Dark energy or modified gravity?

Thesis

Reference

Dark energy or modified gravity?

SAPONE, Domenico

Abstract

Durant ma thèse, j'ai étudié le problème de l'accélération de l'Univers du point de vue observationnel et théorique, et ceci dans le but de comprendre l'accélération de l'Univers à partir de l'étude des structures à grande échelle. Deux pistes sont actuellement explorées pour comprendre ce comportement inattendu. La première solution consiste à introduire dans l'Univers une nouvelle forme d'énergie, appelée énergie sombre. La seconde solution implique la modification de la théorie de la Relativité Générale à grande échelle. En particulier, nous avons étudié différents modèles. Je les ai comparés avec les observations et nous avons montré qu'il n'était pas possible de distinguer entre les deux pistes principales en utilisant la théorie des perturbations linéaires.

SAPONE, Domenico. Dark energy or modified gravity?. Thèse de doctorat : Univ. Genève, 2009, no. Sc. 4070

URN : urn:nbn:ch:unige-34174

DOI : 10.13097/archive-ouverte/unige:3417

Available at:

http://archive-ouverte.unige.ch/unige:3417

Disclaimer: layout of this document may differ from the published version.

UNIVERSIT´E DE GEN`EVE FACULT´E DES SCIENCES D´epartement de physique th´eorique Dr. Martin KUNZ Prof. Ruth DURRER

Dark Energy or Modified Gravity?

TH ESE `

pr´esent´ee `a la Facult´e des sciences de l’Universit´e de Gen`eve pour obtenir le grade de

Docteur `es sciences, mention physique

par

Domenico Sapone

de

Reggio Calabria (Italie)

Th`ese N^◦ 4070

GEN`EVE

Atelier de reproduction de la Section de physique

2009

R´ esum´ e

La cosmologie est l’´etude de l’Univers consid´er´e dans son ensemble.

Cette ´etude implique `a la fois recherche th´eorique et observationnelle.

Durant ces derni`eres d´ecennies, la confrontation entre th´eorie et ex- p´erience a mis `a jour l’existence de questions concernant notre com- pr´ehension de l’Univers. Durant ma th`ese, j’ai ´etudi´e diff´erents aspects de l’une de ces questions fondamentales, `a savoir celle de l’expansion acc´el´er´ee de notre Univers. Ce comportement, observ´e pour la pre- mi`ere fois en 1998, est en effet en contradiction avec les pr´edictions th´eoriques, puisque l’attraction gravitationnelle agissant sur le contenu (mati`ere et radiation) de l’Univers est suppos´ee ralentir l’expansion.

Trois pistes sont actuellement explor´ees pour comprendre ce comporte- ment inattendu. La premi`ere solution consiste `a introduire dans l’Univers une nouvelle forme d’´energie, appel´ee ´energie sombre. La seconde so- lution implique la modification de la th´eorie de la Relativit´e G´en´erale

`a grande ´echelle. La troisi`eme solution explore l’id´ee que les inho- mog´en´eit´es de la densit´e de mati`ere pr´esente dans l’Univers pourraient produire une acc´el´eration effective.

Durant ma th`ese, j’ai ´etudi´e le probl`eme de l’acc´el´eration de l’Univers du point de vue observationnel et th´eorique, et ceci dans le but d’identifier quelle information est susceptible d’ˆetre obtenue sur l’acc´el´eration de l’Univers `a partir de l’´etude des structures `a grande ´echelle. Les pro- jets sur lesquels j’ai travaill´e m’ont permis de construire une image exhaustive des probl`emes et d´efis de la cosmologie actuelle.

Mon premier projet a consist´e en une ´etude d´etaill´ee la th´eorie des perturbations de l’´energie sombre. En particulier ce projet s’int´eresse `a ce qui est appel´e”Phantom divide”, une limite pour laquelle la densit´e d’´energie de l’´energie sombre souffre de divergences. Dans ce travail, nous avons discut´e les conditions pour lesquelles ces divergences peu- vent ˆetre ´evit´ees. Une ´etude plus approfondie a mis en ´evidence que

les perturbations de la densit´e d’´energie sombre d´ependent de mani`ere cruciale de la perturbation de sa pression. Nous avons aussi montr´e que les perturbations adiabatiques ne sont pas exactes pour d´ecrire le fluide d’´energie sombre, et qu’il est d`es lors n´ecessaire de recourir aux perturbations non-adiabatiques de la pression afin de stabiliser l’´energie sombre [1].

R´ecemment, il a ´et´e declar´e qu’il ´etait possible de distinguer deux type de th´eories comologiques dans la mesure o`u il ´etait possible de mesurer lefacteur de croissance des structures `a grande ´echelle (growth factor).

Nous avons montr´e qu’il n’´etait pas simple de distinguer entre diff´erents mod`eles en utilisant la th´eorie des perturbations lin´eaires.

En particulier, nous avons ´etudi´e deux mod`eles capables d’expliquer l’acc´el´eration de l’Univers: l’´energie sombre et le mod`ele de Dvali- Gabadadze-Parroti (DGP). Le premier est simplement l’introduction d’une nouvelle forme d’´energie capable de provoquer l’acc´el´eration de l’Univers; l’autre vise `a modifier la th´eorie de la Relativit´e G´en´erale en affirmant que cette th´eorie n’est pas correcte `a grande ´echelle. En principe, les deux mod`eles pr´edisent deux valeurs diff´erentes concer- nant le facteur de croissance de la mati`ere. Toutefois, de telles pr´edic- tions diff´erentes proviennent de mod`eles d’´energie sombre qui ne tien- nent pas compte des perturbations de cette forme d’´energie. Pourtant, il n’est pas certain a priori que ces perturbations soient n´egligeables, et en [2], en prenant en compte ces perturbations, nous avons montr´e qu’il est toujours possible de trouver un mod`ele d’´energie sombre qui imite le comportement du facteur de croissance du mod`ele DGP, et que par cons´equent, la diff´erence entre les deux mod`eles n’est pas bien d´efinie.

Dans le papier [3], nous avons travaill´e sur une m´ethode servant `a distinguer diff´erents mod`eles capables d’expliquer la phase d’acc´el´eration actuelle de l’Univers. Nous avons pour cela consid´er´e l’effet de Weak

Lensing dˆu aux perturbations de la mati`ere sombre, et nous avons re- li´e ce travail aux possibilit´es de mesures des exp´eriences futures, en particulier l’exp´erience Euclid, [5], [7], [8]. L’effet de Weak Lensing est dˆu `a la defl´ection de la lumi`ere par un champs gravitationnel. La lumi`ere est ´emise par une source, `a partir de laquelle elle se propage dans l’espace. Lorsque la lumi`ere approche un corps massif, i.e. des galaxies, le champs gravitationnel agit sur elle. Il en r´esulte deux ef- fets importants : distorsion de l’image et changement de la luminosit´e apparente. L’effet de Weak Lensing est un outil puissant dans l’´etude de l’Univers. En effet, l’augmentation de la luminosit´e apparente de la source permet un acc`es `a l’observation pour un nombre plus impor- tants d’objets plus lointains. Ainsi, il est possible de d´eterminer des contraintes suppl´ementaires sur les param`etres cosmologiques.

