EXPLORING THE COSMOLOGY OFMODIFIED GRAVITY:

THEORY AND PHENOMENOLOGY OF MODELS

EXHIBITING WEAKLY BROKEN GALILEON

INVARIANCE

THESIS

submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCEin

PHYSICS

Author : Carlos Mateos HidalgoStudent ID : s2076861Supervisor : Dr. A. Silvestri

J . PapadomanolakisS. Peirone

2nd corrector : Prof. Dr. A. Achucarro

Leiden, The Netherlands, July 16, 2019

EXPLORING THE COSMOLOGY OFMODIFIED GRAVITY:

THEORY AND PHENOMENOLOGY OF MODELS

EXHIBITING WEAKLY BROKEN GALILEON

INVARIANCE

Carlos Mateos Hidalgo

Huygens-Kamerlingh Onnes Laboratory, Leiden UniversityP.O. Box 9500, 2300 RA Leiden, The Netherlands

July 16, 2019

Abstract

The problem of the cosmological constant together with the tension in theobservations of the present value of the Hubble parameter has broughtabout 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 brokengalileon (WBG) transformations. We have derived the background cos-mology and found attractor solutions that track a De Sitter Universe atlate 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 LCDMat small scales, WBG can lower the ISW tail of the TT models on the CMBpower spectrum, making it potentially favoured by observations.

Contents

1 Preface 6

2 Introduction 82.1 A homogeneous and isotropic universe . . . . . . . . . . . . . . . . . 8

2.1.1 Dynamics of the Universe . . . . . . . . . . . . . . . . . . . . 92.2 The cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 The problem of the cosmological constant . . . . . . . . . . . 132.3 The Standard Model of Cosmology . . . . . . . . . . . . . . . . . . . 142.4 Beyond the Standard Model of Cosmology . . . . . . . . . . . . . . . 16

2.4.1 Dark energy as a modified form of matter . . . . . . . . . . . 172.4.2 Dark energy as a modification of gravity . . . . . . . . . . . . 18

3 The effective field theory of dark energy 203.1 Cosmological perturbations . . . . . . . . . . . . . . . . . . . . . . . 213.2 A unifying language . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Mapping to EFT: two examples . . . . . . . . . . . . . . . . . 233.3 EFT mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 A simple example . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Stuckelberg mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 263.5 Viability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Numerical tools for the study of DE 304.1 The Einstein-Boltzmann equations . . . . . . . . . . . . . . . . . . . 30

4.1.1 The Boltzmann equations . . . . . . . . . . . . . . . . . . . . 314.1.2 The Einstein equations . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Cosmological observables . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Implementation of an Einstein-Boltzmann

solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3.1 Structure of eftcamb . . . . . . . . . . . . . . . . . . . . . . 354.3.2 Viability conditions . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Weakly broken galileons 385.1 Galileons in flat space . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Galileon model with gravity . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.1 Weakly broken symmetry . . . . . . . . . . . . . . . . . . . . 405.3 Most general theory with WBG invariance . . . . . . . . . . . . . . . 43

2

CONTENTS 3

6 Results I: background evolution 466.1 Dynamical equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.2 Tracker solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.3 Expansion history along the tracker . . . . . . . . . . . . . . . . . . . 54

7 Results II: numerical analysis 567.1 Theoretical set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567.2 Numerical results of WBG . . . . . . . . . . . . . . . . . . . . . . . . 58

8 Conclusions and outlook 70

A Full EFT mapping for Galileons and GLPV theories 74A.1 Generalized galileon mapping . . . . . . . . . . . . . . . . . . . . . . 74A.2 GLPV mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

References 76

List of Figures

2.1 Distribution of galaxies as measured by the 2dF Galaxy Redshift Survey. 92.2 CMB map reconstructed from WMAP observations. . . . . . . . . . . 142.3 Linear matter power spectrum measured by several astronomical sur-

veys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 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. . . . . . 415.2 1PI vertices with one graviton line and two or three scalar fields. . . . 41

7.1 Background expansion history of WBG along the tracker solution . . 597.2 χ evolution for different choices of s and p. . . . . . . . . . . . . . . . 597.3 Stability map for WBG with c5 = 0 . . . . . . . . . . . . . . . . . . . 607.4 Background EFT functions for WBG: Horndeski sector . . . . . . . . 617.5 Second order EFT functions for WBG: Horndeski sector . . . . . . . 627.6 Power spectra for WBG, case I: sampling s. . . . . . . . . . . . . . . 637.7 Power spectra for WBG, case I: sampling p. . . . . . . . . . . . . . . 637.8 Power spectra for WBG, case I: sampling c4. . . . . . . . . . . . . . . 647.9 Power spectra for WBG, case I: cT = 1. . . . . . . . . . . . . . . . . . 647.10 Stability map for WBG with c5 6= 0, s = 2 . . . . . . . . . . . . . . . 657.11 Stability map for WBG with c5 6= 0, s 6= 2 . . . . . . . . . . . . . . . 667.12 Background EFT functions for WBG: beyond Horndeski sector. . . . 667.13 Second order EFT functions for WBG: beyond Horndeski sector. . . . 677.14 Power spectra for WBG: case II. . . . . . . . . . . . . . . . . . . . . . 677.15 Summary of WBG models against Planck ’s TT modes of the CMB

anisotropies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4

A mis padres.

5

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, werely on observations to learn from the Cosmos and describe it. As plenty of evidencesupports, our Universe is currently undergoing an epoch of accelerated expansion,what we call cosmic acceleration. Cosmologists rely on dark energy to explain thisfact, for which the simplest candidate is the cosmological constant (CC): a constantenergy density across the Universe that is driving the acceleration. The StandardModel of Cosmology (SMC) states that we live in a flat Universe in which about70% of its energy content is in the form of this CC. Despite its simplicity, the CCis still a puzzle for physicists. The lack of theoretical understanding worrisome andthe tensions in the SMC, has brought cosmologists to search for different approachesto dark energy.

In this thesis we explore the cosmology of modified gravity (MG), one of themost promising alternatives to the cosmological constant. The idea behind MGis simple: introduce modifications to general relativity that can generate a selfaccelerating universe at cosmological scales without altering the cosmology at smallscales. The number of MG models that has been proposed is overwhelming. In orderto deal with such a plethora of theories the Effective Field Theory (EFT) of darkenergy was proposed. This powerful approach offers a universal language to mapevery particular model to the EFT framework. Our project studies the cosmologyof 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 theStandard Model of Cosmology in chapter 2 where we offer a self contained descriptionof the topic, from the cosmological principle to theories beyond the Standard Modelof Cosmology. Chapter 3 is devoted to the EFT of dark energy, its language andthe 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 thatimplements 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 chapter 6, where we study the background

6

7

dynamics. The results coming from the implementation of the model into eftcambare discussed in chapter 7. We end up with the conclusions of the project and futureprospect in chapter 8.

2

Introduction

In this chapter, we present a self contained introduction to Cosmology. We willstart by the foundation of the standard model of cosmology (SMC), the cosmologicalprinciple, in section 2.1. That will takes us to the mathematical description of auniverse that satisfies this principle and the study of its dynamics, given by theFriedmann equations. The expansion of the universe is explained by the inclusionof the cosmological constant as dark energy. We will devote section 2.2 to the studyof the cosmological constant and its problems. The SMC is properly described insection 2.3. We finish this chapter with section 2.4, where we review the mainalternatives beyond the SMC.

2.1 A homogeneous and isotropic universe

The cosmological principle is at the heart of the standard model of cosmology andstates 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 inspace, 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, thecosmological principle translates into the so-called Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, that describes a homogeneous, isotropic and expandinguniverse:

ds2 = gµνdxµdxν = −dt2 + a(t)2 γijdx

idxj , (2.1)

where ds2 is the line-element, gµν is the metric, the indices denotes with Greekcharacters are space-time indices, µ, ν = 0, 1, 2, 3, while the latin ones are spatialonly indices, i, j = 1, 2, 3. The spatial part of the metric, γij, corresponds to athree dimensional space with a constant curvature, K. In spherical coordinates:

γijdxidxj =

dr2

1−Kr2+ r2

(dθ2 + sin2 θφ2

), (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 = 0corresponds to a flat universe; K = +1, a closed universe and K = −1 is an open

8

2.1. A HOMOGENEOUS AND ISOTROPIC UNIVERSE 9

Figure 2.1: Distribution of galaxies as measured by the 2dF Galaxy Redshift Survey. Inthis image, taken from (1), it is manifest that the universe is isotropic and homogeneousat large scales.

universe. The FLRW metric defines a maximally symmetric universe consequenceof assuming homogeneity and isotropy. We say a spacetime is maximally symmetricwhen it has the maximum number of killing vectors: in this case homogeneity canbe thought as invariance under space time translation while the latter invarianceunder rotations.

It is usually useful to do the following conformal parametrization of these coor-dinates

ds2 = a(t)2(−dτ 2 + γijdx

idxj), (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, theforce governing the cosmological dynamics. Even though is is the weakest of thefour fundamental forces, gravity is the one with largest range. Einstein’s theory ofgeneral relativity (GR) describes the gravitational interactions as the consequence ofthe curvature of spacetime. The acceleration is thus caused by the geodesic motionin a curved spacetime. GR has been successfully tested in solar system scales aswell 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 andGµν is the Einstein tensor given by

Gµν = Rµν −1

2Rgµν (2.6)

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,

∇µTµν = 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 theuniverse while the left hand side comprises the information about the geometryof spacetime. Therefore, given the FLRW metric and the energy content of theuniverse, the Einstein equations yield the dynamics of our universe. Let us look ateach side individually.

Geometry

It is given by the left hand side of equation (2.5). We can calculate the Einsteintensor from the definition of the Ricci tensor in terms of the Christoffel symbolsthat are computed from the metric. The definitions of these geometrical quantitiescan be found in the literature, see reference (2). The components of Gµν are:

G00 = −3

(H2 +K/a(t)2

),

Gi0 = G0

i = 0 , (2.8)

Gji = −3

(3H2 + 2H +K/a(t)2δji

).

where H = a/a is called the Hubble parameter and describes the expansion rate ofthe 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, weassume that all the energy content of the universe behaves as a perfect fluid describedby the energy-momentum tensor,

T µν = (ρ+ P )uµuν + Pδµν , (2.9)

where uµ = (−1, 0, 0, 0) is the 4-velocity of the fluid in comoving coordinates, ρ isthe desnsity and P the pressure. For the way this tensor is constructed we can seethat the (00) component corresponds to the density,

T 00 = −ρ , (2.10)

while the spatial part (ij) corresponds to the pressure of the fluid,

T ji = Pδji . (2.11)

2.1. A HOMOGENEOUS AND ISOTROPIC UNIVERSE 11

Friedmann equations

Putting this information all together on the Einstein equation yields:

(00)→ H2 =8πG

3ρ− K

a2, (2.12)

(ii)→ 3H2 + 2H = −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 Friedmannequation to give:

a

a= −4πG

3(ρ+ 3P ) , (2.14)

known as the second Friedmann equation or acceleration equation. Differentiatingequation (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 =ρiρcr

, (2.16)

where ρcr is the critical density of the universe and it is given by

ρcr =3H2

0

8πG, (2.17)

where H0 is the value of the Hubble parameter at present. With this notation, thefirst Friedmann equation can be written as

Ωm + Ωr + Ωk = 1 . (2.18)

Interpreting curvature as other energy fluid with Ωk = KH2a2

.

The first Friedmann equation (2.25) and the continuity equation (2.15), describethe 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 itscomponents. Regarding the latter, it is usual to consider the equation of state of aperfect 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 ofthe 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 ofmatter 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 ofthe 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 isdue to the fact that not only is radiation diluted as the volume of the universeexpands, but also it gets redshifted. This effect accounts for the decrease ofthe frequency of the radiation when space expands.

