Relativistic effects in galaxy clustering

Thesis

Reference

Relativistic effects in galaxy clustering

MONTANARI, Francesco

Abstract

This thesis investigates new cosmological information that can be obtained from present and future large-scale galaxy surveys, in order to constrain our theory of gravity. We introduce directly observable cosmological probes and discuss model independent measurements, useful to put complementary constraints on cosmological parameters. We compute, through cosmological perturbation theory, fully relativistic expressions describing the galaxy distribution in the Universe. All relevant observational effects are included, and a code is provided for fast and accurate numerical evaluations. We forecast the capability of future experiments to constrain sub-leading terms contributing to the galaxy spectrum and bi-spectrum. New promising probes are associated in particular with the weak lensing effect on galaxy statistical distributions, which represents an alternative constraint on the lensing potential, therefore on our gravity theory.

MONTANARI, Francesco. Relativistic effects in galaxy clustering. Thèse de doctorat : Univ. Genève, 2015, no. Sc. 4820

URN : urn:nbn:ch:unige-755131

DOI : 10.13097/archive-ouverte/unige:75513

Available at:

http://archive-ouverte.unige.ch/unige:75513

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

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UNIVERSIT´E DE GEN`EVE FACULT´E DES SCIENCES

Section de Physisque Professeure Ruth DURRER

D´epartement de Physique Th´eorique

Relativistic Effects in Galaxy Clustering

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

Francesco MONTANARI de

Tregnago (Italie)

Th`ese N^○ 4820

Gen`eve

Atelier de reproduction de la Section de Physique 2015

Abstract

This thesis investigates new cosmological information that can be obtained from present and future large-scale galaxy surveys, in order to constrain our theory of gravity. We introduce directly observable cosmological probes and discuss model independent mea- surements, useful to put complementary constraints on cosmological parameters. We compute, through cosmological perturbation theory, fully relativistic expressions de- scribing the galaxy distribution in the Universe. All relevant observational effects are included, and a code is provided for fast and accurate numerical evaluations. We fore- cast the capability of future experiments to constrain sub-leading terms contributing to the galaxy spectrum and bi-spectrum. New promising probes are associated in par- ticular with the weak lensing effect on galaxy statistical distributions, which represents an alternative constraint on the lensing potential, therefore on our gravity theory.

iii

Contents

Acknowledgements xi

Jury Members xiii

List of Publications xv

R´esum´e xvii

1 Introduction 1

I Theoretical Aspects 7

2 Galaxy Number Counts 9

2.1 Relativistic Number Counts . . . 10

2.1.1 Redshift Perturbations . . . 12

2.1.2 Volume Perturbations. . . 14

2.1.3 Additional Effects and Final Form . . . 18

2.2 Kaiser Approximation . . . 20

2.3 Lensing Magnification . . . 22

2.4 Sub-Leading Contributions . . . 24

3 Observables 27 3.1 Angular Power Spectra . . . 28

3.1.1 Number Counts Transfer Functions . . . 29

3.2 Correlation Function . . . 32

3.2.1 Expansion in Tripolar Spherical Harmonics . . . 35

3.2.2 Coordinate System . . . 36

3.2.3 Plane-parallel limit . . . 37 v

3.2.4 Observable Correlation Function . . . 39

II Applications 41

4.2 Forecasts with relativistic correlations . . . 47

4.3 Relativistic bispectrum . . . 52

5 New Method for Alcock-Paczynski Test 55 5.1 Introduction . . . 55

5.2 Generalities . . . 56

5.3 The Alcock-Paczy´nski test . . . 64

5.3.1 Transverse correlation function. . . 66

5.3.2 Longitudinal correlation function . . . 68

5.4 Conclusions . . . 72

5.5 Appendix: Wide-angle RSD using the tripolar expansion . . . 73

6 CLASSgal 79 6.1 Introduction . . . 79

6.2 CLASSgal, a code for LSS . . . 81

6.3 Power spectra and correlation functions . . . 87

6.4 Example . . . 92

6.5 Conclusions and outlook . . . 95

6.6 Appendix: Differences between class and classgal. . . 96

6.6.1 Conventions and notations used in the code . . . 96

6.6.2 Modifications to the input module. . . 97

6.6.3 Modifications to the perturbation module . . . 98

6.6.4 Modifications to the transfer module . . . 99

6.6.5 Modifications to the spectra module . . . 102

6.7 Appendix: Luminosity fluctuations . . . 103

7 Parameter Estimation with LSS Observations 105 7.1 Introduction . . . 105

7.2 Number counts versus the real space power spectrum . . . 107

7.3 The Fisher matrix and the nonlinearity scale . . . 110

7.3.1 Fisher matrix forecasts . . . 110

7.3.2 The nonlinearity scale . . . 111

7.4 Results . . . 113

7.4.1 A Euclid-like catalog . . . 113

CONTENTS vii

7.4.2 A DES-like catalog . . . 119

7.4.3 Measuring the lensing potential . . . 122

7.4.4 The correlation function and the monopole . . . 125

7.5 Conclusions . . . 129

7.6 Appendix: The galaxy number power spectrum . . . 130

7.7 Appendix: Basics of Fisher matrix forecasts. . . 130

8 Measurements with Relativistic Correlations 135 8.1 Introduction . . . 136

8.2 Galaxy correlations in General Relativity . . . 137

8.3 Future galaxy surveys . . . 139

8.3.1 Survey specifications . . . 139

8.3.2 Fisher matrix analysis . . . 142

8.4 Parameter Estimation with Relativistic Correlations. . . 143

8.4.1 Dynamical Dark Energy . . . 145

8.4.2 Initial conditions from Inflation . . . 146

8.5 Cosmological information in relativistic corrections . . . 150

8.6 Summary of results . . . 151

8.7 Conclusions . . . 152

8.8 Appendix: primordial NG in Class . . . 154

8.8.1 Definitions and notation . . . 154

8.8.2 Modifications of class . . . 156

9 Lensing Potential with Galaxy Number Counts 159 9.1 Introduction . . . 159

9.2 Determining the lensing potential with number counts . . . 161

9.2.1 Galaxy number counts . . . 162

9.2.2 The contributions to the number counts. . . 164

9.2.3 Measuring the cross correlation ⟨D(zi)κ(zj)⟩. . . 165

9.3 Lensing constraints from tomographic number counts . . . 170

9.3.1 Euclid forecasts . . . 172

9.3.2 SKA forecasts . . . 175

9.4 Conclusions . . . 177

9.5 Surveys specifications . . . 179

9.5.1 Euclid . . . 179

9.5.2 SKA . . . 181

9.6 Euclid magnification bias . . . 182

9.7 Spin weighted spherical harmonic decomposition of the lens map . . . . 184

9.8 The Fisher matrix . . . 185

10 Galaxy Number Counts Bispectrum 189

10.1 Introduction . . . 189

10.2 Galaxy Number Counts . . . 191

10.3 From the Geodesic Light-Cone to the Poisson gauge. . . 192

10.4 Evaluation of the number counts to second order. . . 196

10.4.1 Redshift density perturbation . . . 196

10.4.2 Volume perturbation . . . 198

10.4.3 Number Counts . . . 207

10.4.4 Number Counts: leading terms . . . 211

10.5 The Bispectrum and its numerical evaluation . . . 212

10.5.1 Density . . . 213

10.5.2 Redshift space distortions . . . 215

10.5.3 Lensing . . . 216

10.5.4 Numerical Results . . . 217

10.6 Conclusions . . . 220

10.7 Appendix: Useful relations . . . 222

11 Conclusions and Outlook 225 A Notation 229 B Cosmological Perturbation Theory 233 B.1 The Background Universe. . . 233

B.1.1 Friedmann Equations . . . 234

B.1.2 Distances . . . 236

B.2 The Perturbed Universe. . . 238

C Scalar Harmonic Functions 243 D Expansion in Tripolar Spherical Harmonics 247 D.1 Tripolar spherical harmonics . . . 247

D.1.1 Special case: L=0 . . . 249

D.2 Wigner 3-j Symbol. . . 250

D.3 Coefficients of the Tripolar Expansion . . . 251

So my mind sinks in this immensity:

and floundering is sweet in such a sea.

