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(1)Thesis. Precision cosmology in the light of upcoming radio and galaxy surveys. JALILVAND, Mona. Abstract We live in the era of precision cosmology, upcoming optical and radio surveys will probe high redshifts, large volumes of the sky, and non-linear scales with high precision. As a result, future surveys open up the opportunity to test cosmological and gravity models in a wider range of distances and scales. In order to use the potential of the new precise data, we need to include a more precise modeling in our model constraints and take into account the effects that have been neglected so far. The precision of the new data will be sufficient to detect some of these effects, like lensing magnification. If they are taken into account, then they will help to constrain cosmological models, otherwise, they will show up as systematic biases in the results. In this thesis, we have studied some of these effects.. Reference JALILVAND, Mona. Precision cosmology in the light of upcoming radio and galaxy surveys. Thèse de doctorat : Univ. Genève, 2020, no. Sc. 5491. DOI : 10.13097/archive-ouverte/unige:145058 URN : urn:nbn:ch:unige-1450583. Available at: http://archive-ouverte.unige.ch/unige:145058 Disclaimer: layout of this document may differ from the published version..

(2) UNIVERSITÉ DE GENÈVE Section de Physique Département de Physique Théorique. FACULTÉ DES SCIENCES Professeur Martin Kunz. Precision cosmology in the light of upcoming radio and galaxy surveys. THÉSE présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention physique. par Mona Jalilvand de Téhéran (l’Iran). Thése N 5491. GENÈVE Atelier de reproduction de la Section de Physique 2020.

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(4) Abstract. We live in the era of precision cosmology. We have access to precise mappings of the large scale structures and the cosmic microwave background radiation. Based on the data from these surveys, scientists have established the standard model of cosmology known as the ⇤CDM model that describes the composition and the evolution of the Universe with 6 parameters. ⇤ stands for cosmological constant (a simple model for the dark energy), and CDM stands for Cold Dark Matter. According to this model ⇠ 69% of the Universe is made up of dark energy, ⇠ 27% is made up of cold dark matter, and the remaining ⇠ 4% is made of baryonic matter. Dark energy is the reason behind the accelerated expansion of the Universe. The standard model of cosmology agrees very well with the data, however, we do not fully understand the origin and nature of some of its ingredients like the dark energy and the dark matter. This is the motivation for running future surveys with the hope of distinguishing between different models that exist or to exclude some of them. Upcoming optical and radio surveys will probe high redshifts, large volumes of the sky, and non-linear scales with high precision. As a result, future surveys open up the opportunity to test cosmological and gravity models in a wider range of distances and scales. In order to use the potential of the new precise data, we need to include a more precise modeling in our model constraints and take into account the effects that have been neglected so far. The precision of the new data will be sufficient to detect some of these effects, like lensing magnification. If they are taken into account, then they will help to constrain cosmological models, otherwise they will show up as systematic biases in the results. In this thesis we have studied some of these effects which we describe below. In Chapter 2 we study the contribution of the second order lensing of the 21cm intensity mapping (IM) to the angular power spectrum. Like the CMB the lowest order lensing of the IM is second order. We found new lensing terms that are higher order than the second-order lensing and are negligible in the CMB lensing but should be taken into account in the 21cm intensity mapping lensing. These new terms double the signal-to-noise ratio (SNR). We found that for large scales ` < 700, one can safely neglect the effect of second order lensing and therefore should take it into account as noise. In Chapter 3, we study the possibility of combining the future galaxy surveys like Euclid with the 21cm IM surveys like Square Kilometer Array (SKA) and Hydrogen III.

(5) Intensity and Real-time Analysis eXperiment (HIRAX) to measure magnification bias. For this purpose we introduced a new estimator, Galaxy Intensity Mapping cross-COrrelation estimator (GIMCO) which reduces cosmic variance and enhances the signal-to-noise by a factor of 4 compared to the standard estimator which is based on galaxy-galaxy cross correlation. And finally in Chapter 4 we study the effect of non-linearities on the angular power spectrum of galaxy number counts in redshift space. Non-linearities have been studied extensively in P (k). In this work we study how they affect the C` for several models available in the literature. None of these models agree well in redshift space with the results of the N-body simulations. To compute the C` we used a flat sky approximation. For the linear power spectrum the result of this flat sky approximation for auto-correlations agrees with CAMB within 1%. We found that for narrow redshift bins, at very low `’s of the order of ` ' 10, we can see the effect of non-linearities. This is one of the interesting results of this paper. We concluded that for narrow redshift bins, we need to model non-linearities very well in P (k) in order to be able to use the C` safely..

(6) To Farbod, my parents, and my brothers, Ali and Hamed.

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(8) Acknowledgements. I want to thank my supervisor Prof. Martin Kunz for guiding me through the hard path of PhD and for all of his supports and encouragements. I also want to thank my other academic parents from whom I learnt a lot, Prof. Ruth Durrer, Prof. Camille Bonvin, Dr. Elisabetta Majerotto, and Dr. Fabien Lacasa. I want to thank my collaborators Basundhara Ghosh and Benjamin Bose for the amazing experience of working I had with them. I want to thank my husband, Farbod Hassani, for supporting me through hard times of PhD and for fruitful discussions we had on our scientific projects. I want to thank my colleagues at the department of theoretical physics who created a friendly environment especially Elisabetta Majerotto who was supportive like an older sister, Basundhara Ghosh, Jérémie Francfort, Andreas Finke, Francesa Lepori, Benjamin Bose, Goran Jelic-Cizmec, Antonia Frassino, Giulia Cusin, Azadeh Moradinezhad, Viraj Nistane, Yves Dirian, David Racco, Charles Dalang, Pierre Fleury, Michael Sonner, and others. I want to thank Prof. Jean-Pierre Eckmann for his supports and for the fruitful discussions in the coffee sessions and for revising the French abstract of my thesis. I want to thank Prof. Kavilan Moodley for hosting me during my visit to the University of Kwa-Zulu Natal, Durban, South Africa. I want to thank Dr. Alkistis Pourtsidou and Dr. Phil Bull for hosting me during my visit to Queen Mary University of London and to thank Dr. David Alonso and Pedro Ferreira for hosting me at Oxford University, UK. I want to thank my Iranian friends who were always there to help me whether by listening to my practice seminars or by emotionally supporting me, especially Razieh Masoumi, Alireza vafaei, Farida Farsian, Golshan Ejlali, Hasti Khorraminezhad, Dr. Aghile Ebrahimi, Dr. Hajar Vakili, Amirnezam Amiri, Abdolali Banihashemi, and others. I also want to thank Dr. Shant Baghram, Dr. Nima Khosravi, and Mohammad Zhoolide for inviting me as speaker at Sharif University of Technology, Shahid Beheshti University, and IPM in Tehran, Iran. I want to thank Mrs. Francine Gennai-Nicole for helping me with bureaucratic works during my PhD and helping me to move in to Geneva by finding an apartment on behalf of me. I want to thank my M.Sc. supervisor, Prof. Rahvar, and Dr. Shant Baghram and Dr. Nima Khosravi for their mentorship. I want to thank all members of Theoretical Physics department at Alzahra University for excellent Physics education especially Prof. VII.

(9) Mohammad Khorrami. And finally I want to thank my parents for supporting my studies and always encouraging me to do science..

(10) Jury Members. List of Jury members in alphabetical order • Prof. Ruth Durrer Département de Physique Théorique & Center for Astroparticle Physics, Université de Genève, 24 Quai E. Ansermet, CH-1211 Genève 4, Switzerland • Prof. Martin Kunz Département de Physique Théorique & Center for Astroparticle Physics, Université de Genève, 24 Quai E. Ansermet, CH-1211 Genève 4, Switzerland • Prof. Kavilan Moodley Astrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Durban 4000, South Africa • Dr. Alkistis Pourtsidou School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK I would like to thank all the Jury members for accepting to be part of it.. IX.

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(12) List of Publications. • Non-linear contributions to angular power spectra Mona Jalilvand, Basundhara Ghosh, Elisabetta Majerotto, Benjamin Bose, Ruth Durrer, Martin Kunz Phys.Rev.D, 101, (2020), arXiv:1907.13109. Chapter 4 is based on this paper. • A new estimator for gravitational lensing using galaxy and intensity mapping surveys Mona Jalilvand, Elisabetta Majerotto, Camille Bonvin, Fabien Lacasa, Martin Kunz, Warren Naidoo, Kavilan Moodley Phys. Rev. Lett. 124, 031101, (2020), arXiv:1907.00071. Chapter 3 is based on this paper. • Intensity mapping of the 21cm emission: lensing Mona Jalilvand, Elisabetta Majerotto, Ruth Durrer, Martin Kunz JCAP 1901, 020 (2019), arXiv:1807.01351. Chapter 2 is based on this paper.. XI.

