David Bacon

James Bartlett

Vincenzo Fabrizio Cardone

Françoise Combes

Sophie Maurogordato

Alexandre Refregier

Marco Kostas

Mel

Joana Sandrine

Remy Denise

Benjamin C Zac

Alex Calum

Marie Sylvain

DES-Y1 LSST Euclid

for accepting to be members of my thesis committee. It is an honour for me to have you as my Jury. A particular thanks to my reviewers David Bacon and Alexandre Refregier, for taking the time to thoroughly read my manuscript and provide comments to improve it.

My deepest thanks go to my two advisors Valeria and Jean-Luc, whose careful and constant supervision during the three years made it possible for me to accomplish this milestone. Thank you for all our scienti c discussions and for your precious advice and guidance concerning the PhD projects as well as my professional future. Thank you for being always present and ready to answer any of my questions and for never stop motivating me, both in normal and pandemic times. Thank you for everything you taught me and for the thoughts and care you have dedicated to this PhD. Moreover, I would like to thank Valeria also for involving me in the Euclid IST: Likelihood team: observing the organisation and management of this fantastic team has been inspiring and has taught me so much in terms of team interactions,

S

Higher order statistics for cosmology: likelihood development for future surveys like Euclid

Weak gravitational lensing by the large-scale structure is the e ect of bending of light emitted by background galaxies due to the presence of foreground matter. It represents a powerful tool for estimating cosmological parameters as it is sensitive to the large-scale structure of the universe. Past, present, and future cosmological surveys, like the upcoming European Space Agency's Euclid mission, will use it as one of the main physical probes for investigating unsolved questions in current cosmology, such as: what the properties of the dark components of the universe are, what the origin of its accelerated expansion is and what the sum of neutrino masses is. As weak lensing surveys become deeper, they reveal more non-Gaussian features of the matter density eld, requiring statistics beyond the second order to properly extract cosmological information. This has motivated the introduction of several statistics of order higher than the second, such as Minkowski functionals, higher-order moments, the bispectrum, peak counts and, most recently, the scattering transform, wavelet phase harmonic statistics, and machine learning-based methods to account for non-Gaussian information in cosmological analysis. The aim of this thesis is to investigate and develop statistical methods to optimally extract the information encoded in the data in the context of higher order statistics, that can eventually help to improve the constraints on cosmological parameters in current and future cosmological analysis. In a rst study, we have compared di erent summary statistics (from power spectrum to peak counts) and quanti ed the impact of di erent multi-scale ltering techniques on cosmological forecasts obtained in an ideal setting. Speci cally, we employ a starlet lter, which is an isotropic undecimated wavelet transform that naturally decomposes the original image in several di erent images of the same size, enabling to extract cosmological information from the maps at di erent scales simultaneously. The performed study is tomographic, and we compare di erent summary statistics, assuming Euclid-like shape noise. The performance of the starlet is compared with a concatenation of Gaussian lters using peak counts and with the state of the art of summary statistics in weak lensing in an ideal setting without systematics e ects. The ndings are that in both multi-scale settings, peak counts result to perform better than the state of the art for second order statistics and single scale peak counts. Moreover, when using a multi-scale approach, joining power spectrum and peaks does not add any relevant information over considering just the peaks alone. While the performance and behaviour of both multi-scale lters are similar, for the starlet lter we nd that the majority of the information of the data covariance matrix is encoded in the diagonal elements. This can be advantageous when inverting the covariance matrix, speeding up the numerical implementation. Based on the promising performance of the multi-scale approach in the context of peak counts found in the rst study, we propose a new high order summary statistics called the starlet 1 -norm. This statistics provides a multi-scale calculation of the full voids and peaks distribution and avoids the problem of the de nition of what i is a peak and what is a void. Indeed, it is built so that it carries the information encoded in all pixels of the map and not just the one in local maxima and local minima. The outcome is that, in a tomographic ideal setting (without further systematics) with the assumption of Euclid shape noise, this new summary statistics outperforms commonly used ones, such as weak lensing peak counts, minimum counts, or combination of the two in terms of constraining power. The starlet 1 -norm represents a new promising uni ed framework to account for the information encoded in peak counts and voids simultaneously, preserving the advantages of the multi-scale approach. In order to accomplish these two studies, I developed a pipeline written in python that takes as input weak lensing convergence maps and provides as output cosmological parameters constraints. In a second part, the thesis presents preliminary results and procedures of the current work in progress. We aim at extending the previous ndings to di erent survey settings and cosmological probes, by applying high order statistics to galaxy clustering simulations for the Kilo-Degree Survey (KiDS); as the two previous works were carried in an ideal setting (without systematics), we show preliminary results of a rst application to real data from the Canada-France Imaging survey (CFIS) to test the robustness of these statistics.

Keywords:Weak gravitational lensing ; higher order statistics ; cosmological parameters ; wavelet transform ; Euclid ; Dark energy ; Cosmology ; multi-scale analysis ; Large-scale structure Statistiques d'ordre supérieur pour la cosmologie : développement de la fonction de vraisemblance pour des missions comme Euclid

L' e et de lentille gravitationnelle faible lié aux structures à grande échelle est l'e et de la courbure de la lumière émise par les galaxies d'arrière-plan en raison de la présence de matière au premier plan. Il représente un outil puissant pour estimer les paramètres cosmologiques car il est sensible aux structures à grande échelle de l'univers. Les études cosmologiques passées, présentes et futures, comme la prochaine mission Euclid de l'Agence spatiale européenne, l'utiliseront comme l'une des principales sondes physiques pour enquêter sur les questions non résolues de la cosmologie actuelle, comme les propriétés des composants sombres de l'univers, l'origine de son expansion accélérée ou de la somme totale des masses des neutrinos. Au fur et à mesure que les relevés par lentilles faibles deviennent plus profonds, ils révèlent davantage les caractéristiques non Gaussiennes du champ de densité de matière, nécessitant ainsi l'utilisation de statistiques au-delà du second ordre pour mieux extraire des données les informations cosmologiques. Ceci a motivé l'introduction de di érentes mesures statistiques d'ordre supérieur, telles que les fonctions de Minkowski, le calcul des moments d'ordre supérieurs à 2, le bispectre, les comptages de pics des vides ou, plus récemment, la scattering transform, la decomposition wavelet phase harmonic statistics et des techniques d'intelligence arti cielle. Cette thèse étudie le potentiel d'e ectuer des mesures statistiques multi-échelles dans le contexte des statistiques d'ordre superieur pour contraindre les paramètres cosmologiques. Le manuscrit commence en fournissant les outils généraux nécessaires pour comprendre les résultats présentés pendant la thèse. En particulier, le modèle cosmologique standard, les principes de formations des structures et certains des dé s actuels de la cosmologie moderne sont présentés dans le premier chapitre. Un aperçu des principales sondes cosmologiques qui seront utilisées par Euclid est présenté avec les outils statistiques bayésiens qui seront utilisés dans les di érents travaux présentés dans la thèse. Le deuxième chapitre traite ensuite plus en détail les lentilles gravitationnelles faibles. La présentation commence avec la dérivation des équations de lentille à partir de la déviation lumineuse. Puis, après avoir introduit la convergence et le cisaillement, le lien entre observations et modèles cosmologiques est traité. Ensuite, des statistiques de second ordre du cisaillement cosmique sont décrites et un bref aperçu des e ets systématiques tels que les e ets des baryons, l'alignement des ellipticités intrinsèques des galaxies, le biais sur la mesure du cisaillement dans les lentilles faibles et certaines des contraintes actuelles sont présentés. Après, je décrit l'état de l'art des statistiques d'ordre supérieur comme par exemple le bispectre, les fonctions des Minkowski, les moments d'ordre supérieur à 2, le comptage de pics et des vides et en n des techniques qui utilisent l'apprentissage automatique. L'objectif du troisième chapitre est d'introduire quelques dé nitions utiles dans le cadre du formalisme en ondelettes. En particulier, je commence par donner une brève introduction de la notion de parcimonie, de transformée en iii ondelettes continue et d'ondelette discrète. Grâce à ces outils, la transformée en starlette, qui sera largement utilisée dans les chapitres 4, 5 et 6 est introduite. Ensuite, la procédure suivie pour estimer la bruit dans les images utilisées dans le manuscript et calculer les coe cients d'ondelettes est illustrée. En n, un bref aperçu des récentes méthodes de reconstruction des cartes de masse basées sur des représentations parcimonieuses, comme DeepMass et MCALens, est fourni. Dans le quatrième chapitre je présente la motivation, la méthodologie et les résultats de notre étude présentés dans mon premier papier (Ajani et al. 2020). Nous avons étudié l'impact des approches de ltrage multi-échelles sur les paramètres cosmologiques en employant un ltre starlette (ondelette) et une concaténation de Filtres Gaussiens pour contraindre la somme des masses de neutrinos M ν , le paramètre de densité de matière Ω m et l'amplitude du spectre de puissance primordial A s en utilisant quatre redshifts tomographiques en simulant un bruit pour une mission comme Euclid.

Nous avons comparé di érentes statistiques: le spectre de puissance, le comptage de pics à une seule échelle et à plusieurs échelles et évalué leurs performances dans un cadre idéal (sans inclure e ets systématiques). Nous considérons d'abord deux techniques d'analyse multi-échelles sur des simulations de cartes convergence de lentilles faibles fournies par les manipulations MassiveNus. Nous ajoutons du bruit qui simule Euclid à ces cartes et nous comparons les deux techniques, a n de quanti er l'impact du choix du ltre sur la capacité à contraindre les paramètres. La première approche est la concaténation de ltres Gaussiens suivie de comptages de pics et la seconde consiste à utiliser une transformée starlette qui est une transformée en ondelettes isotrope non décimée et qui décompose une image en plusieurs bandes de même taille, permettant d'extraire simultanément des informations cosmologiques des cartes à di érentes échelles. Cette étude exploite l'information tomographique et considère un bruit d'une amplitude de l'ordre du bruit attendu avec la mission Euclid. Nous trouvons que ces deux approches de comptage de pics multi-échelles performent mieux des méthodes couramment utilisées qui sont les statistiques de second ordre comme le spectre de puissance ou le comptage de pics à une seule échelle pour ce qui concerne la puissance de contraindre. De plus, nous observons que dans le cas d'une analyse multi-échelle, combiner les pics multi-échelles avec le spectre de puissance n'ajoute aucune information supplémentaire par rapport à la prise en compte uniquement des pics. L'avantage de la transformée starlette par rapport aux ltres multi-Gaussiens est que la matrice de covariance est quasiment diagonale, ce qui facilite son inversion. Plus spéciquement, nous trouvons que: a) Pour les comptages de pics, une approche de ltrage multi-échelle des cartes bruitées conduit à un facteur d'amélioration de plus de deux par rapport à une approche monoéchelle (noyau Gaussien unique pour ltrer la carte de convergence) pour les contraintes conjointes sur (M ν , Ω m ), (M ν , A s ) et (Ω m , A s ) lors de l'utilisation d'un noyau starlette, et un facteur de plus de trois lors de l'utilisation d'un ltre multi-Gaussien. Ceci est encore plus évident dans les contraintes marginalisées, où l'amélioration est respectivement de 43% sur M ν , 25% sur Ω m et 34% sur A s pour la starlette, alors que pour le setting multi-Gaussien c'est 54% sur M ν , 25% sur Ω m et 45% sur A s . L'utilisation d'un ltre multi-Gaussien au lieu d'un ltre starlette dans le contexte v du nombre de pics pourrait améliorer les contraintes de 19% sur M ν et de 18% sur A s , alors qu'aucune amélioration est remarquée pour Ω m . En n, les deux comptages de pics multi-échelles fonctionnent mieux que le spectre de puissance sur l'ensemble des paramètres [M ν , Ω m , A s ] respectivement de 63%, 40% et 72% lors de l'utilisation d'un ltre starlette et de 70%, 40% et 77% lors de l'utilisation d'un ltre Gaussien multi-échelle.

b) Lors de la combinaison des pics multi-échelle avec le spectre de puissance, c'est-à-dire en utilisant une concaténation du nombre de pics et du spectre de puissance comme vecteur de données observé, nous constatons que l'information est principalement contenue dans les seuls pics (pour certains paramètres, tels que Ω m dans le cas de la starlette, elle est encodé complètement). Cela suggère que lors de l'adoption d'une approche multi-échelle, il pourrait être su sant de travailler avec les pics seuls.

c) L'inclusion de la carte coarse dans la décomposition en starlette lors du comptage des pics préserve des informations cruciales. De plus, pour les cartes avec une taille de pixel de 0.4 arcmin, il existe une résolution minimale (c'est-à-dire la plus petite échelle nécessaire) pour les échelles de starlette correspondant à θ ker = 3.2 arcmin pour atteindre une puissance de contrainte maximale. Cela nous permet d'exclure les deux premières échelles les plus nes de la décomposition en starlette, qui correspondent aux fréquences les plus élevées et sont les plus sujettes à l'impact du bruit, permettant une analyse plus rapide et plus e cace.

d) Nous remarquons que l'utilisation d'un ltre starlette conduit à une matrice de covariance de données très diagonale, tandis que pour le ltre multi-Gaussien, les termes hors diagonale sont prédominants et les corrélations entre les di érentes échelles sont non négligeables. En d'autres termes, la majorité des informations dans le cas du ltre starlette est codée dans les éléments diagonaux de la matrice de covariance. C'est un aspect intéressant du ltre starlette qui pourrait s'avérer utile lorsqu'il s'agit de données de grande dimension où la matrice de covariance peut être di cile à inverser.

En résumé, dans mon premier papier, nous con rmons que le nombre de pics dans le contexte des lentilles faibles est un outil puissant pour dériver contraintes sur les paramètres cosmologiques, en particulier lors de l'étude du régime non linéaire où l'impact de paramètres tels que les masses de neutrinos devient important. Nous soulignons également l'importance d'adopter une approche multi-échelle dans le contexte de comptages de pics de lentilles faibles, qui présentent l'avantage d'analyser les informations encodées à di érentes échelles simultanément, conduisant ainsi à des contraintes plus strictes que l'analyse à une seule échelle. Ensuite, dans le cinquième chapitre, je présente l'étude qui à mené à mon deuxième papier (Ajani et al. 2021). Dans cet étude, nous proposons ensuite de remplacer le comptage de pic multi-échelle par un nouveau descripteur statistique, que nous appelons compte de toute l'information contenue dans les coe cients de notre analyse. Outre le fait d'élégamment uni er l'analyse des pics et des vides, cette nouvelle méthodologie est plus e cace non seulement que chacune des deux indépendamment mais aussi de l'analyse jointe pics-vides. Motivé par les avantages des approches multi-échelles pour l'inférence de paramètres cosmologiques comme discuté dans le chapitre 4 et avec les résultats récents de Coulton et al. 2020a en combinant pics et minima, nous proposons la starlette 1 -norm comme nouvelle statistique d'ordre supérieur multi-échelles. La mesure des amplitudes des pics à plusieurs échelles peut être considérée comme une mesure de la norme 1 d'un sousensemble de coe cients d'ondelettes positifs. De même, la mesure des amplitudes des vides peut être considérée comme une mesure de la norme 1 d'un sous-ensemble de coe cients d'ondelettes négatifs. Les ondelettes fournissent donc un excellent cadre pour une analyse conjointe des pics et des vides, dans laquelle les informations de tous les coe cients d'ondelettes sont incluses. Nous proposons donc d'utiliser une statistique 1 -norm très simple dé nie comme la somme de la 1 -norm de tous les coe cients dans une case du rapport signal sur bruit S/N (pixel) donnée pour chaque échelle d'ondelettes, dé nie comme:

l j,i 1 = #coef (S j,i ) u=1 | S j,i [u] |= ||S j,i || 1 (1)
si B i et B i+1 sont positifs, cela correspond à la dé nition de l'ensemble des coe cients S j,i à l'échelle j et dans le bin i tel que S j,i = {w j,k /B i < w j,k < B i+1 }, où k est le pixel index. On peut alors calculer la somme cosmologique. La nouvelle sonde envisagée est le clustering photométrique des galaxies, qui, avec la lentille faible et le clustering spectroscopique des galaxies, sera l'une des principales sondes des relevés de galaxies de prochaine génération comme Euclid. Di érentes sondes cosmologiques sont a ectées de manière di érente par les redshifts considérés et par les ranges dynamiques de la structure à grande échelle sous-jacente. Ceci, ainsi que les di érences dans les instruments employés et les approches de mesure, permettent d'obtenir des informations cosmologiques complémentaires et de traiter facilement les e ets systématiques car ils sont généralement distincts et non corrélés dans di érentes sondes. La combinaison de sondes peut donc exploiter ces avantages. De plus, comme discuté en détail tout au long de cette thèse, les statistiques d'ordre supérieur au second ordre peuvent fournir des informations non Gaussiennes supplémentaires qui sont généralement perdues lors de l'utilisation de statistiques du second ordre uniquement. En utilisant donc une nouvelle suite de simulations fournies par le premier auteur de Harnois-Déraps, J., Giblin, B., and Joachimi, B. 2019, je décris les paramètres et la procédure suivie jusqu'à présent et que nous envisageons de suivre pour étudier l'impact de la combinaison de sondes dans le cadre des statistiques d'ordre supérieur. Simultanément, j'illustre les questions ouvertes que nous devons encore aborder pour étendre le pipeline pour atteindre cet objectif et l'extension déjà accomplie. Dans le septième chapitre, je fournis la description de la première application de mon pipeline à des données réelles du relevé Canada-France Imagerie (CFIS). Le relevé d'imagerie Canada-France (CFIS), est un relevé en bandes u et r couvrant 5000 deg2 en l'hémisphère Nord. Sa haute résolution et sa profondeur en font un excellent candidat aux sondages pour la science des lentilles faibles à ce jour. Le CFIS est également un élément essentiel des données nécessaires pour redshifts photométriques pour Euclid. Nous pro tons donc de la disponibilité des données du CFIS et commençons à inclure les systématiques une à une, pour quanti er leurs e ets d'abord séparément et éventuellement conjointement.

Nous commençons par l'inclusion des e ets baryoniques et fournissons une première description qualitative de leur impact dans trois scénarios di érents pour une analyse des comptage de pics. Maintenant, dans le but nal d'inclure autant de systématiques que possible, nous nous concentrons sur une deuxième systématique, à savoir le biais sur la mésure de cisaillement. 2.1 Illustration of the propagation of two light beams emitted by a galaxy in the source plan through a cluster located in the lens plan. As described in the text θ is the angular separation between the observed apparent positions of the source, β is the angular separation between the true (unlensed) positions of the source, and α is the de ection angle, representing the difference between the two. χ is the comoving distance between the observer and the source plan and χ is the comoving distance between the observer and the lens plan. Image credits to Guinot 2020. . . . . . . . . . . . . . . .

2.2

Illustration of the mapping between unlensed and lensed space thruogh the distortion matrix A. Given an original circular source image S the e ect of weak lensing leading to the observed image I can be summarised in the action of 1) the shear γ whose e ect is to imprint an anisotropic stretching (example shown in blue) where its norm |γ| characterizes the atness of the ellipse and the rotation angle φ indicates the distortion direction;

2) the convergence κ that isotropically magni es or contracts the original shape (example shown in red). This is formalised in the distortion matrix A, being the Jacobian between the original coordinates β and the observed coordinates θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Comparison between a convergence map from the MassiveNus suite of simulations (left top panel) with a fake convergence map (right top panel) built such that its power spectrum matches the power spectrum of the convergence map. The bottom panel shows the power spectrum measured on the convergence map (continuous line) with respect to the power spectrum computed on fake convergence map (dashed line). The message of this illustration is to show how a large number of high value peaks (as the big blob at the top right of κ map 1 detectable also by naked eye or the several structures at the center of the same map) are not accounted for when extracting information with the power spectrum, indicating the presence of information beyond Gaussian statistics that is not present in the information brought by the power spectrum. summarising the current state of the art of high order statistics where an estimate of the constraining power and its performance with respect to second order statistics have been provided by the authors. I would like to stress that this table does not include all studies ever performed with statistics of order higher than the second but the studies where cosmological inference has been performed and an explicit comparison with second order statistics is provided in the paper. . As desribed in the text, time series analysis provide good resolution in time, but none in frequency; Fourier transform, convert the time information in frequency information, loosing all information on time; STFT provide both frequency-time resolution, with the drawback that the window is xed, and inevitable loss of information is present because of this. The wavelet transform enables a multiresolution analysis, i.e. to analyse the signal into di erent frequency ranges at di erent time resolutions. . . . . . . . . . . . . . . 78

3.3 Illustration of the multilevel decomposition of the signal f . By passing through a low pass lter, the approximation coe cients { a 1 , a 2 , a 3 , .., a j ,..a J } are generated. Each approximation coe cient is iteratively passed through a low pass lter generating the a j s and through a high pass lter that generates the detail coe cients { d 1 , d 2 , d 3 , ..,d j ,..d J }. If a decimated DWT is employed, at each step the size of the coe cients is halved because the output is downsampled by a factor of 2. . . . . . . . . . . . . . . . . . . . . 79

3.4 Illustration of the starlet decomposition with j max = 4 scales. As the j index increases, larger scales are considered and the resolution decreases. . . . . 83 3.5 We show the 2D starlet function (top panel) as de ned in Eq. (3.15) and its 1D pro le (bottom panel). Being a wavelet it is a compensated function, i.e. it integrates to zero over its domain. This comes from the admissibility condition for the wavelet function ψ: 2 Convergence maps κ are noiseless. We apply Gaussian noise and then lter the map using either the Gaussian or starlet ltering. For illustration purposes, we show the Gaussian ltering with θ G ker = 1.6 arcmin of one map realisation for the ducial model {M ν , Ω m , 10 9 A s }={0.1, 0.3, 2.1}. The colour bar on the right of each map describes values of the convergence eld κ. For each realisation of the 10000 maps provided for each redshift we generate 10000 noise maps as described in Section 4.1.3 corresponding to the di erent value of n gal respectively for z s = [0.5, 1.0, 1.5, 2.0]. . . . . . . . . 95 4.3 We show the matching between the Gaussian lter and the starlet at the di erent scales. We chose the kernel for the multi-Gaussian concatenation such that the maximum of the two pro les match. From left to right are the nest scale to the smoothest scale where [ w 1 , w 2 , w 3 , c 4 ] = [ 1.6, 3.2, 6.2, 12.8 ] .6, 3.2, 6.4 ] arcmin and the coarse maps (red dotted line) and the Gaussian case (black dashed line). Due to the decomposition at di erent scales, for each map ltered with the starlet there are 4 di erent distributions. Indeed, the number of counts depends on the resolution: the larger the smoothing size (the lower the frequency) the smaller the number of peaks. 97 4.5 We show the correlation coe cients respectively for the starlet (left) and the multi-Gaussian (right) peak counts. The white dashed lines split the contribution of the di erent redshifts: the bin range [1,164] refers to the correlations at z s = 0.5, the bin range [165,328] refers to the correlations at z s = 1.0, the bin range [328,492] refers to the correlations at z s = 1.5 and the bin range [493,656] refers to the correlations at z s = 2.0. Each redshift contribution is split in the four di erent scales (the four mini-blocks inside the white boxes) in increasing order, i.e. [1.6, 3.2, 6.4, coarse] arcmin for the starlet and [1. 2, 2.7, 5.5, 9.5] arcmin for the multi-Gaussian, where each scale is binned in 41 values of S/N in the range [-0.6, 6]. We notice how the starlet decomposition has a tendency to make the observable correlation matrix more diagonal while the o -diagonal terms of the multi-Gaussian matrix show more correlations along the scales and among S/N values in the mini-blocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.6 Visualisation of Gaussian processes predictor input and output: the training set, constituted by the cosmological parameters and corresponding peaks distributions obtained from the simulations are represented in blue. The output for the prediction is given by the red curves: for 200 new given cosmologies in parameter space fed to the Gaussian processes prediction function we get the corresponding peaks distribution. . . . . . . . . . . . 102 4.7 95 % con dence contours tomography with source redshifts z s = [ 0.5,1.0,1.5, 2.0 ] and corresponding galaxy number density: n gal =[ 11. 02, 11.90, 5.45, 1.45]. The black dotted line is the ducial model: [ m ν , Ω m , 10 9 A s ] = [0.1, 0.3, 2.1]. Left panel: constraints from power spectrum (blue contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1 arcmin, constraints from Gaussian Peak counts (green contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1.6 arcmin, constraints from Starlet Peak counts (red contours) computed on noisy maps smoothed with a Starlet kernel with corresponding resolutions [1.6, 3.2, 6.4] arcmin + coarse map, constraints from multi-Gaussian Peak counts computed on noisy maps smoothed with a Starlet kernel with corresponding resolutions [1. 2, 2.7, 5.5, 9.5] arcmin. Right panel: constraints from power spectrum (blue contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1 arcmin, constraints from Gaussian Peak counts (green contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1.6 arcmin and the two statistics jointly (violet contours). . . . . . . 108 4.8 95 % con dence contours tomography with source redshifts z s = [ 0.5,1.0,1.5, 2.0 ] and corresponding galaxy number density: n gal = [ 11. 02, 11.90, 5.45, 1.45 ]. The black dotted line is the ducial model: [ m ν , Ω m , 10 9 A s ] = [ 0.1, 0.3, 2.1 ]. Left panel: constraints from power spectrum (blue contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1 arcmin, constraints from Starlet Peak counts (red contours) computed on noisy maps smoothed with a Starlet kernel with corresponding resolutions [1.6, 3.2, 6.4] arcmin + coarse map and constraints from the two statistics joint (orange contours) . Right panel: constraints from power spectrum (blue contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1 arcmin, constraints from multi-Gaussian Peak counts (black contours) computed on noisy maps smoothed with a multi-Gaussian kernel with corresponding resolutions [1. 2, 2.7, 5.5, 9.5] arcmin and the two statistics jointly (light blue contours). . . . . . . . . . . . . . . . . . . . . . 109 4.9 95 % con dence contours using tomography with redshifts z s = [ 0.5,1.0,1.5, 2.0 ] and corresponding galaxy number densities n gal = [ 11.02, 11.90, 5.45, 1.45 ]. The black dotted line is the ducial model: [ m ν , Ω m , 10 9 A s ] = [ 0.1, 0.3, 2.1 ]. Left panel: we show the impact of the di erent starlet scales and we prove there exists a minimum resolution θ ker = w 2 = 3.2 arcmin that allows us to obtain constraints comparable to what is achieved with the full wavelet decomposition and that the information contained in the coarse map cannot be neglected. Starting from the rst starlet scale alone θ ker = w 1 = 1.6 arcmin (dashed big contours in light blue) we add scale by scale in light blue until [w1, w2, w3]. In dark blue we show the constraints corresponding to [w 2 , w 3 , c 4 ] that almost match with the constraints provided by the full starlet decomposition (in red). Right panel: we show here the constraints obtained by adding each tomographic redshift at the time: the dashed pink corresponds to the contours relative to z s = 0.5, the lighter red to z s = [ 0.5, 1.0 ], the darker pink to z s = [ 0.5, 1.0, 1.5 ] and the dark red to the full set of redshifts in the starlet case. We plot this against the power spectrum (blue contours) to show how each source redshift contribution to shrinks the contours and helps break the degeneracy with respect to the power spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 95 % con dence contours tomography with redshifts z s = [0.5, 1.0, 1.5, 2.0] and corresponding galaxy number density: n gal = [11. 02, 11.90, 5.45, 1.45].

The black dotted line is the ducial model:

[ m ν , Ω m , 10 9 A s ] = [ 0.1, 0.3, 2.1 ].
Left panel: constraints from starlet peak counts (continuous red contours) obtained employing the full covariance matrix against constraints from starlet peak counts (dashed red contours) obtained employing the diagonal elements only of the covariance matrix in the likelihood analysis. Right panel: constraints from multi-Gaussian peak counts (continuous black contours) obtained employing the full covariance matrix against constraints from multi-Gaussian peak counts (dashed gray contours) obtained employing the diagonal elements only of the covariance matrix in the likelihood analysis . . , w 0 and h for ve redshift bins with weak lensing power spectrum (blue contours) and four redshift bins for photometric galaxy clustering power spectrum (violet contours). As mentioned above, the constraining power of GCph seems to be actually more constraining than expected and we expect the error contours to signi cantly in ate in more realistic conditions, starting with a varying galaxy bias. Right panel: joint forecast for Ω m and S 8 ≡ σ 8 Ω m /0.3 for weak lensing power spectrum (blue contours) and galaxy clustering photometric power spectrum (violet contours) and their combination (red contours). We stress also here that the constraining power of GCph seems to more constraining than expected and that the reason for this (if it is due to the simulations or to some approximations we have made) is currently under investigation. What we know for sure is that the value for the galaxy bias is xed at b g = 1.72 and as the bias is actually largely unknown in the data, we expect that once applied to real data, the error contours for GCph will in ate signi cantly. . . . . . . . . . . . . . . . . . . 144 6.9 Starlet decomposition for a galaxy map. The scales shown in the gure are the ones included in the analysis. The value in arcmin corresponds to the resolution of the corresponding starlet scale, considering that the resolution of the original map is 1 arcmin. The corresponding values in are also shown. 145 6.10 This plot represents a preliminary test for the extension of our method to GCph second order statistics and starlet peaks. As discussed above, the contours for the power spectrum seem to be more constraining than expected and we are currently investigating if this might be due to some limitations of the employed simulations or to some approximations we made throughout the analysis. I illustrate here a rst attempt to get cosmological forecasts for Ω m , σ 8 , w 0 and h for four redshift bins with galaxy clustering photometric power spectrum (violet contours) and starlet peaks (green contours). The scales considered for the starlet decomposition are chosen to roughly match the range in for the power spectrum. We notice that the constraints for w 0 tend to slightly shift the contours. It is to be investigated weather this is caused by some assumption made in the analysis as noise estimation or 4.11 for the di erent parameters pairs (α, β) for each observable employed in the likelihood analysis: the power spectrum, the peaks counted on maps smoothed with the kernel in consideration and Peaks + PS always refer to the constraints obtained with the peaks relative to some lters and the power spectrum while the term diagonal refers to the contours obtained with a likelihood analysis performed by only considering the diagonal elements of the data covariance matrix. We provide in the last column the 3D FoM given as the inverse of the volume in (M ν , A s , Ω m ) space. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Value of the Figure of

Correlation for each pair of cosmological parameters corresponding to the di erent tomographic observables: the power spectrum alone (PS), the Peaks alone for di erent lters and the two statistics combined (Peaks + PS). As explained in the text, FoC = 1 corresponds to uncorrelated parameters, while the further the FoC is to 1, the more correlations are present. Qualitatively, this can be appreciated by looking at the inclination of the contours: by looking at Figure To answer these questions, the rst step to undertake is to investigate which physical e ects and observables are sensitive to dark matter, dark energy and the sum of neutrino masses. We know that dark energy for example, is associated with the accelerated expansion of the universe (Riess et al. 1998;[START_REF] Perlmutter | Measurements of Ω and Λ from 42 High-Redshift Supernovae[END_REF]) and that the amount and the behaviour of dark matter and dark energy have an impact on how cosmic structures grow. Combining information coming from the expansion history with the one from the growth of structures will enable to understand whether the current discrepancies between theory and observations lie in how dark energy is modelled (testing the validity of considering it is a constant or if it is actually a uid evolving in time) or if it is a sign that our current theory of gravity needs to be modi ed. Next year, in 2022, the European Space Agency Medium Class Euclid (Laureijs et al. 2011) satellite will be launched in space to survey the sky for 6 years in the near infrared and visible part of the electromagnetic spectrum. In practice, Euclid will collect ∼ 2 billions of galaxy shapes and ∼ 30 millions of galaxy positions that will make it possible to investigate the expansion history of the universe and the growth of structure in a redshift range 0 < z < 2, namely covering the period of time during which dark energy played a crucial role in accelerating the expansion of the Universe. This will allow to further test the ΛCDM paradigm at low redshifts, providing complementary information to the CMB, as Planck was designed to test the physics of the very early Universe, which does not bring a lot of information on dark energy. The galaxy shapes and positions that will be collected by Euclid are the cornerstone of the primary cosmological probes used by the Euclid mission. These are the gravitational lensing by the large-scale structure and galaxy clustering. Gravitational lensing is the phenomenon of bending of light emitted from some background sources due to the presence of massive bodies between the sources and the observer. This bending induces a distortion of the original images of the sources and it represents a powerful tracer of the total foreground matter causing the distortion. Therefore, it can be used to directly probe the distribution of dark matter. In the regime in which these deformations are small, this phenomenon is called weak lensing and it is possible to directly map the dark matter distribution and the measured lensing quantities. To make the connection between the physical observable that we will obtain and the theoretical models that need to be tested, statistical techniques to extract the cosmological information present in the data are needed. Generally speaking, this means extracting features that characterise the data with a summary statistics and then compare the summary statistics relative to the observed data with the ones predicted by a given model. A common summary statistics that has been used in the past and current cosmological analysis is the two-point correlation function of these data (or its Fourier transform, the power spectrum). Despite being widely employed in current analysis, it is known that second order statistics fail to capture the non-Gaussian information encoded in the data due to non-linear regime e ects of structure formation. Indeed, to overcome this information loss, several statistics of order higher than the second have been proposed in the literature, as the bispectrum, Minkowski functionals, higher order moments, peak counts, minima counts and very recently and in parallel to this thesis work, the scattering transform and wavelet phase harmonic statistics. This long list will be discussed more in details in a dedicated chapter. [START_REF] Spergel | Wide-Field InfrarRed Survey Telescope-Astrophysics Focused Telescope Assets WFIRST-AFTA 2015 Report[END_REF] will measure the matter distribution with unprecedented precision, will enable to reach very non-linear scales and will provide an enormous amount of data. Hence, a crucial, parallel challenge of modern cosmology is to master this unprecedented precision, to be able to use and optimise the huge amount of information we will receive to extract useful information, with high accuracy. The path from input data to the nal cosmological parameters includes many challenges. Among them, we need to be able to extract the useful features encoded in the data, to optimise the gain in cosmological information. This can be done at di erent levels of the analysis.

For instance, to be able to appreciate the weak lensing distortions, one needs to smooth or lter the shape noise that is dominating over the lensing signal. Some questions arising at this level are then: how does the ltering of the noise impact the information retrieved on cosmological parameters? Are there lters that are more suited than others to extract the information? More in general, which is the statistics that best allows us to retrieve cosmological information and impact our understanding of the universe? Depending on the knowledge that one has on the physical eld to be analysed the choice of the summary statistics employed plays a crucial role. For example, if we know that the physical eld is very close to a Gaussian eld, then the power spectrum can be a good choice to extract features. Or, if it is not far from a Gaussian eld one can use n-point correlation functions. At late times, we know that an important contribution of the density maps is composed by dark matter halos and clusters, so peak counts can be an optimal choice. Moreover, it has been proved that combination of certain summary statistics as for example peaks and voids can bring an improvement in the constraining power, so this is also another aspect to keep in mind.

Beyond the combination of summary statistics of the same probe, it is known that combination of di erent cosmological probes adds additional information to cosmological analysis with respect to a single probe. As explained before, di erent cosmological probes are sensitive to the underlying large-scale structure eld in di erent ways as well as to systematics e ects.

Exploit the advantages coming from both combination of summary statistics of the same probe and of di erent probes combined, represent an important challenge for next generation galaxy surveys.

The goal of this thesis is to investigate these aspects, to nd ways to optimally extract the information encoded in the data in the context of higher order statistics that can eventually help to improve the constraints on cosmological parameters in current and future cosmological analysis.

The content of the thesis is structured as follows: a rst part composed by Chapter 1, Chapter 2 and Chapter 3 provides the background tools and the state of the art needed to understand the rest of the thesis. In particular, in Chapter 1 I introduce some basic concepts in General Relativity and cosmology, de ne cosmological parameters and the current cosmological model and principles of structure formation. Then a brief overview of Euclid primary probes is illustrated, followed by some main tools used in Bayesian inference. Chapter 2 outlines the weak lensing formalism, second order summary statistics and current constraints from some of the main current galaxy surveys. Then, the state of the art of statistics of order higher than the second is presented. Chapter 3 describes some concepts in signal processing useful for the purposes of this work, as sparsity, the wavelet starlet transform and some example of usage of wavelet for mass mapping aims. A second part composed by Chapter 4, Chapter 5, Chapter 6 and Chapter 7 illustrates the contributions of my PhD work. In particular, in Chapter 4, I present the analysis and results of a work where we investigate the impact of the multi-scale ltering techniques using weak lensing peak counts. This led to the paper: Ajani et al. 2020. In Chapter 5 I present a new summary statistics that we have introduced motivated from the ndings of the rst paper along with the search for a uni ed framework for peaks and voids. This new statistics is called the starlet 1 -norm and we presented it for the rst time in a second paper: Ajani et al. 2021. Chapter 6 and Chapter 7 describe instead my current ongoing work, so the nature of the results and discussion of these last two chapters is more preliminary and at a earlier stage than Chapter 4 and Chapter 5. Speci cally, in Chapter 6 I describe the work I am following to extend the results of this thesis work to photometric galaxy clustering. Chapter 7 presents a rst attempt of application on real data. The nal discussion and future prospects are then summarised in Chapter 8.

5

B

The purpose of this rst chapter is to provide the general tools needed to understand the rest of this thesis. In particular the standard cosmological model, principles of structure formation and some of the current challenges of modern cosmology are presented. A brief overview of the primary cosmological probes that will be used by Euclid is presented along with the Bayesian statistical tools that will be used in the di erent works presented throughout the thesis.

Cosmological Principle

Our current understanding of the structure of the universe is based on the Friedmann-Lemaître-Robertson-Walker (FLRW) model. In this model the universe is homogeneous and isotropic on large-scales. Isotropy is rotational invariance around any point. Homogeneity implies that the space is the same in every point and it can be seen as translational invariance. The universe's homogeneity and isotropy constitute the Cosmological Principle, i.e. there is no privileged position nor preferential direction in our universe. The present section summarises the main results of modern cosmology which will be useful for the following chapters.

Homogeneity and isotropy

In order to undertake a description consistent with the Cosmological Principle, it is necessary to work with a metric built assuming homogeneity and isotropy. Friedmann-Lemaître-Robertson-Walker (FLRW) metric is then built using invariants in order to satisfy the Cosmological Principle criteria. Also, it is convenient to work in a comoving reference frame and to choose spherical coordinates which are very well suited for the isotropy requirement. The resulting line-element is:

ds 2 = dt 2 -a 2 (t) dr 1 -Kr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 (1.1) = dt 2 -a 2 (t) dχ 2 + f K (χ) 2 dθ 2 + sin 2 θdφ 2 , (1.2)
where a(t) is the scale factor, a parameter that is free to vary as a function of time that physically provides information on how distances between points change as the universe contracts or expands (Hubble 1929); (r, θ, φ) the spherical coordinates, χ and f K (χ) the comoving radial and transverse distance, de ned in Section 1.1.3 and K is a parameter of curvature such that

K =      1 Closed universe (S 3 ) 0 Flat universe (R 3 ) -1 Open universe (H 3 ) (1.3)
where S 3 , R 3 , H 3 are respectively the three-dimensional sphere, hyperboloid and at Euclidean space embedded in four-dimensional Euclidean space R 4 . All three possible spatial geometries are homogeneous and isotropic, hence satisfying the Cosmological principle. More in general, the local geometry of space-time can be entirely described by a metric tensor g µν that de nes the in nitesimal distances of a trajectory in a curved space through the line element

ds 2 = g µν dx µ dx ν .
(1.4)

The kinematics and dynamics of universe's evolution are described by Einstein's Field Equations

R µν - 1 2 g µν R = 8πGT µν . (1.5)
The left hand side (l.h.s.) contains the information relative to the geometry and curvature of space-time, as the Ricci curvature tensor R µν and Ricci scalar R are expressions of the metric g µν and its derivatives. The right hand side (r.h.s) provides information about the source of such curvature, encoded in the energy-momentum tensor T µν . Einstein's Equations can be obtained as eld equations of General Relativity by applying the principle of least action to

S = S HE + S M = d 4 x √ g - 1 16πG R + L M , (1.6)
where S HE is the Hilbert-Einstein action, that minimised leads to Equation 1.5 while S M is the action for matter, which leads exactly to the source of space-time curvature when minimised, i.e.

T µν = - 2 √ g δS M δg µν (1.7)
to the energy-momentum tensor.

As mentioned above, through Equation 1.5, it is possible to derive information about the content of the universe and its evolution. In particular, thanks to the cosmological principle, working in a comoving inertial frame, the energy-momentum tensor can be treated as a perfect uid, completely de ned by its rest-frame energy density ρ and momentum p as

B T µ ν = T µα g αν =     ρ 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p     (1.8)
where ρ is the energy density of the perfet uid and p its momentum. This can be formally obtained by considering the four momentum p µ = mU µ , where U µ is the four velocity U µ = dx µ dτ with respect to proper time τ . Hence, T µν represents the ux of the four momentum p µ , and in a non comoving reference frame their relation is given by (Hobson, Efstathiou, and Lasenby 2006):

T µν = (ρ + p)U µ U ν -pg µν (1.9)
and its expression of Equation 1.8 is then found by considering a comoving observer in U µ = (1, 0, 0, 0) and by lowering the index with the metric.

Dynamical evolution of the universe

Thanks to how the metric and the energy-momentum tensor can be written under the Cosmological Principle, it is possible to describe the dynamical evolution of the universe and the di erent contributions of its components through the Einstein eld equations as follows. The resulting equations have been rst derived by A. Friedman in 1922(Friedmann 1922) and are known as Friedmann Equations. Practically, this translates in considering the zeroth-component of Equation 1.5 and in substituting for the l.h.s the expressions obtained for the Ricci tensor and the scalar curvature in the FLRW metric, while using T 00 = ρ for the r.h.s, that leads to

ȧ a 2 + K a 2 = 8πGρ 3 , (1.10)
which is the rst Friedmann equation, where ȧ indicates the derivative with respect to cosmic time t. It describes the evolution of the scale factor a(t) stating that expansion depends only on the energy density ρ. The rst ratio of the l.h.s. represents the Hubble parameter

H(t) = ȧ(t) a(t) [H] = [t -1 ] (1.11)
whose value today H 0 = ȧ(t 0 ) (given that a(t 0 ) = 1) is given by the Hubble Constant

H 0 = 100 h km s -1 Mpc -1 , (1.12)
expressed above in terms of the dimensionless reduced Hubble parameter h. Equation 1.10 does not provide information on a possible expansion, but it depicts the velocity of the expansion in terms of the energy density and in dependence on the curvature. The same procedure for the ij th -component of Equation 1.5 always assuming a perfect uid approximation, leads to

ä a = - 4πG 3 (ρ + 3p) = - 4πG 3 ρ(1 + 3w), (1.13)
which is the second Friedmann equation, where the parameter w = p/ρ relates the pressure of the fuid to its energy density, representing its equation of state. Equation 1.13 encodes the crucial information on whether the expansion is accelerating or decelerating. The rst situation arises for ä > 0 hence requiring the condition 1+3w < 0, that is w < -1/3 , while the second implies w > -1/3. This consideration provides important information about which component is predominant. In fact, from observations, today's expansion appears to be accelerated (Riess et al. 1998).

Universe energy-matter content

To describe the evolution of ρ with the scale factor one can combine Friedmann equations.

In particular, by taking the derivative of Equation 1.10 and then using Equation 1.13 one can nd

ρ = -3(1 + w) ȧ a .
(1.14) Then, considering di erent i components, each one of them is characterised by its equation of state, given by the solution of Equation 1.14: 3(1+w) , (1.15) where ρ i,0 is the energy density of the i th component at present time t 0 . The FLRW model assumes these components to be three non-interacting components. Speci cally, considering a system consituted by N particles with mass m, each one characterised by the fourmomentum p µ n (t) = mu µ n , with n = 1, .., N :

ρ i = ρ i,0 a -
Non-relativistic components: if the uid is formed uniquely by non-relativistic matter, the ki-B netic energy is much smaller than the rest energy. Hence, the energy

E n = |p 2 n + m 2 | ∼ m + p 2 n
2m and consequently its density becomes

ρ = n m + |p n | 2 2m δ 3 (x -x n (t)) = mn + n |p n | 2 2m δ 3 (x -x n (t)) (1.16)
where n = n δ 3 (xx n (t)) is the number density of particles. Using then the fact that for a perfect uid p = n |pn| 2 3m δ 3 (xx n (t)), that leads to ρ = mn + 3 2 p. However, since the thermal energy, which is the source of pressure, is much less than the rest mass, in a non-relativistic uid the pressure p is considered negligible with respect to the energy density ρ, i.e. p ρ. For a non-relativistic uid one can consider p ∼ 0 and consequent equation of state w = 0.

Relativistic components: for a relativistic uid, rest masses are null or su ciently small compared to the kinetic energy to be considered negligible. This corresponds to the limit p n ∼ E n , which leads to

ρ = n E n δ 3 (x -x n (t))
(1.17)

p = n |p n | 2 3m δ 3 (x -x n (t)) ∼ 1 3 n E n δ 3 (x -x n (t)) (1.18)
and consequently to p = 1 3 ρ, namely to an equation of state given by w = 1 3 .

