RESSALVA Atendendo solicitação do(a) autor(a), o texto completo desta tese será disponibilizado somente a partir de 21/02/2026. TESE DE DOUTORAMENTO IFT–T.001/2025 Testing cosmological models using state-of-the-art galaxy surveys João Victor Silva Rebouças Orientador Rogério Rosenfeld Janeiro de 2025 Rebouças, João Victor Silva R292t Testing cosmological models using state-of-the -art galaxy surveys / João Victor Silva Rebouças. – São Paulo, 2025 237 f.: il. color. Tese (doutorado) – Universidade Estadual Paulista (Unesp), Instituto de Física Teórica (IFT), São Paulo Orientador: Rogério Rosenfeld 1. Energia escura (Astronomia). 2. Supernova (Estrela). 3. Cosmologia. I. Título Sistema de geração automática de fichas catalográficas da Unesp. Biblioteca do Instituto de Física Teórica (IFT), São Paulo. Dados fornecidos pelo autor(a). TESTING COSMOLOGICAL MODELS USING STATE-OF-THE-ART GALAXY SURVEYS Tese de Doutorado apresentada ao Instituto de Física Teórica do Câmpus de São Paulo, da Universidade Estadual Paulista “Júlio de Mesquita Filho”, como parte dos requisitos para obtenção do título de Doutor em Física, Especialidade Física Teórica. Comissão Examinadora: Prof. Dr. ROGÉRIO ROSENFELD (Orientador) Instituto de Física Teórica - UNESP Prof. Dr. MARCO RAVERI University of Genova Prof. Dr. TANVI KARWAL University of Chicago Profa. Dra. MARIANA PENNA-LIMA Universidade de Brasília Prof. Dr. FELIPE ANDRADE OLIVEIRA University of Zurich Conceito: Aprovado São Paulo, 21 de fevereiro de 2025. Dedico esta tese a minha mãe, Maria da Conceição Silva Rebouças. i Agradecimentos Agradeço primeiramente a meus pais e meu irmão por darem todo o carinho e suporte necessários pra que eu alcance meus sonhos e objetivos. Sem eles, certamente não teria conseguido chegar até aqui. Agradeço a Duda Barbosa, por me apoiar nos momentos mais difíceis dessa jornada, por todo o amor, carinho, escuta e compreensão. Agradeço aos meus amigos - Marcus, Brisma, Bia, Loredo, Carlos e outros que terão que me perdoar - por serem espaços seguros onde eu posso descansar e me desenvolver enquanto pessoa. Todos vocês foram fundamentais para que eu me tornasse quem eu sou hoje. Agradeço a meu orientador, Rogério Rosenfeld, por ser um grande exemplo pra mim e por todo o apoio nessa trabalhosa e por vezes dolorosa jornada acadêmica. Sem o Rogério, eu teria desistido desse doutorado. Agradeço a Vivian Miranda por ter acreditado em mim, por todas as discussões, ensinamentos e broncas. Rogério e Vivian enxergaram o meu potencial, e sou eternamente grato a ambos. Agradeço a meus colegas e colaboradores - Diogo Souza, Lucas Faga, Kunhao Zhong, Jonathan Gordon, Felipe Falciano, Guilherme Brando, Victoria Lloyd, entre outros - por errarem comigo: afinal de contas, ciência é feita de erros. Agradeço também a todos os meus colegas da UFPE e do IFT-UNESP, por compartilharem as alegrias e sofrimentos da ciência e da academia. Ao longo da minha jornada, conheci várias pessoas dispostas a tornar o ambiente acadêmico mais inclusivo, diverso e seguro. Agradeço a todas as pessoas que lutam para isso, e que sigamos lutando. Agradeço à Fapesp pelo apoio financeiro. Agradeço ao IFT-UNESP por proporcionar toda a infraestrutura necessária, promover eventos e outras iniciativas científicas. Agra- deço à equipe do GridUnesp por todo o suporte em questões computacionais. Agradeço ao Laboratório Nacional de Computação Científica e à equipe do cluster Santos Dumont por fornecer recursos computacionais e pelo suporte. ii “O peito é infantaria do espírito” Negro Leo, Marcha para Longe iii Resumo As últimas décadas viram gigantes avanços na cosmologia, tanto teóricos quanto experimentais. Há pouco mais de cem anos, Edwin Hubble obtia as primei- ras evidências da expansão do Universo, através da luminosidade de "nébulas". Hoje, conseguimos modelar a abundância de elementos atômicos primordiais usando a teoria da Nucleosíntese do Big Bang, que usa a termodinâmica de fluidos cosmológicos para inferir a razão de massa entre elementos atômicos em estre- las. Conseguimos também prever anisotropias da Radiação Cósmica de Fundo, previsões confirmadas por experimentos de alta resolução angular. Conseguimos fazer o levantamento de milhões de galáxias, estudar correlações estatísticas na população levantada, observando efeitos cosmológicos previstos utilizando teoria de perturbação cosmológica. O sucesso de tais experimentos, entretanto, levantou enormes questões teóricas. Existem evidências de uma matéria escura, que se comporta como uma partícula massiva que não interage sob nenhuma interação fundamental conhecida no Modelo Padrão de Física de Partículas. Ainda mais, também existem evidências de que a expansão do Universo atualmente é acele- rada. De acordo com a Relatividade Geral, isso só é possível devido a uma energia escura, um fluido de pressão negativa, que é a forma de energia mais abundante no Universo. A natureza dessa energia escura é desconhecida. O modelo mais simples que a descreve é de uma constante cosmológica, uma densidade de ener- gia constante e igualmente distribuída pelo espaço. Finalmente, também existem evidências de uma época de expansão acelerada do Universo logo após o Big Bang, período conhecido como inflação, e cujo mecanismo ainda é desconhecido. Estas e outras perguntas motivam uma ampla gama de experimentos cosmoló- gicos com o intuito de entender melhor a natureza da matéria e energia escuras, além da inflação. Hoje em dia, contamos com uma ampla miríade de dados obser- vacionais cada vez mais precisos, cada qual sensível a diferentes aspectos da teoria cosmológica e cada qual com sua previsão para parâmetros do modelo. Com o aumento gradual da precisão dos experimentos, alguns começaram a discordar entre si