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(14) Résumé. La cosmologie est l’étude de l’origine et de l’évolution de l’univers à grande échelle. Par grandes échelles, nous entendons des échelles d’amas de galaxies qui contiennent des milliers de galaxies qui ont une taille de l’ordre de 10Mpc, et des structures encore plus grandes comme des filaments de galaxie (avec une taille de ⇠ 50 Mpc) et des vides (régions à très faible teneur en densité dematière). Au 20ème siècle, nous avons appris que nous vivons dans un univers en expansion et que cette expansion est accélérée. Ce fut une motivation pour étudier l’évolution de l’Univers de proposer des modèles qui expliquent cette expansion accélérée. La force principale agissant à ces échelles est la gravité. Sur la base de diverses observations, y compris le CMB qui signifie le rayonnement de fond cosmique micro-ondes ( the Cosmic Microwave Background radiation en Anglais) et la structure à grande échelle de l’Univers, et en utilisant la théorie de la relativité générale d’Einstein pour étudier la dynamique de l’Univers depuis l’Antiquité jusqu’à nos jours, les scientifiques ont établi le modèle standard de la cosmologie connu sous le nom de modèle ⇤CDM, où ⇤ représente la constante cosmologique. décrivant l’expansion accélérée de l’univers, et CDM signifie froid matière noire (en Anglais, Cold Dark Matter). Selon ce modèle, aujourd’hui, ⇠ 69% de l’univers est constitué d’énergie noire, ⇠ 27% est constitué de matière noire froide et les ⇠ 4% restants sont constitués de matière baryonique. Dans le chapitre 1, nous présentons une introduction générale à la cosmologie à partir de l’histoire chronologique des observations importantes et des modèles théoriques proposés pour expliquer ces observations. Selon les observations, l’Univers est homogène et isotrope à grande échelle, à des échelles de ⇠ 100Mpc, et peut être décrit par la métrique FLRW bien connue (FLRW signifie Friedmann, Lemaître, Robertson, Walker qui a proposé la métrique). Nous donnons une introduction à cette métrique et à la dynamique de l’univers isotrope homogène. Il y a de petites perturbations au-dela de cet univers isotrope homogène qui sont créées au début de la phase inflationniste de l’univers qui sont les germes des structures. Nous étudions la croissance des structures à partir de ces petites perturbations initiales. Ensuite nous présentons une introduction de base à la lentille gravitationnelle et aux distorsions de l’espace du décalage vers le rouge qui sont les deux principaux effets intermédiaires qui affectent nos observations des galaxies et du CMB. Finalement nous introduisons le rayonnement de 21 cm. Dans le chapitre 2, nous reproduisons la référence [1]. Dans ce travail, nous étuXIII.

(15) dions l’importance de la lentille de second ordre de la cartographie d’intensité de 21 cm. Comme le CMB, la lentille d’ordre le plus bas de la cartographie d’intensité est de second ordre. Nous avons trouvé de nouveaux termes de lentille d’ordre supérieur à la lentille de second ordre et négligeables dans la lentille CMB, mais devraient être pris en compte dans la lentille de cartographie d’intensité de 21 cm. Ces nouveaux termes doublent le rapport signal sur bruit (SNR). Nous avons constaté que pour les grandes échelles ` < 700, on peut sans risque négliger l’effet de la lentille du second ordre et donc, il faut le prendre en compte comme du bruit. Dans le chapiter 3, nous reproduisons la référence [2]. Nous étudions la possibilité de combiner les futurs levés de galaxies comme Euclid avec des études cartographiques d’intensité de 21cm comme Square Kilometer Array (SKA) et Hydrogen Intensity and Real-time Analysis eXperiment (HIRAX) pour mesurer le biais de grossissement. Dans ce but, nous avons introduit un nouvel estimateur, en Anglais Galaxy Intensity Mapping cross-COrrelation estimator (GIMCO) qui réduit la variance cosmique et améliore le signal sur bruit d’un facteur 4 par rapport à l’estimateur standard basé sur la galaxie corrélation croisée entre les galaxies. Dans le chapiter 4, nous reproduisons la référence [3]. Nous étudions l’effet des non-linéarités sur le spectre de puissance angulaire (Angular power spectrum en Anglais). Les non-linéarités ont été largement étudiées dans P (k). Dans ce travail, nous étudions comment ils affectent le C` pour plusieurs modèles disponibles dans la littérature. Aucun de ces modèles ne concorde bien dans l’espace de décalage vers le rouge avec les résultats des simulations à N corps. Pour calculer le C` , nous avons utilisé une approximation du ciel plat. Pour le spectre de puissance linéaire, le résultat de cette approximation du ciel plat pour les auto-corrélations est en accord avec le CAMB à l’intérieur de 1%. Nous avons trouvé que pour des bins étroits du décalage vers le rouge, nous pouvons voir l’effet des non-linéarités à des `’s très bas de l’ordre de ` ' 10. C’est l’un des résultats intéressants de cet article. Nous avons conclu que pour les bins é troits du décalage vers le rouge, nous devons très bien modéliser les non-linéarités dans P (k) afin de pouvoir utiliser le C` en toute sécurité..

(16) Notations. List of symbols used in the thesis GN a H H h ⌘, ⌘0 t ,t0 z n,n̂ , D1 (z) f (z) v. P (k), Pm (k) PN L (k) ⇠(✓, z1 , z2 ) C` (z1 , z2 ) b s  1 and 2 ; P` (X) Y`m (✓, ) j` (x) D ij , Abbreviations (⇤)CDM FLRW. Gravitational constant scale factor Hubble parameter (H0 denotes Hubble parameter today) conformal Hubble parameter H = aH reduced Hubble parameter h = H0 /100 conformal time (today) cosmic time (today) comoving distance = ⌘0 ⌘ redshift line-of-sight direction Bardeen potentials growth function growth rate velocity divergence redshift-space power spectrum, linear matter power spectrum non-linear redshift-space power spectrum two-point correlation function angular power spectrum galaxy bias magnification bias lensing potential convergence shear components; complex shear Legendre polynomials Spherical harmonics Spherical Bessel functions Kronecker and Dirac delta functions (Cosmological constant) Cold dark matter Freidmann-Lemaître-Robertson-Walker XV.

(17) GR CMB LSS BAO RSD IM SPT LPT EFT TNS. General relativity the Cosmic Microwave Background Large-scale structures Baryon acoustic oscillations Redshift Space Distortions Intensity Mapping Standard Perturbation Theory Lagrangian Perturbation Theory Effective Field Theory Taruya-Nishimichi-Saito.

(18) Contents. 1 Introduction 1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Friedmann–Lemaître–Robertson–Walker (FLRW) metric . . . 1.3 Structure formation . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Perturbed FLRW universe and perturbation equations 1.4 Gravitational lensing . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Weak lensing . . . . . . . . . . . . . . . . . . . . . . . 1.5 Redshift space distortions . . . . . . . . . . . . . . . . . . . . 1.6 Statistical tools . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Correlation function: ⇠(r) . . . . . . . . . . . . . . . . 1.6.2 Power spectrum: P (k) . . . . . . . . . . . . . . . . . . 1.6.3 Angular power spectrum: C` . . . . . . . . . . . . . . . 1.6.4 Likelihood function and Fisher matrix . . . . . . . . . 1.7 The 21cm radiation . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Brightness temperature . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. 2 Intensity mapping of the 21cm emission: lensing 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Weak Lensing Corrections to Intensity mapping . . . . . . . . . . . 2.2.1 Higher order intensity mapping . . . . . . . . . . . . . . . . 2.2.2 The lensing terms . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 The lensed HI power spectrum . . . . . . . . . . . . . . . . . 2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Higher order lensing contribution . . . . . . . . . . . . . . . 2.3.2 Comparing lensing terms with gravitational potential terms 2.4 Is lensing of intensity mapping observable? . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.A Derivation of the new lensing terms for HI intensity mapping . . . . 2.B The computation of C` [3] . . . . . . . . . . . . . . . . . . . . . . . 2.C Halofit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.D Cancellation of C`[1] and C`[2] in IM and in the CMB . . . . . . . 2.E The windowed C`[3] . . . . . . . . . . . . . . . . . . . . . . . . . . XVII. . . . . . . . . . . . . . .. 1 1 5 7 7 11 12 12 14 14 16 16 17 18 19. . . . . . . . . . . . . . . .. 21 21 23 23 26 27 29 29 30 31 35 46 47 49 49 50.

