
• fR< 0.
Scalar-tensor theories
This class of theories rely on the addition of a scalar field coupled to the metric tensor to mediate the acceleration at late times. Within this category we can find the so called Brans-Dicke theory (14) that connects the gravitational constant to the cosmic field. The general action for a scalar tensor can be written as
S = Z d4x√−g MPl 2 f (φ, R) − 1 2ξ(φ)(∇φ) 2 + Sm, (2.44)
where f and ξ are general functions. The model includes a huge variety of theories. For instance, we can recover f (R) by setting f (φ, R) = f (R) and ξ = 0 or Brans-Dicke by setting f (φ, R) = φR and ξ = wBD/φ. Other interesting scalr-tensore
thoery ara galileons, that we will extensively study in chapter 5.
Braneworlds
In braneworld scenarios our Universe is understood as a three+one dimensional brane, embedded in a higher dimensional bulk space. This model of universe was proposed by Arkani-Hamed, Dimopulous and Dvali (15;16) and it has been named Brane World (or ADD scenario). In this model, general relativity is recovered at short distances while at larger scales modifications to gravity appear.
Other approach is the one proposed by Randall and Sundrum (17) in which the three-dimensional brane is embedded into a 5-dimensional Anti De Sitter space bulk. The Anti De Sitter radius defines the scales above which gravity is modified.
3
The effective field theory of dark
energy
The discovery of cosmic acceleration has motivated the proposal of a tremendous number of theories that could explain this phenomenon. In the light of this over-whelming landscape of dark energy models, the EFT approach has proved to be the best framework for two main reasons. Firstly, it provides with an unifying language to analyze all DE models. That way we can have a global view and work in a model independent framework that can implement general theories and particular cases (18). On the other hand, this approach easily connects the theory with observa-tions. Since this treatment is model independent, there is no need to confront every model to data. Data can constrain the general formalism from which we can get the implications for particular models. This approach was firstly proposed in cosmology in the context of inflationary models where they proved to be valuable (19;20; 21). Another prevalent feature of modified gravity theories is the inclusion of scalar fields as new degrees of freedom. In inflation, this field is responsible for the accelerated expansion and mediates its termination. Addressing cosmic acceleration, we can also find a great number of models that invoke a scalar field as the mediator of the acceleration. It is essential then, not to rely on the particularities of the added degrees of freedom, to keep the framework as general as possible. As we will see, in the EFT framework the scalar field arises naturally as the Goldstone boson of time diffeomorphism symmetry breaking.
In this section we introduce the EFT framework for dark energy and set up all the technical machinery we will need in following sections of this thesis. We will start by considering cosmological perturbations since they lie at the heart of this formalism. Taking the perturbations around the background as the relevant degree of freedom will allow us to write the unifying action for dark energy in the EFT framework in section (3.2).
3.1. COSMOLOGICAL PERTURBATIONS 21
3.1
Cosmological perturbations
As for the anisotropies of the Cosmic Microwave Background (CMB) and the large scale structures (LSS), cosmological perturbations play an important role in the dynamics of the universe. It is then natural to identify the perturbations about the homogeneous Friedmann-Robertson-Walker (FRW) as the relevant degree of freedom when building the action for the EFT of dark energy (22). However, as we have mentioned above, we do not want to rely on specific details of the scalar field nor fix a background cosmology. Writing an action in terms of the perturbations requires to explicitly write the background evolution, φ0, and the perturbation of
the field, since
φ → φ0+ δφ. (3.1)
In principle, we cannot write the perturbed action without specifying φ0, which
would require solving its background evolution. However, we can deal with this issue by considering the unitary gauge, equivalent to choosing the field as the time coordinate. In the unitary gauge, the dynamics of the field are eaten by the metric, equivalent to setting the scalar field to zero. Thus, we can construct the perturbed action easily without fixing the evolution of the scalar field. As a consequence of fixing the gauge, time diffeomorphism invariance is broken. The strategy to build the lagrangian would be to write the most general action invariant under the group of residual symmetries −in this case, three-dimensional diffeomorphisms− for the metric perturbations, and all other geometrical quantities. We shall go into detail about this concept.
The unitary gauge
We will now argue that the scalar field can arise as a Goldstone boson due to the symmetry breaking under time diffeomorphism. Let us consider a field, φ, that is invariant under a transformation γ as follows,
φi(x) → γijφj(x). (3.2)
The symmetry group, G, spontaneously breaks when we choose a minimum −vacuum expectation value, < φi >0, different than zero−. A transformation along
the vacuum that respects the symmetry will take us to an equivalent vacuum. These massless excitations of the field are called Goldstone bosons. However, a transfor-mation along the vacuum that breaks the symmetry will produce massive fields. At a low energy regime, as the one we are interested in for our EFT the massive excitations will decouple and we are left with the Goldstone bosons. As it is very well known, making a local gauge theory introducing covariant derivatives can add new terms to the lagrangian that provide the field with mass.