Dans un projet suivant [4], nous avons consid´er´e le facteur de crois- sance des fluctuations lin´eaires, ce dernier ´etant l’une des quantit´es les moins connues en cosmologie observationnelle, et nous avons ´evalu´e les contraintes que les oscillations baryoniques mesur´ees par les futures ex- p´eriences peuvent imposer sur le facteur de croissance ainsi que sur les param`etres de l’´energie sombre. Les oscillations baryoniques apparais- sent en cons´equence des effets oppos´es de l’attraction gravitationnelle et de la pression du gaz dans le plasma primordial. Ces oscillations lais- sent leur empreinte dans les structures `a toute ´epoque de l’histoire de l’Univers, fournissant une mesure sur les distances `a partir desquelles il est possible d’inf´erer l’histoire de l’Univers.

Dans le dernier projet [6], et ceci toujours dans le but de distinguer entre des mod`eles de gravit´e modifi´ee et d’´energie sombre, nous avons focalis´e notre attention sur la mani`ere la plus g´en´erale de param´etriser les observables cosmologiques, tels les potentiels gravitationnels et le facteur de croissance. L’int´erˆet de la d´emarche provient du fait que seuls deux param`etres devraient ˆetre mesur´es afin de couvrir l’ensemble de tous les mod`eles de gravit´e modifi´ee et d’´energie sombre.

Dans l’ensemble de ma th`ese, je me suis attach´e `a comprendre les probl´ematiques de la cosmologie contemporaine. Tous mes projets mettent en ´evidence diff´erentes ´enigmes en lien avec la construction de mod`eles cosmologiques coh´erents et capables d’expliquer l’acc´el´eration de l’Univers.

Acknowledgements

I would like to thank firstly my supervisor Martin Kunz, for his com- petence and his kindness; he always injected me enthusiasm into the so dunkel Cosmology. He has been really a pleasure to have worked with him in the last four years and I hope I will have the chance to keep working with him.

I will never stop thanking Luca Amendola who was my supervisor in my Diploma Thesis in Rome and he is here being a member of the jury in my PhD thesis; he helped me in carrying on the passion to Cosmology. I would also like to thank him because he was the first one who relied on me.

It is a great pleasure to thank Robert Crittenden, who has warmly welcome me during the three months I spent at the ICG in Portsmouth.

It was a great opportunity for me to benefit from his advertisements and from his great knowledge.

I want to thank Ruth Durrer because she has been extremely kind to me and she is a fantastic person. I regret to have not spent more time discussing together.

I am particularly thankful to Michele Maggiore for his nice adver- tisements during all my PhD.

I would like to thank also the people who helped me in the last four years of this ”journey” here in Geneva: Camille Bonvin, Umberto Cannella, Chiara Caprini, Stefano Foffa, Marco Lista, Chicca Morvay, Enrico Pomarico, Syky R¨as¨anen, Max Rinaldi, Marcus Ruser, Hillary Sanctuary, Janine Splettstoesser, Richard Sturani, Antti Va¨ıhk¨onen, Marc Vonlanthen; and a particular Grazie to Elisa Fenu who tried to keep my nerves calm when I was particular charged up in the last months.

Merci chaleureaux also to Andreas Malaspinas, Nathalie Chadu- iron, Dani`ele Chevalier, Francine Gennai-Nicole, C´ecile Jaggi e Chris-

tine Schaffter, for all precious and fundamental help.

I want also to thank my sister Mariangela. She has always been next to me in the most difficult situations in my life. I want to give to her the chance to be here, because, somehow, she is also part of this thesis. Grazie di cuore.

The jury members are:

• Dr. Martin Kunz, Sussex University, Brighton, United Kingdom.

• Prof. Ruth Durrer, Universit´e de Gen`eve, Suisse.

• Dr. Luca Amendola, INAF, Osservatorio Astronomico di Roma, Italia.

• Dr. Robert Crittenden, Institute of Cosmology and Gravitation, Portsmouth, United Kingdom.

• Prof. Michele Maggiore, Universit´e de Gen`eve, Suisse.

I want to thank them again for having accepted to be part of the jury for my thesis.

Contents

1 Introduction 15

1.1 Accelerated phase . . . 20

1.2 The role of perturbations . . . 23

1.3 Growth of perturbations . . . 29

1.4 Weak lensing in cosmology . . . 30

1.5 BAO in the galaxy power spectrum . . . 38

2 Crossing the phantom divide 45 2.1 Introduction . . . 47

2.2 First order perturbations . . . 49

2.3 Barotropic fluids . . . 53

2.4 Non-adiabatic fluids . . . 60

2.5 The Quintom model as explicit example . . . 67

2.6 Conclusions . . . 72

2.7 acknowledgments . . . 77

2.8 Appendix . . . 79

2.8.1 Equivalence between scalar fields and fluid mod- els . . . 79

2.8.2 Effective perturbations in two barotropic fluids 81 2.8.3 Effective perturbations in the Quintom model . 85 3 Dark Energy vs Modified Gravity 87 3.1 Introduction . . . 89