quanto à previsão de parâmetros. Tais discordâncias, conhecidas como tensões, podem algumas vezes ser aliviadas a partir de diferentes tratamentos de erros sistemáticos. Algumas tensões, como a tensão de Hubble, são tão graves que não possuem tratamento sistemático que consiga aliviá-las. A falta de soluções iv às tensões motivou a investigação de modelos de energia escura alternativos, que consigam acomodar os dados discordantes. Nesta tese, apresento o modelo cosmológico ΛCDM, amplamente utilizado para prever observações de distâncias, das anisotropias da CMB, modelar a formação da estrutura em grande escala do Universo, dentre vários efeitos observáveis. Apresento também análise Bayesiana de dados cosmológicos, o procedimento padrão de como obter informação sobre o modelo cosmológico a partir da grande miríade de diferentes dados cosmológicos. Com o arcabouço teórico desenvolvido e as ferramentas em mão, investigo propri- edades da energia escura, como sua equação de estado, sua abundância ao longo da história cósmica, suas possíveis interações com a matéria escura, seu efeito na formação de estruturas e sua capacidade de resolver tensões cosmológicas. Mais especificamente, eu discuto dois trabalhos originais desenvolvidos ao longo do meu doutorado: o primeiro investiga o paradigma de "Early Dark Energy"com uma modificação adicional na equação de estado da energia escura recente, e o segundo discute indicações de energia escura dinâmica usando dados recentes de medidas de distâncias. Palavras Chaves: Energia Escura; Levantamentos de Galáxias; Radiação Cósmica de Fundo; Supernovas; Inferência Estatística. Áreas do conhecimento: Física; Física Geral: Relatividade e Gravitação; Astrofísica Extragaláctica: Cosmologia. v Abstract The last few decades have seen significant advances in cosmology, both theore- tical and experimental. Just over a hundred years ago, Edwin Hubble obtained the first evidence of the expansion of the Universe through the luminosity of "nebulae". Today, we can model the abundance of primordial atomic elements using the Big Bang Nucleosynthesis theory, which uses the thermodynamics of cosmological fluids to infer the mass ratio of atomic elements in stars. We can also predict anisotropies in the Cosmic Microwave Background (CMB), predictions confirmed by high angular resolution experiments. We have conducted surveys of millions of galaxies, studying statistical correlations in the surveyed population and observing predicted cosmological effects using cosmological perturbation theory. The success of such experiments, however, has raised enormous theoretical questions. There is evidence of dark matter, which behaves like a massive particle that does not interact through any known fundamental interaction in the Standard Model of Particle Physics. Furthermore, there is evidence that the expansion of the Universe is currently accelerating. According to General Relativity, this is only possible due to dark energy, a negative pressure fluid that is the most abundant form of energy in the Universe. The nature of this dark energy is unknown. The simplest model that describes it is a cosmological constant, a constant and evenly distributed energy density throughout space. Finally, there is also evidence of a period of accelerated expansion of the Universe shortly after the Big Bang, known as inflation, whose mechanism is still unknown. These and other questions motivate a wide range of cosmological experiments aimed at better understanding the nature of dark matter, dark energy, and in- flation. Today, we have a vast array of increasingly precise observational data, each sensitive to different aspects of cosmological theory and each with its predic- tion for model parameters. As experimental precision gradually increases, some parameters’ predictions have begun to disagree with one another. These disagre- ements, known as tensions, can sometimes be alleviated by different treatments of systematic errors. Some tensions, such as the Hubble tension, are so severe that no systematic treatment can alleviate them. The lack of solutions to these tensions has motivated the investigation of alternative dark energy models that can accommodate the conflicting data. In this thesis, I present the Lambda-CDM vi (ΛCDM) cosmological model, widely used to predict distance observations, CMB anisotropies, model the large-scale structure formation of the Universe, among various observable effects. I also present Bayesian analysis of cosmological data, the standard procedure for obtaining information about the cosmological model from the vast array of different cosmological data. With the developed theoretical framework and tools in hand, I investigate properties of dark energy, such as its equation of state, its abundance throughout cosmic history, its possible interacti- ons with dark matter, its effect on structure formation, and its capacity to resolve cosmological tensions. In particular, I discuss two original works done during my PhD: the first investigates Early Dark Energy with late-time equation of state modifications, and the second discusses hints of dynamical dark energy from recent geometric measurements. Keywords: Dark Energy; Galaxy Surveys; Cosmic Microwave Background; Super- novae; Statistical Inference. Knowledge areas: Physics; General Physics: Relativity and Gravitation; Extraga- lactic Astrophysics: Cosmology. vii Index 1 Introduction and Motivation 1 2 The ΛCDM Model 11 2.1 Einstein Equations and The Cosmological Principle . . . . . . . . . 11 2.2 The Homogeneous and Isotropic Cosmological Theory . . . . . . . 12 2.2.1 The Geometry of the Universe . . . . . . . . . . . . . . . . . 12 2.2.2 Kinematics in an Expanding Universe . . . . . . . . . . . . . 