The unitary gauge is equivalent to setting the Goldstone bosons to zero. In this sense, it selects, among any field configuration, the one without fluctuations around the direction of the symmetry. Fixing this gauge for the EFT function will be tremendously helpful since we do not have to explicitly write the field in
22 3. THE EFFECTIVE FIELD THEORY OF DARK ENERGY
the lagrangian. This idea is extremely powerful since it naturally introduces the scalar field is a consequence of the time diffeomorphism symmetry breaking rather than a postulated field (22). The resulting theory is only gauge invariant under the unbroken three-dimensional diffeomorphism. Therefore it must contain: (1) three dimensional geometrical objects defined for t = constant, (2) four-dimensional geo-metrical quantities with free time indices and (3) scalar quantities such as the Ricci scalar.
In order to restore the invariance under full diffeomorphism we can apply the Stuckelberg trick, that consists on reintroducing the Goldstone fields by forcing the time diffeomorphism invariance,
t → t + π, (3.3)
we will expand on the Stuckelberg mechanism in subsection (3.4).
3.2
A unifying language
Once we have identified cosmological perturbations as the relevant degree of free-dom, we can write the action as an expansion in the perturbations of the geometrical quantities. This treatment is especially interesting since perturbations will be small at cosmological scales, assuring the validity of the EFT. To do so, we must write the most general action that is invariant under the unbroken symmetry group con-taining all possible contractions of the perturbations of the geometrical quantities aforementioned. Therefore, the action must contain the perturbations of the metric, the Ricci scalar, the Ricci tensor and the extrinsic curvature. The action written in the EFT framework for dark energy was proposed in (23; 24) assuming the weak equivalence principle (WEP) −all matter fields will be universally coupled to the metric− as follows, S = Z d4x√−g MPl 2 (1 + Ω(t))R + Λ(t) − c(t)δg 00 +M 4 2(t) 2 (δg 00)2+M 4 3(t) 3! (δg 00)3+ . . . − M¯ 3 1(t) 2 δg 00δKµ µ− ¯ M22(t) 2 (δK µ µ)2− ¯ M32(t) 2 δK µ νδKνµ+ . . . + λ1(t)(δR)2+ λ2(t)δRµνδRµν+ α1(t)CµνσλCµνσλ+ . . . + α(t)µνσλCρθµνCσλρθ+ m21(t)n µnν∂ µg00∂νg00+ . . . + m22(t)(gµν+ nµnν)∂µg00∂νg00+ . . . + Sm[gµν], (3.4)
where Cµνρσ is the Weyl tensor, δg00, δR, δR
µν, and δKµν are the perturbations
of the above mentioned geometrical quantities and nµ is the normal vector to the
3.2. A UNIFYING LANGUAGE 23
the function Ω dimensionless. This action has been constructed by considering all possible contractions of the geometrical quantities up to second order. Note that, as the action must enjoy spatial diffeomorphism invariance, all spatial indices are contracted, unlike the temporal ones that are written explicitly. For the same rea-son, every term has a time dependent coefficient, namely, Ω, Λ, c Mi; that are called
EFT functions.
Regarding the very first line on (3.4), it contains the functions that are relevant to the background evolution,
Ω(t), Λ(t), c(t) (3.5) In the case of a minimal coupling to the metric Ω = 0. The remaining operators are relevant for the evolution of the perturbations and can be rewritten as dimensionless operators as follows, γ1 = M4 2 m20H02, γ2 = ¯ M3 1 m20H0 , γ3 = ¯ M2 2 m20, γ4 = ¯ M2 3 m2 0 , γ5 = ˆ M2 m2 0 , γ6 = m2 2 m2 0 . (3.6)
3.2.1
Mapping to EFT: two examples
As we argue in the first part of this section, the main advantage of the EFT frame-work is that is model independent. We can constrain the EFT functions Ω, Λ, c, Mi, ¯Mi, m1 and λi and infer conclusion to other models by mapping them into the
EFT language. The most simple example is LCDM, whose action is given by
S = Z d4x√−g M 2 Pl 2 R − M 2 PlΛ , (3.7)
can be mapped by a simply comparison with (3.4), yielding:
Ω(t)LCDM = 0, Λ(t)LCDM = MPl2Λ. (3.8) In the case of Quintessence (10), the action is given by
S = Z d4x√−g M 2 Pl 2 R − (∇φ)2 2 − U (φ) . (3.9)
The mapping to the EFT language in not as direct as in the previous example. In order to find the mapping we have to expand action (3.9) in perturbations. Re-garding the second one, the kinetic term, we should recall: g00 → −1 + δg00 and
φ(t) → φ0(t) + δφ(t). Since field is only dependent on time −we are working at the
background level− we have that (∇φ)2 = g00φ˙2. Putting everything together, and
fixing the unitary gauge, yields:
S = Z d4x√−g M 2 Pl 2 R − 1 2 ˙ φ2δg00+ 1 2 ˙ φ2− U (φ0) . (3.10)
Comparing (3.10) and (3.4) we find the mapping
ΩQ(t) = 0, cQ(t) = φ˙ 2 0 2 , Λ Q(t) = φ˙20 2 − U (φ0). (3.11)
24 3. THE EFFECTIVE FIELD THEORY OF DARK ENERGY
3.3
EFT mapping
Finding the mapping from a specific model to the EFT work can be easily done by employing the Arnowitt Deser Misner formalism (ADM). The ADM decomposition is useful because it describes spacetime as if it were foliated into a family of space-like hypersurfaces labeled by the time coordinate. The metric in ADM formalism reads
ds2 = −N dt2+ hij(dxi+ Nidt)(dxj+ Njdt), (3.12)
where hij is the induced metric on the 3-dimensional hypersurfaces, N the lapse
function and Ni the shift vector.