CONTENTS CONTENTS

3.2 Setting the stage . . . 91

3.3 The importance of dark energy perturbations . . . 93

3.4 Anisotropic stress and modified gravity models . . . 95

3.5 Conclusions . . . 99

3.6 acknowledgments . . . 100

4 Measuring the dark side (with weak lensing) 101 4.1 Introduction . . . 103

4.2 Defining the dark Side . . . 105

4.2.1 Parametrisation of the expansion history . . . . 105

4.2.2 Parametrisation of the first order quantities . . 107

4.3 Observables . . . 111

4.3.1 Constraining the expansion history . . . 111

4.3.2 Growth of matter perturbations . . . 112

4.3.3 Weak lensing . . . 113

4.3.4 Other probes . . . 115

4.4 Dark energy models . . . 116

4.4.1 Lambda-CDM . . . 116

4.4.2 Quintessence . . . 116

4.4.3 A generic dark energy model . . . 117

4.4.4 DGP . . . 118

4.4.5 ΛDGP . . . 123

4.4.6 Scalar-tensor theories . . . 124

4.5 Forecasts for weak lensing large-scale surveys . . . 126

4.6 Conclusions . . . 134

4.7 Appendix . . . 138

4.7.1 Perturbation equations . . . 138

4.7.2 The lensing Fisher matrix . . . 140

5 Constraining the growth factor with baryon oscillations143 5.1 Introduction . . . 145

CONTENTS CONTENTS

5.2 Background equation . . . 147

5.3 Fisher matrix formalism . . . 149

5.4 growth factor . . . 153

5.4.1 Case 1 . . . 153

5.4.2 Case 2 . . . 154

5.5 Results and Conclusions . . . 157

5.6 acknowledgments . . . 161

6 Constraints on early dark energy from CMB lensing and weak lensing tomography 163 6.1 Introduction . . . 166

6.2 Early dark energy and structure growth . . . 169

6.2.1 Parameterisation of dark energy . . . 169

6.2.2 The growth of fluctuations . . . 171

6.2.3 The growth index parameterisation . . . 174

6.2.4 Impact on the density power spectrum . . . 175

6.3 CMB, CMB lensing and cosmic shear . . . 178

6.3.1 Information from the CMB . . . 178

6.3.2 Information from CMB lensing . . . 179

6.3.2.1 CMB Fisher matrix calculations . . . . 181

6.3.3 Information from cosmic shear . . . 184

6.3.3.1 Cosmic shear Fisher matrix calculations 185 6.4 Results . . . 187

6.4.1 Results for CMB and CMB lensing . . . 187

6.4.2 Results for cosmic shear . . . 188

6.4.3 Combined results . . . 188

6.4.4 EDE figure of merit . . . 190

6.5 Summary & conclusions . . . 190

7 Conclusions and outlook 205

CONTENTS CONTENTS

A Dark Energy Phenomenology 209

A.1 Introduction . . . 211

A.2 The dark sector . . . 213

A.3 Dark energy phenomenology . . . 214

A.4 Forecasts for future experiments . . . 216

A.5 Conclusions . . . 217

Chapter 1

Introduction

The modern cosmology was born in the 20s of the last century as one of the first applications of General Relativity for studying the properties of thecosmosat large scales. If the traditional Astrophysics and Astronomy were studying single objects (e.g. stars) or cluster of such objects (e.g. galaxies), characterizing the local properties of the Universe, with a more sophisticated instrumentation it was possible to observe deeper in the sky and at larger scales. More in general, if the global properties of the cosmos are to be described, we refer to the Universe as the first object under investigation of cosmology.

Since the days when Einstein first introduced relativistic models, cosmology has always been characterised by theoretical considerations that have found, over the years, observational proofs that made pos- sible to understand more accurately the physical processes underling cosmological phenomena.

It can be said that cosmology combines theoretical calculations with more and more accurate observations.

The Universe is filled by matter which forms structures like galaxies and cluster of galaxies and they evolve in time; so cosmology requires a theory of matter, in order to describe the structures that are seen in the Universe, and a theory of gravity able to explain their evolu-

CHAPTER 1. INTRODUCTION tion. Nowadays the theory of matter is the Standard Model of particle physics and the well accepted theory of gravity is General Relativity.

However, recent observations have put serious doubts on the com- pleteness of these two theories, the theoretical and experimental con- siderations are:

• Primordial nucleosynthesis: it predicts the baryonic abundance which is in agreement with the observations but it is not enough to produce the observed gravitational effects.

• Inflation: (needed to solve the curvature and horizon problems) requires an extension of standard model of particle physics.

• Supernovae type Ia (SNIa): show the Universe is in a phase of accelerated expansion.

• Large Scale Structure (LSS): show there is a lack of matter.

• Cosmic Microwave Background Anisotropies (CMB): suggest the total energy density in the Universe has to be equal to a partic- ular critical value.

The result of these observations has built a scenario in which the Universe is in a phase of accelerated expansion (SNIa) and its matter density (LSS) is not sufficient to set the curvature equal to zero (CMB).

General Relativity theory has led to an increasing development in the understanding of our Universe. The most interesting aspect of this theory is that the dynamic of every object can be expressed by a single tensorial equation:

Gµν = 8πG

c⁴ Tµν (1.1)

known as Einstein’s equation. Here Gµν represents Einstein’s tensor, which defines the geometry of the Universe, G is Newton’s constant,

CHAPTER 1. INTRODUCTION Tµν denotes the energy momentum tensor, which describes the matter and the energy content of the Universe and c is the speed of light. In other words, Einstein’s equation relates the geometry of the Universe to its content.

Then under the conditions of homogeneity and isotropy the most general line of element for an expanding Universe is the Friedmann- Lemaˆıtre-Robertson-Walker metric (FLRW):

ds² = −dt²+a²

dr²

1−Kr² +r²dθ²+r²sin²θdφ²

(1.2) where a(t) is the scale factor and represents the evolution in time of all the physical scales (every scale `0 will change in time as `(t) = a(t)`₀); K is the curvature parameter (it can be −1, 1 or 0 meaning an open, closed or flat Universe respectively); t is the cosmic time and the coordinates r, θ and φ are spherical comoving coordinates.

Einstein’s equation gives rise then to the Friedmann equations:

a˙ a

2

+ K

a² = 8πG

3 ρ (1.3)

a¨ a

= −4πG

3 (ρ+ 3p) (1.4)

where the energy density of the Universe is parameterized as ρ and its pressure byp; the dot denotes the derivative with respect to the cosmic time t; and K represents the geometrical curvature of the Universe.

In both the Friedmann equations the left hand side is given by the left hand side of Einstein’s equation, in other terms the geometry of a homogeneous and isotropic fluid is fully characterized by the scale factor a(t), whereas the right hand side of Friedmann equations come directly from the energy momentum tensor T_µν, hence geometry is related to matter.

Observations (mainly SNIa) show that the Universe is in a phase

CHAPTER 1. INTRODUCTION of accelerated expansion, which means that ¨a > 0. Eq. (1.4) requires that the pressure p < −¹₃ρ but this cannot be achieved by ordinary matter and radiation because they both have ρ+ 3p > 0.

Several solutions have been proposed to solve the acceleration prob- lem; the first one consists in modifying the right hand side of Einstein’s equations simply adding a new component in the Universe with a neg- ative pressure (namely dark energy), leading to ¨a > 0.

The second approach tends to modify instead the left hand side of Einstein’s equation (modification of gravity) saying that the laws of gravity fail to describe the Universe at large scales.

The third approach suggests that we just do not understand prop- erly how to apply usual General Relativity, and that the acceleration might be explained without introducing anything new. The Universe expansion and the scale factor are supposed to describe the average dynamics of the Universe, so first one should solve the Einstein’s equa- tion and then averaging; but the Friedmann equations are obtained on averaging first the metric and the energy density and then solving the Einstein’s equation. Of course the two approaches give different results since Einstein’s equation is non-linear. This idea is of course the most appealing and economical one might think of but it remains very controversial.

We are in an epoch at which physics is not able to explain most of the matter content in the Universe and then its evolution. The only thing that can be done is to test the consistency of the three approaches. There are two ways for doing this:

• Exploit current and future data in search of signatures of unex- pected phenomena that may signal new physical effects.

• Construct new cosmological models, confronting them with fun-

CHAPTER 1. INTRODUCTION damental principles and, immediately afterward, with observa- tional constraints.

In this thesis, we present some works on two aspects of cosmological research: dark energy models and modified gravity models, trying to understand whether the two scenarios can be distinguished.

First we assume the existence of a dark energy component and we study the behaviour of density perturbations in the so called phantom divide limitp ' −ρ, region at which divergencies in the energy density appear. We test models where w crosses −1 and we find that in many realistic cases the divergencies are only apparent and can be avoided.

We then show up to the first order perturbation theory that in gen- eral it is not so easy to distinguish between modified gravity models and dark energy models. We demonstrate explicitly that a generalised dark energy model can match the growth rate of the Dvali-Gabadadze- Porrati model and reproduces the 3 + 1 dimensional metric perturba- tions. As a consequence cosmological observations seem to be unable to distinguish the two cases. Then we focus on weak lensing experiments in search of new unexpected phenomena; we use such experiments be- cause weak lensing makes use of both background and perturbation dynamics helping us to break the degeneracy (with complementary works) between different models.