(19) 2.F. class settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51. 3 A new estimator for gravitational lensing using galaxy and intensity mapping surveys 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Contamination and signal-to-noise ratio . . . . . . . . . . . . . . . . . 3.4 Forecasts on the lensing amplitude AL . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.A Covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.B Contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.C Interferometer noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.D Galaxy and IM specifications . . . . . . . . . . . . . . . . . . . . . . 3.E Redshift pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55 56 57 59 61 63 64 64 65 66 66. 4 Non-linear contributions to angular power spectra 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The flat sky approximation . . . . . . . . . . . . . . . . . . . . 4.3 Non-linear correction to the angular power spectrum . . . . . 4.4 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . 4.A Non-linear corrections to the power spectrum in redshift space 4.A.1 COLA . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.A.2 SPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.A.3 pr-LPT . . . . . . . . . . . . . . . . . . . . . . . . . . 4.A.4 EFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.A.5 TNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.A.6 Comparisons . . . . . . . . . . . . . . . . . . . . . . . 4.B Derivation of the 1-loop terms . . . . . . . . . . . . . . . . . . 4.C TNS model, the A, B and C correction terms . . . . . . . . . 4.D Fitting Procedure for EFT and TNS model . . . . . . . . . . . 4.E Neglecting the lensing term . . . . . . . . . . . . . . . . . . . 4.F Fisher forecast . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69 70 72 74 84 85 86 87 87 87 88 88 91 93 94 95 97. 5 Conclusion and summary. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. 99.

(20) 1 Introduction. 1.1. History. In the 20th century we learned that we live in an expanding Universe and that this expansion is accelerated. In 1929 Hubble observed that distant galaxies are receding from us and the velocity of recession is linearly proportional to the distance, i.e., v = H0 DL , this is known as Hubble law and H0 is called the Hubble constant. Hubble measured the flux of the standard candles and their redshifts. Standard candles have a known intrinsic Luminosity, therefore by measuring the flux, F , we can get the Luminosity distance, DL , by the following relation F =. L . 4⇡DL2. (1.1). On the other hand we can relate redshift to receding or approaching velocity. We know from the theory of special relativity that the wavelength of a photon emitted from a source that has relative velocity v with respect to the observer changes as s 1 + vc v o = , (1.2) v ' 1+ 1 c c e where o is the observed wavelength and e is the emitted wavelength. If we define redshift, z, as o / e ⌘ (1 + z), we have z = v/c. Therefore by measuring the flux and the redshifts he could find the relation between velocity and distance. This observation indicated that we could live in an expanding universe, however there were scientists who believed in a steady state model of the Universe. In 1965, Penzias and Wilson, two radio astronomers in New Jersy observed a persistent isotropic noise which was larger than what they expected and corresponded to a temperature of 3.5K [4]. Such a radiation was theoretically predicted by Dicke et al. [5] in the same year. The hot Big Bang model states that the Universe started from a hot, highly dense state and expanded over time. According to this model, we should see a remnant radiation from that early phase of the Universe. The observation of this isotropic radiation known as the Cosmic Microwave Background 1.

(21) Chapter 1. Introduction. 2. radiation (the CMB) excluded the steady state model of the Universe and confirmed the hot Big Bang model. The CMB which is a radiation from early times of universe evolution is like a childhood photo that will help us understand its evolution better. On the theoretical side, Friedmann [6] and Lemaître [7] had found solutions to the Einstein’s equations which showed that the Universe could be expanding. We will study the Friedmann equations in details in the Section 1.2. The discovery of the CMB was a milestone in the field of cosmology. After the observation of an isotropic background radiation by Penzias and Wilson, cosmologists moved one step further and started to look for the anisotropic perturbations on top of the isotropic background as they expected the structures we see today have grown out of the small initial density perturbations; which then collapsed into the structures due to the gravitational instability. For a long time cosmologist could only detect a dipole and they were able to put an upper limit of T /T . 10 4 [8] on the magnitude of the small scale perturbations. This value is not big enough for the baryons alone to form the structures. The reason is that the baryons interact with the photons and therefore there is not enough time after photon-baryon decoupling (as the Universe expands and cools) until today to form the structures out of those small initial perturbations [8]. We need an extra matter content that does not interact with the photons and is able to start growing earlier. On top of this, there were astrophysical evidences for the presence of this unknown matter content. In the 1970s, Rubin et al. [9, 10] observed a flat rotation 1 curve for several galaxies, which indicates that the total mass within a radius r from the center of the galaxy, should grow linearly as a function of r. Since luminous matter does not have such a behavior, they introduced the concept of dark matter to explain the flat rotation curves. The Cold Dark Matter (CDM), a pressure-less fluid that does not interact with the baryons, was established as an essential component to explain the structure formation. In 1989, the NASA satellite COBE (COsmic Background Explorer) was launched with the goal of observing the anisotropies of the CMB map. The COBE satellite detected the temperature fluctuations of the order of 30µK ( T /T . 10 5 )[11] and provided the first evidence of the initial matter fluctuations that seeded all the structures. This survey was later followed by the NASA satellite WMAP and the ESA satellite Planck. In Fig. (1.1), we see an illustration showing the evolution of the map’s resolution from the COBE satellite to Plank. One surprising result of these observations was that the CMB temperature is the same (putting aside the small anisotropies) in every direction. If we trace back the patches separated by the angles bigger than 1 , assuming the General Relativity (GR) and that the Universe is filled with matter and radiation, we find that those patches were outside the horizon at the time of decoupling of the photons. The photons outside horizon could not interact and being thermalized, so how is it that the CMB is homogeneous at large scales? This is known as ’horizon problem’ and the standard Big Bang model can not explain it. Cosmological inflation, a rapid exponential expansion of the Universe after the big bang solves this problem. On top, the quantum fluctuations of the inflation field can explain the initial seeds of 1. A constant relation between rotational velocity and distance from the center of the galaxy..

(22) Chapter 1. Introduction. 3. Figure 1.1: An illustration of the evolution of the CMB map resolution from COBE to Plank. Left: The COBE satellite map. Middle: The WMAP satellite map. Right: The Plank satellite map. As we see the resolution has increased tremendously from COBE to Plank. Image credit: NASA/JPL-Caltech/ESA. the perturbations. In 1998, two groups (Riess et al. [12] and Perlmutter et al. [13]) found that the relation between the fluxes and the redshifts of the supernovae can be explained by an accelerated expansion of the Universe, while, a clump of normal matter would collapse under the effect of gravity and should result in a decelerated expansion. Therefore the accelerating expansion implies that whether the GR is wrong and we need to modify it, or the Universe is filled with an unusual matter that has a negative pressure, which cosmologists call it ’dark energy’. The simplest model of the dark energy is a fluid that has negative pressure with the equation of state P = ⇢ (where P is the pressure and ⇢ is the energy density) and the energy density is a constant function of time and space. This fluid is equivalent to a constant added to the geometry part of the Einstein’s equations which is known as the ‘cosmological constant’ and is denoted by ⇤. The energy density of the dark energy becomes dominant at low redshifts (z . 0.7). The standard model of cosmology known as ⇤CDM is a parametrization of the Big Bang cosmological model with an early phase of exponential expansion that has three ingredients: 1) cosmological constant denoted by ⇤, 2) cold dark matter denoted by CDM, and 3) baryonic matter. Other than the cosmological constant, several modified gravity theories and dynamical dark energy models have been proposed by cosmologists to explain the late accelerating expansion of the Universe. Other than the CMB (radiated at z = 1400), study of the large scale structure of the Universe (LSS) including the distribution of galaxies, clusters of galaxies, filaments, and voids allows us to test the cosmological models at lower redshifts. The goal of the upcoming LSS surveys is to put tighter constraints on cosmological and gravitational models and hopefully exclude some of them. Various existing or.