Following (25; 26; 24), we write a general lagrangian in ADM formalism and compare to the most general action for the EFT of dark energy written in ADM formalism as well. Said comparison will be the mapping from any arbitrary theory in ADM to the EFT language. This is done in reference (25) where the authors consider up to sixth order derivatives to be able to map models with HD operators, such as generalized galileons (27) or GLPV (28) theories.
General lagrangian in ADM formalism
We must write all geometrical quantities up to six spatial derivatives. The action can be espressed as,
S = Z d4x√−gL(N, K, S, R, Z, U , Z1, Z2, α1, α2, α3, α4, α5, ; t), (3.13) where, K = Kµµ, R =(3) Rµ µ, S = KµνKµν, Z =(3) RµνRµν. (3.14) U = RµνKµν, Z1 = ∇iR∇iR , Z2 = ∇iRjk∇iRjk, α1 = aiai, α2 = ai∆a1, α3 = R∇iai, α4 = ai∆2ai, α5 = ∆R∇iai. (3.15)
The normal vector to the spacial hypersurfaces, nµ, and the extrinsic curvature can
be defined as
nµ= N δµ0, Kµν = hλµ∇λnν, (3.16)
∆ = ∇a∇aand ai is the acceleration vector. These operators are the ones needed to
write the action up to sixth order in spatial derivatives. The perturbed lagrangian can be written as δL = LNδN + LKδK + LSδS + LRδR + LZδZLUδU +1 2 δN ∂ ∂N + δK ∂ ∂K + δS ∂ ∂S + δR ∂ ∂R + δZ ∂ ∂Z + δU ∂ ∂U 2 L +O(3). (3.17)
In reference (25) the authors write the perturbations of all geometrical quantities, resultng in:
3.3. EFT MAPPING 25 SADM = Z d4x√−g¯ L + ˙F + 3HF + (LN − ˙F )δN + F +˙ 1 2LN N(δN ) 2+ L SδKµνδKνµ +1 2A(δK) 2+ BδN δK + CδKδR + DδN δR + E δR + 1 2G(δR) 2+ L ZδRµνRνµ +Lα1∂iδN ∂ iδN + L α2∂iδN ∇k∇ k∂iδN + L α3R∇i∂ iδN + L α4∂iδN ∆ 2∂iδN +Lα5∆R∇i∂ i δN + LZ1∇iδR∇ i δR + LZ2∇iδRjk∇ i δRjk (3.18) where A = LKK + 4H2LSS − 4HLSK, B = LKN − 2HLSN, C = LKR− 2HLSR+ 1 2LU − HLKU + 2H 2L SU, D = LN R+ 1 2 ˙ LU − HLN U, (3.19) E = LR− 3 2HLU − 1 2 ˙ LU, F = LK− 2HLS, G = LRR+ H2LU U − 2HLRU.
EFT action in ADM formalism
Writing the EFT action in equation (3.4) simply consists of perturbing all the geo-metrical quantities about the FLRW metric.
g00= −1 + δg00. (3.20)
In reference (25) they compute aforementioned perturbed quantities and compare the resulting EFT action in ADM language with equation 3.18. The outcome is a mapping form the general action to the EFT framework. A direct cross check in number of perturbations yields the following:
Ω(t) = 2 m2 0 E, c(t) = 1 2( ˙F − LN) + (H ˙E − ¨E − 2E ˙H), Λ(t) = ¯L + ˙F + 3HF − (6H2E + 2 ¨E + 4H ˙E + 4 ˙HE ) , M¯22(t) = −A − 2E , M24(t) = 1 2l(LN + LN N 2 ˚) − c 2, ¯ M13(t) = −B − 2 ˙E, M¯32(t) = −2LS+ 2E , m22(t) = Lα1 4 , m¯5(t) = 2C, ˆ M2(t) = D, λ1(t) = G 2, λ2(t) = LZ, λ3(t) = Lα3 2 , λ4(t) = Lα2 4 , λ5(t) = LZ1, λ6(t) = LZ2, λ7(t) = Lα4 4 , λ8(t) = Lα5 2 . (3.21)
26 3. THE EFFECTIVE FIELD THEORY OF DARK ENERGY
We refer the interested reader to (25) for a complete derivation of these expres-sions. The mapping to any theory can be obtained by writing the action of certain model in ADM notation and then compute the EFT functions from above using definitions in (3.19).
3.3.1
A simple example
The process of finding a mapping might be clearer if applied to an specific example. In this section, I would like to derive the mapping of the quadratic covariant galileon (CG) whose lagrangian is given by
LCG2 = G2(X), (3.22)
where X = −(∇φ)2. The first step consists on expanding the lagrangian in
pertur-bations up to quadratic order and expressing it in ADM notation.