As we can see form the previous equations, the expansion of the universe quicklydilutes all energy constituents. Non relativistic matter will be the first species tobecome subdominant followed by matter. As we will see below, the late time universeis 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 conservedquantity. Since the metric satisfies∇µg

µν = 0, Einstein equations admit the additionof the term Λgµν sice ∇µg

µν = 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 onthe idea that the Universe should be static. As we saw in expression (2.20), theFriedmann equations predict an expanding, dynamical universe. To fix this problemEinstein added the cosmological constant to the equations that result in the followingFriedmann equations:

H2 =8πG

3ρ− K

a2+

Λ

3, (2.25)

3H2 + 2H = −8πGP − K

a2+

Λ

3. (2.26)

These equations provide a static universe (a = a = 0) given the domination ofa pressureless fluid of constant density. As shown by Lemaitre, such a solution isunstable 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 whenEinstein 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 withequation 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 canbe written as

H2 = H20

(Ωm,0a(t)−3 + Ωr,0a(t)−4 − K

H20a

2+

Λ

3H20

). (2.27)

De Sitter universe

W. De Sitter found in 1917 a solution to the Einstein equations that representedthe first model of an accelerated Universe. The De sitter universe is given by thefollowing metric,

ds2 = dt2 + eHtdxidxi , (2.28)

for a empty universe. This metric gives a universe that grows exponentially witha(t) = eHt yielding a expansion rate:

H(t) =a

a=

√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 andradiation 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, difficultproblems that make cosmologists discontent. We interpret Λ as the energy densityof vacuum and its value can be predicted from Quantum Field Theory. An estima-tion can be made by considering the contribution of all the known particles in theStandard Model to the cosmological constant. To do so, we consider these particlesas fields that can be describe as a series of harmonic oscillator. The vacuum energywould be given by

〈ρ〉 ∼∫ ΛUV

O

d3k

(2π)3

1

2~Ek ∼

∫ ΛUV

0

dk k2√k2 +m2 ∼ Λ4

UV , (2.30)

where ΛUV is the cutoff of the theory, namely the highest energy at which we cantrust predictions. This is given by the weak interaction having ΛUV ∼ TeV . 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 aworrisome 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 thatdescribes the cosmology of our universe tested against the following cosmologicalobservations.

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 perfectlyisotropic radiation comes from the time when the universe became transparent. Af-ter the Big Bang, photons and matter particles were coupled, evolving together andforming what we know as the primordial plasma. When the Universe was 380 000years old it was cool enough for photon to decouple from the primordial plasma andtravel freely throughout space. Today, we receive this redshifted radiation as theCMB.

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 hasdetermined that the universe is flat, namely K = 0.

Abundances of light elements

The Big Bang Nucleosynthesis (BBN) is a process that formed the light elementsthat we observe today, such as deuterium and helium, when the primordial was coolenough. Not only do the predictions of BBN on light element abundances agree withobservations, but also place strong constraints on the energy density parameter ofdark energy. An upper bound can be found form the fact that the expansion rateof the universe modify the proton to neutron ratio at freeze out inevitably changingthe 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 strongevidence in favour of dark energy and the cosmological constant. As shown in (4),the measured power spectrum of galaxy distribution depends on the matter densityparameter 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 theobservation of supernovae of supernovae by the High Redshift Supernovae SearchTeam (HSST) (6). The survey determined that the stellar objects were receding fromus and placed constraints on the value of H0, and the energy density parameter ofmatter and dark energy.

In Figure (2.4) we can see the Hubble diagram created form the SN observationsat different redshifts. The quantity m − M is directly related to the luminositydistance.

These observations support the SMC that describes a flat universe with thefollowing energy content distribution: 4% baryonic matter, 23% cold dark matterand around 73% of dark energy understood as the cosmological constant. Preciselydue to the two main energy components, the SMC is usually called LCDM standingup for the cosmological constant (L) and cold dark matter (CDM). The model isfully 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 DescriptionΩb 0.02233 ± 0.00015 Baryonic energy densityΩc 0.1198 ± 0.0012 CDM energy densityH0 67.37 ± 0.54 Hubble parameterτ 0.0540 ± 0.0074 Optical depth

ln(1010As) 3.043 ± 0.014 Scalar amplitudens 0.9652 ± 0.0042 Spectral index

Table 2.1: Parameters of the SMC, the values correspond to the PlanckTT,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. Inparticular, the value of the Hubble parameter evaluated today observed by Planckis in tension with its local measurements (8). The amplitude of the linear powerspectrum 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 ofintroducing dark energy as a dynamical energy fluid in the energy-momentum tensor,Tµν , in the Einstein equations. The second approach consists of modifying generalrelativity at large scales to account for the cosmic acceleration, what we call modifiedgravity. At the end of the day this is just a convenient way of categorizing thetheories: the two strategies are completely equivalent and cannot be distinguished.Any change of the energy-momentum tensor can be absorbed by the Einstein tensorand vice versa. In this section we briefly discussed the most important models withineach 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 dynamicalequations of motion that changes with time. The main representatives of dynamicaldark 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 ofthe universe. Their solutions have trackers, attractors in which the field falls, thatmitigates the coincidence problem. Other models that introduce modified matterare phantom and coupled dark energy. We refer to (4) for a detail description of theaforementioned models.

Quintessence

The action of this model is given the the usual one form general relativity with acanonical scalar field, as below,

S =

∫d4x√−g(M2

Pl

2R + Lφ

)+ SSM , (2.33)

where M2Pl = (8πG)−1 and

Lφ = −1

2gµν∂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 aregiven by:

ρφ =1

2φ2 + V (φ) , (2.35)

Pφ =1

2φ2 − V (φ) , (2.36)

Therefore the equation of state, defined as the ratio between pressure and density,yields to:

wφ =Pφρφ

=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 actionwith respect to the field,

φ+ 3Hφ+∂V (φ)

∂φ= 0. (2.38)

The form of the potential, V (φ), must be fixed so that the energy density of thefield becomes dominant at late times. Namely, ρφ ρm at early times. It is in thissense that we say the field track ρm. Also, in order to have acceleration the equationof 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 =

∫d4x√−g(M2

Pl

2R + P (φ,X)

)+ SSM , (2.39)

where

X = −1

2gµν∂µΦ∂νΦ . (2.40)

The tracker behaviour of these models is such that can account for the expansionas the cosmological constant without introducing it. During radiation dominateduniverse the k-essence will track radiation equation of state. The latter will drop atthe start of matter dominated era to track a De Sitter universe. This tracker willremain subdominant until the matter component is diluted and the scalar field cancatch up.

2.4.2 Dark energy as a modification of gravity

Models within this category introduce modifications to the geometrical sector of theEinstein 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 =

∫d4x√−gM

2Pl

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 =

TµνMPl

, (2.42)

where fr = ∂f∂R

. Note that these equations are fourth order. The trace equationyields

(1− fR)R + 2f − 3fR =T

MPl

. (2.43)

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 familyof f(R) models labelled by a parameter that is related to the length scale of thescalaron. The construction of f(R) must account for the following conditions ofviability (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 metrictensor to mediate the acceleration at late times. Within this category we can findthe so called Brans-Dicke theory (14) that connects the gravitational constant tothe cosmic field. The general action for a scalar tensor can be written as

S =

∫d4x√−g(MPl

2f(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-tensorethoery ara galileons, that we will extensively study in chapter 5.

Braneworlds

In braneworld scenarios our Universe is understood as a three+one dimensionalbrane, embedded in a higher dimensional bulk space. This model of universe wasproposed by Arkani-Hamed, Dimopulous and Dvali (15; 16) and it has been namedBrane World (or ADD scenario). In this model, general relativity is recovered atshort distances while at larger scales modifications to gravity appear.

Other approach is the one proposed by Randall and Sundrum (17) in which thethree-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 darkenergy

The discovery of cosmic acceleration has motivated the proposal of a tremendousnumber 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 thebest framework for two main reasons. Firstly, it provides with an unifying languageto analyze all DE models. That way we can have a global view and work in a modelindependent 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 everymodel to data. Data can constrain the general formalism from which we can get theimplications for particular models. This approach was firstly proposed in cosmologyin 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 fieldsas new degrees of freedom. In inflation, this field is responsible for the acceleratedexpansion and mediates its termination. Addressing cosmic acceleration, we canalso find a great number of models that invoke a scalar field as the mediator ofthe acceleration. It is essential then, not to rely on the particularities of the addeddegrees of freedom, to keep the framework as general as possible. As we will see, inthe EFT framework the scalar field arises naturally as the Goldstone boson of timediffeomorphism symmetry breaking.

In this section we introduce the EFT framework for dark energy and set up allthe technical machinery we will need in following sections of this thesis. We willstart by considering cosmological perturbations since they lie at the heart of thisformalism. Taking the perturbations around the background as the relevant degreeof freedom will allow us to write the unifying action for dark energy in the EFTframework in section (3.2).

20

3.1. COSMOLOGICAL PERTURBATIONS 21

3.1 Cosmological perturbations

As for the anisotropies of the Cosmic Microwave Background (CMB) and the largescale structures (LSS), cosmological perturbations play an important role in thedynamics of the universe. It is then natural to identify the perturbations aboutthe homogeneous Friedmann-Robertson-Walker (FRW) as the relevant degree offreedom when building the action for the EFT of dark energy (22). However, as wehave mentioned above, we do not want to rely on specific details of the scalar fieldnor fix a background cosmology. Writing an action in terms of the perturbationsrequires to explicitly write the background evolution, φ0, and the perturbation ofthe field, since

φ→ φ0 + δφ. (3.1)

In principle, we cannot write the perturbed action without specifying φ0, whichwould require solving its background evolution. However, we can deal with thisissue by considering the unitary gauge, equivalent to choosing the field as the timecoordinate. 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 perturbedaction easily without fixing the evolution of the scalar field. As a consequence offixing the gauge, time diffeomorphism invariance is broken. The strategy to buildthe lagrangian would be to write the most general action invariant under the groupof residual symmetries −in this case, three-dimensional diffeomorphisms− for themetric perturbations, and all other geometrical quantities. We shall go into detailabout this concept.

The unitary gauge

We will now argue that the scalar field can arise as a Goldstone boson due to thesymmetry breaking under time diffeomorphism. Let us consider a field, φ, that isinvariant 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 alongthe vacuum that respects the symmetry will take us to an equivalent vacuum. Thesemassless 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 massiveexcitations will decouple and we are left with the Goldstone bosons. As it is verywell known, making a local gauge theory introducing covariant derivatives can addnew terms to the lagrangian that provide the field with mass.

The unitary gauge is equivalent to setting the Goldstone bosons to zero. Inthis sense, it selects, among any field configuration, the one without fluctuationsaround the direction of the symmetry. Fixing this gauge for the EFT functionwill 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 thescalar field is a consequence of the time diffeomorphism symmetry breaking ratherthan a postulated field (22). The resulting theory is only gauge invariant under theunbroken three-dimensional diffeomorphism. Therefore it must contain: (1) threedimensional geometrical objects defined for t = constant, (2) four-dimensional geo-metrical quantities with free time indices and (3) scalar quantities such as the Ricciscalar.

In order to restore the invariance under full diffeomorphism we can apply theStuckelberg trick, that consists on reintroducing the Goldstone fields by forcing thetime 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 geometricalquantities. This treatment is especially interesting since perturbations will be smallat cosmological scales, assuring the validity of the EFT. To do so, we must writethe most general action that is invariant under the unbroken symmetry group con-taining all possible contractions of the perturbations of the geometrical quantitiesaforementioned. Therefore, the action must contain the perturbations of the metric,the Ricci scalar, the Ricci tensor and the extrinsic curvature. The action writtenin the EFT framework for dark energy was proposed in (23; 24) assuming the weakequivalence principle (WEP) −all matter fields will be universally coupled to themetric− as follows,

S =

∫d4x√−g[MPl

2(1 + Ω(t))R + Λ(t)− c(t)δg00

+M4

2 (t)

2(δg00)2 +

M43 (t)

3!(δg00)3 + . . .

− M31 (t)

2δg00δKµ

µ −M2

2 (t)

2(δKµ

µ)2 − M23 (t)

2δKµ

νδKνµ + . . .

+ λ1(t)(δR)2 + λ2(t)δRµνδRµν + α1(t)CµνσλC

µνσλ + . . .

+ α(t)εµνσλCρθµνCσλρθ +m2

1(t)nµnν∂µg00∂νg

00 + . . .

+m22(t)(gµν + nµnν)∂µg

00∂νg00 + . . .

]+ Sm[gµν ], (3.4)

where Cµνρσ is the Weyl tensor, δg00, δR, δRµν , and δKµν are the perturbationsof the above mentioned geometrical quantities and nµ is the normal vector to thespatial hypersurfaces. The constant m0 has mass dimensions and it is used to make

3.2. A UNIFYING LANGUAGE 23

the function Ω dimensionless. This action has been constructed by considering allpossible contractions of the geometrical quantities up to second order. Note that,as the action must enjoy spatial diffeomorphism invariance, all spatial indices arecontracted, unlike the temporal ones that are written explicitly. For the same rea-son, every term has a time dependent coefficient, namely, Ω, Λ, cMi; that are calledEFT functions.