We take then in consideration the growth factor of linear fluctua- tions, since is one of the least known quantity in observational cosmol- ogy and we discuss the constraints that baryon acoustic oscillations in galaxy power spectra can put on a conveniently parametrized growth factor.

At the end we report a note showing that the dark energy cannot be fully constrained unless the dark matter is found, arguing that two functions are needed in order to characterise the observational proper-

Accelerated phase

ties of dark sector for cosmological probes.

1.1 Accelerated phase

The discovery by supernova surveys ([11], [12]) that the expansion of the Universe is currently accelerating came as a great surprise to cos- mologists. As we said, within the standard cosmological framework of a nearly isotropic and homogenous Universe and an evolution described by General Relativity, a new component with a negative parameter of equation of state (w <−1/3) is required to explain such a behaviour.

The idea is to insert in the content of the Universe an extra fluid, apart from matter and radiation, which can lead to the observed accelera- tion, which in turn means to add an extra term in right hand side of Einstein’s equation. The Hubble parameter then looks like:

H² = a˙

a 2

∼ 8πG

3 (ρm +ρx) (1.5)

where ρm is the energy density of matter in the Universe and ρx is the energy density of dark energy (here we are neglecting the curvature term for sake of clarity).

The first cosmological model makes central use of the cosmological constant as the most confirmed model (ΛCDM); which is character- ized by a parameter of equation of state w = −1. Particle physics has brought a new concept for the cosmological constant: it represents the energy density of the vacuum. Given a scalar field with a par- ticular potential then the corresponding energy momentum tensor Tµν

can be reconstructed. According to this, the configuration of lowest energy state will be given by setting the kinetic term equal to zero; as a consequence the energy momentum tensor will depend only on the potential at its minimum and there is no reason to believe that this

Accelerated phase value has to be zero. Anyway this model suffers of serious problems, the first one is known as cosmological constant problem: the value of ground state was evaluated to be 10¹²⁰ bigger than the value required to fit the data. There are several alternative solutions trying to reduce this discrepancy between theory and observation, but none of them seems to be a convincing one. The cosmological constant has another problem, known as coincidence problem: we are in an epoch at which its energy density and the energy density of matter are approxima- tively equal. Or, phrase it differently, why has the acceleration of the Universe started in the recent past?

These difficulties have led people to consider that the dark energy in the Universe is not just a constant but it may be more complicated, leading to an evolving parameter of equation of state w(t). So dark energy now can evolve on time and the coincidence problem can be partially solved. Indeed, the relation ΩDE ∼ 2Ωm suggests that there is a connection between dark energy and dark matter and motivates the construction of models of coupled dark energy and dark matter.

Though a host of models have been constructed based on this hope, none of them provides a satisfactory solution to the problem of fine- tuning.

Because of the unfeasibility of solving such theoretical problems, cosmologists have started to look at other models. An alternative ap- proach postulates that General Relativity is only accurate on small scales and has to be modified on cosmological scales. Such a modifi- cation of gravity leads to a modification of the left hand side of the Einstein’s equation. One of the best studied example is the Dvali- Gabadadze-Parroti brane-world model (DGP), in which matter is con- fined in the 4-dimensional brane and only gravity can propagate in the 5-dimensional bulk. Gravity leaks off the 4-dimensional brane into the 5-dimensional bulk at large scales, say λ > rc. On small scales, λ < rc, gravity is effectively bound to the brane and 4-dimensional dynamic is

Accelerated phase recovered with very good approximation. Then gravity leakage leads to the observed late-time acceleration of the expansion of the Universe.

As we said, this implies a modification of the left hand side of Einstein’s equation; then the Hubble parameter can be evaluated as:

H²− H

rc ≈ ρ_m (1.6)

where rc is the cross-over scale and it separates the 5D and the 4D regimes. We have now two different theories which can explain the observed acceleration of the Universe; then a question arises: can DGP model be distinguished from one invoking dark energy? Since matter is conserved on the brane, ρm satisfies the usual conservation equation;

comparing eq. (1.6) with eq. (1.5) it is easy to see that we can move the cross-over term to the right hand side and think of it as a dark energy contribution: consequently the effective parameter of equation of state can be evaluate. In this case we treat a model of modification of gravity as a 4D model with an effective dark energy. The answer then is no:

it is impossible to rule out dark energy, in favor of a modified gravity model, only measuring the expansion history, because any expansion history can be reproduced choosing a suitable parameter of equation of state.

Lately there have been claims that if we are able to measure the growth rate of structures then we can rule out different models. This because different theories predict different growth of structure, then fixing the parameter of equation of state w (effective or not) we can predict the evolution of the growth rate; if we observe a different growth then the theory is wrong and can be ruled out. In chapter 3 we demon- strate that this may not be so easy, but to show this we require the first order perturbation theory.

The role of perturbations

1.2 The role of perturbations

The conventional paradigm for the formation of structures in the Uni- verse is based on the growth of small perturbations due to the grav- itational instabilities. In this picture, some mechanism is required to generate small perturbations in energy density in the very early phase of the Universe. The central quantity that it is needed to describe the growth of structures is the density contrast, defined as δ(t, ~x) = [ρ(t, ~x)−ρ¯(t)]/ρ¯(t) which describes the change in energy density (ρ(t, ~x)) compared to the background (¯ρ(t)).

The growth of perturbations depends on the characteristics of the fluid itself, namely pressure, pressure perturbation and maybe anisotropic stress. For ordinary matter this quantities are known, so the growth of the density contrast can be easily evaluated. For dark energy these parameters are a priori unknown and need to be measured. Usually it is taken for granted that dark energy does not cluster on small scales because it is assumed that its sound speed is too high to allow per- turbations to gravitationally collapse. For some specific models, i.e.

scalar field models, this is a good approximation but for a general dark energy fluid this may not be the case and perturbations in energy density have to be studied.

The dynamical evolution of the scalar field can affect a number of observables, including the spectrum and the growth of large scale struc- tures, weak gravitational lensing, SNIa apparent luminosity, and CMB anisotropies. One particular manifestation occurs in the late-time In- tegrated Sachs-Wolfe (ISW) effect, which measures the evolution of the gravitational potential as the Universe enters a phase of dark en- ergy domination. This effect is only significant on large scales (low multiples), since small scale fluctuations in the gravitational potential smooth out along the line of sight. And it is only significant at late time since potentials evolve the same as the background during matter

The role of perturbations domination. The ISW effect has been detected in cross-correlations be- tween CMB temperature anisotropies and surveys of large scale struc- ture [30].