14 2.2.3 The Components of the Universe . . . . . . . . . . . . . . . . 15 2.2.4 The Friedmann Equations . . . . . . . . . . . . . . . . . . . . 18 2.2.5 Measuring and Predicting Distances . . . . . . . . . . . . . . 20 2.3 The Inhomogeneous Universe . . . . . . . . . . . . . . . . . . . . . . 24 2.3.1 Metric Perturbations . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.2 The Scalar-Vector-Tensor Decomposition . . . . . . . . . . . 25 2.3.3 Energy-Momentum Tensor Perturbations . . . . . . . . . . . 30 2.3.4 Perturbed Conservation and Einstein Equations . . . . . . . 31 2.3.5 The Gauge Problem . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.6 Fluid Microphysics . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.7 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.3.8 Inflation and the Primordial Power Spectrum . . . . . . . . . 62 2.3.9 Matter Power Spectrum . . . . . . . . . . . . . . . . . . . . . 67 2.3.10 Cosmic Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.3.11 CMB Temperature Power Spectrum . . . . . . . . . . . . . . 74 3 Cosmological Observations and Tensions 76 3.1 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.1.1 Luminosity Distances and Type-Ia Supernovae . . . . . . . . 76 3.1.2 Local Distance Ladder . . . . . . . . . . . . . . . . . . . . . . 77 3.1.3 Cosmic Microwave Background Anisotropies . . . . . . . . 78 3.1.4 CMB Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.1.5 Cosmic Shear, Galaxy Clustering and Cross-Correlations . . 81 3.1.6 Baryonic Acoustic Oscillations . . . . . . . . . . . . . . . . . 84 3.1.7 Redshift Space Distortions . . . . . . . . . . . . . . . . . . . . 87 viii 3.1.8 Big Bang Nucleosynthesis . . . . . . . . . . . . . . . . . . . . 88 3.1.9 Standard Sirens . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.1.10 Cosmic Chronometers . . . . . . . . . . . . . . . . . . . . . . 89 3.2 Cosmological Tensions . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.2.1 Quantifying Tensions . . . . . . . . . . . . . . . . . . . . . . 90 3.2.2 The Hubble Tension . . . . . . . . . . . . . . . . . . . . . . . 91 3.2.3 The S8 Tension . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4 Beyond ΛCDM Models 93 4.1 Remarks about ΛCDM . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.1.1 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . 93 4.1.2 Constant Dark Energy Density . . . . . . . . . . . . . . . . . 94 4.1.3 Negligible Dark Energy at Early Times . . . . . . . . . . . . 94 4.1.4 No Interactions in the Dark Sector . . . . . . . . . . . . . . . 95 4.1.5 Smooth Dark Energy . . . . . . . . . . . . . . . . . . . . . . . 95 4.1.6 Presureless Dark Matter . . . . . . . . . . . . . . . . . . . . . 95 4.2 Alternative Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2.1 Late-Time Dynamical Dark Energy . . . . . . . . . . . . . . . 95 4.2.2 Early Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . 102 4.2.3 Modified Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.2.4 Dark Sector Interactions . . . . . . . . . . . . . . . . . . . . . 106 5 Cosmological Data Analysis 110 5.1 The Bayesian Interpretation of Probability . . . . . . . . . . . . . . . 110 5.2 The Bayes’ Theorem: likelihood, prior and posterior distributions . 111 5.3 Markov Chain Monte Carlo and the Metropolis-Hastings algorithm 113 5.3.1 The Gelman-Rubin Convergence Criterion . . . . . . . . . . 115 5.3.2 Confidence Contours and Marginalized Statistics . . . . . . 116 5.3.3 Optimizing MCMCs . . . . . . . . . . . . . . . . . . . . . . . 116 5.3.4 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . 118 5.4 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.5 Cosmological Analysis Pipeline . . . . . . . . . . . . . . . . . . . . . 120 6 Early Dark Energy Constraints with Late-Time Expansion Marginaliza- tion 123 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.2 Dark Energy Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 ix 6.2.1 Early Dark Energy (EDE) . . . . . . . . . . . . . . . . . . . . 125 6.2.2 Late Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.3 Datasets and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.3.1 CMB data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.3.2 SNe Ia data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.3.3 Large-Scale Structure data . . . . . . . . . . . . . . . . . . . . 130 6.3.4 Strong Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.3.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.4.1 Early Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . 135 6.4.2 Marginalization over Late-time Expansion . . . . . . . . . . 140 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7 Late-time Dark Energy Dynamics and Massive Neutrinos In Light of DESI Y1 BAO 146 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.2 Dark Energy Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.2.1 Transition model . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.2.2 Binned w(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.2.3 Monodromic k-essence . . . . . . . . . . . . . . . . . . . . . . 151 7.3 Datasets and Analysis Methodology . . . . . . . . . . . . . . . . . . 154 7.3.1 DESI 2024 BAO . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.3.2 Type Ia Supernovae Catalogs . . . . . . . . . . . . . . . . . . 155 7.3.3 Planck 2018 CMB . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.3.4 ACT DR4 CMB Temperature and Polarization Anisotropies 156 7.3.5 ACT DR6 CMB Lensing Power Spectrum . . . . . . . . . . . 156 7.3.6 Dataset Combinations . . . . . . . . . . . . . . . . . . . . . . 157 7.3.7 Analysis Methodology . . . . . . . . . . . . . . . . . . . . . . 158 7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.4.1 Model Parameter Constraints . . . . . . . . . . . . . . . . . . 