TESE DE DOUTORAMENTO IFT-T.004/24 Dark Energy and Neutrinos in Cosmology Diogo Henrique Francis de Souza Advisor: Rogério Rosenfeld Co-advisor: Vivian Miranda August, 2024 Souza, Diogo Henrique Francis de S729d Dark energy and neutrinos in cosmology / Diogo Henrique Francis de Souza. – São Paulo, 2024 129 f.: il. color. Tese (doutorado) – Universidade Estadual Paulista (Unesp), Instituto de Física Teórica (IFT), São Paulo Orientador: Rogério Rosenfeld Coorientador: Vivian Miranda 1. Cosmologia. 2. Energia escura - Astronomia. 3. Neutrinos. 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). Dark Energy and Neutrinos in Cosmology 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 Doutor em Física , especialidade Física Teórica. Comissão Examinadora: Prof. Dr. ROGÉRIO ROSENFELD (Orientador) Instituto de Física Teórica/UNESP Prof. Dr. MIGUEL BOAVISTA QUARTIN Universidade Federal do Rio de Janeiro Prof. Dr. FELIPE TOVAR FALCIANO Centro Brasileiro de Pesquisas Físicas Profa. Dra. MARIANA PENNA LIMA VITENTI Universidade de Brasília Prof. Dr. FELIPE ANDRADE OLIVEIRA Physik-Institute - University of Zurich Conceito: Aprovado São Paulo, 09 de Agosto de 2024. Em memória de Maria Francisca e José Souza iii Agradecimentos A realização deste trabalho de doutorado foi possível devido às várias pessoas que me apoiaram ao longo da minha trajetória por estes quatro anos. Neste espaço limitado de linhas, expresso aqui os meus agradecimentos a cada uma destas pessoas. Ao Rogério Rosenfeld, agradeço pela orientação e apoio em todos os momentos durante o doutorado. Sua assistência e dedicação comigo foram essenciais para tornar este trabalho possível. Também agradeço imensamente seu suporte para que eu fosse realizar o doutorado sanduíche. O estímulo do Rogério foi parte fundamental para que o meu intercâmbio se tornasse possível. Agradeço a Vivian Miranda por ter feito parte do meu doutorado desde o início, e por também ter me recebido na Stony Brook University. Também agradeço o seu suporte fundamental para que o meu intercâmbio de fato pudesse ter se tornado realidade. Foi uma experiência incrível e de valor inestimável. Assim como o Rogério, Vivian foi muito importante para a minha formação. Agradecer imensamente aos meus familiares que sempre estiveram comigo em todos os momentos; Adelina, Elizângela, Felipe e Sirlene, Mônica, Euvério e Fabíola. Este trabalho definitivamente não teria sido possível sem um nome em especial; Letícia. Agradeço imensamente por sempre ter estado comigo nos momentos felizes, e também desafiadores, que apenas superei graças ao seu amor, carinho e suporte. Expresso minhas lembranças para meu pai e meus irmãos; Vinícius e Miguel. Agradeço João Rebouças, meu colega de doutorado no IFT e parceiro de pesquisa desde o início desta jornada. Sou muito grato por ter compartilhado estes quatro anos com João, com quem pude aprender muito e tive ótimas conversas. Agradeço também ao Hugo Camacho, Felipe Oliveira e Lucas Faga que sempre de alguma forma estiveram envolvidos durante o meu processo e que portanto foram muito importantes para mim. Também agradeço Raphael e a Renata pelo companheirismo e ótimos momentos durante meu período em Stony Brook. I want to thank my colleagues at Stony Brook University: Jonathan, Evan, Josh, Mark, and Yijie. I’m very grateful to Jonathan for his assistance with my transition from Brazil to New York and for his help during my early days at Stony Brook. Agradeço a todos os funcionários do IFT que pude conhecer; ao pessoal da parte técnica e de administração; Rosane, Edna, Ana, Marcus, Rafael, Walter e Heitor; a dona Jô, e ao pessoal da limpeza; Rita, Mayara e Sandra. Às recepcionistas; Edilene, Lauane e Geane. Aos porteiros Adailton, Valdemiro e Genildo. Obrigado a todos vocês. Expresso os meus agradecimentos a todos os professores de todas as escolas públicas por onde estudei; à professora Cláudia, por suas aulas maravilhosas, à bibliotecária Riva iv pela companhia. Ao professor de Física Luslan que teve um papel muito importante na minha trajetória! E claro, agradeço à UFMG e à UFRGS, instituições com professores que me inspiraram com suas aulas incríveis. Sou uma grata consequência de cada um destes lugares que passei e pessoas que conheci. Por fim, expresso meu agradecimento à CAPES pela bolsa de doutorado, e ao programa CAPES-Print que possibilitou a realização do meu doutorado sanduíche. v Resumo O paradigma cosmológico predominante é o modelo ΛCDM com curvatura espacialmente plana e com condições iniciais adiabáticas e gaussianas. A letra grega “Λ” designa a constante cosmológica, enquanto a sigla “CDM” significa a Matéria Escura Fria e sem pressão. Até hoje, tal modelo tem sido a descrição mais econômica do Universo que é consistente com grande variedades de dados observacionais. Apesar do seu sucesso, o modelo cosmológico padrão não está isento de barreiras. O problema do horizonte e a planicidade são bem conhecidos com possíveis soluções quando trazemos à cena a teoria da Inflação Cósmica. Além disso, os levantamentos cosmológicos alcançaram uma melhorias significativas na precisão das suas medidas de sinais, o que nos permitiu impor restrições mais rigorosas aos parâmetros do modelo ΛCDM. Quando diferentes conjuntos de dados são combinados, os principais problemas que surgem são as chamadas tensões Hubble e S8. Juntamente com a matéria escura, a energia escura e os neutrinos são dois componentes que merecem atenção especial pelo seu papel importante na física de partículas e na cosmologia. Este trabalho está separado em duas partes. Na primeira parte, forneço uma breve introdução à cosmologia, fornecendo referências para maiores detalhes ao leitor interessado. Inicio com as equações de background e de perturbação derivadas da teoria geral da relatividade de Einstein, e termino com as condições iniciais necessárias para resolver essas equações. Na segunda parte desta Tese, me concentro mais especificamente nos neutrinos e no seu efeito na formação de estruturas do universo, e na energia escura como campo escalar. Os dois últimos capítulos tratam de um artigo publicado e de um trabalho em progresso iniciado na Universidade de Stony Brook, respectivamente. Palavras chaves: cosmologia; neutrinos massivos; energia escura primordial; energy escura em tempos tardios. Field: física; cosmologia. vi Abstract The prevailing cosmological paradigm is the spatially flat ΛCDM model with adiabatic and Gaussian initial conditions. The Greek letter “Λ” designates the cosmological constant while the acronym “CDM” means the pressureless Cold Dark Matter. Until today, this has been the more economical description of the Universe consistent with an assortment of observational probes. Although its success, the benchmark cosmological model is not absent of issues. The horizon and flatness are well-known long-term problems with possible solutions when we bring the Cosmic Inflation theory. Furthermore, cosmological surveys have attained significant precision improvement in their signal measurements, which have enabled us to impose more stringent constraints on the ΛCDM parameters. When different probes are combined, the main challenges are the so-called Hubble and S8 tensions. Alongside dark matter, dark energy and neutrinos are two components that deserve special attention for their important role in particle physics and cosmology. This work is separated into two parts. In the first one, I provide a brief introduction to cosmology, providing references for further details to the interested reader. This starts with the background and perturbation equations derived from Einstein’s general theory of relativity and ends with the initial condition required to solve those equations. In the second part of this Thesis, I focus more specifically on neutrinos and their effect on structure formation and dark energy as a scalar field. The last two chapters are about a published paper and a work in progress started at Stony Brook University, respectively. Key words: cosmology; massive neutrinos; early dark energy; late-time dynamical dark energy. Field: physics; cosmology. vii Contents I The Basis of Cosmology 1 1 Background and perturbed universe 2 1.1 Robertson-Walker metric . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Cosmological distances . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Luminosity and angular diameter distance . . . . . . . . . . 6 1.3 Friedmann equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 Equation of state and background evolution . . . . . . . . . 13 1.4 Linear perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.1 From smooth to perturbed metric . . . . . . . . . . . . . . . 18 1.4.2 Gauge transformations for metric tensor . . . . . . . . . . . 19 1.4.3 From smooth to perturbed energy-momentum tensor . . . . 21 1.4.4 Perturbed equations of motions in Newtonian gauge . . . . 22 1.4.5 Proof of some equations . . . . . . . . . . . . . . . . . . . . . 25 1.4.6 Evolution of the gravitational potential . . . . . . . . . . . . 29 1.5 Cosmic Microwave Background . . . . . . . . . . . . . . . . . . . . . 34 1.5.1 Primordial spectrum and CMB . . . . . . . . . . . . . . . . . 35 2 Gauge invariant perturbations 42 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2 Setting equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3 Building gauge invariant perturbations . . . . . . . . . . . . . . . . 44 2.4 Applying the method . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4.1 Bardeen’s potentials . . . . . . . . . . . . . . . . . . . . . . . 45 2.4.2 Comoving curvature perturbation . . . . . . . . . . . . . . . 48 2.4.3 Lukash variable . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.4.4 Non-adiabatic pressure perturbation . . . . . . . . . . . . . . 49 2.4.5 Comoving gauge density perturbation . . . . . . . . . . . . . 50 3 Initial conditions 52 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 viii 3.1.1 Cold Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1.2 Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1.3 Photons and baryons . . . . . . . . . . . . . . . . . . . . . . . 53 3.1.4 The Einstein-Field equations . . . . . . . . . . . . . . . . . . 54 3.2 Early-times and super-horizon initial conditions . . . . . . . . . . . 56 3.2.1 Zeroth order and higher order correction . . . . . . . . . . . 60 3.2.2 First order correction . . . . . . . . . . . . . . . . . . . . . . . 62 3.2.3 Adiabatic initial conditions . . . . . . . . . . . . . . . . . . . 63 II Dark Energy and Neutrinos in Cosmology 64 4 Mass Varying Neutrinos and the Hubble tension 65 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 Neutrino-assisted EDE . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3 A background analysis of the neutrino-assisted EDE . . . . . . . . . 71 4.4 Initial value of the EDE field . . . . . . . . . . . . . . . . . . . . . . . 73 4.4.1 Field initially frozen . . . . . . . . . . . . . . . . . . . . . . . 74 4.4.2 Field initially dynamical . . . . . . . . . . . . . . . . . . . . . 75 4.5 Field displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.6 Minimum of the effective potential . . . . . . . . . . . . . . . . . . . 82 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5 Dark energy parametrizations 86 5.0.1 Constraining EDE with late DE marginalization . . . . . . . 87 5.1 General polynomial dark energy equation of state . . . . . . . . . . 