(23) Chapter 1. Introduction. 4. Figure 1.2: SDSS map of the Universe in redshift space. Each dot is a galaxy. The dark regions are shielded by the Milky Way disk and we can not see galaxies in those regions. We are lucky to be born in a disk galaxy not an elliptical galaxy. Image credit: M. Blanton and SDSS. planned LSS optical surveys include Dark Energy Survey (DES), Sloan Digital Sky Survey (SDSS), Euclid (to be lunched), the Wide Field Infrared Survey Telescope (WFIRST), and the Large Synoptic Survey Telescope (LSST). These surveys will observe galaxies and measure their redshifts and angular positions in the sky, therefore they are able to make a 3D map of the galaxies which can be used to study the dynamics of the dark energy at low redshifts ( here we mean z . 2.5). Euclid as an example will do more than a billion photometric redshift measurements and several million spectroscopic redshift measurements. Euclid can constrain any deviation from cosmological constant at a percent level [14]. In Fig.(1.2), we can see the distribution of the galaxies observed by SDSS. This week (week starting at 20th July of 2020, SDSS published the result of 20 years of spectroscopic measurements and the implications on cosmology [15].) Other than optical galaxy surveys, the observation of the 21cm radiation from neutral hydrogen clouds is very promising. Intensity mapping technique specifically will allow us to observe high redshifts and large volumes of the sky as it does not aim to detect individual galaxies but to measure intensity of 21cm radiation from large pixels in the sky, so therefore opens up the opportunity to test cosmological models.

(24) Chapter 1. Introduction. 5. in a wider range of distances. Some of the radio surveys include the Canadian Hydrogen Intensity Mapping Experiment (CHIME), Hydrogen Intensity mapping and Real time Analysis eXperiment (HIRAX), and Square Kilometer Array (SKA). In the following sections of the introduction, we give a brief introduction to the cosmological topics used in the rest of the thesis including structure formation, weak lensing, redshift space distortions, and statistical tools.. 1.2. Friedmann–Lemaître–Robertson–Walker (FLRW) metric. At large scales, ⇠ 100 Mpc, our universe is homogeneous and isotropic as the Big Bang model predicts. Of course we have anistropies and inhomogeneities due to presence of structures, but we can consider them as perturbations on top of a background homogeneous and isotropic universe. Here we make an assumption that perturbation theory works at least on large scales. Therefore we split the metric into a background part plus a perturbations part as: gµ⌫ = ḡµ⌫ + gµ⌫ .. (1.3). We choose the time coordinate such that at each constant time we have an isotropic and homogeneous 3D hyper-surface; this time is called cosmic time. For the spatial coordinate we choose the spherical coordinate and we call comoving coordinate. The FLRW metric in this coordinate is:  dr2 2 µ ⌫ 2 2 ds = ḡµ⌫ dx dx = dt + a (t) + r2 d✓2 + sin2 ✓d 2 (1.4) 2 1 Kr. where function a(t) is called scale factor and K is the spatial curvature of 3D hypersurfaces at constant t. The case K = 0 corresponds to a flat spatial part, K = 1 to a closed universe (like the geometry of a 3D sphere), and K = 1 corresponds to an open universe (like the geometry of a 3D hyperbolic space). The distance between any two points in this comoving coordinate is fixed but the physical distance increases by scale factor as universe expands. For the flat universe, let us call physical distance dp and comoving distance , we have dp (t) = a(t) , where comoving distance is given by Z t0 Z 0 Z a0 dt0 dr0 da0 p = = = , 0 a02 H(a0 ) 1 Kr02 te a(t ) re a. (1.5). (1.6). where te is the emission time of the photon, t0 is the receiving time, re is the radial coordinate of emission, and H is the Hubble parameter defined as: H = (da/dt)/a . For a matter dominated flat universe we will see from Eq.(1.11) that H = H0 a 3/2 , so we can take the integral analytically and obtain   r 2 p a 2 p 1 = a0 1 ⌘ a0 1 p , (1.7) H0 a0 H0 1+z.

(25) Chapter 1. Introduction. 6. where we have defined z ⌘ a0 /a 1. Since the ratio of scale factors at two different times is an observable, we normalize it such that a0 = a(t = 0) = 1. For low redshifts we have ⇠ z/H0 . If we take derivative of the physical distance with respect to cosmic time, we have the following relation for the radial velocity v=. d(dp ) = ȧ + a ˙ = Hdp + a ˙ = vH + vp , dt. (1.8). the first term on the right hand side of the equation above is the receding velocity due to Hubble expansion and the second term is the velocity due to the local movements of galaxies and structures with respect to the Hubble flow and is called the ’peculiar velocity’. As the overdensities grow, they become separated from the local Hubble flow. For the flat space time (K=0), we normalize the scale factor such that at present time, a(t = 0) = 1. To derive the evolution of this maximally symmetric universe we should solve Einstein equations: (1.9). Gµ⌫ = 8⇡GN Tµ⌫. where Tµ⌫ is the energy momentum tensor of matter ingredients of the Universe and Gµ⌫ describes the geometry of the Universe. The content of the Universe determines its dynamics. Due to homogeneity and isotropy, the energy momentum tensor of FLRW universe should have the form of a perfect fluid where pressure and density are not functions of space and we do not have anistropic pressure. The energy momentum tensor for a single perfect fluid with pressure P and density ⇢ is: T⌫µ = (⇢ + P )uµ u⌫ + P. µ ⌫. (1.10). .. In the comoving coordinate we have u = ( 1, 0, 0, 0), T00 = ⇢, and Tji = P ji . After solving Einstein equations from 00 and ii components, we get H2 =. 8⇡G ⇢ 3. 3H 2 + 2Ḣ =. K a2 8⇡GN P. K . a2. (1.11). The first line in Eq. (1.11) is one of the Friedmann equations. If we eliminate the term K/a2 between the first line and second line we get the second Friedmann equation: ä 4⇡G = (⇢ + 3P ) (1.12) a 3 where ä determines the acceleration of the expansion of the background. We can see that in order to have an accelerated expansion we need a fluid with which has an equation of state, P = w⇢ with w < 1/3. The case w = 1 corresponds to cosmological constant. The continuity equation, i.e., rµ T⌫µ = 0 reads ⇢˙ + 3H(⇢ + P ) = 0 ,. (1.13).

(26) Chapter 1. Introduction. 7. If we solve for a fluid with w = 1, we get a constant energy density for this fluid as the Universe expands. A fluid with constant energy density is equivalent to adding a constant, ⇤, to left hand side of the Einstein equation (the geometry part): Gµ⌫ + ⇤gµ⌫ = 8⇡GN Tµ⌫ ,. (1.14). where ⇤ is called the cosmological constant. A Universe filled with a fluid with w = 1 (and therefore constant energy density), expands forever as we can find from first Friedmann equation (first line of Eq. (1.11) ) that a / eHt for this universe where H is constant with time. The concordance model of cosmology assumes a cosmological constant for the recent accelerated phase of the Universe.. 1.3. Structure formation. In this section, we provide a brief review on structure formation. The stochastic quantum fluctuations of the inflation field create the initial curvature perturbations which are the initial seeds of structures in the Universe. In the following subsection, we review the growth of structures out of these initial small perturbations.. 1.3.1. Perturbed FLRW universe and perturbation equations. As we have seen in Section 1.2, we split the metric into the homogeneous isotropic background metric and the perturbed metric. Up to first order we have gµ⌫ = ḡµ⌫ + gµ⌫. (1.15). We should note that this split is not unique up to the gauge transformations, however the full theory (background plus perturbations) should be invariant. We can decompose the 4 ⇥ 4 tensor gµ⌫ as ✓ ◆ 2 wi 2 gµv = a (1.16) wi 2 ij + hij where hij is a symmetric traceless 3 ⇥ 3 tensor (5 degrees of freedom), and are two scalars (two degrees of freedom) and wi is a vector (3 degrees of freedom). In total we have 10 degrees of freedom of which 4 go away by gauge transformations. The contribution from the vector and tensor parts are small at least in ⇤CDM and thus, we only focus on the scalar parts of the metric. In the Newtonian gauge the two scalar degrees of freedom of the tensor and vector parts are set to zero due to gauge choice and we have ⇥ ⇤ ds2 = gµ⌫ dxµ dx⌫ = a2 (⌘) (1 + 2 )d⌘ 2 + (1 + 2 ) ij dxi dxj (1.17). whereR ij dxi dxj is the flat spatial metric and ⌘ is the conformal time defined as ⌘ = dt/a(t). In this coordinate, the background metric given in Eq.(1.4) for K = 0 is ds2 = a2 (⌘) d⌘ 2 + ij dxi dxj (1.18).