In chapter 2 we report the details of the cosmological perturbation theory up to the first order. The motivation of this work comes from the idea that we need to consider a general model for dark energy until we do have some information from observations. In particular, we consider the growth of density perturbations for a general perfect fluid model close to w = −1; this because, as long as the cosmological data indicates the presence of a dark energy with an effective equa- tion of state p ∼ −ρ it will be necessary to consider models with the same equation of state, then in analysing the data we therefore have to be able to use general self-consistent models at the level of linear perturbation theory. We want to investigate only the scalar modes of the perturbation equations, we choose the Newtonian gauge which is very simple for scalar perturbations because they are characterized by two scalar potentials ψ and φ. We also assume that the Universe is filled with a perfect fluid so the energy momentum tensor Tµν takes a simple form. We evaluate the perturbation equations for a density fieldδ and for the velocity fieldθ. These equations contain terms going like 1/(1 +w) and will generally diverge if w → −1; these are only apparent divergencies and can simply be avoided if θ is replaced with a new variable V = (1 +w)θ; this variable turns out to have a more intuitive meaning since it is directly related to the off diagonal term of the energy momentum tensor and it represents the energy flow in one spatial direction. There is now the problem of defining the pres- sure perturbation which is the only unknown quantity. We start to consider barotropic fluids (where pressure p is a function of the energy density ρ only). We find again that the pressure perturbation diverges if w → −1. The physical problem here is that we demand the pressure to be a unique function of the energy density. Now if the fluid crosses

The role of perturbations w = −1 the energy density ρ will first decrease and then increase, while the pressure p will decrease around the crossing. It is therefore impossible to maintain one-to-one relationship between pressure and energy density.

As subsequent step, we consider non-adiabatic fluids and we ex- press the pressure perturbation in the rest frame coordinates of the fluid so its velocity becomes irrelevant, we define the rest frame sound speed ˆc²_s that represents the speed with which fluctuations propagate.

We find out that also this parameterization fails to describe the fluid when w → −1. The reason is that the gauge transformation relating the pressure perturbations in the different gauge contains the adiabatic sound speed c²_a = ˙p/ρ˙ which diverges if w → −1 (because ˙ρ → 0); so the dark energy at rest-frame becomes unphysical at crossing. Then the problem is to find a way of characterizing the pressure perturba- tions in a physical way; we give an explicit example of a Quintom model. Such a model considers two perfect fluids with constant equa- tion of state parametersw, one being a quintessence (w > −1) and the second one being a phantom (w < −1), and both with constant and equal rest frame sound speed. The idea of considering this model lies on the fact that we want to express the composition of two well de- fined fluids as a single effective fluid that can cross the limit ofw = −1.

Evaluating again all the quantities we see that everything stays finite except the pressure perturbation in which there are diverging terms but they cancel out together giving a finite result; these parameters act as internal and relative pressure perturbations leading to an appar- ent sound speed which goes to infinity but the sum of all contribution to δp remains finite. We show that although models with purely adia- batic perturbations cannot cross w = −1 without violating important physical constraints (like causality or smallness of perturbations), it is possible to rectify the situation by allowing for non-adiabatic sources of pressure perturbations. However, the parameterisation of δp in terms

The role of perturbations of the rest frame perturbation of the energy density cannot be used as this frame becomes unphysical at w = −1. By parameterising δp in- stead in any other frame the divergencies can be avoided. Even though the region in whichw ≤ −1 may be unphysical at the quantum level, it is still important to be able to probe it, not least to test for alternative theories of gravity which can give rise to an effective phantom energy or an effective cosmological constant.

The dark energy models considered so far have a strong assump- tion. We are imposing that dark energy has no anisotropic stress which implies that the metric perturbations variables are the same φ = ψ.

In general this is not true or better, there is no evidence to set the anisotropic stress equal to zero. This new quantity increases the de- grees of freedom of dark energy since it depends on the intrinsic charac- teristic of the fluid itself and needs to be measured. We show in chapter 3 that the anisotropic stress is a crucial quantity for the growth of per- turbations moreover if we want to distinguish between dark energy and alternative gravity theories.

As we previously said there has been the trend to say that if one is able to measure the growth factor of structures then cosmological models can be ruled out. We show how the dark energy perturbations influence the dark matter and the metric perturbations providing an explicit example in which a general dark energy model can reproduce the metric perturbation of the DGP scenario; as a consequence the growth rate of structures is not sufficient to distinguish between dark energy and modifications of gravity.

In this work we want to investigate only the scalar modes of the perturbation equations. We choose the Newtonian gauge (also known as longitudinal gauge) which is very simple for scalar perturbations because they are characterized by two scalar potentialsψ and φ; so we consider linear perturbations about a spatially-flat background models,

The role of perturbations

defined by the line of element:

ds² = gµνdx^µdx^ν = a²

−(1 + 2ψ)dτ²+ (1−2φ)dxidxⁱ

(1.7) where dτ = dt/a is the conformal time; ψ is the scalar potential and φ is the scalar perturbation to the spatial curvature. The advantage of using the Newtonian gauge is that the metric tensor g_µν is diagonal and this simplifies the calculations.

It is important to say that the perturbations in density in different fluids are linked via the perturbations in the metric ψ and φ through:

k²φ = −4πGa²X

i

¯

ρi∆i (1.8)

k²(φ−ψ) = 12πGa²X

i

(1 +wi) ¯ρiσi (1.9) where the sum runs over matter and dark energy; ∆ = δ + 3aHV /k² is the longitudinal density contrast and σ is the anisotropic stress (for matter σm = 0). The important quantity for us is the growth factor G(a) = ∆_m/a, normalised to 1 for a 1 (using that ∆_m ∝ a during matter dominated era and on sub-horizon scales). The growth factor is not uniquely determined by the expansion history of the Universe (even if it remains the main effect) but also by the contribution from the gravitational potentials because different dark energy perturbations will contribute differently on the evolution ofψandφwhich can modify the behaviour of ∆m. Dark energy has now three parameters that need to be defined: parameter of equation of state w which can be inferred from the observed expansion history; pressure perturbation and anisotropic stress. These two quantities are important for the growth of dark energy perturbations: for instance if a large sound speed ˆ

c²_s (or equivalently δp) is assumed then dark energy cannot cluster at sufficient small scales. As we decrease the sound speed, the dark

The role of perturbations energy is able to cluster more and more, then the increased dark energy perturbations lead to enhanced metric perturbations. The dark matter in turn falls into the potential wells created by the dark energy, leading to an increase of the growth factor G(a).

An important aspect of DGP and/or other brane-world models is that the dark matter does not see the higher-dimensional aspects of the theory as it is bound to the brane. Its evolution is then the same as in the standard model. The modifications appear only in the gravitational sector, represented by the metric perturbations:

k²φ = −4πGa²

1− 1 3β

¯

ρm∆m (1.10)

k²ψ = −4πGa²

1 + 1 3β

¯

ρm∆m (1.11)

whereβ depends on the Hubble parameter. The dark matter then does not care if the metric perturbations are generated (in addition to its own contribution) by a modification of gravity or by an additional dark energy fluid: its response to them is identical. Or, to put it differently, if the dark energy and dark matter together can create the sameφ and ψ of eqs. (1.10) and (1.11) then the growth factor (and indeed all other cosmological observables) will be the same as in DGP scenario. We see immediately that in order to generate the same metric perturbations in DGP model we require an anisotropic stress for dark energy different from zero so that φ 6= ψ. Of course a question arises: is it possible to match both ψ and φ of the DGP model within a generalised dark energy model? Yes, it is: the metric perturbations have two degrees of freedom, and we do have two degrees of freedom of the dark energy to adjust, σ and δp. Indeed we play with these two parameters and we find that it possible to mimic exactlythe growth of matter in the DGP model. Although the construction of a matching dark energy model seems very fine tuned, we are here more concerned with the question

Growth of perturbations to what degree this is possible at all. Just measuring a growth of matter perturbation, at least up to the first order perturbation theory, is not enough to rule out General Relativity in favor of a modification of gravity.