L’infinito - G. Leopardi

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Acknowledgements

It is a pleasure to thank Ruth for her teachings, support, and friendship since when she accepted to supervise my master thesis, and then my doctorate. Her example gave me the possibility to learn a lot and to appreciate even more physics.

I had the opportunity to spend fantastic years in Geneva. I will keep especially the good moments passed at the Department of Physics, for which I thank the cosmology group. I also acknowledge the help of the secretaries Francie Gennai-Nicole, who hosted me during my first year, and C´ecile Jaggi, as well as the aid of Andreas Malaspinas with hardware and software.

I am grateful to Alexandre Refregier for having hosted me at ETH Zurich, Camille Bonvin at the University of Cambridge, Luca Amendola and Valeria Pettorino at the University of Heidelberg and Alvise Raccanelli, Marc Kamionkowski and Alex Szalay at the Johns Hopkins University. Each visit has been very valuable.

I would like to thank my collaborators: Daniele Bertacca, Enea Di Dio, Olivier Dor´e, Ruth Durrer, Julien Lesgourgues, Giovanni Marozzi and Alvise Raccanelli. I am also grateful to Enea Di Dio for his fundamental contribution to the computation presented in section 2.1, and to Yves Dirian for having accepted the hard work of correcting my French introduction.

A special thanks goes to the friends who made pleasant my staying in Geneva:

Alba, Enea, Giovanni, Guillermo, Julian, Kwan, Matteo, Pietro and Wilmar.

My achievements would have not been possible without the constant support, de- spite the distance, from my parents and my sister Laura. I also thank G¨on¨ul and Kˆazım for their invaluable presence.

I am extremely grateful to my wife for her patience and confidence. Thanks, ˙Ipek, for having always been there to encourage me.

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Jury Members

ˆ Dr. Camille Bonvin

CERN, Theory Division, 1211 Geneva, Switzerland

ˆ Prof. Ruth Durrer

D´epartement de Physique Th´eorique & Center for Astroparticle Physics, Universit´e de Gen`eve, Quai E. Ansermet 24, CH-1211 Gen`eve 4, Switzerland

ˆ Dr. Martin Kunz

D´epartement de Physique Th´eorique & Center for Astroparticle Physics, Universit´e de Gen`eve, Quai E. Ansermet 24, CH-1211 Gen`eve 4, Switzerland and

AIMS, 6 Melrose Road, Muizenberg, 7945, South Africa

ˆ Prof. Alexandre Refregier

Institute for Astronomy, Department of Physics,

ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland

I would like to thank them for having accepted to be part of the jury for my Thesis.

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List of Publications

The following works have been considered as part of the thesis:

[200] F. Montanari and R. Durrer, “A new method for the Alcock-Paczynski test,”

Phys. Rev. D 86 (2012) 063503 [arXiv:1206.3545[astro-ph.CO]].

[106] E. Di Dio, F. Montanari, J. Lesgourgues and R. Durrer, “The CLASSgal code for Relativistic Cosmological Large Scale Structure,” JCAP 1311 (2013) 044 [arXiv:1307.1459 [astro-ph.CO]].

[105] E. Di Dio, F. Montanari, R. Durrer and J. Lesgourgues, “Cosmological Parameter Estimation with Large Scale Structure Observations,” JCAP 1401 (2014) 042 [arXiv:1308.6186 [astro-ph.CO]].

[224] A. Raccanelli, F. Montanari, D. Bertacca, O. Dor´e and R. Durrer, “Cosmological Measurements with General Relativistic Galaxy Correlations,”arXiv:1505.06179 [astro-ph.CO].

[201] F. Montanari and R. Durrer, “Measuring the lensing potential with galaxy clus- tering,” arXiv:1506.01369[astro-ph.CO].

[103] E. Di Dio, R. Durrer, G. Marozzi and F. Montanari, “Galaxy number counts to second order and their bispectrum,” JCAP 1412(2014) 017 [arXiv:1407.0376 [astro-ph.CO]].

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Additional publications related to the thesis:

[199] F. Montanari and R. Durrer, “An analytic approach to baryon acoustic oscilla- tions,” Phys. Rev. D 84 (2011) 023522 [arXiv:1105.1514 [astro-ph.CO]].

[221] A. Raccanelli, E. Di Dio, F. Montanari, R. Durrer, M. Kamionkowsky and J. Les- gourgues, “Curvature constraints from Large Scale Structure,” in preparation.

[104] E. Di Dio, R. Durrer, G. Marozzi and F. Montanari, “The bispectrum of rela- tivistic Galaxy number counts,” in preparation.

R´esum´e

Le dernier si`ecle a ´et´e caract´eris´e par des progr`es remarquables quant `a la descrip- tion de l’´evolution de notre univers. Sur le point de vue th´eorique, la publication des travaux d’Albert Einstein sur la th´eorie de la Relativit´e G´en´erale en 1917 a jet´e les bases qui ont donn´e naissance `a la cosmologie moderne. Le principe fondamental est celui de l’´equivalence entre les masses inertielles et gravitationnelles, ce qu’Einstein a su traduire au niveau g´eom´etrique. Ceci a permis d’obtenir les ´equations fondamentales d´ecrivant une m´etrique homog`ene et isotrope, dite de Friedmann-Lemaˆıtre-Robertson- Walker (FLRW), qui est le fondement de la th´eorie des perturbations cosmologiques actuelle. En 1929, Hubble a publi´e des r´esultats sur la relation entre vitesse et dis- tance qui ont confirm´e l’id´ee d’un univers en expansion. La d´ecouverte du fond diffus cosmologique (ou CMB, issu de l’anglais) par Penzias et Wilson en 1964 a ´egalement permis d’importants d´eveloppements sur le point de vue observationnel. Le succ`es de l’´etude du CMB `a travers des satellites tels que COBE, WMAP et, plus r´ecemment Planck, a donn´e la possibilit´e de contraindre fortement notre mod`ele cosmologique.

Le CMB a ´et´e ´emis 400 000 ans apr`es le Big Bang, son spectre de radiation suivant celui d’un corps noir avec une pr´ecision meilleure qu’un sur 10⁵, ceci nous empˆeche encore aujourd’hui de pouvoir observer ses distorsions spectrales. Toutefois, l’´etude des anisotropies spatiales du CMB a permis de conclure que les conditions initiales de l’univers, qui ont permis la naissance des structures cosmiques observ´ees aujourd’hui, indiquent une p´eriode d’expansion acc´el´er´ee primordiale qu’on appelle l’inflation cos- mique.

D’autre part, d’importants probl`emes sont encore pr´esents dans notre mod`ele cos- mologique. Premi`erement, une th´eorie quantique de la gravit´e, utile par exemple pour mieux comprendre la phase inflationnaire de l’univers, n’a pas encore ´et´e ´elabor´ee.