(27) Chapter 1. Introduction. 8. The general form of the energy momentum tensor is [16] Tµ⌫ = (⇢ + P )uµ u⌫ + P gµ⌫ + [qµ u⌫ + q⌫ uµ + ⇡µ⌫ ] ,. (1.19). where P and ⇢ are the pressure and density of the fluid, ~u is four velocity, and ~q and ⇡µ⌫ are the heat flux vector and viscous shear tensor respectively. The four velocity is given by ! µ i dx d⌘ dx uµ = = p ,p (1.20) d⌧ a2 (⌘)(1 + 2 )d⌘ 2 a2 (⌘)(1 + 2 )d⌘ 2. where ⌧ is the affine parameter and we chose it such that d⌧ = ds. Up to first order in perturbations we have ✓ ◆ 1 vi µ u = (1 ), , (1.21) a a i. where v i = dx . Here we stick to a perfect fluid and neglect ~q and ⇡µ⌫ . The absence d⌘ of ⇡µ⌫ yields that Tij = 0 for i 6= j. For a perfect fluid, first order perturbed energy momentum tensor reads ⇥ ⇤ Tvµ = ⇢¯ 1 + c2s uv uµ + (1 + w) ( uv uµ + uv uµ ) + c2s vµ , (1.22) where. is the density contrast and is defined as ⌘. ⇢(x, t) ⇢¯(t) , ⇢¯(t). (1.23). and cs is speed of sound and is defined as c2s ⌘. P . ⇢. (1.24). If we have a barotropic fluid (where P is only a function of ⇢), then we have c2s =. dP Ṗ = . d⇢ ⇢˙. (1.25). This is called adiabatic sound speed. In general pressure is a function of density and internal degrees of freedom or entropy and we have an extra term called nonadiabatic sound speed that comes from taking derivative with respect to entropy. Einstein equations at first order are given by (1.26). Gµ⌫ = 8⇡G T⌫µ. To compute the left hand side, we need to compute the perturbed Ricci tensor and the perturbed Ricci scalar. Here we don’t give the details of the calculations. By equating the left hand side and right hand side of the above equation, for a flat universe with K = 0, we get the first order perturbed Einstein equations: 0 3H (H ) + r2 = 4⇡Ga2 ⇢ r2 ( 0 H ) = 4⇡Ga2 (1 + w)⇢✓ = 00 + 2H 0 H 0 H2 + 2H0 =. 4⇡Ga2 c2s ⇢ ,. (1.27).

(28) Chapter 1. Introduction. 9. where ✓ = ri v i , H is the conformal Hubble parameter (H ⌘ (da/d⌧ )/a = aH), and derivatives are with respect to conformal time. In Eq.(1.27), first to fourth line respectively come from (00), (0i), (ij), and (ii) components of the Einstein’s equations. Note that if we had shear tensor not equal to zero, we would not have µ = . If we look at continuity equations, i.e., T⌫;µ = 0, the ⌫ = 0 component reads: 0 + 3H c2s w = (1 + w) (✓ + 3 0 ) , (1.28) which for a fluid with w = 0 and c2s = 0 (dark matter for example) reduces to 0. =. ✓. 3. 0. (1.29). ,. This equation is known as perturbed continuity equation. The term 0 which is new compared to Newtonian dynamics is small at small scales. Then this equation is the familiar continuity in Newtonian dynamics that shows that if we have a negative divergence (inward flow of matter) in a position in our fluid the density grows in that position. The continuity equation for ⌫ = i component for a fluid with w = 0 and c2s = 0 reads ✓0 + H✓ = r2 r2 c2s . (1.30). If we Fourier expand perturbation fields, we see that in linear theory modes are decoupled, therefore we can solve the equations in Fourier space for each mode separately. After going to Fourier space, we can easily solve equations for two regimes, super horizon scales (k ⌧ H) and sub-horizon scales (k H) where horizon scale is ⇠ 1/H. Here we focus on scales smaller than horizon. For these scales with some gymnastics we can show that perturbed continuity equation and Euler’s equation read: 0. ✓. 0. = =. ✓ H✓ + c2s k 2. k2 .. (1.31). Taking the derivative of the first line and replacing it in the second line we get ✓ ◆ 3 2 00 0 2 2 + H + cs k H = 0, (1.32) 2 which determines the evolution of density perturbations as a function of time. We note that Eq.(1.32) is valid for the case where K = 0 and ⇤ = 0. In a Minkowski background ( a(t) = 1 at any time or H = 0 ), this equation is simply a harmonic oscillator equation and in an expanding universe (H 6= 0), it is a harmonic oscillator with friction (the term H 0 acts like friction). If c2s k 2. 3 2 H >0 2. (1.33). the solution is damped oscillations and density perturbations do not grow. If we substitute H from Friedmann equation H2 = (8⇡G/3)⇢a2 and recall the definition.

(29) Chapter 1. Introduction of the Jean’s scale to. J. = cs. 10 p ⇡/GN ⇢, the condition for structures to grow translates. 2⇡ > J, (1.34) k where p is the physical scale. The smaller the sound speed, the smaller the scale above which structures can grow. Dark matter has small speed of sound and therefore structures can form at very small scales. For photons we have p. =a. 1 P = ⇢c2 3 so cs =. s. Ṗ c =p ⇢˙ 3. (1.35). (1.36). therefore. r r c ⇡ c ⇡(8⇡) =p ⇡ H, (1.37) J = p 3H 2 3 G⇢ 3 where H is the Hubble horizon, so for photons the growth of structures is damped for all scales within the Hubble horizon. Before decoupling of baryons from photon, the speed of sound of baryons is comparable to photons, so baryon structures can not grow at sub Hubble horizon scales. However, after decoupling, baryons start to fall into dark matter potential well and their density perturbations grow. Scales bigger than Jean’s scale grow freely. We call it gravitational instability regime for which gravitation counterweights pressure. For the regime p J or equivalently c2s k 2 ⌧ 3/2H2 , we can separate the time and scale dependence of as (k, a) = f (k)g(a) and we will find two solutions, one growing, + , and one decaying, , as (k, a) = B(k)a 3/2 (1.38) + (k, a) = A(k)a where coefficients A and B are set by initial conditions. If we translate scale factor to time we have + (k, t) = A(k)t2/3 and (k, t) = A(k)1/t. We neglect the decaying mode and define growth factor, D1 (t), as (k, t) =. D1 (t) (k, tini ) . D1 (tini ). (1.39). In the regime that we can not separate scale and time dependence of we should solve Eq.(1.32) numerically and find transfer function, T (k), defined as (k, t) = T (k). D1 (t) (k, tini ) . D1 (tini ). (1.40). Several Boltzman codes have been developed to solve for the evolution of the density perturbations including the code CLASS [17] and camb. The Poisson equation in sub-horizon scale reads as k 2 = 4⇡Ga2 ⇢ , (1.41) we can see that for growing mode, gravitational potential is constant in matter dominated universe. The field (x, t) is a random field with zero mean by definition..

(30) Chapter 1. Introduction. 11. We assume that the distribution of (x, t) is Gaussian as standard model of inflation predicts. Up to here we have reviewed the linear growth of structures. At small scales and at late times, structures become nonlinear and different modes do not evolve independently anymore. Perturbative approaches have been developed to deal with non-linearities of large scale structure including Standard Perturbation Theory (SPT), Lagrangian Perturbation Theory (LPT), and Effective Field Theory (EFT) which will be discussed in more details in Chapter (4). Another way of dealing with non-linearities is doing N-body simulations.. 1.4. Gravitational lensing. According to GR, light bends near massive objects, this effect is known as gravitational lensing as this bending acts like a lens and magnifies objects that are lensed. In the cosmological context, photons undergo gravitational lensing by the matter overdensities along their geodesics which distorts their path. Therefore the measurement of these distortions can be a probe of matter overdensities in the Universe. There are three kind of gravitational lensing: 1. Microlensing happens when the lens is a compact light object in the mass range of 10 6  m/M  106 such as planets, dwarf stars, primordial black holes, and etc., and rises to the angular separations of the two images that are of the order of milli-arcsecond. The source in the galactic microlensing events are stars. Since the separation of the two image is small and we can not observe individual images, the brightness of the two images add and we see the lensing effect as an increase in the brightness of the source. A cosmological application of microlensing is to detect primordial black holes. Here I mention two images because if we solve the lens equation 2 , we will see that it has two solutions (for singular lenses), in a special case when the source is exactly behind the lens along the line of sight, we see a ring known as Einstein ring. 2. Strong lensing happens under rare circumstances by massive objects as the lens such as galaxies, groups and clusters of galaxies and as a result we see multiple images of the source. The time delays of different images in the strong lensing regime can also be used to measure the Hubble constant, H0 [19]. 3. Weak lensing where the deflection angles are very small and the effect of lensing is small distortions of the photons geodesics which will eventually distort the shape of galaxies and magnify them. We will discuss these two effects in more details in this section. On top of these two effects, lensing changes the number density of galaxies which is known as magnification bias. In cosmology, we are mostly interested in weak lensing especially if we are looking for a probe of the LSS. The shapes of galaxies will be distorted due to weak lensing caused by total matter overdensities (baryon plus dark matter). 2. We will not discuss the lens equation here. It can be found here [18] for example..