89 5.1.1 Constant w(z) . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.1.2 Constant w(z) in bins of redshift . . . . . . . . . . . . . . . . 91 5.1.3 Linear w(z) in bins of redshift . . . . . . . . . . . . . . . . . 92 5.1.4 Quadratic w(z) in bins of redshift . . . . . . . . . . . . . . . 93 5.1.5 Cubic w(z) in bins of redshift . . . . . . . . . . . . . . . . . . 96 5.2 CAMB for Late Dark Energy: . . . . . . . . . . . . . . . . . . . . . . 102 5.2.1 Late-Time Dark Energy and Massive Neutrinos with DESI 2024 BAO data . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Conclusion 115 Bibliography 117 ix Part I The Basis of Cosmology 1 Chapter 1 Background and perturbed universe 1.1 Robertson-Walker metric Cosmology is the study of the universe as a whole, this means that we should be able to describe how the universe evolves, predict its expansion rate through cosmic time, estimate positions and distances, as well as velocities of astrophys- ical objects (e.g. galaxies) with respect to an observer (e.g. us), etc. This can be understood as the “kinematics” study of the universe. Naturally, the next step is to ask who are the agents (e.g. photons, neutrinos, baryons, dark matter, and dark energy) and forces (e.g. gravity and the repulsive nature of dark energy) responsible for making the universe evolve as it does. In this case, it is essential to know the equations that describe the abundances, evolution, and interactions (that depend on the epochs of the universe) between the various components of the cosmos. Similarly to the previous kinematic case, this can be seen as the “dy- namical” study of the universe. This chapter doesn’t intend to exhaust the content of each mentioned topic. Still, it provides the basic elements for an introduction to cosmology and further references where details are far beyond the scope of this thesis. This chapter closely follows the textbook [1] by Daniel Baumann. We start defining basic quantities used through this text: a coordinate system is denoted by xµ = (x0, xi) = (t, xi) = (t, x). While Greek indices (µ, ν, etc) can assume integer values between 0 and 3, Latin indices (i, j, etc) run over the values 1, 2, 3. Still about the coordinate system, the first component of xµ, i.e. x0, represents time (there are many ways that we can measure time in cosmology, some of the most common are: cosmic time denoted by t, conformal time τ, scale factor a, redshift z and the cosmic microwave background temperature T. When necessary, we will see how these quantities relate to each other). The three- component xi describes the usual three-dimensional space part of the spacetime coordinate system. An infinitesimal shift in the components of xµ, represented by dxµ, can be translated into the invariant line element, ds2, through the (spacetime) 2 Chapter 1. Background and perturbed universe 3 metric tensor gµν ds2 = 3 ∑ µ,ν=0 gµν(xα)dxµdxν ≡ gµν(xα)dxµdxν, (1.1) where we assume the signature (−,+,+,+) for the metric and Einstein’s notation for the sum over repeated indices as indicated in Eq. (1.1). In addition, note that the metric is, in general, a function of the coordinate system xα = (t, x). We assume the Cosmological Principle (CP), i.e. that the space is homogeneous and isotropic, which means that the three-space is maximally symmetric (the conversely also is true, i.e., a metric space that is maximally symmetric implies homogeneity and isotropy). By definition, the space is said homogeneous if there exist infinitesimal isometries that carry any given point xµ into any other point in its immediate neighborhood. The space is said isotropic about a given point xµ if there exist infinitesimal isometries that do not change the point xµ and for which the first derivative exists (see Weinberg’s textbook for details on this matter [2]). The previous definition guarantees the existence of maximally symmetric three- space under homogeneity and isotropic conditions. Ultimately, the CP simplifies the four-dimensional line element, Eq. (1.1), for ds2 = −dt2 + a2(t)dl2, (1.2) dl2 ≡ γij(xk)dxidxj, (1.3) where Eq. (1.3) defines the line element of the three-dimensional space and a(t) is the scale factor that parameterize the expansion or contraction of the universe. CP also implies in the uniqueness of this property, i.e., that exists a coordinate transformation that carries two maximally symmetric metric spaces into the other if the two maximally symmetric metrics have the same constant intrinsic curvature k for the three-dimensional space and the same numbers of eigenvalues of each sign [2]. We can deplete the possibilities with just three options: zero curvature or flat space (k = 0, e.g. the 3-dimensional Euclidean space E3), positive curvature (k = +1, e.g. the 3-sphere S3), or negative curvature (k = −1, e.g. the 3-hyperboloid H3). The line element for the spatial part of the metric in Eq. (1.3), dl2, for each Chapter 1. Background and perturbed universe 4 case of the constant intrinsic curvature k is given by [1] zero curvature: dl2 = dx2 = δijdxidxj, (1.4) positive curvature: dl2 = dx2 + du2, u2 + x2 = R2 0, udu = −x · dx, (1.5) negative curvature: dl2 = dx2 − du2, u2 − x2 = R2 0, udu = +x · dx, (1.6) where u is an auxiliary coordinate that allows us to embed a three-sphere in a four-dimensional Euclidean space in the case of Eq. (1.5) with R0 being the radius of the sphere. Switching the sign in front of u we also switch the geometry for the case of a three-hyperboloid embedded in a four-dimensional Lorentzian space represented by Eq. (1.6). We can turn the three Eqs. (1.4) - (1.6) into just one at the price of having explicitly the intrinsic constant curvature k [1] dl2 = dx2 + k (x · dx)2 R2 0 − kx2 = γijdxidxj, γij ≡ δij + k xixj R2 0 − k(xmxm) , k = 0,+1,−1. (1.7) The observer usually sets a sphere around the Earth when looking into the sky. Therefore, it is useful to write Eq. (1.7) in terms of the spherical coordinate system. In this case, we have dx2 = dr2 + r2dΩ2, x · dx = rdr, x2 = r2 and the line element on a unit sphere is dΩ2 ≡ dθ2 + sin2θdϕ2. Simplifying the resulting expression, we obtain dl2 = dr2 1 − k(r/R0)2 + r2dΩ2. (1.8) The spacetime is not maximally symmetric, but it can be spanned into a family (i.e. sub-spaces) of maximally symmetric metric spaces. The way to do this is to fo- liate the spacetime into three-dimensional space of constant time, all sewn together by the parameterization function called scale factor a(t) defined in Eq. (1.2) [1, 2]. This procedure leads to the Robertson-Walker (RW) metric ds2 = −dt2 + a2(t) [ dr2 1 − kr2/R2 0 + r2dΩ2 ] . (1.9) In Eq. (1.9) we have to determine the form of the scale factor (i.e. the why the universe expands or contracts), whether the three-space is flat (k = 0), positively (k = +1) or negatively (k = −1) curved and if not flat, we also have to determine Chapter 1. Background and perturbed universe 5 the intensity of the curvature given by the curvature scale R0. In other words, we no longer need to determine the ten components of the symmetric metric tensor gµν, just a time-dependent function, a(t), and two numbers, k and R0 (if k ̸= 0). The RW metric in Eq. (1.9) is left invariant under rescaling symmetry: a → λa, r → r/λ, R0 → R0/λ, or a → a/λ, r → λr, R0 → λR0. A position given by r = (r, θ, ϕ) is called a comoving coordinate system with comoving distance r and the combination rph = a(t)r is called physical coordinate system with physical distance rph = a(t)r because under the rescaling (a → λa, r → r/λ or a → a/λ, r → λr) rph is left unchanged. If we take the time derivative of the physical coordinate we have vph ≡ ṙph = ȧ(t)r + a(t)ṙ = H(t)rph + vpec. Where we have defined the Hubble parameter by H ≡ ȧ a , (1.10) where overdot is the derivative with respect to time t. The physical velocity have the contribution of the Hubble flow, Hrph, and the peculiar velocity vpec ≡ aṙ. We can rewrite the RW metric in Eq. (1.9) in a more economical way by redefin- ing the metric component grr as follows dχ ≡ dr/ √ 1 − kr2/R2 0. To solve this inte- gral we do the change of variable √ k(r/R0) = sin(z) ⇒ √ k(dr/R0) = cos(z)dz, therefore we have dχ = R0√ k cos(z)dz√ 1 − sin2(z) = R0√ k dz ⇒ χ = R0√ k z. (1.11) Going back to the comoving coordinate r we have χ = R0√ k sin−1 (√ k r R0 ) , (1.12) r = R0√ k sin (√ k χ R0 ) . (1.13) We can split Eq. (1.13) into three cases according to the value of k. If k = +1 then r = R0 sin(χ/R0). For k = 0 we have limk→0[(R0/ √ k) sin( √ kχ/R0)] = χ. Finally, for k = −1 we have r = (R0/i) sin(iχ/R0) where i = √ −1. Using that sin(iθ) = i sinh(θ) we can write the negative curvature case as r = R0 sinh(χ/R0). Chapter 1. Background and perturbed universe 6 In summary: Sk(χ) ≡ r = R0 ×  sin (χ/R0) if k = +1 χ/R0 if k = 0 sinh (χ/R0) if k = −1 . (1.14) In this way, the RW metric Eq. (1.9) can compactly be written as follows ds2 = a2(τ) [ −dτ2 + ( dχ2 + S2 k(χ)dΩ2 )] , (1.15) where we have defined the conformal time dτ ≡ dt a(t) . (1.16) 1.2 Cosmological distances Determining how far and faster are objects moving with respect to us is not an easy task and may lead to some misconceptions [3]. Despite its complexity, defining and estimating distances it is an important and essential source of cosmo- logical information. For example, evidence that the universe is not static and could actually be expanding, came through Hubble’s paper [4] in 1929 that showed a linear relation between velocity and distance of extra-galactic nebulae (Cepheids). The confirmation of the accelerated expansion of the universe using type Ia super- novae led the 2011 Nobel Prize shared by Saul Perlmutter et al. [5], A. D. Riess et al. [6] and Brian P. Schmidt et al. [7]. In all these works, cosmological distances played an important role. Due to its importance, in the following, we discuss the definition of luminosity distance and angular diameter distance. 