Regarding the very first line on (3.4), it contains the functions that are relevantto 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 arerelevant for the evolution of the perturbations and can be rewritten as dimensionlessoperators as follows,

γ1 =M4

2

m20H

20

, γ2 =M3

1

m20H0

, γ3 =M2

2

m20

,

γ4 =M2

3

m20

, γ5 =M2

m20

, γ6 =m2

2

m20

. (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 theEFT language. The most simple example is LCDM, whose action is given by

S =

∫d4x√−g(M2

Pl

2R−M2

PlΛ

), (3.7)

can be mapped by a simply comparison with (3.4), yielding:

Ω(t)LCDM = 0, Λ(t)LCDM = M2PlΛ. (3.8)

In the case of Quintessence (10), the action is given by

S =

∫d4x√−g(M2

Pl

2R− (∇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 andφ(t)→ φ0(t) + δφ(t). Since field is only dependent on time −we are working at thebackground level− we have that (∇φ)2 = g00φ2. Putting everything together, andfixing the unitary gauge, yields:

S =

∫d4x√−g(M2

Pl

2R− 1

2φ2δg00 +

1

2φ2 − U(φ0)

). (3.10)

Comparing (3.10) and (3.4) we find the mapping

ΩQ(t) = 0, cQ(t) =φ2

0

2, ΛQ(t) =

φ20

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 byemploying the Arnowitt Deser Misner formalism (ADM). The ADM decompositionis 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 formalismreads

ds2 = −Ndt2 + hij(dxi +N idt)(dxj +N jdt), (3.12)

where hij is the induced metric on the 3-dimensional hypersurfaces, N the lapsefunction and Ni the shift vector.

Following (25; 26; 24), we write a general lagrangian in ADM formalism andcompare to the most general action for the EFT of dark energy written in ADMformalism as well. Said comparison will be the mapping from any arbitrary theoryin ADM to the EFT language. This is done in reference (25) where the authorsconsider 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 actioncan be espressed as,

S =

∫d4x√−gL(N,K,S,R,Z,U ,Z1,Z2, α1, α2, α3, α4, α5, ; t), (3.13)

where,K = Kµ

µ , R =(3) Rµµ, S = KµνK

µν , 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 canbe defined as

nµ = Nδµ0 , Kµν = hλµ∇λnν , (3.16)

∆ = ∇a∇a and ai is the acceleration vector. These operators are the ones needed towrite the action up to sixth order in spatial derivatives. The perturbed lagrangiancan 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 =

∫d4x√−g[L+ F + 3HF + (LN − F)δN +

(F +

1

2LNN

)(δN)2 + LSδK

νµδK

µν

+1

2A(δK)2 + BδNδK + CδKδR+DδNδR+ EδR+

1

2G(δR)2 + LZδRµ

νRνµ

+Lα1∂iδN∂iδN + Lα2∂iδN∇k∇k∂iδN + L

α3R∇i∂

iδN + Lα4∂iδN∆2∂iδN

+Lα5∆R∇i∂iδN + LZ1∇iδR∇iδR+ LZ2∇iδRjk∇iδRjk

](3.18)

where

A = LKK + 4H2LSS − 4HLSK ,

B = LKN − 2HLSN ,

C = LKR − 2HLSR +1

2LU −HLKU + 2H2LSU ,

D = LNR +1

2LU −HLNU , (3.19)

E = LR −3

2HLU −

1

2LU ,

F = LK − 2HLS ,

G = LRR +H2LUU − 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 comparethe resulting EFT action in ADM language with equation 3.18. The outcome is amapping form the general action to the EFT framework. A direct cross check innumber of perturbations yields the following:

Ω(t) =2

m20

E , c(t) =1

2(F − LN) + (HE − E − 2EH),

Λ(t) = L+ F + 3HF − (6H2E + 2E + 4HE + 4HE) , M22 (t) = −A− 2E ,

M42 (t) =

1

2l(LN +

LNN2

)− c

2, M3

1 (t) = −B − 2E , M23 (t) = −2LS + 2E ,

m22(t) =

Lα1

4, m5(t) = 2C, M2(t) = D, λ1(t) =

G2,

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 certainmodel in ADM notation and then compute the EFT functions from above usingdefinitions 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

2G2XX(X −X0)2. (3.23)

In ADM language, X = X0

N2 , therefore

G2(X) = G2(X0) +G2X(X0)X0(1

N2− 1) +

1

2G2XXX

20 (

1

N2− 1)2. (3.24)

By using the map (3.21) with definitions (3.19) we can get the mapping. TheEFT 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

M42 = G2XX(X0)X2

0 , (3.27)

since LNN = 4X0G2X(X0).

3.4 Stuckelberg mechanism

As we previously mentioned, we can restore the full-diffeomorphism by introducingtime perturbations, namely by the transformation (3.3). This way, functions of timewould 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

δR = R−R(0)(t)→ δR− ˙R(0)π + ... (3.29)

3.5. VIABILITY CONDITIONS 27

Applying this transformations to the action (3.4) we can find (23):

S =

∫d4x√−g[MPl

2Ω(t+ π)R + Λ(t+ π)

− c(t+ π)

(δg00 − 2π + 2πδg00 + 2∇iπg

0i − π2 +gij

a2∇iπ∇jπ + . . .

)+M4

2 (t+ π)

2

(δg00 − 2π + . . .

)2

− M31 (t+ π)

2

(δg00 − 2π + . . .

)(δKµ

µ + 3Hπ +∇2π

a2+ . . .

)

− M22 (t+ π)

2

(δKµ

µ + 3Hπ +∇2π

a2+ . . .

)2

− M23 (t+ π)

2

(δK i

j + Hπδij +gik

a2∇k∇jπ + . . .

)(δKj

i + Hπδji +gjl

a2∇l∇iπ + . . .

)+M2(t+ π)

2

(δg00 − 2π + . . .

)(δR(3) + 4H

∇2π

a2+ 12H

k0

a2π + . . .

)

+m22(t+ π)(gµν + nµnν)∂µ(g00 − 2π + . . .)∂ν(g

00 − 2π + . . .) + . . .

]+ Sm[gµν ]

(3.30)

3.5 Viability conditions

High derivative terms are usually avoided in physics since they are prompted tointroduce instabilities in the model. Therefore, working in the EFT frameworkof dark energy, where HD operators have relevant contributions, requires studyingstability conditions that ensures the robustness of the theory.

Ghost instability

The presence of HD operators can result in equations of motion with more thansecond time derivatives acting on the scalar field, which translates into extra modesof propagation. If the mass of the ghost fields lies above the energy cutoff of theEFT, we can interpret them as nonphysical modes produced by the truncation ofan infinite series that are fatal for our EFT.

Ghost fields are characterized for having negative kinetic quanta given by a awrong 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 signthen infinite number of pair ghost field - healthy field could be created with noenergy 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 causeinstabilities. See, for instance, the following lagrangian

L =1

2φ2 +

1

2∇φ2. (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 thatcontribute more to the instability are those with high energy (29) which makes nosense 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 +

m2

2φ2. (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 theorycharacterized by the timescale m−1 and indicates that the theory has not beenperturbed around the true vacuum.

In the context of EFTs, the theory is predictive at shorter scales than m−1. Incosmology we can think of it as an EFT that on small scales can be described as aMinkowski space, ignoring the cosmic expansion.

3.5. VIABILITY CONDITIONS 29

4

Numerical tools for the study ofDE

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 orderto get information out of the data and constrain the theoretical models we neednumerical tools that can compute cosmological observables. This task is done byEinstein-Boltzmann (EB) solvers, codes that solve the full set of Einstein-Boltzmannequations. The original work done by (35; 36; 37; 38) for the study of cosmologicallinear perturbations required solving sets of differential equations. These tediouscalculations were facilitated by the line of sight method (39) that considerably re-duced the computational time. The method was implemented into the cmbfastcode. Currently, CAMB (Code for Anisotropies in the Microwave Background) (40)and CLASS ( Cosmological Linear Anisotropy Solving System) (41; 42) are the mostdeveloped and spread out EB solvers commonly used in cosmology. Both of themcount 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 provedto agree up to a great level of precision (45).

In this chapter we introduce the EB equations that describe the interactionbetween the different constituents of the Universe and with the metric. We showhow they relate to cosmological observables and discuss the implementation into anEB solver.

4.1 The Einstein-Boltzmann equations

Even though homogeneity is a good approximation at cosmological scales, it is verywell known that the Universe is not perfectly uniform. In fact,, inhomogeneities arethe responsible of anisotropies in the CMB and the formation of LSS. The cosmolog-ical evolution of matter and radiation perturbations is complicated because of thegreat number of interactions between them. On one hand, every species −photons,neutrinos, dark matter and baryons− directly interact with the metric, described

30

4.1. THE EINSTEIN-BOLTZMANN EQUATIONS 31

by the Einstein equations. On the other hand, there are also interactions betweenthem that are described by the Boltzmann equations: photons and electrons interactby Compton scattering and protons and electrons present Coulomb scattering. TheEinstein-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 Boltzmannequation is

df

dt= C[f ], (4.1)

where f is the distribution function of a particle and C[f ] is collision term containingthe information about creation and annihilation of the particle. It is natural that thecollision term depends on the distribution function, since the number of collisions−interactions− are conditioned by the abundance of the species involved in theinteraction. When C[f ] = 0, meaning there are no interactions, the distributionfunction is constant and abundances do not change. The total derivative can bewritten as

d

dt=

∂

∂t+ ~v∇~x + ~F ∇~p. (4.2)

where v = ∂x∂t

and ~F = ∂~p∂t

.

To calculate C[f ] we have to integrate over the energies of all species the squareamplitude 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 willbe translated into overdensities that will generate the formation of LSS. Baryonicmatter 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 thedynamics of these systems in fourier space. In this formalism, non-relativistic matterwill depend on the wavenumber k and conformal time η.

The description of photon and neutrinos is less straightforward. The distributionof 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 = pk. 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

∫ 1

−1

dµ

2Pl(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 = −3Φ

v +a

av = −ikΨ (4.4)

δb + ikvb = −3Φ

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 alsowith the metric. The Einstein equations account for that relation between the metricperturbation and the matter and radiation inhomogeneities. The Einstein equationsread

Gµν = 8πGTµν . (4.7)

In Fourier space, for the perturbed metric in equation (4.6) the Einstein equationscan be written as

k2Φ + 3a

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 darkmatter, and r stands for radiation, including photons and neutrinos; as follows

ρmδm = ρDMδ + ρbδb ,

ρmvm = ρDMv + ρbvb , (4.9)

ρrΘr,0 = ργΘ0 + ρrN0 ,

ρrΘr,1 = ργΘ1 + ρrN1 .

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 differentialequations to be solved together with the initial conditions. The solutions can berelated to cosmological observables. In particular, we can get the power spectra ofthe CMB anisotropies and matter overdensities.

CMB anisotropies

The power spectrum of temperature anisotropies describes the information presentin the CMB temperature map. We can relate the temperature anisotropies withthe function Θl describing the radiation anisotropies. The temperature field in theuniverse can be wrtiten as

T (~x, p, η) = T (η)[1 + Θ(~x, p, η)]. (4.10)

Following expression (4.10), we can state

Θ(~x, p, η) =δT

T. (4.11)

The function Θ(~x, p, η) can be written as a series of harmonic oscillators, Ylm asfollows

Θ(~x, p, η) =∞∑l=1

l∑m=−l

alm(~x, η)Ylm(p). (4.12)

Which can be inverted to yield the coefficient alm:

alm(~x, η) =

∫d3k

(2π)3ei~k~x

∫dΩYl

∗m(p)Θ(~x, p, 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 thedistribution from its mean and its variance, that we call Cl:

< alma∗l′m′ >= δll′δmm′ Cl (4.14)

34 4. NUMERICAL TOOLS FOR THE STUDY OF DE

Rewriting expression (4.13) we can relate alm with other conventional quantitiesin cosmology: the transfer function, ∆T l(k), and the scalar modes, R:

alm(~x, η) = 4π(−i)l∫

d3k

(2π)3∆T l(k)R~kYl

km. (4.15)

Equations (4.15) and (4.14) together with the following relation (48)

l∑m=−l

Ylm(k)Yl∗m(k′) =

2l + 1

4πPl(k)k′, (4.16)

yield to

CTlT =

2

π

∫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 theoreticallycalculated by EB solvers.

It is important to note that for a given l, every alm has the same variance drawnfrom 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 informationabout the distribution, However, for smaller values of l, let us say l = 2, we onlyhave 5 a5m. In the latter case, we have a large uncertainty on the distribution. Thisuncertainty on low values of l is named cosmic variance (46), it scales as(

∆ClCl

)cosmic variance

=

√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.