1.3 Growth of perturbations

The central quantity needed to describe the growth of structures is the density contrast δ(t, ~x). Since one is often interested in statistical behaviour of structures in the Universe, it is conventional to assume that δ and other related quantities are elements of an ensemble. The most accredited model of structure formation suggests that the initial density perturbations in the early Universe can be represented as a Gaussian random variable and a given initial power spectrum. The last quantity is defined through the relation P (k) (2π)³δD

~k −k~⁰

= hδ(~k)δ^∗(k~⁰)i where δ(k) is the Fourier transform of δ(t, ~x) and h...i means averaging over the ensemble. Though gravitational clustering will make the density contrast non Gaussian at late times, the power spectrum (and the correlation function) remains the most important quantity that needs to be studied in structure formation.

When the density contrast is small, its evolution can be studied by linear perturbations theory and each Fourier mode δk(t) will grow in- dependently. We can take the dark matter perturbation equation and evaluate its growing mode. In the standard ΛCDM model of cosmol- ogy, the dark matter perturbations on sub-horizion scales grow linearly with the scale factoraduring matter domination era. During radiation domination they grow logarithmically, and also at late times, when the dark energy starts to dominate, their growth is slowed. The growth fac- tor G(a) = ∆m/a is therefore expected to be constant at early times (but after matter-radiation equality) and to decrease at late times.

Weak lensing in cosmology Unfortunately we do not have any analytic solution from the second order differential equation of dark matter if we account for dark en- ergy. But over the years many parameterizations and fitting formula were evaluated. The main effect for the slowed of the growth factor is due to the expansion history of the Universe (Hubble parameter). In addition to this effect there is also the possibility that fluctuations in the dark energy can change gravitational potentials and so affect the dark matter clustering. A particularly nice formula was found for the ΛCDM model and it seems to work properly also for many different dark energy and modified gravity models. This expression is simply the integral over redshift of the energy density contrast of dark matter with an exponent γ, called growth index. This parameter turns out to be one the most important parameter because it encodes information about dark energy.

1.4 Weak lensing in cosmology

Light rays are deflected when they propagate through an inhomoge- neous gravitational field. It was Einstein’s theory which elevated the deflection of light by masses from a hypothesis to a firm prediction.

Assuming light behaves like a stream of particles, its deflection can be calculated within Newton’s theory of gravitation, but General Rel- ativity predicts that the effect is twice larger. The first experiment was in the 1919 when the deflection of a light ray, coming from a star, due to the gravitational field of the Sun was measured. The confirma- tion of the larger value was the most important step towards accepting General Relativity as the correct theory of gravity.

Cosmic bodies more distant, more massive, or more compact than the Sun can bend light rays from a single source sufficiently strongly so that multiple light rays can reach the observer. The observer sees

Weak lensing in cosmology an image in the direction of each ray arriving at their position, so that the source appears multiply imaged. Tidal gravitation fields lead to differential deflection of light bundles. The size and shape of the sources are therefore changed. Since photons are neither emitted nor absorbed in the process of gravitational light deflection, the intrinsic surface brightness of lensed sources remains unchanged. Changing the size of the cross section of a light bundle therefore changes the flux.

Since astronomical sources like galaxies are not circular, this defor- mation is generally very difficult to identify in individual images. In some cases, however, the distortion is strong enough to be recognised (this is the case of Einstein rings and arcs in galaxy clusters). Anyway such strong effects are not so common in nature and most of the time we are dealing with very weak distortions which are hard to be seen.

Although weak distortions in individual images cannot be recognised, the net distortion averaged over an ensemble of images can still be detected. This is what is called weak gravitational lensing.

There are two reason why gravitational lensing has became nowa- days a powerful tool in modern cosmology:

• The deflection angle of a light ray is determined by the gravi- tational fields (ψ +φ) of the matter distribution along its path.

According to General Relativity, the gravitational fields are de- termined by the energy momentum tensor of matter distribution.

So information about the matter density distribution can be ob- tained.

• Once the deflection angle is given, gravitational lensing can be easily reproduced. Since most of the lens system involves sources at moderate and high redshift, lensing can probe the geometry of the Universe.

The two main distortion effects here that can be used in weak lensing to probe the statistical properties of the matter distribution

Weak lensing in cosmology between the observer and an ensemble of distant sources are: shear and magnification.

The images are distorted in shape and size. The shape distortion is due to the tidal component of the gravitational field, described by the shear, whereas the magnification is given by both isotropic focusing caused by the local matter density and anisotropic focusing due to the shear, hence lensing changes the apparent brightness of a source.

For cosmological use the weak lensing equation takes a simple form because of different assumption that can be made:

• Density perturbations are well localised in an homogeneous and isotropic background (each perturbation can be surrounded by a spatially flat neighbourhood).

• The Newtonian potential of the perturbations is small and typical velocities are much smaller than the speed of light.

Light rays are deflected by a gravitational potential due to a mass distribution, then it can be defined an effective surface mass density (and it will define the impact parameter). The power spectrum of the effective surface mass density is closely related to the power spectrum of the matter fluctuations, and it forms the central physical object in cosmology. Any two point statistics of cosmic magnification and cosmic shear can then be expressed in terms of the effective convergence power spectrum.

Briefly we can say that the convergence κ is related to the first derivative of the deflection angle α which is proportional to the first derivative of the gravitational potential. The effective convergence κef f then involves the Laplacian of the potential which is the Poisson’s equation. The effective convergence along a light ray is therefore an integral over the density contrast along the light path, weighted by a combination of comoving angular diameter distances (sources and lens

Weak lensing in cosmology at different redshift). Of course we are interested on the statistical properties of the effective convergence, especially the power spectrum.

According to the definition above the convergence power spectrum P_κ(`) is a weighted integral of the matter power spectrum; then any weak lensing experiment will give information on the mass distribution (more correctly the gravitational potential) in the Universe.

In chapter 4 we investigate the extent to which additional parame- ters can be used to detect signatures of new cosmology in future weak lensing surveys (having in mind a setup similar to Euclid), producing Fisher matrix confidence regions for the relevant parameters.

In General Relativity the gravitational potential is given by eq.

(1.8). All the fluids with non-zero perturbations will contribute to it. If we assume for instance that dark energy does exist we cannot demand that its perturbations are too small. In addition it might be that we are dealing with some modified gravity model or something more complicated; in order to account for all these effects we parametrise the Poisson’s equation as:

k²φ = −4πGa²Q(k, a) ¯ρm∆m. (1.12) Here Q(k, a) is a phenomenological quantity that, in General Relativ- ity, is due to the contributions of the non-matter fluids (and in this case depends on theirδp andσ). But it is more general, as it can describe a change of the gravitational constantG due to a modification of gravity (see eq. (1.10)). It could even be apparent: If there is non-clustering early quintessence contributing to the expansion rate after last scatter- ing then we added its contribution to the total energy density during that period wrongly to the dark matter, through the definition of Ωm. In this case we will observe less clustering than expected, and we need to be able to model this aspect. This is the role of Q(k, a).