1.2.1 Luminosity and angular diameter distance The luminosity distance is defined using the known energy emitted per unit time or simply luminosity L (hence the name) of an astronomical object. Then we measure the observed flux F. The relation between L and F is mediated by distance, however, the flux F has a unit of energy per unit of time per area, therefore we should be concerned about how these three quantities (area, time, and energy) change in an expanding universe. Chapter 1. Background and perturbed universe 7 In a Euclidean space, the comoving distance r is simplified to χ, then a sphere has an area equal to 4πχ2, however, according to Eq. (1.14) the comoving distance depends on the spatial curvature of the universe. Therefore we should generalize to include the possibility of curved space, in this case, we have Ak ≡ 4πS2 k(χ). There are other two corrections that we should be concerned about. In addition, due to the universe’s expansion and assuming a fixed time interval, photons are able to travel smaller distances at late times on the comoving coordinate system than at early times because the associated physical distance at late times is bigger. Therefore the rate at which photons arrive us (or equivalently the number of photons crossing a shell) is smaller than at early times (at the emission epoch) by a factor of 1/(1 + z). Lastly, we should remember that the photon wavelength is related to its energy by the equation E = h/λ where h is Planck’s constant. In an expanding universe the wavelength is stretched out (or redshifted) then the energy today is smaller than at emitted time by a factor of 1/(1 + z). These two last effects produce Ak → Ak(1 + z)2 then we have the right relation between the flux and the luminosity F = L 4πS2 k(χ)(1 + z)2 ≡ L 4πd2 L , (1.17) where we define the luminosity distance dL = (1 + z)Sk(χ). (1.18) Note that dL depends on Sk(χ), hence is a function of the intrinsic constant spatial curvature k. The value of k is determined by the total amount of energy density. Therefore, ultimately the luminosity distance is a function of the cosmological parameters. The small angle δθ ≪ 1 subtended by an object of physical size l (thus of comoving size l/a = l(1 + z)) where a is the scale factor when the light is emitted is related to its comoving distance by the following δθ = l(1 + z) Sk(χ) ≡ l dA , (1.19) where we define the angular diameter distance by dA = Sk(χ) 1 + z . (1.20) Chapter 1. Background and perturbed universe 8 1.3 Friedmann equations The Einstein field equation gives the dynamics of the universe. It describes how the Einstein tensor, Gµν, responds to the presence of matter and energy distribution Tµν Gµν ≡ Rµν − 1 2 Rgµν = 8πGTµν, (1.21) where gµν is the metric tensor that translates the coordinates system, xµ = (t, xi), into an invariant line element, ds2 (see Eq. (1.9)). Rµν and R are the Ricci tensor and the Ricci scalar, respectively Rµν = ∂λΓλ µν − ∂νΓλ µλ + Γλ λρΓρ µν − Γρ µλΓλ νρ, (1.22) R = Rµ µ = gµνRµν. (1.23) The relation between the Christoffel symbols and the metric tensor is Γµ αβ = 1 2 gµλ ( ∂αgβλ + ∂βgαλ − ∂λgαβ ) . (1.24) Once equipped with the metric tensor, gµν, it is possible to compute the left- hand side (l.h.s) of Eq. (1.21) through the set of Eqs. (1.22)-(1.24). However, we still need to specify the right-hand side (r.h.s) of Eq. (1.21), i.e. the stress energy- momentum tensor Tµν. Provided this source term for the Einstein equation, we ended with a differential equation that (given appropriate initial conditions) can be solved to describe the dynamics of the universe. Under the assumption of the CP, it is possible to show [2] that a general expression for the energy-momentum tensor is highly constrained, reducing to the form of a perfect fluid Tµν = (ρ + p)UµUν − Pgµν, (1.25) where ρ and P are the energy density and the pressure of the fluid, respectively. Uµ is the 4-velocity. Now we have all the ingredients to compute the Friedmann equations, which describes the universe’s dynamics. In order to do this, we start computing the l.h.s of Eq.(1.21). Considering a flat universe (k = 0) we can evaluate Eq. (1.24) for the RW metric. Chapter 1. Background and perturbed universe 9 In this scenario, Eq. (1.9) becomes Null curvature: ds2 = dt2 − a2dx2, (1.26) Cartesian coordinates: dx2 = dx · dx = dx2 + dy2 + dz2. (1.27) Therefore, the components of the RW metric tensor are g00 = 1, gij = −a2δij, gµλgλν = δ µ ν , (1.28) where δij e δ µ ν are the Kronecker delta, and gµλ is the inverse of the gλν. In total, we have 4 × 4 × 4 = 64 Christoffel symbols. Therefore, it is convenient to set one of the indices to zero and compute the spatial part for the others. Using Eq. (1.24), it follows that Γ0 ij = 1 2 g0λ ( ∂igjλ + ∂jgiλ − ∂λgij ) = −1 2 g0λ∂λgij = −1 2 g00∂0 ( −a2δij ) = aȧδij or = a2Hδij, (1.29) where we define the Hubble parameter H ≡ 1 a da dt = ȧ a . (1.30) Conversely, from Eq. (1.24), we have Γi 0j = 1 2 giλ (∂0gjλ + ∂jg0λ − ∂λg0j ) = 1 2 giλ∂0gjλ = 1 2 gij∂0gjj = 1 2 ( − 1 a2 δij ) ∂0 ( −a2δjj ) = 2aȧ 2a2 δijδjj = ȧ a δi j or = Hδi j. (1.31) Allowing the three indices to be only spatial, we obtain Γi jk = 1 2 giλ (∂jgkλ + ∂kgjλ − ∂λgjk ) = −1 2 giλ∂λgjk = −1 2 gi0∂0gjk = 0 ∀ i, j, k. (1.32) Setting simultaneously α = β = 0 we have Γµ 00 = 1 2 gµλ(2∂0g0λ − ∂λg00) = 0 ∀ µ. (1.33) Chapter 1. Background and perturbed universe 10 Analogously Γ0 0β = 1 2 g0λ ( ∂0gβλ + ∂βg0λ − ∂λg0β ) = 0 ∀ β. (1.34) Finally, if we switch α → β and β → α and using gαβ = gβα, from Eq. (1.24) it follows that Γµ αβ = Γµ βα. (1.35) Therefore, we have for a flat universe with a RW metric the following Christoffel symbols Γ0 ij = a2Hδij , (1.36) Γi 0j = Hδi j , (1.37) Γi jk = Γµ 00 = Γ0 0β = 0 . (1.38) We illustrate those 64 Christoffel symbols distributed in Eqs. (1.36), (1.37), and (1.38) in Fig. 1.1. As indicated by the figure, most part of the symbols are nulls (represented by the light gray circles). µ ( Γµ 00 ) β ( Γ0 0β ) α ( Γ0 α0 ) 0 1 2 3 1 2 3 1 2 3 Figure 1.1: Christoffel symbols visualization for the RW metric and flat universe. We proceed to compute the Ricci tensor and the Ricci scalar for the RW met- ric. Firstly, consider the following decomposition for the Ricci tensor Rµν ≡ {R00, Rij, R0i} and lets compute each of these terms separately. Using the Christof- fel symbols displayed in Eqs. (1.36)-(1.38) and the relation between the Ricci tensor Chapter 1. Background and perturbed universe 11 and the Christoffel symbols by Eq. (1.22) we have R00 = ∂λΓλ 00 − ∂0Γλ 0λ + Γλ λρΓρ 00 − Γρ 0λΓλ 0ρ = −∂0Γi 0i − Γl 0kΓk 0l = −∂0 ( ȧ a δi i ) − ( ȧ a δl k ȧ a δk l ) = − ( äa − ȧ2 a2 ) δi i − ȧ2 a2 δl l = −3 ( ä a − ȧ2 a2 + ȧ2 a2 ) R00 = −3 ä a . (1.39) The second term in our list is Rij, then by Eq. (1.22) we have Rij = ∂λΓλ ij − ∂jΓλ iλ + Γλ λρΓρ ij − Γρ iλΓλ jρ = ∂0Γ0 ij + Γl l0Γ0 ij − ( Γ0 imΓm j0 + Γk i0Γ0 jk ) = ∂0 ( aȧδij ) + ȧ a δl l aȧδij − ( aȧδim ȧ a δm j + ȧ a δk i aȧδjk ) = aäδij + ȧ2 ( δij + δl l δij − δimδm j − δk i δjk ) = aäδij + ȧ2 (δij + 3δij − δij − δij ) = ( aä + 2ȧ2 ) δij= ( ä a + 2 ȧ2 a2 ) a2δij Rij = − ( ä a + 2H2 ) gij, where H = ȧ/a and gij = −a2δij. (1.40) Finally, from the RW metric we have that g0i = 0 and T0i = 0 for i = 1, 2, 3. Therefore, by the Einstein equation we obtain that R0i is zero, as shown bellow G0i = R0i − 1 2 Rg0i = R0i = 8πGT0i = 0. (1.41) We advance to compute the Ricci scalar given by Eq. (1.23) R = Rµ µ = gµνRµν = g00R00 + gijRij = 1 · ( −3 ä a ) − 1 a2 δij ( aä + 2ȧ2 ) δij = −3 ä a − ( ä a + 2 ȧ2 a2 ) 3δijδij = −6 ( ä a + ȧ2 a2 ) (1.42) R = −6 ( ä a + H2 ) . (1.43) Armed with the Ricci tensor (Eqs. (1.39)-(1.41)) and the Ricci scalar (Eq. (1.43)) for the RW metric and the energy momentum-tensor for the perfect fluid (Eq. (1.25)), we are able to compute the time-time component (i.e. µ = ν = 0) of the Einstein Chapter 1. Background and perturbed universe 12 equation (1.21) G00 = R00 − 1 2 Rg00 = 8πGT00. (1.44) Using Eq. (1.39) for R00, Eq. (1.43) for R and T00 = ρ, Eq. (1.44) becomes G00 = −3 ä a − 1 2 [ −6 ( ä a + ȧ2 a2 )] · 1 = 8πGρ ⇒ ȧ2 a2 = 8πG 3 ρ which can be rewritten as follows H2 = 8πG 3 ρ . (1.45) Equation (1.45) is known as the First Friedmann equation for the dynamics of the universe. It says that an infinitesimal change (e.g. an expansion or contraction) in the scale factor, da, in an infinitesimal time interval, dt, is proportional to the square root of the product between the scale factor by the energy density √ aρ at the current time t. Analogously, we do the same above procedure for the space-space component (i.e. µ = ν = i) of the Einstein equation Gii = Rii − 1 2 Rgii = 8πGTii. (1.46) Plugging the RW metric, Eq. (1.9), into the energy-momentum tensor, Eq. (1.25), we have Tii = −Pgii = Pa2δii. In addition, using the Ricci tensor in Eq. (1.40) and again the Ricci scalar in Eq. (1.43) we have Gii = ( aä + 2ȧ2 ) δii − 1 2 [ −6 ( ä a + ȧ2 a2 )] ( −a2δii ) = 8πG ( Pa2δii ) . (1.47) Simplifying the above expression, we have 2ä/a + ȧ2/a2 = −8πGP which is the Second Friedmann equation ä a = −4πG 3 (ρ + 3P) . (1.48) Chapter 1. Background and perturbed universe 13 1.3.1 Equation of state and background evolution Inspecting the Friedmann (1.45) we should conclude that to find how scale factor evolves with time, it is necessary to know the energy density ρ, i.e. the expansion history of the universe depends on its energetic content. We can find ρ as follows: in general relativity, the energy-momentum tensor for a perfect fluid, expressed by Eq. (1.25), satisfies the covariant conservation equation ∇µTµ ν = ∂µTµ ν + Γµ µλTλ ν − Γλ µνTµ λ = 0. (1.49) If we set ν = 0 in Eq. (1.49) and use the energy-momentum tensor in Eq. (1.25), we ended with a differential equation for the energy density known as continuity equation ρ̇ + 3 ȧ a (ρ + P) = 0 ⇔ ρ̇ + 3 ȧ a ρ (1 + w) = 0, (1.50) where in Eq. (1.50) we have defined an important parameter called equation of state (EoS) w ≡ P ρ . (1.51) The main elements of the universe that contribute to the energy momentum- tensor (thus being important to the universe’s expansion history through the Friedmann equations) are photons, neutrinos, cold dark matter (CDM), baryons and dark energy. If the pressure, P, and the energy density, ρ, are known for some of these listed components, then it should be possible to access its equation of state, w, by Eq. (1.51). In the following, we show an example of how to compute the EoS for photons. We start with the number density, n, weighted by the density of states, g/h3, where g is the internal degree of freedom (e.g. spin) and h = 2πh̄ is the Planck’s constant n = g (2πh̄)3 ∫ d3p f (p). (1.52) where f (p) is the Fermi-Dirac or Bose-Einstein distribution function f (p, T) = 1 eE(p)/KBT ± 1 , (1.53) Chapter 1. Background and perturbed universe 14 where the +, − signs represent fermions and bosons, respectively. Neglecting the interaction energy between particles, the relativistic energy of a particle with mass m and momentum p is E(p) = √ p2c2 + m2c4. (1.54) We use the number density to find ρ and P. While the energy density is weighted by E(p), the pressure is weighted by p2/(3E(p)) (for further details see [1]), then we have ρ = g (2πh̄)3 ∫ d3 f (p)E(p) (1.55) P = g (2πh̄)3 ∫ d3 f (p) p2 3E(p) (1.56) Now we can write the integrals for n and ρ only in terms of the momentum, which will allow us to solve them n = g 2π2h̄3 ∫ ∞ 0 dp p2 exp [√ p2c2+m2c4 kBT ] ± 1 , (1.57) ρ = g 2π2h̄3 ∫ ∞ 0 dp p2 √ p2c2 + m2c4 exp [√ p2c2+m2c4 kBT ] ± 1 . (1.58) Before to compute the pressure, it is useful to define new variables x ≡ mc2 kBT , ξ ≡ pc kBT , (1.59) Also is convenient to remember the following identity ∫ ∞ 0 dξ ξn eξ − 1 = ζ(n + 1)Γ(n + 1) = n!