< δ∗m(η,~k′) δm(η,~k) >= (2π)3 P (k, η)δ3(~k − ~k′). (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 closeto linear regime. It can be computed via

P (k) =2π2

k3PR(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)(4.21)

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, butsince its creation it has been developed to do much more. CAMB solves the EinsteinBoltzmann equations, for a given set of cosmological parameters, to compute transferfunctions, CMB and matter power spectra. The code is written on Fortran90 andhas 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 (equations.f90, equations 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 mostconvenient 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 beenmainly developed by B. Hu, M. Raveri, N. Frusciante and A. Silvestri. The modelindependence of the EFT framework allows the study of modified gravity theories.The code does not rely on the quasistatic approximation (51) and checks for thestability conditions of the models.

4.3.1 Structure of eftcamb

The structure of eftcamb is based on a flag system, which reflects the differentways in which modified gravity can be treated within the EFT framework. Themain 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 languageconsists 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 selectedby setting EFTflag=1. Along with the EFT functions, the user must also specifythe background expansion history by defining a parametrization of the equation ofstate of dark energy.

A different EFT parametrization can be included by selecting EFTflag=2. Thisoption 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 equationof state of dark energy while the pertirbations are evolved accordingly to a specifictheory.. Via the EFT framework, eftcamb computes the EFT functions. For thebackground they are

ca2

m20

=(H2 − H

)(Ω +

aΩ′

2

)− a2H2

2Ω′′ +

1

2

a2ρDE

m20

(1 + wDE) . (4.22)

36 4. NUMERICAL TOOLS FOR THE STUDY OF DE

Figure 4.1: eftcamb flag structure

Λa2

m20

= −Ω(

2H +H2)− aΩ′

(2H2 + H

)− a2H2Ω′′ + wDE

a2ρDE

m20

, (4.23)

The designer mapping approach can be selected with EFTflag=3 and it currentlyincludes 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 solvedseparately and then used to compute the EFT functions that must be implementedas well in the EFT mapping procedure. The current public version of eftcambincludes the cases of Horava gravity, covariant galileons (up to fifth order) amongothers. Furthermore, as one of the results of this project, weakly broken galileonswill be included to the public release.

A detailed description of eftcamb can be found in reference (51), where thereader can find all the equations implemented into eftcamb.

4.3.2 Viability conditions

As discussed in section 3.5, we need to ensure that no instabilities arise in the EFT.eftcamb has implemented the conditions to avoid ghost instability and gradientstability and can check the theoretical stability for a range of scale factor. This isextremely 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

W2

[4W1W2 − W 2

3

]> 0, (4.24)

W0W23 + aH

(W2W3W

′6 + W6W3W

′2 − W6W2W

′3

)+ 2HW3W2W6 >

9

2W 2

6

a2

m20

(ρm + Pm),

where

W0 = −Ω,

W1 =ca2

m20

+ 2H20a

2γ1 − 3H2(1 + Ω)− 3aH2Ω′ + 3H2γ4 − 3aH0Hγ2,

W2 = −3[Ω− γ4],

W3 = 6HΩ + 3aHΩ′ − 6Hγ4 + 3aH0γ2, (4.25)

W6 = −4[12

Ω + γ5

],

W ′2 = −3[Ω′ − γ′4]

W ′3 = 6

HaH

Ω + 9HΩ′ + 3HH

Ω′ + 3aHΩ′′ − 6Hγ′4 − 6HaH

γ4 + 3aH0γ′2 + 3H0γ2

W ′6 = −4

[12

Ω′ + γ′5].

5

Weakly broken galileons

Galileon models were proposed as a modification of gravity in (53; 54), wherethey were firstly studied in the context of Dvali-Gabadadze-Porrati (DGP) model(Braneworld) and extended later to other well-behaved models, such as massivegravity (55). Due to their ability to produce self acceleration, galileons have beenused 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 thegalilean 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 againstquantum corrections. On the other hand, galileons are ghost free theories whoseequations of motion have only second order derivatives.

In this chapter, we will introduce Galileons as scalar-tensor modified gravitytheories. We will start by considering galileons in flat space in section 5.1. Section5.2 is devoted to the study of galileons coupled to gravity. As we will see, couplingto gravity tends to break the galileon shift invariance. Dealing with this issue willtake us to the concept of weakly broken galileon. Finally, in section 5.3, we willbuild 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 Lorentztranformations was derived in (53). Its lagrangian can be written as:

L = (∂φ)2 +5∑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 Wilsoncoefficients. 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 symmetryup to a total derivative. Therefore, it is the action that presents said invariance. Asaforementioned, this action was built so that the equation of motion for the scalarfield 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 fourthorder, the equation of motion is

(φ)3 + 2 ∂µ∂σφ ∂µ∂νφ ∂ν∂

σφ− 3φ ∂µ∂νφ ∂µ∂νφ

+2 ∂µφ ∂ν∂σφ (∂ν∂σ∂µφ− ∂µ∂ν∂σφ) + 4 ∂µφ ∂µ∂νφ (∂σ∂

σ∂νφ− ∂ν∂σ∂σφ)

+2φ ∂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 ofthe field. Moreover, galileons in flat space satisfy the non-renormalization theo-rem meaning that they do not take quantum corrections from loops at any orderin perturbation theory. In other words, they are stable under quantum correctionsthanks 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 aboutwhat will happen in curved space. When gravity comes into the picture with aminimal coupling, we must replace partial derivatives by covariant ones that wontcommute. As a consequence, third and fourth derivatives appear in the equationsof motion fostering the presence of ghost fields. A non-minimal coupling could bechosen so its contribution to the equations of motion cancels the HD terms. As wewill 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 thegalileon shift symmetry (5.1). This is problematic since we can no longer ensurethat the resulting covariant model will be robust and ghost-free. However, as itwas shown in (53), we can couple galileons to gravity with a non-minimal couplingin 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 whoseinvariance under (5.1) is not exact but approximate. We can illustrate this conceptwith 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 shifttransformation, up to a total derivative. However, the third operator wil unavoidablybreak the galileon symmetry. We say the symmetry is weakly broken if Λ2 Λ3.Furthermore, the quantum corrections generated by the quartic operator, that aresymmetry breaking operators as well, will be also suppressed at the energy scale ofthe toy model, Λ3. This special kind of symmetry breaking is called weakly brokensymmetry and will preserve the robustness of the theory and protect it against ghostinstabilities.

In the following, we will define more rigorously in what sense the symmetry mustbe 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 ofmotion 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 breakingvertices 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 correctionsare of the same order of magnitude as the original symmetric operators. In orderto ensure that that the symmetry is approximately conserved −in other words, thatthe symmetry is only weakly broken− we must counteract those symmetry breakingoperators.

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, wehave the energy cutoff of the EFT, Λ3, which defines the energy scale at which themodel breaks down. On the other hand, each symmetry breaking operator will besuppressed at another energy scale. The smallest suppression energy scale for thelatter 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 cutoffof the theory. Then, loop generated operators (62) can be written as:

(∂φ)2n

Λ4(n−1)n

, (5.6)

where

Λk,n ≡[Mk

P lΛ4n−k−43

] 14n−4 . (5.7)

5.2. GALILEON MODEL WITH GRAVITY 41

For the symmetry breaking operators to be sufficiently suppressed, Λk,n must begreater than Λ2. This allows us to define the suppression scale Λ2 as the asymptoticlimit of Λk,n. Therefore,

Λ2 = (MPlΛ33)1/4 (5.8)

With this definition, Λ2 is parametrically higher than Λ3 for Λ3 MPl. Letus now study how this energy scales can be implemented into the action to meetour goal. We can write a vertx as (∂φ)k (∂mφ)n ∂l hp). In a Feynman diagram, anexternal leg, ∂φ, will be represented as a solid line; scalars with more than onederivative acting on them, ∂mφ, are depicted by a dashed line and the graviton, ∂h,by a wiggly line.

Figure 5.1: Conventions for the external legs in the Feynmann diagram.

Pure galileons will only interact with external legs with double derivatives, orcan be written in such way, since those satisfy the galileon shift symmetry. Actually,that is why pure galileons have a non-renormalization property (61).

Let us firstly consider diagrams with one graviton line, that comes from covari-antizing the model: substituting partial by covariant derivatives in the lagrangian(5.2). We have six vertices of such kind, depicted in figure 5.2.

Figure 5.2: 1PI vertices with one graviton line and two or three scalar fields.

As it can be seen, vertices with three solid lines are possible even if they arenot explicitly present in the lagrangian. In particular, terms like ∇2φ can be ex-panded yielding ∂2φ and ∂φ ∂h, that together with other ∂2φ with produce thesediagrams.Let us consider the symmetry breaking term produced by a vertex withone graviton and three solid lines. Such a vertex can be written as

(∂φ)3 ∂2φ ∂h

MPl Λ63

(5.9)

42 5. WEAKLY BROKEN GALILEONS

Vertex 5.9 corresponds to the second vertex of the lowest line in figure 5.2. Notethat the suppression scale is given by purely dimensional analysis, where the gravi-ton contributes with a factor of MPl and the scalar field with the energy cutoff.The above mentioned operator generates quantum corrections whose scale can becalculated by setting k = 2a and n = 3a in equation (5.7), for large a, we have(MPl Λ5

3)1/6. This scale is parametrically lower than Λ2 meaning that its contribu-tion will be relevant below the energy cutoff of the EFT. Given that this operatorbreaks the symmetry this behaviour is unwanted. The same analysis can be donefor all vertices with 3 and 5 solid lines. We must provide the theory with a mecha-nism that prevent these vertices to appear. As we have already commented, a nonminimal coupling will meet this goal: cancel all vertices with odd number of solidlines so symmetry breaking operator are suppressed.

The counterterms that come in the form of a non-minimal coupling are found in(62), as follows

L3 →√−gLmin

3 ,

L4 →√−g((∇φ)4R− 4Lmin

4

),

L5 →√−g(

(∇φ)4Gµν∇µ∇νφ+2

3Lmin

5

), (5.10)

where g is the determinant of the metric, Lmini is the minimally coupled lagrangian

with covariant derivatives from (5.3), ∇φ is the covariant derivative of the scalarfield φ, R is the Ricci scalar and Gµν is the Einstein tensor. Interaction lagrangiansin (5.10) define the so called covariant galileon, the most symmetric generalizationof pure galileons coupled to gravity.

However, covariant galileons are not the most general theory exhibiting weaklybroken galileon invariance. We have built it by covariantizing a model with exactgalileon invariance. In fact, we can find a more general theory it by considering aHorndeski class:

LGG2 = Λ42G2(X),

LGG3 =Λ4

2

Λ33

G3(X)φ ,

LGG4 =Λ8

2

Λ63

G4(X)R + 2Λ4

2

Λ63

G4X(X)((φ)2 − [φ2]

),

LGG5 =Λ8

2

Λ93

G5(X)Gµν∇µ∇νφ− Λ42

3Λ93

G5X(X)((φ)3 − 3φ[φ2] + 2[φ3]

), (5.11)

where [φ2] ≡ ∂µ∂νφ∂ν∂µφ and [φ3] ≡ ∂µ∂νφ∂

ν∂ρφ∂ρ∂µφ. Gi are dimensionless func-

tions in terms of the dimensionless variable

X = − 1

Λ42

∂µφ ∂µφ. (5.12)

5.3. MOST GENERAL THEORY WITH WBG INVARIANCE 43

The fact that we found WBG models to belong to a subclass of Horndeski theoriesshould not be surprising. Horndeski lagrangian comprises the most general scalartensor theory in four dimensions with second order equations of motion (63), yet itdoes not posses a WBG invariance in its most general form. An EFT that exhibitsWBG symmetry will be the most invariant generalization of pure galileons in curvedspace. Therefore, the resulting model will have only second order derivatives in theirequation of motion. The main difference of model (5.11) with Horndeski theories isthat in the latter the free functions Gi are functions of both the field, φ, and thekinetic term, X. In the case of WBG, we cannot have an explicit dependence of Gi

on φ since that would inevitably spoil the approximate galileon invariance.

5.3 Most general theory with WBG invariance

Now that we have defined in what sense the galileon shift symmetry should beweakly broken, we are in a position to find the most general lagrangian exhibitingsuch symmetry. In the previous subsection we have covariantized galileons witha non-minimal coupling and checked that they correspond to a particular case ofHorndeski class models. However, this does not result in the most complete set ofoperators exhibiting WBG invariance. Finding the broadest WBG model is nontrivial at all since there are many combinations of the fields that contribute at thesame order of magnitude. To deal with this problem, Santoni et al. introduced apower counting rule for the EFT of WBG in (64) that is structurally robust. At theend of this section, we will see that the the final subset of operators are a sub-classof Horndeski and beyond-Horndeski theories.