160 7.4.2 Goodness-of-Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.4.3 Constraints on ∑ mν . . . . . . . . . . . . . . . . . . . . . . . 174 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 8 Conclusions 180 x A Derivations 182 A.1 The Homogeneous and Isotropic Spatial Metric . . . . . . . . . . . . 182 A.2 Christoffel Symbols, Ricci and Einstein Tensors for FLRW Metric . 184 A.3 Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 187 A.4 Klein-Gordon Equation in FLRW . . . . . . . . . . . . . . . . . . . . 191 B Perturbed Cosmological Equations using xPand 192 C Appendices to Chapter 6 199 C.1 Constraints for Dataset Combinations 3, 4 and 5 . . . . . . . . . . . 199 C.2 Principal Component Analysis of Late Dark Energy . . . . . . . . . 199 D Appendices to Chapter 7 205 D.1 Appendix: Full Reconstruction of the Equation of State . . . . . . . 205 D.2 Appendix: Continuous Binned w(z) . . . . . . . . . . . . . . . . . . 205 D.3 Appendix: Transition Model: Varying σ . . . . . . . . . . . . . . . . 207 References 210 xi Chapter 1 Introduction and Motivation This chapter qualitatively explains what is cosmology, its main objects of study, the current state of the field, and the relevance of the research developed during my PhD. The ideas presented in this chapter will be carefully developed throughout the thesis. Cosmology is the study of the Universe’s evolution as it expands. The ex- pansion of the Universe may be a difficult concept to grasp: the effects of the Universe’s expansion can only be measured at very large scales, in distances ty- pically of the order of 1Mpc (one megaparsec), or approximately 3 × 1022m. For comparison, the distance between the Milky Way and Andromeda, the nearest galaxy, is approximately 765kpc away. The expansion of the Universe is a gravitational effect that makes the physical spatial distance between two comoving points (i.e. points that have no relative motion) increases with time. This phenomenon affects different physical processes such as the formation of the first atomic elements, stars, galaxies, galaxy clusters and voids. It affects, for instance, how we perceive light from distant sources. The hypothesis of an expanding Universe was raised in the early twentieth century by physicists Alexander Friedmann and Georges Lemaître [1], and was developed by many great physicists such as James Peebles [2]. Since then, multiple predictions based on the Universe’s expansion have been confirmed, such as the Hubble law for nearby astronomical objects [3], the existence of an almost isotropic Cosmic Microwave Background radiation [4] - as well as statistical properties of its ani- sotropies [5] -, the weak gravitational lensing of light in the large scale structure of the Universe [6], the relative abundance of light elements in stars [7], statisti- cal properties of the spatial distribution of galaxies [8] - in particular the effect of Baryonic Acoustic Oscillations, or BAOs [9] - among many other observable effects. Today, the expansion of the Universe is unanimously accepted by the scientific community and continuously proven by new experiments. General Relativity (GR) provides a theoretical framework that can describe the cosmological expansion effect in the spacetime geometry through the history of 1 Chapter 1. Introduction and Motivation 2 the Universe. In GR, the geometrical properties of space and time are summarized by a metric tensor, whose components are gµν(r, t) and are functions of the spatial position r and time t. The equations of motion for the metric tensor are the Einstein’s field equations, Gµν = 8πGTµν. (1.1) The left hand side of Equation 1.1 is a complicated, nonlinear function of the metric components, its partial derivatives with respect to space and time, and its inverse. In the standard four spacetime dimensions, the components gµν are symmetric, leaving 10 independent components. The Einstein’s field equations are a set of 10 nonlinear, coupled, second-order partial differential equations for the metric components. Due to the enormous complexity of the problem, in order to solve the Einstein’s field equations, physicists rely on approximations and (sometimes bold) assumptions. The main assumption in cosmology is the so-called cosmological principle. To explain it, I use an analogy: the liquid content in a glass of water seems homogeneous. We know that water is formed by molecules, which in turn are formed by atoms, which are formed by more elementary particles. If we visualize a droplet of water in scales of nanometers, 1nm = 10−9m, we would observe that water is not homogeneous at all, but rather is made of a complex structure. Molecules interact with each other, forming intricate structures. Nevertheless, in scales much larger than the nanometer, a glass of water looks homogeneous. In the same way, the description of a single water molecule, composed by two hydrogen atoms and one oxygen atom, is intrinsically anisotropic: the molecule, for instance, has a nonzero electric dipole. However, in an ensemble of molecules, the dipoles are randomly distributed, in such a way that the average electric dipole over the ensemble is zero. The description of the average over the ensemble of particles is, therefore, isotropic: it shows no preferential spatial direction. In the same way, the space is filled with numerous distinct components such as planets, stars, galaxies, black holes and an infinity of astronomical objects, all of them interacting gravitationally. However, at very large scales, the distribution of astronomical objects in the Universe seems homogeneous. The cosmological principle is an assumption that can be stated as: the properties of the