ζ(n + 1) = { 2ζ(3) if n = 2, π4/15 if n = 3, (1.60) where ζ and Γ are the Zeta and the Gamma functions, respectively. Now we proceed to compute the EoS for relativistic particles, in particular for photons. In this case, we set the particle’s mass to zero (m = 0) in the pressure’s Chapter 1. Background and perturbed universe 15 equation (1.56) P = g (2πh̄)3 ∫ d3p f (p) p2 3E(p) = g 6π2 k4 B c5h̄3 T4 ∫ ∞ 0 dξ ξ3 eξ − 1 = 1 3 ( π2 30 g k4 B c5h̄3 T4 ) . It is possible to show that ρ = π2 30 g k4 B c5h̄3 T4, which in natural units (kB = c = h̄ ≡ 1) reduces to the result of [1]. Then we finally have P = ρ 3 . (1.61) To estimate the number density and the energy density of photons today we rewrite the equations for n (Eq. (1.57)) and ρ (Eq. (1.58)) in terms of the new variables, x and ξ, defined in Eq. (1.59) n = g 2π2 T3 ( kB h̄c )3 I±(x), I±(x) ≡ ∫ ∞ 0 dξ ξ2 exp [√ ξ2 + x2 ] ± 1 , (1.62) ρ = g 2π2 T4 ( kB h̄c )3 kB c2 J±(x), J±(x) ≡ ∫ ∞ 0 dξ ξ2 √ ξ2 + x2 exp [√ ξ2 + x2 ] ± 1 . (1.63) Considering the relativistic limit, x → 0, and using for the photon temperature T0 = 2.73 K, we have nγ,0 = 2ζ(3) π2 T3 0 × (kB/h̄c)3 ≈ 412 photons/cm3, (1.64) likewise for the energy density ργ,0 = π2 15 T4 0 × (k4 B/h̄3c5) ≈ 4.7 × 10−34 g/cm3. (1.65) Using the critical energy density today ρcr0 = 3H2 0 8πG = 1.29 × 10−29h2g/cm3, (1.66) then the density parameter for the CMB is Ωγ,0h2 = ργ,0 ρcr,0 h2 ≈ 2.46 × 10−5. (1.67) Chapter 1. Background and perturbed universe 16 Equation (1.61) shows the EoS for photons. However, the universe is made of elements other than photons. Bellow we just summarize the EoS for these components: photons (γ), cold dark matter (c), and dark energy as cosmological constant (Λ) (for further details see [1, 2, 8]) wγ = 1 3 , wc = 0, wΛ = −1. (1.68) The previous equation, (1.68), shows that w is time independent for photons, CDM and the cosmological constant, but this is not a rule in general. The EoS for neutrinos and dark energy as a scalar field, for example, varies over time. 1 2 3 4 5 x 0.20 0.25 0.30 w Fermion Boson Figure 1.2: Equation of state for fermions and bosons. Note that when the particle’s mass is much smaller than the particle’s temperature, m/T ≪ 1 ⇔ x ≪ 1, w approaches to the value of the equation of state for photons, w → 1/3 ≈ 0.333, Lets look into more details for the case of neutrinos. From the oscillation experiment, we know that neutrinos have mass, then we can not set to zero the mass term in Eq. (1.58), however, we still can study the asymptotic behavior for the EoS for neutrinos (henceforth wν) in the very early/late time regime of the universe. The neutrino temperature was bigger (the comparison makes sense in natural units) than its mass, therefore we have the ratio m ≪ T ⇔ x/ξ ≪ 1. This regime is known as relativistic where the neutrino mass can be neglected, and we have wν ≈ 1/3. On the opposite side, at later times, neutrinos are non- relativistic. In the non-relativistic regime, neutrinos behave like CDM, and in this case, we have wν ≈ 0. Indeed, we can verify this statement numerically by solving the ratio of Eq. (1.56) by Eq. (1.55), the result is shown in figure 1.2. Note that x → 0 ⇒ w → 1/3, conversely x → ∞ ⇒ w → 0. Once we have the equation of state, we can integrate the continuity equation Chapter 1. Background and perturbed universe 17 10 9 10 7 10 5 10 3 10 1 a 10 4 10 3 10 2 10 1 100 CDM baryon photon m 1 = 0 eV m 2 = 0.05 eV m 3 = 0.01 eV Figure 1.3: The evolution of the density parameter for each component: cosmologi- cal constant (Λ), cold dark matter (CDM), baryons, photons, and neutrinos, where for the latter we are assuming one massless neutrino and two massive neutrinos. (1.50) to obtain the energy density as a function of the scale factor ∫ ρ0 ρ dρ′ ρ′ = ln ρ0 ρ = −3 ∫ a0 a da′ a′ (1 + w(a′)). (1.69) If the equation of state is constant, Eq. (1.69) can be integrated analytically ρ = ρ0 ( a a0 )−3(1+w) . (1.70) We can compute Eq. (1.70) for each equation of state in Eq. (1.68) ργ ∝ a−4, ρc ∝ a−3, ρΛ ∝ constant. (1.71) The energy density for neutrinos, ρν, should be computed numerically since its equation of state changes with time. However, notice by Figure. 1.3 that in the limit of high redshift, neutrinos behave like radiation with the equation of state ≈ 1/3 and in the regime of low redshift, neutrinos behave like dark matter with the equation of state ≈ 0. To eliminate the proportionality constant in the previous equations (1.71), we Chapter 1. Background and perturbed universe 18 define the density parameter Ωi0 ≡ ρi0 ρcr0 . (1.72) Using this definition, we have Ωγ = Ωγ0a−4, Ωm = Ωm0a−3, ΩΛ = ΩΛ0, Ων = ρν/ρcr0. (1.73) In figure 1.3 we show the evolution for each component. 1.4 Linear perturbation theory 1.4.1 From smooth to perturbed metric The cosmological principle stated in section 1.1, says that the universe can be represented by spatially flat hyper-surfaces, Στ, of constant time. This translates into the Robertson-Walker metric ds2 = a2(τ) [ dτ2 − δijdxidxj ] . (1.74) Therefore, homogeneity and isotropy are implicit in the RW metric. However, the universe is not perfectly homogeneous and isotropic, for example: galaxies, voids (regions of space with under-density of matter), clusters, and super-clusters of galaxies form filaments that spread through space. In addition, the temperature fluctuations in the cosmic microwave background are of the order 10−5, which is small but not zero. Therefore, it is important to modify the metric in Eq. (1.74) to take into account those small fluctuations. We can do this by introducing functions that represent small perturbations in the metric (1.74). Then, the modified metric can be written as ds2 = a2(τ) [ (1 + 2A)dτ2 − 2Bidτdxi − (δij + hij)dxidxj ] . (1.75) In this equation, the perturbation functions are: A(t, x), called temporal lapse function or the potential function, Bi(t, x) named metric shift or vector function, and hij(t, x) that introduce perturbations in the curvature and the metric shear. It is possible to write Bi and hij into scalar and vector components that satisfy certain properties. Using the Scalar Vector Tensor (SVT) decomposition [1] we can write Chapter 1. Background and perturbed universe 19 the perturbation functions as follows Bi = ∂iB + B̂i (∂iB̂i = 0), (1.76) hij = 2Cδij + 2∂⟨i∂j⟩E + 2∂(iÊj) + 2Êij, (∂iÊi = ∂iÊij = δijÊij = 0), (1.77) where ∂⟨i∂j⟩E ≡ ( ∂i∂j − 1 3 δij∇2 ) E, ∂(iÊj) ≡ 1 2 ( ∂iÊj + ∂jÊi ) . (1.78) In table 1.1 we summarize the scalars, vectors and tensors part of the perturbed RW metric, and also the requirements that must be satisfied Scalars Vectors Tensor Divergenceless Traceless A, B, C, E B̂i, Êi Êij ∂iB̂i = ∂iÊi = ∂iÊij = 0 δijÊij = 0 Table 1.1: The SVT decomposition and the requirements that must be satisfied. 1.4.2 Gauge transformations for metric tensor The RW metric expressed by Eq. (1.75) is implicitly written in terms of a specific time slice and, consequently a specific three-dimensional space. To allow the description of these perturbations in other1 coordinate system we need to perform a gauge transformation. To prevent introducing unphysical perturbations in this process, we can perform the gauge transformation under the requirement that the spacetime interval, ds2, remains invariant regardless of the adopted gauge. Such a transformation is performed by the gauge generator function: ξµ. Starting at the gauge G, we can generate a new one, G̃, by the following xµ 7→ x̃µ ≡ xµ + ξµ(τ, xi), (1.79) ξ0 ≡ T, (1.80) ξ i ≡ Li = ∂iL + L̂i, (∂i L̂i = 0), (1.81) where in Eq. (1.81) we have used the SVT decomposition of the spatial part of ξµ. 1The description of these perturbations in other gauges can be more convenient depending on the problem, e.g., in the CMB analysis it is more practical to choose the synchronous gauge, but the spatially-flat gauge is more suitable to study inflation. The Newtonian gauge is more adequate to study the evolution of gravitational potential. Chapter 1. Background and perturbed universe 20 The requirement that the spacetime interval should be invariant under a gauge transformation implies ds2 = gµν(x)dxµdxν = g̃αβ(x̃)dx̃αdx̃β. (1.82) Using the components of the metric given by Eq. (1.75) into Eq. (1.82) it is possible to show that the perturbations functions transform as à = A − T′ −HT, (1.83) B̃i = Bi + ∂iT − L′ i, (1.84) h̃ij = hij − 2∂(iLj) − 2HTδij. (1.85) In terms of the SVT decomposition, these perturbations can be rewritten as follows • Scalars à = A − T′ −HT, (1.86) B̃ = B + T − L′, (1.87) C̃ = C −HT − 1 3 ∇2L, (1.88) Ẽ = E − L (1.89) • Vectors ˜̂Bi = B̂i − L̂′ i, (1.90) ˜̂Ei = Êi − L̂i. (1.91) • Tensor ˜̂Eij = Êij (1.92) We can combine the Eqs. (1.86)-(1.92) in order to eliminate the components of gauge generator functions ξµ. This provides us with quantities that are gauge invariant. For example, the tensor component, Eq. (1.92), already is a gauge invariant quantity because the gauge generator is absent. Following this idea, it is possible to show that the following relations are gauge Chapter 1. Background and perturbed universe 21 invariant Ψ ≡ A +H(B − E′) + (B − E′)′, (1.93) Φ ≡ −C −H(B − E′) + 1 3 ∇2E, (1.94) R ≡ C − 1 3 ∇2E +H(B + v), (1.95) ζ ≡ C − 1 3 ∇2E − H ρ̄′ δρ, (1.96) ρ̄′∆ ≡ δρ + ρ̄′(B + v), (1.97) δPnad ≡ δP − P̄′ ρ̄′ δρ. (1.98) The first two equations, (1.93) and (1.94), are known as the Bardeen’s potentials. Eq. (1.95) is known as comoving curvature perturbation and is useful to study inflation. 1.4.3 From smooth to perturbed energy-momentum tensor In the previous section, we modified the RW metric to take into account small fluctuations in the spacetime, however, the spacetime and the matter-energy distribution are connected by the Einstein equation (1.21), which means that perturbations in the RW metric induce and are induced by perturbations in the matter distribution. Therefore we also need to describe small fluctuations in the energy-momentum tensor [1] Tµ ν = T̄µ ν + δTµ ν , (1.99) T̄µ ν = (ρ̄ + P̄)ŪµŪν − P̄δ µ ν , (1.100) δTµ ν = (δρ + δP)ŪµŪν + (ρ̄ + P̄)(δuµŪν + Ūµδuν)− δPδ µ ν − Πµ ν , (1.101) where δ() represents the perturbation. For example, δTµ ν is a small deviation from the background energy-momentum tensor T̄µ ν , and so forth. Equation (1.100) is the same as Eq. (1.25) except that we have changed the notation and left the upper bar to denote the unperturbed quantity. Using the perturbed RW metric, Eq. (1.75), Chapter 1. Background and perturbed universe 22 it is possible to obtain the components for δTµ ν , these are given by [1] δT0 0 = δρ, (1.102) δTi 0 = (ρ̄ + P̄)vi, (1.103) δT0 j = −(ρ̄ + P̄)(vj + Bj), (1.104) δTi j = −δPδi j − Πi j, (1.105) where vi ≡ aδUi. It is important to note that the perturbations (δρ, δP, etc) have the contributions of several components (e.g. photons, neutrinos, baryons, dark matter, dark energy, etc.). This implies that we should sum over all possible components, i.e. δρ = ∑I δρI , and the same for δP, vi and Πij. In the same way as the gauge transformations for the metric, we can perform the analogous transformation for the energy-momentum tensor. The coordinate transformation to avoid unphysical perturbations in the energy-momentum tensor is Tµ ν (x) = ∂xµ ∂x̃α ∂x̃β ∂xν T̃α β (x̃). (1.106) Using the Eqs. (1.102)-(1.105) into Eq. (1.106) we obtain the relations between the perturbations in the gauge G and G̃ δ̃ρ = δρ − Tρ̄′, (1.107) ˜δP = δP − TP̄′, (1.108) q̃i = qi + (ρ̄ + P̄)L′ i, (1.109) ṽi = vi + L′ i, (1.110) Π̃ij = Πij, (1.111) where qi is know as momentum density and is defined by qi ≡ (ρ̄ + P̄)vi. 1.4.4 Perturbed equations of motions in Newtonian gauge Once we have the perturbed metric and the perturbed energy-momentum tensor, in principle should be possible to derive the perturbed equations of motions. We start with perturbations in the Einstein equation, Eq. (1.21), as follows Ḡµν + δGµν = 8πG(T̄µν + δTµν). (1.112) Chapter 1. Background and perturbed universe 23 In order to present the main ideas, we use the Newtonian gauge and we adopt the notation of [1] as follows B = E = 0, A ≡ Ψ, C ≡ −Φ. (1.113) Assuming the Newtonian gauge, we can rewrite the perturbed RW metric in terms of the new variables, Eq. (1.113), which simplifies the metric to ds2 = a2(τ) [ (1 + 2Ψ) dτ2 − δij (1 − 2Φ) dxidxj ] . (1.114) Once the gauge and the metric are specified, it is possible to derive all geometric quantities like the Chistoffel symbols, the Ricci tensor, and the Ricci Scalar. Perturbed stress-energy conservation Using the perturbed energy-momentum tensor into the conservation equation2 ∇µTµ ν = 0 (1.115) it is possible to derive the perturbed versions of the continuity and Euler equations. • Continuity equation The first one is the continuity equation and is obtained considering ν = 0. The result is δ′ + ( 1 + P̄ ρ̄ ) ( ∇ · v − 3Φ′)+ 3H ( δP δρ − P̄ ρ̄ ) δ = 0, (1.116) where δ ≡ δρ/ρ̄ is the dimensionless fractional overdensity3. • Euler equation The Euler equation is obtained considering ν = i. The result is v′ +Hv − 3H P̄′ ρ̄′ v = − ∇δP ρ̄ + P̄ −∇Ψ (1.117) Perturbed Einstein equations • Traceless space-space component 2Note that you will need the perturbed Christoffel symbols. 