The power count will be defined so all the operators that are equally importanton the background. This is essential because we want HD to play a role in themodel since they can achieve self acceleration. We are not introducing any potentialso a kinetic term would be incapable of accelerating the universe. We can writethe terms building the lagrangian as a combination of powers of (∇φ)2 and ∇∇φ ,with their corresponding suppressing scales. For the latter, the genuine scale of themodel, Λ3, since it is invariant under transformation (5.1). For the former, Λ2, asthe terms generated by loops that we previously discussed. As determined by (64)the power count is given by

(∇φ)2n

Λ4n−42

(∇∇φ)m

Λ3m3

. (5.13)

A theory built with operators satisfying this power count will not necessarily berobust. As we have been presenting throughout the chapter, in order to have WBGinvariance we need a specific combination of operators. Terms that do not satisfya non-renormalization property will have to be suppressed in comparison with therelevant terms. In table 5.1 above, taken from (64), the power counting rule is shownwith its corrections for different groups of operators.

The operators that fall within category OI are of the same order of magnitudeare the energy cutoff of the theory and must be considered. On the contrary, those

44 5. WEAKLY BROKEN GALILEONS

OI (∇φ)2n

Λ4(n−1)2

(∇∇φ)m

Λ3m3

δcI

cI∼ Λ3

MPl

OII (∇φ)2n

Λ4n2

(∇∇φ)m

Λ3m−43

δcII

cII∼ O(1)

OIII ∇m(∇∇φ)n

Λ3n+m−43

δcIII

cIII∼ O(1)

Table 5.1: The power counting of higher derivative EFTs.

combinations of operators in groups OII and OIII are also present in the EFT butstrongly suppressed.

The EFT power count (5.13) allows us to write the most general action up toquadratic order in ∇∇φ for operators in the class OI (64). To write all the termswe will write all possible contractions for terms satisfying the power count.

Starting with m = 0, we have:

S2 = Λ42

∫d4x√−g G2(X) , (5.14)

here G2(X) is a function of the parameter X.

The next term is given by m = 1. In this case, we could consider severalcontractions of (∇φ)2n∇∇φ. However, combinations like ∇µφ∇νφ∇µ∇νφ can berecast to (∇φ)2φ. Therefore, only the following term remains.

S3 = Λ42

∫d4x√−g G3(X)

φΛ3

3

, (5.15)

where G3(X) is a free function of X.

Finally, we will write the terms for m = 2. In the case, many contractions arepossible so we should look at them carefully to write a set of operators that areentitled to be a WBG. These terms are:

S4 =

∫d4x√−g[

Λ82

Λ63

G4(X)R +Λ4

2

Λ63

Cµν,ρσ∇µ∇νφ∇ρ∇σφ

], (5.16)

where,

Cµν,ρσ =α1(X)

2

(∇φ)2

Λ42

(gµρgνσ + gµσgνρ) + α2(X)(∇φ)2

Λ42

gµνgρσ

+α3(X)

2Λ42

(gρσ∇µφ∇νφ+ gµν∇ρφ∇σφ)

+α4(X)

4Λ42

(gνσ∇µφ∇ρφ+ gµσ∇νφ∇ρφ+ gνρ∇µφ∇σφ+ gµρ∇νφ∇σφ)

+α5(X)

Λ82

∇µφ∇νφ∇ρφ∇σφ , (5.17)

5.3. MOST GENERAL THEORY WITH WBG INVARIANCE 45

It can be seen that the tensor Cµν,ρσ satisfies the symmetry structure Cµν,ρσ =Cνµ,ρσ = Cµν,σρ = Cρσ,µν . As opposed to the actions (5.14) and (5.15), wherethe simple combination of terms that belong to the power count of the group OI

are WBG, this time we should check which combinations of the αi are valid verycarefully. Indeed, only a particular combination of operators will define a WBG. Inparticular, for loop with no graviton lines not to diverge αi(X) should verify thefollowing conditions:

α1 = −α2 , α3 = −α4 , α5 = 0 , (5.18)

On the other hand, going to loops with more graviton lines will require imposing:

4G4X + 2Xα2 +Xα3 = 0 (5.19)

6

Results I: background evolution

Expressions (5.14) - (5.17) together with conditions (5.18) and (5.19) define the mostgeneral theory exhibiting WBG invariance. For the sake of clarity, it is helpful toabsorb the energy scales into the Gi functions that from now on will be dimensionful.Now, variable X will be given by

X = −1

2∇µφ∇µφ . (6.1)

Under these considerations, we can write the most general WBG theory up toquadratic order in ∇∇Φ as:

S =

∫d4x√−g

(1

2M2

PlR +4∑i=2

Li + LSM

). (6.2)

Where g is the determinant of the metric gµν , MPl is the Planck mass, LSM isthe Lagrangian for matter and radiation and Li are given by

L2 = G2(X) , (6.3)

L3 = G3(X)φ , (6.4)

L4 = RG4(X) +G4X(X)((φ)2 − (∇µ∇νφ)(∇µ∇νφ)

)−F4(X)

((∇φ)2

((φ)2 − (∇µ∇νφ)(∇µ∇νφ)

)−2∇µφ∇νφ (∇µ∇νφφ−∇µ∇σφ∇σ∇νφ)

). (6.5)

Here, the subscript X on a function f denotes the partial derivative of f with re-spect to X. We have considered it convenient to rename function α3(X) by 2F4(X)in order to agree with the common notation found in literature. In fact, the termF4 corresponds to the usual beyond-Horndeski operator.

6.1 Dynamical equations

We are interested in the background evolution for which the scalar field is onlydependent on time. The equation of motion for the scalar field can be obtainedvarying the action with respect to φ,

46

6.1. DYNAMICAL EQUATIONS 47

δS

δφ= 0, (6.6)

that yields

36H3φ3F4(X) + 24HHφ3F4(X) + 36H2φ2φF4(X) + 9H3φ5F4X(X)

+ 6HHφ5F4X(X) + 27H2φ4φF4X(X) +3

2HφG2X(X) +

1

2φG2X(X)

− 9

2H2φ2G3X(X)− 3

2Hφ2G3X(X)− 3HφφG3X(X) + 9H3φG4X(X)

+ 6HHφG4X(X) + 3H2φG4X(X) + 3H2φ6φF4XX(X) +1

2φ2φG2XX(X)

− 3

2Hφ3φG3XX(X) + 9H3φ3(G4XX(X) + 6HHφ3G4XX(X)

+ 12H2φ2φG4XX(X) + 3H2φ4φG4XXX(X) = 0 , (6.7)

where dots represent derivatives with respect to cosmic time. The variable X corre-sponds to the one defined in equation (6.1) on the background. Typically that couldbe noted as X0, but we will omit the subscripts for the sake of clarity. It is worthmentioning that this equation is, as expected, second order in time derivatives, thusensuring we have only one propagating mode in the model.

On the other hand, varying the lagrangian with respect to the metric

δS

δgµν= 0, (6.8)

we obtain the Friedmann equations, as follows

3H2M2Pl = ρDE + ρm + ρr (6.9)

(3H2 + 2H)M2Pl = −pDE − ρr/3 , (6.10)

where

ρDE = −G2(X) + 2XG2X(X)− 6√

2HX3/2G3X(X)− 6H2G4(X)

+24H2XG4X(X) + 24H2X2G4XX(X)

+48H2X3F4X(X) + 120H2X2F4(X), (6.11)

pDE = G2(X) +√

2√XXG3X(X) +G4(X)(6H2 + 4H)− 12H2XG4X(X)

−4HXG4X(X)− 8HXG4X(X)− 8HXXG4XX(X)− 24H2X2F4(X)

−16HX2F4(X)− 32HXXF4(X)− 16HX2XF4X(X). (6.12)

These equations provide us with the background dynamics of the field and theexpansion history of the Universe.

48 6. RESULTS I: BACKGROUND EVOLUTION

6.2 Tracker solution

To study the cosmology of the WBG invariant theories we need to solve equations(6.7), (6.9) and (6.10). However, we are interested in a particular kind of solutions.In order for a solution to be observationally admissible it must behave as a De Sitteruniverse at late times. In this section we find attractor solutions that track a DeSitter cosmology, meaning, the equation of state of dark energy is wDE = −1 at latetimes.

In (65), De Sitter tracker solutions for galileons have been found under thecondition

Hφ2q = constant, (6.13)

with q a real constant, for a parametrization of the free functions as powers of X.These powers were chosen so that all terms in the Friedmann equations where pro-portional to Xp, being p another real constant.

Since the WBG theory can be recast to the extended Galileon by setting F4 tozero, it is reasonable to follow a similar approach when looking for tracker solutions.Let us firstly parametrize the free functions of our action (6.2) as

G2(X) = −c2Xp2M

4(1−p2)2

G3(X) = −c3Xp3M1−4p3

3

G4(X) = −c4Xp4M2−4p4

4

F4(X) = c5Xp5M−6−4p5

5 . (6.14)

Here, Mi are constants with dimensions of mass and ci and pi are dimension-less constants. Following the approach for extended galileon models, we can makeequation (6.9) proportional to Xp by setting

p2 = p, p3 = p+ q − 1

2, p4 = p+ 2q, p5 = p+ 2q − 2. (6.15)

To ensure there is a De Sitter fixed point, we can impose a new constraint on the freefunctions. Solving the Friedmann equations (6.9) and (6.10) at a De Sitter universe,where H = 0, X = 0, we obtain the following results

c2 = 3(1− 8c5 + 2c4(−1 + 2p+ 4q)), (6.16)

c3 =

√2

−1 + 2p+ 2q

(p+ 4c4(2p2 + q(−1 + 4q) + p(−1 + 6q))− 16c5(p+ q)

).

In order to get c2 and c3 non dependent on HDS, or XDS, we have defined themass terms, Mi, as follows:

6.2. TRACKER SOLUTION 49

M4(p−1)2 =

XpDS

H2DSM

2Pl

,

M4(p+q)−33 =

Xp+qDS

HDSM2Pl

,

M2−4(p+2q)4 =

M2Pl

Xp+2qDS

,

M2−4(p+2q)5 =

M2Pl

Xp+2qDS

. (6.17)

Once the values for the constants Mi have been defined we can get on the theconstraints we talked about in the previous section. An extra condition that relatesG4 and F4 can be imposed when setting the speed of tensor to 1. For shift symmetricoperators, that conditions reads (66):

2G4(X) = XF4. (6.18)

For the functions G4 and F4 above defined we find that

c5 = −2p

(1 +

2

s

)c4. (6.19)

The variables H, X, ΩDE and Ωr define the dynamics of the background. ΩDE

and Ωr are the dark energy and radiation energy density parameters that shall bedefined below.

Ωr =ρr

3H2M2Pl

, ΩDE =ρDE

3H2M2Pl

. (6.20)

At the De Sitter era, when ΩDE = 1, the values of the remaining parameters aregiven by (H,X,Ωr) = (HDS, XDS, 0). However, instead of working with the variablesH and X it is convenient to define the following parameters

r1 =HDSX

qDS

HXq, r2 =

(X

XDS

)(p+2q)

. (6.21)

In terms of these parameters, the dark energy density (6.11) and pressure (6.12)can be rewritten as

ρDE = − 1

r21

(3H2DSM

2Plr

pp+2q

2 (2c4(r1 − 1)(2p+ 4q − 1)(2p(r1 − 1)− 4q − r1 − 1)

+ 8c5(r1 − 1)(−2p(r1 − 1) + 4q + r1 + 1) + r1(2p(r1 − 1)− r1))) (6.22)

pDE = − 1

r31(p+ 2q)

H2DSM

2Plr− 2q

p+2q

2 (2c5(2p+ 4q − 1)(p(3r31r2 + 2r2

1r′2 − 3r1

r2 − 2r1r′2 + 2r′1r2) + 2q(3r3

1r2 + r21r′2 − r1(3r2 + r′2) + 2r′1r2))

−8c5(p(3r31r2 + 2r2

1r′2 − 3r1r2 − 2r1r

′2 + 2r′1r2) + 2q(3r3

1r2 + r21r′2

−r1(3r2 + r′2) + 2r′1r2)) + r21(p(3r1r2 + r′2) + 6qr1r2) (6.23)

50 6. RESULTS I: BACKGROUND EVOLUTION

The De Sitter era corresponds to (r1, r2,Ωr) = (1, 1, 0). Dark energy densityparameter can be written as

ΩDE = (−2c4(−1 + 2p+ 4q)(−1− 4q + 2p(−1 + r1)− r1)(−1 + r1)

+r1(−2p(−1 + r1) + r1)− 8c5(−1 + r1)(1 + 4q − 2p(−1 + r1)

+r1))r2. (6.24)

As it can be checked easily, along the tracker, when r1 = 1, we find that ΩDE → r2.Therefore, at the De Sitter fixed point ΩDE = 1.