Universe, when averaged over sufficiently large (but finite) scales, don’t depend on the position or direction: they are homogeneous and isotropic. Properties of the Universe include the metric components, matter densities and velocities. There is evidence supporting the cosmological principle. The Cosmic Mi- Chapter 1. Introduction and Motivation 3 crowave Background is a radiation signal that can be detected from any part of the Earth, pointing a detector to any direction in the sky. This electromagnetic radiation follows very perfectly a black body radiation distribution at temperature TCMB ≈ 2.725K [10]. The temperature of this radiation is almost isotropic, with very small fluctuations of order ∆T/TCMB ≈ 10−5 [11, 5]. This indicates that the CMB photons, primordial photons created right after the Big Bang, were in thermal equilibrium everywhere, an indication of homogeneity. While there are tests of the cosmological principle using type Ia supernovae [12] and galaxy surveys [13, 14], these are very sensitive to astrophysical effects, and there is no decisive evidence against the principle. The cosmological principle is also closely related to the Einstein equivalence principle, which states that the Earth is not a privileged reference frame in the Universe. In practice, the Cosmological Principle greatly reduces the complexity of the Einstein’s field equations: under the homogeneity assumption, the left and right-hand sides of Equation 1.1 depend only on time, and not on position. Under the cosmological principle, the metric components gµν reduce to a single degree of freedom a(t): the scale factor, a quantity that describes how the distance between any two given points change with time. We can interpret the scale factor in the following way: let A and B be two fixed points in space, and DAB be the physical distance between the points. The expansion of the universe makes this distance change with time. The scale factor is defined such that, for arbitrary time instants t1 and t2, the distance DAB obeys the relation DAB(t1) DAB(t2) = a(t1) a(t2) ∀ A, B. (1.2) It’s common practice to normalize the scale factor today as a(t0) = 11. There- fore, when specializing the Einstein’s field equations to an isotropic, homogeneous and expanding space, we can determine how the scale factor varies in time. The solution of the Einstein’s field equations reveals the expansion history of the Universe. The energy-momentum tensor Tµν, appearing in the right-hand side of Equa- tion 1.1, describes the distribution of energy and momentum across the Universe. For instance, part of the Universe is made by particles described by the Standard 1Quantities with a 0 underscript, such as t0 and a0, are evaluated "today", in the present time. Since the cosmological timescale is much larger than our human time scale, there is no issue with the fact that the definition of "today" or "now" varies continuously. Chapter 1. Introduction and Motivation 4 Model of Particle Physics. Quantum field theory provides a way to calculate the tensor Tµν associated to the particle. At the largest scales, under the cosmological principle, much of the microphysics of the particles is "washed out", and only spatial-averaged quantities such as energy density ρ and pressure P affect the cosmological expansion. To describe cosmological contents of the Universe under the cosmological principle, we employ a coarse-grained fluid description in which particles are homogeneously distributed in space with an energy density and pressure that varies in time. The relation between energy density and pressure is dependent on the specific component. Ultrarelativstic particles (i.e. particles whose kinetic energy pc is much bigger than their rest mass energy mc2) and mas- sless particles are known to follow the relation P = ρ/3. Non-relativistic particles (i.e. particles whose kinetic energy is small compared to their rest mass energy) are pressureless, P = 0. Today, all visible matter in the universe is non-relativistic. Right after the Big Bang, when the temperature of the Universe was extremely high, these particles were ultrarelativistic. The transition between ultrarelativistic and non-relativistic behaviors occurs roughly when the temperature of the Uni- verse cools down below the threshold kBT/mc2 ≈ 1, where m is the mass of the particle. The cosmological principle reduces the Einstein’s field equations in 1.1 to two equations, named Friedmann equations, relating a(t), ρ(t) and P(t), ( ȧ a )2 = 8πGρ 3 , (1.3) ä a = −4πG 3 (ρ + 3P). (1.4) Due to energy conservation, energy density and pressure must obey the conti- nuity equation, dρ dt + 3H(ρ + P) = 0. (1.5) A cosmological model must specify the components of the Universe, that is, the fluids that contribute for the total energy density ρ. Most of the Standard Model particles, which we will call "baryonic matter"2, have become non-relativistic very early in the history of the Universe. Massive neutrinos could still be relativistic if their mass is around mν ≈ 2 × 10−4eV, the mass corresponding to the CMB temperature. To this day, no constraints on the neutrino mass can rule out such 2In cosmology, baryonic matter refers to all Standard Model particles except for photons and neutrinos. Chapter 1. Introduction and Motivation 5 a small value. Photons and massless neutrinos - jointly referred to as radiation - obey the equation P = ρ/3 throughout all cosmological history, although the cosmological community is abandoning the massless neutrino paradigm. The current consensual cosmological model, called ΛCDM, adds two extra components in the Universe in addition to radiation and baryonic matter: cold dark matter (CDM) and a cosmological constant (Λ), also known as dark energy. CDM