Cosmological constant Λ: writing Einstein's equations in the case in which the cosmological constant Λ is not absorbed into the energy-momentum tensor yields

R µν - 1 2 g µν R = 8πG T µν + Λg µν 8πG , (1.19)
where along with the matter-radiation tensor described so far, the energy-momentum tensor of the cosmological constant T (Λ) µν = Λg µν plays the role of a source term, characterised by an energy density ρ Λ = T 00 (Λ) = Λ 8πG g 00 = Λ 8πG and pressure p such that T ij (Λ) = -pg ij . This leads to an equation of state for Λ given by w = -1. Now, combining the above discussion with Equation 1.15, it is possible to describe the evolution of each component i = (m, r, Λ), where m stands for matter (the non-relativistic component), r indicates radiation (the relativistic component), Λ describes the cosmological constant Λ. Hence, by making the equations of state explicit in Equation 1.15, one nds:

ρ m (t) = ρ m,0 a -3 (t), ρ r (t) = ρ r,0 a -4 (t), ρ Λ = ρ Λ,0 = const. (1.20)
As already mentioned when describing Equation 1.13, a uid with an equation of state of w < -1/3 would drive an acceleration of the universe expansion. In particular, the cosmological constant Λ can be interpreted as a uid with w Λ = -1, in which case its energy density remains constant with time. However, in order to investigate the nature of dark energy, one can let its equation of state vary with time, to have a more general expression where ρ de is di erent from ρ Λ . In particular, its evolution can be expressed as a function of the more general equation of state w de = f (a) that presents several parametrisations. A common parametrisation is given by [START_REF] Linder | Cosmic structure growth and dark energy[END_REF]:

w de (a) = w 0 + (1 -a)w a , (1.21)
that includes the case w de < -1/3 which is compatible with the expansion of the universe and also recovers the simpler cosmological constant case for w 0 = -1 and w a = 0. Now, by manipulating the rst Friedmann equation Equation 1.10 to write it as

1 + K H 2 a 2 = 8πG 3H 2 ρ, (1.22)
and de ning the quantity in the. r.h.s as

ρ c ≡ 3H 2 8πG , (1.23)
called the critical density, it is possible to introduce for each component i = (matter, radiation, Λ) its corresponding density parameter, de ned as

Ω i = ρ i ρ c . (1.24)
This leads to a total density parameter de ned as

Ω(a) = Ω m a -3 + Ω r a -4 + Ω de a f (a) (1.25)
obtained by dividing the energy densities for each component by the critical density and by considering Equation 1.20 with a more general equation of state for dark energy: w = f (a).

B

Then, Equation 1.22 expressed in terms of the density parameter Ω becomes K H 2 a 2 = Ω -1, telling us that for Ω > 1 we expect a universe with positive curvature K > 0, for Ω < 1 a universe with positive curvature K < 0 and for Ω = 1 a at universe with K = 0.

Kinematics in an expanding universe

To deal with the evolution of the motion of freely-falling particles in a FLRW space-time, geodesic equations in a curved space-time can be obtained. From the General Relativity Equivalence Principle, or by applying the principle of least action to the action

S = -g µν dx µ dx ν , (1.26)
it is possible to derive the geodesic equations of motion (Weinberg 1972):

d 2 x µ dλ 2 + Γ µ αβ dx α dλ dx β dλ = 0, (1.27)
representing the evolution of the shortest possible trajectories of a particle in free fall between two points with respect to the a ne parameter λ. The study of the radiation emitted by the objects populating the universe provides crucial information about their characteristics, hence representing a fundamental tool in cosmology. Speci cally, when investigating the emission and propagation of such radiation, the expansion of the universe needs to be taken into account. Indeed, considering that photons are massless, their four-momentum p µ = (E, p i ) follows the equation of motion p µ p µ = 0. This, combined with the Friedmann equations, relates the energy of the photon E the scale factor as E ∝ a -1 and hence λ ∝ a (consistently with the solution for the energy density of radiation in Equation 1.14). So, in the presence of expansion, the relation between the light emitted at a given time of emission t e from a given position r e , characterised by the wavelength λ e is related to the wavelength of the same photon observed at time t 0 through

λ e λ 0 = a e a 0 .
(1.28)

In the absence of expansion a e = a 0 and the wavelength of the light stays the same as it propagates through the universe. In the presence of expansion, the scale factor at t 0 is larger than it was when the light was emitted by the distant source, in which case λ 0 > λ e , namely its spectrum is shifted towards the red. This shift towards the red is called redshift, and it is quanti ed by the parameter

z ≡ λ 0 -λ e λ e .
(1.29)

Along with Equation 1.28, Equation 1.29 sets the relation between the measured redshift z of a distant source to the scale factor at the time of emission t e :

a e = 1 1 + z , (1.30)
considering that a(t 0 ) = 1. Hence, the presence of a cosmological redshift is due to the expansion of the universe (which is encoded in the scale factor) and from the knowledge of the emission of well-known objects (such as the standard candles) at a certain redshift z it is possible to trace the value of the scale factor at the time of emission.

Distances

The redshift of distant sources can be linked to their distance through the cosmological model. Hubble 1929 observed a roughly linear relation between the velocity and the distance of extra-galactic nebulae. Such relation is known as Hubble's law : v = H 0 r. It can be recovered from Equation 1.12 and it is only valid for the local universe. On cosmological scales, the evolution of the Hubble parameter as a function of time needs to be taken into account.

Proper distance: The proper distance d p is the distance seen by a photon. It describes the collection of in nitesimal light paths despite the expansion of the universe. Mathematically, it is de ned as d p = cdt. Since the light speed is a constant, the proper distance is simply given by

d p (z 0 , z e ) = c[t(z 0 ) -t(z e )], (1.31)
where z 0 and z e are respectively the redshift at which the light ay if observed and emitted.

Comoving distance: The comoving radial distance (or simply comoving distance) χ is dened as dχ = cdt a , namely is the rescaling by the scale factor of the physical distance, such that two particles initially at rest in the evolving FLRW metric stay equidistant in comoving coordinates. As mentioned before, the cosmological model enables to relate the observed redshift of a distant source to its comoving distance with respect to the observer. In particular, by manipulating the Hubble parameter de nition as dt = da aH , it is possible to re-write

dχ = cdt a as B χ(a) = 1 a cda a 2 H(a ) , (1.32)
where the extreme of the integral indicate that χ(a) is the comoving distance of an object given that we observe now (at t 0 , when holds a(t 0 ) = 1) light that was emitted when the scale factor of the universe was equal to a. Through the scale factor-redshift relation Equation 1.30, this can be explicitly expressed with respect to the redshift as

χ(z) = z 0 cdz H(z ) . (1.33)
Comoving transverse distance: The comoving transverse distance f K is the ratio of the comoving separation between two points at χ to their separation angle. It only depends on the comoving distance χ and it is related with the curvature K as

f K (χ) =      1 √ K sin ( √ Kχ) for K > 0 χ for K = 0 1 √ -K sinh ( √ -Kχ) for K < 0.
(1.34)

Angular diameter distance: The angular diameter distance D A is the ratio of a physical separation at the emission epoch (with comoving radial distance χ) to its angular separation from an observer. It can be obtained by scaling the comoving transverse distance to the emission epoch z e . So, for an observer at z 0 = 0 and calling the emission epoch z e = z, the angular diameter distance is de ned as

D A (z) = f K [χ(z)] 1 + z . (1.35)
Luminosity distance: The luminosity distance is based on the scaling of the ux F of a distant source with its distance. It is de ned so that the ux of the source at distance with an intrinsic luminosity L corresponds to:

D L = L 4πF
.

(1.36)

It can be expressed in terms of the transverse comoving distance f K or in terms of the angular diameter distance as:

D L (z) = (1 + z)f K [χ(z)] = (1 + z) 2 D A (z) (1.37)
for any observer at z 0 = 0 and sources at z e = z.

Cosmological parameters

Given the tools introduced in the previous sections, it is now possible to de ne some cosmological parameters and their evolution as function of redshift.

Age of the universe

The rst Friedmann Equation can be rewritten in terms of today's (at t 0 ) cosmological parameters and it is known as Hubble equation. In particular, it can be rephrased as

ȧ2 = H 2 0 i Ω i (t 0 )a -(1+3w i ) + [1 -Ω(t 0 )] .
(1.38) By integrating the above equation, the evolution for the scale factor a(t) is obtained and it is consequently possible to give an estimation of the expansion rate of the universe today.

Considering instead the inverse function of the scale factor t(a), one can derive the age of the universe, given a certain cosmological model. For instance, the calculation of t can be performed by considering the change of variables dt = da/ ȧ and by integrating it starting from an initial time (imposed to be t i = 0 ) and supposing the scale factor at that time to be a point-like singularity, i.e. a(t i ) = 0. In this way the following expression for t is obtained as

t = 1 H 0 ∞ z dz(1 + z) -2 [ i Ω i (t 0 )(1 + z) (1+3w i ) + 1 -Ω(t 0 )] 1/2 , (1.39)
where the scale factor-redshift relation as da = -dz(1 + z) -2 has been used. Equation 1.39 returns the age of the universe today when integrated between zero and in nity (corresponding to an integration between 0 and 1 with respect to the scale factor). The estimated (Planck Collaboration et al. 2020) age of the universe today is 13.801 ± 0.024 Gyr.

(1.40) B

Hubble parameter

By simply using that ȧ2 = H 2 a 2 in Equation 1.38, the following explicit expression for the Hubble Parameter as a function of the redshift is also obtained

H 2 (z) = H 2 0 i Ω i (t 0 )(1 + z) 3(1+w i ) + [1 -Ω(t 0 )](1 + z) 2 ,
(1.41) so that the expansion rate at a given redshift z can be related to the energy density contribution of each component Ω i , knowing its value today.

Density Parameter

The connection between the density parameter Ω i and the redshift leads to the signi cant consideration that if the universe is at today will remain so for any value of the redshift and, more generally, if the universe has a certain curvature, this will stay the same for every time. Indeed the density parameter can be formulated as

Ω i (z) = Ω i (t 0 )(1 + z) 3(1+w i ) [ i Ω i (t 0 )(1 + z) 3(1+w i )+(1-Ω(t 0 ))(1+z) 2 ]
,

(1.42)

where Equation 1.23 and Equation 1.28 have been used.

Primordial power spectrum

The current theory explaining how structures as galaxies, galaxy clusters or the cosmic web that we observe today formed assumes the existence of small primordial perturbations, namely deviations from the mean density of the universe which slowly increased in amplitude because of the phenomenon of gravitational instability and formed the virialised structures observed at the present time. Among the theories proposed to explain the existence of these density perturbations, in ation (Guth 1981; Baumann 2012) predicts that these deviations from the background density eld are the product of processes in the very early universe, corresponding to quantum uctuations which expanded to macroscopic proportions during an extremely fast period of expansion. During this process, these quantum uctuations reach macroscopic scales and will eventually act as seeds of large-scale structure formation. The scale dependence of such perturbations are described by the primordial power spectrum, de ned as

P ζ (k) = A s k k p ns-1 , (1.43)
and parametrised as a power law of amplitude A s and spectral index n s , that measures the deviation from scale invariance (n s = 1) and k p is a pivot scale. In the large-scale structure convention, it is common to normalise the primordial power spectrum of matter uctuations at present time by requiring that the root mean square (r.m.s.) variance on a sphere of radius R 8 = 8 h -1 Mpc is equal to a normalisation factor, called σ 8 .

Sum of neutrino masses

Thanks to the discovery of oscillations among neutrino avor eigenstates, neutrino masses m ν are now known to be non-zero. This can be understood intuitively by considering the minimal neutrino mixing scenario where the existence of three species of neutrinos is assumed. The neutrino avor eigenstates ν α (with α = e, µ, τ ) can be expressed as a superposition of neutrino mass eigenstates ν i (with i = 1, 2, 3 ) through the unitary transformation:

ν α = Σ α U αi ν i , (1.44)
and the concept of neutrino mixing is encoded in the unitary matrix U αi : if U αi is the identity matrix, the avor eigenstates coincide with the mass eigenstates and there is no oscillation. However, from solar and atmospheric neutrino experiments, compelling evidences for neutrinos oscillation have been provided (Group, Zyla, Barnett, et al. 2020) as it is observed that the massive states propagate di erently between the source (in which neutrinos are produced) and the observer. One of the main quantities considered by these experiments to describe and quantify this phenomenon, is given by the probability that a neutrino of avor α is later detected as a neutrino with avor β. Such probability can be derived explicitly and its expression depends on the neutrino mass squared di erences ∆m 2 ij ≡ |m 2 im 2 j | [START_REF] Maki | Remarks on the uni ed model of elementary particles[END_REF]Pontecorvo 1968;Kajita 1999). Hence, neutrino oscillation experiments can provide information about the di erence of the squared masses while remaining blind with respect to the absolute value of the neutrino mass and to the way in which these masses are ordered. This allows two scenarios in which the masses can be distributed: a normal hierarchy where the masses are ordered as m 1 < m 2 < m 3 and an inverted hierarchy corresponding to m 3 < m 1 < m 2 . Moreover, from the combination of solar and atmospheric results, a hierarchy among the mass splittings can be derived as well, namely ∆m 2 21 |∆m 2 31 | ∼ |∆m 2 32 |. Assuming then that the lightest neutrino is massless, it is possible to obtain a lower bound from oscillation experiments on the sum of neutrino masses ν m ν ≡ M ν by taking the sum of the mass di erences: in a normal hierarchy setting this value correspond to M ν > 0.06 eV and in a inverted hierarchy setting to M ν > 0.1 eV. Particle physics experiments (Wolf 2010;Esfahani, Asner, Böser, et al. B 2017) are only sensitive to the mass of the lightest neutrino. Fortunately, massive neutrinos leave an imprint on the expansion history and growth of structure in the universe and this can be used to infer the sum of their masses M ν . Speci cally, the fact that neutrinos with mass less than a few eV remain relativistic and behave like radiation at the time of matterradiation equality and became then non-relativistic behaving like matter today, delays the beginning of linear growth by moving the matter-radiation equality at a later time. Moreover, neutrinos with small masses have large thermal velocities, and can then stream out of cold dark matter potential wells, causing a suppression of the growth of structure below a certain scale. This scale is approximately equal to the Hubble radius when neutrinos are relativistic while when neutrinos become non-relativistic this scale corresponds to the point in which gravitational instability overcomes the thermal pressure of neutrinos. The strongest upper bound we have today on the sum of neutrino masses comes from cosmology, and it is speci cally M ν < 0.12 eV, from the combination of temperature anisotropies in the cosmic microwave background (CMB), CMB lensing and Baryon Acoustic Oscillation (BAO) (Planck Collaboration et al. 2020).

ΛCDM model

Depending on the cosmological parameters, various cosmological models with di erent characteristics emerge. The one that best describes current observations is the Λ Cold Dark Matter model (ΛCDM). The name itself gives us information about its two main components: Λ being the cosmological constant appearing in Einstein Fields Equations of General Relativity, related to the simplest parametrisation of a uid called dark energy (DE); and CDM standing for cold dark matter. This model is a mathematical parameterisation of Big Bang cosmology, as described by General Relativity and the Friedman-Lemaître-Roberson-Walker (FLRW) equations introduced in Section 1.1.1. It is based on the assumptions that the universe is composed of elementary particles, i.e. ordinary matter (baryons, leptons, photons, neutrinos), and cold (non-relativistic at the time of decoupling) dark matter, which only interacts gravitationally along with dark energy, responsible for the observed acceleration in the Hubble expansion. Dark energy is assumed to take the form of a constant vaccuum energy density, which should coincide with the cosmological constant introduced by Einstein in 1917. It requires only a small set of parameters (illustrated in Table 1.1), to reproduce all cosmological observations and completely specify the cosmological model. The requirement of space atness comes from these parameters and it is in agreement with the observational data. Indeed, by rewriting the r.h.s. of Equation 1.22 in terms of the density parameter Ω TOT = i Ω i it can be seen how for a at universe (K = 0), Equation 1.22 requires that Ω TOT = 1. By looking at the values of the energy density parameters for the di erent components in Table 1.1, one can see how in the ΛCDM paradigm, the vast majority of the total energy-matter density budget is dominated by the cosmological constant Λ, accounting for ∼ 69% of the total energy-matter density budget. The remaining 31% is encoded in the matter density Ω m , whose contributions are respectively given by a 26.2%

of cold dark matter (encoded in the parameter Ω c ) and 4.8% of ordinary matter, usually called in cosmology baryonic matter (encoded in Ω b ). The contribution of Ω Λ is associated in the ΛCDM framework with the dark energy, responsible for the accelerated expansion of the universe (Riess et al. 1998;[START_REF] Perlmutter | Measurements of Ω and Λ from 42 High-Redshift Supernovae[END_REF], because of its negative pressure, introduced through its equation of state in Section 1.1.2. The necessity for dark matter, the second largest contribution to the total energy content of the universe, was historically invoked to explain the mass de cit of visible matter from observations (Zwicky 1937) and the observations of at galaxy rotation curves to large distances [START_REF] Rubin | Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions[END_REF]. Also, the combined weak lensing and X-ray observation of the Bullet Cluster provided as well a strong hint of the presence of dark matter through the o set found between the center of the total mass and the center of the baryonic mass peaks of the clusters system [START_REF] Clowe | A Direct Empirical Proof of the Existence of Dark Matter[END_REF].

Parameter Symbol Value

Hubble parameter H 0 (67.66 ± 0.42) km s These two dark components, dark matter and dark energy, dominate the current energymatter content of the universe. However, their nature and properties are still unknown and one of the main goals of the current and future cosmological surveys is exactly to try to determine them, by comparing theory and observations and by performing cosmological parameter inference, as will be described in the next sections, through the imprints these components leave on the cosmological observables.

Principles of structure formation

The previous section dealt with a homogeneous and isotropic universe, constituted by a perfectly smooth cosmological uid. However, from both simulations an observations we know that structures as galaxies, galaxy clusters or the cosmic web actually exist. As brie y B anticipated in Section 1.2.4, the current theory explaining how those structures formed assumes the existence of small primordial deviations from the mean density which slowly increase in amplitude because of the phenomenon of gravitational instability and form the virialised structures observed at the present time.

Density contrast in the linear regime

The above mentioned deviations from the background can be considered already at the level of the FLRW metric de ned in Equation 1.2 by adding to it a perturbative term, in the limit where these perturbations are su ciently small. This perturbative term propagates in the matter density which becomes the sum of a background density ρm (t) term and a perturbation term δρ m (r, t). The deviation from the background density is then summarised in the quantity

δ(r, t) = δρ m (r, t) ρm (t) = ρ m (r, t) -ρm (t) ρm (t) , (1.45)
called the matter density contrast. At early times and on large scales, this deviation is very small and it is fair to work in the limit δ 1 that allows to derive the evolution of the density eld using linear perturbation theory. The dynamical evolution of the density ρ can be described through a Vlasov-Poisson system of equations (Peebles 1993) and consequently the density contrast δ as well. By combining these equations and working under the assumption of an ideal pressureless uid, in the linear regime the evolution for the density contrast is given by (Dodelson 2003)

δ + 2H δ = 4πGρ m δ, (1.46)
which provides a good description of the density uctuations evolution in the early epochs of the universe, when density uctuations are still small. The r.h.s. comes from the Poisson equation and its the source term governing the evolution of the density contrast, while the second term of the l.h.s. takes into account the universe expansion through the Hubble parameter H. This ordinary second order di erential equation admits the general solution of the form (Peebles 1980; Peacock 1999):

δ(t, r) = D + (t)δ + (r) + D -(t)δ -(r), (1.47)
where the spatial and temporal coordinates can be decoupled and D + and D -are respectively a growing and decaying mode. Since the decaying mode will disappear as time goes by, it is not relevant at late times and in this context it can be neglected. Hence, the density contrast can then be expressed using the growing mode whose general solution in terms of the scale factor is given by

D + (a) = 5Ω m H(a) 2H 0 a 0 da a 3 H 0 H(a ) 3 ,
(1.48) and it is conventionally normalised such that D + (a) = a during the matter-dominated era and it is referred to as the growth factor. It is convenient to consider its normalised expression as .49) to ensure that its value at present time is D + (a = 1) = D + (z = 0) = 1 so that the density contrast in the linear regime is given by δ(r, z) = D(z)δ(r).

D(a) = D + (a) D + (a = 1) , ( 1

Matter power spectrum

A consequence of the cosmological principle is that the density contrast de ned in Equation 1.45 is a statistically isotropic and homogeneous eld. To be able to perform a quantitative comparison between the theory of structure formation and the actual distribution of the structures that we observe, statistical measures of the density eld are needed. A natural way to do it is to invoke the two-point correlation function (2PCF) of the eld, which is given by

δ(r )δ(r + r ) r = R 3 δ(r + r )δ(r )dr , (1.50)
and describes the probability of nding two objects separated by the distance r at the positions r and r +r with respect to the probability one would have if those objets were simply scattered independently with uniform probability. Considering the 3D Fourier transform of the density eld δ

(k) = 1 (2π) 3 d 3 re -ik•r δ(r)
, it is possible to write the 2PCF of this eld in Fourier space as

δ(k) δ (k ) = (2π) 3 P δ (k)δ (3) D (k -k ),
(1.51)

where δ (k) is the complex conjugate and δ

D is the Dirac delta and P δ (k) is the 3D matter power spectrum. Given the assumption of homogeneity and isotropy, the dependence on the scale is actually only P δ (k) = P δ (|k|), with |k| = k. Then, it is possible to relate the matter power spectrum P δ (k) to the primordial power spectrum of initial uctuations B de ned in Equation 1.43, through a transfer function T (k) as for example (Eisenstein and Hu 1998)

T (k) = δ(k, a = 1) δ(k = 0, a = 0) δ(k, a = 0) δ(k = 0, a = 1) , (1.52)
such that the processed power-spectrum P δ (k) is related to its primordial form via P δ (k) ∝ T 2 (k)P ζ (k), accounting for the modulation of each Fourier mode between their original value at a = 0 to their current value at a = 1. The actual computation of the the power spectrum at su ciently large scales can be performed by solving linear perturbation theory equations. In particular, it is possible to derive some analytical expressions for the transfer [START_REF] Smith | Stable clustering, the halo model and non-linear cosmological power spectra[END_REF] whose main idea is based on the assumption that all the matter content in the universe is assumed to be bound in dark matter halos. Then the power spectrum can be decomposed into two terms, one accounting for the correlation between two distinct halos, that is dominant at large scales, and another term describing matter correlations within the same dark matter halo, which is dominant at small scales. Boltzmann solvers as CLASS can then be used to compute the linear power spectrum as mentioned before and also to compute the nonlinear power spectrum including non-linear corrections computed through the HALOFIT code. It is very important to remind that the functional form for the power spectrum is then based on the halo model, but the model parameters are calibrated from the N-body simulations in consideration. As can be seen by looking at Figure 1.1, the impact of these non linear correction leads to an enhancement of the power at small scales. Moreover, as will be described more in details in Section 2.2.5, for future surveys such as Euclid and Rubin Observatory Legacy Survey of Space and Time (LSST) that will have a very high precision and very low statistical uncertainties, it is very important to control and be able to properly model the non-linearities in the matter power spectrum with high accuracy to get accurate and unbiased cosmological constraints from weak gravitational lensing measurements.

Brief overview of cosmological probes

As described in Section 1.2.6, the nature and properties of the vast majority of the components of our universe are still unknown and from Section 1.2.5 one can see how even in the Standard Model of particle physics there are still some open questions that need an answer. Fortunately, the behaviour and characteristics of dark energy, dark matter and the sum of neutrino masses, leave an imprint on cosmological observables and this can be used to infer cosmological parameters, with the aim of trying to answering those questions.

10 -3 10 -2 10 -1 k [M pc -1 ]
As an example, Figure 1.2 shows the fractional di erence matter power spectrum between power spectra of models including massive neutrinos (the di erent colours corresponding to di erent values of the masses) and the power spectrum with massless neutrinos as a function of comoving wavenumber k. The same plot indicates also the sensitivity in the power spectrum for di erent probes depending on the scale.

For instance, the Cosmic Microwave Background radiation (CMB) (Penzias and Wilson 1965), that is the result of the decoupling of photons from electrons at the epoch of recombination, can be used as a probe to constrain the neutrino mass. Indeed, for low enough masses (such that they became non-relativistic after recombination), the angular diameter distance that is degenerate with decreasing H 0 changes, impacting the CMB power spectrum. Small secondary temperature uctuations induced in the CMB due to the evolution of the gravitational potentials right after recombination can be used to constrain relativistic species, hence also the number of neutrinos and their masses. Moreover, Planck Collaboration et al. 2014 moved the analysis to a regime where the dominant e ect comes from the gravitational lensing of the CMB by the large scale structure. Indeed, CMB lensing leads to a direct measurement of the matter power spectrum on scales where the e ects of neutrino masses become relevant. More generally, the decrease in temperature due to the expansion of the universe makes the neutrinos transition from being a ultra-relativistic gas behaving as radiation to a cold gas behaving as dark matter. This results in a suppression of power on scales smaller than the neutrinos free-streaming scale. This scale is associated with the scale at which the matter power spectrum peaks, caused by the transition from the radiation to the matter dominated era, when perturbations cross the cosmological horizon. This can be "seen" through the CMB at large scales and by large scale structures probes at small scales.

In general, measurements of tracers of the matter clustering in the large-scale structure can be used to constrain cosmological parameters. The combination with absorption lines in the spectra of distant galaxies and quasars arising from the Lyman-α electron transition of the neutral hydrogen due to intervening low density gas, more simply referred to as Ly-α forest (Weinberg et al. 2003) has enabled to provide one of the strongest inferred constraints on neutrino mass for example [START_REF] Seljak | Cosmological parameters from combining the Lyman-alpha forest with CMB, galaxy clustering and SN constraints[END_REF]. The detection of the 21 cm line from high-redshift neutral hydrogen (HI) represents a powerful probe that with the potential of mapping the distribution of matter until the epoch of reionisation and is the goal of several present and upcoming radio telescopes as for instance the Square Kilometer Array (SKA) (Santos et al. 2015;[START_REF] Combes | Science with SKA[END_REF]. The next two sections, are dedicated to two speci c probes, weak lensing and galaxy clustering, that I have used in some of the works described in this thesis and that represent the two main cosmological probes that will be employed by Euclid.

Galaxy Clustering

Another tracer of the underlying matter distribution is given by the distribution of galaxies. Indeed, galaxies as well as dark matter haloes are sampling the matter density eld, however not in a uniform way, in the sense that the spatial clustering of galaxies do not precisely represent the clustering of the totality of the matter in the universe. Galaxies are hence considered as a biased tracer of the underlying total matter density eld (Kaiser 1984;Bardeen et al. 1986). Following the same philosophy of de ning a matter density contrast to investigate the evolution of the overdensity with respect to the background to eventually compute a statistical measure of the eld as the 2PCF or the power spectrum, it is possible to de ne the uctuations in the galaxy number density eld n(x; z) in the linear regime as

δ g (x; z) = n(x; z) -n(z) n(z) = b g (z)δ(x; z), (1.53)
related to the matter density contrast de ned in Equation 1.45 through the galaxy bias b g (z).

At large scales, it is fair to assume the bias to be scale independent and to model it uniquely as a function of the redshift. Under these conditions, from Equation 1.53 follows that the galaxies power spectrum can be expressed in terms of the matter power spectrum as

P g (k, z) = b 2 g (z)P δ (k, z). (1.54)
It is important to underline that more complex bias modelling are required, in order to B include e ects such as primordial non-gaussianities, the presence of massive neutrinos, dark energy, modi ed gravity and redshift space distortions, as these factors can lead to a more complicated expression for the bias, making it scale-dependent (Desjacques, Jeong, and Schmidt 2018). The discussion of these e ects goes beyond the purposes of this work, however their role is non-negligible in current and future cosmological analysis and a huge scienti c e ort from the community to properly model them is currently ongoing. To map the distributions of the galaxies it is possible to measure their frequency distribution of observed photons per galaxy. Then, through spectroscopy, knowing the absorption and emission lines in the rest-frame and obtaining the redshifted ones, it is possible to determine the position of each galaxy in terms of right ascension (RA), declination (DEC) and redshift z spec . Thanks to spectroscopic instruments as the Sloan Digital Sky Survey (SDSS), it has been possible to create an extremely detailed 3D map of the universe, using deep multicolor images for one third of the sky, and spectra for more than 3 millions astronomical objects have been retrieved (Ahn et al. 2012) and the future Dark Energy Spectroscopic Instrument (DESI) data releases will obtain spectra and measure the positions and receding velocities of ∼ 40 million galaxies (DESI Collaboration et al. 2016). The medium-class ESA mission Euclid will enable to measure up to ∼ 30 millions spectroscopic redshifts (Pozzetti, L. et al. 2016;Euclid Collaboration et al. 2020b) through the spectroscopic channel of its Near Infrared Spectrometer and Photometer (NISP) instrument. An alternative way to estimate the galaxy redshifts without involving the spectra, consists in measuring the galaxy ux in a number of standard lters that select a relatively broad passband of colours to determine the redshift that is then related to the distance through the Hubble law, de ned in Section 1.1.3. The redshifts estimated in this way are referred to as photometric redshifts. These measurements lead to the probes for the clustering of galaxies that depending on the technique used are referred to as Spectroscopic Galaxy Clustering (GCsp) and Photometric Galaxy Clustering (GCph) that along with weak gravitational lensing (see Section 1.4.2) constitute the primary probes of the Euclid mission. Section 6.5.2 provides a slightly more detailed description of Photometric Galaxy Clustering, used in one of the projects of this thesis (Chapter 6).

Weak Gravitational Lensing

Another powerful cosmological probe to constrain cosmological parameters is the bending of light emitted by distant background galaxies due to the presence of foreground largescale structure between the sources emitting the light and the observer. This phenomenon is widely used in current cosmology and is called Weak Gravitational Lensing (WL). Indeed, the distortions induced by the presence of the foreground large-scale structure on background galaxy shapes provide a direct probe of the matter power spectrum P δ (k) (Kaiser 1992). This is because the distortions, even if small in the weak regime, are induced by the matter distribution along the path of the emitted photons. Hence, it is possible to retrieve information on the properties of this matter distribution, given enough galaxies across the sky. Moreover, since both baryonic and dark matter contributes to this e ect, weak lensing traces directly all matter, in contrast for example to the distribution of galaxies, that as seen in the previous section, needs to invoke a galaxy bias to retrieve the dark matter distribution using galaxies. The observable signal related to this e ect encodes an anisotropic deformation, called the shear along with an isotropic magni cation, called the convergence, of the original source galaxy shape. So what it is actually observed is galaxy shapes, precisely, the observed ellipticity of the galaxy that in the weak regime can be expressed as the sum of its intrinsic ellipticity plus the shear induced by weak lensing. Details of weak lensing formalism, de nitions and state of the art results are illustrated in Chapter 2, which is entirely dedicated to weak lensing, as it represents one of the main focus of this thesis in the framework of cosmological parameter inference. As mentioned above, weak lensing constitutes one of the primary probe of future galaxy survey, like Euclid. Its visible (VIS) instrument will measure the shapes of 1.5 billions galaxies covering 15000 deg 2 of the sky, up to redshifts of about z s = 2. The near infrared photometry introduced in the previous section, combined with the VIS data will enable to derive photometric redshifts and rough estimates of these galaxies. Past and current galaxy surveys as the Canada-France-Hawaii Telescope Lensing Survey (CFTHLenS) (Heymans et al. 2012), the Kilo-Degree Survey (KiDS) (Hildebrandt et al. 2016) or the Dark Energy Survey (DES) (Zuntz et al. 2018) have also used and are at present using weak lensing as main cosmological probe and released the associated data products that led to the delivery of some of the most recent results in cosmology (DES Collaboration, Abbott, et al. 2021). In the coming years, along with Euclid, also the Vera C. Rubin Observatory Large Synoptic Survey Telescope (LSST) will survey a remarkable fraction of the sky (∼ 20000 deg 2 ). Both experiments will deliver source galaxy samples consisting of billions of galaxies and with the combination of deep and wide-eld optical photometry that will be provided by LSST along with the wide-eld optical photometry, NIR photometry and high resolution of Euclid it is crucial to optimise and try to maximise the amount of information they will bring.

Parameter inference for future galaxy surveys

In order to exploit the richness in information provided by the cosmological probes described in the previous section and be able to measure and estimate the parameters of a certain cosmological model, given some observed data, statistical methods are needed. This section presents an introduction to the main tools in a Bayesian inference framework useful for the purposes of this thesis based on (Heavens 2009; Verde 2010) 1 . B

Bayesian inference

In the Bayesian framework, probabilities are interpreted as degree of belief in a hypothesis. Given a random variable (event) x and denoting the probability density of such event to take a speci c value as P(x), then the probability that the random variable x takes a value between x and x+dx is given by P(x)dx. Being a probability, P(x) has to be de ned within the range 0 ≤ P(x) ≤ 1 and it has to satisfy the condition ∞ -∞ P(x)dx = 1. Moreover, for mutually exclusive events x 1 and x 2 holds the relation: P(x 1 ∪ x 2 ) = P(x 1 ) + P(x 2 ). Considering now the probability of x 1 and x 2 to happen, i.e. the joint probability density function (PDF) P(x 1 , x 2 ), this is given by the probability of x 1 times the conditional probability of x 2 given that x 1 has already happened P(x 2 |x 1 ), namely P(x 1 , x 2 ) = P(x 1 )P(x 2 |x 1 ). If x 1 and x 2 are independent events, then P(x 2 |x 1 ) = P(x 2 ) and the joint PDF reduces to P(x 1 , x 2 ) = P(x 1 )P(x 2 ). Based on P(x 1 , x 2 ) = P(x 2 , x 1 ) the following identity is a direct consequence:

P(x 1 , x 2 ) = P(x 1 )P(x 2 |x 1 ) = P(x 2 )P(x 1 |x 2 ).
(1.55)

For the purposes of this work, it is now interesting to embed the previous discussion in a potential application for cosmology. If we consider a given cosmological model M characterised by a set of cosmological parameters Θ = {θ 1 , θ 2 , ..., θ n }, assuming that we have observed a set of data d, we can rely on Equation 1.55 to write the following identity for the conditional probability density function of the cosmological parameters Θ, given the data d:

P(Θ|d, M) = P(Θ|M)P(d|Θ, M) P(d|M) .

(1.56)

This identity is known as Bayes Theorem and it regulates the relation between the conditional probability density function of the parameters Θ given the observed data d, with the conditional probability density function of observing the data d given a model M characterised by the parameters Θ and given a precedent knowledge on the the parameters Θ, that might be given for instance by some previous experiments, and on the data. P(Θ|d, M) is called the posterior probability distribution, P(Θ|M) the prior, P(d|Θ, M) the likelihood and P(d|M) the evidence. With the nal goal of determining the posterior distribution for the parameters of a given cosmological model, one needs to de ne a prior on the parameters, assume a likelihood and de ne the evidence. The latter, that is the probability of the data before it is known, is fundamental for model comparison but it can be considered as a normalisation constant for parameter inference, in a way that the problem to determine the posterior distribution becomes:

P(Θ|d) ∝ P(Θ)P(d|Θ), (1.57)
where to simplify the notation the assumption of the model M has been omitted. Concerning the prior, it can provide a de nite information about a certain variable or parameters of a given model or general information about such variable. In the rst case it is called informative prior and in the second uninformative prior. To give an example of the rst scenario, in many situations we have knowledge of the parameters from some previous experiment and this information can enter the inference analysis through P(Θ). For instance, the state-of-the art of cosmological parameters inference from previous experiments such as COBE [START_REF] Smoot | Summary of Results from COBE[END_REF], WMAP (Hinshaw et al. 2013), Planck (Planck Collaboration et al. 2020), SDSS (Tegmark, Strauss, Blanton, et al. 2004), provides a rich and solid theoretical cornerstone for current and future analysis. In the absence of any previous information, the prior can be assumed to be a constant and it is called a at prior. In summary, the prior represents the state of belief on the parameters that will be inferred, prior to any in uence by the new observed data d.

Likelihood

Once the prior choice is made, the fundamental ingredient that needs to be determined to evaluate the posterior probability distribution for the parameters is the likelihood function P(d|Θ), as one can see from Equation 1.57. The likelihood can have an analytical expression when this is known or it can be estimated through density estimation problems belonging to the class of methods known as likelihood free inference (LFI) (Papamakarios and Murray 2018;Alsing et al. 2019), Section 1.5.4 brie y discusses this point. A very common choice for the likelihood function in cosmological analysis is the assumption of a Gaussian likelihood relying on arguments such as the central-limit theorem or the principle of maximum entropy. In these cases, when the likelihood is assumed to be Gaussian, it is convenient to work with its logarithm, namely

log L(d|Θ) = 1 2 [(d -µ(Θ)) T C -1 (d -µ(Θ))] ,
(1.58)

where d indicates the observed data, Θ the cosmological parameters de ning the model, C is the covariance matrix encoding the information about the data errors and the correlation among observables at di erent scales, and µ(Θ) is the observable coming from the model as a functions of the parameters Θ. From this expression of the likelihood, one can see how the key ingredients needed to build the likelihood are a) the covariance matrix, b) the model.

B

Covariance matrix estimation

Concerning the covariance matrix, this can be computed analytically (following again some assumptions as for example a Gaussian covariance as for example in Euclid Collaboration et al. 2020b) or it can be computed from N-body simulations as for example in Harnois-Déraps et al. 2018, when a Gaussian assumption does not hold anymore for example and the non Gaussian terms become more di cult to estimate. For the purposes of this work, I will focus on the case where the covariance is assumed to be cosmology independent, namely when it is computed for a xed set of cosmological parameters. Nevertheless, it is important to brie y mention that a way to accurately assess the uncertainty on the measurements relies on the computation of the covariance using a large ensemble of independent N-body simulations at each of the cosmologies that can be sampled in parameters space. However, this requires that the number of simulations per ensemble is signi cantly larger than the data vector and this scenario needs computing resources far exceeding those currently available.

As mentioned before, an alternative approach is to run an ensemble of full N-body simulations at a single cosmology and ignore the variation of the covariance with cosmology. This case also requires a large number of independent simulations with respect to the dimension of the summary statistics and the data array, as in practice it enters the likelihood function with a correction factor, known as the Hartlap factor (Hartlap, Simon, and Schneider 2007) that takes into accounts the loss of information due to the nite number of bins n bins and realisations N as

C -1 = N -n bins -2 N -1 C -1 * , (1.59)
where the element of the covariance matrix C * are given by

C ij * = N r=1 (x r i -xi )(x r j -xj ) N -1 , (1.60)
and x r i is the value of the observable in the i th bin for a given realisation r and xi = 1 N r x r i is the mean of such observable in a given bin over all the realisations. By looking at Equation 1.59 one can see then how the more the pre-factor is close to one (i.e. N n bins ) the more the estimated covariance is close to the truth and the uncertainty is smaller. Sellentin and Heavens 2017 have quanti ed the loss of information associated to this pre-factor for DES, KiDS and Euclid settings and in particular they shown how for KiDS-450 and DES SV, the uncertainty due to the covariance matrix is subdominant with respect to parameter degeneracies and the nite data set that are the dominating drivers of the uncertainties of cosmological parameters; for Euclid they found a loss of 1% of information for 1900 simulations, as compared to knowing the true the covariance matrix. In the works presented in this manuscript, the condition N n bins is always satis ed, thanks to the large suite of independent N-body simulations employed in the di erent studies. However, it is important to point out that this is not always the case for weak lensing analysis, depending on the available simulations. When the condition N n bins is not satis ed, the formalism to follow to properly account for all biases to be removed from the parameter inference is the one presented in Sellentin and Heavens 2016 and Sellentin and Heavens 2017. Moreover, as at the late-times tracers of the large-scale structure enter the non-linear regime, deviations from a Gaussian distribution of their PDF at smaller scales increase and it is necessary to properly consider in the analysis the non-Gaussian contributions such as the supersample covariance (SSC) (Hu and Kravtsov 2003). This extra term arises from the non-linear impact caused by density uctuations that have wavelengths that are greater than the survey size and modify the local observables such that the background density averaged over the survey size is not representative of the averaged density in the universe. This correction is crucial when performing cosmological analysis as it is expected to be the dominant source of statistical error or cosmic variance in the context of weak lensing weak lensing (Barreira et al. 2018). In the works presented in this thesis, this term is not included, but it is important to be aware of this correction for future extensions, considering that ways to take it into account exist or are currently in development, as for example the approximation presented by Lacasa and Grain 2019, that makes fast numerical computation of the SSC possible.

Building the model with Gaussian Processes

The next ingredient needed to build the likelihood de ned in Equation 1.58 is the model µ(Θ), namely a prediction for the observable used to perform the analysis as a function of the cosmological model. When an analytical prescription of the observable is possible to compute, one can use it to directly predict the observable as a function of cosmological parameters. There are several codes that provide an estimate for certain observable and enable to get a theoretical prediction, for example CAMB (Lewis, Challinor, and Lasenby 2000b), CLASS (Lesgourgues 2011), pyccl [START_REF] Chisari | Core Cosmology Library: Precision Cosmological Predictions for LSST[END_REF] for the matter power spectrum or lensing power spectrum, NICAEA (Kilbinger et al. 2017), for calculating second-order correlation functions, and in general pipeline and softwares are available to obtain prediction for cosmological observable as a function of cosmology (e.g. CosmoSIS (Zuntz et al. 2015), nbodykit (Hand et al. 2018), PyCosmo (Refregier et al. 2018)). However, there are a number of situations in which an analytical prescription is not always available or presents some drawbacks with respect to e ects that can impact the prediction (systematics, masks..). Weak lensing peak counts, that will be introduced and discussed in Section 2.3.5, for example can be used as cosmological observable to derive constraints on parameters. Analytical models to obtain an estimation of their predictions exist, but present some downsides as it will be described in Section 2.3.5. Hence, one of the alternative ways to get the theoretical predictions in these situations relies in employing Gaussian Processes Regression (GPR). Gaussian processes can be seen as an in nite-dimensional multivariate Gaussian distribution or as Gaussian distributions over functions such that they are completely de ned by a B mean function m(θ) and a covariance function K(θ, θ ), that in this context is dubbed the kernel and f (θ) ∼ GP (m(θ), K(θ, θ )). Then, the Gaussian Processes act by an assumption of smoothness, i.e. by considering that for a new point in parameter space θ * which is su ciently close to a known point θ belonging to the training set, the corresponding observable will be described by a joint normal distribution along with the known observable. This can be summarised by:

f f * ∼ N m m * , K(θ, θ) + σ 2 n I K(θ, θ * ) K(θ * , θ) K(θ * , θ * ) ,
where K(θ, θ ) is the kernel of the Gaussian processes that assesses the smooth relation among points in parameter space and for the works presented in this thesis we take its form to be an anisotropic squared exponential function. σ n is the standard error of the noise level in the targets. Then, the prediction can be retrieved by making use of the Multivariate Gaussian Theorem, so that it is simply de ned by its posterior conditional

P(f * |θ * , θ, f ) = N (f * |m * , Σ * )
, completely de ned by its conditional mean and covariance (Rasmussen and Williams 2005):

m = m(θ * ) + K(θ * , θ)[K(θ, θ) + σ 2 n I] -1 (f -m(θ)), Σ * = K(θ * , θ * ) -K(θ * , θ)[K(θ, θ) + σ 2 n I] -1 K(θ, θ * ).
The predictions are then exactly given by the conditional means m , and variances can be obtained directly taking the diagonal elements of the covariance matrix Σ * .

Explore parameter space with MCMC

Now that what is needed to build the likelihood is de ned, the next step is to explore the parameter space to nd the parameter range that provides theoretical predictions µ(θ) that best t the data d, with the nal goal of deriving the posterior probability distribution dened in Equation 1.56. One way to do this relies on Markov Chain Monte Carlo (MCMC). Given a generic probability distribution p(θ), called the target distribution, the MCMC samples the parameter space through a random walk that is initialised at a given point θ 0 . A common choice for such point is a value (or a set of n values for n-dimensional parameter space) very close to where it is expected that the likelihood is centred. In the case of forecasts for instance, given a ducial model, one can initialise the random walk in a Gaussian ball with a tiny radius around the ducial parameters values. Now, to start the parameter space exploration one needs to determine the next position in parameter space θi+1 . To do this, one can follow the Metropolis-Hastings (MH) algorithm [START_REF] Metropolis | Equation of State Calculations by Fast Computing Machines[END_REF]Hastings 1970). The main idea is based on the fact that a new point in parameter space θ i+1 can be explored by de ning a proposal probability distribution function q( θi+1 , θ i ) that only depends on the previous point θ i . Then, by de ning the following quantity

a = min 1, r ≡ p( θi+1 )q(θ i | θi+1 ) p(θ i )q( θi+1 |θ i ) , (1.61)
if the acceptance ratio parameter r < 1, the step to the value θi+1 is accepted and θ i+1 = θi+1 with probability a; if r > 1, the proposed value θi+1 is rejected and θ i+1 = θ i . Then, the procedure is repeated iteratively by doing i → i + 1 until the set {θ i } i=1,..N converges to the a sample of the target p(θ), in the limit N → ∞. While being a cornerstone of MCMC methods, the MH algorithm presents some limits when dealing with high dimensional parameter space as well as many di erent models (Akeret et al. 2013). Alternative ways within the MCMC framework have hence been proposed, such as the A ne invariant ensemble sampler. First introduced by Goodman and Weare 2010 and then implemented by Foreman-Mackey et al. 2013, this method considers an ensemble of walkers spread across the parameter space of the target distribution to explore the parameter space. At each iteration, the walkers are split into groups and are randomly assigned to a partner subgroup chosen from the walkers ensemble and what is proposed as the next step is a random point on a line that links their positions. Then the walkers of one subgroup are updated using the other half of the walkers as reference. Formally, following Goodman and Weare 2010, the proposed position θi+1 is given by

θi+1 = θ m i + z(θ k i -θ m i ) (1.62)
where the indexes m, k indicate walkers belonging to the di erent subgroups and z is a value sampled from a xed distribution q(z) = 1 √ z for z ∈ [ 1 a , a] and 0 elsewhere, with a being a tuning parameter. Then, the proposed position is accepted if

z n-1 p( θi+1 ) p(θ m i ) ≥ r, (1.63)
where r ∈ [0, 1] is a random number and n is the dimension of the target. The implementation of this method by Foreman-Mackey et al. 2013 led to the python package emcee. One of the advantages of this a ne invariant ensemble sampling of the MCMC is that it can be parallelised since a large number of positions are updated simultaneously at each iteration of the process. In the works presented in this thesis, as some of the simulations used have a vast number of di erent cosmological models, parameter inference is performed with the package emcee to exploit the advantages of parallelisation. A variety of di erent methods to perform MCMC are however present such as Gibbs sampling, Hamiltonian Monte Carlo and others. A detailed review can be found in Sharma 2017; Robert and Changye 2020.