In order to ensure observational compatibility, we require that the dark energydensity is not relevant at early times, when radiation is dominant. At early times,when 0 r1 1, the following condition must be verified (67):

p− 1

2q + 1≥ 0. (6.25)

Given the new variables, let us rewrite the dynamical equations (6.7), (6.9) and(6.10) in terms of r1, r2 and Ωr.

The dynamical evolution or r1 is given by

r′1r1

= − H

H2− qX

HX. (6.26)

where H and X can be obtained by solving equations (6.7) and (6.10). The primedenotes a derivative with respect to lna.

r′1r1

= [(−1 + r1)2c4(−1 + 2p+ 4q)(−2q + p(−1 + r1)) + pr1 + 8c5(p+ 2q − pr1)

(Ωr − 2Ωr p+ 3(1− 4q − 2c4r2 − 8c5r2 − 32c5qr2 + 32c4q2r2

+8c4p2(−1 + r1)2r2 + 8c4qr1r2 + 32c5qr1r2 − 32c4q

2r1r2 − r21r2 + 2c4r

21r2

+8c5r21r2 − 8c4qr

21r2 − 2p(1− 4c4(2q(−2 + r1)− r1)(−1 + r1)r2

+8c5(−1 + r1)2r2 + r1r2 − r21r2)))]/∆, (6.27)

where

∆ = 2[pr1(−1 + 2q + r1 + p(2 + r1(−2 + r2))) + 64c25(2q(−1 + 2q(−2 + r1))

+2p2(−1 + r1) + p(1 + 8q)(−1 + r1))(−1 + r1)r2 + 4c24(−1 + 2p+ 4q)2

(2q(−1 + 2q(−2 + r1)) + 2p2(−1 + r1) + p(1 + 8q)(−1 + r1))(−1 + r1)r2

−2c4(−1 + 2p+ 4q)(−2q(−1 + r1)(−1 + 16c5r2) + 4q2(−2 + r1)(−1

+16c5(−1 + r1)r2) + 2p2(1 + r1)(−1 + r1 − 16c5r2 + (−1 + 16c5)r1r2)

+2pq(4 + 64c5r2 + 2(−1 + 32c5)r21r2 + r1(−4 + (3− 128c5)r2)) + p(−1 + r1)

(1− 16c5r2 + r1(−1 + (−1 + 16c5)r2)))− 8c5(2q(1 + 2q(−2 + r1)− r1)

+2p2(−1 + r1)(1 + r1(−1 + r2)) + p((−1 + r1)(−1 + r1 + r1r2)

+q(−8 + 8r1 − 6r1r2 + 4r21r2)))]. (6.28)

6.2. TRACKER SOLUTION 51

A crucial factor we ought to take into consideration is the stability of the solutionto check that it is, indeed, a tracker. As we can easily check, the evolution equationfor the parameter r1 has a fixed point for r1 = 1 that defines the tracker. For thissolution to be a real attractor we must check that the solution does not abandonthe tracker when perturbed. A homogeneous perturbation of equation (6.28) aboutthe fixed point yields

δr1

δr1

= −6(p+ 2q)− 3 + (2p− 1)Ωr + 3r2

2(pr2 + 2q). (6.29)

Since we require the theory to be local and, thus, p, q > 0, the tracker solutionwill be stable only if

6(p+ 2q)− 3 + (2p− 1)Ωr + 3r2 > 0. (6.30)

On the other hand, the first Friedmann equation, written as the continuity equa-tion gives the following

Ω′rΩr

=2qr′2

r2(p+ 2q)− 4. (6.31)

Finally, the second Friedmann equation yields

0 = (4c4p2r2(−3r1r2 + 3r3

1r2 + 2r′1r2 − 2r1r′2 + 2r2

1r′2) + pr2(3(1− 8c5

+2c4(−1 + 8q))r31r2 + 2r′1(1− 8c5r2 + 2c4(−1 + 8q)r2) + (1− 16c5

+4c4(−1 + 6q))r21r′2 − r1(3 + Ωr − 24c5r2 − 16c5r

′2

+c4(−6r2 + 48qr2 − 4r′2 + 24qr′2))) + 2q(3(1− 8c5 + c4(−2 + 8q))r31r

22

+2r′1r2(1− 8c5r2 + 2c4(−1 + 4q)r2)− 2(c4 + 4c5 − 4c4q)r21r2r

′2

+r1(6(c4 + 4c5 − 4c4q)r22 + r′2 − r2(3 + ΩR − 2c4r

′2 − 8c5r

′2

+8c4qr′2)))) (6.32)

Along the De Sitter tracker we find

r1 = 1 , (6.33)

r′2r2

= −(p+ 2q)(−3− Ωr + 3r2)

2q + pr2

, (6.34)

Ω′rΩr

=2(q(−1 + Ωr − 3r2)− 2pr2)

2q + pr2

(6.35)

Equations (6.35) and (6.35) are of great relevance since they determine the cos-mological dynamics along the tracker. They can be solved to find out the evolutionof the cosmological parameters. Equations (6.34) and (??) can be combined to give

r′2r2

− (1− s

2)Ω′rΩr

= 4− 2s , (6.36)

wheres =

p

q. (6.37)

52 6. RESULTS I: BACKGROUND EVOLUTION

The evolution of the parameter r2 yields

r2 = C a4+2s Ω1+s/2r , (6.38)

where C is a constant and a is the scalar factor.

We should impose that the parameter r2 is subdominant at early times sinceit will behave as dark energy along the tracker. Considering that Ωr ∼ ρrH

−2 ∼a−4H−2 and recalling expression (6.38) a simple calculation will show r2 ∼ H−2−s.Then, for r2 to be subdominant at early times we require

2 + s > 0 (6.39)

Taking the result in equation (6.38), the evolution for Ωr can be obtained fromequation (??)

C a4(1+s)Ω1+sr = 1− Ωr(1−Da) , (6.40)

where D is a constant.

The background solution we have obtained for the WBG model does not differat all from the general extended galileon model extensively studied in (65), (67) onthe background. This coincidence is due to the fact that all terms proportional tothe constants c4 and c5, where we have introduce all our modifications, are cancelledalong the De Sitter tracker.

Regarding the equation of state of dark energy, wDE = ρDE/pDE along thetracker, it can be computed by setting r1 = 1 and r′1 = 0 in equations (6.22) and(6.23), which yields

wDE = −3 + (3 + Ωr)s

3(sr2 + 1). (6.41)

We can check that at early times, when r2 → Ωr 1, we have wDE '−1− s(1 + Ωr/3). Note that at the De Sitter fixed point ((r1, r2,Ωr) = (1, 1, 0)) theequation of state of dark energy tracks a De Sitter universe given by wDE = −1.

One final consideration we must consider regards observational compatibility ofthe equation of state of dark energy. Many modified gravity models are charac-terized by having phantom dark energy, wDE < −1 at some times. In reference(65) the authors study different models of extended galileons for several values ofthe parameter s. By analyzing the behaviour of wDE with several initial conditionsthey conclude that some values of s will produce phantom dark energy that is notallowed observationally. Despite being stable along the tracker, this phantom darkenergy abruptly changes its equation of state at late times what is disfavoured byobservations. Therefore, we must impose a final constraint

s < 2. (6.42)

6.2. TRACKER SOLUTION 53

Summary of the constraints on parameters s and p

As we can see, the background dynamics are merely defined by the parameters p andq, which can be reshaped in the parameter s = p/q. The parameter space can beconstrained by making some considerations regarding observational compatibility.in the following list we sum up all the necessary condition p and s must satisfy.

• Local theory

p, q > 0

• Dark energy density not relevant at early times, condition (6.25).

p− 1

2q + 1≥ 0

• Parameter r2 subdominant at early times, condition (6.39).

s > −2

• Tracker stability, condition (6.30).

6(p+ 2q)− 3 + (2p− 1)Ωr + 3r2 > 0.

• Observational compatibility of the equation of state of dark energy, condition(6.42)

s < 2

Taking all of these constraints into consideration we find:

0 < s < 2 (6.43)

p ≥ 1 (6.44)

Summary of the constraints on parameters c2, c3, c4, c5, XDS

There are four constraints that can be used to constrain the parameter space:

1. The expansion history of the universe is characterized for having a De Sitterattractor. Therefore, the Friedmann equations set two constraints for thiscondition to be satisfied, shown in equation (6.16).

2. The field, φ, has scaling degeneracy: it can be arbitrarily rescaled by a con-stant. This allows us to fix on of th parameters above to an arbitrary number.

3. The speed of tensor can be set to unity using relation (6.18), yielding condition(6.19).

54 6. RESULTS I: BACKGROUND EVOLUTION

6.3 Expansion history along the tracker

The expansion history of the universe is given by the Hubble parameter evolutionwhich we can derive from the first Friedmann equation,

3H2M2Pl = ρm + ρr + ρDE. (6.45)

To study the phenomenology of the model, we are interested in solutions alongthe tracker, r1 = 1. Therefore, taking equation (6.22) for the dark energy densityevolution and applying the condition (6.17) for a De Sitter fixed point we find

Ωφ =

(HDS

H

)2+s

. (6.46)

Thus,3H2M2

Pl = 3H−sH2+sDS M

2Pl + ρm + ρr (6.47)

Imposing the flatness condition, Ωm + Ωr + ΩDE = 1, at present time allows usto find an expression for HDS in terms of cosmological parameters, as follows.

HDS = H0 (1− Ωm,0 − Ωr,0)1

2+s (6.48)

Recalling that ρm ∼ ρm,0a−3 and ρr ∼ ρr,0a

−4, together with expression (6.48)the expansion history for the background will be given by the algebraic equation

1−(H0

H

)2(Ωm,0

a+

Ωr,0

a2

)−(H0a

H

)2+s

(1− Ωm,0 − Ωr,0) = 0. (6.49)

Where H is the Hubble parameter in conformal time, τ , namely,

H =da

dτ

1

a(6.50)

6.3. EXPANSION HISTORY ALONG THE TRACKER 55

7

Results II: numerical analysis

In chapter 6 we derived the equations of motion of the scalar field and the modifiedFriedmann equations. We have found a particular set of solutions with a De Sitterattractor whose stability and observational compatibility have allowed us constrainthe parameters of the theory. The next step in our analysis is to study the phe-nomenology of the model by computing cosmological observables with numericaltools, as described in chapter ??. As we introduced then, the Einstein-Boltzmannsolver eftcamb has already implemented the EFT of dark energy approach. Inorder to study the phenomenology of WBG we have created a new patch to thecode that follows the EFT full mapping approach described in chapter ??.

In this chapter we recap on the theoretical set up needed for the implementationof WBG into eftcamb and discuss the results obtained with the code.

7.1 Theoretical set up

eftcamb works in the EFT framework of dark energy. Working in the full mappingapproach requires providing the code with an EFT mapping and the background evo-lution, which will allow us to compute the cosmological observables. The expansionhistory was derived in the previous chapter and it is shown in equation (6.49). Thatalgebraic equation must be solved numerically for the Hubble parameter for eachvalue of the parameter s, the only parameter that enters in the background solution.Let us recall that the aforementioned expression already includes the imposition ofhaving a De Sitter tracker.

Regarding the mapping to the EFT language, we will set our starting point atthe general one computed in (25) for Horndeski and GLPV theories, that can befound in Appendix A. Since the WBG action is made of a Horndeski sector, consist-ing of well-known galileons, and a beyond Horndeski term we can use the generalmapping and specify it for our particular choice of the free functions, Gi, in (6.14)with conditions (6.15) and (6.16).