For a dark energy model we have to admit an arbitrary anisotropic

Weak lensing in cosmology

stress σ (see eq. (1.9)) and we use it to parametrise ψ:

ψ = [1 +η(k, a)]φ. (1.13) There is no sign for a non-vanishing anisotropic stress beyond that generated by the free-streaming of photons and neutrinos. However, it is expected to be non-zero in the case of topological defects [69] or very generically it is expected to be non-zero for modified gravity models (see eq. (1.11)). With Q and η then we can recover all the possible models. For ΛCDM model we have Q= 1 and η = 0, if these parame- ters have a different value then the evolution of matter perturbations changes, so that we need to take into account the modified growth of linear perturbations, parmeterised by γ, called growth index. How- ever, the weak lensing effect is proportional to the lensing potential Φ = φ+ ψ, the quantities that enter in the weak lensing calculations are the growth index γ and the combination Σ = Q(1 +η/2). Weak lensing will therefore constrain the set of parameters{γ,Σ}. We derive the sensitivity of typical next-generation tomographic weak lensing sur- veys to the non-standard parameters introduced above with the help of Fisher matrix formalism. All the characteristics of the survey are well within the range considered for Euclid satellite proposal. We plot the confidence regions for significant parameters obtaining small values for the errors indicating that weak lensing survey is a powerful tool for constraining parameters. The most important result here is that weak lensing surveys seem to be able to differentiating the DGP model from ΛCDM at more than 7 standard deviations.

Different models have been constructed in order to explain the ob- served acceleration of the Universe, even though none of them is really appealing on theoretical grounds. A general approach must be taken in order to account for all the class of models: that is to construct a gen- eral parameterisation of the dark energy, in the hope that measuring

Weak lensing in cosmology these parameters will give us insight of new physical mechanisms. A successful example for the dark energy context is the equation of state parameter w = p/ρ. This quantity can be inferred from the expansion history (i.e. the Hubble parameter) and this is the only observable that can be measured at background level.

In Appendix A we highlight that including experiments of weak lensing (or also CMB data) to other experiments, like BAO or SNIa, one can have full information about dark energy. This is because such experiments constrain both the expansion history and the clustering properties of the fluid. In this case we are dealing now with a set of parameters which fully describes the properties of dark energy (up to the first order perturbation theory): parameter of the equation of state (w), pressure perturbation (δp) and anisotropic stress (σ). Con- sequently we have a set of observables: Hubble expansion which can inferr the dark energy equation of state parameter and two gravita- tional potentials ψ and φ (if we work in the Newtonian gauge) which can give information on the dark energy pressure perturbation and anisotropic stress. We now have a full system of equations describing our Universe and the same number of ”variables” which describe the fluid.

We show that this set of parameters is more general and it can be used also for modified gravity models as we can reconstruct a set of effective parameters also for this class of models. This is also the case addressed in chapter 4 where we use (γ,Σ) as our set of parameter. In Appendix A we point out that such a reconstruction leads to exactly the same observational properties for both dark energy and modified gravity models. Even though this seems to be bad news for cosmology because we will never be able to distinguish dark energy from modified gravity, we find here a clear target for future experiments: together with w, only two extra quantities (δp and σ or equivalently γ and Σ) are needed to be measured in order to span the complete model space

Weak lensing in cosmology for both modified gravity and dark energy models.

In chapter 6 we consider a particular model, called early dark en- ergy, which belongs to the tracking quintessence model. We use here lensing effects from CMB and galaxy shear, in particular we forecast the constraints that next generation satellite experiments can put on cosmological parameters. For CMB and CMB lensing we consider a setup similar to Planck and CMBpol while for the weak lensing to- mography we calculate the constraints for a space based mission like Euclid.

Quintessence has been proposed as the missing energy density that must be added to the baryonic and matter densities in order to reach the critical density in the Universe. Quintessence is a dynamical, slowly-evolving component with negative pressure. An example is a scalar field χ slowly rolling down its potential. For this class of models the parameter of the equation of state wχ lies between −1 and 0; and depending on the potential V(χ), w_χ can be constant, slowly-varying, rapidly-varying or oscillatory. A key problem with the quintessence model (as previously mentioned) is explaining why its energy density ρχ and the matter density should be comparable today. There are two sides to this problem. First of all, throughout the history of the Universe, the two densities decrease at different rates, so it appears that the conditions in the early Universe have to be set very carefully in order for the energy densities to be comparable today. A second aspect is that the value of the quintessence energy density is very tiny compared to typical particle physics scales.

Recently, a form of quintessence, called tracking field was intro- duced which avoids the coincidence problem. Tracker fields have an equation of motion with attractor like solutions in which a very wide range of initial conditions rapidly converge to a common track. The initial value then of ρχ can vary by nearly 100 order of magnitude without altering the cosmic history. The equation of state here varies

Weak lensing in cosmology according to the background equation of state wB. When the Universe is radiation dominated (wB = 1/3), then wχ is less or equal to 1/3 and ρχ decreases less rapidly than radiation. When the Universe is matter dominated (w_B = 0), then w_χ is less or equal to 0 and ρ_χ decreases less rapidly than matter. Eventually ρχ overtakes the matter density and start to dominate and the Universe is driven into an accelerat- ing phase. Also this mechanism is still unknown and often requires a fine-tuning; however a tracking solution seems to be a natural choice for quintessence models because it avoids the problem of the initial conditions.

In this case the cosmology is independent of the initial conditions, as a consequence both the equation of state parameter w_χ and energy density Ωχ only depend on the potential V(χ). Hence, for any given potential, once Ωχ is measured, wχ is determined. We emphasize here that the tracker solution is one which undergoes a long period of at- traction, which means that solutions of the equation of motion are drawn to a common solution.

As previously mentioned, a dark energy fluid seems to be the dom- inant component in our Universe only at late time; i.e. if its equation of state remains close to−1 at early times, such as a cosmological con- stant, its density evolves slowly and was negligible relative to matter and radiation fluid.

However, in dynamic models of dark energy, such as scalar field, this may be different and dark energy could have a non-negligible influence on earlier stages of the growth history. The presence of an early fraction of dark energy in our Universe could have had an observable impact on the probes of the early Universe, such as nucleosynthesis, CMB physics and the growth rate of structure. These models are collectively known as early dark energy models.

Early dark energy models may help to resolve some apparent dis- crepencies in the amplitude of density fluctuations, usually parame-

BAO in the galaxy power spectrum terised by σ8, the r.m.s. of the density field on scales of 8 h⁻¹Mpc. If the excess in the CMB power spectrum at high multipoles is explained by the Sunyaev-Zel’dovich (SZ) effect of unresolved clusters of galaxies one needs a value of σ₈ close to one, which is significantly higher than currently estimated from combining CMB and lensing data. Secondly, the observed strong lensing cross section is predicted by ΛCDM or tra- ditional dark energy models to be too small by an order of magnitude and one would be forced to increase σ₈, again to values close to unity.