Brief mention of likelihood-free inference and ABC methods

Even if the analysis presented in this thesis are all based on a MCMC approach, it is important to mention that exist alternative ways that during the recent years have drawn the attention of the community to perform inference. The main motivation behind the introduction of these methods is to nd a solution to situations in which the likelihood function is intractable. Indeed, current cosmological analyses are typically based on the assumption that the likelihood is Gaussian. However, it has been shown that non-Gaussianities are present even for underlying Gaussian elds (e.g. the lensing eld) and if neglected by assuming a Gaussian likelihood, they can induce biases on the parameter constraints (Sellentin and Heavens 2017). Moreover, MCMC methods can present some limitations in terms of speed when dealing with very high-dimensional parameter space. In general, to try to overcome the problems arising when the likelihood is unknown and/or too many (in terms of speed) likelihood calls are necessary to obtain well sampled posteriors, the concept of Approximate Bayesian computation (ABC) (Rubin 1984) and density estimation Likelihood Free-Inference (DELFI) [START_REF] Papamakarios | Fast e-free Inference of Simulation Models with Bayesian Conditional Density Estimation[END_REF], have been introduced in cosmological analysis. The main idea behind ABC methods is to draw parameters from some proposal density, then forward simulate data that are compared to the observed data under some distance metric. The proposed parameters are then accepted if this distance is below some threshold value and the accepted samples provide the samples from an approximate posterior, that recovers the exact posterior one only under the limit → 0.

While making it possible to probe the likelihood without a direct evaluation of it, this accept/reject criterion present some disadvantages. As example, as said before these methods recover the exact posterior only in the limit → 0, but they critically slow down in this limit, and what happens in practice is that they draw samples from a distribution that is broader than the posterior [START_REF] Papamakarios | Fast e-free Inference of Simulation Models with Bayesian Conditional Density Estimation[END_REF]. Hence, DELFI methods have been introduced to overcome this limitation. In this framework, inference is treated as a density estimation task and the sampling distribution of the data is learned by neural density estimators (NDEs) as a function of the parameters of the model and by adaptively acquiring simulations with active learning. Very schematically, rst forward simulations are run to obtain a set of parameters θ and the corresponding data d. Then, following some compression procedure (e.g. Alsing and Wandelt 2018), the D dimensional data are compressed to informative summaries d → t, where t has the same dimension of the parameter space. From the simulated mocks, the sampling distribution P(t|θ) is then learned by the NDE and evaluated at the observed data t 0 , providing the likelihood. Some examples of LFI implementations that can be used in cosmological analysis are PyDelfi, BOLFI and others, that have been proven to potentially be a robust alternative for cosmological infer-ence (Je rey, Alsing, and Lanusse 2020; Alsing, Wandelt, and Feeney 2018). Both classes of methods (ABC and LFI) have reached a vast extension in their theoretical development and applications that goes beyond the purposes of this thesis, and a single section won't su ce to provide an extensive overview. However, it is worth to mention these approaches, as even if they are still in a early epoch of their usage (validations and diagnostics of such methods are a very topical object of current research) their potential applications on the enormous amount of data that we will receive from next generation cosmological surveys could bring many advantages.

Summary

In this Chapter, I de ned some background tools that are needed to understand the following Chapters. Starting from the cosmological principle, main cosmological quantities have been derived and our current cosmological model, ΛCDM, has been introduced. Along with its de nition, the unsolved questions concerning the two main components of our universe, dark matter and dark energy have been introduced. Then statistical tools as the two point correlation function and the matter power spectrum have been de ned, and their connection with cosmological probes as the CMB, weak lensing, galaxy clustering.. has been presented. Lastly, the Bayesian inference framework has been outlined, to show how given a model and some observed data, it is possible to infer the parameters of the model through techniques such as MCMC. The next Chapter will zoom in one particular cosmological probe, weak gravitational lensing and will provide more details on how this probe is employed to constraints cosmological parameters.

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37

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This Chapter discusses more in details weak gravitational lensing. The presentation starts with the derivation of lensing equations from light de ection. Then, after introducing the convergence and the shear, the link between observations and cosmological models is illustrated. Second order statistics of cosmic shear are then de ned, followed by a brief overview of systematics e ects in weak lensing and some of the current constraints. Next, I provide an overview of the state of the art of summary statistics in weak lensing.

Lensing formalism

The following summarises the most relevant quantities for the purposes of this thesis for weak gravitational lensing based on [START_REF] Seitz | Light propagation in arbitrary spacetimes and the gravitational lens approximation[END_REF]Bartelmann and Schneider 2001;Bacon, Refregier, and Ellis 2000;Refregier 2003; Kilbinger 2015.

Light de ection and the lens equation

There are di erent ways to derive the equations describing the bending light due to the presence of massive bodies. For example one can derive them directly from the rst principles of General Relativity (GR). In particular, we know that in GR the trajectory of light is de ned by null geodesics. Hence, given any metric, light de ection from gravitational lensing can be derived from the geodesic equations (introduced in Equation 1.27 and illustrated again for convenience)

d 2 x µ dλ 2 + Γ µ αβ dx α dλ dx β dλ = 0, (2.1)
where Γ µ αβ are the Christo el symbols for a given metric and λ the a ne parameter. Now, following Seitz, Schneider, and Ehlers 1994, we can consider two arbitrary light rays, where we set γ µ fid (λ) to be the ducial and γ µ (λ, θ) to be the one separated by the angular coordinates θ = (θ 1 , θ 2 ) from the ducial. Given this setting, the quantity of interest to formalise the phenomenon of gravitational lensing is the physical separation between the two rays ξ(θ, λ) = (x 1 , x 2 ) as it depends on the contents of the Universe they pass through, at any arbitrary point λ > 0. Speci cally, the components of ξ(θ, λ) are the transverse components, perpendicular to the tangential direction x 3 of the reference ray in the physical 3D space, as shown in Figure 2.1. The evolution of the physical separation ξ(θ, λ) is then described through the geodesic deviation equation, that reads:

d 2 ξ(θ, λ) dλ 2 = T (x 1 , x 2 , x 3 )ξ(θ, λ), (2.2)
where T = T bg + T cl is the optical tidal matrix, accounting for the deviation due to the isotropic background Universe encoded in T bg and the contribution to the deviation due to local inhomogeneities, or clumps, described in T cl . By assuming a metric, these two last terms can take an explicit form. We work in the weak eld limit that implies that the potential is small Φ c 2 as well as the angles through which the light's path is deviated. Under this assumption, in the case of a FLRW metric, Seitz, Schneider, and Ehlers 1994 found the explicit expressions for the two terms constituting the optical tidal matrix to be

(T bg ) ij = 4πG c 2 ρ0 a 5 δ ij , (2.3)
(T cl ) ij = - 1 c 2 a 2 2 ∂ 2 ∂x i ∂x j + δ ij ∂ ∂x 3 Φ, (2.4)
with i, j = {1, 2} being the indeces of the transvere components x 1 , x 2 and x 3 being the l.o.s. direction. From now on, we work under the thin-lens approximation, namely we consider that inhomogeneities are geometrically thin and the variation along the x 3 direction vanishes, following Seitz, Schneider, and Ehlers 1994. To obtain the lens equation one can de ne the comoving separation x = (x 1 , x 2 ) ≡ ξ/a, and by explicitly plugging the sum of Equation 2.3 and Equation 2.4 into Equation 2.2 one can obtain the evolution for the comoving separation x in comoving coordinates χ as:

d 2 x dχ 2 + Kx(χ) = - 2 c 2 [∇ ⊥ Φ(x, χ) -∇ ⊥ Φ(0, (χ))] , (2.5)
where

∇ ⊥ = (∂/∂x 1 , ∂/∂x 2 )
is the comoving transverse gradient, χ the comoving coordinate, and the space-time curvature K arise from the use of Friedmann equation before passing to the comoving reference frame. The solution of Equation 2.5 obtained with boundary conditions x(χ = 0) = 0 and (∂x/∂χ)| χ=0 = θ is given by

x(χ) = f K (χ)θ - 2 c 2 χ 0 f K (χ -χ ) [∇ ⊥ Φ(x(χ ), χ ) -∇ ⊥ Φ(0, χ )] dχ . (2.6)
Now that we have the expression for the comoving separation x(χ) we can derive the lens equation by considering that in absence of lensing the comoving separation would be seen by the observer under an angle

β(θ, χ) = x(χ)/f K (χ).
Hence, the di erence between the angular separations of two light rays in unperturbed and perturbed Universe, at comoving distance χ gives us the de ection angle α:

α(θ, χ) = θ -β(θ, χ) = 2 c 2 χ 0 f K (χ -χ ) f K (χ) [∇ ⊥ Φ(x(χ ), χ ) -∇ ⊥ Φ(0, χ )] dχ , (2.7) Figure 2
.1: Illustration of the propagation of two light beams emitted by a galaxy in the source plan through a cluster located in the lens plan. As described in the text θ is the angular separation between the observed apparent positions of the source, β is the angular separation between the true (unlensed) positions of the source, and α is the de ection angle, representing the di erence between the two. χ is the comoving distance between the observer and the source plan and χ is the comoving distance between the observer and the lens plan. Image credits to Guinot 2020.

that using β(θ, χ) = x(χ)/f K (χ) basically corresponds to dividing Equation 2.6 by the comoving angular diameter distance f K (χ). Assuming that that the potential evaluated on the perturbed light path does not di er substantially from the one on the unperturbed line of sight, i.e. working under the Born approximation1 , it is possible to substitute the comoving separation x(χ) in Equation 2.6 with its 0 th order solution x 0 (χ) = f K (χ)θ and Equation 2.6 becomes:

α(θ, χ) = θ-β(θ, χ) = 2 c 2 χ 0 f K (χ -χ ) f K (χ) [∇ ⊥ Φ(f K (χ )θ, χ ) -∇ ⊥ Φ(0, χ )] dχ . (2.8)
The above expression is the lens equation under the Born approximation and it encodes the information on the deviation of the apparent (observed) position of a source at θ with respect to its true position β(θ, χ) due to the gravitational potential Φ between the source and the observer. An illustration of the above description can be found in Figure 2.1.

Shear and convergence elds

Given the lens equation derived in the previous section, it is possible to quantify the e ect of lensing by mapping the lensed space to the unlensed space. This can be done by de ning a linear distortion matrix A that describes the variation of the true position β with respect to the observed position θ trhough the Jacobian A ≡ ∂β/∂θ, whose elements are explicitly given by

A ij (θ, χ) = δ ij - ∂α i ∂θ j = δ ij - 2 c 2 χ 0 f K (χ -χ )f K (χ ) f K (χ) ∂ 2 ∂x i ∂x j Φ[f K (χ )θ, χ ]dχ , (2.9)
where it has been used that ∇ ⊥ = (∂/∂x 1 , ∂/∂x 2 ) and that x 0 (χ) = f K (χ)θ to express the partial derivatives with respect to the comoving transverse components x 1 and x 2 . It is useful now to de ne the lensing potential as

ψ(χ, θ) = 2 c 2 χ 0 f K (χ -χ ) f K (χ)f K (χ ) Φ[f K (χ )θ, χ ]dχ , (2.10)
such that the distortion matrix can be expressed as .11) We can already see from the equation above how the mathematical formalism starts to give us an intuition of the physics behind the distortion: the impact of lensing on the unlensed scenario that corresponds to the identity encoded in the Krönecker delta δ ij is quanti ed by the variation of the lensing potential ψ that physically de nes how much the gravitational potential Φ arising from a mass distribution changes the direction of the light path.

A ij (θ, χ) = δ ij - ∂ 2 ψ(χ, θ) ∂θ i ∂θ j . ( 2
It is possible now to do a step further and de ne some fundamental quantities in weak lensing that make even more intuitive the e ect of lensing. More speci cally, the distortion matrix A can be parametrised as

A = 1 -κ -γ 1 -γ 2 -γ 2 1 -κ + γ 1 = (1 -κ) 1 -g 1 -g 2 -g 2 1 + g 1 , (2.12)
where (γ 1 , γ 2 ) are the components of a spin-2 eld γ = γ 1 + γ 2 = |γ|e 2iφ called shear, and κ is a scalar eld called convergence and the two are related through the reduced shear g i (θ, χ) ≡ γ i (θ, χ)/(1κ). The reduced shear is a complex eld that can as well be expressed as a function of its modulus and its phase as g = g 1 +g 2 = |g|e 2iφ . The shear and the

S I

! " ! # $ " $ # % & '"
convergence ( and shear ) convergence ( Figure 2.2: Illustration of the mapping between unlensed and lensed space thruogh the distortion matrix A. Given an original circular source image S the e ect of weak lensing leading to the observed image I can be summarised in the action of 1) the shear γ whose e ect is to imprint an anisotropic stretching (example shown in blue) where its norm |γ| characterizes the atness of the ellipse and the rotation angle φ indicates the distortion direction; 2) the convergence κ that isotropically magni es or contracts the original shape (example shown in red). This is formalised in the distortion matrix A, being the Jacobian between the original coordinates β and the observed coordinates θ.

convergence describe respectively the anisotropic stretching and the isotropic magni cation of the galaxy shape due to lensing. Equation 2.11 de nes the shear and the convergence elds as second-order derivatives of the lensing potential:

γ 1 ≡ 1 2 (∂ 1 ∂ 1 -∂ 2 ∂ 2 )ψ; γ 2 ≡ ∂ 1 ∂ 2 ψ (2.13) κ ≡ 1 2 (∂ 1 ∂ 1 + ∂ 2 ∂ 2 )ψ = 1 2 ∇ 2 ψ. (2.14)
With the tools introduced so far, it is now possible to understand the physical meaning of the shear and the convergence, the relation between the two, and how they are related with the lensing potential. This can be intuitively understood by looking at Figure 2.2. Following Lin 2016 and looking at Figure 2.2, given a circular source S of radius β, the actual observed distorted image I can be obtained through the local mapping of the source light distribution to image coordinates given by the inverse of the Jacobian distortion matrix A -1 , as

I = A -1 S = β (1 -κ)(1 -|g| 2 ) cos t + |g| cos (2φ -t) sin t + |g| sin (2φ -t) (2.15)
with t ∈ R parametrising the circumference S. Applying now a rotation of -φ, the rotated image is given by: (1-κ+|γ|) . In the absence of shear (γ = 0), the equation of a circumference is recovered, telling us that the scalar eld κ acts as isotropic transformation. If the shear is non zero, its norm |γ| characterizes the atness of the distortion and the rotation angle φ gives the distortion direction.

I rot = cos (-φ) sin (-φ) -sin (-φ) cos (-φ) I = cos (t + φ) β (1-κ-|γ|) sin (t + φ) β (1-κ+|γ|) , ( 2

From observation to cosmology

Now that the formalism to depict the main weak lensing quantities useful for this thesis has been outlined, the next step is to understand the connection between what we actually observe through galaxy surveys with what we can say about cosmology. This can be done by nding the relation between the galaxy observed ellipticities, that is what we actually measure with a survey like Euclid for example, with the shear, that allows to quantify the e ect of lensing. Then, by linking the shear to the convergence, that as will be illustrated in the following section can be related to the matter density eld, it is possible to infer information about the matter distribution between the observer and the source and nally, about the underlying cosmological model.

Estimating shear from galaxy shapes

In the weak regime, the values for the convergence and the shear are of the order of a few percent or less (|κ| 1 and |γ| 1). There are no multiple images caused by lensing and each source is mapped uniquely into one image and the matrix A is invertible. The quantity that is actually measured by galaxy surveys is the observed galaxy ellipticity obs and it's related to the intrinsic ellipticity of the source galaxy s for |g| ≤ 1 through [START_REF] Seitz | Steps towards nonlinear cluster inversion through gravitational distortions. III. Including a redshift distribution of the sources[END_REF]):

obs = s + g 1 + g * s for |g| ≤ 1,
(2.17) where g * is the complex conjugate of the reduced shear g = g 1 + ig 2 . In the weak regime this relation can be approximated to obs ≈ s + γ. The variance of the observed ellipticity is then given by σ 2 ( obs ) = σ 2 ( s ) + σ 2 (γ). The variance of the intrinsic ellipticity of the source is usually dominant and it is referred to as shape noise. In order to isolate the lensing signal and reduce the shape noise, a very large number of galaxies are required. Under the assumption that the intrinsic ellipticity of galaxies has no preferred orientation (that implies that the expectation value of the intrinsic ellipticities vanishes s = 0), the expectation value of the observed ellipticity becomes:

obs ≈ g ≈ γ, (2.18)
meaning that averaging the ellipticity of many galaxies in a circular area and assuming that the shear is constant over this area, one should obtain an unbiased estimate of the shear itself (provided that there are no systematics in the shape measurement). For the moment, we work under the approximation described above, leading to Equation 2.18. Hence, what is directly measurable is the shear g ≈ γ, while the convergence κ is not, and inversion methods are necessary to retrieve κ from γ.

Projected overdensity and convergence

As mentioned at the beginning of this section, we are walking through the steps to understand the path that starting from the observations lead us to the cosmological information.

Having introduced one unbiased estimator of the observed ellipticity through Equation 2.18, the following step is now to provide an interpretation for the other fundamental weak lensing quantity vastly used in weak lensing analysis as well as in this work: the convergence κ. From Equation 2.16 and Figure 2.2 we have seen how the convergence κ is related to an isotropic increase or decrease of the observed size of a source image. Moreover, Equation 2.14 shows how the convergence eld can be obtained as the second derivative of the lensing potential. This last point was mentioned en passant, but it actually contains all we need to connect the convergence to the cosmological information that we want to extract. Indeed, if we look more carefully into it, we see that the equation relating the lensing potential and the convergence is a Poisson equation. We remind here few useful steps to underline why this is important for the aim of this section. By making Equation 2.10 explicit in Equation 2.14 we get: (2.19) and using now the fact that the gravitational potential Φ is related to the matter density contrast δ = ∆ρ/ρ through the Poisson equation ∇ 2 Φ = 4πGa 2 ρm δ, the convergence eld can be rewritten as (2.20) where the mean matter density has been written in terms of the critical density, de ned in Equation 1.23, in comoving space.

κ(θ, χ) ≡ 1 2 ∇ 2 ψ(θ, χ) = 1 c 2 χ 0 f K (χ -χ ) f K (χ)f K (χ ) ∇ 2 Φ[f K (χ )θ, χ ]dχ ,
κ(θ, χ) = 3H 2 0 Ω m 2c 2 χ 0 f K (χ -χ )f K (χ ) f K (χ) δ[f K (χ )θ, χ ] a(χ ) dχ ,
It is more explicit now how the convergence is connected to the matter distribution: Equation 2.20 depicts the convergence as an integration of the density contrast along the line of sight, weighted by comoving transverse distances and the scale factor. Hence, the convergence κ can be interpreted as the projected overdensity, or the projected mass. Now, given a population of source galaxies with source redshift distribution n(χ), the mean convergence from such population can be obtained as

κ(θ) = χ lim 0 n(χ)κ(θ, χ)dχ, (2.21)
where χ lim is the limiting comoving distance of the galaxy sample. Putting everything together:

κ(θ) = 3H 2 0 Ω m 2c 2 χ lim 0 g(χ)f K (χ) δ[f K (χ) θ, χ] a(χ) dχ, (2.22)
where the lensing strength at a distance χ of the combined background galaxy distribution at χ has been summarised in the term

g(χ) = χ lim χ n(χ ) f K (χ -χ) f K (χ ) dχ , (2.23)
known as lens e ciency or lensing kernel. Back to the initial intent of this section, we see how the convergence is a measure of the total matter density projected along the line of sight accounting for lensing and the geometry of the universe via the distance ratios, and the source galaxy distribution, given a cosmological model.

Relation between shear and convergence: mass mapping methods

It has been described so far how the shear is related to what is the actual observed quantity (galaxy shapes) and how the convergence provides rich information about the total matter density projected along the line of sight. The last step consists in considering how the shear and the convergence are related, to complete the link between observation and the information of matter distribution along the line of sight. From Equation 2.13 and Equation 2.14 one W can see how both the shear and the convergence are related through the lensing potential. Considering these expressions in Fourier space, one gets:

γ1 = (k 2 1 -k 2 2 ) 2 ψ; γ2 = k 1 k 2 ψ (2.24) κ = (k 2 1 + k 2 2 ) 2 ψ. (2.25)
Then, one way to obtain the convergence from the shear is to follow Kaiser-Squire (KS) inversion method (Kaiser and Squires 1993) that combines these equations to build a minimum variance estimator of the convergence

κest γ = (k 2 1 -k 2 2 ) k 2 γ1 + 2k 1 k 2 k 2 γ2 or κest γ = k 2 1 + k 2 2 (k 1 + ik 2 ) 2 γ, (2.26) with k = k 2 1 + k 2 2
. This is equivalent to write in direct space:

κ(θ) = 1 π D(θ -θ )γ(θ )dθ + κ 0 , (2.27)
where the convolution kernel D is the Fourier transform of the inverse of the pre-factor in Equation 2.26. This equation tells us that the convergence can be obtained as a convolution of the shear eld, meaning that there exists a linear relation between the shear and the convergence. Nevertheless, Kaiser-Squires inversion is characterised by some complications due to di erent factors. On one side, the shape noise (de ned in Section 2.1.3) is larger than the shear induced by lensing and the random sampling of background galaxies propagates errors through this reconstruction. Therefore, smoothing to reduce the shape noise per pixel and ensure that there are no empty pixels is required. This imprints a loss information at small scales and requires accounting for masks. On the other side, this method su ers from the so called mass-sheet degeneracy: a constant convergence does not induce a shear, and this can be seen by noticing that Equation 2.26 is not de ned for k 1 = k 2 = 0. In real space, this is encoded by the constant term κ 0 in Equation 2.27 and it implies that the global level of the convergence map is undetermined by this method.

Despite some advantageous properties as linearity (in the weak regime), minimum variance and the ability to accommodate reduced shear of this inversion method, the di erent shortcomings mentioned above led to the introduction in the literature of alternative mass mapping techniques. Some examples are approaches aimed to remove the errors caused by the nite size of the eld in weak lensing maps through inpainting [START_REF] Pires | FAst STatistics for weak Lensing (FASTLens): fast method for weak lensing statistics and map making[END_REF] or based on estimating the convergence from the gradient of the convergence ( exion) as proposed in Bacon et al. 2006. Weak-lensing reconstructions of the potential and mass distribution in three dimensions are also possible. In the recent years, powerful methods based on sparsity have also been introduced, such as GLIMPSE (Lanusse et al. 2016) where to preserve all available small-scale information, any binning of the irregularly sampled input shear and exion elds are avoided and the mass mapping problem is treated as a general ill-posed inverse problem, which is regularised using a robust multi-scale wavelet sparsity prior. In Je rey et al. 2020 the rst reconstruction of dark matter maps from weak lensing observational data using deep learning was presented, with a method called DeepMass. Such method estimates the posterior mean of the convergence map from observed weak lensing shear measurements. More speci cally, it estimates the convergence maps through a function F Θ that maps the pixelised shear to the convergence map such that

κ = F Θ (γ),
where what is learned by the CNN are the parameters Θ by minimising a mean-squareerror (MSE) cost function

J(Θ) = ||F Θ (γ)-κ true || 2 2 .
This cost function is evaluated on a set of training data which consists of pairs of realistic shear and truth convergence maps and the truth maps are drawn from a prior distribution P (κ), so that the corresponding noisy shear map is drawn from the likelihood P (γ|κ). This provides a generalised method where the simulations are trained over a broad prior distribution of cosmological parameters and it has proven to be substantially more accurate than existing mass-mapping methods: compared to Wiener ltering with a xed power spectrum, the DeepMass method improved the mean square error (MSE) by 11%.

Another novel approach introducing a new convergence map model based on a mixture of a sparsity-based component to account for the non-Gaussian structure of the eld and a Gaussian random eld to represent the linear characteristics of the eld was presented in Starck et al. 2021 and it is called MCALens. As it uses concepts that need an understanding of sparsity, it will be treated more in detail in Section 3.6. Moreover, very recently, another method, called KaRMMa for performing mass map reconstruction based on a Bayesian approach has been introduced (Fiedorowicz et al. 2021) in the literature. It is based on a forward model of the observed shear eld as a noisy realsation of an underlying noiseless convergence eld, that is assumed to be a non-linear transformation of a Gaussian random eld parametrised by a shift parameter, being the minimum possible value that the map can take in a pixel. This method outperforms the Kaiser-Squires reconstruction by exhibiting a narrower distribution of residuals when compared to some true mass maps from simulations. It also appears to be potentially powerful for aspects such as generating void or peak catalogues, as otherwise other methods as KS will also identify prior-sourced structures in peak/void nding algorithms. However, its current implementation does not allow for cosmological inference analysis as a speci c cosmology is required to be able to specify the prior; also, it can only perform reconstruction for one single tomographic bin, hence making a joint reconstruction across multiple tomographic bins impossible. Lastly, it is still restricted to relatively low resolutions, therefore, an improved version of it would be needed at this stage in the interests of performing a cosmological inference analysis.

W

Cosmic shear

Now that the building blocks to understand the main weak lensing quantities have been described, the next milestone is to understand how to extract cosmological information from weak lensing observables. At this point, we know that the shear is strictly related with what is actually measured, i.e. the galaxy shape, that the convergence can be interpreted as the 2D mass projection along the line of sight providing information about the matter distribution, and that the shear and the convergence are related through the lensing potential and can be obtained with inversion methods such as the ones mentioned in the previous section.

The shear two-point correlation function

In particular, through Equation 2.22, we have seen how the convergence is related to the density contrast and via Equation 2.26 the link between shear and convergence. Since the average density contrast δ is zero, following Equation 2.22 the expected value of convergence is also zero. In order to retrieve cosmological information, the rst non trivial cosmic shear observable is the real-space shear two-point correlation function (2PCF). It can be built by decomposing the shear in a tangential component γ t and in a cross component γ × :

γ t = -(γe -2iφ ); γ × = -(γe -2iφ ) (2.28)
where φ is the polar angle of the direction θ. Then three two-point correlators can be built from these two shear components: γ t γ t , γ × γ × , γ t γ × leading to the three components of the 2PCF ξ(θ):

ξ ± (θ) ≡ γ t γ t (θ) ± γ × γ × (θ); ξ × (θ) ≡ γ t γ × (θ). (2.29)
These correlation functions should be invariant under parity transformation, namely:

γ t → γ t , γ × → -γ × .
Then ξ × vanishes. Reminding now that in the weak regime Equation 2.18 holds, i.e. the average of the observed ellipticity provides an unbiased estimate of the shear (assuming that the shear is constant over the averaged area and provided there are no systematics in shape measurement), an estimator of the 2PCF is given by [START_REF] Schneider | Analysis of two-point statistics of cosmic shear -I. Estimators and covariances[END_REF]:

ξ± (θ) = ij w i w j [ t (θ i ) t (θ j ) ± × (θ i ) × (θ j )] ij w i w j , (2.30)
where t and × are respectively the tangential and cross component of the observed ellip-ticity obs (the apex obs has been omitted) and each galaxy has a measured ellipticity obs and an attributed weight w i to account for the measurement uncertainty. The sum extends over pairs of galaxies (i, j) at corresponding positions on the sky ϑ i and ϑ j whose separation |ϑ iϑ j | lies in an angular distance bin around θ. More details on this formalism and notation can be found in [START_REF] Schneider | Analysis of two-point statistics of cosmic shear -I. Estimators and covariances[END_REF].

The convergence power spectrum

Analogously to the previous section, it is possible to de ne a 2PCF for the convergence eld. In particular, working in Fourier space, the two-point correlation function de nes the convergence power spectrum C κ ( ) through:

κ( )κ * ( ) = (2π) 2 δ D ( -)C κ ( ), (2.31)
where δ D is the Dirac delta, and following Schneider 2006, κ is the Fourier transform of the convergence eld κ:

κ( ) = d 2 θκ(θ)e i •θ . (2.32)
Then, using the expression found in Equation 2.22 for the convergence in Equation 2.32 to compute Equation 2.31 and working under the following assumptions:

• Limber approximation: correlations along the line of sight are neglected, namely only modes that lie in the plane of the sky are considered [START_REF] Limber | The Analysis of Counts of the Extragalactic Nebulae in Terms of a Fluctuating Density Field[END_REF]Kaiser 1992;[START_REF] Simon | How accurate is Limber's equation?[END_REF] • Small-angle approximation: approximating to rst order the expansion og trigonometric functions of the angle

• Flat-sky limit: replacing the expansion in spherical harmonics by an expansion in Fourier modes (Lemos, Challinor, and Efstathiou 2017), it is possible to get an explicit expression for the convergence power spectrum as (2.33) where P δ is the matter power spectrum, arising from the the density contrast in the right hand side of Equation 2.22 and g(χ) is the lensing kernel introduced in Equation 2.23. Reminding also the relation that holds between the shear and the convergence through Equation 2.26, this implies that |κ| = |γ|. Hence, the shear power spectrum de ned through

C κ ( ) = 3H 2 0 Ω m 2c 2 2 χ lim 0 g 2 (χ) a 2 (χ) P δ k = f K (χ) , χ dχ,
W γ( )γ * ( ) = (2π) 2 δ D ( -)C γ (
), is actually equivalent to the convergence power spectrum, namely C κ ( ) = C γ ( ). Thus, while it is possible to obtain the convergence from shear measurements through inversion methods as described in Section 2.1.3, we can also work directly with the 2-point statistics of shear and still retrieve the cosmological information contained in Equation 2.33 and viceversa.

Systematics

There is a range of systematics e ects that can mimic a shear signal, namely, multiplicative shear bias, mean redshift bias, baryonic feedback, intrinsic alignment, and boost factor. This section provides a brief overview of these systematics e ects, by de ning them and by presenting some of the issues that characterise them and some of the current approaches to try to alleviate them.

Baryonic e ects

As mentioned in Section 1.3.2 and as will be outlined more in details in Section 2.2.5, predicting the matter distribution in the non-linear regime with high accuracy still represents a challenge. For instance, for Fourier modes as large as k ∼ 10 h Mpc -1 baryonic physics can di er from the dynamics of dark matter. In particular, it has been shown that to understand the fact that the star formation rate is higher than expected, the feedback from active galactic nuclei (AGN), i.e. the re-deposition of energy and momentum into the interstellar medium of a galaxy through the out ows and radiation, needs to be taken into account during galaxy formation (Dubois et al. 2013). Hence, as baryonic physics modi es the clustering property of matter, it impacts the non-linear matter power spectrum and, as a direct consequence, it also in uences the lensing signal. If the baryonic physics is not properly accounted for, biases are induced on cosmological constraints. In particular, Daalen et al.

2011 have shown that this impacts the matter power spectrum to a level which is higher than the accuracy required for current and future weak lensing surveys, that require an accuracy of a few per cent for a large range of scales.

This becomes more important at more non-linear scales; however, already at the level of second order statistics, it has been shown that ignoring the e ects of baryonic physics can induce a bias as large as ∼ 40% on the dark energy equation of state for the AGN scenario [START_REF] Semboloni | Quantifying the e ect of baryon physics on weak lensing tomography[END_REF]). The convergence power spectrum is signi cantly altered specically at angular scales corresponding to ≤ 1000 (or to ≤ 3000, depending on the assumed statistical uncertainty (Kilbinger 2015)). With this awareness, a lot of e ort has been put in the attempt of modelling such complex physics in the ensemble of baryonic processes as radiative cooling, star formation, supernovae, AGN and their associated feedback on the matter density eld through hydro-dynamical simulations such as the Horizon-AGN and it allows to perform many fast realisations of the non-linear cosmic density eld with varying baryonic parameters. As already mentioned above, being able to account for baryonic e ects becomes more and more important when willing to probe non linear scales. In the next section, dedicated to higher order statistics, some recent studies and results with baryonic e ects in non-Gaussian statistics will be discussed. In our rst application on real data using CFIS that will be presented in Chapter 7, we consider this e ect, alone and in combination with the multiplicative shear bias.

Intrinsic alignment

The interpretation of the lensing signal and convergence maps as introduced in this work so far is based on the hypothesis that the intrinsic galaxy orientation should not have any privileged direction, i.e. on the assumption made in Equation 2.18. In reality, galaxies are continuously exposed to the gravitational interaction with the surrounding matter distribution, during their formation and evolution, and this leads to a coherent alignment of their intrinsic shapes, induced by the underlying tidal eld on physically near galaxies and by a complex relation with their host halos. This phenomenon is called intrinsic alignment (IA) (Heavens, Refregier, and Heymans 2000). Extensive reviews on this topic are present in Kirk et al. 2015;Kiessling et al. 2015;Joachimi et al. 2015.

At the level of second order statistics, this e ect can be included by forgetting the assumption that s = 0 and by considering the 2PCF for the observed galaxy ellipticities obs (indicating with i and j a pair of galaxies and the average is over all pairs):

obs i obs j = γ i γ j + s i s j + γ i s j + s i γ j , (2.34)
where the γ i γ j can be be referred to as "GG", describing the cosmic shear correlation (the only contribution that survives, when working under the assumption that there is no preferred orientation for intrinsic galaxy shapes); the s i s j term that can be referred to as "II", encodes the correlations between intrinsic ellipticities; and the last two terms γ i s j , s i γ j can be summarised in the notation "GI", and describe the correlations between the shear for one galaxy and the intrinsic ellipticity of the other. Neglecting the IA e ects can intro-W duce important biases on weak lensing signal and consequently on parameters estimation. Indeed, in order to properly model this e ect in weak lensing analysis, a wide number of models has been proposed in the literature, such as the Linear Alignment (LA) model (Catelan, Kamionkowski, and Blandford 2001), based on the tidal alignment of galaxies taken at linear order or the Non-Linear Alignment (NLA), an empirical modi cation of the LA based on the consideration that it better ts the data, as motivated in Hirata and Seljak 2004;[START_REF] Bridle | Dark energy constraints from cosmic shear power spectra: impact of intrinsic alignments on photometric redshift requirements[END_REF]King 2007. Kirk et al. 2012 noted that, if IA is imprinted in the primordial phases of galaxy formation, perturbations are still in the linear regime: they then claim that the linear power spectrum should be used for the "II" correlation. Along with this, shear depends on the non-linear power spectrum which must be taken into account in the correlation with IA. Therefore, they proposed to use a Freeze -in model (FM) where the overall e ect of IA is suppressed at low redshifts and large k because of the use of linear rather than non-linear matter power spectrum.

Many other models that try to account for what is neglected in the LA and NLA models (such as astrophysical e ects beyond the lowest order) can be mentioned, such as the Full tidal alignment (FTA), obtained by carrying on the computation of the intrinsic-matter power spectrum to the next leading orders (Blazek, Vlah, and Seljak 2015) or the Tidal alignment and torquing (TAT) including also terms originating from the tidal torquing expected in rotation supported systems. In Fortuna et al. 2020, a uni ed framework to account for di erent IA signatures at large and small scales that also incorporates luminosity and radial dependence has been presented. In this context, the total IA power spectra are given by the sum of the behaviour at large scales (where the majority of the IA signal is due to red central galaxies) and the behaviour at small scales (dominated by the satellite alignment signal). The fact that there is such a vast number of di erent models is due to the fact that some of these models are not able to represent accurately the real physical e ects due to the large number approximations (LA) or they are too hard to compute (as for example FTA and TAT) or not completely understood (FM), or with too many parameters (full NLA).

Given the di culties to reproduce the physical properties of galaxies coherently with the observations employing DM-only simulations to predict the intrinsic alignment signal, the use of hydro-dynamical simulations has been proven powerful to better predict the complex relation between halo shape and spin and galaxy shape and spin [START_REF] Codis | Intrinsic alignment of simulated galaxies in the cosmic web: implications for weak lensing surveys[END_REF]Bate et al. 2019). An alternative way to include IA in weak lensing analysis, consists in treating it as an additive component to the cosmological signal, directly obtained from the particle shells of the simulations in a similar fashion as the cosmological convergence signal κ itself, as performed in Zürcher et al. 2021 with the Ufalcon pipeline [START_REF] Sgier | Fast generation of covariance matrices for weak lensing[END_REF]).

Multiplicative shear bias

The multiplicative shear bias is a systematics arising at the level of the shear measurement that can be caused by various e ects as noise bias, a systematics error whose origin is identi ed in high-order noise terms in the measurement of the shape parameters of galaxies, due to the non-linear relation between image pixels and galaxy shape (Refregier et al. 2012); model bias in the context of model tting methods, if the model is not able to represent realistic galaxy morphologies well enough, then the shape estimator will be biased (Kacprzak et al. 2014); or imperfect PSF corrections (Paulin-Henriksson, S., Refregier, A., and Amara, A. 2009). This systematics can change the overall amplitude of the cosmic shear auto-correlation and its cross-correlation with other probes of the large scale structure and consequently cause biased estimations of cosmological parameters [START_REF] Liu | Constraining multiplicative bias in CFHTLenS weak lensing shear data[END_REF]. The simplest way to deal with shear estimation biases consists in estimating the shear errors by processing simulated data with known input shear through the same pipeline as the real data. These calibration methods require simulations that need to be carefully matched to the properties of the given data in order to have correct calibration factors. An alternative way to calibrate the shear measurement is the metacalibration (Hu and Mandelbaum 2017). This method does not rely on simulations, and allows to avoid the challenge of creating a large number of time-consuming image simulations and it has proven to be very e ective both when applied to the DES Year 1 (Y1) catalogue (Zuntz et al. 2018) and to the Canda-France Imaging Survey (CFIS) r-band data (Guinot et al. in prep.). In our rst application on real data using CFIS that will be presented in Chapter 7, we consider this e ect, alone and in combination with baryonic e ects.

Mean redshift bias

As one can see from Equation 2.21 and Equation 2.23, it is required to have a knowledge of the the source galaxy distribution n(z), in order to be able to interpret weak lensing observables from a cosmological point of view. In particular, for a given desired accuracy to estimate cosmological parameters, the mean redshift z has to be determined at least to that accuracy. When performing a tomographic analysis, this applies to each individual redshift bin, as the knowledge of the centers of each bin needs to be better than a per cent to avoid a decrease in the accuracy of dark-energy parameters to 50% as was shown by Huterer et al. 2006. The estimation of galaxy redshifts through photometry ( brie y mentioned in Section 1.4.1), or photometric redshifts (photo-zs) is hence needed. The requirements of the accuracy of photometric redshifts depend on the survey. For example, for KiDS, it was found that even a Gaussian 1σ uncertainty on the measured mean redshift of each tomographic bin can degrade the statistical errors on relevant cosmological parameters by ∼ 25% (Hildebrandt et al. 2016). Since the galaxy samples photometric redshift distributions typically present highly non-Gaussian tails, analytic estimates based on Gaussian redshift errors don't represent the optimal way to obtain accurate calibrations. In Hildebrandt et al. 2016, di erent methods, based on weighted direct calibration obtained by a magnitude-space re-weighting or on angular cross-correlation based calibration, that rely on spectroscopic redshift (spec-z) training samples are compared. Their favoured technique relies on a weighted direct calibration obtained by a magnitude-space re-weighting of spectroscopic redshift catalogues that overlap with KiDS (Lima et al. 2008). They also conclude that for future cosmic shear surveys bringing considerably large data sets, it will be essential to reduce the statistical uncertainty in the redshift calibration. For Euclid, Euclid Collaboration et al. 2020a presented a report on the performance of 13 photometric redshift codes for single value redshift estimates and redshift probability distributions on a common set of data, focusing particularly on the redshift range that the Euclid will probe. Their nding is that while all methods are able to provide reliable single value estimates, several machine-learning methods do not manage to produce useful redshift probability distributions. Moreover, they nd that no machine-learning method provides good results in the regions of galaxy color-space that are sparsely populated by spectroscopic-redshifts, while generally performing better than template-tting [START_REF] Bolzonella | Photometric redshifts based on standard SED tting procedures[END_REF] methods at low redshift, indicating that template-tting methods do not use all of the information contained in the photometry. They conclude that further work in identifying how to best select between machinelearning and template-tting approaches for each individual galaxy should be pursued as a priority for future galaxy surveys like Euclid.

Boost factor

Another consequence of the uncertainty of photometric redshift estimates is the contamination of the source galaxy catalogue with galaxies associated with the foreground cluster, namely counting galaxies residing at the lens redshift in the n(z) of the sources. As this contamination dilutes the measurement, in order to recover the true signal, the raw signal has to be boosted. Hence, this e ect is referred to as the boost factor. This systematics is mostly relevant for higher order statistics such as peak counts as the strength of the shear signal varies with the distance of source galaxies to the lens and this causes the detection probability of a peak to depend on the distribution of the redshifts of source galaxies at the position of a peak (Kacprzak et al. 2016). In the context of peak counts, the boost factor can be estimated in di erent ways. For instance, Kacprzak et al. 2016 corrected the peak function found in their cosmological training set as a function of S/N by evaluating the fractional over-crowding and over-blending rates in peaks of di erent S/N from a separate catalogue that matches the DES-SV n(z) and by restricting their measurement to S/N < 4 where they found that the impact of this e ect is minimal (a shift in S 8 of about 0.01) and can then be neglected. For KiDS-450, Shan et al. 2017 used a cluster catalogue that overlaps with the KiDS-450 survey, and evaluated the boost factor from the excess source density around these massive objects. Their nding is that the contamination to the peak function reaches 27% for peaks with S/N = 5, being however less than 6% for S/N < 4, consistently with Shan et al. 2017.

Constraints with cosmic shear second order statistics

The detection of cosmic shear and its interpretation as direct measure of the mass power spectrum (Bacon, Refregier, and Ellis 2000;Kaiser, Wilson, and Luppino 2000) and the consequent proof of the technical feasibility of using weak lensing surveys to measure dark matter clustering to constrain cosmological parameters (Waerbeke et al. 2000), led to the use of cosmic shear as a primary probe for past, current and future galaxy surveys. The most popular approach adopted by these surveys to perform cosmological parameter inference relies on second order statistics, in the form of correlation functions (or its Fourier equivalent, the power spectrum), or estimated using pseudo-C , band powers or quadratic estimators. For example, Kilbinger, Fu, Heymans, et al. 2013 [START_REF] Spergel | Wide-Field InfrarRed Survey Telescope-Astrophysics Focused Telescope Assets WFIRST-AFTA 2015 Report[END_REF] will also use cosmic shear as a primary probe and will give access to small scales than ever before. Modelling challenges in cosmic shear will become hence more and more important on non-linear scales. This last point is further discussed in the next section, with a focus on the Euclid mission.

Non-Linear regime

This paragraph aims to provide a concise description of the impact on the modelling of cosmic shear of the scales at the which matter density eld perturbations become larger and linear theory fails to predict the evolution of large-scale structures, mostly based on Martinelli et al. 2020 andEuclid Collaboration et al. 2020b. A wide contribution to the constraining power of the cosmic shear signal lies in non-linear scales, speci cally in correspondence of scales up to k ≈ 7 h Mpc -1 (Taylor, Kitching, and McEwen 2019). This requires the capability of modelling the matter power spectrum at high-k. Indeed, if nonlinear scales not properly modelled of the shear signal are included in the analysis, this can lead to biases or underestimation of the contours size. For instance, Copeland, Taylor, and Hall 2018 have shown that the impact of baryonic e ects on the non-linear power spectrum can lead to a decrease of the FoM by up to 40%, depending on the model and the prior. Hence, it is very important to model the behaviour at high-k and be aware of the choices adopted when modelling the non-linear power spectrum.

A straightforward way to avoid the problematic of dealing with non-linear scales that might be di cult to model and interpret is to apply a cut in -modes by setting a maximum multipole value max . This method is the most commonly employed in weak lensing analysis as it yields unbiased results. For example, Euclid Collaboration et al. 2020b identi ed cuts in -modes corresponding to an optimistic and a pessimistic settings in the context of Fisher forecasts. Speci cally, for the pessimistic setting the cut is set at max = 1500. This choice is motivated by the fact that the information content of the shear power spectrum, de ned as the total signal-to-noise ratio, decreases by 30% at max = 5000 when in the covariance matrix is considered also the non-Gaussian contribution (Takada and Hu 2013).

The optimistic setting sets max = 5000, to quantify the gain that could be obtained extending to the non-linear regime, if one were able to include such high multipoles. As also discussed in Euclid Collaboration et al. 2020b, this approach is practical, but has limited applicability and can lead to imprecise parameter constraints, especially at low redshifts. Indeed, parameter sensitivities are generally scale dependent [START_REF] Copeland | English[END_REF] and respond di erently to di erent cuts and to non-Gaussian correlations in the covariance. In the analysis presented in this thesis, we follow the optimistic setting we of Euclid Collaboration et al. 2020b when employing the MassiveNus simulations, while using a value closer to the pessimistic setting of Euclid Collaboration et al. 2020b (we use max ∼ 2000) when using the cosmo-SLICS simulations. To overcome these issues, methods to optimally remove sensitivity to poorly modelled scales based on the Bernardeau-Nishimichi-Taruya (BNT) transform (Bernardeau, Nishimichi, and Taruya 2014), where the observed tomographic angular power spectrum can be reweighted in such a way that each redshift bin retains only the information about lenses within a small redshift range, have been proposed. In particular, employing these methods can reduce the error on S 8 by 32% relative to a correlation function analysis with the same priors but with angular scale cuts and can alleviate baryons modelling uncertainties, as shown when working in con guration space by Taylor, Bernardeau, and Hu 2021. At the same time, Deshpande, Taylor, and Kitching 2020 showed how the optimistic scenario of max = 5000 mentioned for Euclid Collaboration et al. 2020b is more achievable when the k-cut cosmic shear technique, that consists in applying a redshift-dependent -cut after making the BNT transform, is applied.

Besides the complexity of modelling the power spectrum at non-linear scales, the fact that the uctuations of the cosmic density eld are Gaussian in the early Universe, but then grow into a complex structure containing walls, laments, nodes, and voids, makes second order statistics like the power spectrum blind to non-Gaussian features of the large-scale structure eld due to non-linear evolution at late times.