G2(X) = G2(X0) + G2X(X0) + 1 2G2X X(X − X0) 2 . (3.23) In ADM language, X = X0 N2, therefore G2(X) = G2(X0) + G2X(X0)X0( 1 N2 − 1) + 1 2G2X XX 2 0( 1 N2 − 1) 2 . (3.24)
By using the map (3.21) with definitions (3.19) we can get the mapping. The EFT function c(t) would be:
c(t) = −1
2LN = G2X(X0)X0, (3.25) because F , E = 0 and LN =
−2X0G2X(X0)
N3 and N = 1 for the background.
Regardgin Λ,
Λ = ¯L = G2(X0) (3.26)
again, because F , E = 0. Finally, the only other non trivial EFT function is
M24 = G2X X(X0)X02, (3.27)
since LN N = 4X0G2X(X0).
3.4
Stuckelberg mechanism
As we previously mentioned, we can restore the full-diffeomorphism by introducing time perturbations, namely by the transformation (3.3). This way, functions of time would transform as
f (t)(t + π(x))(t) + ˙f (t)π(x) + ... (3.28) As by definition, scalar will not get affected by this time translation, R(x) = R(¯x). For perturbations it works differently since
3.5. VIABILITY CONDITIONS 27
Applying this transformations to the action (3.4) we can find (23):
S = Z d4x√−g MPl 2 Ω(t + π)R + Λ(t + π) − c(t + π) δg00− 2 ˙π + 2 ˙πδg00+ 2 ˜∇ iπg0i− ˙π2+ ˜ gij a2∇˜iπ ˜∇jπ + . . . +M 4 2(t + π) 2 δg 00− 2 ˙π + . . .2 −M¯ 3 1(t + π) 2 δg 00− 2 ˙π + . . . δKµµ+ 3 ˙Hπ + ˜ ∇2π a2 + . . . ! −M¯ 2 2(t + π) 2 δK µ µ+ 3 ˙Hπ + ˜ ∇2π a2 + . . . !2 −M¯ 2 3(t + π) 2 δKij + ˙Hπδij+ ˜g ik a2∇˜k∇˜jπ + . . . δKji + ˙Hπδij+ ˜g jl a2∇˜l∇˜iπ + . . . + ˆ M2(t + π) 2 δg 00− 2 ˙π + . . . δR(3)+ 4H ˜ ∇2π a2 + 12H k0 a2π + . . . ! + m22(t + π)(gµν+ nµnν)∂µ(g00− 2 ˙π + . . .)∂ν(g00− 2 ˙π + . . .) + . . . + Sm[gµν] (3.30)
3.5
Viability conditions
High derivative terms are usually avoided in physics since they are prompted to introduce instabilities in the model. Therefore, working in the EFT framework of dark energy, where HD operators have relevant contributions, requires studying stability conditions that ensures the robustness of the theory.
Ghost instability
The presence of HD operators can result in equations of motion with more than second time derivatives acting on the scalar field, which translates into extra modes of propagation. If the mass of the ghost fields lies above the energy cutoff of the EFT, we can interpret them as nonphysical modes produced by the truncation of an infinite series that are fatal for our EFT.
Ghost fields are characterized for having negative kinetic quanta given by a a wrong sign in the kinetic term. Of course, this sign is simply conventional. However, if the ghost field is coupled to a healthy field whose kinetic term has opposite sign then infinite number of pair ghost field - healthy field could be created with no energy cost.
28 3. THE EFFECTIVE FIELD THEORY OF DARK ENERGY
Gradient instability
Equivalently to ghosts being created due to a wrong sign in the time-time compo-nent of the kinetic term, spatial gradient terms with opposite sign can also cause instabilities. See, for instance, the following lagrangian
L = 1 2 ˙ φ2+1 2∇φ 2. (3.31)
The equation of motion of this filed in Fourier space grows exponentially as φ ∼ exp ±kt, where k−1 indicates the instability timescale. Therefore, the modes that contribute more to the instability are those with high energy (29) which makes no sense for an EFT. In general, an EFT with gradient instability will not be predictive.
Tachyonic instability.
Tachyons are fields whose mass squared is negative. Consider the action
L = 1 2(∂φ) 2+m 2 2 φ 2. (3.32)
In this case the solution of the equation of motion for the field is given by φ±mt. This growing mode solution indicates the presence of the instability in the theory characterized by the timescale m−1 and indicates that the theory has not been perturbed around the true vacuum.
In the context of EFTs, the theory is predictive at shorter scales than m−1. In cosmology we can think of it as an EFT that on small scales can be described as a Minkowski space, ignoring the cosmic expansion.
4
Numerical tools for the study of
DE
The vast amount of data currently available has brought us in the era of the pre-cision cosmology. In particular, cosmological surveys such as Planck (30), SDSS (31), DES (32), LSST (33) or Euclid (34) provide with high precision data. In order to get information out of the data and constrain the theoretical models we need numerical tools that can compute cosmological observables. This task is done by Einstein-Boltzmann (EB) solvers, codes that solve the full set of Einstein-Boltzmann equations. The original work done by (35; 36; 37;38) for the study of cosmological linear perturbations required solving sets of differential equations. These tedious calculations were facilitated by the line of sight method (39) that considerably re-duced the computational time. The method was implemented into the cmbfast code. Currently, CAMB (Code for Anisotropies in the Microwave Background) (40) and CLASS ( Cosmological Linear Anisotropy Solving System) (41;42) are the most developed and spread out EB solvers commonly used in cosmology. Both of them count with further extensions of the code that allow the study of scalar-tensor the-ories, namely eftcamb (43) and hi class (44). These two EB solvers have proved to agree up to a great level of precision (45).