Cette limitation est aussi li´ee au probl`eme de la constante cosmologique. On observe aujourd’hui un univers vieux de 13.8 milliard d’ann´ees, compos´e de mati`ere bary- onique `a 4.9%, de mati`ere sombre (qui interagit seulement gravitationnellement) `a 26.6%, et `a 68.5% d’une composante appel´ee ´energie sombre qui cause une r´ecente expansion acc´el´er´ee de l’univers. Cette derni`ere peut ˆetre mod´elis´ee par une constante cosmologique Λ dans les ´equations d’Einstein. Toutefois, sa valeur est tr`es sensible aux corrections quantiques. Dans le contexte du Mod`ele Standard des Particules ceci

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n’est pas un probl`eme en soi, car la valeur physique de Λ ne peut pas ˆetre fix´ee par la th´eorie mˆeme, mais n´ecessite une mesure. Le probl`eme se situe plutˆot dans le fait qu’une future th´eorie fondamentale qui peut pr´edire la valeur de Λ peut requ´erir un ajustement fin des param`etres pour donner lieu `a la valeur de Λ observ´ee. De plus, il existe un probl`eme de co¨ıncidence: pourquoi aujourd’hui la valeur de la densit´e d’´energie sombre est du mˆeme ordre que celle de la densit´e de mati`ere, compte tenu du fait que Λ ´etait n´egligeable durant la plus grande partie de l’histoire de l’univers, et sera dominante dans le future? Des modifications de la gravit´e `a basse ´energie ont ´et´e envisag´ees comme solutions possibles `a ces deux derniers probl`emes, mais un candidat, alternatif `a la Relativit´e G´en´erale, n’a pas encore ´et´e trouv´e.

Compte tenu de ces probl`emes, il est donc d’une importance extrˆeme de contraindre le mod`ele cosmologique, en y incluant la th´eorie de la gravit´e elle-mˆeme. Mises `a part les observations du CMB, la structure `a grande ´echelle de l’univers est ´egalement une autre observable importante. Celle-ci est accessible, par exemple, grˆace `a l’observation de la position des galaxies dans le ciel et de leur d´ecalage spectral vers le rouge (red- shift en anglais). Des observations telles que DES, Euclid, LSST, SKA vont fournir des donn´ees tr`es pr´ecises. La th´eorie doit donc ´egalement s’adapter pour atteindre une telle pr´ecision. Par rapport `a la physique lin´eaire et bi-dimensionnelle du CMB, le probl`eme est que les catalogues de galaxies contiennent une information tridimen- sionnelle et, `a moyennes et petites ´echelles, non-lin´eaire. `A grandes ´echelles la th´eorie des perturbations cosmologiques peut encore ˆetre utilis´ee.

Le but de cette th`ese est d’´evaluer la capacit´e des prochaines observations de galax- ies, afin de contraindre les effets relativistes `a grandes ´echelles. Dans la premi`ere partie on montre comment tenir compte, au premi`ere ordre de la th´eorie des perturbations, des effets relativistes qui influencent la densit´e de galaxies en fonction des angles et du redshift. Leurs origines se situe dans le changement de coordonnes de la m´etrique de r´ef´erence FLRW, par rapport aux vraies coordonn´ees observ´ees (angles et redshifts).

On calcule aussi le spectre de puissance et la fonction de corr´elation angulaire, preuves observationnelles fondamentales pour la cosmologie, en termes des coordonn´ees ob- serv´ees.

Dans la deuxi`eme partie de la th`ese, on expose nos r´esultats principaux. Dans le chapitre 5, reproduisant la r´ef´erence [200], on calcule la fonction de corr´elation `a deux points qui tient compte de certains termes de vitesse importants `a grandes ´echelles, nor- malement n´eglig´es dans les analyses statistiques standard de la distribution de galax- ies. Le r´esultat est ´ecrit en termes d’angles et de redshifts directement observables, et repr´esente donc une analyse qui ne requiert pas l’utilisation d’un mod`ele cosmologique pour convertir ces observables en termes de coordonn´ees de la m´etrique FLRW de r´ef´erence. Ce dernier passage est aujourd’hui effectu´e dans la majeure partie des anal- yses, notre m´ethode est donc une analyse compl´ementaire. En tant qu’application, on consid`ere la r`egle cosmologique donn´ee par l’´echelle des Oscillations Acoustiques

xix des Baryons g´en´er´es durant l’univers primordial, qui peuvent ˆetre consid´er´ees comme des distributions statistiquement sph´eriques. Avec une mesure radiale (en termes de redshift) et angulaire de cette ´echelle par la fonction de corr´elation, on peut effectuer le test de Alcock-Paczi´nski pour contraindre le produit de la constant d’Hubble par la distance angulaire.

Dans le chapitre6, reproduisant la r´ef´erence [106], on d´ecrit un code qui calcule effi- cacement le spectre de puissance angulaire des galaxies et leurs fonctions de corr´elation

`

a deux points. Le code,Classgal, a ´et´e obtenu `a partir d’un code d´ej`a existant pour le CMB, Class. Toutes les corrections relativistes sont inclues au niveau lin´eaire, ainsi que des effets observationnels qui permettent d’effectuer des pr´evisions pr´ecises sur les capacit´es des prochaines observations de galaxies.

Dans le chapitre7, reproduisant la r´ef´erence [105], on utiliseClassgal pour analyser en d´etail les capacit´es des prochaines observations telles que DES et Euclid pour con- traindre le spectre de puissance angulaire. On utilise des techniques bas´ees sur les ma- trices de Fisher, pour lesquelles on ´elabore des algorithmes qui excluent soigneusement les ´echelles non-lin´eaires de l’analyse. On compare le spectre angulaire au spectre dans l’espace de Fourier, plus souvent employ´e pour ce type d’analyses. Pour des catalogues dont le redshift n’est pas d´etermin´e avec une grande pr´ecision, au profit d’un nom- bre plus grand de galaxies dans le catalogue, le spectre angulaire donne des r´esultats comp´etitifs, qui fournissent donc des observations compl´ementaires importantes. On inclue tous les termes relativistes dans l’analyse. `A part les termes de densit´e et de vitesse usuels, on trouve aussi que l’effet de lentille gravitationnelle peut ˆetre important.

Par contre, ceci est normalement exclus lors de l’analyse des corr´elations du nombre de galaxies. Enfin, on montre ´egalement que le monopˆole et le dipˆole du spectre angulaire peuvent ˆetre combin´es pour obtenir de l’information accessible perturbativement. Ces termes sont normalement exclus de toute analyse cosmologique, car ils sont affect´es par le potentiel gravifique et la vitesse non-lin´eaires de l’observateur.

Dans le chapitre8, reproduisant la r´ef´erence [224], on s’int´eresse `a savoir si n´egliger les effets relativistes dans l’analyse de la corr´elation du nombre de galaxies peut affecter s´erieusement les contraintes des param`etres cosmologiques, ou si cela ne donne pas de biais. Grˆace `a une analyse bas´ee sur des matrices de Fisher pour des observations telles que Euclid et SKA, on compare les mesures des param`etres cosmologiques dans deux univers hypoth´etiques. Dans le premier les effets relativistes ne sont pas pr´esents, alors que dans le deuxi`eme ils sont inclus. Du fait que les deux cas donnent des r´esultat tr`es diff´erents, cela indique que certains termes ne sont pas n´egligeables. On identifie que la contribution la plus importante vient du terme qui tient compte de l’effet de lentille gravitationnelle, consid´erable `a grandes s´eparations radiales. Lorsque ces s´eparations sont pr´esentes dans l’analyse, ce terme ne peut pas ˆetre n´eglig´e.