Figure 2.3 shows this with the comparison between a convergence map of size 12.25 deg 2 from the MassiveNus suite of simulations (left top panel) with a fake convergence map of the same size (right top panel) built such that its power spectrum matches the power spectrum of the convergence map. By looking at both maps, it is possible to notice how many structures (such as the big blog at the right top of the left map, or the structures located at its center) are not present in the map of the right and yet, the two maps are described at second order by the same power spectrum. Indeed, by de nition, second order statistics can potentially capture all cosmological information encoded in linear, Gaussian density elds. However, as shown in Figure 2.3, a large number of high value convergence are seen in the rst convergence map, but are not encoded in the information brought by the power spectrum indicating rich non-Gaussian information and the need to go beyond second order statistics. The next section is dedicated to the state of the art of statistics of order higher than the second, introduced to account for non-Gaussianities.

State of the art of higher-order statistics

In this section, I will present an overview of the state of the art of current higher-order statistics with the corresponding de nitions and main recent results on forecasts and real data constraints. As mentioned at the end of the previous section, in the context of weak lensing, second-order statistics as the two-point correlation function or its Fourier transform (the power spectrum) do not capture the non-Gaussian information encoded in the non-linear features of weak lensing data (Weinberg et al. 2013). This has motivated the introduction of several higher-order statistics, such as Minkowski functionals (Kratochvil et al. 2012;[START_REF] Petri | Emulating the CFHTLenS weak lensing data: Cosmological constraints from moments and Minkowski functionals[END_REF]Vicinanza et al. 2019;Marques et al. 2019;Parroni et al. 2020), higher-order moments [START_REF] Petri | Cosmology with photometric weak lensing surveys: Constraints with redshift tomography of convergence peaks and moments[END_REF]Vicinanza, Cardone, Maoli, et al. 2018;Peel, Pettorino, Giocoli, et al. 2018;[START_REF]Dark Energy Survey Year 1 results: curved-sky weak lensing mass map[END_REF]Gatti et al. 2020), the bispectrum (Takada and Jain 2004; [START_REF] Chan | Assessment of the information content of the power spectrum and bispectrum[END_REF]Coulton et al. 2019), peak counts (Kruse and Schneider 1999;Kratochvil, Haiman, and May 2010;Dietrich and Hartlap 2010;Maturi, Fedeli, and Moscardini 2011;[START_REF] Pires | Cosmological constraints from the capture of non-Gaussianity in weak lensing data[END_REF]Hamana et al. 2012;Hilbert Figure 2.3: Comparison between a convergence map from the MassiveNus suite of simulations (left top panel) with a fake convergence map (right top panel) built such that its power spectrum matches the power spectrum of the convergence map. The bottom panel shows the power spectrum measured on the convergence map (continuous line) with respect to the power spectrum computed on fake convergence map (dashed line). The message of this illustration is to show how a large number of high value peaks (as the big blob at the top right of κ map 1 detectable also by naked eye or the several structures at the center of the same map) are not accounted for when extracting information with the power spectrum, indicating the presence of information beyond Gaussian statistics that is not present in the information brought by the power spectrum. et al. 2012;Marian et al. 2012;Marian et al. 2013;Martinet et al. 2015;Lin and Kilbinger 2015;Giocoli, Moscardini, Baldi, et al. 2018;Martinet et al. 2018;Peel, Pettorino, Giocoli, et al. 2018;Li, Liu, Zorrilla Matilla, et al. 2019), the scattering transform [START_REF] Cheng | A new approach to observational cosmology using the scattering transform[END_REF], wavelet phase harmonic statistics (Allys et al. 2020), and machine learning-based methods (Fluri et al. 2018;Peel et al. 2019;Gupta, Matilla, Hsu, et al. 2018;Ribli et al. 2019;[START_REF] Shirasaki | Decoding Cosmological Information in Weak-Lensing Mass Maps with Generative Adversarial Networks[END_REF], to account for non-Gaussian information in cosmological analysis. Focusing on peak counts, it has been shown that this statistics is particularly powerful in breaking degeneracy between the standard model and fth forces in the dark sector (Peel, Pettorino, Giocoli, et al. 2018) as well as in constraining cosmological parameters when employed in a multi-scale setting [START_REF] Liu | Cosmology constraints from the weak lensing peak counts and the power spectrum in CFHTLenS data[END_REF][START_REF] Lin | A new model to predict weaklensing peak counts. III. Filtering technique comparisons[END_REF]Fluri, Kacprzak, Sgier, et al. 2018;Ajani et al. 2020;Zürcher et al. 2021). In particular, as it will be discussed in Chapter 4, in Ajani et al. 2020 we have advanced the state of the art by showing that multi-scale peak counts signi cantly outperform the weak lensing power spectrum, improving the constraints on the sum of neutrino masses m ν ≡ M ν by 63% when using a starlet lter. In the same study, we have also found that multi-scale peak counts are so constraining that the addition of the power spectrum does not further improve constraints. A very interesting feature that we have found for multi-scale peaks, when they are obtained using the starlet transform [START_REF] Starck | The Undecimated Wavelet Decomposition and its Reconstruction[END_REF], is the behaviour of the covariance matrix that tends to encode all information in its diagonal elements. Another weak lensing probe of large-scale structure is represented by cosmic voids, namely under-dense regions of the large-scale matter eld (Colberg, Pearce, Foster, et al. 2008;[START_REF] Pisani | Cosmic voids: a novel probe to shed light on our Universe[END_REF]. Local minima of weak lensing convergence maps, namely pixels with values smaller than their eight neighbouring pixels, have been proposed as tracers of the matter distribution voids to infer cosmological parameters, both in a mono-scale setting (Coulton et al. 2020b;Martinet, Nicolas et al. 2021;Davies, Cautun, Giblin, et al. 2020) and in a multi-scale setting (Zürcher et al. 2021). More speci cally, Coulton et al. 2020b found that lensing minima alone are slightly less constraining than the peaks alone and, in agreement with Martinet, Nicolas et al. 2021 andZürcher et al. 2021, that the combination of the two statistics produces signi cantly tighter constraints than the power spectrum.

One-point probability distribution function (PDF)

A summary statistics alternative to the power spectrum is the one-point probability distribution function (PDF), which is obtained by taking the histogram of projected weak lensing convergence in pixel values (Thiele, Hill, and Smith 2020). Using the one-point PDF, Liu and Madhavacheril 2019 found that the non-linear growth generates non-Gaussianity in the PDF, demonstrating additional information encoded in the tomographic PDF beyond the power spectrum and that the PDF alone outperforms the power spectrum in constraining cosmology as previously found by Patton et al. 2017. Recently, Boyle et al. 2020 provided a theoretical PDF model to perform forecasts and eliminate the need for relying on expensive N-body simulations, allowing for performing forecasts with the convergence PDF on the full set of ΛCDM parameters and established with Fisher forecasts that the constraining power of the convergence PDF compares favourably to the two-point correlation function for a Euclid-like survey area at a single source redshift.

Bispectrum

Another way to go beyond the second-order convergence power spectrum and include the non-Gaussian information encoded at non-linear scales consists in considering the nextleading order statistic de ned as the bispectrum B κ . Analogously to how the 2PCF is related to the power spectrum, by measuring the shear three-point correlation function (3PCF) (Takada and Jain 2003), one can relate the 3PCF to the bispectrum, and then to the matter distribution which depends on cosmology [START_REF] Schneider | The three-point correlation function of cosmic shear -I. The natural components[END_REF]. The bispectrum measures three-point correlations of the convergence de ned on a closed triangle in Fourier space:

κ( 1 )κ( 2 )κ( 3 ) = (2π) 2 δ D ( 1 + 2 + 3 ) [B κ ( 1 , 2 ) + B κ ( 2 , 3 ) + B κ ( 1 , 3 )] . (2.35)
It is not trivial to de ne the relation between the 3PCF and the bispectrum and to e ciently evaluate it numerically. One di culty of directly using the Fourier bispectrum is its large number of terms, which generally must be reduced in some way. Recent works that used the bispectrum to perform parameter inference relied on bispectrum estimators as the binned bispectrum estimator based on Bucher, Racine, and Tent 2016 and adapted to the at-sky regime. Moreover, under the assumption of homogeneity and isotropy, the bispectrum can be written as a function that only depends on the magnitude of the modes and the binned bispectrum estimator can be implemented as a simple sum over products of lter maps. Using this assumptions, Coulton et al. 2019 found that the bispectrum and power-spectrum measurements are highly complementary and analysing them jointly produces 30% tighter parameter contours when using the MassiveNus simulations in an ideal setting, namely, neglecting systematic e ects, baryonic e ects, intrinsic alignment, multiplicative bias.

Minkowski functionals

Another powerful tool to account for deviations from Gaussianity is represented by Minkowski functionals (MFs), rst introduced by Mecke, Buchert, and Wagner 1994 for galaxy clusters applications and then widely used in astrophysics and cosmology [START_REF] Saar | Multiscale morphology of the galaxy distribution[END_REF]Hikage, Komatsu, and Matsubara 2006;Ducout et al. 2012). Minkowski functionals are a complete family of morphological measures invariant under rotations and translations. Their usage allows one to morphologically characterise a given random eld in a D dimensional space by a set of D + 1 functionals V j , with j = 0.., D. For a 2D eld as for example when working with a weak lensing convergence map κ, setting a threshold ν and de ning σ 0 as the standard deviation of the convergence eld, the three Minkowski functionals V 0 , V 1 , V 2 describe respectively the area, the perimeter and the genus of the excursion set Σ(ν):

V 0 (ν) = 1 A Σ(ν) da, (2.36) V 1 (ν) = 1 4A ∂Σ(ν) dl, (2.37) V 2 (ν) = 1 2πA ∂Σ(ν)
Kdl, (2.38) where A is the total area of the map, Σ(ν) is the set of pixels with κ such that κ > κ 0 = νσ 0 and ∂Σ(ν) is the boundary of the excursion set. Given these de nitions, the area V 0 is equivalent to the cumulative distribution function, the perimeter V 1 describes the length of its boundary and the genus V 2 is a measure of the total number of connected regions above κ 0 minus number of connected regions below κ 0 , measured by the integrated geodesic curvature K along the boundary. Employing MFs for weak lensing cosmological constraints, Kratochvil et al. 2012 found that a substantial amount of information from non-Gaussian features is encoded by MFs of convergence maps and that this statistic is particularly well suited to break degeneracies and to constrain the dark energy equation of state parameter w. They identi ed that the non-Gaussian information is brought in part by the one-point function of the convergence through V 0 , and in part through combining di erent smoothing scales for V 0 , and through V 1 and V 2 . Vicinanza et al. 2019 presented a method to match the theoretical predictions with measured MFs taking into account shape noise, imperfections in the map reconstruction and inaccurate description of the non-linearities in the matter power spectrum and bispectrum and validated their method against simulated maps reconstructed from shear elds generated by the MICE simulations. With a Fisher matrix analysis they forecasted the accuracy on cosmological parameters from a joint MFs and shear tomography analysis and found that MFs are helpful to break the Ω m -σ 8 degeneracy but leading to an overall increase of the Functionals for a stage-3-like weak lensing survey, and found that Minkowski functionals increase the constraining power by a factor of about 2 with respect to the power spectrum, and can help add some robustness against galaxy intrinsic alignment in a non-tomographic setting. [START_REF] Petri | Emulating the CFHTLenS weak lensing data: Cosmological constraints from moments and Minkowski functionals[END_REF] applied Minkowski functionals to perform parameter inference using CFHTLenS data, nding that Minkowski functionals can constrain the non-degenerate direction in parameter space, Σ 8 , where the e ects of degeneracy are mitigated. However, W they also nd that constraints on the (Ω m , σ 8 ) plane su er from a strong bias that might be due to uncorrected systematics in the CFHTLenS data when using this statistics.

Higher-order moments

The non-Gaussianity of the convergence eld can also be quanti ed through higher than second order moments. Following Vicinanza et al. 2016, the moment of order n of the convergence κ can be de ned as

κ n = 1 N pix κn i , (2.39)
where i runs over the number of pixels in the map N pix and κn i is the value of the convergence in the i-th pixel. Focusing on moments up to 4th order, one can construct the skewness S 3 and the kurtosis S 4 of the convergence eld de ned as

S 3 = κ 3 /( κ 2 ) 3/2 , S 4 = κ 4 /( κ 2 ) 2 , (2.40)
that quantify deviations from a zero centred Gaussian distribution. In particular, using aperture mass computed on weak lensing convergence maps, Peel, Pettorino, Giocoli, et al. 2018 found that these observables can be powerful to discriminate between f(R) models and ΛCDM. Howerver, they also found that peak counts computed in aperture mass maps outperform third and fourth-order moments for this purpose. Using CFHTLenS data [START_REF] Petri | Emulating the CFHTLenS weak lensing data: Cosmological constraints from moments and Minkowski functionals[END_REF], obtained constraints on the parameter set (Ω m , w, σ 8 ) using a set of moments up to 4 th order and nd that high-order moments can break the (Ω m , σ 8 ) degeneracy and provide tight constraints with no apparent bias. Nevertheless, high-order moments are very sensitive to outliers and may hence su er high empirical variance meaning that if the distribution of eld intensity has a long tail, the amount of information accessible to n-point functions quickly decrease.

Weak lensing peaks

A high order statistics that has become very popular and more and more adopted in the recent years in weak lensing analysis is peak-count statistics. It can be obtained by counting the local maxima of shear or convergence maps speci cally by selecting all pixels that have a higher value than their eight neighbour pixels in 2D. Such local maxima correspond to high signal regions, that are closely associated with massive structures along the line-ofsight. This subsection is dedicated to peak counts in a more detailed way, as peaks are one of the main protagonists of this thesis work, as will be described in Section 4.1.

Models for predicting peak counts

Providing an explicit expression for the peak function, namely a peak number density as a function of the pixels, is a research issue that gave rise to a rich literature in the last decades. At the same time, being able to provide a complete and satisfying theoretical prediction for weak lensing peak counts as a function of the cosmological model, still represents a challenge. Jain and Waerbeke 2000 measured the distribution of peaks from simulated lensing surveys with a peak model based on Bardeen et al. 1986;[START_REF] Bond | The statistics of cosmic background radiation uctuations[END_REF]; Waerbeke 2000 and showed that the peaks distribution from lensing data contains information on the projected mass function of dark matter halos and represents a probe of non-Gaussian information beyond second order statistics. Later, Bartelmann, M., Perrotta, F., and Baccigalupi, C. 2002 proposed weak-lensing halo counts with the aperture-mass technique as a new method for constraining the nature of dark energy.

Then, Maturi, M. et al. 2010 introduced a new approach based on the peak theory of the Gaussian random eld (Bardeen et al. 1986;[START_REF] Bond | The statistics of cosmic background radiation uctuations[END_REF] providing a simple analytical recipe to compute the signal-to-noise distribution of detections caused by chance projections of large-scale structures as well as the shape and shot noise contributions by the background galaxies. The same year, Fan, Shan, and Liu 2010 proposed a model to calculate the number of peaks in large-scale reconstructed convergence maps from weak-lensing observations, including both true peaks corresponding to clusters of galaxies and false peaks from intrinsic ellipticities of background galaxies based again on the Gaussian peak theory. However, while in this study peaks are identi ed as local maxima in the map, in Maturi, M. et al. 2010 a di erent de nition simplifying the analytical expression is given, according to which peaks are determined as contiguous areas with values above a given threshold. [START_REF] Cardone | Weak lensing peak count as a probe of f(R) theories[END_REF] computed the number density of haloes based on the theoretical mass function to constrain f (R) theories and then the total number of peaks by integrating over the full redshift range, smoothing out the dependence of the mass function on the redshift.

Whilst being the only way to provide explicit expressions for the peak function, analytical models present some downsides. Speci cally, when working in realistic conditions, such as in the presence of mask, or the bias from photometric redshifts, as well as the errors from shape measurement, the performance of these models can be strongly limited. Moreover, the link between cosmology and observed lensing peaks is made even more complicated because of some complexity from halo geometry, projections e ect and shape noise. To correct for large-scale structure projection e ects, Yuan et al. 2018 extended the model of Fan, Shan, and Liu 2010 by incorporating these projection e ects by modelling them as a random eld, i.e. by adding the stochastic large-scale structure contribution to the convergence map from the halo plus the shape noise. However, being able to model realistic survey settings, including additional cosmological or astrophysical features as for example IA and errors in general still remains challenging for analytical models.

A way to circumvent the di culties that arise when trying to predict peak counts with an-W alytical models, relies on generating weak lensing simulations densely sampled in cosmological parameter space. For example, Dietrich and Hartlap 2010 generated a set of N-body simulations in the (Ω m , σ 8 ) plan for 158 di erent cosmologies and interpolated using bilinear smoothing splines their shear peak function across parameter space. It consists in assuming that the main contribution to the lensing signal comes from halos and it discretises the space between the lensed galaxies and the observer in redshift bins, populating them with halos following a Navarro-Frenk-White (NFW) density pro le and placing them randomly within its redshift bin. This method was used by Peel et al. 2017a to assess the constraining power of peak counts in a simulated Euclid-like survey on the cosmological parameters (Ω m , σ 8 , w 0 ). They found that using CAMELUS, peak statistics yield a tight but signi cantly biased constraint in the (Ω m , σ 8 ) plane, indicating the need to better understand and control the model's systematics before applying it to a real surveys. Furthermore, Zorrilla Matilla et al. 2016 investigated how CAMELUS predicts the mean abundance of high-S/N peaks compared to N-body simulations, and found reasonable agreement, but that further development is needed to accurately estimate CAMELUS's variance and to probe the low-S/N tail.

Considered the above discussion, despite being costly to run, N-body simulations present several advantages with respect to analytical or semi-analytical models such as the possibility to incorporate the exact survey mask and shape noise in an easy way when one needs to deal with real data. Next section presents a brief overview of the results obtained using peak counts on real data.

State of the art of peaks on real data

The rst cosmological constraints from peak analyses were performed approximately at the same time and in an independent way by Liu ) plane. They found that, including systematics, peaks perform roughly as good as the power spectrum in a non-tomographic analysis. For the KiDS 450 deg 2 data release, two analysis were performed using peak counts: one obtaining the theoretical prediction using Fan's analytical model [START_REF] Shan | KiDS-450: cosmological constraints from weak lensing peak statistics -I. Inference from analytical prediction of high signal-to-noise ratio convergence peaks[END_REF]) and one employing N-body simulations (Martinet et al. 2018). Both studies give consistent estimates of the parameter S 8 ≡ σ 8 Ω m /0.3, however, I think it's important to consider that the analytical predictions of the rst study, were calibrated on the same simulations employed in the study based on N-body simulations. Considering the combination of shear peaks with non-tomographic measurements of the shear two-point correlation functions brought to an improvement of 20% in the uncertainty on the parameter S 8 in the simulation based analysis, compared to the shear two-point correlation functions alone, highlighting the great potential of peaks as a cosmological probe. The rst tomographic peak counts data analysis was recently performed by Harnois-Déraps et al. 2020, with a joint cosmic shear two-point shear correlation function and peak-counts analysis for DES-Y1 data release. To calibrate the cosmological dependence of peak counts they used the cosmo-SLICS suite of simulations (details on these simulations are also discussed in Chapter 6) covering the (Ω m , σ 8 , h, w 0 ) parameter space. They assessed the impact of baryons with the Magneticum hydrodynamical simulations, quanti ed the e ect of nite particle mass from high-resolution light-cones, and investigated the impact of intrinsic alignment by infusing intrinsic shapes to mock galaxies following a physical model based on the shape of dark matter halos. They also modelled source-lens clustering by comparing galaxy excess in peaks of di erent height, and found to have a negligible impact on their results given their peak selection criteria. They found a ∼ 20% gain in precision compared to the 2PCF analysis for the constraints on S 8 when using peak counts. Being the rst tomographic analysis with peak counts on real data, they con rm that including cross-redshift bins reinforces the constraints, once systematics such as IA are properly calibrated and conclude that for the peaks statistics to remain competitive with the 2PCF, further development is required in the modelling of baryon feedback.

In summary, it becomes more and more clear that peak counts represent a powerful tool, given their complementarity to second order statistics as the 2PCF or the power spectrum to constrain cosmological parameters, and this has been proven on real data applications. Now, in order for peaks statistics to remain competitive with the 2PCF, one challenge consists in further developing the modelling of baryon feedback. The baryoni cation method described in Section 2.2.3 could be one possible way to further model this, as well as metholodogies based on hydrodynamical simulations. Another crucial aspect that still has to be addressed concerns intrinsic alignment, as the inclusion in peaks analysis is still critical, and improving the IA modelling is necessary for future analyses. The model for IA adopted in Harnois-Déraps et al. 2020 remains too simple, and as they demonstrated, this e ect has an impact on the inferred cosmology at the same level as baryonic e ect that can reach 1σ level, limiting in a severe way the analysis. Further exciting work is hence needed to be able to properly model these systematics e ects to maximise the exploitation of peak counts potential in cosmological inference.

Weak lensing minima

A complementary probe of non-Gaussian information is given by weak lensing minima that is the number counts of local minima in a lensing convergence map as a function of their depth. In practice, for a 2D map, weak lensing minima are computed as the pixels with values smaller than their eight surrounding pixels. Physically, they are associated to underdense regions (voids) in our universe, being hence complementary to the lensing peaks which are typically associated with massive regions.

Cosmological forecasts with weak lensing minima

The projected mass distribution around voids can be used as a tool for constraining models of dark matter, dark energy, and modi ed gravity (Hamaus, Sutter, and Wandelt 2014). Deriving forecasts on the parameters (M ν , Ω m , A s ) for an LSST-like survey, Coulton et al. 2020a proposed lensing minima as a novel statistic to extract cosmological information in weak lensing data as they probe the emptiest regions (voids) in of the universe and also motivating their choice with the fact that baryonic e ects are expected to impact voids differently than overdense regions (peaks). Their results show how in their settings, lensing minima alone are slightly less constraining than lensing peaks but that when the two are combined, they produce signi cantly tighter constraints with respect to the power spectrum. In the same study, the BAHAMAS hydrodynamical simulations are used to study the e ects of baryons on lensing minima and the nding is that baryonic processes tend do decrease the number of very negative minima and high, positive minima, while increasing the number of minima values closer to zero. The bias induced by baryonic e ects appears to have a small impact on minima (∼ 0.5σ), while a more signi cant one on peak counts (∼ 4σ). Another recent study that forecasted the cosmological power of weak lensing voids was presentend by Davies et al. 2020. Using the cosmo-SLICS source catalogue down-sampled to match LSST speci cations, they provide forecasts on the (Ω m , S 8 , h, w 0 ) parameters and nd that these are tighter than the constraints obtained with the 2PCF with the same simulated observational speci cations.

These results show how weak lensing minima (voids) represent a promising probe for upcoming cosmological experiments and how when combined with peak counts they represent a very competitive summary statistics to constrain cosmological parameters.

Void de nition debate

Since voids may have irregular shapes, the lensing signal depends on the identi cation scheme considered. A large number of quite di erent void-nding algorithms is present in the literature, constructed using particles, haloes, and semi-analytical model galaxies extracted from simulations. For example, Bos et al. 2012 used a scheme called Watershed Void Finder based on halos and galaxy distributions to investigate whether void shapes can be used as a probe of the properties of dark energy. They found that the potential of voids as probe for dark energy, when they are identi ed through halos and galaxies distributions is limited by the discrete, sparse and biased nature of these objects. Another example is the study performed by Davies, Cautun, and Li 2018, where a higher lensing signal for voids is identi ed using lensing peaks, compared to the one found for voids identi ed using galaxies. Indeed, since the full mass distribution is not easily observable, observational studies make use of the galaxy distribution to identify voids. This approach can lead to galaxy voids less underdense than one would have using the full matter density eld. This was one of the reason of Coulton et al. 2020a for introducing weak lensing minima avoiding void tracers such as halos or lensing peaks. However, this method is based on the assumption that lensing minima are mostly associated with negative lensing signal, but in reality, a small number have imprints from slightly overdense regions as discussed by

Machine learning based methods

In the recent years, with the massive advances in the eld of machine learning (ML), machine learning and deep learning (DL) techniques have been proposed as analysis tools for cosmology. In particular [START_REF] Schmelzle | Cosmological model discrimination with Deep Learning[END_REF] , Ω m ) and showed that their neural network could yield ve times tighter constraints than the power spectrum, and four times tighter than the lensing peaks. A further step was then achieved by Ribli, Pataki, and Csabai 2018 that improved the convolutional neural network architecture to recover cosmological parameters more accurately from simulated weak lensing maps. Then, Fluri et al. 2018 explored the constraining power of CNNs compared to a standard power spectrum analysis for varying levels of noise and smoothing scales applied to convergence maps for the same set of parameters. They showed how the network is able to signi cantly improve the constraints with respect to the power spectrum, but also how the gain tends to decrease as the noise level and smoothing scales adopted increase. Peel et al. 2019 explored whether a neural network can discriminate better than peak counts between ΛCDM and modi ed gravity models with massive neutrinos and found that it can potentially discriminate between such models better than conventional higher-order statistics. Merten et al. 2019 investigated observational degeneracies between nine models of modi ed gravity and massive neutrinos using three di erent machine learning techniques and compared the performance with classical summary statistics as power spectrum, peak counts and Minkowski functionals. Their outcome is that the CNN provides the most discriminating representation of the data in a optimal noise-free maps setting. Matilla et al. 2020 analysed the neural network employed in Ribli et al. 2019 and compared its performance with constraints obtained from a combination of three commonly used WL statistics (power spectrum, lensing peaks, and Minkowski functionals), nding that the network can improve the inferred parameter constraints relative to this combination by 20% even in the presence of realistic levels of shape noise. However, the study needs to be veri ed under more realistic treatments that include realistic galaxy distributions and noise, as well as systematic errors. The rst study that used a CNN to measure the cosmological parameters from observed weak lensing data has been performed for KiDS-450 (Fluri et al. 2019). By comparing the results obtained with the CNN to the ones obtained with the power spectrum analysis on the same maps they found an improvement of about 30% on S 8 for the CNN.

In summary, machine learning based methods can be promising tools to retrieve cosmological information beyond second-order statistics e ciently. Moreover, the advancement in parallelised training strategies opens the gates to the possibility of training of larger networks so that the geometry of large survey (as next generation surveys) can be taken into account. Despite this enormous potential, these techniques still need a huge work of investigation to become a robust and accepted tool by the community as when applied to real data for example, systematic errors that are not taken into account in the training process are di cult to get checked and controlled. Moreover, the large number of tted parameters involved, and the depth of the many layers of non-linearities, on one side enable the learning of complex non-linear relationships between data and the parameters, while on the other side are still very di cult to interpret from a physical point of view.

The scattering transform, wavelet-phase harmonic statistics

Very recently, simultaneously and independently of the work presented in the thesis, two high order summary statistics based on wavelet transform were introduced: the scattering transform [START_REF] Cheng | A new approach to observational cosmology using the scattering transform[END_REF]) and Wavelet Phase Harmonics (WPH) statistics (Allys et al. 2020). The rst one shares ideas with convolutional neural networks but doesn't require training nor tuning. More speci cally, it employs the scattering transform introduced by Mallat 2012 and consists in a wavelet convolution of the weak lensing input image followed by a pixelwise modulus. The scattering operation is then iterated at di erent scales and nally the spatial average of every scattering image is taken. The constraining power of this statistic for the (Ω m , σ 8 ) plane has been compared to the one of the power spectrum, peak counts and the CNN adopted in Ribli et al. 2019. Their outcome is that on convergence maps with and without galaxy shape noise, the scattering transform outperforms the power spectrum and peak counts, and performs as well as the state-of-the-art CNNs. Then, Cheng and Menard 2021 have tested its performance extending the parameter space also to the dark energy equation of state parameter w 0 using the set of Dark Energy simulations from the Columbia Lensing Group and to the sum of neutrino masses M ν using the MassiveNus simulations and compared their performance with respect to the power spectrum and the bispectrum. They found an improvement for all parameters, and their results on M ν are consistent with the improvement we have found in both our studies for multi-scale higher order statistics (Ajani et al. 2020;Ajani et al. 2021), as they also state in Cheng and Menard 2021.

The second study from Allys et al. 2020 applied Wavelet Phase Harmonics (WPH) [START_REF] Mallat | Phase harmonic correlations and convolutional neural networks[END_REF] to constrain the parameters (Ω m , Ω b , h, n s , σ 8 ) using the Qujote simulations (Villaescusa-Navarro et al. 2020). The computation of this statistics consists in applying the phase harmonic operator to the 2D projection of the large-scale structure matter density eld provided from the simulations and convolved with the bump steerable wavelets introduced in Mallat, Zhang, and Rochette 2019. By computing Fisher information matrices, they nd that the WPH statistics lead to stringent constraints on all parameters except for σ 8 when compared to statistics based on the combination of the power spectrum and bispectrum.

As in our works (Ajani et al. 2020;Ajani et al. 2021), that are presented in details in Chapter 4 and Chapter 5, these two statistics exploit the advantages of performing multi-scale analysis, thanks to the properties of wavelet functions. From these results, is becoming more and more clear that the possibility of extracting cosmological information from different scales simultaneously can bring great improvement to the cosmological constraints and further e ort should be put in testing the robustness of such multi-scale statistics with respect to systematics, to prepare them to be applied to real data for future galaxy surveys.

Summary

As second-order statistics fail to fully capture the non-Gaussian information encoded in the non-linear features of weak lensing data, an increasing number of statistics of order higher than second have been proposed by the weak lensing community. Depending on the survey settings, systematics, noise level and cosmological parameter considered, they can perform in di erent ways with respect to each other in constraining cosmological parameters. I show here an attempt of summary of the state of the art of studies that have provided clear quanti cation of the performance of these statistics with respect to second order statistics for forecasts as well as for real data analysis, when available.

• Statistics: lists the names of the summary statistics employed to perform cosmological parameter inference.

• Tomography: this column indicates whether a tomographic study has been performed or not.

• Systematics: indicates if systematic e ects have been included in the analysis and if so which ones.

• Parameters: indicates which cosmological parameters have been constrained in the analysis. Notice that some analysis might have taken into account certain parameters but couldn't constrain them (e.g. w 0 ) for CFHTLenS peak counts. In this case the nonconstrained parameters are not listed. Only parameters that have been constrained are listed. For clarity and for the purposes of this summary, only references that present a clear quanti cation of the improvement in constraining power with respect to second order statistics are shown.

• Forecasts: speci es the improvement for forecasts analysis (i.e. without the use of real data) for the high order statistics combined with second order statistics w.r.t. the performance of second order statistics alone. The percentages indicates the improvement for the marginalised 1D probability of the corresponding parameter, the summarising the current state of the art of high order statistics where an estimate of the constraining power and its performance with respect to second order statistics have been provided by the authors. I would like to stress that this table does not include all studies ever performed with statistics of order higher than the second but the studies where cosmological inference has been performed and an explicit comparison with second order statistics is provided in the paper.

. integer numbers illustrate the improvement of the gure of merit. If a generic "≥", "∼" is shown, it means that the exact improvement has not been mentioned by the authors.

• Real data: report the results in terms of constraining power found in real data analysis.

• Survey: indicates the name of the survey if the survey settings used in the analysis are explicitly associated by the authors to a speci c survey, otherwise the sky coverage and galaxy number density employed are reported.

• References: indicates the corresponding study, identi ed by rst author name + year. The aim of this chapter is to introduce some de nitions useful for the purposes of this thesis in the context of wavelet formalism. In particular, I start by giving a brief introduction of the concept of sparsity, of the continuous wavelet transform and of the discrete wavelet transform. Thanks to these tools, the starlet transform, that will be widely use in Chapter 4, Chapter 5 and Chapter 6 is introduced. Then, the procedure followed to estimate the noise and compute the wavelet coe cients is illustrated. Finally, a brief overview of recent mass mapping methods based on sparse representations is provided.

W

Sparse signal representation

In a context where the goal is to try to maximise the information that we can extract from cosmological data, there are di erent representations of the data that can be considered. Indeed, any given signal can be represented in a variety of di erent domains without losing information, but depending on the properties of the signal and on the application, a given representation may be preferred.

Sparsity data model

A signal X, X = [x 1 , ..., x N ] T is de ned sparse if most of its entries are equal to zero. More speci cally, a k-sparse signal is a signal with only k non-zero samples values. It is also possible to give a less stringent de nition when considering a signal where only a few of its entries have a large magnitude and the majority of them are close to zero: in this case, one can de ne the signal as weakly sparse or compressible. Hence, in a sparse signal, either only a nite number of coe cients is non zero, or the coe cients decrease fast when rankordered. If a signal is not sparse, it can be sparsi ed using a given data representation.

Intuitively, this can be understood by thinking to periodic signals as for example sin(x) and by looking at Figure 3.1: in real space, sine functions are not sparse, but they are sparse in Fourier space, in particular "1-sparse", as they become the Dirac delta functions.

! +! -! $ %

Real space

Fourier space Having this in mind, a signal X is de ned sparse in the Fourier domain if it has sparse Fourier coe cients, de ned as

X[u] = 1 N k X[k]e 2π uk N
. This concept can be generalised by introducing a dictionary Φ = [φ 1 , ..., φ M ], whose elements are T elementary waveforms that enable to model the signal X ∈ R N as the linear combination:

X = Φα = M i=1 α[i]φ i , (3.1)
where α ∈ R M are the decomposition coe cients of the signal X in the dictionary Φ. This is of interest for this work because, if one can nd a good representation that allows to separate the signal from the noise e ciently, this leads to a gain of extracted information.

The representation basis treated in this work is the wavelet starlet transform, as it is very well suited for astronomical images that often contain isotropic sources [START_REF] Starck | The Undecimated Wavelet Decomposition and its Reconstruction[END_REF].

Continuous Wavelet transform

This section provides an introduction to the fundamentals of the continuous and discrete wavelet transforms, in order to have the tools to introduce and understand the starlet transform, that will be used in the works presented in this thesis. A wavelet is a function ψ(x) satisfying the admissibility condition:

∞ 0 | ψ(k)| 2 dk k < ∞, (3.2)
where ψ(k) is the Fourier transform of ψ(x). In order for Equation 3.2 to be satis ed, ψ has to satisfy the condition ψ(x)dx = 0. This condition implies that wavelets are highly localized functions with vanishing mean. Wavelet transforms decompose a signal into a family of scaled and translated functions by generating a family of daughter wavelets:

ψ a,b (x) = 1 √ a ψ x -b a , (3.3)
where ψ is the complex conjugate of ψ, a is a scaling parameter and b is a translation (or position) parameter. Hence, ψ a,b are scaled and translated versions of ψ, called the mother wavelet. Given a scale parameter, a > 0, and the translation parameter b ∈ R, the continuous wavelet transform (CWT)(Grossmann and Morlet 1984) of a function f (x) can be de ned as:

W ψ (a, b)(f ) = ∞ -∞ f (x)ψ a,b (x)dx = 1 √ a ∞ -∞ f (x)ψ x -b a dx. (3.4)
W By varying continuously the scale parameter a and the position parameter b one can obtain the wavelet coe cients for a given scale and position. Intuitively, one can think at this transform by considering how Fourier transforms measure the similarity between a signal and an analysing function. In the Fourier transform case, the analysing functions are complex exponential e jωt , while in the case of the continuous wavelet transform, the analysing function is a wavelet, ψ, that compares the signal to shifted and compressed or dilated versions of the mother wavelet. In other words, as the Fourier transform outputs coe cients as a function of the frequency, the wavelet transform outputs coe cients which are a function of their scale and translation. Another nice, canonical way, commonly used in lectures on wavelets, to understand the properties of the wavelet function, is to consider the information in terms of time and frequency that can be extracted through time series analysis, Fourier transforms, short-time Fourier transforms (STFT) and wavelet transforms, as shown in Figure 3.2. From time series analysis (top left panel of Figure 3.2), namely the analysis of a sequence of data points (signal) that occur in successive order over some period of time, information on the signal is provided for every single point in time, missing however information about its frequency. The Fourier transform (top right panel) converts the time information in the frequency domain, providing great frequency resolution and being well suited for stationary signals, but loosing track of the time information. To overcome the poor time resolution of the Fourier transform, the short-time Fourier transform (STFT) (Gabor 1946) was introduced. The STFT (bottom left panel of Figure 3.2) provides a timefrequency representation of the signal and it is based on the assumption that a portion of the non-stationary signal is stationary. In this way, a small portion of the signal is analysed at a time. Then the Fourier transform of each stationary portion is taken and the results are added up, to recover the entire signal. With this approach, one can recover frequency resolution for a given time resolution. This is done by taking a window function of xed length that is moved along the signal. The drawback is hence that once the window size is chosen, it remains unchanged during the entire analysis. This leads to a trade-o between the time and frequency resolution that it is possible to derive: by choosing a narrow windowing in time, the time resolution will be good but the frequency resolution will decrease; by choosing instead a wider windowing in the time axis, the time resolution will decrease as the frequency resolution increases. This leads to an inevitable loss of information, as low frequency components usually last a longer time, requiring good resolution in frequency, while high frequency components are often shorten in time, invoking the need of good resolution in time. Wavelets transform improve this context, as they allow to analyse the signal into di erent frequencies at di erent resolutions (bottom right panel of Figure 3. 2). By looking at the frequency-time plane in Figure 3.2, one can see how in the high frequencies regime, the resolution is good in time and poor in frequency. In other words, where the frequencies are high, the vertical lines are much denser than the horizontal lines. At the same time, at low frequencies, there is better resolution in frequencies and poorer resolution in time. In contrast to the STFT, the width and the central frequency of the wavelet can be changed by changing the scale parameter a, while moving across the signal. In this way, expanded wavelets, namely with large values of a, will be good in resolving low frequencies components of the signal with bad time resolution. Similarly, narrower wavelets, characterised by a smaller values of a, will be better in resolving high frequency components with better time resolution. So, when the frequency of the wavelet matches part of the signal with a very similar frequency, the output of the wavelet coe cients W ψ (a, b)(f ) is high.

In practice, starting from smaller to increasing scales a, the signal is anaysed from high frequencies to low frequencies: high frequency wavelets will trace the high frequencies of the signal, and low frequencies wavelets will detect low frequency of the signal.

Discrete Wavelet transforms

In the case of CWT, the scale parameter a and the translation parameter b are continuous, implying that a huge amount of wavelet coe cients is generated. To reduce the amount of data generated, one can consider discrete values for the parameters a and b. With this in mind, a discrete wavelet transform (DWT) can be de ned through the concept of multiresolution analysis (MRA) [START_REF] Mallat | A theory for multiresolution signal decomposition: the wavelet representation[END_REF][START_REF] Meyer | Wavelets and Operators[END_REF]. In this framework, the scale and translation parameters are chosen so that they are based on powers of two (dyadic). The main idea of MRA is to build a sequence of approximations f j of a function f ∈ L 2 (R) by smoothing it with a kernel of increasing width proportional to 2 j . Each of these approximations then belong to a subspace that contains all possible approximations at scale 2 j . For each approximation subspace is then possible to build an orthogonal subspace containing all the details that are lost between two consecutive approximations. Performing this procedure recursively enables to decompose the entire L 2 (R) space into a direct sum of the orthogonal subspace containing all the details that were lost between two consecutive approximations. Then for each j ∈ Z, it is possible to build a wavelet ψ, such that the family of wavelets:

ψ j,n = 1 √ 2 j ψ t -2 j n 2 j n∈Z , (3.5)
is an orthonormal basis of the subspace containing all the details that are lost between two consecutive approximations. The scale parameter is now a = 2 j and the translation (position) parameter b = 2 j n. In practice, what happens when analysing a signal f with a DWT, can be understood considering multilevel decomposition, reading Figure 3.3, as follows. The input signal f is passed into a low pass lter h and a high pass lter g, so that low pass lters will select the low frequency components while rejecting the high frequency components and vice-versa. The low pass portions of the signal are called approximation coe cients a j and they are iteratively ltered by the low pass lter and the high pass lter. The output of the high pass lters are instead the detail (or wavelet) coe cients d j , corresponding to high frequencies information. At each level of this process the output is downsampled in a way that its size is halved. This last step is called decimation and takes place when a decimated DWT is used to analyse the signal. The result of the entire process is a set of approxima- To give a more formal description, a consequence of the multiresolution approximation

! " # $ # $ % " % $ & " & $ ' " ' $ ( " ( ↓ 2 ↓ 2 ↓ 2 ↓ 2 ↓ 2 ↓ 2 ↓ 2 ↓ 2
Figure 3.3: Illustration of the multilevel decomposition of the signal f . By passing through a low pass lter, the approximation coe cients { a 1 , a 2 , a 3 , .., a j ,..a J } are generated. Each approximation coe cient is iteratively passed through a low pass lter generating the a j s and through a high pass lter that generates the detail coe cients

{ d 1 , d 2 , d 3 , ..,d j ,..d J }.
If a decimated DWT is employed, at each step the size of the coe cients is halved because the output is downsampled by a factor of 2. [START_REF] Mallat | A theory for multiresolution signal decomposition: the wavelet representation[END_REF][START_REF] Meyer | Wavelets and Operators[END_REF]) is that it is possible to build a scaling function φ that constitutes an orthonormal basis in the approximation subspace, such that

φ j,n = 1 √ 2 j φ t -n 2 j , (3.6)
for each j ∈ Z, that enables to compute the approximations f j of f at the scale 2 j as the orthogonal projection of f in the approximation subspace as follows:

f j = n∈Z a j [n]φ j,n , (3.7)
where a j [n] are the approximation coe cients given by a j [n] = f, φ j,n . Now, looking at the scaling functions de ned in Equation 3.6, one can consider the scaled function φ 1,0 = 1 √ 2 φ t 2 , and by exploiting again the implications of the multiresolution approximation [START_REF] Mallat | A theory for multiresolution signal decomposition: the wavelet representation[END_REF][START_REF] Meyer | Wavelets and Operators[END_REF], it is possible to introduce the discrete lter h such that

1 √ 2 φ t 2 = n∈Z h[n]φ(t -n), (3.8)
where h

[n] = 1 √ 2 φ t 2 , φ(t -n)
, namely the φ 1,0 has been decomposed on the basis {φ 0,n } n∈Z . This can be generalised to any scale 2 j+1 and basis {φ j,p } p∈Z in a way that the approximation coe cients at the scale j+1 can be obtained recursively from the coe cients at the previous scale j, as

W a j+1 [p] = n∈Z h[n -2p]a j [n].
(3.9) Furthermore, it can be shown [START_REF] Mallat | A theory for multiresolution signal decomposition: the wavelet representation[END_REF][START_REF] Meyer | Wavelets and Operators[END_REF]) that the projection of f onto the family of wavelets de ned in Equation 3.5 forming an orthonormal basis of the subspace containing all the details that were lost between two consecutive approximations, enables to decompose f as

f = j∈Z n∈Z d j [n]ψ j,n , (3.10)
through the details (wavelet) coe cients d j [n] = f, ψ j,n , with scale parameter 2 j and position parameter 2 j n. Then, analogously to Equation 3.9, it is possible to de ne a lter g to obtain the details coe cients at the scale j + 1 from the approximation coe cients of the previous scale j:

d j+1 [p] = n∈Z g[n -2p]a j [n] (3.11)
as schematically summarised in Figure 3.3. Depending on the properties of the lters g and h, the wavelet employed in the analysis has di erent properties that can be more or less convenient depending on the goal of the analysis. For the purposes of this thesis, we are interested in a particular type of DWT, called the starlet transform, introduced in the next section.

The starlet transform

The Isotropic Undecimated Wavelet Transform (IUWT) [START_REF] Starck | The Undecimated Wavelet Decomposition and its Reconstruction[END_REF], also called starlet transform, is an undecimated DWT, meaning that the decimation step introduced in the previous section is removed from its implementation. Moreover it is isotropic, meaning that -the lters h and g must be symmetric, namely h

[k] = h[-k] = h[k] and ḡ[k] = g[-k] = g[k],
-in dimensions ≥ 2, the lters h, g, the scaling function φ and the wavelet ψ must be nearly isotropic.

À trous algorithm

The starlet decomposition can be implemented through the à trous algorithm (Holschneider et al. 1989) as follows. The approximations and details coe cients introduced in Equation 3.9 and Equation 3.11 are recursively computed as

a j+1 [k] = l∈Z h[l]a j [l -2 j k], (3.12
)

d j+1 [k] = l∈Z g[l]a j [l -2 j k], (3.13)
where at each scale j di erent lters h (j) and g (j) are derived from h and g by inserting 2 j -1 zeros between each of the values, so that for j = 1 there will be one zero between each value of the lter, for j = 2 there will be three zeros.. and so on as

h (1) = (..., h[-2], 0, h[-1], 0, h[0], 0, h[1], 0, h[2], ...) h (2) = (..., h[-1], 0, 0, 0, h[0], 0, 0, 0, h[1], 0, 0, 0, h[2], ...)
so that they become sparse. Then, as mentioned before, the scaling function φ in D ≥ 2 must be isotropic and it is chosen to be the spline of order 3 (Daubechies 1992), i.e.

φ 1 (x) = 1 12 (|x -2| 3 -4|x -1| 3 + 6|x| 3 -4|x + 1| 3 + |x + 2| 3 ) (3.14)
with φ(x, y) = φ 1 (x)φ 1 (y) and the wavelet function is built as the di erence between two successive approximations of this isotropic scaling function as

1 4 ψ x 2 , y 2 = φ(x, y) - 1 4 φ x 2 , y 2 , (3.15)
so that ψ is also isotropic. This is re ected also in the relation between the lters h and g, that is

g[k, l] = δ[k, l] -h[k, l], (3.16)
where δ[0, 0] = 1 and δ[k, l] = 0 otherwise and

h[k, l] = h (1D) [k]h (1D) [l], (3.17)
with

h (1D) = [1, 4, 6, 4, 1] 16 . (3.18)
Hence, when convolved with a 2D starlet, an original image I is decomposed by this transform into a coarse version of it a J plus several images of the same size at di erent resolution scales j, as shown in Figure 3.4:

I[k, l] = a 0 [k, l] = a J [k, l] + J j=1 d j [k, l], (3.19)
where the wavelet images d j represent the details of the original image at dyadic (powers of two) scales corresponding to a spatial size of 2 j pixels. The maximum j is determined by the map size as log 2 N for an N × N image. In practice, this is done by implementing the above discussion, by rst considering the lter bank h 1D = { 1 16 , 1 4 , 3 8 , 1 4 , 1 16 } and initializing j = 0. Each coe cient a j+1 [k, l] is hence smoothed by convolving the previous coe cient a j [k, l] with the corresponding sparsi ed weight h 1D at the scale j. After this smoothing, the discrete wavelet transform is given by the di erence a j [k, l]a j+1 [k, l]. More formally, each wavelet coe cient is obtained as:

a j+1 [k, l] = m n h 1D [m]h 1D [n]a j [k + 2 j m, l + 2 j n] (3.20)
and

d j+1 [k, l] = a j [k, l] -a j+1 [k, l] (3.21)
so that the ensemble of {d 1 , d 2 , ....a J } are the wavelet coe cients, i.e. the starlet transform of the original image.