3Maybe this isn’t a good name, because δ represent regions of overdensity but also regions of lack of density (voids). A more appropriate name in common use is density contrast. Chapter 1. Background and perturbed universe 24 The trace-free part of the spatial component of the Einstein tensor is given by ∂⟨i∂j⟩ (Φ − Ψ) = 0, (1.118) If Πij = 0, then Φ = Ψ. (1.119) where the r.h.s of Eq. (1.118) is zero because we are neglecting the anisotropic stress. If we consider this term, then the r.h.s of Eq. (1.118) is sourced by the term −8πGa2Πij. • time-time component Considering the 00 component of the Einstein equation, G00 = 8πGT00, we have at zeroth order H2 = 8πG 3 a2ρ̄, (1.120) and at first order we have ∇2Φ = 4πGa2ρ̄δ + 3H ( Φ′ +HΦ ) . (1.121) • time-space component Considering the 0i component of the Einstein equation, G0i = 8πGT0i, we are left with Φ′ +HΦ = −4πGa2 (ρ̄ + P̄) v. (1.122) Combing (1.122) and the (1.121) we have ∇2Φ = 4πGa2ρ̄∆, (1.123) where ρ̄∆ ≡ δ̄ − 3H(ρ̄ + P̄)v. • Trace space-space component Finally, we consider the trace part of the Einstein equation, that is, Gi i = 8πGTi i . The result of these computations is, at zeroth order given by 2H′ +H2 = −8πGP̄ (1.124) and at first order, we are left with Φ′′ + 3HΦ + (2H′ +H2)Φ = 4πGa2δP. (1.125) Chapter 1. Background and perturbed universe 25 1.4.5 Proof of some equations In this section, we want to demonstrate the equations (1.84), (1.85) and prove that the spatial part of the perturbed Christoffel symbol, Γ0 0i, in equation (1.38) is not zero. In summary, we want to prove the following equations Bi 7→ B̃i = Bi + ∂iT − L′ i, (1.126) hij 7→ h̃ij = hij − 2∂(iLj) − 2HTδij, (1.127) Γ0 0i = ∂iΨ. (1.128) Proof 1 We start considering the following coordinate transformation Xµ 7→ X̃µ ≡ Xµ + ξµ(τ, x), (1.129) Xµ → {X0 = τ, Xi → x}, (1.130) ξ0 ≡ T, (1.131) ξ i ≡ Li = ∂iL + L̂i. (1.132) By requesting the invariance of the spacetime interval we can avoid unphysical perturbations ds2 = gµνdXµdXν = g̃αβdX̃αdX̃β. (1.133) The spacetime interval in the coordinate system without/with “tilde” is given by ds2 = a2(τ) [ (1 + 2A)dτ2 − 2Bidxidτ − (δij + hij)dxidτ ] (1.134) = a2(τ̃) [ (1 + 2Ã)dτ̃2 − 2B̃idx̃idτ̃ − (δij + h̃ij)dx̃idτ̃ ] . (1.135) For clarity, it is very useful to write explicitly each component of the metric tensor. First for frame without “tilde” g00 = a2(τ)(1 + 2A), (1.136) g0i = gi0 = −a2(τ)Bi, (1.137) gij = gji = −a2(τ)(δij + hij). (1.138) Chapter 1. Background and perturbed universe 26 Analogously for the frame with “tilde” g̃00 = a2(τ̃)(1 + 2Ã), (1.139) g̃0i = g̃i0 = −a2(τ̃)B̃i, (1.140) g̃ij = g̃ji = −a2(τ̃)(δij + h̃ij). (1.141) The relation of metric tensor between this coordinates system is given by gµν(X) = ∂X̃α ∂Xµ ∂X̃β ∂Xν g̃αβ(X̃). (1.142) Let us study the components µ, ν → 0, i. It is useful to decompose them α and β as follows: α, β → {0, 0; 0, i; i, 0; i, j}. With Eqs. (1.129)-(1.132) and (1.139)-(1.141), the r.h.s of Eq. (1.142) can be sepa- rated in the following terms ∂X̃0 ∂τ ∂X̃0 ∂Xi g̃00 = (1 + T′)(∂iT)a2(τ̃)(1 + 2Ã), (1.143) ∂X̃0 ∂τ ∂X̃ j ∂Xi g̃0j = (1 + T′)(δi j + ∂jξ i)(−a2(τ̃)B̃j), (1.144) ∂X̃ j ∂τ ∂X̃0 ∂Xi g̃j0 = (∂τξ j)(∂iT)(−a2(τ̃)B̃j), (1.145) ∂X̃ j ∂τ ∂X̃k ∂Xi g̃jk = (∂τξ j)(δk i + ∂iξ k)(−a2(τ̃)(δjk + h̃jk)), (1.146) where T′ := ∂τT. Now we combining the Eqs. (1.165)-(1.146) in the r.h.s of the (1.142) and eliminating the terms of O2 in the perturbations terms, we are left with g0i = a2(τ̃) [ ∂iT − B̃i − L′ i ] , (1.147) where L′ i := ∂τ Li. The last step is to expand the scale factor a2(τ̃) = [a(τ + T)]2 where in this equality we have used the 0 component of Eq. (1.129), then we have a2(τ̃) = [a(τ + T)]2 = [ a(τ) + a′(τ)T + 1 2 a′′(τ)T2 + ... ]2 = a2(τ) [ 1 +HT + 1 2 a′′(τ) a′(τ) T2 + ... ]2 ≈ a2(τ) [1 +HT]2 first approximation ≈ a2(τ) [1 + 2HT] second approximation. (1.148) Chapter 1. Background and perturbed universe 27 where H = a′/a is the comoving Hubble parameter. Substituting Eq. (1.148) in r.h.s of Eq. (1.147), and using (1.137) in the l.h.s of (1.147) we have −a2(τ)Bi = a2(τ) [1 + 2HT] [ ∂iT − B̃i − L′ i ] . (1.149) Expanding and eliminating terms of the order O2 in the previous equation, we are left with −Bi = ∂iT − B̃i − L′ i which shows that B̃i = Bi + ∂iT − L′ i . (1.150) Proof 2 The demonstration of h̃ij is analogous to the demonstration for B̃i. Following the same procedure, we start decomposing the r.h.s of Eq. (1.142), but now for µ, ν → i, j ∂X̃0 ∂Xi ∂X̃0 ∂X j g̃00 = (∂iT)(∂jT)(a2(τ̃)(1 + 2Ã)), (1.151) ∂X̃0 ∂Xi ∂X̃k ∂X j g̃0k = (∂iT)(δk j + ∂jLk)(−a2(τ̃)B̃k), (1.152) ∂X̃k ∂Xi ∂X̃0 ∂X j g̃k0 = (δk i + ∂iLk)(∂jT)(−a2(τ̃)B̃k), (1.153) ∂X̃k ∂Xi ∂X̃l ∂X j g̃kl = (δk i + ∂iLk)(δl j + ∂jLl)(a2(τ̃)(δkl + h̃kl)). (1.154) Combining Eqs. (1.151)-(1.154) into the r.h.s of Eq. (1.142), and eliminating the terms that are of second order in perturbation, O2, we are left with gij = a2(τ̃) [ δij + h̃ij + ∂iLj + ∂jLi ] . (1.155) Let’s remember the notation for the vector field perturbation: ∂(iLj) ≡ 1 2 ( ∂iLj + ∂jLi ) . (1.156) Using Eqs. (1.138), (1.148) and (1.156) in Eq. (1.155) we are left with −a2(τ)(δij + hij) = −a2(τ)(1 + 2HT) [ δij + h̃ij + 2∂(iLj) ] . (1.157) Expanding the r.h.s of the previous equation and eliminating terms of O2 we get Chapter 1. Background and perturbed universe 28 hij = h̃ij + 2∂(iLj) + 2HTδij. Finally, we have the desired equation h̃ij = hij − 2∂(iLj) − 2HTδij . (1.158) Proof 3 The final task is to derive the perturbed connection coefficient (or the perturbed Christoffel symbol) as shown by Eq.(1.128). The procedure developed here can be easily applied to obtain the other Christoffel symbols. Let’s start with the perturbed RW metric tensor gµν = a2 ( 1 + 2Ψ 0 0 −(1 − 2Φ)δij ) . (1.159) The inverse of gµν is gµν = 1 a2 ( 1 − 2Ψ 0 0 −(1 + 2Φ)δij ) . (1.160) The relation between the connection coefficients and the metric tensor is Γµ νρ = 1 2 gµλ(∂νgλρ + ∂ρgλν − ∂λgνρ). (1.161) We want the case µ = ν = 0 and ρ = i, then the relation (1.161) reduces to Γ0 0i = 1 2 g0λ(∂0gλi + ∂igλ0 − ∂λg0i). (1.162) The above equation is non-zero only for λ = 0. Then, Eq. (1.162) becomes Γ0 0i = 1 2 g00 (∂ig00) (1.163) Finally, using g00 = (1 − 2Ψ)/a2, g00 = a2(1 + 2Ψ) and eliminating terms second order in Ψ, we obtain the derided expression Γ0 0i = ∂iΨ . (1.164) Chapter 1. Background and perturbed universe 29 1.4.6 Evolution of the gravitational potential In this section, we numerically solve a differential equation to obtain the evolution for the gravitational potential Φ. The complete solution is separated into three steps: (i) First we need the general ordinary differential equation governing the evolution of the gravitational potential, (ii) then we need to find the non- constant coefficients of this differential equation. (iii) Finally we need to rewrite this equation in order to be easily numerically integrated. (i) General equation: the gravitational potential: The first-order in perturba- tion theory for the components 0,0 and i,i in the Einstein equations are, respectively, given by ∇2Φ = 4πGa2δρ + 3H(Φ′ +HΦ), (1.165) Φ′′ + 3HΦ′ + (2H′ +H2)Φ = 4πGa2δP. (1.166) Consider the following: ∇2 → −k2 and Eq.(1.165) × ( − P̄ ρ̄ ) + Eq.(1.166), then we have Φ′′ + 3H ( 1 + P̄ ρ̄ ) Φ′ + [ 2H′ +H2 + 3H2 P̄ ρ̄ ] Φ + P̄ ρ̄ k2Φ (1.167) = 4πGa2δρ ( δP δρ − P̄ ρ̄ ) ≈ 0. (1.168) Analogously, we also consider the zeroth order in the Einstein equations of the components 0,0 and i,i. These correspond to the first and second Friedmann equations, respectively H2 = 8πG 3 a2ρ̄, (1.169) 2H′ +H2 = −8πGa2P̄. (1.170) Performing the following operation Eq.(1.169) × ( 3 δP δρ ) + Eq.(1.170) we have 2H′ +H2 + 3H2 δP δρ = 2H′ +H2 + 3H2 P̄ ρ̄ = 8πGa2ρ̄ ( δP δρ − P̄ ρ̄ ) ≈ 0. (1.171) Substituting Eq. (1.171) back to Eq. (1.168), we have Φ′′(τ) + 3H (1 + weff)Φ′(τ) + Φ + k2weffΦ(τ) = 0 . (1.172) Chapter 1. Background and perturbed universe 30 Equation (1.172) is an ordinary differential equation with non-constant coefficients. Therefore, to solve this equation we need to find these terms. Firstly, we define the effective equation of state as weff ≡ P̄ ρ̄ . (1.173) (ii) Early times: radiation + pressureless matter: for this first solution, we simplify the problem and solve the differential equation (1.172) in the regime of radiation and cold dark matter (pressureless matter) domination. Therefore, the energy density and pressure can be written as follows ρ̄ = ρ̄r + ρ̄m = ρ̄r,0 a4 + ρ̄m,0 a3 (1.174) P̄ = P̄r + P̄m ≈ P̄r = wrρr = 1 3 ρr (1.175) ΩI,0 ≡ ρ̄I,0 ρcrit,0 , I = r or m. (1.176) Using Eqs. (1.174)-(1.176), the effective equation of state, Eq. (1.173), can be rewrit- ten as follows weff = 1 3 ( 1 + Ωm,0 Ωr,0 a ) . (1.177) For convenience, we define a new variable y as follows aeq = Ωr,0 Ωm,0 , y ≡ a aeq . (1.178) Using these changes of variable, the effective equation of state can be written as weff = 1 3 (1 + y) . (1.179) The final task is to find the form of H as a function of y in these simplified radiation + CDM universe models. The relation between the conformal Hubble parameter, H, and Hubble parameter, H, is H2 = a2H2. (1.180) Chapter 1. Background and perturbed universe 31 Considering only radiation and pressureless matter, the Hubble parameter is H2 = H2 0 [ Ωr,0 a4 + Ωm,0 a3 ] ⇒ H2 = H2 0 [ Ωr,0 a2 + Ωm,0 a ] . (1.181) Using the change of variable is not difficult to show that the conformal Hubble parameter is given by H2 = H2 0Ωm,0 aeq [ 1 y + 1 y2 ] , or H = H0 √ Ωm,0√aeq √ 1 + y y . (1.182) Taking the derivative of the conformal Hubble parameter with respect y we have Ḣ = − H0 √ Ωm,0√aeq 2 + y 2y2 √ 1 + y . (1.183) (iii) Gravitational potential: change of variable: finally, we need to rewrite the Eq. (1.172) in terms of the new variable y. The first step is to express dτ in terms of dy, we have d dτ = a a da dτ d da = Ha d da = H(yaeq) d d(yaeq) , (1.184) then we have d dτ = Hy d dy . (1.185) It is going to be useful for the second derivative of Eq. (1.185) d2 dτ2 = d dτ d dτ = Hy d dy ( Hy d dy ) , then (1.186) d2 dτ2 = Hy [ Ḣy d dy +H d dy +Hy d2 dy2 ] . (1.187) Just to recap, we are using the following notation d dτ ≡ ( )′, d dy ≡ ˙( ). (1.188) Applying the Eqs. (1.185) and (1.188) into Eq. (1.172), after some algebraic manip- Chapter 1. Background and perturbed universe 32 ulations, we are left with Φ̈(y) + 1 Hy [ Ḣy +H (4 + 3weff) ] Φ̇(y) + k2weff H2y2 Φ(y) = 0. (1.189) Finally, using Eqs. (1.179), (1.182) and (1.183) into Eq. (1.189) we have the desired equation of the potential in terms of the new variable y Φ̈k(y) + [ 8 + 7y 2y(1 + y) ] Φ̇k(y) + [ aeqk2 3H2 0Ωm,0(1 + y)2 ] Φk(y) = 0 . (1.190) We use the new notation Φ(y) → Φk(y) to remember that this equation depends on the Fourier mode k. (iv) Numerical results: To compute equation (1.190) numerically, we can use any programming language that allows us to solve ordinary differential equations (ODE), for example, Python. ODEs in Python can be solved with the function called odeint from the package Scipy. The function odeint only solves first- order ODEs, or a system of coupled ODEs. For this reason, we need to transform the second order Eq. (1.190) into a system of two coupled ODEs.4 To break the ODE into two other ones, we pay the price of gaining a new variable, here defined by v, so Eq. (1.190) becomes vk ≡ Φ̇k, (1.191) v̇k(y) = −A(y)vk(y)− Bk(y)Φk(y). (1.192) The coefficients are defined by A(y) ≡ 8 + 7y 2y(1 + y) , Bk(y) ≡ aeqk2 3H2 0Ωm,0(1 + y)2 . (1.193) We have assumed natural units where the speed of light in a vacuum is 1, but here we need to reintroduce this constant for a correct numerical computation. By dimensional analysis we see that [H] = km/s/Mpc and [k] = 1/Mpc, therefore we need multiply k by c, i.e. we re-scale the value of k by k → kc. Python also return to us the rate of the gravitational potential with respect to the variable y. The result is shown in figures 1.4 and 1.5. 