(31) Chapter 1. Introduction. 1.4.1. 12. Weak lensing. To calculate the deflection angle of a photon, we need to solve geodesic equation for a single photon. If the angular position of the source in the plane perpendicular to line of sight is ✓S and the observed angular position is ✓ then we will find that [20] ✓ ◆ Z s 0 i i 0 0 ✓S = ✓ + 2 d x ( )) 1 (1.42) ,i (~ s. 0. where ~x ( 0 ) = 0 (1, ✓) and the derivatives are with respect to the transverse (per~ If we define deflection pendicular to the line of sight) components of ~x given by 0 ✓. angle of the photon as ↵ ~ = ✓~ ✓~S , we have ✓ ◆ Z s 0 0~ 0 ↵ ~= 2 d r? (~x ( )) 1 , (1.43) s. 0. where s is the comoving distance of the source and 0 is the comoving distance along the photon geodesic which we integrate over. We can define lensing potential ~ n̂ , so lensing potential is simply equal to such that ↵ ~ ⌘r ✓ ◆ Z s 0 1 0 0 = 2 d (~x ( )) 0 1 (1.44) s. 0. We should notice that Eq.(1.43) is not an exact expression for the deflection angle ~ ? we should use the true perturbed and is an approximation. In the argument of r geodesic of the photons instead of the background geodesic. This approximation is known as Born approximation. We should also note that instead of we should integrate over the Weyl potential defined as W ⌘ /2 where and are the Bardeen potentials. Here we have assumed that we have no anisotropic stress and therefore = . If we look at the Jacobian of the transformation defined as. Aij. @✓i ⌘ Sj ⌘ @✓ =. ij. ✓. 1 0 0 1 R +2 0 d. ◆ 0. +. ✓. ,ij. . 1 2. (~x ( 0 )). 0. 2. ⇣. + 1. 0. 1. ⌘. ◆. (1.45). where  is called the convergence and describes how much the shape of galaxies is magnified. The vector ~ = ( 1 , 2 ) is called the shear and its components describe how the shapes of the galaxies are distorted.. 1.5. Redshift space distortions. On sub-horizon scales, the perturbed continuity equation of Eq.(1.29) can be written in Fourier space as, ˙ + ikv = 0 (1.46).

(32) Chapter 1. Introduction. 13. The evolution of overdensities on the other hand is coded in the growth factor parameter D1 (t) as in Eq. (1.39). Thus according to the previous expression, the velocity reads,  i d i (k, ⌘) dD1 v(k, ⌘) = D1 (t) = (1.47) k d⌘ D1 (tini ) kD1 (tini ) d⌘ It is useful to define a dimensionless parameter to relate the linear velocity and growth of density as a dD1 f⌘ , (1.48) D1 da where f is called the dimensionless linear growth rate. Rewriting the Equation. 1.47 in terms of the scale factor and the linear growth rate, we have v(k, a) =. if aH (k, a) , k. (1.49). d d d where we have used the relation d⌘ = (dt/d⌘)(da/at) da = a2 H da . The number of galaxies in a certain volume in redshift space and the corresponding volume in real space are equal, so we can write. (1.50). ns (~xs ) d3 xs = n(~x)d3 x. where n and ns are the density of galaxies in real space and redshift space respec3 tively. The Jacobian of transformation defined as J ⌘ dd3 xxs is given by d3 x dx x2 = d 3 xs dxs x2s. (1.51). where we have used the fact that the angular parts sin ✓d✓d are similar in both redshift and real spaces. The number densities in redshift and real space are then related with the Jacobian as following (1.52). ns (~xs ) = n(~x)J In order to compute the Jacobian, first we use. (1.53). z = H0 x + ~v · x̂. which comes from the fact that the observed redshift is the sum of two terms, a term from the Hubble expansion of the Universe and a term from the peculiar velocity along the line of sight. In the redshift space the galaxy’s redshift is equal to its distance from us, so we can write [20] xs = x +. ~v · x̂ H0. As a result the Jacobian can be written as, ✓  ◆ @ ~v · x̂ J = 1+ @x H0. 1. ✓. (1.54). ~v · x̂ 1+ H0 x. ◆. 2. (1.55).

(33) Chapter 1. Introduction. 14. The number densities in redshift and real space can be written as a background part plus an overdensity, i. e. n = n̄(1 + ) and ns = n̄ (1 + s ) , where n̄ is the average number density. Using the equation. 1.52 the overdensities are related as ✓  ◆ @ ~v · x̂ 1 + s = [1 + ] 1 (1.56) @x H0 In Fourier space, using the distant observer approximation i. e. x̂ · ~v ! ẑ · ~v where ẑ is a radial vector pointing to the center of the galaxies, we can write ⇥ ⇤ ˜s (~k) = 1 + f µ2 ˜(~k) (1.57) k where µk is the cosine of the angle between the line of sight and k̂ which is ẑ · k̂. From the above equation the power spectrum immediately reads as, ⇥ ⇤2 Ps (~k) = P (k) 1 + µ2k (1.58). where = f /b and b is the linear bias factor. In fact the power spectrum in redshift space depends on both the magnitude and the direction of ~k.. 1.6. Statistical tools. The 2D map of the CMB and 3D map of the LSS (through radio and optical surveys) are the main probes of cosmology and structure formation. The main question is how to compare theory predictions with observations and what statistical descriptors to use. In this section, we give an introduction to the two main categories of statistical tools used in this thesis. First we review concisely correlation function, power spectrum and angular correlation function which try to ’describe’ the data that we get from the sky and second, we give an introduction to likelihood analysis and Fisher forecast which try to infer information from data about our model parameters.. 1.6.1. Correlation function: ⇠(r). In this subsection, I introduce correlation function. Since we can not predict what perturbation value we observe in an arbitrary direction in the sky at a given redshift, i.e. x (n, z), where the superscript x corresponds to a random variable which can be the CMB temperature fluctuations, galaxy number density fluctuations, and etc., we need to compare statistics of these variables with what our theory predicts. First statistical descriptor that we can think of is the mean value. For the rest of the discussion to make explanations more specific, let us consider galaxy number count as the observable we want to study. If we consider the mean number of galaxies at a redshift z as our descriptor, it can not distinguish between a uniform distribution and a clustered distribution of galaxies at that redshift. We need to build a descriptor that depends on the distance between number count of galaxies at two different.

(34) Chapter 1. Introduction. 15. rab rab. Figure 1.3: A schematic distribution of galaxies in a volume V. Two pairs of voxels are shown in different positions of the sample with distance rab . points of the sky, this descriptor is useful to study whether galaxies cluster at a certain distance (scale) or not. The average number density is equal to n̄ = N/V. (1.59). where N (z) is the total number of galaxies in the volume V . If we take two voxels dVa and dVb that are separated in space by rab , and count number of galaxies in each voxel and multiply the number of counted galaxies and average over all such voxels that are separated by rab in the sky as shown schematically in Fig. (1.3), we get dNab = hna nb i = h(n̄ + na )(n̄ + nb )i 0 : 0 ⇠⇠⇠ = hn̄n̄i + ⇠ hn̄⇠n⇠a⇠ i +⇠ hn̄ nb: i + h na nb i ⌘ n̄2 (1 + ⇠(rab )) ,. (1.60). where we have shown the averaging by h i and we have defined correlation function any additional correlation to hn̄n̄i: 1 h na nb i (1.61) n̄2 where = n/n̄. If ⇠(rab ) is positive, it means that in average we have positive correlations at distance rab . If it’s zero it means we have a uniform distribution up to random fluctuations around n̄ such that in average they cancel. The averaging can be done in two ways, whether we fix the voxels dVa and dVb and average over different realizations of our sample (ensemble average) or we average over different positions in one sample (sample average) [16]. Since we only have access to one universe, observationally we do the sample average. Let us write the correlation function in a form more familiar to cosmologists. The correlation function as a function of separation r in real space and between two constant time hypersurfaces t1 and t2 is defined as ensemble average of density perturbations as ⇠(rab ) = h. a bi. =. h⇠ (|r|, t, t0 ) = ⇠ (x, t, x0 , t0 ) = h (x, t) (x0 , t0 )ii .. (1.62).