The EFT mapping we have implemented in eftcamb is the following:

• LWBG2

56

7.1. THEORETICAL SET UP 57

Λ(a)WBG = −c2H2DSM

2PlX

−pDSH(a)2pφ′(a)2p ,

c(a)WBG = −c2H2DSM

2PlpX

−pDSH(a)2pφ′(a)2p ,

M42 (a)WBG = −c2H

2DSM

2Pl(−1 + p)pX−pDSH(a)2pφ′(a)2p ,

• LWBG3

Λ(a) = − 1

as

√2c3HDSM

2Pl(−s+ 2p(1 + s))X

−p(1+ 1s

)

DS

H(a)−1+(2p(1+s))/sφ′(a)−1+2p(1+s)

s (H(a)φ′(a)

+aH(a)2φ′′(a)) ,

c(a)WBG = − 1√2as

c3HDSM2Pl(−s+ 2p(1 + s))X

−p(1+ 1s

)

DS H(a)−1+2p(1+s)

s

φ′(a)−1+2p(1+s)

s (H(a)φ′(a) +H(a)2(−3φ′(a) + aφ′′(a))) ,

M42 (a)WBG =

1

2√

2as2c3HDSM

2Pl(−s+ 2p(1 + s))X

−p(1+ 1s

)

DS H(a)−1+2p(1+s)

s

φ′(a)−1+2p(1+s)

s (sH(a)φ′(a) +H(a)2(6(p− s+ ps)φ′(a)

+asφ′′(a))) ,

M31 (a)WBG = −

√2c3HDSM

2Pl(−s+ 2p(1 + s))X

− p(1+s)s

DS H(a)2p(1+s)

s φ′(a)2p(1+s)

s

s

• LWBG4,H (Horndeski sector)

Ω(c)WBG = −1− 2c4X− p(2+s)

sDS H(a)

2p(2+s)s φ′(a)2p(2+s)s ,

c(a)WBG =1

a2s22c4M

2Plp(2 + s)X

− p(2+s)s

DS H(a)−2+2p(2+s)

s φ′(a)−2+2p(2+s)

s ((−s

+2p(2 + s))H(a)2φ′(a)2 + sH(a)H(a)φ′(a)2 +H(a)2H(a)φ′(a)

((−2s+ 4p(2 + s))φ′(a) + a(s+ 4p(2 + s))φ′′(a)) +H(a)4((s

−6p(2 + s))φ′(a)2 + a2(−s+ 2p(2 + s))φ′′(a)2 + aφ′(a)((−3s

+4p(2 + s))φ′′(a) + asφ′′′(a)))) ,

Λ(a) =1

a2s24c4M

2Plp(2 + s)X

− p(2+s)s

DS H(a)−2+2p(2+s)

s φ′(a)−2+2p(2+s)

s ((−s

+2p(2 + s))H(a)2φ′(a)2 + sH(a)H(a)φ′(a)2 + (s+ 4p(2 + s))

H(a)2H(a)φ′(a)(φ′(a) + aφ′′(a)) +H(a)4(sφ′(a)2 + a2(−s+2p(2 + s))φ′′(a)2 + aφ′(a)(4p(2 + s)φ′′(a) + asφ′′′(a)))) ,

(7.1)

58 7. RESULTS II: NUMERICAL ANALYSIS

M42 (a)WBG = − 1

a2s3c4M

2Plp(2 + s)X

− p(2+s)s

DS H(a)−2+2p(2+s)

s φ′(a)−2+2p(2+s)

s (s(−s

+2p(2 + s))H(a)2φ′(a)2 + s2H(a)H(a)φ′(a)2 + sH(a)2H(a)φ′(a)

((−2s+ 4p(2 + s))φ′(a) + a(s+ 4p(2 + s))φ′′(a)) +H(a)4(2(2s2

−9ps(2 + s) + 6p2(2 + s)2)φ′(a)2 + a2s(−s+ 2p(2 + s))φ′′(a)2 +

asφ′(a)((−3s+ 4p(2 + s))φ′′(a) + asφ′′′(a)))) ,

M31 (a)WBG = (1/(as2))4c4M

2Plp(2 + s)X

− p(2+s)s

DS H(a)−1+2p(2+s)

s φ′(a)−1+2p(2+s)

s

(sH(a)φ′(a) +H(a)2((−2s+ 4p(2 + s))φ′(a) + asφ′′(a))) ,

M22 (a)WBG =

1

s

(4c4M

2Plp(2 + s)X

− p(2+s)s

DS H(a)2p(2+s)

s φ′(a)2p(2+s)s

),

M23 (a)WBG = −M2

2 (a)WBG ,

M24 (a)WBG =

1

2M2

2 (a)WBG

• LWBG4,bH (beyond Horndeski sector)

c(a)WBG = − 1

a2s8c5M

2PlX

− p(2+s)s

DS H(a)2p(2+s)

s φ′(a)−1+2p(2+s)

s ((s+ 2p(2 + s))

H(a)φ′(a) +H(a)2(−7sφ′(a) + 2ap(2 + s)φ′′(a))) ,

Λ(a)WBG = − 1

a2s8c5M

2PlX

− p(2+s)s

DS H(a)2p(2+s)

s φ′(a)−1+2p(2+s)

s (2(s+ 2p(2 + s))

H(a)φ′(a) +H(a)2(sφ′(a) + 4ap(2 + s)φ′′(a))) ,

M42 (a)WBG =

1

a2s4c5M

2PlX

− p(2+s)s

DS H(a)2p(2+s)

s φ′(a)−1+2p(2+s)

s ((s+ 2p(2 + s))

H(a)φ′(a) + 2a(−2s+ p(2 + s))H(a)φ′′(a) + sH(a)2(11φ′(a)

+4aφ′′(a))) ,

M22 (a)WBG = −8c5M

2PlX

− p(2+s)s

DS H(a)2p(2+s)

s φ′(a)2p(2+s)

s ,

M23 (a)WBG = −M2

2 (a)WBG ,

M24 (a)WBG =

1

2M2

2 (a)WBG

7.2 Numerical results of WBG

In this section we show the numerical results of the implementation of WBG intoeftcamb. We will start by considering the evolution of the Hubble parameter,H(a), and the variable χ(a).

The evolution of the Hubble parameter is given by equation (6.49). Given achoice of cosmological parameters, the background evolution only depends on thevalue of s. As one can see in figure 7.1, the background expansion history can beslightly modified by WBG with respect to LCDM. We can see that WBG can gen-erate lower values of H, which could mitigate the H0 tension. At early times, when

7.2. NUMERICAL RESULTS OF WBG 59

a→ 0, the Hubble parameter evolution mimics LCDM very well, as dark energy isa subdominant component. Departures from there are more relevant as the scalefactor grows but it always ends up tracking the current value of H0 for a LCDMUniverse. Note that the position of the minimum, the point in time when the ac-celeration starts, does not move.

Figure 7.1: Hubble parameter evolution for different background solutions, on the left, andits time derivative on the right.

Regarding the evolution of χ, shown in figure 7.2, we can see that it stronglydepends on both s and p.

Figure 7.2: χ evolution for different choices of s and p.

The remaining results can be listed as follows: study of stability in the param-eter space, computation of the EFT functions and of power spectra of cosmologicalobservables.

We separate the numerical results into two main cases. Firstly, we will addressthe cosmology for the WBG without the beyond Horndeski sector, namely c5 = 0,that we call Case I. The results including c5 6= 0 are discussed afterwards as Case II.

60 7. RESULTS II: NUMERICAL ANALYSIS

For each of them we will discuss the stability of the model, the EFT functions andcompute cosmological observables. The effect of the additional constraint, cT = 1,is also studied.

Case I: Horndeski sector of WBG.

In this case we set c5 = 0 to switch off the beyond Horndeski operators. There-fore, the parameters that are relevant in Case I are s, p and c4.

Stability map

Since the HD operators of the WBG are prompted to introduce instabilities, wewill start our discussion with the numerical analysis of the stability of WBG. Inparticular, we will look at the dependence of the stability of the models with theparameters s and p for fixed values of c4. The stability map is shown in figure??. One can observe that stable models gather in regions of low p. For instance,c4 = 0.05 and c4 = 0 have the following stability maps.

Figure 7.3: Stability map for WBG models with c5 = 0. The figure on the left has setc4 = 0.05, while the figure on the right has c4 = 0, value that corresponds to the case whencT = 1. Points outside this map and below p=10 are either unstable tracker solutionss,disfavoured by observations or produce ghost or gradient instabilities.

The study of the stability map for different values of c4 shows that the largerthe value of c4 is, the smaller the area of the stable parameter space becomes. Infact, we have determined that models with c4 > 1 do not present stable models,at least for the explored values of s and p. It is interesting to note that modelswith c4 = c5 = 0, corresponding to the right plot, belong to the set of WBG thatsatisfies the relation ct = 1 given by equation (6.19). In that case we can also observethat the dependence of the stability is not that sensitive to s as in the case whenc4 = 0.05.

7.2. NUMERICAL RESULTS OF WBG 61

EFT functions.

We have computed the evolution of the EFT functions for a choice of parameterswhich gives a stable theory. Figures 7.4 and 7.5 depict the background and secondorder EFT functions, respectively, for different values of s and c4. These plots tellus how the time-dependent coefficients of the action in EFT language (3.4) evolve.The solid lines correspond to models with c4 6= 0 for different backgrounds whilethe dashed line corresponds to the case c4 = 0. The latter is particularly interestingbecause it satisfies the cT = 1 constraint.

Figure 7.4: Background EFT functions for WBG sampling the parameter s with fixedp = 1 and c5 = 0.

62 7. RESULTS II: NUMERICAL ANALYSIS

Figure 7.5: Second order EFT functions for WBG sampling the parameter s with fixedp = 1 and c5 = 0.

Regarding the background EFT functions in figure 7.4, we can see that differentvalues of s modify their evolution slightly. Unlike s, the value of c4 is more sensitiveand can introduce remarkable differences as it can be observed for c4 = 0 plottedwith a dashed line. For the sake of clarity, we have not included variations of theparameter s for the latter since they only represent small deviations.

Departures of the EFT functions from zero indicate deviations from general rel-ativity. In fact, at early times, when dark energy is subdominant, all EFT functionsapproach zero. It is interesting to note that by looking at the EFT functions we canidentify some of the features of the model. For example, Ω modifies the minimalcoupling to gravity to a non minimal one. As we can see from the contribution of L4

in expression (6.5), the non minimal coupling for WBG is controlled by c4. Indeed,the EFT function Ω in figure 7.12 grows when c4 6= 0 but stays flat at zero for c4 = 0as consequence of switching off the non-minimal coupling.

Power spectra

We focus on the CMB anisotropies of TT modes and matter power spectrum andstudy how each of the parameters affect the observables. Figure 7.6 shows how thepower spectra behaves when changing the value of parameter s, the only parameterthat enters in the background, while keeping p = 1 and c4 = 0.05.

At large scales, this plot reveals that values of s close to s = 2 could be favouredby observations since WBG can lower the ISW tail with respect to LCDM. Thereader may notice that at small scales the percentage error oscillates and deviatesfrom LCDM remarkably. Of course, this behaviour is unwanted since our EFT is

7.2. NUMERICAL RESULTS OF WBG 63

Figure 7.6: Phenomenology of the WBG for different values of the parameter s, thatcontrols the background. The remaining parameters are fixed to p = 1, c4 = 0.05 and c5 =0. The figure on the left represents the CTTl of the power spectrum of the CMB anisotropies.On the right we can see the matter power spectrum. The dashed line represents LCDMbest fit with Planck 2015 data (30).

designed to modify gravity in the cosmological scales and general relativity describesaccurately those scales. This disagreement at large modes is due to the fact that weare using Planck ’s best fit to LCDM for both LCDM and WBG. We claim that theWBG best fit will agree on those scales with LCDM. This reasoning applies to theremaining power spectra analyzed in this chapter.

Not only modifications of the background can introduce changes. If insteadwe vary the parameter p while setting s and c4 fixed, we can also see importantdepartures from LCDM.

Figure 7.7: Phenomenology of the WBG for different values of the parameter p. Theremaining parameters are fixed to s = 1.8, c4 = 0.05 and c5 = 0. Figure on the leftrepresents the CTTl of the power spectrum of the CMB anisotropies. On the right we cansee the matter power spectrum. The dashed line represents LCDM best fit with Planck2015 data (30).

Finally, we computed the power spectra for fixed s = 1.8, p = 1 and change thevalue of c4, displayed in figure 7.8.

64 7. RESULTS II: NUMERICAL ANALYSIS

Figure 7.8: Phenomenology of the WBG sampled by parameter c4. The remaining param-eters are fixed to s = 1.8, p = 1 and c5 = 0. Figure on the left represents the CTTl ofthe power spectrum of the CMB anisotropies. On the right we can see the matter powerspectrum. The dashed line represents LCMD best fit with Planck 2015 data (30).

If we look again at the ISW tail, we can see that c4 = 0.05 can account for a25% difference with respect to LCDM, considerably lowering the tail at those scales.Note that the line for c4 = 0 is the case that satisfies condition (6.19) and thusverifies cT = 1. We have studied this particular case, that corresponds to a cubicWBG, in more detail. In the figure below we show the power spectra for c4 = 0 andc5 = 0 for several combination of s and p.

Figure 7.9: Phenomenology of the WBG for different background evolution sampled byparameter s and p. The remaining parameters are fixed to c4 = 0 and c5 = 0. Figure onthe left represents the CTTl of the power spectrum of the CMB anisotropies. On the rightwe can see the matter power spectrum. The dashed line corresponds to LCDM best fit withPlanck 2015 data (30).

7.2. NUMERICAL RESULTS OF WBG 65

The interesting feature about the left plot is that the percentage error differs foreach WBG, unlike in the previous cases, due to the fact that we consider differentbackgrounds.

Case II: Beyond Horndeski sector of WBG.

Let us consider models with a beyond Horndeski term. These theories are char-acterized the parameters s, p c4and c5.