behaves like non-relativistic matter, with P = 0, and the cosmological constant has a rather strange behavior: it follows P = −ρ. In the subsequent chapters, I will explain in more detail the motivations for having such strange components, as well as their behavior. Evidence for cold dark matter was first observed in the rotational speed of astronomical objects [15]. The mass of a galaxy, for instance, can be estimated from its brightness and from the rotational speeds of stars in the galaxy due to gravity. In the first half of the twentieth century, many observations pointed out that the mass inferred from the gravitational effects is several times bigger than the mass inferred from the object’s luminosity. This "non-luminous"mass was then called dark matter [16]. Since then, the existence of dark matter was confirmed by other methods. The CMB, for instance, can distinguish between dark matter and baryonic matter: the latter interacts with light via Thomson scattering, the former doesn’t. This interaction leaves an imprint in the CMB temperature correlations. CMB measurements from the Planck collaboration point that approximately 30% of the Universe’s energy density is due to non-relativistic matter, but only 5% is due to baryonic matter [5]. Gravitational lensing from galaxies is another effect able to measure the total mass of a galaxy. Since masses distort space, photon trajectories can make curves and circumvent massive objects. The more massive the lens is, the more light will bend: therefore, this lensing effect can also be used to infer the mass of a lens object, such as a galaxy. Finally, one main evidence for dark matter is the relative abundance of light elements in stars and galaxies, which can be predicted by the theory of Big Bang Nucleosynthesis [7]. Using the Boltzmann equation, we can compute, for instance, the abundance of helium that was formed from hydrogen atoms in the early Universe. We obtain the Helium mass fraction of YHe ≈ 0.24, compatible with an Universe with around 5% of baryonic matter. Strong evidence for dark energy was reported in 1998 by two research groups: the High-z supernovae search team led by Adam Riess [17], and the Supernovae Cosmology Project, led by Saul Perlmutter [18]. Type-IA Supernovae are explo- Chapter 1. Introduction and Motivation 6 sions of white dwarf stars. The physical mechanism behind these supernovae is well known, and the intrinsic luminosity from the explosions is roughly the same for all events. Therefore, type-IA supernovae can be treated as standard candles: we can infer the distance DL to the supernovae based on their luminosity. We can also measure its redshift: the change in the light frequency due to the expansion of the Universe. Thus, we can study the relation between z and DL. For nearby supernovae (i.e. those with redshift z ≪ 1), this relation is the Hubble law, DL = 1 H0 z, (1.6) where H0 is the Hubble constant, the current expansion rate of the Universe, defined as H0 = ȧ(t0)/a(t0). For more distant objects, we can see a deviation from this linear approximation, DL = 1 H0 (z + q0z2 + ...), (1.7) where q0 is the current deceleration parameter of the Universe, defined as q0 = −a0 ä0/(ȧ0) 2. An Universe composed by radiation and non-relativistic matter has a strictly positive deceleration parameter: this is a consequence from the Friedmann Equations 1.3, as both the energy density and pressure for radiation and matter are positive. However, both teams found that distant supernovae pointed to a negative deceleration parameter. From the Friedmann equations, this can only happen if ρ + 3P < 0. The final piece in all cosmological models, not just ΛCDM, is the existence of an accelerated expansion period, called inflation, right after the Big Bang [19]. Such an accelerated expansion solves the so-called horizon problem in the CMB. This problem can be stated shortly as: since the CMB photons were emitted early in the history of the Universe, two photons coming from opposite directions could not be in causal contact with each other, and thus could not be in thermal equilibrium. This goes against the evidence that the CMB is very isotropic. However, a period of accelerated expansion would solve this problem, putting all CMB photons in causal contact. The simplest inflation models are realized by a scalar field, named inflaton. Inflation is also a mechanism that provides initial inhomogeneities in the Universe. By calculating the quantum fluctuations of the inflaton in a de Sitter (i.e. exponentially expanding) spacetime, it can be shown that the power spectrum of the field fluctuations is nearly scale-invariant. These fluctuations are Chapter 1. Introduction and Motivation 7 imprinted in the inhomogeneities of the Universe, and can grow or decay during the expansion. CMB anisotropy power spectrum measurements are compatible with initial fluctuations from inflation. Today, we are upon a data-driven era of precision cosmology. The great success of the ΛCDM model in describing such a myriad of data has led to increasingly ambitious and precise experiments able to pinpoint model para- meters with percent-level precision. For instance, the PantheonPlus catalog of type Ia supernovae, calibrated with cepheid variable stars, give a current abun- dance of dark energy of ΩΛ = ρΛ,0/ρtot,0 = 0.666 ± 0.018 and the current ex- pansion rate of H0 = 73.3 ± 1.1 km s−1 Mpc−1 [20]. CMB anisotropy data from the Planck collaboration also constrain the current baryonic matter density to be Ωbh2 = (ρb,0/ρtot,0)h2 = 0.02233 ± 0.000153, the amplitude of inflation-induced perturbations to be As × 109 = 2.101+0.031 −0.034 and its spectral index, or tilt, as ns = 0.9649 ± 0.0042 [5]. Finally, the reionization optical