Dans le chapitre 9, reproduisant la r´ef´erence [201], on ´etudie plus en d´etail le terme causant l’effet de lentille gravitationnelle. Celui-ci donne une amplification de la

dimension des sources qui ont ´et´e d´etect´ees dans les corr´elations du nombre de galaxies

`

a bas redshift, avec des quasars en arri`ere plan `a haut redshift. On montre que cette amplification est certainement importantes sur de grandes s´eparations radiales. Ceci n’est pas vrai seulement pour la corr´elation des deux populations bien s´epar´ees, mais aussi dans des analyses tomographiques qui consid`erent des groupes de galaxies `a plusieurs redshifts. D’apr`es notre analyse bas´ee sur des matrices de Fisher pour Euclid et SKA, cet effet d’amplification peut ˆetre mesur´e avec la mˆeme pr´ecision que d’autres param`etres cosmologiques. Cet observation est compl´ementaires `a d’autres mesures de lentilles gravitationnelles.

Dans le chapitre10, reproduisant la r´ef´erence [103], on calcule les effets au deuxi`eme ordre sur la statistique de la distribution des galaxies. Ceci est fait avec une approche innovante, bas´ee sur un syst`eme de coordonn´ees observable r´ecemment propos´e en litt´erature. Le r´esultat est valable pour des th´eories g´en´eriques de gravit´e, tant que les galaxies se d´eplacent sur des g´eod´esiques. On calcule le bi-spectre angulaire pour les termes les plus importants. L’´etude montre que les effets de lentille gravitationnelle, normalement n´eglig´e dans ce type d’analyse, peuvent aussi ˆetre importants dans ce cas, ce qui justifie de futures travaux plus d´etaill´es.

On conclue que l’´etude des fonctions de corr´elations en termes des quantit´es directe- ment observables (redshifts et angles) fournit des mesures compl´ementaires importantes pour estimer les param`etres cosmologiques avec une pr´ecision de l’ordre du pourcent, et donc pour contraindre notre mod`ele cosmologique, notamment notre th´eorie de gravit´e,

`

a grandes ´echelles.

Chapter 1

Introduction

The last century witnessed remarkable advances in the description of the evolution of our Universe. From the theoretical side, Albert Einstein’s publication in 1917 about his modifications to the theory of gravity was a fundamental breakthrough [114]. The underlying Principle of the Equivalence of Gravitation and Inertia governs the response of a physical system to an external gravitational field and has been tested through the search of variation of physical constants [286,265,280]. Einstein elaborated the Equiv- alence Principle in terms of a geometrical approach providing a rigorous and testable theory. Pioneering applications of General Relativity (GR) to cosmology was studied independently by Friedman and Lamaˆıtre, who derived a set of equations governing the expansion of space in a homogeneous and isotropic model of the Universe, starting from Einstein’s fields equations [125, 164]. Lamaˆıtre also derived what is known as the Hubble’s law, one of the most studied phenomena in relativistic cosmology. These equations rely on the Cosmological Principle—the distribution of matter in the uni- verse is homogeneous and isotropic when viewed on a large enough scale—and are the basis of the standard Big-Bang model. The metric representing the solution of Ein- stein’s field equations in a homogeneous and isotropic Universe was further thoroughly studied by Robertson and Walker who proved its uniqueness [235, 284]. We refer to this metric as Friedmann-Lamaˆıtre-Robertson-Walker (FLRW).

Hubble published in 1929 his velocity-distance relation confirming the idea of an expanding Universe [144]. The discovery by Penzias and Wilson in 1964 of the Cosmic Microwave Background (CMB) [214] allowed dramatic advances from the observational side. The success story of the CMB through experiments such as the COBE, WMAP and Planck satellites proved that cosmology has entered a precision era [43, 262, 8].

Data showed that we live in a 13.8 billion years old Universe composed by 4.9% of baryonic matter, 26.6% of Cold Dark Matter (CDM) interacting only gravitationally, and 68.5% of Dark Energy (DE) behaving like a cosmological constant Λ. The fact that the CMB is a nearly 2-dimensional surface governed by linear physics on a wide range

1

of scales, allowed theoretical cosmological prediction to be tested [110, 112]. In par- ticular, constraints have been established about inflation—the exponential expansion of space in the early Universe. According to the modern paradigm, quantum fluctua- tions in the microscopic inflationary region, magnified to cosmic size and become the seeds for the growth of structure in the Universe [202]. The lack of a quantum grav- ity theory prevent us to describe the unphysical Big Bang singularity. Nevertheless, inflation solves important issues like the horizon problem—why the Universe appears statistically homogeneous and isotropic?—, the flatness problem—why the Universe is so fine-tuned close to an Euclidean geometry?—and the missing magnetic-monopole problem.¹

The study of CMB anisotropies also put strong constraints on the thermal history of the Universe. One of the main feature of the temperature spectrum is the presence of acoustic oscillations originating from the primordial plasma in which photons and baryons were tightly coupled by Thomson scattering. Their detection is an excellent probe of the matter content of the Universe and of the amplitude of primordial fluc- tuations, down to scales where Silk damping dominates, putting a lower limit to the accessible scales. Temperature and polarization data have been used to constrain in- flationary model through the study of the bispectrum [31], and last results from the Planck satellites suggest that the structure we observe today was sourced by adiabatic, Gaussian, and primordial seed perturbations [13]. CMB lensing, a second-order effect, is particularly useful to break the degeneracy between the reionization optical-depth and the amplitude of the temperature spectrum [170, 11]. The challenging study of primordial polarization B-modes is an exciting opportunity for an indirect detection of gravitational waves from inflation [252,150], and observations are planned to reach this aim (e.g., the BICEP3 telescope²). Primordial CMB spectral distortions have also proven elusive so far, confirming that CMB spectrum is extremely close to a Black Body up to a precision of 10⁻⁵. Their analysis would reach scales not accessible to other CMB probes and it would investigate more accurately the thermal history of the Universe [83].

Important theoretical and observational advances also concern the Large Scale Structure (LSS) of the universe. Gravitational instability drives initial overlapping density fluctuations of random Gaussian fields into a structure characterized by walls, filaments, clumps of matter, which are separated by voids [256]. Differently from the CMB physics, LSS requires a full 3-dimensional analysis. Larger theoretical and computational efforts are needed, but more modes are available providing richer in- formation. Numerical simulations are required to model the deep non-linear regime of structure formation. In particular, galaxy formation is still a poorly understood

1We stress that unlike the horizon and flatness problems, which rely on compelling observational evidence and are fundamental issues, the monopole problem concerns the lack of detection of a hypo- thetical particle.

2https://www.cfa.harvard.edu/CMB/bicep3/

3 process lacking of a fundamental and predictive theory [179]. The understanding of how galaxies are related to the underlying Dark Matter (DM) potential wells is a chal- lenging issue that goes under the name of galaxy bias. Nevertheless, the Halo Model has been successfully been employed to describe virialized DM halos. The distribution of DM within each halo is used to provide estimates of statistical properties like the large scale density and velocity fields evolution, weak gravitational lensing as well as estimates of secondary contributions to temperature fluctuations in the CMB [87]. The accuracy of this analytic approach can be tested against exact results from numerical simulations of nonlinear gravitational clustering, and it is crucial for our understand- ing of the Universe on scales up to O(10 Mpc). Weak lensing mass reconstruction and shear measurements allows fine constraints on cosmological parameters [231,249].

Lyman-α forest observations investigate the early epoch of structure formation, al- lowing constraints on the neutral fraction of hydrogen even before the universe was completely re-ionized [285].