In this chapter we introduce the EB equations that describe the interaction between the different constituents of the Universe and with the metric. We show how they relate to cosmological observables and discuss the implementation into an EB solver.
4.1
The Einstein-Boltzmann equations
Even though homogeneity is a good approximation at cosmological scales, it is very well known that the Universe is not perfectly uniform. In fact,, inhomogeneities are the responsible of anisotropies in the CMB and the formation of LSS. The cosmolog-ical evolution of matter and radiation perturbations is complicated because of the great number of interactions between them. On one hand, every species −photons, neutrinos, dark matter and baryons− directly interact with the metric, described
4.1. THE EINSTEIN-BOLTZMANN EQUATIONS 31
by the Einstein equations. On the other hand, there are also interactions between them that are described by the Boltzmann equations: photons and electrons interact by Compton scattering and protons and electrons present Coulomb scattering. The Einstein-Boltzmann equations describe this complicated set of interactions.
4.1.1
The Boltzmann equations
The Boltzmann equation describes the rate of change of the abundance of one par-ticle species due to certain interactions. It accounts for the production and anni-hilation of the particles. The most primitive, differential version of the Boltzmann equation is
df
dt = C[f ], (4.1) where f is the distribution function of a particle and C[f ] is collision term containing the information about creation and annihilation of the particle. It is natural that the collision term depends on the distribution function, since the number of collisions −interactions− are conditioned by the abundance of the species involved in the interaction. When C[f ] = 0, meaning there are no interactions, the distribution function is constant and abundances do not change. The total derivative can be written as d dt = ∂ ∂t+ ~v ∇~x+ ~F ∇~p. (4.2) where v = ∂x∂t and ~F = ∂~∂tp.
To calculate C[f ] we have to integrate over the energies of all species the square amplitude of the corresponding Feynmann diagram taking into account the distri-bution functions of the particles (fermionic, bosonic).
Regarding non relativistic particles, such as baryons and dark matter, this will be translated into overdensities that will generate the formation of LSS. Baryonic matter will be characterized by the overdensity δb and the peculiar velocity vb.
For dark matter those will be simply noted as δ and v. It is useful to express the dynamics of these systems in fourier space. In this formalism, non-relativistic matter will depend on the wavenumber k and conformal time η.
The description of photon and neutrinos is less straightforward. The distribution of radiation, unlike non-relativistic matter, depends on the direction of propagation. Thus, in Fourier space there will be a dependence on k, η and µ = ˆpˆk. There-fore radiation anisotropies are characterized by a monopole, Θ0, and dipole, Θ1,
perturbation (analogous to δ and v) and higher order moments:
Θl= 1 (−i)l Z 1 −1 dµ 2 Pl(µ)Θ(µ), (4.3) where Pl is the Legendre polynomial of order l.
32 4. NUMERICAL TOOLS FOR THE STUDY OF DE
The Boltzmann equations for both matter and radiation are calculated in (46). In fourier space they read:
˙ Θ + ikµΘ = − ˙Φ − ikµΨ − ˙τ Θ0− Θ + µvb− 1 2P2(µ)Π Π = Θ2+ Θp2+ Θp0 ˙ ΘP + ikµΘP = − ˙τ −ΘP + 1 2(1 − P2(µ))Π ˙δ + ikv = −3 ˙Φ ˙v + ˙a av = −ikΨ (4.4) ˙δb+ ikvb = −3 ˙Φ ˙vb+ ˙a avb = −ikΨ + ˙τ R(vb+ 3iΘ1) ˙ N + ikµN = − ˙Φ − ikµΨ),
where ΘP is the strenght of polarization, R is the ratio baryon to photon
R = 3ρ
(0) b
4ρ(0)γ
, (4.5)
N denotes the (massless) neutrino distribution and Φ and Ψ are the metric pertur-bations defined as:
ds2 = a(η)[(1 + 2Ψ)dη2 − (1 − 2Φ)δijdxidxj]. (4.6)
4.1.2
The Einstein equations
As we discussed above, not only do the particles interact with each other but also with the metric. The Einstein equations account for that relation between the metric perturbation and the matter and radiation inhomogeneities. The Einstein equations read
Gµν = 8πGTµν. (4.7)
In Fourier space, for the perturbed metric in equation (4.6) the Einstein equations can be written as k2Φ + 3˙a a ˙ Φ − Ψ˙a a = 4πGa2[ρmδm+ 4ρrΘr,0], k2(Φ + Ψ) = −32πGa2ρrΘr,2, (4.8)
where the subscript m stands for non-relativistic matter, including baryons and dark matter, and r stands for radiation, including photons and neutrinos; as follows
ρmδm = ρDMδ + ρbδb,
ρmvm = ρDMv + ρbvb, (4.9)
ρrΘr,0 = ργΘ0+ ρrN0,
4.2. COSMOLOGICAL OBSERVABLES 33
These equations can be solved together with the adiabatic initial conditions (47) that single-field inflation provides:
Θ0(k, ai) = 1 2Φ(k, ai) , Θ1(k, ai) = − 1 6 k aiHi Φ(k, ai) , δ(k, ai) = 3Θ0 = 3 2Φ(k, ai) , 3Θ1 = − 1 2 k aiHi Φ(k, ai),
where ai and Hi are the scale factor and Hubble parameter at initial time.