Therefore, the IUWT is entirely de ned by the choice of the scaling function φ and associated lter h. One advantage of employing a starlet lter is provided by its multi-scale analysis, namely its ability to investigate and extract the information encoded at di erent scales at the same time (Starck, Murtagh, and Bijaoui 1998). Moreover, compared to the DWT, it presents useful properties such as invariance per rotation and translation. Furthermore, as we are interested in optimising the extraction of information from weak lensing maps, the shape of this lter emphasises round features making it very e cient when dealing with for example with peak counts, which are point-like features. Moreover, peak counts (the signal) are sparse in the starlet domain, as they are described by a few number of non-zero coe cients, while white noise is not sparse in any representation. Hence, as we are employing the starlet to smooth the noise to access the cosmological information from the maps at smaller scales where the noise is more dominant, it represents a powerful tool to separate the noise from the signal. Further details on the advantages of wavelet starlet analysis can be also found in [START_REF] Lin | A new model to predict weak-lensing peak counts -III. Filtering technique comparisons[END_REF], Peel et al. 2017band Peel, Pettorino, Giocoli, et al. 2018. Figure 3.5 shows the starlet 1D and 2D pro les.

+ + + + =

Wavelet coe cients and noise estimation

The summary statistics that we compute on the noisy weak lensing maps in Chapter 4 and Chapter 5, except for the power spectrum that is computed as function of the angular multipoles , are expressed as a function of the signal to noise ratio ν ≡ S/N . Therefore, it is necessary to be able to estimate the noise at each scale. We de ne the signal to noise eld ν = S/N as the ratio between the noisy convergence κ convolved with the lter W over the smoothed standard deviation of the noise for each of the map we consider in our analysis:

ν = (W * κ)(θ) σ filt n (θ) . (3.22)
If W is a Gaussian kernel, the denominator in the above equation will be given by the standard deviation of the noise smoothed with a Gaussian kernel of the same size as the one used to smooth the signal. For instance, if the noisy convergence map is smoothed with a Gaussian kernel of size θ ker = 1 arcmin, then σ filt n (θ) is the standard deviation of the noise smoothed with a Gaussian kernel of size θ ker = 1 arcmin. When W is a starlet kernel instead, one needs to take into account the fact that the noise level varies at each scale. This can be done using the automatic noise estimation from the multiresolution support presented by Starck and Murtagh 1998. Speci cally, the appropriate value of the noise at the scale j in the succession is assessed from the standard deviation of the noise in the original image σ I and from the behavior of the noise in the wavelet space. The procedure to estimate the noise level σ j at each starlet scale is the following:

• we de ne the number of scales for the starlet decomposition,

• we compute the coe cients σ e j by taking the starlet transform of a Gaussian distribution of the same size of the original image with standard deviation one,

• to estimate σ I we take the median absolute deviationof the noisy convergence map.

The estimate of the noise level at each wavelet scale, is then given by σ j = σ e j σ I . An example for these values is provided in Table 3.1.

Wavelets and Dark Matter Mass Map Reconstruction

As described in Section 2.1.3, it is essential to be able to recover the convergence from the shear in a proper way. Mass mapping methods solve an ill-posed problem due to the irregular sampling of the lensing eld and the low signal to-noise ratio on small scales (Lanusse et al. 2016), that consists in nding the unknown underlying convergence eld κ (the signal) from observed observed shear data (noisy measurements), γ:

γ = Aκ + n, (3.23)
where n is the statistical uncertainty vector associated with the data. As mentioned in Section 2.1.3, there exist mass mapping methods based on sparsity. For example, the Gravitational Lensing Inversion and MaPping using Sparse Estimators (GLIMPSE) (Leonard, Lanusse, and Starck 2014;Lanusse et al. 2016) relies on the assumption that the density eld can be sparsely represented in an appropriate dictionary in order to regularise the inversion of the lensing operator. Being based on sparse representation, GLIMPSE is well suited for recovering piece-wise smooth features. A di erent approach, that relies on a Bayesian framework, is the one of Wiener ltering (Wiener 1949). In this approach a Gaussian random eld is assumed as a prior for the convergence map, which is responsible for inserting some bias that prevents the estimate from over-tting (Zaroubi et al. 1995). It has been shown (Lanusse et al. 2016;Price et al. 2019) that the solutions derived using a wavelet regularization signi cantly outperform standard techniques such as Wiener ltering for peaks recovery, while Wiener ltering performs better on the Gaussian content of the map (Je rey et al. 2018). As explained in Section 2.3, since the density eld is inhomogeneous, it encompasses Gaussian-type large-scale structures, as well as non-Gaussian features. It is therefore essential for mass mapping methods to preserve both Gaussian and non-Gaussian features during the reconstruction process. Motivated by the above considerations, Starck et al. 2021 proposed a method, called MCALens, to model the matter density eld using both linear and non-linear characteristics. More speci cally, the idea is to consider that modelling the convergence map as a Gaussian random eld leads to a good recovery of the large scales of the convergence map but not the non-Gaussian structures, while assuming the convergence map is compressible in the wavelet domain (i.e. sparse modelling) recovers non-Gaussian structures as peaks extremely well but poorly the Gaussian content. Hence, the sparse recovery is somehow complementary to the Wiener solution. To address these limitations, Starck et al. 2021 introduce a novel approach where the convergence eld κ is given by a Gaussian and a non-Gaussian part:

κ = κ G + κ NG . (3.24)
The non-Gaussian part of the signal κ NG undergoes a sparse decomposition in a wavelet dictionary, while the component κ G is assumed to be inherently non-sparse and Gaus-W sian. These two components are separated through the Morphological Component Analysis (MCA) proposed in Starck, Elad, and Donoho 2004 that allows to separate two components mixed in a single image when these have di erent morphological properties. If one can exploit the morphological di erences between the components, it was shown that it is possible to extract and separate them.

This requires to de ne the functions C G and C NG , such that the following is minimised

min κ G ,κ NG γ -A(κ G + κ NG ) 2 Σn + C G (κ G ) + C NG (κ NG ) , (3.25)
where Σ n is a pre-de ned covariance matrix of the noise, A is the inversion matrix from Equation 3.23 and γ is the shear.

Morphological Component Analysis solves this by performing an alternating minimization scheme, as follows:

• estimate κ G assuming κ NG is known:

min κ G (γ -Aκ NG ) -Aκ G ) 2 Σn + C G (κ G ) , (3.26)
• estimate κ NG assuming κ G is known:

min κ NG (γ -Aκ G ) -Aκ NG ) 2 Σn + C NG (κ NG ) . ( 3

.27)

A recent range of MCA applications in astrophysics can be found in [START_REF] Melchior | SCARLET: Source separation in multi-band images by Constrained Matrix Factorization[END_REF]Joseph et al. 2019 andWagner-Carena et al. 2020 The Gaussian component

The Gaussian component is then modelled following the standard Wiener modelling where κ G is assumed to be a Gaussian random eld: (3.28) and the solution of Equation 3.26 is obtained using the iterative Wiener ltering method described in Je rey et al. 2018.

C G (κ G ) = κ 2 Σ k ,

The non-Gaussian component

There are di erent ways to use a sparse model in the MCA framework. One way consists in using the standard 1 or 0 -norm regularisation in a wavelet-based sparsity model, as it is done in the GLIMPSE algorithm (Lanusse et al. 2016). This results in: (3.29) where p = 0 or 1, Φ is the wavelet matrix, and λ is a Lagrangian parameter. However, it seems that this approach leads to large wavelet scales and low Fourier frequencies that are relatively close, making it di cult to separate the information.

C N G (κ NG ) = λ Φ κ NG p ,
For this reason, Starck et al. 2021 implement another approach, that involves rst estimating the set of scales and positions where wavelet coe cients are above a given threshold, typically between 3 and 5 times the noise standard deviation relative to each wavelet coefcient. This set is composed by some coe cients named Ω, called active coe cients. These coe cients can be interpreted as a mask in the wavelet domain, where Ω j,x = 1 if a wavelet coe cient detected at scale j and position x and 0 otherwise. The noise σ j,x at scale j and position x can be determined using noise realizations.

The noise is whitened, since the noise factor considered is Gaussian with standard deviation equal to one and with a uniform power spectrum. Once this wavelet mask Ω is estimated, the non Gaussian component κ NG can be estimated through

min κ NG {|Ω Φ ((γ -Aκ G ) -Aκ NG )) | 2 + C NG (κ NG )}, (3.30) with C N G (κ NG ) = i R (κ N G ).
Once Ω is xed, the algorithm is almost linear and only a positivity constraint remains. Such positivity constraint is applied on the non-Gaussian component κ NG . Peaks in the map can be on top of voids, and therefore have negative pixel values. As peaks are captured by the non-Gaussian component, they are positive by construction, but the convergence map κ = κ G + κ NG can still be negative at peaks positions. The larger the non-Gaussianities, the more MCALens is expected to improve over linear methods such as the Wiener one. The experiments run using this method on simulated data show a signi cant improvement compared to the state of art.

Example

Here is shown an example performed in Starck et al. 2021 of application of MCAlens to reconstruct a convergence map of the 1.64 deg 2 HST/ACS COSMOS survey [START_REF] Scoville | COSMOS: Hubble Space Telescope Observations[END_REF]. The bright galaxies shape catalogue produced for Schrabback et al. 2010 is employed.

Summary

To summarise, wavelets are functions that allow to decompose data into di erent frequency components, and then study each component with a resolution matched to its scale. The goal of this chapter is to illustrate the methodology and results of our study presented in the paper Ajani et al. 2020. We investigated the impact of multi-scale ltering approaches on cosmological parameters by employing a starlet (wavelet) lter and a concatenation of Gaussian lters to constrain the sum of neutrino masses M ν , the matter density parameter Ω m and the amplitude of the primordial power spectrum A s for four tomographic redshifts in a Euclid-like setting. We have compared di erent summary statistics: power spectrum, single-scale peak counts and multi-scale peak counts and assessed their performance in an ideal setting (no systematics). 91

4.1 Forecasts on neutrino masses using multi-scale peak counts

Overview

As described in Section 2.3, second order statistics such as the two-point correlation function or the power spectrum fail to capture the non-Gaussianities of the large-scale structure eld and it is necessary to consider statistics of order higher than the second to distinguish the cosmological information hidden in non-linear modes that can represent new signals beyond the standard ΛCDM model. Moreover, as illustrated in Section 3.1, multi-scale approaches based on wavelets can be used for several applications in astrophysics. In the context of weak lensing peak counts, [START_REF] Liu | Cosmology constraints from the weak lensing peak counts and the power spectrum in CFHTLenS data[END_REF] have shown that a combination of certain smoothing scales can tighten the errors on the cosmological constraints, using the publicly available data from the 154 deg 2 CFHTLenS survey. However, they claim that the combination that they nd is only a limited investigation of the bene t of using multiple smoothing scales and leave the open question of identifying optimal lter shapes, sizes, and combinations that can help further tighten constraints using peak counts. An investigation in this direction has then been performed by [START_REF] Lin | A new model to predict weaklensing peak counts. III. Filtering technique comparisons[END_REF], where they compared linear ltering with a Gaussian and two compensated lters (the starlet wavelet and the aperture mass), and the non-linear ltering method MRLens when inferring the matter density parameter Ω m , the power spectrum normalisation σ 8 , and the dark-energy parameter w 0 . While they nd that a compensated lter function with counts included separately from di erent smoothing scales yields the tightest constraints on cosmological parameters from weak lensing peaks, their non-tomographic con guration does not allow to impose constraints on w 0 . The goal of the study presented in this chapter is to compare and quantify the performance of di erent ltering techniques in a multi-scale framework and to compare di erent summary statistics in this context: as second order statistics we employ the convergence power spectrum, as higher order statistics we employ peak counts of lensing convergence maps. We use as synthetic data the lensing convergence maps from MassiveNus simulations [START_REF] Petri | Mocking the weak lensing universe: The LensTools Python computing package[END_REF][START_REF] Liu | MassiveNuS: cosmological massive neutrino simulations[END_REF]. Using the same suite of simulations, [START_REF] Liu | Constraining neutrino mass with the tomographic weak lensing one-point probability distribution function and power spectrum[END_REF], Marques et al. 2019, and Coulton et al. 2019 have already shown for a LSST-like survey (LSST Science Collaboration et al. 2009) that combining the lensing power spectrum with higher-order statistics can provide tighter constraints on parameters. For this purpose, we perform our analysis using the lensing power spectrum and peak counts as summary statistics following Li, Liu, Zorrilla Matilla, et al. 2019. We extend the study by considering a survey with Euclid-like noise, and to smooth the noisy convergence maps we employ a multi-scale approach, investigating a concatenation of Gaussian lters and separately a starlet lter (Starck, Murtagh, and Fadili 2010).

Simulations

We 1}. The presence of massive neutrinos is taken into account assuming normal hierarchy and using a linear response method, where the evolution of neutrinos is described by linear perturbation theory but the clustering occurs in a nonlinear cold dark matter potential. The simulations have a 512 Mpc/h box size with 1024 3 CDM particles. They are implemented using a modi ed version of the public tree-Particle Mesh (tree-PM) code Gadget2 with a neutrino patch, describing the impact of massive neutrinos on the growth of structures up to k = 10 h Mpc -1 . A complete description of the implementation and the products is presented in [START_REF] Liu | MassiveNuS: cosmological massive neutrino simulations[END_REF]. We use the simulated convergence maps as mock data for our analysis. When dealing with real data, the actual observable is the shear eld that can be converted into the convergence eld as discussed in Section 2.1.3. We bypass this step from γ to κ and work with the convergence maps directly provided as products from MassiveNus. The maps are generated using the public ray-tracing package LensTools [START_REF] Petri | Mocking the weak lensing universe: The LensTools Python computing package[END_REF] for each of the 101 cosmological models at ve source redshifts z s = {0.5, 1.0, 1.5, 2.0, 2.5}. Each redshift has 10000 di erent map realisations obtained by rotating and shifting the spatial planes. Each κ map has 512 2 pixels, corresponding to a 12.25 deg 2 total angular size area in the range ∈ [100, 37000] with a resolution of 0.4 arcmin. Figure 4.1 shows the 2D parameter space planes for the 100 cosmological models available from the simulations.

Noise and survey speci cations

The method described in this work can be applied to any given survey. For illustration purposes, we perform here a tomographic study using the source redshifts z s = { 0.5, 1.0, 1.5, 2.0 } and mimicking the noise expected for a survey like Euclid 2 (Laureijs et al. 2011; Euclid Collaboration et al. 2020b). Speci cally, at each source redshift we produce 10000 map realisations of Gaussian noise with mean zero and variance

σ 2 n = σ 2 n gal A pix , (4.1)
where we set the dispersion of the ellipticity distribution to σ = 0.3, and the pixel area for these simulations is by A pix 0.16 arcmin2 . The redshift dependence that makes a tomographic investigation possible is encoded in the source galaxy redshift distribution, for which we assume the parametric form

n(z) = C z z 0 α exp - z z 0 β , (4.2)
with α = 2, β = 3/2 z 0 = 0.9/ √ 2 as in (Laureijs et al. 2011;Euclid Collaboration et al. 2020b), and C is the normalization constant to guarantee the constraint zmax z min n(z) dz = 30 gal/arcmin -2 . Then, we compute the galaxy number density at each bin as

n i gal = C z + i z - i n(z)dz, (4.3)
where z - i , z + i are the edges of the i th bin. We adapt the binning choice to the provided simulation settings, assuming that we observe galaxies within a small range around the actual source redshift. This leads to the values for the galaxy number density n gal per source redshift bin z s provided in Table 4.1: z s 0.5 1.0 1.5 2.0 n gal 11.02 11.90 5.45 1.45 Table 4.1: Values of n gal for each source redshift z s . We adapt the binning choice to the provided simulation settings, assuming that we observe galaxies within a small range around the actual redshift. In practice, this means considering as bin edges {0.001, 0.75, 1.25, 1.75, 2.25}, in order to compute the integral in Equation 4.3.

Filtering strategy

In this section, I will describe the strategy used for ltering. Some de nitions introduced in Section 3.1 will be re-established in order to keep reading smooth. In order to access the signal in the convergence maps at small scales, where they are mostly dominated by noise, we lter them, considering a multi-scale analysis compared to a single-scale analysis. First, we use a single Gaussian kernel of size θ ker , de ned as

G(θ; θ ker ) = 1 √ 2πθ ker e -θ 2 /(2θ 2 ker ) , (4.4)
which was also used for a single-scale lter in Li, Liu, Zorrilla Matilla, et al. 2019. We then compare the results with those obtained when applying instead a concatenation of Gaussian lters and an Isotropic Undecimated Wavelet Transform, also known as a starlet transform [START_REF] Starck | The Undecimated Wavelet Decomposition and its Reconstruction[END_REF], introduced in Section 3.4. This allows us to decompose our original image (the noisy convergence map) I as a sum of wavelet coe cient images w j plus a coarse resolution image c J :

I(x, y) = c J (x, y) + jmax j=1 w j (x, y), (4.5)
where the wavelet images w j represent the details of the original image at dyadic (powers of two) scales corresponding to a spatial size of 2 j pixels and J = j max + 1.

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One of the advantages of employing a starlet lter is provided by its multi-scale analysis, namely its ability to investigate and extract the information encoded at di erent scales at the same time (Starck, Murtagh, and Bijaoui 1998). Hence, the starlet transform presents the properties to compute e ciently J scales with a fast algorithm with a complexity of O(N 2 log N ) for an image of N × N pixels and to analyse data with compensated aperture lters with nite support. Further details on the advantages of wavelet starlet analysis have been discussed in Chapter 3 and are also present in Peel et al. 2017;Peel, Pettorino, Giocoli, et al. 2018. The following example illustrates how we can compare results from these two di erent ltering schemes. Applying a starlet transform with j max = 4 to a map with 0.4 arcmin pixel size results in a decomposition of four maps with resolutions [ 0.8, 1.6, 3.2, 6.4 ] arcmin plus the coarse-scale map. For our study, we will consider as nest scale θ St ker = 1.6 arcmin, being a more realistic choice in terms of resolution for convergence maps coming from Euclid-like survey data. We will therefore focus on the set of scales [ 1.6, 3.2, 6.4 ] arcmin plus the coarse map. Concerning the multi-Gaussian lters, to fairly compare them to the starlet, we set the standard deviations of the Gaussian lters such that their maximum matches that of the corresponding single starlet scale pro le as shown in 2, 2.7, 5.5, 9.5] arcmin. Based on the above, in our study we compare cosmological constraints obtained using noisy maps smoothed from a single-Gaussian kernel with the ones obtained from a multi-Gaussian analysis and from a starlet decomposition. We exclude the observables corresponding to 0.8 arcmin in our analysis after having veri ed that this does not cost any loss of information. The starlet transform can be seen as multi-Gaussian ltering where each Gaussian kernel is replaced by a compensated lter. In Figure 4.2 we show the result of the ltering procedure for a Gaussian kernel: given the original convergence map κ, we add white noise as described in Section 4.1.3 and then we lter the noisy map with the chosen kernel. 4.3: We show the matching between the Gaussian lter and the starlet at the di erent scales. We chose the kernel for the multi-Gaussian concatenation such that the maximum of the two pro les match. From left to right are the nest scale to the smoothest scale where .6, 3.2, 6.2, 12.8 ] arcmin. [ 1.6, 3.2, 6.4 ] arcmin and the coarse maps (red dotted line) and the Gaussian case (black dashed line). Due to the decomposition at di erent scales, for each map ltered with the starlet there are 4 di erent distributions. Indeed, the number of counts depends on the resolution: the larger the smoothing size (the lower the frequency) the smaller the number of peaks.

[ w 1 , w 2 , w 3 , c 4 ] = [ 1

Summary statistics: power spectrum and multi-scale peaks

To extract and investigate the cosmological information encoded in the weak lensing convergence maps, we compute the power spectrum (PS) and peak counts as summary statistics.

Convergence power spectrum

In this study, we compute as second order summary statistics the power spectrum as dened in Section 2.1 of the noisy ltered convergence maps. For a given cosmology we add Gaussian noise to each realisation of κ. For each redshift we generate a di erent set of noise maps following Equation 4.1. To lter the maps we employ a Gaussian kernel with smoothing size θ G ker = 1 arcmin and consider angular scales with logarithmically spaced bins in the range = [300, 5000]. We compute the power spectra using the python package LensTools for each of the 10000 realisations per cosmology and then we take the average over the realisations. We parallelise our code using joblib3 to accelerate processing due to the large number of realisations per cosmology.

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Multi-scale peaks

As higher order statistics, we detect and count weak lensing peaks on the noisy ltered maps using our code4 as follows. We compute peaks as local maxima of the signal-to-noise eld ν ≡ S/N i.e. as a pixel of larger value than its eight neighbours in the image. We de ne the signal to noise eld ν as the ratio between the noisy convergence κ convolved with the lter W(θ ker ) over the smoothed standard deviation of the noise for each realisation per redshift:

ν = (W * κ)(θ ker ) σ filt n , (4.6)
where W(θ ker ) can be the single-Gaussian, the multi-Gaussian or the starlet lter. Concerning σ filt n , its de nition depends on the employed lter. For a Gaussian kernel it is given by the standard deviation of the smoothed noise maps, while for the starlet case we need to estimate the noise at each wavelet scale for each image per redshift. To estimate the noise level at each starlet scale we follow Starck and Murtagh 1998. As described in Section 3.5, we use the fact that the standard deviation of the noise at the scale j is given by σ j = σ e j σ I , where σ I is the standard deviation of the noise of the image and σ e j are the coe cients obtained by taking the standard deviation of the starlet transform of a Gaussian distribution with standard deviation one at each scale j. To estimate σ I we take the median absolute deviation5 of the noisy convergence map. We do this for each one of the 10000 realisations for each cosmology and then take the average over the realisations. We consider the peaks distribution for 41 linearly spaced bins within the range ν = [-0.6, 6], based on the outcomes of Li, Liu, Zorrilla Matilla, et al. 2019 where it is shown that including the low (S/N < 1), medium (1 < S/N < 3) and high peaks (S/N > 3) jointly give the best constraints. Moreover, low and medium peaks, typically formed due to multiple much smaller halos than the single halos that cause the high peaks [START_REF] Liu | Origin of weak lensing convergence peaks[END_REF], contain a similar level of information as the high peaks. In Figure 4.4 we show for illustration purposes the peak counts distribution in logarithmic scale for each starlet scale and for a single-Gaussian lter case. We see that the number of counts depends on the resolution: the larger the smoothing size (the lower the frequency) the smaller the number of peaks. We have investigated the impact of the binning on the results by testing di erent boundaries, and we have found that choosing 41 bins instead of 50 decreases the condition number of the data matrix, hence facilitating its inversion during the likelihood analysis. We have also considered the minimum and the maximum values of the S/N maps as bin edges, and we have seen that this choice is not very convenient, since it increases the condition number by two orders of magnitude. To compute the covariance matrices of the data, we use the independent ducial massless neutrino simulation, de ned by [M ν , Ω m , 10 9 A s ] = [0.0, 0.3, 2.1] and obtained from initial conditions di erent from the massive simulations used to perform inference. We consider a parameter-independent covariance to reduce the risk of assigning an excess of information to the observables in the context of a Gaussian likelihood assumption, following the results of Carron, J. 2013. The covariance matrix elements are computed as

C ij = N r=1 (x r i -µ i )(x r j -µ j ) N -1 , (4.7)
where N is the number of observations (in this case the 10000 realizations), x r i is the value R of the power spectrum or the peak counts in the i th bin for a given realisation r and

µ i = 1 N r x r i , (4.8)
is the mean of the power spectrum or the peak counts in a given bin over all the realisations.

In Figure 4.5 we show the correlation coe cients of the multi-scale tomographic peak counts: the starlet case (in the left panel ) and the multi-Gaussian case (in the right panel).

The matrices are organised as follows. From left to right, the four main blocks are for the tomographic redshifts respectively in the order z s = [0.5, 1.0, 1.5, 2.0]. Within each of the main blocks, there are four sub-blocks representing the lter scale, i.e. the scales [1.6 , 3.2 , 6.4 , coarse] for the starlet and [1.2 , 2.7 , 5.5 , 9.5 ] for the multi-Gaussian. Each scale is binned in 41 values of signal to noise in the range S/N = [-0.6, 6]. We see that the starlet decomposition has a tendency to make the matrix more diagonal, while the odiagonal terms for a multi-Gaussian show more correlations between the scales and for small and high values of S/N . Consistently with this, we notice that the most correlated bins in the starlet case are the ones corresponding to the coarse scale (the last mini-block for each of the main blocks) whose pro le indeed closely mimics a Gaussian, as one can see in the last panel of Figure 4.3. Furthermore, we take into account the loss of information due to the nite number of bins and realisations by adopting for the inverse of the covariance matrix the estimator introduced by Hartlap, Simon, and Schneider 2007:

C -1 = N -n bins -2 N -1 C -1 * , (4.9)
where N is the number of realisations, n bins the number of bins, and C * the covariance matrix computed for the power spectrum and peak counts, whose elements are given by Equation 4.7. We also scale the covariance for a Euclid sky coverage by the factor f map /f survey , where f map = 12.25 deg 2 is the size of the convergence maps and f Euclid = 15000 deg 2 .

In using Equation 4.9, we do not expect all biases to be removed from our parameter inference, as this has already been ruled out in Sellentin and Heavens 2016 and in Sellentin and Heavens 2017. Nevertheless, we rely on the fact that the number of realisations that we are using ( 10000) is su ciently large and greater than n bins to consider it a reliable estimator for our purposes 6 . By looking at the correlation matrices of Figure 4.5 it is clear that the starlet (left panel) has the tendency to make the matrix more diagonal while the multi-Gaussian case (right panel) presents non-trivial o -diagonal terms. This nding, namely the fact that the majority of the information seems to be encoded in the diagonal terms of the covariance matrix in the starlet case, will be further investigated in the next section where I illustrate the results we obtained in this study.

Likelihood

To perform Bayesian inference and get the probability distributions of the cosmological parameters, we use a Gaussian likelihood for a cosmology-independent covariance:

log L(θ) = 1 2 (d -µ(θ)) T C -1 (d -µ(θ)), (4.10)
where d is the data array, C is the covariance matrix of the observable, µ the expected theoretical prediction as a function of the cosmological parameters θ. In our case, the data array is the mean over the (simulated) realisations of the power spectrum or peak counts or combination of the two for our ducial model. Cosmological parameters are the ones for which simulations are available, namely {M ν , Ω m , A s }. In order to determine the relation between the observable and the models µ(θ), i.e. to be able to have a prediction of the power spectrum and the peak counts given a new set of points in parameter space, we employ an interpolation with Gaussian Processes Regression (GPR, (Rasmussen and Williams 2005)) using the scikit-learn python package. As discussed in Section 1.5.2, Gaussian Processes are a generic supervised learning method that, via an assumption of smoothness between parameters with close values, allows one to compute the prediction for an observable at a new given point in parameter space. The cosmological parameters and the corresponding observables (power spectrum and peak counts or the two statistics combined) from the simulations are used as a training set, i.e. as the input for the GPR. Then, the Gaussian Processes act by assuming that for a new point in parameter space θ * which is su ciently close to a known point θ belonging to the training set, the corresponding observable will be described by a joint normal distribution along with the known observable. For this interpolation, we employ an anisotropic squared exponential function. We estimate the standard error of the noise level in the targets as the noise given by the fact that we take the mean over 10000 realisations for each observable and each bin. More speci cally, for each bin we compute the observable mean and its corresponding standard error over the 10000 realisations available from the simulations. The tting procedure takes then as input the three cosmological parameters and the re-scaled mean of the observable for each bin, while the standard errors are added as dual coe cients of the training data points in kernel space, i.e. as a regularisation term to the diagonal of the kernel matrix to take into account the noise level on the mean. This results in a number of GP corresponding to the number of bins that are then taken in by a prediction function. The latter reads new points in parameter space and returns the corresponding observable predictive distribution (power spectrum, peaks or the two statistics jointly) and its standard deviation. The hyper-parameters are t with the standard marginal likelihood approach. For the validation we compare the prediction of a given model obtained with the GP emulator excluding the corresponding model from the simulation, for 10 models near the ducial model following Liu and Madhavacheril 2019. We nd di erences at the sub-percent level that always lie within the statistical error consistent with Li, Liu, Zorrilla Matilla, et al. 2019;Coulton et al. 2019;Marques et al. 2019. Intuitively, this can be understood by looking at Figure 4.6: the training set, constituted by the cosmological parameters and corresponding peaks distributions obtained from the simulations are represented in blue. The output for the prediction is given by the red curves: for 200 new given cosmologies in parameter space fed to the Gaussian processes prediction function we get the corresponding peaks distribution. During the likelihood analysis, parameter space is explored in this way: by computing the summary statistics corresponding to each new set of cosmological parameters explored by the MCMC walkers.

Results

In this section I illustrate forecast results on the sum of neutrino masses M ν , on the matter density parameter Ω m and on the primordial power spectrum amplitude A s for a survey with Euclid-like noise in a tomographic setting with four source redshifts z s = [0.5,1.0,1.5,2.0 ], and compare results for di erent observables (power spectrum and peak counts) and lters (single-Gaussian, starlet and multi-Gaussian) to answer the questions lying at the origin of the study performed in Ajani et al. 2020:

A) What is the performance of multi-scale peak counts in terms of constraining power with respect to the state of the art summary statistics in weak lensing?

B) Does the combination of second order statistics and multi-scale higher-order statistics bring an improvement in tightening the constraints? C) What is the impact of the di erent scales on the parameter constraints? D) What is the impact of the lter choice on the covariance matrix? E) What is the impact of the di erent tomographic redshift bins on the parameter constraints?

To answer these questions and compare the di erent contours corresponding to the di erent summary statistics and ltering settings we perform a Bayesian likelihood analysis and use the emcee package7 to explore the parameter space with MCMC. We assume a at prior, speci cally following Coulton et al. 2019, a Gaussian likelihood function as de ned in Equation 4.10, and a model-independent covariance matrix as discussed in Section 4.1.6. The walkers are initialised in a tiny Gaussian ball of radius 10 -3 around the ducial cosmology [M ν , Ω m , 10 9 A s ] = [0.1, 0.3, 2.1] and we estimate the posterior using 120 walkers.

Our chains are stable against the length of the chain, and we verify their convergence by employing Gelman Rubin diagnostics (Gelman and Rubin 1992). To plot the contours we use the ChainConsumer python package (Hinton 2016).

As our results consist in comparing the constraining power for di erent summary statistics and ltering settings, in order to quantify our results, we use estimators common in the literature, whose de nitions are recalled here.

Figure of Merit

To have an approximate quanti cation of the size of the parameter contours that we use to compare their constraining power, we consider the following Figure of Merit (FoM) as de ned in Euclid Collaboration et al. 2020b:

FoM = det ( F ) 1/n , (4.11)
Table 4.2: Values of the FoM as de ned in Equation 4.11 for the di erent parameters pairs (α, β) for each observable employed in the likelihood analysis: the power spectrum, the peaks counted on maps smoothed with the kernel in consideration and Peaks + PS always refer to the constraints obtained with the peaks relative to some lters and the power spectrum while the term diagonal refers to the contours obtained with a likelihood analysis performed by only considering the diagonal elements of the data covariance matrix. We provide in the last column the 3D FoM given as the inverse of the volume in (M ν , A s , Ω m ) space.

where F is the marginalised Fisher submatrix that we estimate as the inverse of the covariance matrix among the set of cosmological parameters {M ν , Ω m , 10 9 A s } obtained with the MCMC chains. In the exponent, n is equal to the parameter space dimensionality, e.g. n = 2 for FoM in a 2-dimensional plane between two parameters while n = 3 if we take the Fisher matrix among the three parameters. We show the values of the FoM four our observables in Table 4.2.

Figure of correlation

To quantify the correlations among the parameters we use the Figure of Correlation [START_REF] Casas | Linear and non-linear Modi ed Gravity forecasts with future surveys[END_REF]Euclid Collaboration et al. 2020b):

FoC = det(P -1 ), (4.12)
where P is the correlation matrix, whose elements are de ned as

P αβ = C αβ / C αα C ββ ,
with C αβ being the covariance between the cosmological parameters α and β as de ned in the previous section. When the parameters are fully uncorrelated FoC = 1, while for FoC > 1 the o -diagonal terms are non-zero, indicating an increasing presence of correlations among parameters as the FoC increases. The values of the FoC for our constraints are shown in Table 4. 3. responding to the di erent tomographic observables: the power spectrum alone (PS), the Peaks alone for di erent lters and the two statistics combined (Peaks + PS). As explained in the text, FoC = 1 corresponds to uncorrelated parameters, while the further the FoC is to 1, the more correlations are present. Qualitatively, this can be appreciated by looking at the inclination of the contours: by looking at Figure 4.7 we can see more 'oblique' contours for the Gaussian peaks in the plane (M ν , Ω m ) compared to the power spectrum, while for the pair (M ν , A s ) the power spectrum shows more correlation than peaks.

FoC (M ν , Ω m ) (M ν , A s ) (Ω m , A s )

Matrix condition number

To estimate how di cult it is to invert our data covariance matrices, we compute the corresponding condition number: if the matrix is singular, the associated condition number is in nite, i.e. matrices with large condition numbers are more di cult to invert. We compute the condition number through the 2-norm of the matrix using singular value decomposition (SVD). As shown in Table 4.4, the condition number depends on the binning choice, especially for the multi-scale analysis. Indeed, we nd that choosing 41 linearly spaced bins for the peak counts instead of 50 reduces the condition number of the starlet peaks of about 10 orders of magnitude. For this reason we choose 41 bins when performing inference using peak counts.

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Marginalised error

To estimate the marginalised 1σ error on a single parameter θ α (which means having included all the degeneracies with respect to other parameters), we use the quantity:

σ αα = C αα , (4.13)
where C αα are the diagonal elements of the parameter covariance matrix. We show the values of the σ αα for our observables in Table 4.5.

A) Multi-scale peak counts performance vs state of the art summary statistics

In the left panel of Figure 4.7 we compare constraints obtained from the single-scale and the multi-scale peak counts analysis against the power spectrum contours. For the single scale we employ a Gaussian lter with θ G ker = 1.6 . For the multi-scale analysis we employ a starlet lter and a concatenation of Gaussian lters with smoothing widths chosen such that the pro les match the starlet scales, as described in Section 4.1.4. More speci cally, we show the comparison among the power spectrum (blue contours), the single-scale peaks (green contours), the starlet peaks (red contours) and the multi-Gaussian peaks (black contours). We con rm that peak counts outperform power spectrum constraints even in the single-Gaussian case, as found in Li, Liu, Zorrilla Matilla, et al. 2019. In addition, we nd that a multi-scale approach leads to a remarkable improvement with respect to a single-scale approach in terms of constraining power, as expected, due to its higher information content concerning structure formation. We quantify these outcomes by considering the Figure of Merit de ned in Equation 4.11. As shown in Table 4.2, the FoM for the single-Gaussian peaks in the parameter space plane (M ν , Ω m ) is more than twice the one given by the power spectrum; the one from starlet peaks is more than twice that obtained with the single-Gaussian peaks, and the multi-Gaussian peaks FoM is more than three times the one corresponding to the single-Gaussian case. Concerning the (M ν , A s ) and (Ω m , A s ) planes, the FoM for Gaussian peaks is about three times that for the power spectrum, the one from starlet peaks is again about twice that obtained with the single-Gaussian peaks, and the multi-Gaussian peaks give again a FoM about three times that seen for single-Gaussian peaks contours.

As further investigation, we compute the Figure of Correlation as de ned in Equation 4.12 to study the correlation among the parameters. By looking at Table 4.3, one can see how values for the power spectrum for the pairs (M ν , Ω m ) and (Ω m , A s ) are close to one, suggesting that correlation among them appears to be very small, while the plane (M ν , A s ) shows more correlation, as its FoC is nearly twice as large. Qualitatively this can be appreciated by looking at the inclination of the contours. More speci cally, concerning the plane (M ν , Ω m ), the power spectrum contours are horizontal and show also visually that these two parameters are not correlated; constraints obtained via peak counts show a slightly larger correlation, increasing by 21% for Gaussian peaks and by about 13-14% for the multi-scale analysis, with respect to the power spectrum. It is interesting to note that in the (M ν , A s ) plane, the correlation decreases by 30% when using single-Gaussian peaks and by about 40% when using a multi-scale approach compared to the power spectrum, suggesting that peak counts can play an important role in breaking the degeneracy for this pair of parameters. Independently of the correlation, all constraints obtained with multi-scale ltering are tighter than the ones obtained via single-scale ltering, and both are tighter than the ones for the power spectrum.

B) Performance of second order statistics and multi-scale peaks combined

Based on the previous result, we are now interested in the constraints obtained when considering the two statistics jointly and on the impact of the di erent lter settings in this context. In particular, by focusing on the right panel of Figure 4.7, where we compare the 95% con dence contours of the power spectrum (blue contours) with the single-Gaussian peaks (green contours) and the two joint statistics (violet contours), we notice how in the single-scale approach the addition of the power spectrum to the peak counts brings a nonnegligible improvement in terms of constraining power with respect to the peaks alone. More speci cally, reading the values presented in Table 4.2, we see that the FoM of the joint contours is roughly 1.6 times that of the peaks alone, and more than three times that of the power spectrum.

Focusing now on the multi-scale approaches, in Figure 4.8 we show the same comparison in the left panel by comparing the power spectrum with the starlet peaks alone (red contour) and the two joint statistics (orange contours) and in the right panel for the multi-Gaussian case with the multi-Gaussian peaks alone (black contours) and the two joint statis- .6, 3.2, 6.4] arcmin + coarse map, constraints from multi-Gaussian Peak counts computed on noisy maps smoothed with a Starlet kernel with corresponding resolutions [1.2, 2.7, 5.5, 9.5] arcmin. Right panel: constraints from power spectrum (blue contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1 arcmin, constraints from Gaussian Peak counts (green contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1.6 arcmin and the two statistics jointly (violet contours).

tics (turquoise contours). We notice in this case that the FoM of the combined statistics are roughly 1.1 -1.2 times the peaks alone in the starlet case and 1.1 times the peaks alone in the multi-Gaussian case, suggesting that the information given by the joint statistics is mostly contained in the multi-scale peak counts alone. Peak counts therefore appear to be competitive and su cient statistics for parameter inference when dealing with weak lensing convergence maps as input data, in our ideal setting. This further con rms that lensing peaks are a powerful tool in the context of cosmological parameter inference, emphasising as well the importance of the role played by the ltering choice. [1.6, 3.2, 6.4] arcmin + coarse map and constraints from the two statistics joint (orange contours) . Right panel: constraints from power spectrum (blue contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1 arcmin, constraints from multi-Gaussian Peak counts (black contours) computed on noisy maps smoothed with a multi-Gaussian kernel with corresponding resolutions [1.2, 2.7, 5.5, 9.5] arcmin and the two statistics jointly (light blue contours).

C) Starlet scales impact on constraints

The left panel of Figure 4.9 shows the impact of the di erent starlet decomposition scales on the constraints. The MCMC chain used for the starlet decomposition results of the analysis has been obtained by considering all starlet scales, i.e. [1.6, 3.2, 6.4] arcmin + coarse scale, shown in red. To check that we are allowed to exclude the nest scale in the entire analysis, namely not to include the resolution corresponding to 0.8 arcmin -which won't satisfy the survey requirements -we compare the constraints relative to [0.8, 1.6, 3.2, 6.4] arcmin + coarse scale with the ones for [1.6, 3.2, 6.4] arcmin + coarse scale and we verify that they overlap. We then investigate the impact of the di erent starlet scales and we obtain that it is su cient to consider the setting [3.2, 6.4] + coarse scale to obtain results competitive with the full set of scales, as shown by the dark blue contours in the gure that match the R full starlet decomposition contours. Hence, we identify w 2 = 3.2 arcmin as the smallest scale needed to obtain the maximal constraints with convergence maps of resolution 0.4 arcmin. We also perform the inference by adding one scale at a time in the observable array to show how the contours shrink as a function of the number of starlet scales. We nd that the only setting that recovers almost the full information is given by [3.2, 6.4, coarse]. We also notice from the contours relative to [w 1 , w 2 , w 3 ] that excluding the coarse scale leads to a loss of information (precisely 28% on M ν , 33% on Ω m and 19% on A s ). we show the impact of the di erent starlet scales and we prove there exists a minimum resolution θ ker = w 2 = 3.2 arcmin that allows us to obtain constraints comparable to what is achieved with the full wavelet decomposition and that the information contained in the coarse map cannot be neglected. Starting from the rst starlet scale alone θ ker = w 1 = 1.6 arcmin (dashed big contours in light blue) we add scale by scale in light blue until [w1, w2, w3]. In dark blue we show the constraints corresponding to [w 2 , w 3 , c 4 ] that almost match with the constraints provided by the full starlet decomposition (in red). Right panel: we show here the constraints obtained by adding each tomographic redshift at the time: the dashed pink corresponds to the contours relative to z s = 0.5, the lighter red to z s = [ 0.5, 1.0 ], the darker pink to z s = [ 0.5, 1.0, 1.5 ] and the dark red to the full set of redshifts in the starlet case. We plot this against the power spectrum (blue contours) to show how each source redshift contribution to shrinks the contours and helps break the degeneracy with respect to the power spectrum.

D) Impact on covariance matrix: Gaussian multi-scale and Starlet comparison

In this section we investigate the impact of the choice of the lter on the data covariance matrix. Indeed, by looking at the correlation matrices of Figure 4.5 it is clear that the starlet (left panel) has the tendency to diagonalise the matrix while the multi-Gaussian case (right panel) presents non-trivial o -diagonal terms as introduced in Section 4.1.6. To further explore this aspect we have run the likelihood analysis considering just the diagonal elements of the covariance matrices in order to compare results with full covariance case.

We nd the constraints illustrated in Figure 4.10: in the left panel we plot the starlet peak counts contours obtained with the full covariance matrix (red) against the diagonal-only version of the data covariance matrix (dashed dark red). On the right panel we show the same comparison for the multi-Gaussian peaks case with the full covariance case (black) against diagonal-only contours (dashed gray). We see that for the starlet lter the majority of the information is indeed encoded in the diagonal elements, while for the multi-Gaussian case the presence of non-trivial correlations among the scales makes the contours slightly larger for Ω m and A s , while it adds some information on M ν with respect to the diagonal case. We can quantify this by taking the ratio between the FoM relative to the full covariance and the diagonal elements cases: for the starlet we nd a ratio of 1.07 and 1.15 for the multi-Gaussian. The same interpretation arises when looking at the 1-σ marginalised error of Table 4.5: excluding the o -diagonal terms in the starlet data covariance matrix implies a loss of information of 4% on M ν , no loss on Ω m and a gain of 7% on A s . For the multi-Gaussian case, the same procedure leads to a loss of 11% on M ν , a gain of 33% on Ω m and a gain of 17% on A s . This is an interesting aspect of the starlet lter that could prove to be useful when dealing with high dimensional data and the covariance matrix can be di cult to invert.

E) Impact of di erent tomographic redshifts

Here we investigate how parameter constraint contours shrink by adding the tomographic redshift bins by one by one. In particular, in the right panel of Figure 4.9 we show the information gain resulting from the addition of each tomographic redshift bin. In blue we show the power spectrum, and in darkening shades of red we plot contours for the starlet peaks as follows. Contours for source redshift z s = 0.5 are dashed pink, they are dashed darker pink for source redshift z s = [0.5, 1.0], and so on until reaching the dark red contours which are obtained by concatenating all source redshifts. As expected, the peaks contours show a di erent degeneracy direction from that of the power spectrum due to the higher-order information they contain. Indeed, the contours at z s = 0.5 already show di erent degeneracy compared to the power spectrum contours between M ν and Ω m with a FoC=1.09, giving though larger marginalised constraints on Ω m . Adding z s = 1.0 provides FoM that are more three times those for z s = 0.5 alone for the planes (M ν , Ω m ) and (M ν , A s ) but increases the correlation between M ν and Ω m to FoC=1.17. Finally, the concatenation of the

: [ m ν , Ω m , 10 9 A s ] = [ 0.1, 0.3, 2.1 ].
Left panel: constraints from starlet peak counts (continuous red contours) obtained employing the full covariance matrix against constraints from starlet peak counts (dashed red contours) obtained employing the diagonal elements only of the covariance matrix in the likelihood analysis. Right panel: constraints from multi-Gaussian peak counts (continuous black contours) obtained employing the full covariance matrix against constraints from multi-Gaussian peak counts (dashed gray contours) obtained employing the diagonal elements only of the covariance matrix in the likelihood analysis source redshifts z s = [0.5, 1.0, 1.5] further adds correlations between the two parameters (FoC=1.19) but shrinks the contours to almost reach those of the complete set of source redshifts.