The statistical analysis of galaxy clustering has been one of the most rewarding probes of the LSS through galaxy redshift surveys such as CfA [131], SDSS [297], 2dF [85], Wigglez [57], 6dF [52]. These observations provided independent cosmolog- ical parameter constraints, complementary to CMB measurements. In particular, the discovery of Baryon Acoustic Oscillations (BAO) [115] confirmed previous theoretical predictions [116], providing a cosmological standard ruler important to improve mea- surements of distances besides information about the physics of the primordial plasma present before the recombination epoch. Another signature of the primordial Universe in the LSS is given by non-Gaussian initial condition, analyzable throughn-points cor- relation functions [100]. Galaxy surveys have also been used to constrain the properties and evolution of Dark Energy (DE), e.g., through measurements of the growth rate.

The development and constraining of Modified Gravity (MG) theories is one of the main tasks for present and future large scale surveys like DES³, Euclid [162,18], LSST [2] and SKA [185]. The standard ΛCDM model, based on General Relativity and the Standard Model of particle physics, is in remarkable agreement with observations from CMB, LSS and supernova luminosities. On the other hand, GR is affected by important theoretical problems [113]. First, it breaks down at high energies, of the order of the Planck scale E ≳ Mpl ≃ 10¹⁹ GeV, which would be probed by a still elusive quantum theory of gravity. Second, the simplest option to explain late-time accelerated expansion is to incorporate a cosmological constant Λ. Since the classical gravitational constant Λ enters the Einstein equations in the same way as the quantum vacuum energy, the two cannot be distinguished in observations, giving rise to thefine tuning problem of the cosmological constant [287]. In Quantum Field Theory (QFT), given the cut-off scale Ec of the theory, the vacuum energy density scales as ρvac∼E_c⁴ bringing the cut-off term Λvac ≃ E_c⁴/M_pl² that has to be added to the immeasurable

3http://www.darkenergysurvey.org/

bare value Λ_bare. IfEc=Mpl, then Λ_vac≃10³⁸ GeV². Cosmological observations fix the physical value Λ=Λ_vac+Λ_bare≃10⁻⁸³ GeV², which requires a cancellation of about 120 orders of magnitude. However, Λ_bare is not a physical quantity and only the effective Λ can be measured, so this cancellation is not a fundamental issue in the strict context of the Standard Model. Indeed, in this case we can not derive the physical Λ parameter from first principles as it is only defined through a process—renormalization—that includes experimental measurements. The worry is that a future theory of fundamental particles, in which Λ will be calculable, should not have excessive fine-tunings. E.g., if supersymmetry is unbroken than it predicts a vanishing expectation value for the vacuum energy [287]. In general a spontaneous symmetry breaking would induce a vacuum energy several orders of magnitude larger than the observed Λ≃10⁻⁸³ GeV². Taking into account also a possible running of Λ due to quantum effects [258], the solution to the cosmological constant fine tuning is still challenging. A third important issue is the coincidence problem [288], namely the fact that ρΛ is of the order of the present matter density ρm(z0), where z is the redshift. Since ρΛ=Λ/8πG is constant and the matter density scales as ρm ∝ (1+z)³, DE was negligible in most of the past and will dominate in the future.

Instead of a cosmological constant, DE energy can be modeled with a scalar field with equation of state w< −1/3 (where w = −1 recovers a cosmological constant). In this case, the coincidence problem may be solved if a particular event in the Universe could be identified as a trigger for DE. More in general, an infrared modification of GR that still accounts for late-time acceleration may be considered. So far, there are no convincing alternatives to GR satisfying solar-system constraints, not being af- fected by instabilities, and that would solve the cosmological constant problems [113].

In the framework of GR, acceleration may also be produced by the backreaction of inhomogeneities on the background, treated via non-linear averaging [71]. Another possibility is that our Universe resembles a Tolman-Bondi-Lemaˆıtre solution with our galaxy cluster at the center, then data would not necessarily imply a late-time acceler- ated expansion. In fact it is important to recall that while isotropy is well constrained by CMB and LSS, homogeneity is still an hypothesis dictated by the Cosmological Principle.

Furthermore, a direct detection of gravitational waves is still missing 100 years after Einstein’s formulation of gravity. Such an observation is a fundamental test of the theory and would help to narrow down the available theories of gravity. Future experiments like the eLisa interferometer⁴ are expected to be sensitive enough to detect gravitational waves originating from massive black holes or binaries of compact stars in our Galaxy.

These issues motivate efforts in constraining gravity also by means of cosmology.

Future constraints will come mainly from LSS analysis. At scales well inside the Hub-

4https://www.elisascience.org/

5 ble horizon, GR is replaced by Newtonian gravity and studied with N-body simulations [247] or analytic methods based on the Zel’dovich approximation [198]. On large scales it is necessary to consider that we observe along our past light-cone and relativistic effects must be included for the correct interpretation of the data [118]. Efforts to include GR corrections to N-body simulations resort to the post-Newtonian approxi- mation, which restore the speed of light c in all equations and then expand in powers of 1/c. The main drawback is its intrinsic limitation to non-relativistic sources, as the expansion do not converge for a contribution from non-relativistic species (e.g., neu- trinos). Another approach consists in assuming that the true metric of the Universe is close to that given by the FLRW symmetries, except in the vicinity of strong field objects such as neutron stars and black holes [132]. The key point is to note that this does not imply that also the derivatives of the metric are close to those of the FLRW metric. Second derivatives (i.e., the curvature) can be very different, so that the actual stress-energy tensor of our Universe is far from possessing the FLRW symmetry. Ap- plications of this approximation showed that deviations from Newtonian physics are small, but may be relevant for precise cosmology especially to analyze models beyond ΛCDM [5].

In the large scale limit and when the growth of structures is at an early time, LSS is more easily studied via perturbation theory around a homogeneous and isotropic FLRW background [45]. The main approaches are the Lagrangian and Eulerian pic- tures. In the former case one considers perturbed trajectories about the initial position of fluid elements, while in the latter framework the density and velocity fields are per- turbed. The Lagrangian picture is therefore more powerful than the Eulerian one as it is intrinsically non-linear in the density field.

The scope of the thesis is to study relativistic effects in galaxy clustering through Eulerian perturbation theory and to forecast their detectability for future surveys like Euclid. The work is divided into two parts. In part I, we review the fully relativistic treatment of galaxy number counts. In chapter 2 we define number counts in terms of the observable angles and redshifts. We compute redshift and volume perturbations, and include all relevant observational effects. We compute, allowing for a generic curvature parameter, the main equation that motivates our applications. We derive the Kaiser approximation to redshift-space distortions. We also review well known weak lensing results describing the magnification effect, and verify that this is consistently included in the fully relativistic number counts. We provide dimensional arguments to estimate the relative importance of all the terms contributing to number counts, and give physical interpretation for each of them. In chapter 3 we compute the fully relativistic angular power spectrum, providing all the transfer functions in a form useful for implementation in numerical codes. We also show the detailed expansion in tripolar spherical harmonics of the angular two-point correlation function, including velocity terms important on large scales.