4.2
Cosmological observables
The Einstein (4.8) and the Boltzmann (4.4) equations define a set of differential equations to be solved together with the initial conditions. The solutions can be related to cosmological observables. In particular, we can get the power spectra of the CMB anisotropies and matter overdensities.
CMB anisotropies
The power spectrum of temperature anisotropies describes the information present in the CMB temperature map. We can relate the temperature anisotropies with the function Θl describing the radiation anisotropies. The temperature field in the
universe can be wrtiten as
T (~x, ˆp, η) = T (η)[1 + Θ(~x, ˆp, η)]. (4.10) Following expression (4.10), we can state
Θ(~x, ˆp, η) = δT
T . (4.11)
The function Θ(~x, ˆp, η) can be written as a series of harmonic oscillators, Ylm as
follows Θ(~x, ˆp, η) = ∞ X l=1 l X m=−l alm(~x, η)Ylm(ˆp). (4.12)
Which can be inverted to yield the coefficient alm:
alm(~x, η) =
Z d3k
(2π)3ei~k~x
Z
dΩYl∗m(ˆp)Θ(~x, ˆp, η). (4.13)
Although no predictions can be made for particular values of alm(~x, η), its
dis-tribution is very informative. In particular, we can extract information about the distribution from its mean and its variance, that we call Cl:
34 4. NUMERICAL TOOLS FOR THE STUDY OF DE
Rewriting expression (4.13) we can relate alm with other conventional quantities
in cosmology: the transfer function, ∆T l(k), and the scalar modes, R:
alm(~x, η) = 4π(−i)l Z d3k (2π)3∆T l(k)R~kYl ˆ k m. (4.15)
Equations (4.15) and (4.14) together with the following relation (48)
l X m=−l Ylm(ˆk)Yl∗m(ˆk 0 ) = 2l + 1 4π Pl(ˆk)ˆk 0 , (4.16) yield to ClTT = 2 π Z k2dkPR(k)∆T l(k)∆T l(k). (4.17)
In the latter expression, PR(k), the primordial power spectrum, is given by inflation.
The transfer functions account for the anisotropies and are usually theoretically calculated by EB solvers.
It is important to note that for a given l, every alm has the same variance drawn
from the same distribution. For instance, the distribution for l = 100 of a100,m
is given by 201 coefficients. These many values will give us insightful information about the distribution, However, for smaller values of l, let us say l = 2, we only have 5 a5m. In the latter case, we have a large uncertainty on the distribution. This
uncertainty on low values of l is named cosmic variance (46), it scales as
∆Cl Cl cosmic variance = r 2 2l + 1. (4.18)
Matter power spectrum
The matter power spectrum, P (k, η), is given by the Fourier transform of the cor-relation function for matter.
< δm∗(η, ~k0) δm(η, ~k) >= (2π)3P (k, η)δ3(~k − ~k0). (4.19)
P (k, η) describes the matter overdensities as a function of scale and time. There-fore, it is the most common way to characterize galaxy clustering in linear and close to linear regime. It can be computed via
P (k) = 2π
2
k3 PR(k)∆T(k) 2
, (4.20)
where the transfer function is defined as (43)
∆T(k) =
δm(k, z = 0)δm(0, z = ∞)
δm(0, z = ∞)δm(k, z = 0)
4.3. IMPLEMENTATION OF AN EINSTEIN-BOLTZMANN SOLVER 35
4.3
Implementation of an Einstein-Boltzmann
solver
CAMB is an EB solver primarily created by Antony Lewis and Anthony Challinor (49). The name stands for Code for Anisotropies in the Microwave Background, but since its creation it has been developed to do much more. CAMB solves the Einstein Boltzmann equations, for a given set of cosmological parameters, to compute transfer functions, CMB and matter power spectra. The code is written on Fortran90 and has a Python wrapper available.
CAMB consists of a series of files that can be categorized in three groups: utilities (initialization files, bessel functions and other subroutines), cosmology files, and pa-rameters file. The ones that compute cosmological quantities include transfer func-tions (equafunc-tions.f90, equafunc-tions ppf.f90), initial power spectrum (power tilt.f90) and CMB Cl’s (cmbmain.f90).
As we have argued in section 3, the EFT of dark energy framework is the most convenient to study the plethora of dark energy models. eftcamb (50) is an exten-sion of CAMB that implements the EFT approach into a numerical tool. It has been mainly developed by B. Hu, M. Raveri, N. Frusciante and A. Silvestri. The model independence of the EFT framework allows the study of modified gravity theories. The code does not rely on the quasistatic approximation (51) and checks for the stability conditions of the models.