4Mathematica is more suitable for this task, but here we use an open source software for illustrative purposes only. Chapter 1. Background and perturbed universe 33 10 3 10 2 10 1 100 101 102 103 y a/aeq 0.0 0.2 0.4 0.6 0.8 1.0 k(y ) k = 0.001 k = 0.01 k = 0.1 k = 2 yeq = 1 Figure 1.4: Evolution of perturbed gravitational potential 10 3 10 2 10 1 100 101 102 103 y a/aeq 50 40 30 20 10 0 10 k(y ) k = 0.001 k = 0.01 k = 0.1 k = 2 yeq = 1 Figure 1.5: Evolution of the rate of the perturbed gravitational potential. Chapter 1. Background and perturbed universe 34 1.5 Cosmic Microwave Background Figure 1.6: Figure from Planck image gallery [9] The light that permeates throughout the universe is known as Cosmic Mi- crowave Background (CMB) formed 370000 years after the Big Bang temporal- singularity. This radiation was first discovered by Penzias and Wilson [10] allow- ing them to share the Physics Nobel Prize in 1978. In Figure 1.6 we show a map of the CMB temperature anisotropy measured by the Planck satellite [9]. We can write a general expression for the temperature as a function of the sky direction: T(n) ≡ T̄0 [1 + Θ(n)] , (1.194) where T̄0 is the average CMB temperature and Θ(n) ≡ δT(n)/T̄0 is the fractional temperature fluctuation. In this part, we closely follow the [1] notation. As we will see, it is possible to write a functional form for Θ(n) (thus for T(n) if T̄0 is known), however, observationally we should turn our finds into a statistical equation known as two-point correlation function: C(θ) ≡ ⟨Θ(n)Θ(n’)⟩, (1.195) where θ is the angular separation in the sky between the unitary vectors n and n’ and defined through cos θ = n · n’. This is because a broad variety of theoretical https://www.cosmos.esa.int/web/planck/picture-gallery Chapter 1. Background and perturbed universe 35 models of the early universe predicts that Θ(n) should satisfy a Gaussian distribu- tion [11]. If this actually is the case, then we should be able to capture all features of the CMB temperature anisotropies, θ(n), with a single function of the angular separation C(θ) given by Eq. (1.195). The same cosmological information in C(θ) is contained in the angular power spectra denoted by Cℓ. To see this, we first expand the CMB fractional temperature anisotropy in terms of spherical harmonics Θ(n) = ∞ ∑ l=2 ℓ ∑ m=−ℓ aℓmYℓm(n). (1.196) Substituting Eq. (1.196) into Eq. (1.195) we obtain C(θ) = ∑ ℓ 2ℓ+ 1 4π CℓPℓ(cos θ) (1.197) where Pℓ are the Legendre polynomials, and we have used the assumption of Gaus- sian statistics (predicted by early universe models) for the fractional temperature anisotropy: δmm′δℓℓ′Cℓ = ⟨aℓma∗ℓ′m′⟩, (1.198) Since the Pℓ’s are known, if we determine Cℓ we automatically find C(θ) by means of Eq. (1.197). In Eq. (1.196) note that m varies in the range ±ℓ, thus for a fixed ℓ there are 2ℓ+ 1 possible values for m. Therefore, summing over all possible values for m, we can obtain an estimator for Cℓ Ĉℓ = 1 (2ℓ+ 1) ℓ ∑ m=−ℓ |aℓm|2. (1.199) Note that the estimator is unbiased, ⟨Ĉℓ⟩ − Cℓ = 0. However, there is an intrinsic error due to the cosmic variance [2/(2ℓ+ 1)]1/2Cℓ [12] which is dominated on larger angular scales. 1.5.1 Primordial spectrum and CMB What was the mechanism (inflation, topological defects, etc) behind the gen- eration of the primordial spectrum density perturbation that 13.8 billion years after the Big Bang led the universe that we see today? In order to use CMB to Chapter 1. Background and perturbed universe 36 find an answer to this question, we first need a connection between the angular power spectra, Cℓ’s, and the primordial spectrum of density perturbations, ∆2 R(k). The first step in this direction is to know how the photons (i.e. a property of the photons: energy E, wavelength λ, etc) evolve after decoupling in a perturbed universe. The geodesic equation governing the free photon propagation is dPµ dη = −Γµ νρ PνPρ P0 . (1.200) Using the Christoffel symbols derived in proof 3, section 1.4.5, we can rewrite Eq. (1.200) for µ = 0 (because it is related with the photon’s energy) as follow dP0 dη = − ( H+ Ψ′) P0 − 2∂iΨPi − [ H− Φ′ − 2H (Φ + Ψ) ] δij PiPj P0 (1.201) Given a particle with mass m, physical momentum p = (gijPiPj)1/2, energy E = (p2 + m2)1/2, the mass-shell constraint is given by (see e.g. [13]) gµνPµPν = (1 + 2Ψ)(P0)2 + p2 = m2. For the photon, m = 0, thus we obtain P0 = E a (1 − Ψ), Pi = E a (1 + Φ)pi (1.202) substituting Eq. (1.202) into (1.201) and solving for the energy at first order in perturbation, we get d ln E dη = −H+ Φ′ − pi∂iΨ. (1.203) The equation (1.203) tells how the photon energy varies in an expanding (due the Hubble term) and inhomogeneous universe (due the metric perturbations). At this point, we are interested solely on the effect of metric perturbations, thus we have to discount the Hubble flow from the right hand side of Eq. (1.203), which leads d ln ϵ dη = Φ′ + Ψ′ − dΨ dη (1.204) where ϵ ≡ aE. The Eq. (1.204) tells us how the photon’s energy (corrected by the redshifting due to the Hubble flow) is affected by local perturbations of the Hubble flow (Φ′) which means that some parts of the universe expand slightly faster, or Chapter 1. Background and perturbed universe 37 lower than others one plus potential wells (the whole term Ψ′ − dΦ/dη = pi∂iΦ) caused by the intervening matter of the large scale structure. To obtain how much was the photon’s energy deviated from its initial value at the decoupling time (η∗) to its value seen today (η0) by our telescopes, we should solve Eq. (1.204) for the energy, a result known as line-of-sight solution [1] or line-of-sight integral [14]. A proper way to perform this calculation needs to take into account several physical processes beyond the gravitation redshift, such as the adiabatic growth of perturbations, baryon velocity-induced fluctuations, and photon diffusion. Those effects are then combined and solved with the Euler, Continuity, and Boltzmann equation with the introduction of the visibility function, which is the probability that a photon last scattered within a time interval of [η, η + dη] (see [15] and references therein). For illustrative purposes, we assume an instantaneous decoupling at η∗ (see chapter 7 of [1] or Appendix B for detailed computation), and the solution for Eq. (1.204) becomes δT T̄ ∣∣∣ 0 = δT T̄ ∣∣∣ ∗ + Ψ∗ − Ψ0 + ∫ η∗ η0 dη ( ∂Φ ∂η + ∂Ψ ∂η ) . (1.205) To obtain Eq. (1.205) we have used that after decoupling the photon’s distribution function have its shape unmodified once they trajectories are along geodesics, and since the Bose-Einstein distribution is a function of E/T, then the proportionality relation holds, E ∝ T ⇒ E ∝ T̄(1 + δT/T̄). The first term in Eq. (1.205) has the contribution from two sources: the temperature fluctuation induced by photon density contrast, δγ/4, and a Doppler shift due to the scattering between photons and electrons at the last scattering surface, which is proportional to the projection of the baryon velocity onto the line-of-sight direction, −n · vb, where the minus sight means that the photon momentum vector has a direction opposite to the unit vector n. The term Ψ∗ − Ψ0 in Eq.(1.204) simplifies to Ψ∗ because the local gravitational potential, Ψ0, modify the CMB power spectra as a whole, thus this term is unobserved [16] and can be ignored in the equation. Rearranging the terms we have δT/T̄|∗ + Ψ∗ − Ψ0 = (δγ∗/4 + Ψ∗)− n · vb∗ where the first term inside the parenthesis is recognized as the Sachs-Wolfe effect due to Sachs and Wolfe (1967) [17]. The second term, −n · vb∗, is the Doppler shift. Finally, the integral in Eq.(1.204) is called the integrated Sachs-Wolfe effect that accounts for temperature fluctuation induced by the intervening distribution of matter through Chapter 1. Background and perturbed universe 38 the universe [1, 16]. In this way, Eq. (1.205) can be written as Θ(n) ≡ δT T̄ (n) = 1 4 δγ∗ + Ψ∗ − n · vb∗ + ∫ η0 η∗ (Φ′ + Ψ′)dη. (1.206) In general, it is easier to work in Fourier space, where plane waves can be decomposed in Fourier modes and evolve independently and in the end, we just need to perform a superposition of these modes. This constitutes the so- called spatial to angular projection approach [1]. Let us define new functions F∗(k)Ri(k) ≡ δγ∗/4 + Ψ∗ and G∗(k)Ri(k) ≡ |vb∗| = vb∗ where Ri is the initial curvature perturbation that takes into account the linear evolution of fluctuations. Therefore, Eq. (1.206) becomes Θ(n) = ∫ d3k (2π)3 eik · x∗Θℓ(k)Ri. (1.207) where we have defined the transfer function Θℓ(k) ≡ F∗ − i(k̂ · n̂)G∗. Note that in this equation we neglected the ISW effect, which is small in comparison with the other ones, with the exception at ℓ < 10 (see e.g. figure 5.2 in [14]) which is a little bit larger than the Doppler effect, but that cause no much prejudice for our example. If we take the Legendre expansion of the exponential, ei(kχ∗)k̂ · n̂ = ∑ℓ iℓ(2ℓ+ 1)jℓ(kχ∗)Pℓ(k̂ · n̂), and its derivative with respect to the module of its argument, we can rewrite the equation (1.207) as follows Θ(n) = ∑ ℓ iℓ(2ℓ+ 1) ∫ d3k (2π)3 Θℓ(k)Ri(k)Pℓ(k̂ · n̂). (1.208) Finally, substituting Eq.