In part II we present our applications of the fully relativistic formalism. After a short overview summarizing the main results in chapter4, we introduce a series of self- contained chapters reproducing our publications. In chapter 5 we show how the two- point correlation function, written in terms of angles and redshifts, can be employed to obtain model independent constraints through an Alcock-Packzi´nski test. In chapter6 we compute the fully relativistic power spectrum including all observational terms, and implement it in a code, Classgal, for fast and accurate computations. In chapter 7 we compare tomographic analysis obtained through the angular power spectra, to more traditional analysis in comoving Fourier space. We also estimate numerically the relative importance of all the terms contributing to the relativistic correlations. Finally, we show that the monopole and dipole of the angular power spectra can in principle be constrained at the cosmological level when considering the information from different redshifts. In chapter 8 we estimate qualitatively the bias introduced when neglecting relativistic terms in angular power spectra analysis. We find that lensing magnification is a substantial contribution when several redshift cross-correlations are included. In chapter 9 we study in detail at what level the lensing potential can be constrained by future galaxy survey, and find that competitive constraints are expected. In chapter10 the computation of number counts at second order in perturbation theory is presented.

The bi-spectrum of the leading terms is calculated, showing that lensing can be an important contribution also in this case. In chapter11we conclude and discuss possible future developments.

In appendixAwe set the notation used in partI. Note that some convention change in part II, where nevertheless each chapter is meant to be self-contained. Appendix B reviews standard cosmological perturbation theory and it is mainly meant to set the notation as well. In appendix C we derive the scalar harmonic functions serving as a basis for expansion in harmonic space. In appendix Dwe present the tripolar spherical harmonics, we recall useful properties of the Wigner 3-j symbols and the show the detailed computation of the coefficients of the two-points correlation function expanded over this basis.

Part I

Theoretical Aspects

7

Chapter 2

Galaxy Number Counts

In this chapter we review the general relativistic description of galaxy clustering at linear order in perturbation theory. The seminal paper of Kaiser [149] first introduced the issue of mapping comoving coordinates, defined on the isotropic and homogeneous background, to the observed redshift taking into account the contribution from galaxy peculiar velocities. The effect of perturbations on the angular coordinates, giving rise to the magnification of sources through weak lensing, was considered, e.g., in [68,197].

A fully consistent computation including all redshift and volume perturbation contri- butions to the observed coordinates was first derived in [293,290] and [64, 80], which also computed the angular power spectrum. A thoroughly discussion of additional ef- fects like galaxy bias, primordial non-Gaussianities and gravitational waves is presented in [148, 145]. Reviews of relativistic effects in the context of perturbation theory are given by [291,63,147]. Useful approaches to include GR terms in non-linear N-bodies codes are discussed in [5,132,229,283].

Applications to the number density of galaxies have been studied in several works.

For standard analysis involving even multipoles of the correlation function, the power spectrum in Fourier space or its generalization to the spherical-Bessel Fourier trans- form, only redshift-space distortions (RSD) in the Kaiser approximation are relevant [294, 292, 295]. Tomographic analysis correlating angular power spectra between dif- ferent redshift bins are less sensitive to RSD, while the contribution from lensing con- vergence is substantial for large radial cross-correlations [106, 224, 16, 201]. Fast and accurate codes have been developed for the computation of the fully relativistic linear power spectra [80, 106]. The possibility of isolating relativistic terms has been inves- tigated in [66, 228] through asymmetric galaxy correlation functions for which odd multipoles do not vanish, or through the imaginary power spectrum [194]. Constraints including modified gravity scenarios are discussed in [178,26]. Other interesting appli- cations are presented in [27, 69, 180, 50, 134, 186]. Computations up to second order in perturbation theory are given in [296, 48, 49, 46] and in [103, 104, 153], where the

9

bispectrum of the main terms is also computed.

Relativistic terms have also been investigated in relation to different observables such as cosmic clocks [146] and cosmic rulers [243]. Standard clocks are defined as set of events whose proper time is accurately known. An observable can be defined as the difference between a constant-proper-time hypersurface and a constant-observed- redshift surface. Examples include cosmic recombination that originated the CMB, defined by a unique time, or the large scale CMB dominated by the Sachs-Wolf effect, whose time evolution is known. A standard ruler is defined as a know physical scale to which we can compare observations. An example is given by the lensing treatment through galaxy ellipticities, sizes and fluxes, or by distortions of galaxy correlation functions. This framework include scalar, vector and tensor degrees of freedom. It is interesting to note that the vector component and the shear admit a decomposition into E/B–modes, where theB–modes are free of all scalar perturbations at the linear level and represents good candidate to look for tensor perturbations (gravitational waves) [145,244,245]. In general standard rulers evolve in time, so that they may also represent standard clocks.

In section 2.1 we extend results previously obtained in the literature for galaxy number counts at first order in perturbation theory, including non-vanishing curva- ture. After a brief introduction and motivation, more technical sections contain the full computation following closely [221]. The final form is given by eq. (2.55), which represents the main subject of this thesis. In section 2.2we recover the Kaiser approx- imation for number counts and compare it to the fully relativistic result. In section 2.3 we review also another well-known term contributing to number counts, the lensing convergence. Together with the intrinsic clustering term, these are the main contribu- tions to eq. (2.55). In section 2.4 we provide physical interpretations and dimensional arguments for relative order of magnitude estimates of all the remaining sub-leading terms.

2.1 Relativistic Number Counts

Galaxy surveys provide catalogs with information about galaxies redshift z, angular coordinates, fluxes and shapes. In this section we discuss how the knowledge about redshifts and positions in the sky is used to understand and constrain the dynamic of the Universe. On large enough scales, cosmological perturbation theory can be applied [110]. We assume that the true metric of the Universe is close enough to that of a fictitious homogeneous and isotropic FLRW background. We discuss how to relate the artificial coordinates of the FLRW background to the observed ones. The displacement between these coordinate systems gives rise to the so-called relativistic effects. In par- ticular, given that perturbations around the background are not uniquely determined, it is important to compute consistently transformations between different coordinates

2.1. RELATIVISTIC NUMBER COUNTS 11 used to describe the perturbed Universe, through their different correspondence to the background universe. These are known as gauge transformations, and observables must be gauge-invariant in order to respect the fundamental general covariance symmetry of GR (see appendix B). The computation presented in this section follow closely [221].

Being the galaxy number counts an observable quantity, and therefore gauge- invariant, we have the freedom to derive it in any arbitrary gauge. We choose to compute it in the Newtonian longitudinal gauge (see appendix B for definitions)

d˜s² =a²[− (1+2Ψ)dτ²+ (1−2Φ)γijdxⁱdx^j] , (2.1) where the spatial part of the metric is

γijdxⁱdx^j = [dr²+S_K² (r) (dθ²+sin²θdϕ²)] , (2.2) and

SK(r) =

⎧⎪

⎪⎪

⎪

⎨

⎪⎪

⎪⎪

⎩

√1 Ksin(

√

Kr) for K >0

r for K =0

√1

∣K∣sinh(

√

∣K∣r) for K <0

. (2.3)

We consider only scalar perturbations, i.e. the Bardeen potentials Φ and Ψ, since to first order vector and tensor perturbations, if generated, are diluted by the expansion of the universe. Nevertheless, they might be important at second order.