Marginalised errors

In Figure 4.11 we show the marginalised constraints on each cosmological parameter corresponding to the di erent observables. To compare the improvement obtained by employing the di erent statistics we compute the 1σ marginalised error for each parameter, summarised in Table 4.5. In particular, we nd an improvement of 35%, 20% and 58% respectively on M ν , Ω m and A s when employing the single-Gaussian peaks instead of the power spectrum, an improvement of 63%, 40% and 72% when employing the starlet peaks instead of the power spectrum alone, and an improvement of 70%, 40% and 77% when em-ploying the multi-Gaussian peaks instead of the power spectrum alone. Namely, the starlet peaks outperform the single-Gaussian peaks by 43% on M ν , 25% on Ω m and 34% on A s , and the multi-Gaussian peaks outperform the single-Gaussian peaks by 54% on M ν , 25%

on Ω m and 45% on A s . Finally, employing a multi-Gaussian instead of a starlet lter in the context of peak counts might improve the constraints by 19% on M ν , and 18% on A s , while no improvement is noticed for Ω m . 4.6.

Conclusions and future prospects

In the presented study, we infer the sum of neutrino masses M ν , the matter density parameter Ω m and the amplitude of the primordial power spectrum A s for a survey with Euclid-like noise using tomographic weak lensing. Our goal is to compare the constraining power of multi-scale ltering approaches, namely the starlet lter and a concatenation of Gaussian lters, with respect to a single-Gaussian one in the context of peak counts. We also compute the constraints with standard second-order statistics, in particular using the lensing power spectrum as a benchmark for the comparison. We compare the outcomes obtained from ltering the lensing convergence maps, which have a resolution of 0.4 arcmin, with a 4.11. In this table we also show the values corresponding to the marginalised constraints obtained using only the diagonal elements of the covariance matrices. They are very similar to the ones obtained by employing the full covariance. We further investigate this aspect in Section 4.1.8.

R Observable M ν + Ω m -Ω m + A s -A s +
Gaussian kernel of smoothing size 1.6 arcmin, a starlet kernel, and a concatenation of Gaussians. More speci cally, the starlet lter is an isotropic undecimated wavelet transform that allows us to extract the information encoded in di erent spatial scales simultaneously. Setting the number of scales in the transform to four, the starlet kernel sizes for our maps correspond to [0.8, 1.6, 3.2, 6.4] arcmin + the coarse map, since the starlet transform returns maps ltered at dyadic scales. In deriving parameter constraints we exclude the rst scale and work with [1.6, 3.2, 6.4] arcmin + coarse map. To fairly compare it with a multi-Gaussian, we set the standard deviations of the Gaussian kernels at each scale such that their pro le peaks match the corresponding starlet scale peaks, resulting in a concatenation of Gaussians with standard deviations of [1.2, 2.7, 5.5, 9.5] arcmin respectively.

We nd the following results: a) For peak counts, a multi-scale ltering approach of the noisy maps leads to an improvement factor of more than two over a single-scale approach (single-Gaussian kernel) for the joint constraints on (M ν , Ω m ), (M ν , A s ) and (Ω m , A s ) when using a starlet kernel, and a factor of more than three when using a multi-Gaussian lter. This is even more evident in the marginalised constraints, where the improvement is respectively 43% on M ν , 25% on Ω m and 34% on A s for the starlet, while for the multi-Gaussian it is 54% on M ν , 25% on Ω m and 45% on A s . Employing a multi-Gaussian instead of a starlet lter in the context of peak counts might improve the constraints by 19% on M ν , and 18% on A s , while no improvement is noticed for Ω m . Finally, both multi-scale peak counts perform better than the power spectrum on the set of parameters {M ν , Ω m , A s } respectively by 63%, 40% and 72% when using a starlet lter and by 70%, 40% and 77% when using a multi-scale Gaussian lter.

b) When combining multi-scale peaks and the power spectrum, i.e. using a concatenation of peak counts and the power spectrum as the observed data vector, we nd that the information is mostly encoded in the peaks alone (for certain parameters, such as Ω m in the starlet case, it is completely encoded). This suggests that when adopting a multi-scale approach, it might be su cient to work with the peaks alone.

c) The inclusion of the coarse map when counting peaks preserves crucial information. Moreover, for maps with a pixel size of 0.4 arcmin, there exists a minimum resolution (i.e. smallest scale needed) for the starlet scales corresponding to θ ker = 3.2 arcmin to achieve maximal constraining power. This enables us to exclude the rst two nest scales of the starlet decomposition, which correspond to the highest frequencies and are the most prone to the impact of noise, allowing for a faster and more e cient analysis.

d) We notice that employing a starlet lter leads to a highly diagonal data covariance matrix, while for the multi-Gaussian lter the o -diagonal terms are prominent, and correlations among the di erent scales are non-negligible. In other words, the majority of the information in the starlet lter case is encoded in the diagonal elements of the covariance matrix. This is an interesting aspect of the starlet lter that could prove useful when dealing with high dimensional data where the covariance matrix can be di cult to invert.

In summary, we con rm that weak-lensing peak counts are a powerful tool to infer cosmological parameters, especially when investigating the non-linear regime where the impact of parameters such as the neutrino masses becomes relevant. We also point out the importance of adopting a multi-scale approach in the context of weak-lensing peak counts, which bring the advantage of analysing the information encoded at di erent scales simultaneously, thereby leading to tighter constraints than single-scale analysis. As we have shown in Figure 4.11, the multi-scale lters considered (the multi-Gaussian lter and the starlet lter) lead to similar constraints. This is expected, as we choose the Gaussian kernels such that each pro le peak matches with a starlet scale. Minimal residual di erences between the two lters may be related to the binning: while this is the same for both, it might be that the two lters are optimal with di erent choices of the binning. We leave the investigation of the optimal binning for both multiscale Gaussian and starlet peaks to future work. There is however an advantage, in using the starlet lter over a multi-Gaussian lter: the starlet has the tendency to remove the o -diagonal terms in the covariance matrix, hence making the matrix more diagonal, easy and faster to invert. Moreover, Leonard, Pires, and Starck 2012 have proved that it o ers a clear and signi cant time advantage over standard aperture mass algorithms for all scales of interest. We implemented a pipeline that allows us to go from simulated lensing convergence maps as input data to constraints on cosmological parameters as nal output, employing di erent ltering techniques with second-order (the power spectrum) and higher-order statistics (peak-counts). Hence, a future project will be to generalise the pipeline in terms of exibility of the input data, including systematics e ects and modelling of the noise. In particular, being able to control systematics errors and baryonic e ects is as important as the statistical power to guarantee a robust analysis. In the context of weak-lensing peak counts, baryons can change the shape of the distribution of peaks by increasing the low S/N end and decreasing the high S/N values by a few percent, as quanti ed by Fong et al. 2019. It has been shown as well by Coulton et al. 2020b that ignoring baryonic e ects can lead to strong biases in inferences from peak counts and that in principle these biases can be mitigated without signi cantly degrading cosmological constraints when baryonic e ects are modelled and marginalised. In Chapter 7 I will present a rst application to real data where we include baryonic e ects. Concerning intrinsic alignment and noise uncertainty, further investigations is needed in future studies with the aim of including such modelling in our pipeline and to ultimately apply our pipeline to real data coming from future galaxy surveys. The previous section outlined the potential of peak statistics in a multi-scale framework for cosmological parameter inference. As previously mentioned, besides peak statistics, several alternative high order statistics have been proven to be very powerful in many studies to account for the non-Gaussian information encoded in weak lensing data. In particular, Coulton et al. 2020a have introduced weak lensing minima as a new probe of non-Gaussian information related to underdense regions (voids) in our universe and have shown how the combination of lensing peaks and minima leads to an improvement of respectively 44%, 11% and 63% on the sum of neutrino masses M ν , the matter density parameter Ω m and on the amplitude of the primordial power spectrum A s with respect to the power spectrum contours. In a study to compare the constraining power of high order statistics in the (Ω m , σ 8 ) plane, Zürcher et al. 2021 found that peak counts, minimum counts and Minkowski functionals yield stronger constraints when compared to the angular power spectrum analysis in a non-tomographic, as well as a in tomographic setup. From the results of these studies, it becomes clearer and clearer that the information encoded in lensing minima is complementary to what we can extract through peak counts and that their combination, namely the full distribution, is more powerful than considering the two statistics alone in terms of constraining power. In this direction, Martinet et al. 2021 have performed forecasts with aperture mass (M ap ) map statistics for future cosmic shear surveys, including peaks, voids, and the full distribution of pixels (1D M ap ) and have shown that for the the structure growth parameter S 8 , the dark energy equation of state parameter w 0 and the matter density parameter Ω m , the full distribution of pixels with aperture mass has a superior constraining power than peaks and voids because it includes both peaks and voids. Motivated by the general context just described and by our previous ndings on the potential of multi-scale wavelet based framework for high order statistics, we have then proposed in Ajani et al. 2021 a new summary statistics called starlet 1 norm that allows to perform very naturally a joint multi-scale peaks and voids analysis, accounting for the information present in all pixels in the weak lensing convergence map. The aim of this chapter is to present the methodology and results of the study Ajani et al. 2021.

Toward the starlet 1 norm Starlet peaks

Multi-scale peaks counts can be derived either using a set of Gaussian kernels of di erent sizes or through a wavelet decomposition such as the starlet transform as done in Ajani et al. 2020. The starlet transform decomposes a convergence map κ of size N × N into a set W = {w 1 , ..., w jmax , c J } of J = j max + 1 bands of same size, where j max is the number of wavelet scales considered, c J is the coarse scale, namely a very smoothed version of the original image κ, and w j are the wavelet bands at scale 2 j pixels. An example to illustrate this is shown in Figure 5.1 and details of this have been outlined in Section 3.4 and in Section 4.1.4. It was also shown in [START_REF] Pires | Cosmological constraints from the capture of non-Gaussianity in weak lensing data[END_REF] that the starlet transform can be interpreted as a fast multiscale aperture mass decomposition where the aperture mass kernels have a compact support and are compensated (namely, the kernel integral is null). Starlet peaks are then derived by considering n bins with bin edges given by the minimum and the maximum values of each band in W. An interesting advantage of such an approach is that each wavelet band covers a di erent frequencies range which leads to an almost diagonal peaks counts covariance matrix (Ajani et al. 2020). This is not the case when a standard multiscale Gaussian analysis is applied on the convergence map.

Starlet extrema

In this section we introduce the notation starlet extrema to refer to peaks computed on positive signal and minima computed on negative signal that we will use as a statistics to compare against the starlet 1 -norm. We employ starlet peaks computed on maps with only S/N > 0 contribution combined with starlet minima computed on maps with only S/N < 0 contribution using a 4 scales corresponding to [1.6 , 3.2 , 6.4 , coarse] based on the resolution of the convergence maps that we use, with 29 linearly spaced bins for each scales between the minimum and maximum values of each S/N map.

As mentioned in the introduction, cosmic voids analysis is an alternative to peaks to study convergence maps, and the combination of both improves the constraints on cosmological parameters. It is interesting to notice how a starlet decomposition can naturally include a multiscale voids analysis. Instead of extracting only maxima (peaks) in each band, we can also extract minima (pixels with values smaller than their eight neighbours), and a joint peaks-voids multiscale is therefore obtained extracting wavelet coe cients extrema (minima + maxima). The starlet decomposition provides therefore a very natural framework for a joint multiscale peaks and voids analysis.

De nition of starlet 1 -norm

A particularity of peak and void statistics is that only a few pixels are considered, while other high-order statistics, such as bispectrum or Minkowski functionals, use all the pixels. In a starlet framework, we should emphasise that starlet peaks have mainly positive values and starlet voids negative values due to the property of the wavelet function. So instead of counting the number of peaks or voids in a given bin i de ned by two values, B i and B i+1 , we could take the sum of all wavelet coe cients with an amplitude between B i and B i+1 . If B i and B i+1 are positive, this corresponds to the de nition of the set of coe cients S j,i at scale j and in bin i such that S j,i = {w j,k /B i < w j,k < B i+1 }, where k is the pixel index. We can then compute the sum #coef (S j,i ) u=1 S j,i [u]. This can be generalised to positive and negative bins using:

l j,i 1 = #coef (S j,i ) u=1 | S j,i [u] |= ||S j,i || 1 (5.1)
where ||.|| 1 is the standard 1 -norm (i.e. ||x|| 1 = k |x k |) and the index u runs from 1 to the number of pixels in a given bin i at scale j (i.e. #coef (S j,i )). The quantity l j,i 1 de ned in Equation (5.1) is nothing more than the 1 -norm of the binned pixel values of the starlet coe cients of the original image κ map. In the following, we will name S 1 , the starlet 1 -norm, as the set S 1 of all l j,i 1 numbers obtained from the di erent scales j and bins i. This approach enables us to extract the information encoded in the absolute value of all pixels in the map instead of characterising it only by selecting local minima or maxima. An interesting advantage is that it avoids the open issue of how to de ne a void (Colberg, Pearce, Foster, et al. 2008), brie y discussed in Section 2.3.6. It is interesting to notice that this S 1 statistic is also directly related to the density probability function of the starlet coe cients at di erent scales.

Methodology

We provide constraints on the sum of neutrino masses M ν , on the matter density parameter Ω m and on the power spectrum amplitude A s by employing ve di erent summary statistics applied to the MassiveNus in the same setting as the one described in Section 4.1.2 and Section 4.1.3. We compute the summary statistics on maps of the signal to noise eld where we de ne the signal to noise as the ratio between the noisy convergence κ convolved with the lter W(θ ker ) over the smoothed standard deviation of the noise for each realisation per redshift. We estimate the noise level at each starlet scale following Starck and Murtagh 1998 as described in Section 3.5. For the purposes of this study, we mimic the shape noise expected for a survey like Euclid1 (Laureijs et al. 2011;Euclid Collaboration et al. 2020b) as de ned in Section 4.1.3. We perform a tomographic analysis using four source redshifts z s = {0.5, 1.0, 1.5, 2.0} with corresponding values for the galaxy number density n gal per source redshift bin n gal = {11.02, 11.90, 5.45, 1.45}. The covariance matrix is computed in the same way as described in Section 4.1.6 and the model to build the likelihood is obtained with the Gaussian Process Regressor as in Section 4.1.7 for the di erent summary statistics. Analogously to the multi-scale peaks study detailed in the previous chapter, we explore and constrain the parameter space with the emcee package, which is a python implementation of the a ne invariant ensemble sampler for Markov chain Monte Carlo (MCMC) introduced by (Foreman-Mackey et al. 2013). We assume a at prior, a Gaussian likelihood function as de ned in Section 4.1.7, and a model-independent covariance matrix as discussed in Section 4.1.6. The walkers are initialised in a tiny Gaussian ball of radius 10 -3 around the ducial cosmology [M ν , Ω m , 10 9 A s ] = [0.1, 0.3, 2.1] and we estimate the posterior using 120 walkers. Our chains are stable against the length of the chain, and we verify their convergence by employing Gelman Rubin diagnostics (Gelman and Rubin 1992). To plot the contours we use the ChainConsumer python package (Hinton 2016). To quantify our results, we employ the same result estimators as the ones introduced in Section 4.1.8.

Summary statistics

In this section I illustrate the di erent summary statistics used for the analysis. a) We compute the power spectrum on noisy convergence maps ltered with a Gaussian kernel with smoothing size θ G ker = 1 arcmin. We consider angular scales with 24 logarithmically spaced bins in the range = [300, 5000]. b) Monoscale peaks + voids: we compute peaks (as pixels with values larger than their eight neighbours) and voids (as pixels with values smaller than their eight neighbours) on noisy convergence maps ltered with a single-Gaussian kernel with smoothing size θ G ker = 2 arcmin with 29 linearly spaced bins for peaks between the minimum and maximum of the map in S/N and 29 linearly spaced bins for voids between the negative maximum and positive minimum of the maps in S/N . c) Starlet peak counts are computed as pixels with values larger than their eight neighbours, computed on noisy convergence maps ltered with a starlet kernel with four A scales corresponding to [1.6 , 3.2 , 6.4 , coarse], with 29 linearly spaced bins for each scales between the minimum and maximum values of each S/N maps.

d) Starlet extrema are obtained combining peaks computed on maps with only S/N > 0 contribution combined with starlet voids computed on maps with only S/N < 0 contribution, using a starlet as in c), with 58 (29 for peaks + 29 for voids) linearly spaced bins.

e) the starlet 1 -norm S 1 is computed following Equation 5.1 on noisy convergence maps ltered with a starlet kernel with four scales as in c) and d).

Statistics (d) and (e) are our new proposals for this study. In all statistics where we employ the starlet decomposition, the nest frequency we consider is θ St ker = 1.6 arcmin, corresponding to the maximum angular scale max = 2149.

Results

In this section, I illustrate forecast results on the sum of neutrino masses M ν , on the matter density parameter Ω m and on the primordial power spectrum amplitude A s for a survey with Euclid-like shape noise in a tomographic setting with four source redshifts z s = [0.5,1.0,1.5,2.0]. We compare results for the di erent observables de ned in Section 5.1.5 and investigate the impact of the choice of the lter on the covariance matrix. In Figure 5.3 we show the comparison between the constraints obtained using di erent summary statistics, as described in Section 5.1.5. As expected, we see that all higher-order statistics are more constraining than the power spectrum. The new result of this study is represented by the starlet 1 -norm: the inclusion of all pixels enables to retrieve tighter constraints than the combination of local minima and maxima (voids + peaks). Speci cally, for all parameter space planes, the starlet 1 -norm FoM values, illustrated in Table 5.1, are about twice the ones for the local minima and local maxima combined (and more than seven times larger than the power spectrum FoM values). 4.11 for di erent parameters pairs for each observable employed in the likelihood analysis. We provide in the last column the 3D FoM given as the inverse of the volume in (M ν , A s , Ω m ) space. are larger than starlet peaks and peaks + voids monoscale suggesting that starlet extrema can be a good candidate as well as multiscale higher order statistics. However, the 1 -norm remains the statistics that performs the best with respect to all the summary statistics we have considered in terms of constraining power. In Figure 5.2 we show the marginalised constraints on each cosmological parameter corresponding to the di erent observables. To compare the improvement obtained by employing the di erent statistics we compute the 1σ marginalised error for each parameter, summarised in Table 5.2. We nd that the starlet

FoM (M ν , Ω m ) (M ν , A s ) (Ω m , A s ) (M ν , Ω m ,

Diagonal Covariances

As recalled in the introduction, in Ajani et al. 2020 we found that starlet peak counts have the interesting feature that the corresponding covariance matrix is nearly diagonal. Motivated by this, we have tested in this work whether this feature is also maintained for starlet extrema and for the 1 -norm. Interestingly, we nd that starlet extrema keep this characteristic while the 1 -norm show more correlations in the o -diagonal term. This can be seen by looking at Figure 5.4: in the left panel we show the constraints for the starlet extrema when using the full covariance matrix (continuous contours) and compare them with the starlet extrema when using only the diagonal elements of the covariance matrix (dashed contours). Analogously, we show the same comparison for the 1 -norm in the right panel. When using starlet extrema as summary statistics, if we employ in the likelihood analysis a covariance matrix with only its diagonal elements, there is no loss of information on M ν , and a slight loss of information of 25% on Ω m and 22% A s . Concerning the 1 -norm, it is su cient to look at the contours to notice how the contours with only diagonal terms are considerably di erent with respect to the contours obtained with the full covariance matrix: they appear shifted and they present a di erent degree of degeneracy for (Ω m , A s ). Hence, we conclude that the 1 -norm outperforms starlet extrema in terms of constraining power when considering the full covariance in the analysis, but presents a less diagonal matrix than starlet extrema. We conclude that depending on the context the 1 -norm could be a convenient choice when the priority is the constraining power while starlet extrema might be more useful when interested in speeding up the analysis or when the covariance matrix can be di cult to invert.

Conclusions and future prospects

In this work we propose to use starlet 1 -norm statistics on weak lensing converge maps to constrain cosmological parameters. The measure of multiscale peak amplitudes can be seen as a measure of the 1 -norm of a subset of positive wavelet coe cients. Similarly, the measure of voids amplitudes can be seen as a measure of the 1 -norm of a subset of negative wavelet coe cients. Wavelets provide therefore a great framework for a joint peaks and voids analysis, in which information from all wavelet coe cients is included. We propose therefore to use a very simple 1 -norm statistics de ned as the sum of the 1 -norm of all coe cients in a given S/N (pixel) bin for each wavelet scale, as de ned in Equation 5.1. We investigate the impact of employing the starlet 1 -norm as summary statistics computed on weak lensing convergence maps to estimate cosmological parameters and we nd that the 1 -norm outperforms the state of the art summary statistics, the power spectrum and the combination of monoscale peaks and voids, respectively of 72% and 24% on M ν , 60% and 50% on Ω m and 75% and 24% on A s . We have further proposed starlet extrema and compared them to 1 -norm: also in this case the latter performs better in terms of constraining power, within the current ideal setting, while starlet extrema present the constraints from starlet extrema with the full covariance matrix (continuous contours) computed on noisy maps ltered with a four scales starlet kernel against constraints from starlet extrema with the only diagonal elements of the covariance matrix (dashed contours).

Right panel: constraints from 1 -norm with the full covariance matrix (continuous contours) computed on noisy maps ltered with a four scales starlet kernel against constraints from 1 -norm with the only diagonal elements of the covariance matrix (dashed contours).

advantage of a more diagonal covariance matrix. We are aware that to serve as a robust probe for precision cosmology, the statistical power alone is not su cient; for their usage, it will be important to test how these statistics react in a non-ideal setting, and how their performance is impacted by systematics in the signal. We will dedicate a future study to this aspect.

Pipeline description and visualisation

This section illustrates a description of the pipeline used in the work presented in this chapter and the one presented in Chapter 4. The input of the pipeline is given by weak lensing convergence maps. These convergence maps are then ltered, with a Gaussian or a starlet lter if they are already noisy, otherwise shape noise mimicking a given survey is added to the maps. Then the summary statistics are computed on the noisy smoothed maps for both simulations at the di erent cosmologies available and for the data (or ducial model). With the same settings as the models and data summary statistics, the covariance matrix is computed by concatenating the di erent observables if tomography or probe combination is taken into account. Then, the observables computed from the simulations (model) are fed to the Gaussian Processes (GP) regressor that learns the relation between the summary statistics and the cosmological parameters and is then able to predict the observable to a new given point in parameter space. Afterwards, the summary statistics for the data, for the model and the covariance, are given as input to the likelihood function and to the MCMC module. The output of this last module provides the chains corresponding to a given summary statistics constraining the parameters that where given as input to the GP training. A visualisation of the procedure is presented in Figure 5.5. Here's a detailed description of the di erent steps numbered in correspondence of the gure below: 1a) Convergence map: The convergence map are provided as initial input in this rst version of the pipeline.

1b) Survey noise: If the convergence map is built directly from observed ellipticity maps, the noise is intrinsically taken into account following the survey speci cations present in the simulations that supply the catalogue. If the convergence map is built directly from γ shear maps, we add the noise to the map by taking the standard deviation of the observed ellipticity of each map for each tomographic bin and follow Equation 4.1.

2) Noise ltering: In order to access the cosmological information, the noisy map if convolved with a 2D Gaussian lter or with a 2D starlet lter. The starlet decomposition with J scales generates J + 1 di erent maps, at the scale 2 J , where the J + 1 map is the coarse map. The Gaussian kernel is computed with the python package scipy. The starlet decomposition is performed using the CosmoStat python package lenspack.

3), 4) Summary statistics: On the smoothed/ ltered maps (output of 2)), di erent summary statistics can be computed.

-Power spectrum -Peak counts: single-scale peak counts, multi-scale peak counts, starlet peak counts -Void counts: single-scale minima counts, multi-scale minima counts

-Starlet extrema -Starlet 1 -norm
The computation of all summary statistics is implemented in parallel using the python package joblib, so that their computation is performed simultaneously for each cosmological model, redshift and realisation.

5) Interpolation with Gaussian Processes:

The interpolation with GP to get the predictions of the observable is implemented for each bin in or S/N and for each redshift when a tomographic analysis is performed. This means that for an observable with 20 bins and 4 tomographic redshift, 80 di erent Gaussian Processes will be generated to get the prediction. The Gaussian Process Regressor, the GP kernel and the prediction function are implemented using the scikit-learn package.

6) Covariance:

The covariance matrix takes as input the summary statistic for a given ducial model and it's computed following Equation 4.9. In practice it is implemented using numpy. Depending on the survey sky coverage and the size of the maps, a prefactor to scale the covariance has to be provided. 7), 8) Likelihood analysis and MCMC: The likelihood, prior and posterior functions implemented take as input the cosmological parameters and the GP prediction function.

The MCMC is performed using the python packages emcee and multiprocessing packages so that parameter space is explored in parallel. The chains are plotted using ChainConsumer, that is also employed to compute the mean acceptance fraction and the mean autocorrelation time.

As will be described in Section 6.3.1 a rst extension of the input of the pipeline is now in place, allowing to take as initial input directly a galaxy catalogue from which shear maps or convergence maps can be built. This is one of the focus of Chapter 6. The above illustration shows a noiseless convergence map. To include observational noise,  noise is added to each pixel, drawn from a random Gaussian distribution centered at zero with variance

noise = < 2 > n gal ⌦ . ( 5
)
This expression shows how the noise level depends on the galaxy shape noise , the galaxy density n gal and the noise smoothing scale. The noise contribution has to be modeled for specific survey designs.

Gaussian smoothing

To remove noises that are dominating the small scales the maps are smoothed. Jia's paper takes into account LSST noise and considers three di↵erent smoothing scales (1,2 and 5 arcmin) to compare the confidence contours from lensing peak counts when computing the contraints on cosmological parameters. The resulting maps for the corresponding gaussian smoothings will appear as Small smoothing scales can extract more non-linear information (as well as considering higher `as maximum multipole), but too small scales would result in a noise dominated map, while too large scales loose the too much non-lineari information. We assume a flat prior, specifically following [28], a Gaussian likelihood function as defined in Eq. ( 17), and

Interpolation with Gaussian Processes

2 t 0 d ! t (4)
N data ! p summaries ( 5)

dim(p) = # parameters O(10 3 ) (6) log L(✓) = 1 2 (d µ(✓)) T C 1 (d µ(✓)) (7)
Likelihood analysis + MCMC Summary statistics for the data This chapter presents the procedure and some preliminary results of an ongoing work (Ajani et al. in prep.[a]) whose aim is to extend the pipeline described in the previous with respect to 1) the considered parameter space 2) the cosmological probe. The new probe in consideration is photometric galaxy clustering, that along with weak lensing and spectroscopic galaxy clustering will be one of the primary probes of next generation galaxy surveys like Euclid. Di erent cosmological probes are a ected in di erent ways by the considered redshifts and by the dynamical ranges of the underlying large-scale structure (Harnois-Déraps, J., Giblin, B., and Joachimi, B. 2019). This, along with the di erences in the instru-131

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ments employed and measurement approaches make it possible to obtain complementary cosmological information and to easily deal with systematic e ects as they are typically distinct and uncorrelated in di erent probes (Harnois-Déraps et al. 2018). Probe combination can therefore exploit these advantages. Moreover, as discussed in details throughout this thesis, statistics of order higher than the second can provide additional non-Gaussian information that is typically lost when using second order statistics only. Using hence a new suite of simulations provided by the rst author of Harnois-Déraps, J., Giblin, B., and Joachimi, B. 2019, I describe the settings and procedure followed so far and that we envisage to follow to investigate the impact of probe combination in a higher order statistics framework. Simultaneously, I illustrate the open questions that we still need to address to extend the pipeline to achieve this goal and the extension already accomplished.

Background de nitions

We employ as second order statistics for photometric galaxy clustering, the angular power spectra in harmonic space, C GG ( ) measured on the galaxy maps, built as described in the next section. Its theoretical description is brie y provided here. Given the galaxy number density n G i (z) in each tomographic bin i of the photometric survey, the angular galaxy density in such redshift bin is given by

nG i = zmax z min dz n G i (z) . (6.1)
Then, the radial weight function for a given tomographic bin i for galaxy clustering can be de ned as

W G i (z) = n G i (z) nG i H(z) c . (6.2)
The observable spectrum is theoretically given by the following expression

C GG ij ( ) = dz W G i (z)W G j (z) H(z)r 2 (z) P photo gg + 1/2 r(z) , z (6.3)
where P gg ( , z) is the galaxy-galaxy power spectrum, linked to the matter power spectrum P δδ ( , z) through the bias b g .4) In this preliminary part of the work, we consider for our power spectrum measurements only the auto-spectra in the same redshift bin and the galaxy bias b g is xed in the simulations.

P photo gg ( , z) = [b photo g (z)] 2 P δδ ( , z). ( 6

SLICS and cosmo-SLICS simulations

The analysis presented in this section is based on two suites of N-body simulations, the Scinet LIght Cones • the size of the maps, that is 100 deg 2 for the cosmo-SLICS and 12.25 deg 2 for the MassiveNus.

I want to underline these di erences, since one of the aimed extensions for the pipeline described in Section 5.2 is to test its performance for a di erent parameter space than the MassiveNus. Indeed, the MassiveNus were the rst simulations I have employed as rst application of the pipeline, so it is crucial to verify that the pipeline works with di erent simulations, to be sure that none of the steps is dependent on the simulations and that it works regardless of this choice. We have therefore chosen to use the cosmo-SLICS, since they also provide a catalogue for the lenses that we will aim to use for combining probes.

Weak lensing 6.3.1 Building the convergence maps from the catalogue

A rst extension that needs to be done concerns the input of the pipeline. Indeed, as the MassiveNus directly provide weak lensing convergence maps, the pipeline used to get the results of Chapter 4 and Chapter 5 is built in such a way that it takes directly as input the convergence maps. One of the products of the cosmo-SLICS instead, is the galaxy catalogue mimicking KiDS-1000 requirements (Asgari, Marika et al. 2021). This section shows the procedure I followed to build the convergence maps that are then used to compute the summary statistics from the galaxy catalogue provided by the simulations. The nal pipeline will hence take as input directly the catalogues and give as output the cosmological constraints.

Galaxy catalogue

The galaxy catalogue provides:

• the positions (x, y) for each galaxy in arcmin

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• values for the redshifts z that match to best t estimates from the data, obtained by a self-organising map method

• values for two components of the shear γ 1 , γ 2 assigned to each galaxy

• values for two components of the observed ellipticities obs 1 , obs 2 assigned to each galaxy.

To get the convergence map, we use Kaiser Squires (KS) inversion: rst, we project the shear onto a Cartesian grid and then we smooth it with a Gaussian lter with width 0.7 arcmin following Giblin et al. 2018 to reduce the impact of potential empty pixels. We chose n pix = 600 pixels per side. Since the size of each catalogue covers 100 deg 2 , the resolution is 1 arcmin. Then, we employ Equation 2.26 to get κest γ . Finally, we perform an inverse Fourier transform to recover κ(θ). This is implemented using our CosmoStat python package lenspack. In Figure 6.2 we show an example of our reconstructed κ maps.

Noise and ltering

Once the convergence map is obtained, the next step is to add a noise component to take into account shape noise. As the catalogues provide the observed ellipticity for each galaxy, the average shape noise σ λ can be obtained by simply taking the standard deviation of the observed ellipticities for each cosmology and for each tomographic redshift. Then, to get the standard deviation of the noise as in Equation 4.1, we take the total number of galaxies N for each catalogue and we divide it by the area of the map in pix 2 to get the galaxy number density per catalogue. For each tomographic bin, it is possible to build the n(z) with the redshifts values provided from the catalogue. Figure 6.3 shows the n(z) for the di erent redshifts and the corresponding number of galaxies N that we employ to compute the noise to add to the convergence maps. The legend shows the number of galaxies N associated to each bin, that is employed to compute the noise that we add to the convergence maps.

As the noisy maps are ready, the next step is to lter them in order to access the cosmological information. We then smooth the noisy convergence maps with a Gaussian kernel of size θ ker = 3 arcmin.

Map validation with power spectrum

Once the maps are ltered, we compute the power spectrum for each map of each cosmology and each redshift, to build the training set for the Gaussian Processes interpolation. We use the power spectrum to also validate our maps. We do this by comparing our measurement for each cosmology, taking the average over the realisations and over the two di erent seeds for each tomographic bin to the theoretical predictions, as shown in the left panel of Figure 6.4. We nd that the power spectra we measure are in good agreement from min = 180 up to max = 4700 with the theoretical predictions. We also perform an extra validation, by comparing the power spectrum measured on our reconstructed map with one map provided by the authors of Giblin et al. 2018, as we follow a similar method to reconstruct the κ map 1 , and we nd very good agreement between the two, as shown in the right panel of Figure 6.4. The power spectra used to validate against the theoretical predictions are measured on noiseless κ maps, built from the two components of the shear (γ 1 , γ 2 ) from the catalogue. The power spectrum used to validate against the map from Giblin+ 2018, is measured on a noisy smoothed map, built from the two components of the observed ellipticities ( obs 1 , obs 2 ) from the catalogue. The power spectra employed as summary statistics to perform the parameter estimation analysis are computed as the latter, namely measured on κ maps that are built from the two components of the observed ellipticities ( obs 1 , obs 2 ), hence intrinsically accounting for the shape noise of each map. We use 33 bins in the range = [180,2040], that lies within the region where we are con dent that the power spectra are in agreement with the theoretical predictions (white region in the left panel of Figure 6.4). 1 The map they provided is built from the same catalogue (for KiDS-1000) we are using. In order to obtain the GCph observables we build galaxy density maps from the d KiDS-1000-like lens catalogues of Luminous Red Galaxies (LRGs) provided by the cosmo-SLICS simulations. This set of photometric redshift galaxy foreground samples of LRGs mock catalogues is available for each of the cosmo-SLICS cosmologies illustrated in Figure 6.1 and for 4 tomographic bins. A recent study employing the same catalogue was performed by [START_REF] Burger | A revised density split statistic model for general lters[END_REF]. Each catalogue is constituted by the positions of each galaxies in RA and DEC and redshift. The redshifts of the LRGs match exactly the redshifts of projected mass sheets, from which they originate. So they appear to be very discrete, separated by about ∆χ = 257 Mpc/h. The catalogues are generated assuming a constant linear galaxy bias b g = 1.72, with a galaxy number density of n 0 = 0.028 gal/arcmin 2 . Starting from this catalogue we build galaxy density maps for each cosmology and each realisation as follows:

• we bin the provided positions in RA and DEC into a 2D grid of 600 × 600 pixels • we smooth it with a Gaussian lter of size 3 arcmin an example of the resulting map is shown in the left panel of Figure 6.5. As for the computation of the high order statistics, we are interested in the S/N , we estimate the noise as the mean of our map. The S/N is then de ned as the galaxy map smoothed over the

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standard deviation of our noise estimate, for each map. One example is shown in the right panel of Figure 6.5.

Map validation with power spectrum

We validate our maps by comparing our measured power spectra against the theoretical prediction obtained with the python library pyccl, corresponding to Equation 6.3. In order to do so we measure the power spectrum on the galaxy density map built as described in the previous section. As the map is intrinsically noisy (due to the presence of Poisson noise), while the theoretical prediction is obtained for a noiseless case, we compute the power spectrum of the noise and then subtract it from the power spectrum of the map. As rst approximation, we estimate the noise of the map as its mean. An example of this comparison for one realisation is shown in Figure 6.6. At high a noisy behaviour can be noticed. This noise probably is intrinsic in the data as the shot noise has already been subtracted as described before. The scales where this happens are in any case excluded from the analysis, as we include multipoles up to max = 780 as explained in Section 6.5.2. Moreover, when averaging over the di erent l.o.s. for each cosmology (which is what we do when performing the analysis) the noisy behaviour disappears.

Preliminary results

This section presents some preliminary results of this project. The nal goal is to obtain cosmological forecasts for combined weak lensing and photometric galaxy clustering for both second and higher order statistics. Here some preliminary outputs are shown, some of them being useful to validate some already implemented extensions of the pipeline presented in Section 5.2 and others to understand what are the next steps to follow to extend it to reach the aim of this project. I would like to stress that these outputs are still preliminary and do not represent nal quantitative results. At this stage of the project, they should be read as rst experiments to understand if some approximations we make have to be modi ed and how. To get the parameter constraints for all summary statistics we follow the same procedure as in Chapter 4 and Chapter 5. To get the model predictions and build the likelihood we employ Gaussian Processes interpolation, in the same fashion we did in Section 4.1.7. We assume at priors, in particular spanning the following the ranges for each parameter: Ω m = [0.001, 0.6], h = [0.5, 0.9], w 0 = [-2, -0.3], σ 8 = [0.5, 1.3] and when we consider S 8 instead, S 8 = [0.1, 1.3]. To compute the covariance matrix we employ the SLICS suite of simulations, that is obtained at a xed cosmology (Ω m = 0.2905, h = 0.6898, w 0 = -1, σ 8 = 0.826, Ω c = 0.2432, Ω Λ = 0.7095). The nal SLICS suites for KiDS-1000 will provide 800 independent realisations. However, at the moment these rst tests were run, we found some missing l.o.s., therefore we used the 715 realisations that are available at the moment. We scale the covariance matrix with the pre-factor f map /f sky = 100/1000, since the size of the maps corresponds to 10 × 10 deg 2 and the catalogue is built for KiDS-1000 covering 1000 deg 2 . We then perform MCMC with thee python package emcee, described in Section 1.5.3.

Weak lensing (WL) power spectrum

We perform a cosmological parameter inference analysis using the power spectrum computed as described in Section 6.3.2. We use 33 bins in the range = [180,2040]. We provide forecasts for the matter density parameter Ω m , the clustering amplitude σ 8 , their combination S 8 ≡ σ 8 Ω m /0.3, the dark energy equation of state parameter w 0 and the reduced Hubble parameter h. We nd that the results obtained with our pipeline are consistent with the cosmic shear analysis of the KiDS fourth data release (KiDS-1000) (Asgari, Marika et al. 2021). In particular, in Figure 6.7, we compare our result with the marginal posterior from mock data for the second order statistics employed by Asgari, Marika et al. 2021 for

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the parameter space plane (Ω m , σ 8 ). We consider this step as an important validation of the rst extension of the pipeline. In particular, the step that goes from the catalogue to the convergence maps was not present in the rst version of the pipeline as it was used for its rst applications (Chapter 4, Chapter 5). After this extension its input is directly the galaxy catalogue, and it provides the possibility to work with shear maps as well as with convergence maps obtained with a KS inversion.

Galaxy clustering photometric (GCph) power spectrum

As a second test, we want to provide forecasts for the matter density parameter Ω m , the clustering amplitude σ 8 , their combination S 8 ≡ σ 8 Ω m /0.3, the dark energy equation of state parameter w 0 and the reduced Hubble parameter h. We perform a cosmological parameter inference analysis measuring the power spectrum on our galaxy maps. The value for the galaxy bias is xed in the simulations at b g = 1.72. For the angular scales, we employ a conservative cut, by removing the contributions of multipoles > 780 from the analysis, following the pessimistic setting of Euclid Collaboration et al. 2020b. Speci cally, We use 14 bin in the range = [120,780]. We show the result in the left panel of Figure 6.8. Employing photometric galaxy clustering instead of weak lensing brings very tight constraints in this setting: the constraining power of GCph seems to be actually more constraining than expected. Whether the reason for this lies in some limitations of the simulations or in some approximations we have made, as we estimate the noise for instance, is currently under investigation. What we know for sure is that the clustering varies with the galaxy bias which is xed in the simulations, but largely unknown in the data. That would in ate the error contours signi cantly in more realistic conditions. On the right panel of Figure 6.8 we also show the constraints on the (Ω m , S 8 ) plane for both probes and their combination.

As expected, the combination brings an improvement on the constraining power for both probes, and the GCph contribution is dominant. However, it is important to stress that this is a rst test of the pipeline on a new probe, as well as the fact that the bias is actually largely unknown in the data, so we expect that once applied to real data, the error contours for GCph will in ate signi cantly. At this stage, we do not treat these outcomes as nal quantitative results but more as rst experiments to understand how to properly include this new probe.

Photometric Galaxy Clustering starlet peaks

The next extension involves the inclusion of higher order statistics computed on the galaxy maps. We therefore attempt a rst experiment by applying the starlet peak counts framework described in Section 4.1. We express the maps in terms of their signal to noise ratio S/N and built them as depicted in Section 6.4.1. The size of the lters for the high order statistics are chosen such that when comparing the results for second order statistics (GCph power spectrum) the scales considered roughly match to have a fair comparison. For example, since in the GCph power spectrum analysis the considered range in scale is = [120, 780], for maps with resolution 1 arcmin, we need to consider that for j max = 6 scales, the larger scale gives θ = 64 arcmin, corresponding to = 54 and the nest scale θ = 2 arcmin, corresponds to = 1719.

Hence, we only include scales corresponding to θ = [4,6,16,32] arcmin leading to = [859,430,215,107]. The scales considered in the analysis and corresponding galaxy maps are shown in Figure 6.9. We then compute the peak counts for the di erent starlet scales and perform the MCMC analysis. We show the corresponding 95% con dence contours in Figure 6.10. Qualitatively we see that for the majority of the parameters, except h and Ω m the starlet peaks for GCph tend to provide tighter contours. However, we also notice that the constraints on w 0 tend to induce a slight shift. The next step is then to investigate whether this is due to some approximation in the analysis such as the way we estimate the noise or in some limits of the simulations as the xed galaxy bias or to other origins. , w 0 and h for ve redshift bins with weak lensing power spectrum (blue contours) and four redshift bins for photometric galaxy clustering power spectrum (violet contours). As mentioned above, the constraining power of GCph seems to be actually more constraining than expected and we expect the error contours to signi cantly in ate in more realistic conditions, starting with a varying galaxy bias. Right panel: joint forecast for Ω m and S 8 ≡ σ 8 Ω m /0.3 for weak lensing power spectrum (blue contours) and galaxy clustering photometric power spectrum (violet contours) and their combination (red contours). We stress also here that the constraining power of GCph seems to more constraining than expected and that the reason for this (if it is due to the simulations or to some approximations we have made) is currently under investigation. What we know for sure is that the value for the galaxy bias is xed at b g = 1.72 and as the bias is actually largely unknown in the data, we expect that once applied to real data, the error contours for GCph will in ate signi cantly.

Ongoing work and future prospects

The results shown in this section are very preliminary as the work on this project is currently ongoing. The goal of this rst part of the project mainly consists in understanding in which aspects the pipeline presented in Section 5.2 needs to be extended to be able to deal with a di erent parameter space and a di erent cosmological probe. Concerning the rst point, using a di erent set of simulations (the cosmo-SLICS instead of the MassiveNus) allowed us to extend the analysis to a 4-dimensional parameter space, and most importantly to add the initial step of deriving the convergence maps from the catalogue so that now, the initial input is the galaxy catalogue. Concerning the second point, by employing the mock lens catalogue for the KiDS-1000 LRGs set, allowed us to start including in the analysis photometric galaxy clustering as a new probe. It is indeed known that probe combination is crucial to increase the gain in information as di erent probes are sensitive to di erent dynamical ranges of the underlying structure formation and to di erent redshifts (Harnois-Déraps et al. 2018). For example, weak lensing and galaxy clustering depend on two independent combinations of the gravitational potentials (the growth of structure and de ection of light) and can therefore help in testing gravity (Amendola et al. 2018;Harnois-Déraps et al. 2018;Euclid Collaboration et al. 2020b;Heymans et al. 2020). Furthermore, as detailed throughout this thesis, statistics of order higher than the second enable to include the non-Gaussian information of physical elds present at non-linear scales and as a consequence to improve the constraining power. Therefore, we are willing to investigate what is the impact of combining probes for statistics of order higher than the second on the constraints, with a particular interest in testing our starlet 1 -norm that has been proven potentially powerful in the context of weak lensing (Section 5.1.2). As one can see from these rst preliminary results, there are many aspects that need to be clari ed and are the focus of my current work. In practice, the next steps involve to better understand why the contours for the GCph power spectrum are more constraining than expected and then to understand the behaviour of starlet peaks on the galaxy maps by quantifying the impact of our approximations (concerning the noise estimation and the xed bias). Once these steps are completed, the goal is to compute the starlet 1 -norm for GCph, WL and study the impact of their combination on cosmological parameter as well as the combination of each single probe with the corresponding second order statistics (the angular power spectra in this case). Depending on the outcome of this analysis in an ideal setting, the natural follow-up will be then to apply this on real data. I would like to underline that, despite the exploratory nature of the results presented so far, the use of a new suite of simulations .10: This plot represents a preliminary test for the extension of our method to GCph second order statistics and starlet peaks. As discussed above, the contours for the power spectrum seem to be more constraining than expected and we are currently investigating if this might be due to some limitations of the employed simulations or to some approximations we made throughout the analysis. I illustrate here a rst attempt to get cosmological forecasts for Ω m , σ 8 , w 0 and h for four redshift bins with galaxy clustering photometric power spectrum (violet contours) and starlet peaks (green contours). The scales considered for the starlet decomposition are chosen to roughly match the range in for the power spectrum. We notice that the constraints for w 0 tend to slightly shift the contours. It is to be investigated weather this is caused by some assumption made in the analysis as noise estimation or the xed galaxy bias. The Canada-France Imaging Survey (CFIS), is a u-and r-band survey covering 5000 deg 2 in the Northern hemisphere. Its high resolution and depth make it a great survey candidates for weak lensing science to date. CFIS is also a critical component of the data needed for photometric redshifts for Euclid. A rst attempt of applying the pipeline to real data from the CFIS survey and preliminary results are presented. This Chapter presents the procedure and some preliminary results of an ongoing work (Ajani et al. in prep.[b]). 149 F

CFIS survey

The Canada-France Imaging Survey (CFIS), is a u-and r-band survey covering 5000 deg 2 in the Northern hemisphere. Since 2017 it has covered 3000 deg 2 , and will cover 4800 deg 2 in 2025. Its high resolution and depth make it a great survey candidates for weak lensing science to date. CFIS is also a critical component of the data needed for photometric redshifts for Euclid. In particular, the survey parameters for CFIS are also chosen to provide essential, complementary, sky coverage to Euclid. The optimal area for CFIS-r to best complement Euclid is given by the the northern 5000 deg 2 of the extragalactic sky. Figure 7.1 illustrates the sky coverage for CFIS relative to some complementary surveys. CFIS-u requires extensive sky coverage away from the Galactic plane, whereas the primary science drivers for CFIS-r require SDSS spectroscopy. An area as large the one of the SDSS region as possible is hence covered, leading to a n gal = 7 gal/arcmin 2 for lensing. This value for the galaxy density is similar to the ones of RCS2, KiDS and DES, but over a much larger area than the rst two, and a better delivered image quality than the latter (CFIS coll. 2017).