(35) Chapter 1. Introduction. 16. In case we do sample average, we can write the correlation function as Z 1 ⇠(|r|, t1 , t2 ) = (x1 , t1 ) (x1 + r, t2 )d3 x1 V1. (1.63). where (x, t) = (n(x, t) n̄(t))/n̄(t) and r = x x0 . Due to statistical homogeneity the 2-point correlation function should not depend on the position in the sample but only on the separation and due to statistical isotropy it shouldn’t depend on direction of the separation vector r. Correlation function is not an observable because we do not have access to constant time hypersurfaces, we only have access to constant redshift hypersurfaces.. 1.6.2. Power spectrum: P (k). As we have seen in the Section (1.3.1), in linear perturbation theory different Fourier modes decouple and we can solve the perturbative equations for each Fourier mode separately. By studying the behavior of the overdensities in Fourier space, we can study the behavior of clustering at different scales. We define power spectrum as h (k, z1 ) ⇤ (k0 , z2 )i = (2⇡)3 P (k, z1 , z2 ) where. D. D. (k. k0 ). (1.64). is the 3D Dirac delta function and (k) is the Fourier transform of (x) Z (k, z) = (x, z)e ik·x d3 x , (1.65). and ⇤ (k, z) is the complex conjugate of (k, z). Due to statistical isotropy and homogeneity3 we demand that P(k) only depends on the magnitude of vector k. The power spectrum is the variance of the overdensity fields in Fourier space. If we have a smooth field, then P (k) is small, while if we have lots of overdensities or underdensities at a specific scale, power spectrum will be large at that scale. We can easily see that P (k) is the Fourier transform of 2-point correlation function. 1.6.3. Angular power spectrum: C`. We can measure angles and redshifts of galaxies independent of model. The angular power spectrum is an observable quantity (by definition) based on the 3D maps of the LSS in angle and redshift space. At each fixed redshift, we have a 2D map of anisotropies in the sky: (n, z) (1.66) that can be expanded in terms of spherical harmonics (n, z) =. 1 X̀ X. a`m (z)Y`m (n) ,. (1.67). `=1 m= `. 3. Meaning that the distribution of. is the same in every point in space and in every directions..

(36) Chapter 1. Introduction. 17. which is kind of a Fourier transform just that instead of using eik.x as the complete set of functions, we use Y`m (n) as the basis of function space on a 2D sphere. Higher `s correspond to smaller angles on the sky, normally surveys have a minimum angular resolution below which they can not distinguish structures, this scale corresponds to an `max in the summation above All the information of the field (n, z) is encoded in the coefficients of the expansion, a`m s. If we have more structures at small scales, then the coefficients of Y`m with higher `s are bigger or if we have an isotropic field, then all a`m s except ` = 0 are zero. If we multiply Eq. (1.67) by Y`0 m0 (n) and integrate over direction and using the orthogonality of spherical harmonics Z d⌦Y`m (n)Y`⇤0 m0 (n) = ``0 mm0 , (1.68) we have a`m =. Z. ⇤ Y`m (~n) (n, z)d⌦. (1.69). Angular power spectrum is defined as the variance of the a`m s: ha`m a⇤`0 m0 i =. ``0 mm0 C`. (1.70). p As we know variance of a random field decreases by 1/ N where N is the number of samplings. In the case of angular power spectrum for each `, we have 2` + 1 samplings of the a`m , therefore we have a higher variance at low `0 s.. 1.6.4. Likelihood function and Fisher matrix. The likelihood function is a function of our theoretical model parameters that determines how probable it is to get an observed data set given certain values for the parameters. For a given data set, likelihood function is defined on a n dimensional hyper-surface where n is the number of parameters. If ✓~ is the parameter vector and d~ is the data vector, likelihood function for m independent data points is equal to multiplication of the probability of each of the data points given the parameters ~ = L(d~ | ✓). m Y i=1. ~ P (di |✓). (1.71). If data points are drawn from a Gaussian distribution with mean value µi predicted by the model and variance i2 we have " # m 2 Y ~i µ ~ 1 ( d ~ ( ✓)) i ~ = p L(d~ | ✓) exp (1.72) 2 2 2 2⇡ i i i=1. If the data points are not independent but Gaussian, we can define the covariance Cij as, h(d~i µ ~ i )T (d~j µ ~ j )i = Cij (1.73).

(37) Chapter 1. Introduction. 18. Likelihood function for the correlated data would be given by [21] L ({di ,. i }). 1. =p exp det C(2⇡)m. (d~i. µ ~ i )Cij 1 (d~j 2. µ ~). !. (1.74). The peak of the likelihood function corresponds to the best value of the parameters ~ We define the Fisher matrix as: (✓~⇤ ) matching with the data d. ⌧ 2 @ ln L Fij ⌘ , (1.75) @✓i @✓j ✓=✓⇤ which corresponds to the curvature of the log of likelihood function at the peak. If this curvature is big, it means the likelihood function drops fast around the peak and we have a tight constraint on the best value of the parameters. For a Gaussian likelihood function we can derive the expression for Fisher matrix analytically as ✓✓ ◆ ✓ ◆ ◆ X 1 @C @C @µ↵ @µ 1 1 Fij = Trace C C + C 1 ↵ (1.76) 2 @✓i @✓j @✓i @✓j ↵. Normally for data that is Gaussian distributed, either the mean value is zero or the covariance matrix does not depend on the parameters, therefore normally, one of the terms in the Fisher matrix relation is zero. For a`m s, for example, our data has a Gaussian distribution with ha`m i = 0, therefore the second term is zero, and for C` s, we have non-zero mean value but we compute covariance matrix at the fiducial value and therefore the first term is zero.. 1.7. The 21cm radiation. In this section we review the basics of 21cm radiation and derive the relation for the brightness temperature. Hydrogen has a simple structure being made of an electron and proton and is the most common element in the Universe. Hyperfine splitting of the 1S ground state of hydrogen occurs as the result of the interaction between the magnetic moments of the electron and proton. The difference between the energy levels of the hyperfine splitting is about 5.9 ⇥ 10 6 eV which corresponds to a radiation with a wavelength of 21cm and frequency of 1420MHz. Traditionally the 21cm radiation have only been detected in the local Universe, however, today with a new generation of radio telescopes we will be able to probe the Universe at cosmological scales and especially high redshifts using the 21cm line. The 21 cm radiation line is narrow, therefore it can be used to measure the redshifts of galaxies precisely. There are two types of techniques used in radio surveys, one is to measure the 21cm radiation from individual galaxies and the other is to measure intensity of the 21cm radiation from large pixels on the sky and look at the brightness temperature fluctuations as a probe of the LSS. Line intensity mapping is more economic than usual galaxy surveys, as we do not need to do spectroscopy to measure the redshift of the sources but by measuring the position of the line frequency, we can determine redshift. As a result we can probe high redshifts and large volumes of the sky easily..

(38) Chapter 1. Introduction. 1.7.1. 19. Brightness temperature. Now consider a cloud of hydrogen with number density nHI . We can write the total number density based on the number densities of the two energy levels of hydrogen, i. e. nHI . = n0 + n1 where here 0 and 1 respectively denote the lower and upper level of the hyperfine splitting. The spin temperature, TS is defined using the number densities in the two energy levels, n1 = n0. g1 e g0. T⇤ TS. 3 ' 3n0 = nHI 4. (1.77). where T⇤ = h⌫21 /kB = 0.0682K, g1 = 3, g0 = 1, and we have assumed TS T⇤ . The emissivity (energy per unit time, solid angle/volume, and frequency) of the clump is A10 h⌫21 j21 = n1 (⌫) (1.78) 4⇡ where A10 ' 2.869⇥10 15 s 1 [22] is the Einstein coefficient for spontaneous emission and (⌫) is the line profile, which is assumed to be very narrow, with width d⌫ (a simple approximation for the line profile is ' 1/d⌫ ). The cloud’s luminosity is then 3 dL = A10 h⌫21 nHI (⌫)d⌫dAdr (1.79) 4 where ⌫ is the frequency in the rest frame of the cloud and dA dr is the volume of the cloud, where dr is the comoving distance along the line of sight. If the spin temperature of the gas is much larger than the background temperature (e. g. the CMB), the absorption can be neglected. So the total intensity against background just follows from the aforementioned luminosity. The total flux dF (against the background radiation) from a source object at redshift z reads, dF =. 3h⌫21 A10 nHI (⌫)d⌫dAdr 16⇡(1 + z)2 r2 (z). (1.80). where ⌫ is the redshifted frequancy ⌫ = ⌫21 /(1 + z). If we define the brightness as dF ⌘ Id⌦d⌫obs , we obtain the brightness temperature as Tb =. 3hc3 A10 (1 + z)2 nHI 2 32⇡kB ⌫21 H(z). (1.81). where the comoving number density of neutral hydrogen is given by nHI = ⌦HI. ⇢c,0 (1 + mp. HI ). (1.82). The structure of the rest of the thesis is as follows: In Chapter 2, we study the second order lensing of 21cm intensity mapping (IM). We find two new terms compared to the CMB lensing and show that one of them is negligible but the other that comes from post-Born approximation should be taken into account. We found that at large scales, lensing of the 21cm IM can not be detected but should be taken into account as noise..