Stability Map

Due to the large number of free parameters we study the stability fixing the valuesof s and p. From the previous cases we have learnt that stable models seem to be inregions of the parameter space close to p = 1 so as a case of study we will fix thatvalue.

Figure 7.10 shows the stability map for WBG in the c4− c5 parameter space fors = 2 and p = 1. Even though tracker solutions are not viable for s = 2, we plot thestability map for that choice of parameters since it corresponds to the case wherethere are the largest number of stable models. One can observe that it extendsinto positive and negative regions for a reduced range of parameters. The straightblue line comprises all the models that satisfy cT = 1 and it strongly constrains theparameter space.

Figure 7.10: Stability map for WBG models with c4 6= 0 and c5 6= 0 for s = 2 and p = 1.Points outside this map are either unstable tracker solutions, disfavoured by observationsor produce ghost or gradient instabilities.

We can check that the stability is very sensitive to the parameters s and p. Inthe left panel of figure 7.11, we show the stability map for a change from s = 2 tos = 1.9 and the reduction of the stable area is remarkable, note the change in theaxes scale. The same is equivalent for a change in p = 1.2. We have checked thatp > 1.1 does not have stable models with c5 6= 0

66 7. RESULTS II: NUMERICAL ANALYSIS

Figure 7.11: Stability map for WBG models with c4 6= 0 and c5 6= 0. The plot on the left isgiven by the parameter choice s = 1.9, p = 1; while the once on the right is characterized bys = 1.9, p = 1.1. Points outside this map are either unstable tracker solutions, disfavouredby observations or produce ghost or gradient instabilities.

EFT functions

In the following figures, we plot the EFT functions of stable models for differentparameter choices. The dashed lines correspond to the cT = 1 case.

Figure 7.12: Background EFT functions for WBG with s = 1.9 and p = 1 for c4 6= 0 andc5 6= 0.

The figures below show that even small changes in the parameter space canintroduce relevant modifications in some of the EFT functions of the model.

7.2. NUMERICAL RESULTS OF WBG 67

Figure 7.13: Second order EFT functions for WBG with s = 1.9 and p = 1 for c4 6= 0 andc5 6= 0.

Power spectra

Finally, we study the phenomenology of WBG with the beyond Horndeski sectorincluded.

Figure 7.14: Phenomenology of the WBG sampled by parameter c4 and c5. The remainingparameters are fixed to s = 1.9 and p = 1. Figure on the left represents the CTTl ofthe power spectrum of the CMB anisotropies. On the right we can see the matter powerspectrum. The dashed line represents LCDM best fit with Planck 2015 data (30).

68 7. RESULTS II: NUMERICAL ANALYSIS

Looking at the CMB anisotropies power spectrum we can notice that certainchoices of WBG free parameters have the ability to lower the ISW tail. This featureis attractive since LCDM best fit at large scales is slightly too high in comparisonwith Planck ’s observations. For this reason, it would be very interesting to fit WBGpredictions to data in future work. To illustrate this idea better, in figure 7.15, weplot Planck 2015 observations of CTT

l together with LCDM best fit and a few WBGmodels.

Figure 7.15: TT modes of the CMB anisotropies. Solid lines correspond to WBG models,the dashed line represents LCDM best fit to Planck 2015 data (30), which is given by thegrey dots and error bars.

7.2. NUMERICAL RESULTS OF WBG 69

8

Conclusions and outlook

There is strong observational evidence that supports the accelerated expansion ofthe universe. To explain cosmic acceleration, the standard model of cosmologyrelies on the simplest candidate for dark energy: a cosmological constant (CC). Thismodel has been named LCDM since L stands for the CC, and CDM for cold darkmatter. However, this model shows some inconsistencies both at the theoreticaland observational level. From the theory side, the cosmological constant suffersfrom fine tuning problems and lacks a theoretical explanation. On the observationalside, the tension in the present value of the Hubble parameter when measured fromlocal surveys and from the CMB indicates that there might be some physics thatwe are missing. Modified Gravity (MG) is proposed to explain cosmic accelerationwithout relying on the CC. In this thesis, we have focused on a particular class ofMG models: scalar-tensor theories, in which a new degree of freedom in the form ofa scalar field is added to the Einstein-Hilbert action. Naturally, this modificationmust be relevant only at large scales, where the expansion is happening. MG theoriesmust be screened at small scales at which many local tests have proved the successof general relativity.

The most useful and convenient language to study dark energy is the effectivefield theory (EFT) framework since it provides with a universal language to whichevery model can be mapped. Therefore, the EFT of dark energy offers a globalview of the overwhelmingly large plethora of MG theories. In this thesis, we haveconstructed a model of MG based on: (1) the addition of a scalar field that mediatesthe expansion and (2) the invariance under galileon transformations. The latter con-dition is imposed in order to have second order equations of motion and being stableunder quantum corrections. As we saw, galileons cannot be, in general, coupled togravity without losing the invariance. In references (62; 64), the authors proposed apower count for theories with higher derivatives that exhibit weakly broken galileonsymmetry. These models are characterized by an energy cutoff above which thesymmetry breaking operators are suppressed, hence the term weakly broken.

In chapter 5 we built the most general action for a WBG that turned out tobelong to a subclass of Horndeski theories with a beyond-Horndeski term. Themodel is written in terms of 4 free functions and 4 parameters that account forthe parametrization of the aforementioned free functions. The equations of motion

70

71

for the scalar field as well as the modified Friedmann equations were derived inchapter 6 were we found attractor solutions that track a De Sitter universe at latetimes. The presence of tracker solutions is very attractive since it prevents fromfine tuning issues and coincidence problem. From this theoretical outcome we wereable to derive the background expansion history of the WBG along the tracker andthe mapping from WBG to the EFT language of dark energy. These two are theonly ingredients we need to implement the model into the Einstein-Boltzmann solvereftcamb.

We have created a new patch in the eftcamb code that we have used in or-der to study the stability of the model, compute the EFT functions and, moreimportantly, cosmological observables such as the power spectra of temperature-temperature CMB and matter distribution. Our theoretical outcome indicates thatWBG can offer realistic predictions of cosmological observables while, at the sametime, account for measurable differences. The results are promising since the modelcan lower the ISW tail potentially favoured by observations, though this claim mustbe carefully studied with the proper statistical analysis.

Future work involves the implementation of the WBG patch of eftcamb into aMarkov Chain Monte Carlo to fit the model to data and constrain the parametersof the theory observationally.

Acknowledgements

I would like to express my immense gratitude to Alessandra. Thank you for beingsuch a friendly and supportive supervisor. Your passion for science is inspirationaland encouraging. Thanks for welcoming me to your group with open arms.

To Jorgos and Simone, whose help has been absolutely essential for the successof this project. Thank you for being there every time that I had questions andfor your kindness and patience. Thanks for making me feel so comfortable and fortreating me as a friend, you have made this project fascinating and fun. I could nothave had better supervisors, you will be great mentors!

To my wonderful office mates, with whom I have shared countless laughs andpanic attacks, evenly distributed. Sara, Banafshe, Koen and Rafa: you have madegood times memorable and stress a bit more bearable. Thanks to my friend Guadalu-pe, for taking care of me when I needed it the most, for being an unconditional friendand, mainly, for discovering zumba to me (no sera lo mismo sin tı).

To all the friends I have made in Leiden: you are the reason this experience hasbeen unforgettable. To my friends Joana, Stefano, Nicolas and Dyon, thanks formaking these two years absolutely amazing!

I would like to thank every single member of the cosmology group for makingme feel part of the group from the very beginning. What I have learned from youis invaluable.

A mis padres: me habeis dado todo lo que estaba a vuestro alcance y mas.Gracias por convertirme en la persona que soy hoy. Gracias por ensenarme en valorde la educacion y a quererme a mi mismo. Gracias por apoyarme en todas lasdecisiones que he tomado en mi vida y recordarme que siempre tendre un lugar alque volver. Jamas podre expresar lo importantes que sois para mı. A mi hermano:aunque no me llamas por telefono se que me quieres igual. Nadie como tu hace quesaque lo mejor de mı mismo.

72

73

Appendix A

Full EFT mapping for Galileonsand GLPV theories

In this appendix we show the mapping to the EFT language of galileons and GLPVmodels. The mapping below has been extracted from (25) but the notation has beenchanged to agree with the one used throughout this document.

A.1 Generalized galileon mapping

Generalized galileons can be mapped to EFT with the following mapping.

where X0 is X0 = −H2φ′20 .

• L2 = G2(X)

Λ(a) = G2, (A.1)

c(a) = G2XX0 , (A.2)

M42 (a) = G2XXX

20 , (A.3)

• L3 = G3(X)φ

Λ(a) = H2φ′20

[G3φ − 2G3X

(Haφ′0 +H2φ′′0

)], (A.4)

c(a) = H2φ′20

[G3X

((3H2 − H

) φ′0a−H2φ′′0

)+G3φ

], (A.5)

M42 (a) =

G3X

2H2φ′20

((3H2 + H

) φ′0a

+H2φ′′0

)− 3H6

aG3XXφ

′50

−G3φX

2H4φ′40 , (A.6)

M31 (a) = −2H3G3Xφ

′30 . (A.7)

• L4 = RG4(X) +G4X(X) ((φ)2 − (∇µ∇νφ)(∇µ∇νφ))

74

A.2. GLPV MAPPING 75

Ω(a) = −1 +2

m20

G4 , (A.8)

c(a) = G4X

[2(H2 +HH + 2H2H − 5H4

)φ′ 20 a

2 + 2(

5H2H +H4)φ′0aφ

′′0

+2H4φ′′ 20 + 2H4φ′0φ′′′0

]+G4Xφ

[2H2φ′20

(Haφ′0 +H2φ′′0

)+ 10

H4

aφ′30

]+G4XX

[12H6

a2φ′40 − 8

H4

aφ′30

(Haφ′0 +H2φ′′0

)−4H2φ′20

(H2

a2φ′20 + 2

HH2

aφ′0φ

′′0 +H4φ′′20

)](A.9)

Λ(a) = G4X

[4(4

+ 52˙+˙2 +)φ′ 20 a

2 + 4(

44 + 52)φ′0aφ

′′0 + 44φ′′ 20 + 44φ′0φ

′′′0

]+8H4

aG4Xφφ

′30 − 8G2

4XXφ′ 20

(φ′0a+2 φ′′0

)(22φ′0a+ φ′0a+2 φ′′0

), (A.10)

M42 (a) = G4Xφ

[4H4

aφ′30 −H2φ′ 20

(Haφ′0 +H2φ′′0

)]− 6H6

aφ′50 G4φXX − 12

H8

a2G4XXXφ

′60

+G24XXφ

′ 20

[2(

94 +˙2 + 22)φ′ 20 a

2 + 2(

22˙+ 24)φ′0aφ

′′0 + 24φ′′ 2

]+G4X

[(− 22 + 24 − ˙2 −

)φ′ 20 a

2 −(4

+ 52)φ′0aφ

′′0 −4 φ′′ 2 −4 φ′0φ

′′′0

], (A.11)

M31 (a) = 4G4Xφ

′0

[(+ 22

)φ′0a+H2φ′′0

]− 16G4XX

H5

aφ′40 − 4G4XφH3φ′30 , (A.12)

M22 (a) = 4H2G4Xφ

′20 = −M2

3 (a) = 2M2(a) . (A.13)

A.2 GLPV mapping

• LGLPV =

c(a) = 2H4

a2φ′40 (H − H2)F4 + 8

H4

aφ′30 F4

(Haφ′0 +H2φ′′0

)−4H6

aF4X

(Haφ′0 +H2φ′′0

)φ′50 + 2H6 F4φ

aφ′50 − 12

H6

a2φ′40 F4 , (A.14)

Λ(a) = 6H6

a2F4φ

′40 + 4

H4

a2(H − H2)φ′40 F4 + 16

H4

aφ′30 F4

(Haφ′0 +H2φ′′0

)−8H6

aF4X

(Haφ′0 +H2φ′′0

)φ′50 + 4

H6

aF4φφ

′50 , (A.15)

M42 (a) = −18

H6

a2φ′40 F4 −

H4

a2φ′40 (H − H2)F4 − 4

H4

aφ′30 F4

(Haφ′0 +H2φ′′0

)+2H6

aφ′50 F4X

(Haφ′0 +Hφ′′0

)− H

6

aφ′ 50 F4φ + 6

H6

a2φ′40 F4 , (A.16)

M22 (a) = 2H4φ′40 F4 = −M2

3 (a) , (A.17)

M31 (a) = 16

H5

aφ′40 F4 . (A.18)

76APPENDIX A. FULL EFTMAPPING FORGALILEONS ANDGLPV THEORIES

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