Analysis setting

This Section describes the setting chosen for the analysis. We employ as data convergence maps built from CFIS catalogue as described in the following and as simulations we use the MassiveNus simulations introduced in Section 4.1.2 that we correct including baryonic e ects following Coulton et al. 2020a. We compute peak counts as summary statistics, we assume a Gaussian likelihood, a cosmology independent covariance, and perform MCMC using the emcee python package.

Data

As starting point of our analysis we take a convergence map built by Guinot 2020 through the pipeline ShapePipe 2) and we choose to match the resolution of the simulations. Given the noiseless κ maps from the simulations, we need to have an estimate of the noise to mimic the CFIS survey. For this, we need a redshift estimate for the source galaxies, or at least the galaxy number density distribution as a function of redshift, n(z), for the lensing sample. We employ the n(z) built by Guinot 2020, shown in Figure 7.4. As of today, the galaxy number density distribution is not available based on CFIS data only, since only data on the r-band are available right now1 . For this reason, the n(z) is derived by matching CFIS galaxies with the ones detected in the W3 region of CFHTLens (Erben et al. 2013), that corresponds to P3. Around 700,000 galaxies have been matched with CFHTLenS by Guinot 2020. Then the n(z) is reconstructed from the photometric redshift measurements in CFHTLenS, using the best-t value z B . This method provides only an approximation, but it gives a rst estimate of the galaxy distribution.

The corresponding mean galaxy number density used in the analysis is n gal = 7 gal/arcmin 2 . The shape noise is σ = 0.44.

Methodology

We perform a likelihood analysis by applying the pipeline I developed, introduced in Section 5.2. In this rst attempt, we adapt the data to the available MassiveNus simulations, in terms of resolution and size of the maps. Ideally, one would do the contrary, in order to exploit the full information contained in the data. However, in the interest of performing a rst experiment, and given the huge amount of maps from the simulations (100 cosmologies × 10000 realisations = 10 6 maps), we start by mainly adapting the data to the simulations, to get an idea of what some preliminary output would be, in this rst application on real data. In order to do so, we cut 13 squared patches of size 512 × 512 from the original P3 convergence map of Figure 7.3. In this way, the maps from which we compute the peak counts that are use to build the theoretical predictions, and the maps from which we compute the peak counts for the data, have the same resolution (∼ 0.4 arcmin) and the same size (512 × 512 pixels 2 ). Concerning the redshift for the sources, in chronological order we have acted as follows. As the redshift planes provided by the MassiveNus simulations are xed at z s = [0.5, 1.0, 1.5, 2.0, 2.5], we looked at the n(z) for the data, shown in Figure 7.4. We noticed that the value z = 0.5 (dashed gray line in Figure 7.4) is actually very close to where the n(z) peaks and decided to perform the analysis employing the convergence maps from the simulations at z s = 0.5. We have in a rst moment performed the entire analysis in this setting, adding Gaussian noise with standard deviation F σ n = σ 2n gal A pix , (7.1)

to the noiseless simulated maps, with σ and n gal being respectively the shape noise and the mean galaxy number density de ned in the previous section and with A pix = 0.16 arcmin 2 being the pixel area. At a later time, we realised we could try to adapt the source redshift of the simulations to the actual value at which the n(z) peaks, which is z = 0.65. Therefore we have used the presence of convergence maps at z s = 0.5 and z s = 1.0 from the simulations and performed an interpolation for all 100 cosmologies and 10000 realisations and derived e ective convergence maps at z s = 0.65. This is done by considering that, as shown in Equation 2.21 the convergence eld as a function of θ is given by κ(θ) = and that given that the available redshifts in our case are basically two Dirac delta functions, we can approximate Equation 7.2 for a discrete case to nd our e ective convergence map at z s = 0.65 as κ eff zs=0.65 = κ zs=0.5 λ + κ zs=1.0 (1λ),

where λ is a parameter that weights the convergence maps at a given redshift so that their combination leads to the e ective convergence map at the desired redshift. The parameter λ is then obtained by asking that the mean redshift z is equal to 0.65, namely by asking that z = n(z)zdz (7.4)

= [δ(z -0.5)λ + δ(z -1)(1λ)]zdz (7.5) = 0.5λ + 1(1λ) = 0.65, (7.6) that leads to a value of λ = 0.7. By performing this interpolation, we obtain the set of e ective κ maps for the simulation at z s = 0.65. We show in Figure 7.5 one realisation example of the output of the interpolation. The e ective convergence map κ eff zs=0.65 is in the central panel. Qualitatively, one can recognise by eye the same structures in the same positions as z s = 0.5 and z s = 1.0. In the bottom panel we compare the angular power spectra for the κ maps at z s = 0.5 and z s = 1.0 with our e ective κ eff zs=0.65 at z s = 0.65 as well as the peak counts. Given the di erence between the summary statistics (in particular for the peak counts) at z s = 0.5 and κ eff zs=0.65 , we expect this di erence to be propagated in the results of the inference analysis in the two settings. To further improve this interpolation, by trying for example to match the entire redshift distribution, we would need simulated convergence maps at more intermediate redshift steps, which are not provided by the employed simulations. As a more detailed follow-up study, it could be interesting to see how the constraints behave with simulations at z s = 0.65±∆z. For this study however, we would again need to use other simulations. 7.3. Bottom panel: angular power spectra (left), peak counts (right) for the di erent redshifts. Given the di erence between the summary statistics (in particular for the peak counts) at z s = 0.5 and κ eff zs=0.65 , we expect this di erence to be propagated in the results of the inference analysis in the two settings.

Peak counts

As a rst result for the constraints on the parameters Ω m and σ 8 has already been obtained by Guinot 2020 for the 3 × 2-point analysis, we focus on peak counts as summary statistics for our analysis as a rst experiment to apply the pipeline. For both settings at z s = 0.5 and z s = 0.65, we consider 30 linearly spaced bins in S/N in the range [-2, 6]. The signal to noise ratio is de ned as in the Gaussian lter case of Section 4.1, namely as the noisy κ map smoothed with Gaussian kernel, over the standard deviation of the noise. The size of the Gaussian kernel is 2 arcmin. To build the theoretical predictions we compute the F peak counts for each realisation and each of the 100 cosmologies from the MassiveNus simulations and then interpolate through the Gaussian Process Regression (GPR) presented in Section 5.2. For the data, we compute the peak counts for the 13 patches extracted from the P3 patch and the average over them before inserting them in the likelihood. We then rescale the covariance matrix to take into account for the fact that we average over 13 patches. The peak counts are computed on a HPC cluster in parallel with the package joblib for each of the 10000 realisations and each of the 100 cosmologies.

Correction for baryonic e ects

As we are dealing with real data but modelling the theoretical predictions with N-body simulations, we should take into account a variety of systematics to consider our analysis robust. For this rst experiment we start by including in the analysis a correction for baryonic e ects. As described in Section 2.2.3, baryonic e ects (which are not described by gravity-only N-body simulations) have a signi cant impact on current and future weak lensing observations. Weiss et al. 2019 have shown that the strength of the baryon suppression on peak counts is strongly dependent on the scale of the Gaussian smoothing applied to the convergences maps. Speci cally, they found that for a DES-like con guration, a smoothing scale of θ ker ∼ 8 arcmin or larger is su cient to wash-out the baryonic e ects so that the suppression signal becomes smaller than the statistical error and for a Euclid-like setting that a smoothing of at least θ ker ∼ 16 arcmin is required to reduce the baryon suppression signal enough so that it falls below the statistical error. If one considers smaller smoothing scales, a proper modelling of baryonic e ects is necessary. We apply a smoothing scale of θ ker ∼ 2 arcmin to our convergence maps. Hence, to model the baryonic e ects impact, as we get the theoretical prediction with the MassiveNus simulations, we apply corrections to the peak counts in our analysis following the results of Coulton et al. 2020a. To include baryonic e ects, they consider a parameter called A baryon , which linearly interpolates the fractional e ect of baryons between no baryonic e ects (A baryon = 0) to the BAHAMAS high-AGN model A baryon = 3, with A baryon = 2 for the ducial model, and A baryon = 1 for the low-AGN model. We use the peak counts for the three models (low-AGN, ducial, high-AGN), kindly provided by the authors of Coulton et al. 2020a and compute the ratio between each baryon feedback model and the peak counts obtained by the DM-only simulations. The peak counts sets they provided are computed for 99 linearly spaced bins, in the range S/N = [-6, 16]. Since for our likelihood analysis we want to use 30 bins in the range S/N = [-2, 6], we perform a 1-D interpolation with the python package scipy, to extract the peak counts in our S/N range of interest for the three baryon feedback models. We show in Figure 7.6 the fractional di erence (left panel) and the ratio with respect to the dark matter only (right panel) peak counts due to di erent feedback strengths, from the arrays provided by the authors of Coulton et al. 2020a. The correction input that we use to get our peak counts accounting for the di erent baryonic feedback corresponds to the right panel. All three baryon feedback models imprint a deviation from the DM only distribution, that increases with the strength of the baryonic e ects. The counts for S/N [-1, 3] are boosted, while being suppressed for S/N ≥ 3. We run the analysis in the following settings:

1) for source redshift z s = 0.5: we run the likelihood analysis using as summary statistics a) DM-only peak counts, b) Low-AGN peak counts, c) Fiducial (A baryon = 2) peak counts, d) High-AGN peak counts using as data the peak counts computed on the 13 squared patches extracted from CFIS P3 patch, shown in Figure 7.3.

2) For source redshift z s = 0.65: we run the likelihood analysis using as summary statistics a) DM-only peak counts, b) Low-AGN peak counts, c) Fiducial (A baryon = 2) peak counts, d) High-AGN peak counts using as data the peak counts computed on the 13 squared patches extracted from CFIS P3 patch, shown in Figure 7.3

First experiment and discussion

We run a rst experiment to get constraints on the sum of neutrino masses M ν , the matter density parameter Ω m and the amplitude of the primordial power spectrum A s . We take into account the baryonic feedback for three di erent models, labeled as Low-AGN, ducial and High-AGN. We assume a Gaussian likelihood where for the model µ(θ) we employ the Gaussian Process Regressor trained on the 100 cosmological models from the MassiveNus simulations, for the data d the peak counts computed on the 13 patches of the CFIS P3 patch, and the covariance is computed employing the massless cosmology from the simulations and re-scaled with respect to the area of the 13 patches. We explore parameter space with MCMC using the python package emcee, with 120 walkers and 6500 steps. We verify the convergence of our chains with Gelman-Rubin (Gelman and Rubin 1992) and Geweke (Geweke 1991) statistics. We start by looking at the di erence induced when employing maps at z s = 0.65 and maps at z s = 0.5, namely the constraints shown in Figure 7.7. In particular, for both the dark matter only simulations (left panel) and the ducial baryonic e ects model (right panel), when considering z s = 0.65 (dashed lines) there is a shift for the matter density parameter Ω m and the amplitude of the primordial power spectrum A s , towards smaller values. As already mention, the reason behind the change in source redshift was due to the fact that the MassiveNus simulations exist at xed redshift z s = 0.5 and z s = 1.0 and not at z s = 0.65, for which we have derived some e ective convergence maps to try to better match the peak in n(z), shown in Figure 7.4. After ascertaining the di erence between the two redshift settings, we consider the setting at z s = 0.65 the most reliable for our analysis, given the above discussion on the distribution of the sources. We then explore the impact of the di erent baryonic correction on the constraints. Speci cally, by looking at Figure 7.8, we notice that the presence of baryonic e ects induces a shift in the (Ω m , A s ) plane towards higher values of both Ω m and A s and this e ect increases as the parameter A baryon increases. The e ect on the sum of neutrino masses M ν is harder to detect, from the joint contours, while from the marginalised constraints in Figure 7.9, the Low-AGN and Fiducial correction seem to increase its value and the High-AGN to slightly decrease it, with respect to the case without correction. We show in Table 7.1 the maximum likelihood values for the case without baryonic correction (DM only), and the three baryonic models for the setting at z s = 0.65. In summary, we see that in all cases, the impact of taking into account or not the baryonic feedback is pretty important. We notice that Ω m and A s are shifted to higher value and M ν becomes constrained narrower than the prior. The three baryonic corrections yield to constraints which are pretty close to each other. Thus, even if we do not know how important is the baryonic feedback on the data, we can conclude that the speci c avour of baryonic feedback (if Low-AGN, Fiducial or High-AGN) is less important than the di erence between baryonic feedback and pure dark-matter models. With the baryonic feedback there is less peaks in the model. Hence, this has to be compensated with a higher Ω m because the data does not change.

Ongoing work and future prospects

The aim of this chapter was to show a rst experiment of application of the pipeline presented in Chapter 4 and Chapter 5 using peak counts as summary statistics on real data. We are aware that the preliminary results shown so far are very limited by the fact that the only systematics that we are including are baryonic e ects in the simulations, while other crucial systematics such as intrinsic alignment, multiplicative shear bias, mean redshift bias, boost factor (previously discussed in Section 2.2.3) have been neglected. Indeed, an ongoing e ort with the collaborators of this project is ongoing, in trying to model each one of these e ects rst independently and then combined with the baryonic e ects. For instance, a further step has been undertaken to extend this work by including multiplicative shear bias using metacalibration (Hu and Mandelbaum 2017) with a residual bias ∆m and explore its impact on the constraints, independently from and combined with baryonic e ects. A rst result of this second step is shown in Figure 7.10: rst, the two systematics e ects (multiplicative shear bias and baryonic e ects) are considered separately to have an idea of how the single e ects act with respect to the constraints where no correction is applied to the simulations. This is shown in the left panel of Figure 7.10. We can see how the baryonic correction tends to shift Ω m and A s to higher values whereas the redshift and metacalilbration bias tends to shift them to lower values. Then the two are combined, to see how they act together with respect to the constraints where no correction is applied. On the right panel of Figure 7.10, we can see the constraints obtained for data from the local calibration at 1 deg 2 with ∆m = 0.007 (Guinot et al. in prep.) for the simulations at z s = 0.65 corrected by the baryonic feedback with the ducial model A baryon = 2. We would like to stress that as of today the exact value for ∆m is still unknown for CFIS. Therefore we have used a preliminary value for it (Guinot et al. in prep.) as our best guess. Next steps will be then to include in the analysis further systematics that have been neglected, with the nal goal of quantifying the impact of each of them and of all of them combined on the nal constraints and quantify the robustness and constraining power of peak counts in this context. This work is ongoing, while a straightforward future work that will be interesting to carry out is to test the starlet 1 -norm, introduced in Section 5.1.5, that has been proven powerful in an ideal setting, but still has not been tested on real data. The main goal of this thesis has been the investigation and development of statistical methods to extract information from cosmological data coming from next generation galaxy surveys like Euclid. To address this topic, I rst focused on a study where we investigated the impact on the parameter constraints of multi-scale ltering techniques at the level of the noisy data. Comparing the performance of a wavelet starlet lter and a concatenation of Gaussian lters with the state of the art techniques when using weak lensing peak counts as summary statistics, we found that in an ideal setting, both multi-scale techniques signi cantly outperform the state of the art. For instance, for the set of parameters M ν , Ω m and A s , we found that starlet peak counts provide an improvement of respectively 63%, 40% and 72% with respect to the standard power spectrum. As second result of this rst study, we found that the power spectrum does not seem to bring any additional information to the constraints when multi-scale peak counts are combined with the power spectrum. This is interesting and quite new outcome, as usually when second order statistics and higher order statistics are combined, an improvement on the constraining power is typically found for single-scale statistics, as each statistics bring complementary information to the other. Lastly, when exploring the e ect of the lter choice on the di erent steps of the analysis, we noticed that the covariance matrix for peak counts when using a starlet lter tends to be more diagonal than the one obtained when using multi-Gaussian lter. Intrigued by this feature, we run the same analysis using only the diagonal elements of the covariance matrix, and basically recovered the same constraints as when the full covariance is used. This can be convenient for situations (actually not rare in cosmology) where the covariance matrix is di cult to invert, or there is interest in speeding up the analysis. In a broader sense, this analysis proved some advantages of the starlet transform to optimise the information gain in cosmological analysis, con rmed the potential of higher order statistics in improving the constraining power and showed how the combination of high order statistics in a multi-scale framework can bring a signi cant improvement in the constraining power of future surveys like Euclid. The results of this rst work also led to the construction of a pipeline skeleton for multi-scale peak counts, having as initial input weak lensing convergence maps and as output constraints on cosmological parameters in an ideal simulations-based setting. Then, trying to preserve the advantages of a multi-scale approach in a higher order statistics framework, we wondered if our choice of peak counts as higher order statistics was limiting in some sense. Indeed, as discussed in Section 2.3, a vast literature of di erent type of higher order statistics is present and it was proven that some combination of these statistics can improve even more the constraining power. For example, using the same sets of simulations that we used, Coulton et al. 2020b found that the combination of peak counts (local maxima in convergence maps) and local minima counts, further tighten the constraints. Inspired by this, we wondered: what happens if, instead of selecting only the local minima and local maxima in the map, we consider the information encoded in all pixels, while preserving the starlet multi-scale approach that has been proven advantageous? This question gave origin to my second study, where we have introduced a new multi-scale higher order summary statistics: the starlet 1 -norm. This statistics is built in such a way that starting from a starlet decomposition of the input data, it accounts for the information present at di erent wavelet scales for all pixels in the map. Hence, it automatically contains the information encoded in both peaks and minima. This is con rmed by our outcome on its performance in constraining parameters: the starlet 1 -norm outperforms the state of the art combination of peaks and voids and it avoids at the same time the problem of de ning what is a void (topic on which there is still no unanimous agreement). This second work also led to an extension of the public CosmoStat python package lenspack, that now contains the implementation of this new statistics to perform cosmological analysis.

The practical contributions of these rst two studies can be summarised has follows:

1) a rst paper (Ajani et al. 2020) quantifying the impact of di erent multi-scale ltering on weak lensing peak counts and between di erent statistics 2) the construction of a pipeline skeleton for multi-scale peak counts, having as initial input weak lensing convergence maps and as output constraints on cosmological parameters in an ideal simulations-based setting

3) the introduction of a new multi-scale higher order statistics, de ned in a second paper (Ajani et al. 2021) 4) an extension of the public CosmoStat python package lenspack 5) tomographic MCMC forecasts mimicking a Euclid-like shape noise for the parameters M ν , Ω m , A s obtained with many di erent statistics such as the power spectrum, single-scale peak counts, multi-scale Gaussian and starlet peaks, voids and starlet These results show that the multi-scale higher order statistics approach we developed represents a very promising tool to reach high levels of precision in constraining cosmological parameters. However, to serve as a robust probe for precision cosmology, the statistical power alone is not su cient and it is very important to guarantee robustness with respect to systematic errors. Furthermore, keeping in mind that the nal aim is to maximise the information coming from surveys like Euclid, we also want to apply these ndings not only to weak lensing, but also to galaxy clustering to fully exploit the information gain deriving from their combination. These re ections led to the ongoing work and preliminary results of Chapter 6 and Chapter 7 where we aim to extend our method to photometric galaxy clustering alone and combined with weak lensing and to real data from the Canada France Imaging Survey (CFIS). This, from a practical point of view, translates in understanding in which parts and in which ways my multi-scale higher order statistics pipeline needs to be extended. First of all, for the extension to photometric galaxy clustering, I had to use different simulations than the MassiveNus on which the pipeline was initially built. This change of simulations, naturally led to two extensions simultaneously. The rst concerns the di erent parameter space, both in terms of speci c parameters (w 0 and h were actually xed in the MassiveNus) and in terms of dimensionality (the parameters space for the cosmo-SLICS is 4-dimensional). The second extension is relative to the input, that is now given by the galaxy catalogues directly instead of the convergence maps. What one deals with in real data analysis is actually a galaxy catalogue, and now the pipeline directly takes as input the catalogue, builds a pixelised shear maps and with Kaiser-Squires inversion gets the convergence maps. Lastly, given a lens catalogue instead of a source one, it is now possible to build galaxy density maps with the idea of performing a photometric galaxy clustering analysis. Running exploratory experiments, we obtained some preliminary results for combined second order statistics (the power spectrum) for both weak lensing and galaxy clustering. As expected, we nd that the combination of the two improves the constraints. However, for photometric galaxy clustering, we are applying some approximations in how we estimate the noise and for the galaxy bias. The investigation on these points is ongoing and we expect the contours for GCph to in ate once using real data, where the galaxy bias is largely unknown. We also apply starlet peaks on the galaxy maps built from the lenses catalogue and nd improvement for the majority of the parameters in the constraining power. However, a slight shift in the w 0 constraints arises and we are currently trying to understand whether this is caused by approximations made in the procedure or if its origin has a di erent nature. Lastly, a natural test for the robustness of the pipeline is to adapt our procedure to real data. Fortunately, about 1600 deg 2 of the CFIS data have been already pre-processed and this has enabled a rst test of the pipeline on a part of these avaiable data. In Chapter 7 some preliminary work in this sense is outlined. In its rst implementation, the pipeline didn't include any systematics e ects. We therefore pro t of the availability of CFIS data and start including the systematics one by one, to quantify their effects rst separately and eventually jointly. We start with the inclusion of baryonic e ects, and provide a rst qualitative description of their impact in three di erent scenarios for a peak counts analysis. Now, with the nal aim of including as many systematics as possible, we are focusing on a second systematics, namely the multiplicative shear bias. To calibrate the shear, we use metacalibration and test the impact of local calibration. Indeed, as the multiplicative shear bias depends on the local density (Hoekstra, Viola, and Herbonnet 2017), a local calibration can be more precise and consequently lead to a smaller multiplicative bias. However, as metacalibration is not perfect, there is a residual bias ∆m that needs to be considered and whose value is a current object of research within the CFIS collaboration. The results obtained so far on CFIS data are still very preliminary. However, we start having an idea of some qualitative behaviour, such as that with these data and simulations, the more constrained of the parameters seems to be Ω m . We also nd that local calibration with a ∆m < 0 shifts this parameter to higher values compared to baryonic corrections.

On the other hand, we can conclude that the speci c avour of baryonic feedback is not so relevant, with respect to the di erence between baryonic feedback and pure dark-matter models. Concerning the local calibration, it is sure that a local one is preferred over a global one, as long as the error and standard deviation are small, because it will account for local e ects of the catalogue. Finally, we need a more precise value of ∆m as changes in this value can lead to large di erences on the parameters. Some questions presented above still need an answer, as for example, what the impact of combining multi-scale high order statistics on combined probe is, or what the impact of many systematics e ects that were neglected is. However, these unsolved points are already structured in ongoing projects whose outcomes hopefully will provide us with the answers that we are looking for. Besides the above discussed projects, many other interesting questions that are still pending can be addressed in this context. For example, by paying attention to some assumptions made throughout this work, we could focus on the fact that we derive our input convergence maps with a standard Kaiser-Squires inversion. However, as shown in Section 2.1.3 and Section 3.6, new powerful approaches as DeepMass (Je rey et al. 2018) or MCALens (Starck et al. 2021) that have overtaken the drawbacks of standard inversion techniques are available to be used. It is then worth exploring how the combination of these mass mapping approaches with multi-scale higher order statistics can improve the analysis. On a di erent level, many other questions concerning the considered parameters can arise. For instance, using the MassiveNus simulations, we provided forecasts and constraints on the sum of neutrino masses M ν . However, strong observational degeneracies between the e ects of modi ed f (R) gravity theories and massive neutrinos have been highlighted on structure formation processes over a wide range of scales and redshifts, both in the linear and in the non-linear regime (Baldi et al. 2014). These degeneracies can be broken with the use of high order statistics as was proved by Peel, Pettorino, Giocoli, et al. 2018. However, what the optimal statistics to do so is, still remains an open question, and testing whether the starlet 1 -norm could be a good candidate for this task could be interesting and in practice doable using the DUSTGRAIN-path nder simulations (Giocoli, Baldi, and Moscardini 2018) that include f (R) models. This work is also currently in progress.

A large number of open questions are of course still present. In the context of this work, a summary of what needs to be addressed can be summed up in the fact that next generation cosmological surveys such as Euclid, DESI, the Vera Rubin Observatory Legacy Survey of Space and Time, the Nancy Grace Roman Space Telescope, will provide us with a huge volume of data, characterised by a high resolution up to the very small scales. The main goal is then to fully exploit these small scales up to k ∼ 10h Mpc -1 , as they contain the cosmological information needed to try to answers the big questions about the dark components of the universe outlined at the beginning. A huge collective e ort in this context is ongoing. For example, the current work carried on by the Interscience Taskforce for Likelihood development (IST:L) in Euclid to which I had the chance to contribute, will provide the tools that we will need to perform a full cosmological analysis including weak lensing, galaxy clustering and their cross-correlation. Concerning the higher order statistics, it is clear that given the large amount of di erent statistics already available now, one crucial challenge at this stage is also to compare what it is already available and nd the optimal statistics in terms of constraining power and robustness to systematics. In this sense, a collective e ort is as well ongoing within the higher order weak lensing statistics (HOWLS) work package within the Euclid collaboration. Besides the preparation for future surveys, this is already a very exciting time to pursue these investigations, as data from current surveys as KiDS and DES are already available and have very recently provided the most precise constraints on the ΛCDM and wCDM models to date from weak lensing probes (DES Collaboration, Abbott, et al. 2021). Taylor, Peter L., Francis Bernardeau, andEric Hu (2021)

  . . . . . . . . . . . . . . . . . . . . .

  . . . . . . . . . . . . . . . . . . . . . . 71 3.1 The periodic function sin(x) is not sparse in real space (left image) but it can be sparsi ed by changing the data representation and work instead in Fourier space (right image). . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2 Illustration of the time-frequency diagram for Time series analysis (top left panel), Fourier transform (top right panel), Short Time Fourier transform (STFT, bottom left panel), the Wavelet transform (bottom right panel).

  k < +∞ which implies that ψ(x)dx = 0 and it has compact support in [-2, 2] × [-2, 2]. Its shape emphasises round features, making it very e cient when dealing with peak counts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.6 COSMOS data: Top, galaxies count map (left) and Wiener map (right) smoothed with a Gaussian having a Full Width at Half Maximum of 2.4 arcmin. Bottom, GLIMPSE(left) and MCAlens(right). Images and caption from Starck et al. 2021. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Parameter space of the 100 cosmological models available from the Massiv-eNus simulations. The red dot shows the ducial model corresponding to { M ν , Ω m , 10 9 A s } = { 0.1, 0.3, 2.1 }. . . . . . . . . . . . . . . . . . . . . . 93 4.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Marginalized constraints on each parameter for forecasts showing the 2.5 and 97.5 percentiles with respect to the ducial model. These marginalised constraints refer to a tomographic setting with z = [0.5, 1.0, 1.5, 2.0] with the ducial model set at [M ν , Ω m , 10 9 A s ] = [0.1, 0.3, 2.1] corresponding to the di erent observables employed within the likelihood analysis. The values are listed in

  the xed galaxy bias. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.11 Extension of the initial input of the pipeline illustrated in Figure 5.5. . . . . 148 7.1 Image from https://www.cfht.hawaii.edu/Science/CFIS/ cfissurvey.html, showing the equatorial projection of the entire sky for CFIS-r (red outline) and CFIS-u (blue outline) with respect to other surveys (listed in the top right caption). The areas that will be observed from the A and B semesters are indicated at the top. Image credit: T. Dwelly. . . 150 7.2 Process patches of the CFIS dataset provided by Guinot 2020. In this analysis, we employ the P3 (red patch) corresponding to a size of 249 deg 2 . . . . 7.3 Illustration of the E-mode convergence map constructed for the CFIS P3 patch, corresponding to 249 deg 2 . The data shown in this image have been provided by A. Guinot (Guinot 2020). The map is built through KS inversion, setting a pixel size of ∼ 0.4 arcmin, to match the resolution of the κ maps from the MassiveNus simulations. . . . . . . . . . . . . . . . . . . . . . 7.4 Galaxy number density distribution (blue line) obtained by matching the galaxies with the CFHTLenS-W3 patch by Guinot 2020. . . . . . . . . . . . 7.5 Illustration of one realisation example result of our interpolation to get the e ective convergence map κ eff zs=0.65 at z s = 0.65. Top panel: at the edges the maps from the simulations employed for the interpolation are shown, z = 0.5 (left) and z = 1.0 (right). The central panel shows the result of Equation 7.3. Bottom panel: angular power spectra (left), peak counts (right) for the di erent redshifts. Given the di erence between the summary statistics (in particular for the peak counts) at z s = 0.5 and κ eff zs=0.65 , we expect this di erence to be propagated in the results of the inference analysis in the two settings. . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Left panel:Fractional di erence between peak counts from Low-AGN (blue), Fiducial (yellow), High-AGN (green) models with respect to the peak counts computed on DM ony simulations. Right panel: Ratio between peak counts from Low-AGN (blue), Fiducial (yellow), High-AGN (green) models with respect to the peak counts computed on DM ony simulations. The arrays plotted in this gure have been kindly provided by the authors of Coulton et al. 2020a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Constraints on M ν , Ω m and A s using as data the CFIS P3 patch. Left panel: constraints obtained with DM-only peak counts for source redshift z s = 0.5 (continuous line) and for source redshift z s = 0.65 (dashed line). Right panel: same as left panel but for the baryonic e ects ducial model. . . . . 7.8 Constraints on M ν , Ω m and A s using as data the CFIS P3 patch. Left panel: comparison between constraints with no baryonic e ects (black contours) and including baryonic e ects with the Low AGN model (blue contours, A baryon = 1). Central panel: comparison between constraints with no baryonic e ects (black contours) and including baryonic e ects with the Fiducial model (orange contours, A baryon = 2). Right panel: comparison between constraints with no baryonic e ects (black contours) and including baryonic e ects with the High AGN model (green contours, A baryon = 3). . 7.9 Maximum likelihood values and marginalised errors for the 3 parameters.

Figure 1 . 1 :

 11 Figure 1.1: Illustration of the linear matter power spectrum (continuous line) and nonlinear power spectrum (dashed line) obtained with pyccl. The linear one is computed with Eisenstein and Hu 1998 approximation and the non-linear one with the halo t model of Takahashi et al. 2012.

BFigure 1 . 2 :

 12 Figure 1.2: Plot from Abazajian et al. 2015, showing thee fractional di erence of the matter density power spectrum as a function of comoving scale k for di erent values of ν m ν .

A

  corresponding variety of codes and softwares are as well available and widely employed for inference in cosmology as for example CosmoHammer (Akeret et al. 2013), CosmoMC (Lewis and Bridle 2002), Cobaya (Torrado and Lewis 2021)..

  .16) that is the equation of a parametrized ellipse with semi-axes a = β (1-κ-|γ|) and b = β

  used the shear two-point correlation functions computed on CFHTLenS data spanning 154 deg 2 to constrain σ 8 and Ω m with cosmic shear only and on Ω K , Ω DE and w 0 by adding WMAP7 and BOSS data. Again, using the shear two-point correlation function Troxel et al. 2018 constrained σ 8 , Ω m , their combination S 8 and w, for DES Y1, covering 1321 deg 2 of the southern sky and found constraints in good agreement with previous cosmic shear results from KiDS-450 (covering 450 deg 2 ) (Hildebrandt et al. 2016). For the fourth data release of the KiDS that doubled the survey area (covering 1000 deg 2 ), Asgari, Marika et al. 2021 compared a COSEBIs (Complete Orthogonal Sets of E/B-Integrals) analysis with complementary analyses of the two-point shear correlation function and band power spectra, nding results to be in great agreement and the sensitivity of the three statistics to systematics to be robust and dominated by statistical errors for the constraints on S 8 . Hamana et al. 2020 employed the shear 2PCF from the HSC rst-year data, covering 136.9 deg 2 nding constraints for S 8 consistent with the latest Planck CMB result while in tension at a ∼ 2σ level KiDS + VIKING-450 (Joudaki,S. et al. 2020). To investigate this tension and understand its origin, they plan in future works to analyse the KiDS public shape catalog with their methodology. The most recent results available from cosmic shear to date are from DES-Y3 cosmological results (Amon et al. 2021), with the data spanning 4143 deg 2 for four redshift bins. The analysis provides constraints in the context of the ΛCDM model and nd a 3% constraint for S 8 , the highest signi cance measurement of cosmic shear to date, with a signal-to-noise of 40. The two low-redshift measurements are found to be statistically consistent with Planck Cosmic Microwave Background result, however, a ∼ 2.3σ tension, slightly dependent on the method used, remains for high-redshift predictions(Lemos, DES Collaboration, et al. 2021). Forthcoming experiments like Euclid (Laureijs et al. 2011), the Vera C. Rubin Observatory LSST (LSST Science Collaboration et al. 2009), and the Nancy Grace Roman Space Telescope

  Figure of Merit. In a companion study, Parroni, Carolina et al. 2020 discussed the use of MFs in Stage IV lensing surveys updating the procedure used in Vicinanza et al. 2019 nding that MFs can provide a valuable help in increasing the FoM of the lensing survey, provided the nuisance parameters are known with a non negligible precision. Recently, Zürcher et al. 2021 forecasted the performance of the Minkowski

Figure 2

 2 Figure 2.4: Tablesummarisingthe current state of the art of high order statistics where an estimate of the constraining power and its performance with respect to second order statistics have been provided by the authors. I would like to stress that this table does not include all studies ever performed with statistics of order higher than the second but the studies where cosmological inference has been performed and an explicit comparison with second order statistics is provided in the paper..

  signal representation . . . . . . . . . . . . . . . . . . . . . 74 3.2 Continuous Wavelet transform . . . . . . . . . . . . . . . . . . . . 75 3.3 Discrete Wavelet transforms . . . . . . . . . . . . . . . . . . . . . 77 3.4 The starlet transform . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.5 Wavelet coe cients and noise estimation . . . . . . . . . . . . . . 83 3.6 Wavelets and Dark Matter Mass Map Reconstruction . . . . . . . 85 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Figure 3 . 1 :

 31 Figure 3.1: The periodic function sin(x) is not sparse in real space (left image) but it can be sparsi ed by changing the data representation and work instead in Fourier space (right image).

Figure 3 . 4 :

 34 Figure 3.4: Illustration of the starlet decomposition with j max = 4 scales. As the j index increases, larger scales are considered and the resolution decreases.

Figure 3 . 5 :

 35 Figure 3.5: We show the 2D starlet function (top panel) as de ned in Eq. (3.15) and its 1D pro le (bottom panel). Being a wavelet it is a compensated function, i.e. it integrates to zero over its domain. This comes from the admissibility condition for the wavelet function ψ: +∞ 0 | ψ (k)| 2 dk k < +∞ which implies that ψ(x)dx = 0 and it has compact support in [-2, 2] × [-2, 2]. Its shape emphasises round features, making it very e cient when dealing with peak counts.

Figure 3 .

 3 Figure 3.6: COSMOS data: Top, galaxies count map (left) and Wiener map (right) smoothed with a Gaussian having a Full Width at Half Maximum of 2.4 arcmin. Bottom, GLIMPSE(left) and MCAlens(right). Images and caption from Starck et al. 2021.

Figure 4 . 1 :

 41 Figure 4.1: Parameter space of the 100 cosmological models available from the Massiv-eNus simulations. The red dot shows the ducial model corresponding to { M ν , Ω m , 10 9 A s } = { 0.1, 0.3, 2.1 }.

  use the Cosmological Massive Neutrino Simulations (MassiveNus), a suite of publicly available N-body simulations released by the Columbia Lensing group 1 . It contains 101 di erent cosmological models obtained by varying the sum of neutrino masses M ν , the total matter density parameter Ω m and the primordial power spectrum amplitude A s at the pivot scale k 0 = 0.05 Mpc -1 , in the range M ν = [0, 0.62] eV, Ω m =[0.18, 0.42] and A s • 10 9 =[1.29, 2.91]. The reduced Hubble constant h = 0.7, the spectral index n s = 0.97, the baryon density parameter Ω b = 0.046 and the dark energy equation of state parameter w = -1 are kept xed under the assumption of a at universe. The ducial model is set at {M ν , Ω m , 10 9 A s }={0.1, 0.3, 2.

Figure 4 . 2 :

 42 Figure 4.2: Convergence maps κ are noiseless. We apply Gaussian noise and then lter the map using either the Gaussian or starlet ltering. For illustration purposes, we show the Gaussian ltering with θ G ker = 1.6 arcmin of one map realisation for the ducial model {M ν , Ω m , 10 9 A s }={0.1, 0.3, 2.1}. The colour bar on the right of each map describes values of the convergence eld κ. For each realisation of the 10000 maps provided for each redshift we generate 10000 noise maps as described in Section 4.1.3 corresponding to the di erent value of n gal respectively for z s = [0.5, 1.0, 1.5, 2.0].

  Figure 4.3, resulting in a concatenation of Gaussians respectively with θ G ker = [1.

Figure

  Figure 4.3: We show the matching between the Gaussian lter and the starlet at the di erent scales. We chose the kernel for the multi-Gaussian concatenation such that the maximum of the two pro les match. From left to right are the nest scale to the smoothest scale where [ w 1 , w 2 , w 3 , c 4 ] =[ 1.6, 3.2, 6.2, 12.8 ] arcmin.

Figure 4 . 4 :

 44 Figure 4.4: Peak Counts distribution in logarithmic scale for each starlet scales resolutions:[ 1.6, 3.2, 6.4 ] arcmin and the coarse maps (red dotted line) and the Gaussian case (black dashed line). Due to the decomposition at di erent scales, for each map ltered with the starlet there are 4 di erent distributions. Indeed, the number of counts depends on the resolution: the larger the smoothing size (the lower the frequency) the smaller the number of peaks.

Figure 4 . 5 :

 45 Figure 4.5: We show the correlation coe cients respectively for the starlet (left) and the multi-Gaussian (right) peak counts. The white dashed lines split the contribution of the di erent redshifts: the bin range [1,164] refers to the correlations at z s = 0.5, the bin range [165, 328] refers to the correlations at z s = 1.0, the bin range [328, 492] refers to the correlations at z s = 1.5 and the bin range [493, 656] refers to the correlations at z s = 2.0.Each redshift contribution is split in the four di erent scales (the four mini-blocks inside the white boxes) in increasing order, i.e.[1.6, 3.2, 6.4, coarse] arcmin for the starlet and [1.2, 2.7, 5.5, 9.5] arcmin for the multi-Gaussian, where each scale is binned in 41 values of S/N in the range[-0.6, 6]. We notice how the starlet decomposition has a tendency to make the observable correlation matrix more diagonal while the o -diagonal terms of the multi-Gaussian matrix show more correlations along the scales and among S/N values in the mini-blocks.

Figure 4 . 6 :

 46 Figure 4.6: Visualisation of Gaussian processes predictor input and output: the training set, constituted by the cosmological parameters and corresponding peaks distributions obtained from the simulations are represented in blue. The output for the prediction is given by the red curves: for 200 new given cosmologies in parameter space fed to the Gaussian processes prediction function we get the corresponding peaks distribution.

Figure 4

 4 Figure 4.7: 95 % con dence contours tomography with source redshifts z s = [ 0.5,1.0,1.5, 2.0 ] and corresponding galaxy number density: n gal =[ 11.02, 11.90, 5.45, 1.45]. The black dotted line is the ducial model: [ m ν , Ω m , 10 9 A s ] = [0.1, 0.3, 2.1]. Left panel: constraints from power spectrum (blue contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1 arcmin, constraints from Gaussian Peak counts (green contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1.6 arcmin, constraints from Starlet Peak counts (red contours) computed on noisy maps smoothed with a Starlet kernel with corresponding resolutions [1.6, 3.2, 6.4] arcmin + coarse map, constraints from multi-Gaussian Peak counts computed on noisy maps smoothed with a Starlet kernel with corresponding resolutions [1.2, 2.7, 5.5, 9.5] arcmin. Right panel: constraints from power spectrum (blue contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1 arcmin, constraints from Gaussian Peak counts (green contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1.6 arcmin and the two statistics jointly (violet contours).

Figure 4 .

 4 Figure 4.9: 95 % con dence contours using tomography with redshifts z s = [ 0.5,1.0,1.5, 2.0 ] and corresponding galaxy number densities n gal= [ 11.02, 11.90, 5.45, 1.45 ]. The black dotted line is the ducial model:[ m ν , Ω m , 10 9 A s ] = [ 0.1, 0.3, 2.1 ]. Left panel:we show the impact of the di erent starlet scales and we prove there exists a minimum resolution θ ker = w 2 = 3.2 arcmin that allows us to obtain constraints comparable to what is achieved with the full wavelet decomposition and that the information contained in the coarse map cannot be neglected. Starting from the rst starlet scale alone θ ker = w 1 = 1.6 arcmin (dashed big contours in light blue) we add scale by scale in light blue until[w1, w2, w3]. In dark blue we show the constraints corresponding to [w 2 , w 3 , c 4 ] that almost match with the constraints provided by the full starlet decomposition (in red). Right panel: we show here the constraints obtained by adding each tomographic redshift at the time: the dashed pink corresponds to the contours relative to z s = 0.5, the lighter red to z s = [ 0.5, 1.0 ], the darker pink to z s = [ 0.5, 1.0, 1.5 ] and the dark red to the full set of redshifts in the starlet case. We plot this against the power spectrum (blue contours) to show how each source redshift contribution to shrinks the contours and helps break the degeneracy with respect to the power spectrum.

Figure 4 .

 4 Figure 4.11: Marginalized constraints on each parameter for forecasts showing the 2.5 and 97.5 percentiles with respect to the ducial model. These marginalised constraints refer to a tomographic setting with z = [0.5, 1.0, 1.5, 2.0] with the ducial model set at [M ν , Ω m , 10 9 A s ] = [0.1, 0.3, 2.1] corresponding to the di erent observables employed within the likelihood analysis. The values are listed in Table4.6.

  1 -norm for weak lensing cosmology . . . . . . . . . . . . 118 5.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.1.2 Toward the starlet 1 norm . . . . . . . . . . . . . . . . . 118 5.1.3 De nition of starlet 1 -norm . . . . . . . . . . . . . . . . . 120 5.1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.1.5 Summary statistics . . . . . . . . . . . . . . . . . . . . . . 121 5.1.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.1.7 Conclusions and future prospects . . . . . . . . . . . . . . 125 5.2 Pipeline description and visualisation . . . . . . . . . . . . . . . . 127 In this chapter I illustrate the methodology and results we presented in Ajani et al. 2021.Motivated by the advantages of multi-scale approaches for cosmological parameter inference as discussed in Chapter 4 along with recent results of Coulton et al. 2020a in combining peaks an minima, we propose a new multi-scale higher order summary statistics that we called the starlet 1 -norm that allows for a multi-scale analysis that naturally encodes the information brought by peaks and minima. At the end of the chapter, I also provide a description and visualisation of the di erent steps of the pipeline employed to obtain the results of this chapter and Chapter 4

Figure 5

 5 Figure 5.1: Visualisation of the starlet decomposition of an original noisy convergence map. The original noisy convergence map is obtained by applying shape noise to a noiseless realisation of the MassiveNus convergence maps products. Then by convolution with a 2D starlet, the original image is decomposed in several di erent images corresponding to the scale 2 j plus a smoother (coarse) version of it. Here we show the decomposition for a map of the ducial cosmology. The resolution for the MassiveNus simulations convergence maps is ∼ 0.4 arcmin.

Figure 5 . 2 :

 52 Figure 5.2: Marginalised errors for the observables described in Section 5.1.5, on each parameter showing the 2.5 and 97.5 percentiles with respect to the ducial model. They refer to a tomographic setting with z = [0.5, 1.0, 1.5, 2.0] with the ducial model set to [M ν , Ω m , 10 9 A s ] = [0.1, 0.3, 2.1]. The last observable refers to monoscale peaks and voids, as described in the text.

Figure 5 . 4 :

 54 Figure 5.4: 95 % con dence contours tomography with source redshifts z s = [ 0.5, 1.0, 1.5, 2.0 ] and corresponding galaxy number density: n gal = [ 11.02, 11.90, 5.45, 1.45 ]. The black dotted line is the ducial model: [ m ν , Ω m , 10 9 A s ] = [ 0.1, 0.3, 2.1 ]. Left panel:constraints from starlet extrema with the full covariance matrix (continuous contours) computed on noisy maps ltered with a four scales starlet kernel against constraints from star-

Figure 5 . 5 :

 55 Figure 5.5: Illustration of the di erent steps of the pipeline.

Figure 2 :

 2 Figure 2: Noisy convergence map of Figure 1, smoothed with a gaussian filter of precisions 1,2 and 5 arcmin respectively.

FIG. 4 .

 4 FIG. 4. We show the correlation coe cients respectively for the starlet (left) and the multi-Gaussian (right) peak counts. The white dashed lines split the contribution of the di↵erent redshifts: the bin range [1,164] refers to the correlations at zs = 0.5, the bin range [165, 328] refers to the correlations at zs = 1.0, the bin range [328, 492] refers to the correlations at zs = 1.5 and the bin range [493, 656] refers to the correlations at zs = 2.0. Each redshift contribution is split in the four di↵erent scales (the four mini-blocks inside the white boxes) in increasing order, i.e. [1.6, 3.2, 6.4, coarse] arcmin for the starlet and [1.2, 2.7, 5.5, 9.5] arcmin for the multi-Gaussian, where each scale is binned in 41 values of S/N in the range [ 0.6, 6]. We notice how the starlet decomposition has a tendency to diagonalise the observable correlation matrix while the o↵-diagonal terms of the multi-Gaussian matrix show more correlations along the scales and among S/N values in the mini-blocks. terms are non-zero, indicating an increasing presence of correlations among parameters as FoC increases. The values of the FoC for our constraints are shown in Table IV and we will comment on them in Sec. IV.

3 .

 3 Matrix condition number chain Monte Carlo (MCMC) introduced by [52]. The pipeline is built in a way that both the computation of the power spectrum and peak counts along with the MCMC are run in parallel to gain computation time.