4.3.1
Structure of eftcamb
The structure of eftcamb is based on a flag system, which reflects the different ways in which modified gravity can be treated within the EFT framework. The main structure, as introduced in (51), is illustrated in figure 4.1. The flag EFTflag select the approach that the code should take.
The fastest way one can adopt to study modify gravity in the EFT language consists of simply choosing a parametrization for the EFT functions in equations (3.5) and (3.6). This approach is called the pure EFT models and can be selected by setting EFTflag=1. Along with the EFT functions, the user must also specify the background expansion history by defining a parametrization of the equation of state of dark energy.
A different EFT parametrization can be included by selecting EFTflag=2. This option uses an alternative parametrization that is already mapped to the EFT lan-guage that includes the ReParametrized Horndeski (RPH) (52).
A third option would be using a theory whose background mimics a specific one. In the designer approach, the user fixes the expansion history through the equation of state of dark energy while the pertirbations are evolved accordingly to a specific theory.. Via the EFT framework, eftcamb computes the EFT functions. For the background they are
ca2 m2 0 =H2− ˙H Ω + aΩ 0 2 − a 2H2 2 Ω 00 +1 2 a2ρDE m2 0 (1 + wDE) . (4.22)
36 4. NUMERICAL TOOLS FOR THE STUDY OF DE
Figure 4.1: eftcamb flag structure
Λa2 m2 0 = −Ω2 ˙H + H2− aΩ0 2H2 + ˙H− a2H2Ω00 + wDE a2ρ DE m2 0 , (4.23)
The designer mapping approach can be selected with EFTflag=3 and it currently includes f (R) and minimally coupled quintessence.
The last selection flag, EFTflag=4, corresponds to the full mapping approach. In this case, the expansion history must be specified. The background is solved separately and then used to compute the EFT functions that must be implemented as well in the EFT mapping procedure. The current public version of eftcamb includes the cases of Horava gravity, covariant galileons (up to fifth order) among others. Furthermore, as one of the results of this project, weakly broken galileons will be included to the public release.
A detailed description of eftcamb can be found in reference (51), where the reader can find all the equations implemented into eftcamb.
4.3.2
Viability conditions
As discussed in section3.5, we need to ensure that no instabilities arise in the EFT. eftcamb has implemented the conditions to avoid ghost instability and gradient stability and can check the theoretical stability for a range of scale factor. This is extremely useful since it helps to constrain the parameter space. We will immedi-ately exclude non viable regions from the parameter space. As follows from (51), the conditions again ghost and gradient instabilities are, respectively,
4.3. IMPLEMENTATION OF AN EINSTEIN-BOLTZMANN SOLVER 37 ˜ W24 ˜W1W˜2− ˜W32 > 0, (4.24) ˜ W0W˜32+ aH ˜W2W˜3W˜60+ ˜W6W˜3W˜20 − ˜W6W˜2W˜30 + 2H ˜W3W˜2W˜6 > 9 2 ˜ W62 a 2 m2 0 (ρm+ Pm), where ˜ W0 = −Ω, ˜ W1 = ca2 m2 0 + 2H02a2γ1− 3H2(1 + Ω) − 3aH2Ω0+ 3H2γ4− 3aH0Hγ2, ˜ W2 = −3[Ω − γ4], ˜ W3 = 6HΩ + 3aHΩ0− 6Hγ4+ 3aH0γ2, (4.25) ˜ W6 = −4 1 2Ω + γ5, ˜ W20 = −3[Ω0− γ40] ˜ W30 = 6 H˙ aHΩ + 9HΩ 0 + 3H˙ HΩ 0 + 3aHΩ00− 6Hγ40 − 6 H˙ aHγ4+ 3aH0γ 0 2 + 3H0γ2 ˜ W60 = −4 1 2Ω 0 + γ50.
5
Weakly broken galileons
Galileon models were proposed as a modification of gravity in (53; 54), where they were firstly studied in the context of Dvali-Gabadadze-Porrati (DGP) model (Braneworld) and extended later to other well-behaved models, such as massive gravity (55). Due to their ability to produce self acceleration, galileons have been used to approach cosmic acceleration of the late time Universe as MG (56; 57; 58) as well as during the early universe driving inflation (59; 60).
Galileons are scalar field theories with HD operators whose action satisfies the galilean shift symmetry
φ → φ + bµxµ. (5.1)
Invariance under transformation (5.1) has remarkable consequences for the model. Firstly, galileons satisfy the non-renormalization theorem that protects them against quantum corrections. On the other hand, galileons are ghost free theories whose equations of motion have only second order derivatives.
In this chapter, we will introduce Galileons as scalar-tensor modified gravity theories. We will start by considering galileons in flat space in section 5.1. Section
5.2 is devoted to the study of galileons coupled to gravity. As we will see, coupling to gravity tends to break the galileon shift invariance. Dealing with this issue will take us to the concept of weakly broken galileon. Finally, in section 5.3, we will build the most general theory that exhibits this particular kind of symmetry.