depth, a parameter describing the effect of ionized gas in the CMB photons, is constrained by CMB polarization measurements to be τ = 0.0544 ± 0.0073. The enormous technical effort put in cosmological experiments has paid out: current experiments can place percent-level constraints on the six ΛCDM parameters. A viable cosmological model must be able to consistently explain the myriad of current observational data. The current experimental precision, however, has revealed discrepancies between different experiments. For instance, the value of H0 inferred the local distance ladder is [21] H0 = 73.3 ± 1.04km s−1 Mpc−1, while the value inferred from Planck 2018 CMB anisotropies is H0 = 67.36 ± 0.54km s−1 Mpc−1. Making a simplistic interpretation of these constraints, the posterior distribution of H0 from the SH0ES collaboration is a Gaussian centered at 73.3 with standard deviation 1.04. The mean of the Planck 2018 posterior distribution for H0 is 67.34, which is more than five standard deviations away from the SH0ES mean. Assuming the posterior distribution from SH0ES, the probability of measuring the Planck 2018 mean value for H0 is astronomically low, of order e−25. There are statistical subtleties in making such an interpretation, but the fact is that the 3h = H0/(100 km s−1 Mpc−1) Chapter 1. Introduction and Motivation 8 two measurements are in severe disagreement. This problem has been named the Hubble tension, and it stands as one of the most pressing problems in current physics. A minor but persistent tension is also present between CMB and cosmic shear experiments. The σ8 parameter represents the RMS of matter fluctuations inside a sphere of radius R = 8Mpc/h. This parameter affects the lensing of photons, being most constrained as the combination S8 = σ8 √ Ωm/0.3. CMB measurements from Planck place a constraint of [5] S8 = 0.846 ± 0.017, while cosmic shear measurements from the Dark Energy Survey constrain S8 to be [8] S8 = 0.775+0.024 −0.026. Repeating the same simplistic interpretation of the H0 tension, the difference in means is approximately 2.8σ. While this tension is less extreme, upcoming cosmic shear and CMB surveys will increase the precision, reducing the error bars and potentially aggravating this S8 tension. From a theoretical standpoint, there is no explanation for the nature of dark energy. A cosmological constant could in principle be explained by the vacuum energy of the fields comprising the Standard Model of Particle Physics. Perfor- ming this calculation, however, would yield a cosmological constant orders of magnitude larger than the current measured value. Much theoretical effort is being put into creating fundamental dark energy models and testing their predictions against state-of-the-art cosmological data. Due to the tensions and the lack of theoretical support for a cosmological constant, it is important, to explore alternative dark energy models. Many of these are able to better fit the observed data and alleviate cosmological tensions [22]. Investigating alternative dark energy behaviors can provide information about the fundamental dark energy model, which could shed light on other problems in current physics. As of the time of writing, though, ΛCDM still stands as the standard model of cosmology. My PhD research investigates alternative dark energy models in the context of cosmological tensions. I study scalar field models, fundamental components that can realize general dark energy dynamics in terms of a potential function V(ϕ), usually connected to particle physics. Phenomenological models try to Chapter 1. Introduction and Motivation 9 parametrize the dark energy density ρ(a) or the dark energy equation of state w(a), exploring determined dynamical behavior but having little theoretical support. Within these classes there are infinite possibilities, some of them able to alleviate cosmological tensions. For instance, the Early Dark Energy (EDE) model assumes a non-neglibigle amount of dark energy near recombination, at redshifts z ≈ 3000: this addition can alleviate the H0 tension without degrading the fit for late-time observables. I have studied them using cosmic shear data from the Dark Energy Survey, culminating in the analysis of Chapter 6 and the original work "Early dark energy constraints with late-time expansion marginalization" [23]. On the phenomenological side, my original work "Investigating Late-Time Dark Energy and Massive Neutrinos in Light of DESI Y1 BAO"tests parametrizations of the dark energy equation of state in the context of recent hints of dynamical dark energy from recent supernovae and BAO surveys [24]. Additionally, I have also contributed to the original work "Modeling nonlinear scales with the Comoving Lagrangian Acceleration method: Preparing for LSST Y1" [25], assessing the viability of building a suite of COLA simulations to construct an emulator of the nonlinear matter power spectrum for beyond-ΛCDM models. While the modelling of nonlinear effects represents a great challenge in the analysis of galaxy surveys with extended cosmological models, this work will not be discussed in this thesis. Furthermore, I am a member of two galaxy survey collaborations: the Dark Energy Survey (DES) and the Legacy Survey of Space and Time Dark Energy Collaboration (LSST-DESC). I am currently leading the analysis of an interacting dark energy model using the full observations from DES, and a project to forecast LSST constraints on the dark energy equation of state. In this thesis, I study the theoretical basis of the ΛCDM model, describe the relevant datasets able to provide constraints on the cosmological model, list some alternative candidates to ΛCDM (non-exhaustively), discuss the data analysis techniques