We generalize the approach of [64] considering the metric in eq. (2.1). Let’s denote ng(n, z) the number density of galaxies per redshift z and per solid angle Ω, in the direction of observation −n (where n is the direction of propagation of photons) and at redshift z. If N(n, z) denotes the total number of galaxies within a given redshift and angular bin, then we have ng(n, z) =dN(n, z)/dz/dΩ. Similarly, given the volume V(n, z)integrated within the redshift and angular bin, we define a volume density per redshift and per solid angle byν(n, z) =dV(n, z)/dz/dΩ. Given the galaxy density per comoving volume ρg, we can define the redshift density perturbation

δz =

ρg(n, z) − ⟨ρg⟩ (z)

⟨ρg⟩ (z)

=

ng(n, z) − ⟨ng⟩ (z)

⟨ng⟩ (z)

−δν(n, z)

⟨ν⟩ (z) , (2.4) where we usedρg(n, z) =ng(n, z)/ν(n, z)and expanded at first order. Volume density perturbations are defined as ν(n, z) = ⟨ν⟩(z) +δν(n, z). Brackets denote ensemble averages of random fields. The observed perturbation in the number density of galaxies is then given by

∆(n, z) ≡ ng(n, z) − ⟨ng⟩ (z)

⟨ng⟩ (z)

=δz(n, z) +δν(n, z)

⟨ν⟩ (z) . (2.5)

We remark that not only the the galaxy number counts ∆(n, z), but also the redshift density perturbationδz(n, z)and the volume perturbationδν(n, z) /⟨ν⟩ (z)are gauge- invariant quantities [64]. We will compute explicitly the contributions of these two terms.

In what follows we assume the ergodic hypothesis. Hence, cosmic random fields are assumed to be ergodic, i.e., averages of the ensemble are equal to spatial averages.¹ At linear level angular averages of perturbations vanish, so we replace brackets by over-bars, e.g. ⟨ρ⟩ =ρ, which denote the background value of a cosmological field.¯

The point of the following computation is to relate the observable number counts

∆(n, z)to quantities defined on the FLRW background in the context of cosmological perturbation theory. E.g., first order perturbation theory allows us to compute the galaxy density in terms of a background value plus perturbations, ρg=ρ¯g(z)+δρg(n, z).

Furthermore, also the direction of propagation n and redshift z are random fields in the context of perturbation theory, as they are only calculable if related to background quantities.² We can compute at linear order the redshift perturbationz=z¯+δz, where the background value is easily related to the FLRW coordinates ¯z=z¯(τ)via eq. (B.24), and similarly for the direction n.

In the following derivation we never use Einstein equations, hence the results are valid for general theories of gravity as long as galaxies follow geodesics.

2.1.1 Redshift Perturbations

We now relate the gauge invariant redshift perturbation defining number counts as in eq. (2.5) to the usual quantity computed in perturbation theory. Expanding in Taylor series ¯ρg(n, z) =ρ¯g(n,z¯) +^d¯_d¯^ρ_z^gδz(n,z¯), wherez =z¯+δz, we obtain

δz(n, z) = ρg(n, z) −ρ¯g(z) ρ¯g(z)

= ρg(n, z) −ρ¯g(z¯) ρ¯g(z)

−dρ¯g(z¯) d¯z

δz ρ¯(z)

= δg(n, z) −dρ¯g(¯z) d¯z

δz(n, z) ρ¯(¯z)

. (2.6)

In the second line we used the definitionδρ(n, z) ≡ρ(n, z)−ρ¯(¯z)andδg≡δρ(n, z)/ρ¯(¯z), and the quantity ¯ρ(z) at the denominator has been replaced by ¯ρ(¯z) when it multi- plies a first order perturbation (as the differences are second-order small). Indeed, as discussed in appendix B.2, perturbations are defined as the subtraction between two space-times, i.e., the one of the true universe (determining ρ(n, z)) and the fictitious

1We recall that mathematically ergodicity holds for Gaussian random fields with continuous power spectra which are statistically homogeneous.

2Note that we talk of perturbed direction of propagation n, rather than perturbed direction of observation−n. The latter is indeed fixed by the observer, while the theory relates it to the direction of propagation of photons and then to the background coordinates.

2.1. RELATIVISTIC NUMBER COUNTS 13 FLRW one (determining ¯ρ(¯z)). Hence, it is important to consider that ¯ρ(z) differs to first order from ¯ρ(z¯). Also, the perturbation δg depends on the fictitious FLRW coordinate ¯z, which is not uniquely related to the truly observed z. In other words, the density contrast in longitudinal gauge δg is not gauge-invariant, hence it is not an observable itself. Finally, we also stress that differences between δg(n, z) and δg(¯n,z¯) are second-order small, so that the density perturbation in eq. (2.6) can be thought as living on the background δg(¯n,z¯) and be determined via numerical codes.

In order to computeδz we still need to derive, to first order, the redshiftz measured by an observer moving with a peculiar velocityv_oand emitted by a source with peculiar velocity v_s. We write the metric in Eq. (2.1) as d˜s² = a²ds² and use the fact that light-like geodesics are conformally invariant, i.e., d˜s² andds² have the same light-like geodesics. We choose the affine parameters in the two metrics ˜λ = a²λ, so that the photon momenta are related by ˜n=n/a². We have ˜u=u/a for the observer 4-velocitiy, where u= (1−ψ,v). Then, the energy of the photon as seen by the observer is (see, e.g., [65]):

n˜^µu˜µ=˜gµν˜n^µu˜^ν =a²gµν(n^µ/a²) (u^ν/a) , (2.7) the redshift is given by

1+z=

(˜n^µu˜µ)_s (n˜^µu˜µ)

o

=ao

as

(n^µuµ)_s (n^µuµ)

o

. (2.8)

Being the spatial component uⁱ, i.e. the peculiar velocity, already at first order, and normalizing the affine parameter λ such that photons trajectory ¯n^µ = (1,n) at back- ground level, we just need to compute n⁰ by solving the geodesic equation,

d δn⁰

dλ =Φ˙ −Ψ˙ +2∂rΨ=Φ˙ −Ψ˙ −2n⋅ ∇Ψ, (2.9) where dots indicate partial derivatives with respect to conformal time τ

˙ ≡ ∂

∂τ . (2.10)

As our normalization of the affine parameter gives τ = λ−λs (hence dτ = dλ = −dr, where r=τ0−τ) at the background level, by applying the chain rule we obtain

dA(τ(λ),x(λ))

dλ = dA(τ,x(τ))

dτ =A˙(τ,x) +n⋅ ∇A(τ,x) , (2.11) where A(τ(λ),x(λ))is an arbitrary first order quantity. We solve the geodesic equa- tion (2.9)

δn⁰_o−δn⁰ = ∫

τo

τ

(Φ˙ +Ψ)˙ d˜τ−2Ψ_o+2Ψ_s. (2.12) This yields

1+z = (1+z¯) (1+Ψ_o−Ψ_s+n⋅v_o−n⋅v_s− ∫

τo

τ

(Ψ˙ +Φ˙)d˜τ). (2.13)

From Eq. (2.13), and neglecting the quantities evaluated at the observer position which lead to an unobservable monopole and a dipole terms only, we find

δz= − (1+z¯) (Ψ_s+n⋅v_s+ ∫

τo

τ

(Ψ˙ +Φ)˙ d˜τ) . (2.14) Then, by assuming that the background comoving galaxy number density is conserved, a³ρ¯g=const., we have

dρ¯g

dz =3 ρ¯g

1+z¯. (2.15)

We find the redshift density perturbation

δz(n, z) = δg(n, z) +3Ψ(n, z) +3n⋅v(n, z) +3∫

τo

τ

(Ψ˙ +Φ)˙ d˜τ

= bDcm(n, z) +3Ψ(n, z) +3n⋅v(n, z) −3Hv+3∫

τo

τ

(Ψ˙ +Φ˙)d˜τ . (2.16) We introduced the gauge invariant variable Dcm that coincides with Dark Matter den- sity perturbation in the comoving gauge (see appendix Bfor more details). We assume that it is related to the galaxy density perturbation in longitudinal gaugeδg by a linear galaxy bias b, which can be function of time and of the scale, and by the term −3Hv coming from the gauge transformation. Indeed, theDcmvariable is related to the Dark Matter density fluctuation in conformal Newtonian (or longitudinal) gauge through

Dcm=δ+3Hv , (2.17)

where v is the (gauge invariant) velocity potential in the Newtonian gauge defined through

v= −∇v , (2.18)

while δ is the Dark Matter density perturbation in Newtonian gauge. As explained in appendix B, we fix the gauge such that Dcm equates the density perturbation in synchronous gauge comoving with Dark Matter. This justifies our galaxy bias pre- scription, since on large enough scales both galaxies and dark matter follow the same velocity field as they experience the same gravitational acceleration [27, 148, 47].