5.1
Galileons in flat space
The most general action in flat space for a scalar field, φ, with second order equa-tions of motion and invariant under the galileon shift symmetry (5.1) and Lorentz tranformations was derived in (53). Its lagrangian can be written as:
L = (∂φ)2 + 5 X I=3 cI Λ3(I−2)3 LI , (5.2) 38
5.2. GALILEON MODEL WITH GRAVITY 39
where Λ3 is a constant with energy dimensions and cI are the dimensionless Wilson
coefficients. The interaction terms are
L3 = (∂φ)2 φ ,
L4 = (∂φ)2 (φ)2− [φ2] ,
L5 = (∂φ)2 (φ)3− 3φ[φ2] + 2[φ3] , (5.3)
where we use the useful notation, [φ2] ≡ ∂µ∂νφ∂ν∂µφ and [φ3] ≡ ∂µ∂νφ∂ν∂ρφ∂ρ∂µφ.
It can be shown that the interaction lagrangians verify the galileon symmetry up to a total derivative. Therefore, it is the action that presents said invariance. As aforementioned, this action was built so that the equation of motion for the scalar field contains only second order derivatives, as it can be seen by varying the action (5.2) with respect to the scalar field. For galileons in flat spacetime, up to fourth order, the equation of motion is
(φ)3+ 2 ∂µ∂σφ ∂µ∂νφ ∂ν∂σφ − 3 φ ∂µ∂νφ ∂µ∂νφ
+2 ∂µφ ∂ν∂σφ (∂ν∂σ∂µφ − ∂µ∂ν∂σφ) + 4 ∂µφ ∂µ∂νφ (∂σ∂σ∂νφ − ∂ν∂σ∂σφ)
+2 φ ∂µ(∂µ∂σ∂σφ − ∂σ∂σ∂µφ) = 0,(5.4)
where φ = ∂σ∂σφ. It can be noticed that in a flat spacetime, where partial
deriva-tives commute, the equation of motion (5.4) has only second order derivatives of the field. Moreover, galileons in flat space satisfy the non-renormalization theo-rem meaning that they do not take quantum corrections from loops at any order in perturbation theory. In other words, they are stable under quantum corrections thanks to the exact invariance under (5.1) (61), ensuring the robustness of the model.
The second and third line of equation (5.4) already gives us some hints about what will happen in curved space. When gravity comes into the picture with a minimal coupling, we must replace partial derivatives by covariant ones that wont commute. As a consequence, third and fourth derivatives appear in the equations of motion fostering the presence of ghost fields. A non-minimal coupling could be chosen so its contribution to the equations of motion cancels the HD terms. As we will see, this is indeed the way to proceed.
5.2
Galileon model with gravity
Studying the cosmology of galileons requires coupling the scalar field to gravity. Unfortunately, both minimal and non-minimal coupling will in general break the galileon shift symmetry (5.1). This is problematic since we can no longer ensure that the resulting covariant model will be robust and ghost-free. However, as it was shown in (53), we can couple galileons to gravity with a non-minimal coupling in such a way that the symmetry breaking operators are suppressed at the energy
40 5. WEAKLY BROKEN GALILEONS
scale of the invariant operators. In this sense, we will build an effective theory whose invariance under (5.1) is not exact but approximate. We can illustrate this concept with a simple example. Consider the following toy model,
L = −1 2(∂φ) 2+ 1 Λ3 (∂φ)2φ + 1 Λ2 (∂φ)4. (5.5)
It can be proven that the first two terms are exactly invariant under galileon shift transformation, up to a total derivative. However, the third operator wil unavoidably break the galileon symmetry. We say the symmetry is weakly broken if Λ2 Λ3.
Furthermore, the quantum corrections generated by the quartic operator, that are symmetry breaking operators as well, will be also suppressed at the energy scale of the toy model, Λ3. This special kind of symmetry breaking is called weakly broken
symmetry and will preserve the robustness of the theory and protect it against ghost instabilities.
In the following, we will define more rigorously in what sense the symmetry must be weakly broken and show that the non-minimal coupling required for the suppres-sion of quantum corrections that provides the model with second order equations of motion belongs to a Horndeski class.
5.2.1
Weakly broken symmetry
A dramatic consequence of the covariantization of galileons is the loss of exact in-variance under galileons shift transformation. In particular, symmetry breaking vertices that contribute with loop-generated operators of the form (∂φ)2n will
ap-pear. It turns out, as we will see, that precisely some of these quantum corrections are of the same order of magnitude as the original symmetric operators. In order to ensure that that the symmetry is approximately conserved −in other words, that the symmetry is only weakly broken− we must counteract those symmetry breaking operators.
The approach is the following. We are going to build an EFT with WBG invari-ance. The EFT model will be characterized by two energy scales. On one hand, we have the energy cutoff of the EFT, Λ3, which defines the energy scale at which the
model breaks down. On the other hand, each symmetry breaking operator will be suppressed at another energy scale. The smallest suppression energy scale for the latter is Λ2. It will only make sense to talk about WBG if Λ2 Λ3: the scale
sup-pressing the symmetry-breaking operators is parametrically higher that the cutoff of the theory. Then, loop generated operators (62) can be written as:
(∂φ)2n Λ4(n−1)n , (5.6) where Λk,n ≡MP lk Λ4n−k−43 4n−41 . (5.7)