(39) Chapter 1. Introduction. 20. In Chapter 3 we introduce a new estimator for detecting magnification bias which is based on the cross correlation of the galaxy number count and the 21cm intensity mapping. The advantage of our estimator is that it reduces the cosmic variance contribution to the noise and has lower contamination from the density terms in the signal. This estimator enhances the signal-to-noise by a factor of 4 compared to the standard estimator based on the galaxy-galaxy cross correlations. In Chapter 4, we study the effect of nonlinearities on the angular power spectrum, C` s. We found that for narrow redshift bins, the effect of non-linearities can be already seen at very low `s of the order of ` ' 10 which is the interesting result of this paper. The result is that for narrow redshift bins, we need to model nonlinearities very well in P (k) in order to be able to use C` s safely. In Chapter 5, we give a summary of the results presented in this thesis..

(40) 2 Intensity mapping of the 21cm emission: lensing. Based on: [1] Mona Jalilvand, Elisabetta Majerotto, Ruth Durrer, and Martin Kunz, “Intensity mapping of the 21 cm emission: lensing”, JCAP 01 , (2019), arXiv:1807.01351 Abstract In this paper we study lensing of 21cm intensity mapping (IM). Like in the cosmic microwave background (CMB), there is no first order lensing in intensity mapping. The first effects in the power spectrum are therefore of second and third order. Despite this, lensing of the CMB power spectrum is an important effect that needs to be taken into account, which motivates the study of the impact of lensing on the IM power spectrum. We derive a general formula up to third order in perturbation theory including all the terms with two derivatives of the gravitational potential, i.e. the dominant terms on sub-Hubble scales. We then show that in intensity mapping there is a new lensing term which is not present in the CMB. We obtain that the signal-to-noise of 21 cm lensing for futuristic surveys like SKA2 is about 10. We find that surveys probing only large scales, `max . 700, can safely neglect the lensing of the intensity mapping power spectrum, but that otherwise this effect should be included.. 2.1. Introduction. After the amazing success of Cosmic Microwave Background (CMB) observations, presently major efforts in cosmology go into the observation and modelling of the distribution of galaxies. As this dataset is three dimensional, it is potentially much richer and may allow us to study the evolution of the formation of cosmic structure. 21.

(41) Chapter 2. Intensity mapping of the 21cm emission: lensing. 22. At high redshifts, the signal from galaxies is however very weak and it is difficult to resolve individual images. Intensity mapping (IM) of a well defined spectral line is a possibility to circumvent this problem. Intensity mapping (see e.g. [23] for a recent overview) is a new technique complementary to galaxy number counts and shear measurements. It will not allow for very high spatial resolution but is mainly sensitive to large scale structure which may reveal the clustering properties of dark energy and other deviations from standard ⇤CDM. The most abundant element in the Universe is hydrogen and the 21 centimetre line of the hyperfine structure of neutral hydrogen is well suited for intensity mapping. In the post-reionisation Universe neutral hydrogen (HI) is most abundant in galaxies and protogalaxies and is therefore assumed to be a good tracer of the matter density, see [24, 25]. Recently, also intensity mapping of the H-alpha line [26] and of other spectral lines (see e.g. [23, 27]) has been proposed. During the complicated reionization process at 6 . z . 10 intensity mapping of hydrogen lines may be used to study reionization, but its intensity is not expected to closely follow the matter distribution. At even higher redshifts, z > 15 it may again be used to study matter fluctuations, but these low frequencies are beyond the scope of presently planned instruments, except for HERA [28]. It has been shown in the past that for galaxy number counts, when going to redshifts of order unity and beyond, lensing by foreground sources cannot be neglected [29, 30, 31, 32]. Also in the CMB lensing is very important, see [21, 33, 34] and references therein. Observationally it was first detected with the help of crosscorrelations in [35, 36], and directly by [37]. In the recent Planck satellite data the lensing signal is present at very high significance, over 40 , which permitted the construction of a low resolution map of the lensing potential to the last scattering surface [38]. It also affects the CMB power spectrum, with a significance of over 10 [39], and neglecting lensing would lead to a strong bias in parameter inference from CMB data. It is therefore reasonable to expect that in the future we shall be able to detect lensing in intensity mapping, a prospect that has led to a significant number of papers that study the lensing of HI intensity mapping [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60] and of intensity mapping of other spectral features [61, 62, 63]. The advantage of IM lensing with respect to the CMB lensing is that in principle this provides access to the lensing potential not only out to the last scattering surface at z⇤ ⇠ 1000, but to arbitrary redshifts outside the reionization window. Most of the lensing signal actually comes from the redshift range z 2 [2, 6], which lensing of intensity mapping can capture. In this paper we study the impact of lensing on the intensity mapping power spectrum, concentrating mainly on this redshift range (z 2 [2, 6]). As we will see, we obtain new contributions to lensing of 21 cm lines in this redshift range, which imply that not only the trispectrum but also the bispectrum does not vanish. This will modify the quadratic estimator used in much of the literature cited above. Even though the example we present is for the 21 cm line, our results are general and applicable also to other lines observed in intensity mapping e.g. CO or H-↵ or Lyman-↵ lines, only the appropriate bias changes. The remainder of this paper.

(42) Chapter 2. Intensity mapping of the 21cm emission: lensing. 23. is structured as follows. In Section 2.2 we first present the formalism to calculate the lensing signal of intensity maps up to third order in cosmological perturbation theory, which is needed to compute the power spectrum up to second order. We take into account all the terms containing the highest number of derivatives of the fully relativistic expression. Apart from the lensing terms these are density and velocity terms which are also present in a non-relativistic treatment. In Section 2.3 we compute the lensing signal numerically and in Section 2.4 we discuss its significance using Fisher matrix estimates of the signal-to-noise (S/N) for a future survey out to z = 6, like e.g. SKA [64, 65]. In Section 2.5 we conclude. Some detailed derivations and intermediate results are deferred to Appendices.. 2.2 2.2.1. Weak Lensing Corrections to Intensity mapping Higher order intensity mapping. It is well known that intensity mapping, like the CMB, does not acquire any corrections from lensing at first order in perturbation theory. Unlike for galaxy number counts, in intensity mapping the increase in the transversal volume due to convergence is exactly compensated by the corresponding increase of the flux. This is a simple consequence of photon number conservation. The contribution from lensing that we compute in this paper appears therefore only at second order. In this paper we use the metric given by the line element h i 2 2 2 i j ds = a (t) (1 + 2 )dt + (1 2 ) ij dx dx (2.1) where and are the Bardeen potentials. We only keep scalar perturbations, even though at higher order in perturbation theory scalar, vector and tensor perturbations mix, but the vector and tensor perturbations lead to negligible contributions to lensing [66, 67]. The fully relativistic expression for the first order fractional perturbations for intensity mapping of neutral hydrogen (HI) has been derived in [68]. It is given by. HI (n, z). =. HI. +. +. 1 2 @ V H(z) r. Ḣ +2 H2. fevo. 3HV + !. @r V +. +H +. 1. Z. ˙ (z). 0. d (˙ + ˙). !. .. (2.2). This result can also be obtained from the corresponding expression for the number counts [69, 70, 71] by setting 2 5s ⌘ 0. Here HI denotes the HI intensity fractional fluctuations in comoving gauge, V is the potential of the matter velocity, v = rV , in longitudinal (Poisson) gauge. H is the comoving Hubble parameter, H = aH and an overdot is a derivative with respect to conformal time t. The parameter fevo parametrizes the physical change in the number density of sources, fevo = d ln(a3 n̄HI )/d ln a..

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