2.1.2 Volume Perturbations

Now, we compute the contribution to the galaxy number counts induced by the volume perturbation. We start by considering an infinitesimal volume element around the source defined by

dV =

√

−gµναβu^µdx^νdx^αdx^β, (2.19)

2.1. RELATIVISTIC NUMBER COUNTS 15 wheregis the determinant of the metric. Since we want to express all the perturbations in terms of observable quantities, we expand the angles at source (θs, ϕs) around the observable angles at the observer (θO, ϕO), besides taking into account that also the length of geodesic is perturbed:

θS=θO+δθ , ϕS=ϕO+δϕ , r=r¯+δr . (2.20) Then we express

dV =ν(z, θO, ϕO)dzdθOdϕO, (2.21) where the volume density is given by

ν(z, θo, ϕ0) =

√

−gµναβu^µ∂x^ν

∂z

∂x^α

∂θs

∂x^β

∂ϕs

∣∂(θs, ϕs)

∂(θo, ϕo)

∣ , (2.22)

and ∣^∂(θ_∂(θ^s,ϕs)

o,ϕo)∣is the determinant of the Jacobian of the coordinate transformation that relates the angles at the observer (θO, ϕO) to the unobservable angles at the source (θS, ϕS). To first order in perturbation theory it becomes

∣∂(θS, ϕS)

∂(θO, ϕO)

∣ =1+∂δθ

∂θ +∂δϕ

∂ϕ . (2.23)

Then, eq (2.22) becomes

ν = a⁴(1+Ψ−3Φ) [S_K²(r)sinθs] (1

a(1−Ψ)dr dz −1

av^rdτ

dz) (1+∂δθ

∂θ +∂δϕ

∂ϕ ). (2.24) We need to compute how the radial coordinate r changes along the light geodesic

dr

dz = d

d¯z(¯r+δr) d

dz (z−δz) = dτ dz¯(d¯r

dτ +dδr dλ −d¯r

d¯z dδz

dλ ) (2.25)

and expand around the background the following term S_K²(r)sinθs = S_K² (¯r+δr)sin(θo+δθ)

= S_K² (r¯)sinθo(1+2 dlnSK

dr ∣

¯ r

δr+cotθoδθ) . (2.26) With these two expressions, we can rewrite (2.24) as

ν = a⁴

HS_K² sinθ[1−3Φ−n⋅v_s+ (cotθ+ ∂

∂θ)δθ+∂δϕ

∂ϕ + (2dlnSK

dr − d

dλ)δr+ a H

dδz dλ ],

(2.27)

where all quantities are evaluated at the source background position. Then, we expand ν¯(z)around the background redshift ¯z obtaining

ν¯(z) = ν¯(z¯) +d¯ν

dz¯δz=ν¯(z¯) [1+ δz 1+z¯(2

H

dlnSK

dr −4+ H˙ H²

)]. (2.28) This leads to the volume perturbation

δν

ν¯ (n, z) = −3Φ−n⋅v+ (cotθ+ ∂

∂θ)δθ+∂δϕ

∂ϕ + (2dlnSK

dr − d dλ)δr + (4− 2

H

dlnSK

dr − H˙ H²

) δz

1+z¯+ 1 H(1+z¯)

dδz

dλ . (2.29)

To completely determine the volume perturbation we need to solve the geodesic equa- tions for δr, δθ, δϕ. We start from the radial coordinate

dr

dτ = dr dλ

dλ dτ =

−1+δn^r

1+δn⁰ = −1+δn^r+δn⁰ = −1+dδr

dτ . (2.30)

From the geodesic equation

dδn^r

dλ =∂rΦ−∂rΨ−2 ˙Φ, (2.31) and Eq. (2.9) we obtain

d²δr

dτ² = ∂rΦ+∂rΨ−Φ˙ −Ψ˙ = − d

dλ(Ψ+Φ) , (2.32)

from which it follows

δr = ∫

τo

τ

(Ψ+Φ)d˜τ , (2.33)

where we have again neglected the contributions at the observer position. So, the radial contribution to the volume perturbation is

(2dlnSK

dr − d

dλ)δr=2dlnSK

dr ∫

τo

τ

(Ψ+Φ)d˜τ+Ψ+Φ. (2.34) After the radial coordinate, we look at the angles θ and ϕ. We have

dθ dτ = dθ

dλ dλ

dτ = δn^θ

1+δn⁰ =δn^θ= dδθ

dτ , (2.35)

dϕ dτ =dϕ

dλ dλ

dτ = δn^ϕ

1+δn⁰ =δn^ϕ= dδϕ

dτ . (2.36)

2.1. RELATIVISTIC NUMBER COUNTS 17 The geodesic equations lead to

dδn^θ

dλ −2( d

drlnSK)δn^θ= − 1

S_K² ∂θ(Ψ+Φ)

⇒ d

dλ(δn^θS_K²) = −∂θ(Ψ+Φ) , (2.37) which is solved by

δn^θ = 1 S_K² ∫

τo

τ ∂θ(Ψ+Φ)d˜τ . (2.38)

Analogously, we have dδn^ϕ

dλ −2( d

drlnSK)δn^ϕ = − 1

S_K² sin²θ∂ϕ(Ψ+Φ)

⇒ d

dλ(δn^ϕS_K² ) = − 1

sin²θ∂ϕ(Ψ+Φ) , (2.39)

which is solved by

δn^ϕ = 1 S_K² sin²θ∫

τo

τ ∂ϕ(Ψ+Φ)d˜τ . (2.40) From eqs. (2.35,2.36) it follows

δθ = − ∫

τo

τ

d˜τ S_K² ∫

τo

˜

τ d˜τ^′∂θ(Ψ+Φ), (2.41) δϕ = − ∫

τo

τ

d˜τ S_K² sin²θ ∫

τo

˜

τ d˜τ^′∂ϕ(Ψ+Φ). (2.42) Hence, the angular contribution to the volume perturbation is described by

(cotθ+ ∂

∂θ)δθ+∂δϕ

∂ϕ = − ∫

τo

τ

d˜τ S_K² ∫

τo

˜

τ d˜τ^′[cotθ ∂

∂θ + ∂²

∂θ² + 1 sin²θ

∂²

∂φ²] (Φ+Ψ)

= − ∫

τo

τ

d˜τ S_K² ∫

τo

˜

τ d˜τ^′∆_Ω(Ψ+Φ), (2.43) where ∆_Ω= (cotθ∂θ+∂_θ²+∂_ϕ²/sin²θ) is the angular part of the Laplacian. This expres- sion can be further simplified by partial integration and using

d dr

dlnSK

dr = − 1

S_K² . (2.44)

As expected, we find that the angular contribution to the volume perturbation de- scribes the lensing effect in the generalized form

− ∫

τo

τ

( 1 SK(˜r)

dSK

dr ∣

˜ r

− 1 SK(r)

dSK

dr ∣

r

)∆_Ω(Ψ+Φ)d˜τ , (2.45)