used to test dark energy models and present results from my PhD using various models and datasets. This PhD thesis is organized as follows. The first chapters review theoretical aspects of cosmological theory, observations and data analysis. Chapter 2 presents the ΛCDM model, the simplest model that explains all cosmological observations; Chapter 3 reviews recent and historical cosmological measurements and explains current cosmological tensions, the biggest challenges in cosmology today; Chapter 4 reviews modifications to the ΛCDM theory and their status; Chapter 5 explains the standard data analysis methods being utilized by the cosmological community today. The subsequent chapters present original Chapter 1. Introduction and Motivation 10 results developed during my PhD: Chapter 6 presents an analysis of combined Early Dark Energy and smooth dark energy at late-times; Chapter 7 presents the analysis of three different dark energy models using DESI 2024 BAO data, assessing constraints on the neutrino mass. Finally, Chapter 8 summarizes the thesis. During my entire PhD, I acknowledge financial support from São Paulo Re- search Foundation (Fundação de Amparo à Pesquisa do Estado de São Paulo - FAPESP) under grants #2020/03756-2 and #2022/13999-5. Chapter 8 Conclusions The field of cosmology has seen remarkable progress over the last few decades. Theoretically, cosmological perturbation theory has provided a robust framework to describe inhomogeneities and anisotropies in the matter distribution, enabling predictions of observable effects such as cosmic microwave background (CMB) anisotropies and cosmic shear. Observationally, ambitious surveys have measu- red cosmological probes with unprecedented precision, yielding vast amounts of data. Statistically, we have developed techniques to control systematic errors in observations and optimize the extraction of cosmological information. Com- putationally, advanced software has been created to produce fast and accurate theoretical predictions, facilitating efficient data analysis. As is often the case in physics, progress has brought new questions and chal- lenges. The standard cosmological model, ΛCDM, relies on the existence of dark matter and dark energy, whose fundamental nature remains unknown. Further- more, tensions between different cosmological analyses, such as the Hubble and S8 tensions, suggest potential shortcomings in the ΛCDM paradigm. These unre- solved issues have motivated an extensive body of research, including the work presented in this thesis. During my PhD at IFT-UNESP, I conducted an extensive study on the cosmo- logical background and linear perturbation theories, detailed in Chapter 2. This theoretical foundation underpins our understanding of cosmological observables, discussed in Chapter 3, as well as extensions to the standard cosmological model, presented in Chapter 4. These extended models were analyzed using advanced statistical techniques and computational algorithms, as described in Chapter 5. This background enabled the original research I conducted in collaboration with researchers from Stony Brook University, Centro Brasileiro de Pesquisas Físicas, University of Arizona, University of Portsmouth, and University of Oslo. In Chapter 6, our group explored whether the Hubble and S8 tensions could be addressed simultaneously by employing Early Dark Energy to adjust the sound horizon at recombination and late-time dynamical dark energy to modify the growth of matter perturbations. We found that late-time smooth dark energy cannot simultaneously accommodate the high values of H0 measured by the local cosmic distance ladder and the low values of S8 inferred from galaxy surveys. Chapter 7 examines recent indications of dynamical dark energy reported by the DESI collaboration and multiple supernova datasets, including Pantheon+, 180 Chapter 8. Conclusions 181 Union3, and DESY5. We analyzed complex dark energy models to identify the dynamics preferred by the data and reassessed neutrino mass constraints under these extended parametrizations. Our findings suggest a non-monotonic dark energy equation of state and relaxed neutrino mass constraints compared to ΛCDM. In addition to the work presented in this thesis, I contributed to other projects during my PhD. In collaboration with Stony Brook University and CBPF, we developed an emulator for the nonlinear matter power spectrum using the Como- ving Lagrangian Acceleration (COLA) algorithm, an approximation for N-body simulations. This work culminated in the article "Modelling Nonlinear Scales with COLA: Preparing for LSST-Y1" [25]. Additionally, I am analyzing interacting dark energy models as part of the Dark Energy Survey collaboration, using the complete dataset from six years of observations. I am also a member of the LSST DESC collaboration, where I am leading a project aiming to forecast constraints on the dark energy equation of state using LSST 3x2pt data. In the coming years, Stage-IV cosmological surveys, such as those conducted by the Rubin Observatory’s LSST, the Euclid mission, and the Roman Space Teles- cope, will provide an unprecedented volume of data about the Universe. These surveys will illuminate current challenges and likely reveal new ones. As precision increases, inconsistencies in the ΛCDM model may become more apparent, neces- sitating careful data analysis to isolate cosmological signals and the development of sophisticated methods to manage the growing data volume. Exploring extended cosmological models has never been more critical, as the potential falsification of ΛCDM could usher in new physics, shedding light on the fundamental nature of matter and gravity. 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