  Simulations (SLICS)(Harnois-Déraps et al. 2018) and the cosmo-SLICS (Harnois-Déraps, J.,Giblin, B., and Joachimi, B. 2019). The SLICS set is used to get the observables employed in the computation of the covariance matrix, while the cosmo-SLICS suite to compute the observables for the model and mock data ( ducial cosmology). Both are based on a series of 100 deg 2 light-cones extracted from N-body simulations with the multiple plane technique. The underlying gravity-only calculations are obtained as in(Harnois- Déraps et al. 2013) and evolve 1536 3 particles in a box of comoving side of 505 h -1 Mpc. speci cally designed for the estimation of covariance matrices: they provide of 800 fully independent ΛCDM runs in which the cosmological parameters are xed to the cosmology [Ω m , σ 8 , h, n s , Ω b ] = [0.2905, 0.826, 0.6898, 0.969, 0.0473] and the random seeds in the initial conditions are varied. The cosmo-SLICS are run with a similar but complementary N-body con guration: the random seeds are xed, while the cosmological parameters [Ω m , S 8 , h, w 0 ] are sampled with a Latin hypercube at 25 points in parameter space, while keeping the parameters n s , and Ω b xed to the value used in the SLICS. For each cosmology, a pair of simulations is provided, obtained with two di erent seeds (seed a and seed f ) to suppress the sampling variance, such that any measurement averaged over the two seeds nearly follows the ensemble mean. For each seed for each cosmology and each bin, 25 di erent realisations are provided. The parameter space is shown in Figure6.1. In total, given the for 25 light-cones for the two seeds, for each realisation covering 100 deg 2 , the simulations provide 5000 deg 2 per cosmology. Further details on these simulations can be also found in[START_REF] Burger | A revised density split statistic model for general lters[END_REF]

Figure 6 . 1 :

 61 Figure 6.1: Illustration of the di erent parameter space planes for the cosmo-SLICS suite of simulations. The red dots indicate the values for the ducial cosmology used to mock the observational data.

Figure 6 . 2 :

 62 Figure 6.2: κ map reconstructed from the catalogue with KS inversion. The catalogue used to reconstruct this corresponds to the cosmology 00 ( the list of cosmologies is available in Table 2 of Harnois-Déraps, J., Giblin, B., and Joachimi, B. 2019) of the cosmo-SLICS, for l.o.s.=74. Left panel: noiseless κ. Right panel: noisy κ, smoothed with a Gaussian kernel of size 3 arcmin.

Figure 6 . 3 :

 63 Figure 6.3: Figure showing the di erent redshift distributions n(z) for each tomographic bin provided by the simulations.The legend shows the number of galaxies N associated to each bin, that is employed to compute the noise that we add to the convergence maps.

Figure 6 . 4 :obs 2 )

 642 Figure 6.4: Validation of our reconstructed convergence maps from the catalogue with the power spectrum. Left panel: comparison between the theoretical predictions (continuous lines) provided by the author of the cosmo-SLICS simulations (Harnois-Déraps et al. 2018) and the power spectra measured on our noiseless κ maps reconstructed from the catalogue shear components (γ 1 , γ 2 ). Right panel: comparison between the power spectrum measured on the map provided by the authors of Giblin et al. 2018 and the power spectrum measured on our map. Speci cally, both maps are built from the observed ellipticities ( obs 1 , obs 2 ) provided by the catalogue for the rst tomographic bin of the SLICS simulations for KiDS-1000, for l.o.s. 74.

6. 4

 4 Photometric Galaxy Clustering 6.4.1 Building the maps from the catalogue

Figure 6 . 5 :

 65 Figure 6.5: Left panel: example of galaxy map built from the LRGs cosmo-SLICS catalogue, smoothed with a Gaussian lter of size 3 arcmin. Right panel: S/N map built from the LRGs cosmo-SLICS catalogue, where we provide as estimate of the noise, the mean of the map.

Figure 6

 6 Figure6.6: Comparison between the power spectrum measured on a galaxy map we built from the LRGs lens catalogue (orange line) and the theoretical prediction obtained with pyccl (blue line). The gray region indicates values for the multipoles > max where max = 780 is the maximum value used for the analysis.

  Figure 6.7: Preliminary forecasts derived employing the power spectrum as summary statistics. The analysis is tomographic, performed with the 5 redshift bins shown in Figure 6.3. We compare our results on the right (blue contours) with the results obtained by (Asgari, Marika et al. 2021) for di erent second order statistics for KiDS-1000 mock data (smaller panel on the left).

Figure 6 . 9 :

 69 Figure 6.9: Starlet decomposition for a galaxy map. The scales shown in the gure are the ones included in the analysis. The value in arcmin corresponds to the resolution of the corresponding starlet scale, considering that the resolution of the original map is 1 arcmin. The corresponding values in are also shown.

Figure 6

 6 Figure 6.11: Extension of the initial input of the pipeline illustrated in Figure 5.5.

F

Figure 7 . 1 :

 71 Figure 7.1: Image from https://www.cfht.hawaii.edu/Science/CFIS/ cfissurvey.html, showing the equatorial projection of the entire sky for CFIS-r (red outline) and CFIS-u (blue outline) with respect to other surveys (listed in the top right caption). The areas that will be observed from the A and B semesters are indicated at the top. Image credit: T. Dwelly.

  (Guinot et al. in prep.). In particular, we consider the patch P3 of the four patches presented in Guinot 2020 of 806, 282, 249 and 358 deg 2 , corresponding to P1, P2, P3, and P4, respectively, shown in Figure7.2.

Figure 7 . 2 :

 72 Figure 7.2: Process patches of the CFIS dataset provided by Guinot 2020. In this analysis, we employ the P3 (red patch) corresponding to a size of 249 deg 2 .

Figure 7 . 3 :

 73 Figure 7.3: Illustration of the E-mode convergence map constructed for the CFIS P3 patch, corresponding to 249 deg 2 . The data shown in this image have been provided by A. Guinot (Guinot 2020). The map is built through KS inversion, setting a pixel size of ∼ 0.4 arcmin, to match the resolution of the κ maps from the MassiveNus simulations.

Figure 7 . 4 :

 74 Figure 7.4: Galaxy number density distribution (blue line) obtained by matching the galaxies with the CFHTLenS-W3 patch by Guinot 2020.

Figure 7 . 5 :

 75 Figure 7.5: Illustration of one realisation example result of our interpolation to get the e ective convergence map κ eff zs=0.65 at z s = 0.65. Top panel: at the edges the maps from the simulations employed for the interpolation are shown, z = 0.5 (left) and z = 1.0 (right). The central panel shows the result of Equation7.3. Bottom panel: angular power spectra (left), peak counts (right) for the di erent redshifts. Given the di erence between the summary statistics (in particular for the peak counts) at z s = 0.5 and κ eff zs=0.65 , we expect this di erence to be propagated in the results of the inference analysis in the two settings.

Figure 7 . 6 :

 76 Figure 7.6: Left panel:Fractional di erence between peak counts from Low-AGN (blue), Fiducial (yellow), High-AGN (green) models with respect to the peak counts computed on DM ony simulations. Right panel: Ratio between peak counts from Low-AGN (blue), Fiducial (yellow), High-AGN (green) models with respect to the peak counts computed on DM ony simulations. The arrays plotted in this gure have been kindly provided by the authors of Coulton et al. 2020a.

F

  µ(θ)) T C -1 (dµ(θ)),(7.7)

Figure 7 Figure 7 . 8 :

 778 Figure 7.7: Constraints on M ν , Ω m and A s using as data the CFIS P3 patch. Left panel: constraints obtained with DM-only peak counts for source redshift z s = 0.5 (continuous line) and for source redshift z s = 0.65 (dashed line). Right panel: same as left panel but for the baryonic e ects ducial model.

FFigure 7

 7 Figure 7.9: Maximum likelihood values and marginalised errors for the 3 parameters.

Figure 7 . 10 :

 710 Figure 7.10: Left panel: constraints without corrections (black contours), constraints with local calibration and ∆m uncertainty (light blue contours), constraints with local calibration and ∆m uncertainty (light blue contours), constraints with ducial baryonic correction (orange contours). Right panel: constraints without corrections (black contours), constraints with local calibration and ∆m uncertainty combined with ducial baryonic correction (violet contours).

  Cosmological Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Homogeneity and isotropy . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Dynamical evolution of the universe . . . . . . . . . . . . . . . . . 1.1.3 Kinematics in an expanding universe . . . . . . . . . . . . . . . . . 1.2 Cosmological parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Age of the universe . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Hubble parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Density Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Primordial power spectrum . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Sum of neutrino masses . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 ΛCDM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Principles of structure formation . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shear and convergence elds . . . . . . . . . . . . . . . . . . . . . 2.1.3 From observation to cosmology . . . . . . . . . . . . . . . . . . . . 2.2 Cosmic shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The shear two-point correlation function . . . . . . . . . . . . . . . 2.2.2 The convergence power spectrum . . . . . . . . . . . . . . . . . . . 2.2.3 Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Constraints with cosmic shear second order statistics . . . . . . . . 2.2.5 Non-Linear regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 State of the art of higher-order statistics . . . . . . . . . . . . . . . . . . . . Illustration of the linear matter power spectrum (continuous line) and nonlinear power spectrum (dashed line) obtained with pyccl. The linear one is computed with Eisenstein and Hu 1998 approximation and the non-linear one with the halo t model of Takahashi et al. 2012. . . . . . . . . . . . . . 1.2 Plot from Abazajian et al. 2015, showing thee fractional di erence of the matter density power spectrum as a function of comoving scale k for different values of ν m ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	L	F
	Introduction
	1 Background 1.1
	Mots clés: Lentillage gravitationnel faible ; Statistiques d'ordre supérieur ; paramètres cosmologiques; Transformée en ondelettes ; Euclid ; Energie sombre ; Cosmologie ; analyse multi-échelle; Grandes structures de l'Univers 1.1 xi

1.3.1 Density contrast in the linear regime . . . . . . . . . . . . . . . . . 1.3.2 Matter power spectrum . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Brief overview of cosmological probes . . . . . . . . . . . . . . . . . . . . . 1.4.1 Galaxy Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Weak Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . 1.5 Parameter inference for future galaxy surveys . . . . . . . . . . . . . . . . 1.5.1 Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Explore parameter space with MCMC . . . . . . . . . . . . . . . . 1.5.

4 Brief mention of likelihood-free inference and ABC methods . . . 1.6 2 Weak gravitational lensing 2.1 Lensing formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Light de ection and the lens equation . . . . . . . . . . . . . . . . 2.1.2 2.3.1 One-point probability distribution function (PDF) . . . . . . . . . . 2.3.2 Bispectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  arcmin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.4 Peak Counts distribution in logarithmic scale for each starlet scales resolu-

tions: [ 1

Table 4

 4 Visualisation of the starlet decomposition of an original noisy convergence map. The original noisy convergence map is obtained by applying shape noise to a noiseless realisation of the MassiveNus convergence maps products. Then by convolution with a 2D starlet, the original image is decomposed in several di erent images corresponding to the scale 2 j plus a smoother (coarse) version of it. Here we show the decomposition for a map of the ducial cosmology. The resolution for the MassiveNus simulations convergence maps is ∼ 0.4 arcmin. . . . . . . . . . . . . . . . . . . . 119 5.2 Marginalised errors for the observables described in Section 5.1.5, on each parameter showing the 2.5 and 97.5 percentiles with respect to the ducial model. They refer to a tomographic setting with z = [0.5, 1.0, 1.5, 2.0] with the ducial model set to [M ν , Ω m , 10 9 A map reconstructed from the catalogue with KS inversion. The catalogue used to reconstruct this corresponds to the cosmology 00 ( the list of cosmologies is available in Table 2 of Harnois-Déraps, J., Giblin, B., and Joachimi, B. 2019) of the cosmo-SLICS, for l.o.s.=74. Left panel: noiseless κ. Right panel: noisy κ, smoothed with a Gaussian kernel of size 3 arcmin. . . . . . 6.3 Figure showing the di erent redshift distributions n(z) for each tomographic bin provided by the simulations. The legend shows the number of galaxies N associated to each bin, that is employed to compute the noise that we add to the convergence maps. . . . . . . . . . . . . . . . . . . . . . . . . . The plots shown in gure are very preliminary and do not represent yet quantitative results. They are illustrated to show some rst tests we run on a new probe, to understand the next step to undertake to properly incorporate the GCph analysis. Left panel: Cosmological forecasts for Ω m , σ 8

	6.7 Preliminary forecasts derived employing the power spectrum as summary
	statistics. The analysis is tomographic, performed with the 5 redshift bins
	shown in Figure 6.3. We compare our results on the right (blue contours)

.6. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 s ] = [0.1, 0.3, 2.1]. The last observable refers to monoscale peaks and voids, as described in the text. . 122 5.3 95 % con dence contours tomography with source redshifts z s = [ 0.5, 1.0, 1.5, 2.0 ] and corresponding galaxy number density: n gal = [ 11.02, 11.90, 5.45, 1.45 ]. The black dotted line is the ducial model: [ m ν , Ω m , 10 9 A s ] = [0.1, 0.3, 2.1]. Left panel: constraints from the power spectrum (light blue contours) computed on noisy maps smoothed with a Gaussian lter with θ ker =1 arcmin compared to constraints from starlet 1 -norm (dark blue contours) computed on noisy maps ltered with a four scales starlet kernel. Right panel: constraints from the combination of peaks and voids (magenta contours) computed on noisy maps smoothed with a Gaussian lter with θ ker =2 arcmin against constraints from starlet peak counts (green contours), starlet extrema (orange contours) and 1 -norm (dark blue contours) computed on noisy maps ltered with a four scales starlet kernel. . . . . . 123 5.4 95 % con dence contours tomography with source redshifts z s = [ 0.5, 1.0, 1.5, 2.0 ] and corresponding galaxy number density: n gal = [ 11.02, 11.90, 5.45, 1.45 ]. The black dotted line is the ducial model: [ m ν , Ω m , 10 9 A s ] = [ 0.1, 0.3, 2.1 ]. Left panel: constraints from starlet extrema with the full covariance matrix (continuous contours) computed on noisy maps ltered with a four scales starlet kernel against constraints from starlet extrema with the only diagonal elements of the covariance matrix (dashed contours). Right panel: constraints from 1 -norm with the full covariance matrix (continuous contours) computed on noisy maps ltered with a four scales starlet kernel against constraints from 1 -norm with the only diagonal elements of the covariance matrix (dashed contours). . . . . . . . . . . 126 5.5 Illustration of the di erent steps of the pipeline. . . . . . . . . . . . . . . . 129 6.1 Illustration of the di erent parameter space planes for the cosmo-SLICS suite of simulations. The red dots indicate the values for the ducial cosmology used to mock the observational data. . . . . . . . . . . . . . . . . . 134 6.2 κ 6.4 Validation of our reconstructed convergence maps from the catalogue with the power spectrum. Left panel: comparison between the theoretical predictions (continuous lines) provided by the author of the cosmo-SLICS simulations (Harnois-Déraps et al. 2018) and the power spectra measured on our noiseless κ maps reconstructed from the catalogue shear components (γ 1 , γ 2 ). Right panel: comparison between the power spectrum measured on the map provided by the authors of Giblin et al. 2018 and the power spectrum measured on our map. Speci cally, both maps are built from the observed ellipticities ( obs 1 , obs 2 ) provided by the catalogue for the rst tomographic bin of the SLICS simulations for KiDS-1000, for l.o.s. 74. . . . . 6.5 Left panel: example of galaxy map built from the LRGs cosmo-SLICS catalogue, smoothed with a Gaussian lter of size 3 arcmin. Right panel: S/N map built from the LRGs cosmo-SLICS catalogue, where we provide as estimate of the noise, the mean of the map. . . . . . . . . . . . . . . . . 6.6 Comparison between the power spectrum measured on a galaxy map we built from the LRGs lens catalogue (orange line) and the theoretical prediction obtained with pyccl (blue line). The gray region indicates values for the multipoles > max where max = 780 is the maximum value used for the analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . with the results obtained by (Asgari, Marika et al. 2021) for di erent second order statistics for KiDS-1000 mock data (smaller panel on the left). . . . . 6.8

  7.10 Left panel: constraints without corrections (black contours), constraints with local calibration and ∆m uncertainty (light blue contours), constraints with local calibration and ∆m uncertainty (light blue contours), constraints with ducial baryonic correction (orange contours). Right panel: constraints without corrections (black contours), constraints with local calibration and ∆m uncertainty combined with ducial baryonic correction (violet contours). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 1.1 Some of the main cosmological parameters de ning the ΛCDM model from Planck Collaboration et al. 2020 obtained by the combination of CMB TT, TE, EE + lowE + CMB lensing + BAO. . . . . . . . . . . . . . . . . . . . . . 3.1 Values of the noise estimates for each starlet scale j for a map with (n pix , n pix ) = (512, 512) convolved with a starlet with j max = 4 scales. . . . . . . 4.1 Values of n gal for each source redshift z s . We adapt the binning choice to the provided simulation settings, assuming that we observe galaxies within

a small range around the actual redshift. In practice, this means considering as bin edges {0.001, 0.75, 1.25, 1.75, 2.25}, in order to compute the integral in Equation

4

.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Values of the FoM as de ned in Equation

  Values of the Condition Number for the data covariance matrices. The smaller the number the easier it is to invert the matrix. In this case we get very large values for this estimator, leading to the conclusion that the data covariance matrices for Starlet Peaks and Gaussian Peaks show very singular behaviours. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Values of the 2.5 and 97.5 percentiles for each cosmological parameter as illustrated in Figure4.11. In this table we also show the values corresponding to the marginalised constraints obtained using only the diagonal elements of the covariance matrices. They are very similar to the ones obtained by employing the full covariance. We further investigate this aspect in Section 4.1.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.1 Values of the Figure of Merit (FoM) as de ned in Equation 4.11 for di erent parameters pairs for each observable employed in the likelihood analysis. We provide in the last column the 3D FoM given as the inverse of the volume in (M ν , A s , Ω m ) space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.2 Values of 1-σ marginalised error as de ned in Equation4.13 for each cos-of current research in cosmology is driven by two fundamental questions, being what the nature and properties of dark matter and dark energy are. Observations of the Cosmic Microwave Background (CMB) at microwave and infrared frequencies through the European Space Agency (ESA) Planck mission enabled to measure several cosmological parameters at the percent level and to establish the standard cosmological model to be the Λ Cold Dark Matter (ΛCDM) model(Planck Collaboration et al. 2020). More precisely, we know that only 5% of the Universe energy-matter content is composed by ordinary visible matter that we are able to describe via the standard model of particle physics. The remaining 95% is composed by two unknown components, referred to as dark matter and dark energy. Also if we take a closer look to the standard model of particle physics, unanswered questions are still present and in search of an answer, such as what the values of the neutrino masses are.

4.7

we can see more 'oblique' contours for the Gaussian peaks in the plane (M ν , Ω m ) compared to the power spectrum, while for the pair (M ν , A s ) the power spectrum shows more correlation than peaks. . . . . . . . . . . . . . . . . . . . . . . 4.4 4.5 Values of 1-σ marginalised error for each cosmological parameter for the di erent observables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv 4.6 mological parameter for the di erent observables. . . . . . . . . . . . . . . 124 7.1 Maximum likelihood values for the three parameters for the di erent settings (no correction, Low-AGN, Fiducial, High-AGN). . . . . . . . . . . . . 158 A large part

  Cosmological Principle . . . . . . . . . . . . . . . . . . . . . . . . Brief overview of cosmological probes . . . . . . . . . . . . . . . . 1.4.1 Galaxy Clustering . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Weak Gravitational Lensing . . . . . . . . . . . . . . . . . 1.5 Parameter inference for future galaxy surveys . . . . . . . . . . . 1.5.1 Bayesian inference . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Explore parameter space with MCMC . . . . . . . . . . . 1.5.4 Brief mention of likelihood-free inference and ABC methods 1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	C	1
	B
	1.1

1.1.1 Homogeneity and isotropy . . . . . . . . . . . . . . . . . . 1.1.2 Dynamical evolution of the universe . . . . . . . . . . . . 1.1.

3 Kinematics in an expanding universe . . . . . . . . . . . . 1.2 Cosmological parameters . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Age of the universe . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Hubble parameter . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Density Parameter . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Primordial power spectrum . . . . . . . . . . . . . . . . . 1.2.5 Sum of neutrino masses . . . . . . . . . . . . . . . . . . . 1.2.6 ΛCDM model . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Principles of structure formation . . . . . . . . . . . . . . . . . . . 1.3.1 Density contrast in the linear regime . . . . . . . . . . . . 1.3.2 Matter power spectrum . . . . . . . . . . . . . . . . . . . 1.4

  Osato15[START_REF] Osato | IMPACT OF BARYONIC PRO-CESSES ON WEAK-LENSING COSMOLOGY: POWER SPECTRUM, NONLOCAL STATIS-TICS, AND PARAMETER BIAS[END_REF], theBAHAMAS (McCarthy et al. 2016), the Magneticum path-finder (Dolag, Komatsu, and Sunyaev 2016), the Illustris-TNG[START_REF] Pillepich | Simulating galaxy formation with the IllustrisTNG model[END_REF]). An e ort on modelling these e ects has been also put in analytical and physically intuitive approaches based on the halo model as for example in[START_REF] Semboloni | Quantifying the e ect of baryon physics on weak lensing tomography[END_REF] or in Fedeli et al. 2014. An alternative method, that does not require hydro-dynamical simulations, consists of modifying the output of N-body simulations by slightly displacing particles around halo centres based on an empirical parametrisation of halo pro les including gas, stars, and dark matter. This approach is called baryoni cation, introduced in Schneider and Teyssier 2015 and[START_REF] Schneider | Quantifying baryon e ects on the matter power spectrum and the weak lensing shear correlation[END_REF]

(Dubois et al. 2016)

, the Illustris (Vogelsberger et al. 2014), the EAGLE (Schaye et al. 2014), the

  [START_REF] Liu | Cosmology constraints from the weak lensing peak counts and the power spectrum in CFHTLenS data[END_REF] used a suite of ray-tracing N-body simulations on a grid of 91 cosmological models, varying the three parameters (Ω m , σ 8 , w) and interpolated from the 91 measured peak counts to other cosmological models using two di erent methods ( multi-dimensional Radial Basis Function and Gaussian Processes). Kacprzak et al. 2016 used the same set from[START_REF] Liu | Cosmology constraints from the weak lensing peak counts and the power spectrum in CFHTLenS data[END_REF] and similar interpolation methods to predict the peak counts. With the advent of the MassiveNus simulations[START_REF] Liu | MassiveNuS: cosmological massive neutrino simulations[END_REF], predictions for peak counts based on simulations with a dependence on the sum of neutrino masses M ν have also become possible, as in Li, Liu, Zorrilla Matilla, et al. 2019 and as we did in Ajani et al. 2020, as will be discussed in Chapter 4, with a set of 100 di erent cosmological models. Another recent set of cosmological simulations is the cosmo-SLICS suite of simulations with 25 cosmologies in the [Ω m , σ 8 , h, w 0 ] parameter space, that has been used in Harnois-Déraps et al. 2020 to predict weak lensing shear peaks. Somewhere in between analytical models and N-body simulations, a halo-based model that generates fast stochastic simulations of convergence maps called CAMELUS was proposed by Lin, Chieh-An and Kilbinger, Martin 2015. It's a semianalytic model, requiring only a halo pro le and a halo mass function as input to generate a lensing catalogue.

  [START_REF] Liu | Cosmological constraints from weak lensing peak statistics with Canada-France-Hawaii Telescope Stripe 82 Survey[END_REF][START_REF] Liu | Cosmology constraints from the weak lensing peak counts and the power spectrum in CFHTLenS data[END_REF][START_REF] Liu | Cosmological constraints from weak lensing peak statistics with Canada-France-Hawaii Telescope Stripe 82 Survey[END_REF] derived constraints on cosmological parameters using weak lensing peak statistics measured on the ∼ 130 deg 2 of the Canada-France-Hawaii Telescope Stripe 82 Survey (CS82) employing the model from Fan, Shan, and Liu 2010 described in the previous paragraph. They found that for at ΛCDM, the cosmological constraints on (Ω m , σ 8 ) from peak analyses are fully consistent with the constraints obtained with both WMAP9 and Planck results and from cosmic shear two-point correlation studies. At the same time, they found that the degeneracy direction of the two parameters is atter than those from the 2PCF analyses and deduced from this the complementarity of peak counts analysis to 2PCF studies.[START_REF] Liu | Cosmology constraints from the weak lensing peak counts and the power spectrum in CFHTLenS data[END_REF] derived constraints on (Ω m , σ 8 , w) for CFHTLenS ∼ 154 deg 2 data.Their studies are based on interpolations from the suite of simulation templates on a grid of 91 cosmological models introduced in the previous paragraph. From the outcome of this study resulted that constraints from peak counts are comparable to those from the power spectrum, and that when the two statistics are combined, the area of the contours in the (Ω m , σ 8 ) plane is reduced by a factor of ∼ 2. Moreover, neither observable can constrain w without further external data. Examining the e ects of additional smoothing scales, binning of peaks, and the robustness of the results under masking, they demonstrated that using multiple smoothing scales can reduce the size of the area of the error contour by a moderate amount and left to more detailed study the quanti cation of the impact of these choices on the parameter constraints. With the same methodology adopted in[START_REF] Liu | Cosmological constraints from weak lensing peak statistics with Canada-France-Hawaii Telescope Stripe 82 Survey[END_REF], Liu et al. 2016 presented constraints on the Hu-Sawicki f (R) theory with CFHTLenS data. For DES SV data, Kacprzak et al. 2016 measured shear peaks using aperture mass maps adopting the simulation approach to produce WL maps (Dietrich and Hartlap 2010) spanning the (Ω m , σ 8

  demostrated how a Deep Convolutional Neural Networks (DCNN) can be used to discriminate among cosmologies by learning the relation between ve cosmological models and the corresponding convergence maps. By comparing the performance of this approach to commonly used non-Gaussian statistics as the skewness and kurtosis (de ned in Section 2.3.4) of the convergence maps, they found that the DCNN implemented in their study outperforms the skewness and kurtosis statistics, especially for high noise levels and can e ciently break the (σ 8 , Ω m ) degeneracy with weak lensing convergence maps alone. Another study performed byGupta, Matilla, Hsu, et al. 2018 applied a convolutional neural network to simulated noiseless lensing maps covering 96 di erent cosmological models with varying (σ 8

  One very well established representation is the sparsity model, that has proven to be very powerful for astrophysical applications(Starck, Murtagh, and Fadili 2016; Lanusse 2015;[START_REF] Lin | A new model to predict weak-lensing peak counts -III. Filtering technique comparisons[END_REF]). The following is based on[START_REF] Starck | The Undecimated Wavelet Decomposition and its Reconstruction[END_REF] Starck, Murtagh, and Fadili 2016; Starck et al. 2021.

  Fourier transform, convert the time information in frequency information, loosing all information on time; STFT provide both frequency-time resolution, with the drawback that the window is xed, and inevitable loss of information is present because of this. The wavelet transform enables a multiresolution analysis, i.e. to analyse the signal into di erent frequency ranges at di erent time resolutions. tion coe cients {a 1 , a 2 , a 3 , .., a j , ..a J } and a set of detail coe cients {d 1 , d 2 , d 3 , .., d j , ..d J }, bringing respectively the low and high frequency components of the input signal.

	Time series	Fourier transform (FT)
	Frequency
	Short Time FT	Wavelet transform
	Frequency
	Time	Time
	Good time resolution, poor frequency resolution
	Good frequency resolution, poor time resolution
	Higher frequency values, same time resolution as
	Earlier time values, same frequency resolution as

Figure 3.2: Illustration of the time-frequency diagram for Time series analysis (top left panel), Fourier transform (top right panel), Short Time Fourier transform (STFT, bottom left panel), the Wavelet transform (bottom right panel). As desribed in the text, time series analysis provide good resolution in time, but none in frequency;

Table 3 .

 3 1: Values of the noise estimates for each starlet scale j for a map with (n pix , n pix ) = (512, 512) convolved with a starlet with j max = 4 scales.

	W
	Starlet scale	σ e j
	d 1	0.8891
	d 2	0.2017
	d 3	0.086
	d 4	0.0414
	a 5	0.0324

  The reason for the success of wavelets is due to the fact that wavelet bases represent well a large class of signals, especially astronomical data where many objects present isotropic features. The starlet transform is very well designed for representing these type of data. Sparse wavelet regularization is very e cient for inverse problems in astrophysics such as denoising or deconvolution. Very nice results have been obtained based on weak lensing mass map recovery with the algorithm MCALens(Starck et al. 2021) and Chapter 4.1.8 will show the potential of starlet-based summary statistics as a tool to extract information from cosmological data and estimate cosmological parameters.

C 4 R 4.1 Forecasts on neutrino masses using multi-scale peak counts . . . 92 4.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.1.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.1.3 Noise and survey speci cations . . . . . . . . . . . . . . . 94 4.1.4 Filtering strategy . . . . . . . . . . . . . . . . . . . . . . . 95 4.1.5 Summary statistics: power spectrum and multi-scale peaks 97 4.1.6 Covariance matrices . . . . . . . . . . . . . . . . . . . . . 99 4.1.7 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.1.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.1.9 Conclusions and future prospects . . . . . . . . . . . . . . 113

Table 4 .

 4 3: Value of the Figure of Correlation for each pair of cosmological parameters cor-

	Power spectrum	1.00	1.98	1.10
	Single-Gaussian Peaks	1.21	1.42	1.01
	Single-Gaussian Peaks + PS	1.19	1.31	1.04
	Starlet Peaks	1.14	1.18	1.14
	Starlet Peaks + PS	1.13	1.13	1.20
	Multi-Gaussian Peaks	1.13	1.16	1.17
	Multi-Gaussian Peaks + PS	1.13	1.14	1.18

Table 4 .

 4 4: Values of the Condition Number for the data covariance matrices. The smaller the number the easier it is to invert the matrix. In this case we get very large values for this estimator, leading to the conclusion that the data covariance matrices for Starlet Peaks and Gaussian Peaks show very singular behaviours.

	Condition Number Single-Gaussian Peaks Starlet Peaks Multi-Gaussian Peaks
	41 bins	10 5	10 6	10 7
	50 bins	10 5	10 16	-

Table 4 .

 4 5: Values of 1-σ marginalised error for each cosmological parameter for the di erent observables.

	σ αα	M ν	Ω m	A s
	Power spectrum	0.127 0.005 0.204
	Single-Gaussian Peaks	0.083 0.004 0.086
	Single-Gaussian Peaks + PS	0.061 0.003 0.066
	Starlet Peaks	0.047 0.003 0.057
	Starlet Peaks diagonal	0.049 0.003 0.053
	Starlet Peaks + PS	0.040 0.003 0.052
	Multi-Gaussian Peaks	0.038 0.003 0.047
	Multi-Gaussian Peaks diagonal 0.042 0.002 0.039
	Multi-Gaussian Peaks + PS	0.035 0.002 0.044

  Figure 4.8: 95 % con dence contours tomography with source redshifts z s = [ 0.5,1.0,1.5, 2.0 ] and corresponding galaxy number density: n gal = [ 11.02, 11.90, 5.45, 1.45 ]. The black dotted line is the ducial model: [ m ν , Ω m , 10 9 A

								Power spectrum							Power spectrum
								Starlet peaks							Multi-Gaussian peaks
								Starlet peaks + PS						Multi-Gaussian peaks + PS
		0.315							0.315
	⌦ m	0.300						⌦ m	0.300
		0.285							0.285
			3.0								3.0
	A s	2.4						A s	2.4
			0.3 P m ⌫ [eV ]	0.6	0.285	0.300 ⌦ m	0.315	2.4 A s	3.0		0.3 P m ⌫ [eV ]	0.6	0.285	0.300 ⌦ m	0.315	2.4 A s	3.0

s ] = [ 0.1, 0.3, 2.1 ]. Left panel: constraints from power spectrum (blue contours) computed on noisy maps smoothed with a Gaussian kernel θ ker = 1 arcmin, constraints from Starlet Peak counts (red contours) computed on noisy maps smoothed with a Starlet kernel with corresponding resolutions

  Figure 4.10: 95 % con dence contours tomography with redshifts z s = [0.5, 1.0, 1.5, 2.0] and corresponding galaxy number density: n gal =[11.02, 11.90, 5.45, 1.45]. The black dotted line is the ducial model

							R
						Starlet peaks full covariance			Multi-Gaussian peaks full covariance
						Starlet peaks diagonal elements			Multi-Gaussian peaks diagonal elements
		0.31
									0.304
	⌦ m	0.30							⌦ m
									0.296
		0.29
		2.25							2.2
	A s								A s
		2.00							2.0
		0.15 P	0.30 m ⌫ [eV ]	0.29	0.30 ⌦ m	0.31	2.00	A s	2.25	0.15 P m ⌫ [eV ]	0.30	0.296	⌦ m	0.304	2.0	A s	2.2

Table 4 .

 4 6: Values of the 2.5 and 97.5 percentiles for each cosmological parameter as illustrated in Figure

	Power spectrum	0.514 0.289 0.308 2.008 2.774
	Single-Gaussian Peaks	0.372 0.295 0.311 1.998 2.340
	Single-Gaussian Peaks + PS	0.290 0.294 0.307 2.016 2.277
	Starlet Peaks	0.238 0.293 0.305 2.029 2.255
	Starlet Peaks diagonal	0.244 0.294 0.305 2.039 2.248
	Starlet Peaks + PS	0.210 0.293 0.305 2.032 2.239
	Multi-Gaussian Peaks	0.201 0.294 0.305 2.037 2.222
	Multi-Gaussian Peaks diagonal 0.218 0.295 0.305 2.054 2.207
	Multi-Gaussian Peaks + PS	0.190 0.294 0.304 2.042 2.216

  The black dotted line is the ducial model: [ m ν , Ω m , 10 9 A s ] =[0.1, 0.3, 2.1]. Left panel: constraints from the power spectrum (light blue contours) computed on noisy maps smoothed with a Gaussian lter with θ ker =1 arcmin compared to constraints from starlet 1 -norm (dark blue contours) computed on noisy maps ltered with a four scales starlet kernel. Right panel: constraints from the combination of peaks and voids (magenta contours) computed on noisy maps smoothed with a Gaussian lter with θ ker =2 arcmin against constraints from starlet peak counts (green contours), starlet extrema (orange contours) and Table 5.1: Values of the Figure of Merit (FoM) as de ned in Equation

								Power spectrum					Peaks + voids mono-scale
								Starlet 1-norm					Starlet peaks Starlet extrema Starlet 1-norm
		0.315							0.315
	Ω m	0.300						Ω m	0.300
		0.285
			3.0								2.4
	A s	2.4						A s	2.1
			0.3 m ν [eV ]	0.6	0.285	0.300 Ω m	0.315	2.4 A s	3.0		0.2 m ν [eV ]	0.4	0.300 Ω m	0.315	2.1 A s	2.4
	Figure 5.3: 95 % con dence contours tomography with source redshifts z s = [ 0.5, 1.0, 1.5,
	2.0 ] and corresponding galaxy number density: n gal = [ 11.02, 11.90, 5.45, 1.45 ].

Table 5 .

 5 2: Values of 1-σ marginalised error as de ned in Equation4.13 for each cosmological parameter for the di erent observables.

	A s )

  Background de nitions . . . . . . . . . . . . . . . . . . . . . . . . 6.2 SLICS and cosmo-SLICS simulations . . . . . . . . . . . . . . . . . 6.3 Weak lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Building the convergence maps from the catalogue . . . . 6.3.2 Map validation with power spectrum . . . . . . . . . . . . 6.4 Photometric Galaxy Clustering . . . . . . . . . . . . . . . . . . . . 6.4.1 Building the maps from the catalogue . . . . . . . . . . . 6.4.2 Map validation with power spectrum . . . . . . . . . . . . 6.5 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Weak lensing (WL) power spectrum . . . . . . . . . . . . 6.5.2 Galaxy clustering photometric (GCph) power spectrum . 6.5.3 Photometric Galaxy Clustering starlet peaks . . . . . . . . 6.6 Ongoing work and future prospects . . . . . . . . . . . . . . . . .

	1
			C	6
	2
	E	P	G	C
	3	6.1
	4	5
		7
		8
	6

  The plots shown in gure are very preliminary and do not represent yet quantitative results. They are illustrated to show some rst tests we run on a new probe, to understand the next step to undertake to properly incorporate the GCph analysis. Left panel: Cosmological forecasts for Ω m , σ 8

											E	P	G	C
					WL Noisy Power Spectrum `max = 2000 GC Power Spectrum `max = 780	WL Power Spectrum GCph Power Spectrum
												Combined
		1.05										0.9
		0.90
	h
		0.75
		0.60
												S8
												0.8
		0.5
	w0	1.0
		1.5
		1.2										0.7
		1.0
	8
		0.8
		0.6										0.15	0.30 Ωm	0.45
		0.15	0.30	0.45	0.60 0.75 0.90 1.05	1.5	1.0	0.5	0.6	0.8	1.0	1.2
			⌦m		h	w0				8
	Figure 6.8:

  Correction for baryonic e ects . . . . . . . . . . . . . . . . . . . . 7.4 First experiment and discussion . . . . . . . . . . . . . . . . . . . 7.5 Ongoing work and future prospects . . . . . . . . . . . . . . . . .

7.1 CFIS survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Analysis setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Peak counts . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3

Table 7 .

 7 1: Maximum likelihood values for the three parameters for the di erent settings (no correction, Low-AGN, Fiducial, High-AGN).

	Model	M ν	Ω m	10 9 A s
	DM only 0.105 +0.313 -0.043	0.358 +0.023 -0.028	1.60 +0.43 -0.30
	Low-AGN	0.21 +0.20 -0.15	0.375 +0.036 -0.021	2.05 +0.57 -0.20
	Hihg-AGN 0.091 +0.229 -0.029	0.418 +0.0091 -0.0449	2.09 +0.41 -0.22
	Fiducial	0.21 +0.21 -0.15	0.393 +0.027 -0.024	2.08 +0.46 -0.21

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		: 978-3-642-10598-2.	: 10.1007/978-3-642-
	10598-2_4.	: https://doi.org/10.1007/978-3-642-10598-
	2_4.

-norm starlette, et qui consiste à sommer la norme 1 des coe cients starlette de la carte de convergence. Ceci évite de devoir dé nir la notion de pic ou de vide, et permet de tenir

#coef (S j,i ) u=1 S j,i [u] et l'indice u va de 1 au nombre de pixels dans un bac donné i à l'échelle j (i.e. #coef (S j,i )). Nous étudions l'impact de l'utilisation de la starlette norme 1 en tant que statistique calculée sur des cartes de convergence à lentilles faibles pour estimer les paramètres cosmologiques et nous trouvons qu'elle est potentiellement très e cace en surpasser les statistiques de pointe, le spectre de puissance et la combinaison de pics et de vides monoéchelle, respectivement de 72% et 24% sur M ν , 60% et 50% sur Ω m et 75 % et 24 % sur A s . Nous avons en outre proposé des starlette extrema comme statistiques et les avons comparés à la 1 -norm: dans ce cas également, cette dernière est plus performante en termes de puissance de contrainte, dans le cadre idéal actuel, tandis que les extrema de starlette présentent l'avantage d'une matrice de covariance plus diagonale. Pour servir de sonde robuste pour la cosmologie de précision, la puissance statistique seul ne su t pas; pour leur utilisation, il sera important de tester comment ces statistiques réagissent dans un cadre non idéal, et comment leurs performances sont impactées par les e ets systématiques du signal. En n de chapitre, je propose également une description et visualisation des di érentes étapes du pipeline utilisé pour obtenir le résultats des chapitre 4 et 5. Dans la suite, je montre dans le chapitre 6 les résultats préliminaires relatif à l'application de notre nouvelle méthodologie pour le relevé de galaxies KiDS (Kilo-Degree Survey) et également pour une autre sonde, le clustering des galaxies. Plus en détail, je présente la procédure et quelques résultats préliminaires d'un travail en cours(Ajani et al. in prep.) dont le but est d'étendre le pipeline décrit dans le chapitre précédent en ce qui concerne 1) l'espace des paramètres considérés 2) la sonde

and on the knowledge about this topic I have acquired during the class on Bayesian Statistics by Professor Benjamin Wandelt.

Within the Born approximation, Shapiro and Cooray

proved using numerical simulations that the corrections up to fourth order in the gravitational potential of large-scale structure on the weak lensing power spectra are unimportant (namely the approximation to be accurate to sub-percent on most scales) for any weak lensing survey, including for a full sky survey limited by cosmic variance.

The simulations are publicly available at http://columbialensing.org.

https://www.Euclid-ec.org

https://joblib.readthedocs.io/

Code publicly available at https://github.com/CosmoStat/lenspack.

For a Gaussian distribution the median absolute deviation (MAD) and the standard deviation are directly related as: MAD/σ = 0.6745. We choose to use this estimator since it is more robust when dealing with nonnormal distributions (being more resistant to outliers in a data set) to have a more general implementation in our pipeline.

Indeed, the value of the correction coe cient is close to 1 for each analysis we perform. However, considering that the results of (Sellentin and Heavens 2017) quantify the loss of information also in the case of a Euclid-like survey, it would be worth it to reproduce our study applying their restoration technique to generalize our analysis.

emcee is a python implementation of the a ne-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) introduced by (Foreman-Mackey et al. 2013) and described in Section 1.5.3.

https://www.Euclid-ec.org

-norm (dark blue contours) computed on noisy maps ltered with a four scales starlet kernel.In this work, we have also introduced starlet extrema as new summary statistics to constrain the parameters. Similarly to the 1 -norm, starlet extrema are computed between the minimum and maximum S/N value of the map, but they are de ned as the combination of local maxima computed on S/N > 0 and local minima computed on S/N < 0 (namely, they do not encode the information present in all pixels). We see that starlet extrema FoM values

-norm outperforms the power spectrum by 72% on M ν , 60% on Ω m and 75% on A s , and the state of the art peaks + voids for a single smoothing scale respectively by

24% on M ν , 50% on Ω m and 24% on A s . Starlet extrema outperform the power spectrum by 72% on M ν , 20% on Ω m and 70% on A s . We also quantify the improvement provided by the 1norm with respect of our previous study(Ajani et al. 2020), nding that the 1 -norm starlet outperforms starlet peaks by 28% on M ν , 33% on Ω m and 20% on A s . We also compare the covariance matrices obtained when using starlet extrema and the 1 -norm and we have found that starlet extrema present a more diagonal covariance for the observable.

An ongoing e ort to obtain redshift distributions with the so-called cluster-z technique as presented inMénard et al.

is currently pursued by the collaboration.

-norm.

Acknowledgments to fundings, simulations and data

Virginia Ajani acknowledges support by the Centre National d'Etudes Spatiales and the project Initiative d'Excellence (IdEx) of Université de Paris (ANR-18-IDEX-0001). Virginia Ajani thanks the Columbia Lensing group (http://columbialensing.org) for making their simulations available. The creation of these simulations is supported through grants NSF AST-1210877, NSF AST-140041, and NASA ATP-80NSSC18K1093. We thank New Mexico State University (USA) and Instituto de Astro sica de Andalucia CSIC (Spain) for hosting the Skies & Universes site for cosmological simulation products. We would like to thank Joachim Harnois-Deraps for making public the SLICS mock data which can be found at http://slics.roe.ac.uk/ and for providing the new set of cosmo-SLICS for KiDS-1000. This work is based on data obtained as part of the Canada-France Imaging Survey, a CFHT large program of the National Research Council of Canada and the French Centre National de la Recherche Scienti que. Based on observations obtained with MegaPrime MegaCam, a joint project of CFHT and CEA Saclay, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l'Univers (INSU) of the Centre National de la Recherche Scienti que (CNRS) of France, and the University of Hawaii. We would like to thank the authors of Coulton et al. 2020a for providing us with the data arrays of baryonic corrections used in Chapter 7.

Statistics

Tomo Systematics Params Forecasts (with II order)

Real data Survey References

Summary statistics employed in the analysis

E P G C

enabled a rst extension of the pipeline in terms of initial input. Indeed, the steps described in Section 5.2 can be extended at the beginning with the following items Sources Galaxy catalogue: The galaxy catalogue is provided usually as fits or dat le. It contains the galaxy position (x, y), the redshift z, the observed ellipticities ( obs 1 , obs 2 ) and the values for the two components of shear (γ 1 , γ 2 ). Public python packages as astropy can be used to read it.

Pixelated shear map: Taking as input the observed ellipticity or the shear values provided by the catalogue, a shear map is built by binning the shear values according to position. This is performed using the CosmoStat python package lenspack.

Convergence map: The convergence map is then computed by Kaiser Squires (KS) inversion. This is performed using the CosmoStat python package lenspack.

Lenses Galaxy catalogue: The galaxy catalogue is provided usually as fits or dat le. It contains the galaxy position in RA and DEC and the redshift z. Public python packages as astropy can be used to read it.

Galaxy map:

The galaxy map is obtained by binning the positions in RA and DEC into a cartesian grid of n pix × n pix . that basically summarise the procedure described in Section 6.3.1 and Section 6.4.1. In this way, the pipeline illustration of Figure 5.5 can be updated by adding the initial step illustrated in Figure 6.11. Starting from a lenses catalogue instead of a sources one, this is automatically extended to photometric galaxy clustering. The procedure make it now possible to obtain constraints on cosmological parameters as output starting from a mock catalogue of lenses or sources as input. As brie y mentioned also in this Chapter, the natural extension of the pipeline is actually to apply it to more realistic settings, to be ready to apply it to real data. Along with the extension for new probes which is the focus of the project introduced in this Chapter, a parallel e ort to extend it to more realistic conditions is described in the next chapter. C B Abazajian, K. N. et al. (2015). "Neutrino physics from the cosmic microwave background and large scale structure". In: Ajani, V. et al. (2021). "Starlet l1-norm for weak lensing cosmology". In: A&A 645 (10), p. L11.

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