In Chapter 6 , we consider the combination of an observation of the primary CMB fluctuations, the CMB lensing signal and weak cosmic shear for following the growth history in early dark energy models at high, intermediate and low redshift. Cosmic structure growth on the relevant scales is close to linear, where the theory is able to provide reli- able predictions. The motivation of this investigation was to follow the cosmic growth history by combining high-precision probes at recom- bination redshifts, at intermediate redshifts of z ' 4 by CMB lensing and at low redshifts around unity from weak cosmic shear. As experi- mental data for the microwave sky we consider the Planck survey, and compare to the corresponding constraints from the CMBPol satellite concept. For the weak lensing power spectra we use the characteristics for a satellite mission like Euclid.

1.5 BAO in the galaxy power spectrum

During the last decade an avalanche of balloons and ground experi- ments, together with the most recent WMAP satellite have measured the small and large angular temperature fluctuations of the Cosmic Microwave Background Radiation. Such measurements have detected a series of acoustic peaks in the anisotropy power spectrum and con- firmed early predictions about the evolution of pressure waves in the

BAO in the galaxy power spectrum primordial photon-baryon plasma. The specific features of such peaks are sensitive to the value of the cosmological parameters, in particular to Ωtot, Ωb and the scalar spectral index ns.

So the CMB power spectrum provides information on combina- tions of fundamental cosmological parameters. The physical processes responsible for the anisotropies are well understood allowing a com- plete prediction on the shape of the CMB power spectrum for a given cosmological model. Generally, before recombination, the baryons in the Universe were locked to the photons (via Thompson scattering) of the CMB, and the photon pressure interacting against gravitational instability, due to matter, produced sound waves in the plasma. After recombination, the baryons and the photons separated, but the effects of the acoustic oscillations remains imprinted in the spatial structures of the baryons and eventually on the dark matter. The physical length scale of the acoustic oscillations depends on the sound horizon of the Universe at the epoch of recombination, where the sound horizon is the comoving distance that a sound wave can travel before recombination and depends only on baryon and matter densities. The relative heights of the acoustic peaks in the CMB anisotropy power spectrum measure these densities with excellent accuracy.

The CMB anisotropies can be thought of as fluctuations in tem- perature δT /T around the mean black body temperature T '2.726K of the cosmological radiation. During the radiation dominated era the equation describing the effective temperature fluctuations δT of the CMB has the form of an harmonic oscillator sourced by a function of the gravitational potential (ψ), the amplitude of these oscillations are modulated by the photon-baryon sound speed c²_s = 1/3(1 + 3ρb/ρr).

Therefore when photons decouple from baryons their energy carries an imprint of such oscillations. The characteristic frequencies of these oscillations is fixed by the size of the sound horizon at the decoupling r = R

c dτ. Therefore we have a series of compressions and rarefac-

BAO in the galaxy power spectrum tions at scales kmrsh = mπ. Today such scales appear at angles that are multiple integers of the angular size of the sound speed horizon at the decoupling θhs = rsh/DK(τlss), where DK(τlss) is the distance to the last scattering surface for a spacetime with curvature K. As a consequence, the position of the acoustic peaks in the power spectrum depends on the geometry of the Universe. For a flat cosmology the peaks will appear at the multipoles `m = m`sh. However the acoustic oscillations are perturbed by the evolution of the gravitational poten- tial which shift the position of the peaks by an amount that depends on the cosmological parameters that are relevant before recombination.

Then, it is easy to understand that the position of the peaks and their amplitudes give information on cosmological parameters.

The acoustic peaks in the CMB power spectrum are predicted to be present in the late-time clustering of galaxies as a series of weak modulations in the amplitude of fluctuations as a function of scale, usually calledwiggles. Mapping the acoustic peaks in the galaxy power spectrum at high precision, matching those already measured in the CMB power spectrum, would provide a spectacular confirmation of the standard cosmological model in which mass over densities grow from the seeds of the CMB fluctuations.

The position of the peaks and troughs in Fourier space are cal- culable form straight-forward linear physics and act like a standard ruler. Therefore a power spectrum analysis of a galaxy redshift sur- vey containing acoustic peaks can be used to measure cosmological parameters.

What is a standard ruler?

The comoving size of an object or a feature at redshift z along the line of sight (r_k) and the transverse (r_⊥) direction are related to the observed sizes ∆z and ∆θ of the object by the Hubble parameter H and angular diameter distance DA. When the true scales, r_k and r , are known, measurements of the observed dimensions, ∆z (deep in

BAO in the galaxy power spectrum space) and ∆θ (wide in angle), give estimates of the Hubble parameter and the angular diameter distance: these are the Standard Rulers.

In turns the Hubble parameter and the angular diameter distance (integral over the redshift of the inverse of Hubble parameter) are re- lated to the matter content of the Universe, and information on the cosmological parameters can be extracted.

The matter power spectrum is simply a product of the spectrum of primordial fluctuations and the modifications of those fluctuations in the later epoch. Linear perturbation theory fixes the matter power spectrum in comoving coordinates and changes only the amplitude as the structures evolve (once matter-radiation equality is fixed). The growth factor G(z) rescales the amplitude of the fixed matter power spectrum to account for the growth of structures from the recombi- nation to a redshift z. As shown, the growth factor depends on the characteristic of the dark energy; this is our central quantity in chap- ter 5 where we investigate the extent to which baryon oscillations can set to the growth factor in future large scale observations at redshift up to 3. The main result of this work is that for a survey with the characteristics well within the next generation experiments (i.e. JDEM and EUCLID) is possible to reach a precision with which is possible to rule out models, say for instance ΛCDM model with respect to DGP model.

However, the observations of the galaxy power spectra are compli- cated by different sources of systematic errors and incorrect mapping.

Examples are:

• Bias factor: in the galaxy survey, the observable is the galaxy over-density which is assumed to trace the underlying matter dis- tribution through a function called bias. In principle this quan- tity could be arbitrary; it could depend either on scale and time.

Usually it is assumed that the bias on large scale is independent

BAO in the galaxy power spectrum on scale hence, in the matter power spectrum, this term appears as a multiplicative factor which modulate the overall amplitude of the galaxy power spectrum. It is worth noticing that there are works which pointed out that this assumption is too strong and it may be wrong, see [13].

• Redshift distortions: an observer can only measure the galaxy power spectrum in redshift space, which is distorted compared to the power spectrum in the real space. Redshift distortions are angle dependent distortion on the power spectrum caused by the peculiar velocities of galaxies. In linear theory these will affect structures along the line of sight; i.e. for a structure which is isotropic in the real space, an observer will measure more power in the radial direction than in the transverse direction. On large scales, these distortions follow a simple form and it appears as a multiplicative angle dependent factor in the power spectrum.

• Reference cosmology: takes in consideration the difference of co- moving volumes in two different cosmologies. In practice, the wave vectors k, measured along the line of sight and the trans- verse direction, depend on the cosmology used. As a conse- quence, the power spectrum will be different.

• Shot noise: it is a Poissonian-like noise coming directly from the number of galaxy counted in the survey volume.

All these terms have to be taken into account when observations are made. Another layer of complication is the non linear evolution of the matter density fields. As time goes by matter density fields start to interact with each other breaking the linearity, linear theory does not describe correctly the evolution of structures if observations are made in the local Universe. As a consequence, this will increase the power spectrum at small scales (large wavenumber) starting to become