(1.208) into the two-point correlation function (1.195) and comparing the result with the CMB temperature power spectra in Eq. (1.197) we have Cℓ = 4π ∫ d(ln k) Θ2 ℓ(k)∆ 2 R(k) , (1.209) where the primordial curvature perturbation, ∆R(k), is defined via the two-point correlation function of the initial curvature perturbation, Ri(k), in the following way: ⟨Ri(n)Ri(n’)⟩ = 2π2/k3∆2 R(k)(2π)3δ(k + k’). As we have mentioned, the proper way to derive the CMB temperature anisotropy power spectra is via Boltzmann’s kinetic theory (coupled with the Chapter 1. Background and perturbed universe 39 Einstein equations) that takes into account deviations of the particle distribu- tion function from thermal equilibrium due to the universe expansion, particle scatterings, annihilations and a sharply peaked, but non-instantaneous, photon decoupling as we have assumed in our derivation of equation (1.209). Although the purpose here is not to provide all the details on CMB calculations, we provide a scheme on how real CMB power spectra are estimated (see [14] for details or appendix B of [1]). In figure 1.7 the horizontal line represents the last scattering surface and a photon that starts its journey through the space until be captured by our telescopes today. This photon carries much information about the physical process on the LSS, and in figure 1.7 we show the essential equations that allow us to relate the CMB photon with the universe’s characteristics such as mass-energy distribution and the physics of recombination. L( f ) = C{ f ; fi}LSS dΘ/dη = d ln ϵ/dη − Γ[Anisotropic Scattering Term] τ(η) ≡ ∫ η0 η Γ(η′)dη′ Θℓ(k) = ∫ η0 ηi dη{g(Θ0 + Ψ) + (gk−2θb) ′ + e−τ(Φ′ + Ψ′)}jℓ[k(η0 − ηi)] aℓm = (−i)ℓ ∫ d3kYℓm(k̂)Θℓ(k) Cℓ = ⟨aℓma∗ℓm⟩ Cℓ = 4π ∫ (d ln k)Θ2 ℓ(k)∆ 2 R(k) γ′ γ′ e Figure 1.7: A schematic representation of how we convert the early universe’s information carried by the CMB photons when it reaches us today. We can interpret figure 1.7 as follows. Initially, photons were tightly coupled with baryons, exchanging energy and momentum in a scattering process with a rate higher than the universe expansion. From the photon’s phase-space point of Chapter 1. Background and perturbed universe 40 view, this implies in an evolution of the number of photons at a given point (Pµ, xi) and time t = x0. This is characterized by the distribution function f (Pµ, xν) with a dynamics described by the Boltzmann equation L( f ) ≡ d f dη (η, x, ϵ, p̂) = ∂ f ∂η + ∂ f ∂x · dx dη + ∂ f ∂ϵ dϵ dη + CMB Lensing = C{ f ; fi} (1.210) where the left side in (1.210) is the Liouville operator which simply tells us how the distribution function varies, with respect to η, in response to external inter- actions with other particles given by each distribution function, fi, where i can be electrons, neutrinos, etc. The “CMB lensing” term describes deviation in the CMB photon direction and is given by (∂ f /∂p̂) · (∂p̂/∂η). This term is second order in perturbation (which is called “secondary” CMB effect) so we drop off its contribution. The distribution function in terms of the fractional temperature fluctuation, Θ, is given by f (η, x, ϵ, p̂) = 1 exp [ ϵ aT̄(η)(1+Θ(η,x,p̂)) ] − 1 . (1.211) Combining Eq. (1.211) with (1.210) and taking into account the Isotropic Thompson Scattering, ITS ≡ Θ − Θ0 − p̂ · vb, and the Anisotropic Thompson Scattering, ATS ≡ −3/(16π) ∫ dp̂iΘ(p̂i)[(p̂i · p̂)2 − 1/3], where Θ0 is the monopole and p̂i is the incoming photon’s momentum in the scattering process with the target electron. This produces the second equation from top to bottom in figure 1.7. The solution of this differential equation was mentioned before, it is known as the “line-of-sight solution”, which is obtained by introducing the optical depth, τ(η), that describes the opacity of the universe; higher the opacity is higher is the probability of photon be scattered between η and η0. The mathematical advantage of this quantity is that it helps to integrate the Boltzmann equation, which turns on into the fourth equation shown in figure 1.7. The fractional temperature fluctuation was defined in equation (1.196) in terms of spherical harmonics Yℓm with weights aℓm. These quantities can be computed just by inverting Eq. (1.196), in terms of Θℓ, as shown in the fifth equation, from top to bottom, of figure 1.7. We take the average of these weights which finally produce the CMB power spectra in terms of the transfer function, Θℓ(k), and the Chapter 1. Background and perturbed universe 41 primordial power spectra. Chapter 2 Gauge invariant perturbations 2.1 Introduction In chapter 1.1 we saw expressions that do not change under a gauge transforma- tion, for example, the Bardeen variables and the comoving curvature perturbation. How can we obtain these quantities? This chapter explains the procedure and performs detailed computations to obtain these and other gauge-invariant pertur- bations. The universe is described by the RW metric gµν. This metric is the contribution of two terms: ḡµν which denotes the metric at background level (or at zeroth order in perturbation theory) and δgµν that represents small perturbation in the metric (or the first order in perturbation theory). Through this chapter we use the notation of [1]. As we saw in chapter 1.1, the perturbation functions (PF) are A, Bi, and hij. In order to avoid the gauge problem, i.e. the generation of unphysical perturbations, we need to perform the gauge transformation requesting the invariance of the spacetime interval, i.e. ds̃2 = ds2. This “jump” from the gauge G to the other gauge G̃ is performed by the gauge generator functions given by ξµ (introduced in chapter 1.1), such that |ξµ| ≪ 1 in order to guarantee that the perturbations can be linearized. 2.2 Setting equations The perturbed RW metric is given by gµν = a2(τ)  [1 + 2A(τ, x)] −Bi(τ, x) −Bi(τ, x) −[δij + hij(τ, x)]  . (2.1) 42 Chapter 2. Gauge invariant perturbations 43 Let’s remember the SVT decomposition for the perturbation functions. These are given by Bi = ∂iB + B̃i (∂iBi = 0), (2.2) hij = 2Cδij + 2∂⟨i∂j⟩E + 2∂(iẼj) + 2Ẽij (Ẽi i = ∂iẼi = ∂iẼij = 0), (2.3) ∂⟨i∂j⟩E = ( ∂i∂j − 1 3 δij∇2 ) E, (2.4) ∂(iẼj) = 1 2 ( ∂iẼj + ∂jẼi ) . (2.5) For convenience, we also repeat the gauge generator function x̃µ = xµ + ξµ(τ, x), (2.6) ξ0 ≡ T, (2.7) ξ i ≡ Li = ∂iL + L̃i (∂i L̃i = 0). (2.8) From the invariance of the spacetime interval, ds̃2 = ds2, it possible to show that the components of the metric in the gauge G, gµν, relates with the components of the metric in the gauge G̃, g̃µν, by the following expression gµν(τ, x) = g̃µν(τ, x) + ∂αgµν(τ, x)ξα + ∂µξρgρν(τ, x) + ∂νξρgρµ(τ, x). (2.9) To demonstrate the Eq. (2.9), we start from the invariance of the line elements between the gauges, i.e. ds2 = ds̃2 ⇒ gµν(x)dxµdxν = g̃αβ(x̃)dx̃αdx̃β. (2.10) The coordinate transformation is given by dx̃α = (∂x̃α/∂xµ)dxµ and dx̃β = (∂x̃β/∂xν)dxν we have gµν(x) = ∂x̃α ∂xµ ∂x̃β ∂xν g̃αβ(x̃) (2.11) Using the gauge transformation given by Eq. (2.6) into Eq. (2.11) we have gµν(x) = ( δα µ + ∂µξα ) ( δ β ν + ∂νξβ ) g̃αβ(x̃). (2.12) Expand Eq. (2.12) and neglect quadratic terms in the gauge generator ξα. In addi- tion, it is useful to consider the expansion of the g̃αβ(x̃) around the neighborhood Chapter 2. Gauge invariant perturbations 44 of x, this is given by g̃αβ(x̃) = g̃αβ(x̃µ) = g̃αβ(xµ + ξµ) ≈ g̃αβ(xµ) + ∂µ g̃αβ(xµ)ξµ(xµ), or (2.13) g̃αβ(x̃) ≈ g̃αβ(x) + ∂µ g̃αβ(x)ξµ(x) (2.14) Substituting Eq. (2.14) into Eq. (2.12) and expanding the expression up to first order in perturbation, we are left with gµν(x) = ( δα µ + ∂µξα ) ( δ β ν + ∂νξβ ) ( g̃αβ + ∂σ g̃αβξσ ) = ( δα µδ β ν + δα µ∂νξβ + δ β ν ∂µξα ) ( g̃αβ + ∂σ g̃αβξσ ) = δα µδ β ν g̃αβ + δα µδ β ν ( ∂σ g̃αβ ) ξσ + δα µ ( ∂νξβ ) g̃αβ + δ β ν ( ∂µξα ) g̃αβ = g̃µν(x) + ∂σ g̃µν(x)ξσ + ∂µξα g̃να(x) + ∂νξβ g̃µβ(x), (2.15) where Eq. (2.15) is the same as Eq. (2.9), what was to be shown. Applying Eq. (2.9) to each metric component, we obtain the relations satisfied the gauge generator between the gauges G and G̃. However, this computation was already performed in section (1.4.5) for Bi and hij, the result is à = A − T′ −HT, (2.16) B̃i = B + ∂iT − L′ i, (2.17) h̃ij = hij − 2∂(iLj) − 2HTδij. (2.18) 2.3 Building gauge invariant perturbations In terms of the SVT decomposition, we can separate the perturbations contri- butions, Eqs. (2.16)-(2.18), into scalars, vectors, and tensors part. Let’s concentrate on the scalar perturbation part. Then we have à = A − T′ −HT, (2.19) B̃ = B + T − L′, (2.20) C̃ = C −HT − 1 3 ∇2L, (2.21) Ẽ = E − L. (2.22) In the same sense, we can decompose the components of the energy-momentum Chapter 2. Gauge invariant perturbations 45 tensor (again, taking just the scalar part) δ̃ρ = δρ − ρ̄′T, (2.23) δ̃P = δP − P̄′T, (2.24) ṽ = v + L′, (2.25) q̃ = q + (ρ̄ + P̄) L′. (2.26) Now we have everything that we need in order to describe the general method to find gauge invariant combinations. Firstly, note that the Eqs. (2.19)-(2.26) can be expressed in a most general way as follows à = Ã(A, T, L), B̃ = B̃(B, T, L), ..., q̃ = q̃(q, T, L). Generically, this can be written as Q̃ = Q̃(Q, T, L). These quantities are dependent of the gauge generator function T and L, this means that the perturbed quantity, Q̃, also is gauge dependent. Then, the conclusion is: If we combine the equations from (2.19) to (2.26) in order to eliminate the gauge generator functions, T and L, then the result is a gauge invariant quantity. 2.4 Applying the method We proceed to apply the previous statement in order to find useful combina- tions that are gauge-independent. 2.4.1 Bardeen’s potentials The first gauge invariant quantity widely used in cosmology is the Bardeen’s potentials (or variables), which we define below. First Bardeen’s potential Consider the following equations à = A − T′ −HT, (2.27) B̃ = B + T − L′, (2.28) Ẽ = E − L. (2.29) Chapter 2. Gauge invariant perturbations 46 In order to eliminate the gauge generator functions T and L (as well T′ and L′), we perform the following manipulations −Eq. (2.29)′ ⇒ −Ẽ′ = −E′ + L′, (2.30) ⇓ Eq. (2.28) + Eq. (2.30) ⇒ B̃ − Ẽ′ = B − E′ + T. (2.31) Consider now the following manipulations with Eq. (2.31) H× Eq. (2.31) ⇒ H(B̃ − Ẽ′) = H(B − E′) +HT, (2.32) Eq. (2.31)′ ⇒ (B̃ − Ẽ′)′ = (B − E′)′ + T′. (2.33) Finally, if we do: Eq. (2.27)+Eq. (2.32)+Eq. (2.33), then we obtain the following à +H(B̃ − Ẽ′) + (B̃ − Ẽ′)′ = A −��T′ −� ��HT +H(B − E′) +� ��HT + (B − E′)′ +��T′ à +H(B̃ − Ẽ′) + (B̃ − Ẽ′)′ = A +H(B − E′) + (B − E′)′. (2.34) The l.h.s of Eq. (2.34) is expressed in the gauge G̃, and the r.h.s is expressed in the gauge G, however, how both sides of Eq. (2.34) are equal, then we conclude that the quantity defined by Ψ ≡ A +H(B − E′) + (B − E′)′ (2.35) is a gauge invariant quantity. The Eq. (2.35) is known as the first Bardeen’s potential. Second Bardeen’s potential Consider the following equations B̃ = B + T − L′, (2.36) C̃ = C −HT − 1 3 ∇2L, (2.37) Ẽ = E − L. (2.38) Chapter 2. Gauge invariant perturbations 47 Performing the following manipulations −Eq. (2.37) ⇒ −C̃ = −C +HT + 1 3 ∇2L, (2.39) 1 3 ∇2Eq. (2.38) ⇒ 1 3 ∇2Ẽ = 1 3 ∇2E − 1 3 ∇2L, (2.40) ⇓ Eq. (2.39) + Eq. (2.40) ⇒ −C̃ + 1 3 ∇2Ẽ = −C + 1 3 ∇2E +HT. (2.41) In addition, consider the following computations −Eq. (2.38)′ ⇒ −Ẽ′ = −E′ + L′, (2.42) Eq. (2.36) + Eq. (2.42) ⇒ B̃ − Ẽ′ = B − E′ + T, (2.43) ⇓ −H× Eq. (2.43) ⇒ −H(B̃ − Ẽ′) = −H(B − E′)−HT (2.44) Finally, summing the equations Eq. (2.41)+Eq. (2.44), we have −C̃ + 1 3 ∇2Ẽ −H(B̃ − Ẽ′) = −C + 1 3 ∇2E +���HT −H(B − E′)−���HT, −C̃ −H(B̃ − Ẽ′) + 1 3 ∇2Ẽ = −C −H(B − E′) + 1 3 ∇2E. (2.45) In the same way of Eq. (2.34), we conclude from the Eq. (2.45) that the quantity defined by Φ ≡ −C −H(B − E′) + 1 3 ∇2E (2.46) is a gauge invariant perturbation. The Eq. (2.46) is known as the second Bardeen’s potential. Chapter 2. Gauge invariant perturbations 48 2.4.2 Comoving curvature perturbation Consider the following equations B̃ = B + T − L′, (2.47) C̃ = C −HT − 1 3 ∇2L, (2.48) Ẽ = E − L, (2.49) ṽ = v + L′. (2.50) Performing the following manipulations −1 3 ∇2Eq. (2.49) ⇒ −1 3 ∇2Ẽ = −1 3 ∇2E + 1 3 ∇2L, (2.51) ⇓ Eq. (2.48) + Eq. (2.51) ⇒ C̃ − 1 3 ∇2Ẽ = C − 1 3 ∇2E −HT. (2.52) In addition, consider the following computation H× [Eq. (2.47) + Eq. (2.50)] ⇒ H(B̃ + ṽ) = H(B + v) +HT. (2.53) Finally, doing Eq. (2.52)+Eq. (2.53) we are left with C̃ − 1 3 ∇2Ẽ +H(B̃ + ṽ) = C − 1 3 ∇2E +H(B + v). (2.54) From equation Eq. (2.54), we conclude that the quantity defined by R ≡ C − 1 3 ∇2E +H(B + v) (2.55) is a gauge invariant perturbation. Eq. (2.55) is known as the comoving curvature perturbation. Chapter 2. Gauge invariant perturbations 49 2.4.3 Lukash variable Let’s consider the following equations C̃ = C −HT − 1 3 ∇2L, (2.56) Ẽ = E − L, (2.57) δ̃ρ = δρ − ρ̄′T. (2.58) Performs the following manipulations −1 3 ∇2Eq. (2.57) ⇒ −1 3 ∇2Ẽ = −1 3 ∇2E + 1 3 ∇2L, (2.59) ⇓ Eq. (2.56) + Eq. (2.59) ⇒ C̃ − 1 3 ∇2Ẽ = C − 1 3 ∇2E −HT. (2.60) In addition, consider the following computation −H ρ̄′ × Eq. (2.58) ⇒ −H ρ̄′ δ̃ρ = −H ρ̄′ δρ +HT. (2.61) Finally, combining Eq. (2.60)+Eq. (2.61) we are left with C̃ − 1 3 ∇2Ẽ − H ρ̄′ δ̃ρ = C − 1 3 ∇2E − H ρ̄′ δρ. (2.62) Therefore, the quantity defined by ζ ≡ C − 1 3 ∇2E − H ρ̄′ δρ (2.63) is a gauge invariant perturbation known as Lukash variable. 2.4.4 Non-adiabatic pressure perturbation Consider the following equations δ̃ρ = δρ − ρ̄′T, (2.64) δ̃P = δP − P̄′T. (2.65) Chapter 2. Gauge invariant perturbations 50 Performing the following manipulation − P̄′ ρ̄′ Eq. (2.64) ⇒ − P̄′ ρ̄′ δ̃ρ = − P̄′ ρ̄′ δρ + P̄′T. (2.66) Finally, consider the following manipulation: Eq. (2.65)+Eq. (2.66), then we are left with δ̃P − P̄′ ρ̄′ δ̃ρ = δP − P̄′ ρ̄′ δρ. (2.67) Finally, the quantity defined by δPnad ≡ δP − P̄′ ρ̄′ δρ (2.68) is a gauge invariant perturbation known as non-adiabatic pressure perturbation. 2.4.5 Comoving gauge density perturbation This chapter ends with the comoving gauge density perturbation. Consider the following equations B̃ = B + T − L′, (2.69) ṽ = v + L′, (2.70) δ̃ρ = δρ − ρ̄′T. (2.71) Performs the following manipulations Eq. (2.69) + Eq. (2.70) ⇒ B̃ + ṽ = B + v + T, (2.72) ⇓ ρ̄′ × Eq. (2.72) ⇒ ρ̄′(B̃ + ṽ) = ρ̄′(B + v) + ρ̄′T. (2.73) Finally, consider the following operation: Eq. (2.71)+Eq. (2.73), then we are left with δ̃ρ + ρ̄′(B̃ + ṽ) = δρ + ρ̄′(B + v). (2.74) Chapter 2. Gauge invariant perturbations 51 Therefore, the quantity defined by ρ̄′∆ ≡ δρ + ρ̄′(B + v) (2.75) is a gauge invariant quantity known as comoving gauge density perturbation. We summarize all the gauge-invariant perturbations obtained in this chapter Ψ ≡ A +H(B − E′) + (B − E′)′ , (2.76) Φ ≡ −C −H(B − E′) + 1 3 ∇2E (2.77) R ≡ C − 1 3 ∇2E +H(B + v) , (2.78) ζ ≡ C − 1 3 ∇2E − H ρ̄′ δρ , (2.79) ρ̄′∆ ≡ δρ + ρ̄′(B + v) , (2.80) δPnad ≡ δP − P̄′ ρ̄′ δρ . (2.81) Chapter 3 Initial conditions 3.1 Introduction In this chapter we describe the initial conditions required to solve the perturbed system of coupled Einstein-Boltzmann equations that describe the evolution of the metric components and for the evolution of the energy density for the components of the universe as photons, cold dark matter, neutrinos, etc. Initial conditions are important and a required step to study the evolution of the perturbed equations of general relativity. We start with the equations of motion for all components in the synchronous gauge according to [18] and then we perform a change of variable to switch to the notation of [19]. This is useful because [19] build an Ordinary Differential Equation system with all variables of interest to derive the initial conditions - here we follow the same steps as in this reference. ˙= d dτ , ′ = d dx (3.1) x = kτ (3.2) δ̃i = δi/x (3.3) t̃i = ti/x2 = θi/(kx2) (3.4) Θ = h′ (3.5) σ̃ν = σν/x = Θν/x (3.6) F̃ν3 = Fν3/x2. (3.7) In the following subsections the equation on the left-hand side of the arrow “=⇒” is in the notation of [18]. Conversely, the equation in the right-hand side of the arrow “=⇒” is the same notation as in the reference [19]. 52 Chapter 3. Initial conditions 53 3.1.1 Cold Dark Matter Since [19] does not set to zero the cold dark matter velocity t̃c in the syn- chronous gauge, for consistency we also kept the term θc in the notation of [18], then we obtain δ̇c = −θc − ḣ 2 =⇒ dδc d ln x = −δ̃c − x2 t̃c − 1 2 Θ, (3.8) θ̇c = − ȧ a θc =⇒ t′c = −2t̃c − x k ȧ a t̃c. (3.9) 3.1.2 Neutrinos The neutrino equations of motion are δ̇ν = −4 3 θν − 2 3 ḣ =⇒ d̃δν d ln x = −δ̃ν − 4 3 t̃νx2 − 2 3 Θ, (3.10) θ̇ν = k2 ( 1 4 δν − Θν ) =⇒ dt̃ν d ln x = −2t̃ν + 1 4 δ̃ν − σ̃ν, (3.11) Θ̇ν = 4θν 15 − 3k 10 Fν3 7 + 2ḣ 15 + 4η̇ 5 =⇒ dσ̃ν d ln x = −σ̃ν + 4x2 t̃ν 15 − 3x2F̃ν3 10 + 2Θ 15 + 4η′ 5 , (3.12) Ḟν3 = 1 7 3k 5 Fν2 = 6k 7 Θν =⇒ dF̃ν3 d ln x = −2F̃ν3 + 6σ̃ν 7 . (3.13) 3.1.3 Photons and baryons The density contrast for photons and baryons is given respectively by δ̇γ = −4 3 θγ − 2 3 ḣ =⇒ d̃δγ d ln x = −δ̃γ − 4 3 t̃γx2 − 2 3 Θ, (3.14) δ̇b = −θb − 1 2 ḣ =⇒ dδ̃b d ln x = −δ̃b − x2 t̃b − 1 2 Θ. (3.15) Note that the reference [19] uses the tight-coupling approximation once it defines a common value for the divergence of the fluid velocity t̃γb instead of two separated values, t̃γ and t̃b as in Eqs. (3.14) and (3.15), respectively. The equations for θ̇γ and Chapter 3. Initial conditions 54 θ̇b are given by θ̇γ = k2 ( 1 4 δγ − Θγ ) + aneσT(θb − θγ), (3.16) θ̇b = − ȧ a θb + c2 s k2δb + 4ρ̄γ 3ρ̄b aneσT(θγ − θb), (3.17) where Θγ = σγ is the photon anisotropic stress, ne is the electron density, σT is the Thomson scattering cross-section and ρ̄γ and ρ̄b are the unperturbed energy densities for photons and baryons. To obtain a single equation for the plasma photon-baryon, we should linearly combine the equations (3.16) and (3.17) in order to eliminate the Thomson scattering cross section and then we set the common value θγ = θb = θγb. This algebra produces the following result ( 1 + 4 3 ρ̄γ ρ̄b ) θ̇γb = k2δγ 3 ρ̄γ ρ̄b − ȧ a θγb, (3.18) and performing the change of variables according to Eqs. (3.1) - (3.7) we have dt̃γb d ln x = −2t̃γb + δ̃γ 4 ( 3 4 ρ̄b ρ̄γ + 1 ) − (ȧ/a)(x/k)( 1 + 4 3 ρ̄γ ρ̄b ) t̃γb (3.19) In the tight-coupling regime, we neglect the photon anisotropic stress Θγ, 3.1.4 The Einstein-Field equations According to [18] the Einstein equations in the synchronous gauge are k2η − 1 2 ȧ a ḣ = 4πGa2δT0 0 (syn) (3.20) k2η = 4πGa2 (ρ̄i + P̄i) θi(syn) (3.21) ḧ + 2 ȧ a ḣ − 2k2η = −8πGa2δTi i (syn) (3.22) ḧ + 6η̈ + 2 ȧ a ( ḣ + 6η̇ ) − 2k2η = −24πGa2 (ρ̄ + P̄)Θ(syn) (3.23) We will show the details of the notation transformation below k2η − 1 2 ȧ a kdh kdτ = −3 2 ( 8πGa2ρtot 3 ) ∑ i=γ,ν,c,b δρi ρtot (3.24) Chapter 3. Initial conditions 55 using the Friedmann equation and expanding the summation over all fluid com- ponents, we have k2η − k 2 ( ȧ a )( dh dx ) = −3 2 ( ȧ a )2 { ρrad ρtot [ δργ ργ ργ ρrad + δρν ρν ρν ρrad ] + ρmat ρtot [ δρc ρc ρc ρmat + δρb ρb ρb ρmat ]} (3.25) where now we use the definition for the density contrast δi = ρi/ρi, and the definitions for the fractions according to [19]: Rγ = ργ/ρrad = Ωγ/Ωrad, and similarly for the other components Rν, Rc and Rb. Therefore, Eq. (3.25) can be written as k2η − k 2 2τ + 1 τ2 + τ h′ = −3 2 (2τ + 1)2 (τ2 + τ) 2 { 1 (2τ + 1)2 [δγRγ + δνRν] + 4 ( τ2 + τ ) (2τ + 1)2 [δcRc + δbRb] } (3.26) Switching from τ to x/k = τ, using h′ = Θ and simplifying the ks, we finally have η − 2x/k + 1 x2 (x/k + 1) = −3 2 [ Rγδγ + Rνδν x2 (x/k + 1)2 + 4 (Rcδc + Rbδb) kx (x/k + 1) ] (3.27) Following an analogous procedure for Eq. (3.21) we have k2η̇ = 3 2 ( 8πGa2ρtot 3 ) 1 ρtot { ρrad ρtot [ ργ ρrad (1 + wγ) θγ + ρν ρrad (1 + wν) θν ] ρmat ρtot [ ρb ρmat (1 + wb) θb + ρc ρmat (1 + wc) θc ]} (3.28) and performing the change of variables from τ to x we have η′ = 2 (Rγtγ + Rνtν) x2 (x/k + 1)2 + 6 ( Rbtγb + Rctc ) kx (x/k + 1) . (3.29) Chapter 3. Initial conditions 56 Following a similar approach we can rewrite the Eq. (3.22) as Θ′ + 2 (2x/k + 1) x/k + 1 Θ − 2η = −3 (Rγδγ + Rνδν) x2 (x/k + 1) , (3.30) and Eq. (3.23) as Θ′′ + 6η′′ + 2 (2x/k + 1) x (x/k + 1) ( Θ′ + 6η′′)− 2η = −12 (Rγσγ + Rνσν) x2 (x/k + 1)2 − 36 ( Rbσγb + Rcσc ) kx (x/k + 1) . (3.31) 3.2 Early-times and super-horizon initial conditions The evolution in time for the perturbed components can be obtained once we provide initial perturbations. The formalism adopted here for the procedure of obtaining these initial conditions (IC) is close to [19, 20]. Firstly, we organize the variables to be integrated into a unified vector notation as follows uT = [ δ̃γ, δ̃ν, δ̃c, δ̃b, t̃γb, t̃ν, t̃c, σ̃ν, F̃(3) ν , Θ, η ] , (3.32) where “T” means transpose. These functions can be simplified if we Taylor-expand all coefficients around x/k = τ = 0. By “coefficient” we mean any quantity that multiply δ̃γ, δ̃ν, ..., η in Eq. (3.32). These procedures allow us to separate terms proportional to x0 (constant), x, x2, and so forth. In summary, we ended with an ODE system that can be written in the matrix form, as shown by the following equation du d ln x = [ A0 + A1x + A2x2 + ... ] u, (3.33) where A0, A1, A2, ... are matrices that contain coefficients of terms of different orders in x [19]. We numerically computed these matrices up to the fourth order and the results are shown in Eqs. (3.34) - (3.38) Chapter 3. Initial conditions 57 AT 0 =  −1 0 0 0 1 4 0 0 0 0 −6Rγ 0 0 −1 0 0 0 1 4 0 0 0 −6Rν 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 −2 0 0 8Rγ 5 0 0 0 0 0 0 0 0 −2 0 8Rν 5 0 0 0 0 0 0 0 0 0 −3 0 0 0 0 0 0 0 0 0 −1 0 −1 6 7 0 0 0 0 0 0 0 0 0 0 −2 0 0 − 2 3 − 2 3 − 1 2 − 1 2 0 0 0 2 15 0 −1 0 0 0 0 0 0 0 0 0 0 0 0  , (3.34) AT 1 =  0 0 0 0 − 3Rb 4kRγ 0 0 0 0 12Rγ k 0 0 0 0 0 0 0 0 0 0 12Rν k 0 0 0 0 0 0 0 0 0 0 − 12Rc k 0 0 0 0 0 0 0 0 0 0 − 12Rb k 0 0 0 0 0 − 3Rb kRγ 0 0 8(3Rb−2Rγ) 5k 0 0 2Rγ 0 0 0 0 0 0 0 − 16Rν 5k 0 0 2Rν 0 0 0 0 0 0 − 1 k 24Rc 5k 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 k 0 0 0 0 0 0 0 0 0 0 0 0  , (3.35) Chapter 3. Initial conditions 58 AT 2 =  0 0 0 0 3Rb(3Rb−Rγ) 4k2R2 γ 0 0 0 0 − 18Rγ k2 0 0 0 0 0 0 0 0 0 0 − 18Rν k2 0 0 0 0 0 0 0 0 0 0 12Rc k2 0 0 0 0 0 0 0 0 0 0 12Rb k2 0 − 4 3 0 0 −1 3Rb(3Rb−2Rγ) k2R2 γ 0 0 − 24(Rb−Rγ) 5k2 0 0 6Rb−4Rγ k 0 − 4 3 0 0 0 0 0 24Rν 5k2 + 4 15 0 0 − 4Rν k 0 0 −1 0 0 0 1 k2 − 24Rc 5k2 0 0 6Rc k 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 3 10 0 0 0 0 0 0 0 0 0 0 0 0 1 k2 0 0 0 0 0 0 0 0 0 0 0 0  , (3.36) AT 3 =  0 0 0 0 9R2 b(2Rγ−3Rb) 4k3R3 γ 0 0 0 0 24Rγ k3 0 0 0 0 0 0 0 0 0 0 24Rν k3 0 0 0 0 0 0 0 0 0 0 − 12Rc k3 0 0 0 0 0 0 0 0 0 0 − 12Rb k3 0 0 0 0 0 − 27R2 b(Rb−Rγ) k3R3 γ 0 0 8(3Rb−4Rγ) 5k3 0 0 − 6(Rb−Rγ) k2 0 0 0 0 0 0 0 − 32Rν 5k3 0 0 6Rν k2 0 0 0 0 0 0 − 1 k3 24Rc 5k3 0 0 − 6Rc k2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 k3 0 0 0 0 0 0 0 0 0 0 0 0  , (3.37) Chapter 3. Initial conditions 59 AT 4 =  0 0 0 0 9R2 b(−9RbRγ+9R2 b+R2 γ) 4k4R4 γ 0 0 0 0 −30Rγ k4 0 0 0 0 0 0 0 0 0 0 −30Rν k4 0 0 0 0 0 0 0 0 0 0 12Rc k4 0 0 0 0 0 0 0 0 0 0 12Rb k4 0 0 0 0 0 9R2 b(−12RbRγ+9R2 b+2R2 γ) k4R4 γ 0 0 −8(3Rb−5Rγ) 5k4 0 0 6Rb−8Rγ k3 0 0 0 0 0 0 0 8Rν k4 0 0 −8Rν k3 0 0 0 0 0 0 1 k4 −24Rc 5k4 0 0 6Rc k3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 k4 0 0 0 0 0 0 0 0 0 0 0 0  . (3.38) An approximate solution can be obtained considering the lowest order in the Taylor-expansion (i.e. neglecting all terms except A0) for the system Eq. (3.33). In this case, we have u0(x) = ∑ λ cλxλu(λ) 0 (3.39) where cλ are weights that indicate the contribution of each normal mode, while uλ are eigenvectors with eigenvalues λ for the matrix A0. Let’s show that Eq. (3.39) is a solution for Eq. (3.33) at lowest order. Taking the derivative of Eq. (3.39) with respect to ln x we have du0 d ln x = ∑ λ λcλxλu(λ) 0 . (3.40) Applying A0 to Eq. (3.39) we have A0u0 = A0 ( ∑ λ cλxλu(λ) 0 ) = ∑ λ cλxλ A0u(λ) 0 = ∑ λ cλxλλu(λ) 0 (3.41) where we can identify the right hand side of Eq. (3.41) as being the rhs of Eq. (3.40) which proves that Eq. (3.39) is the lowest order solution for the equation (3.33). Using Eq. (3.34) we can find the eigenvectors and eigenvalues for A0. The results Chapter 3. Initial conditions 60 are shown in the equations (3.42) and (3.43), respectively 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 Rν Rν−1 1 0 0 0 0 0 4Rν Rν−1 4 0 0 Rν Rν−1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 −7 6 0 7( √ 5 √ 5−32Rν+5) 60 1 0 0 0 0 0 0 0 −7 6 0 −7( √ 5 √ 5−32Rν−5) 60 1 0 0 −1 3 −1 3 −1 4 −1 4 −1 36 −(4Rν+23) 144Rν+540 0 2 12Rν+45 4 84Rν+315 1 0 1 3 1 3 1 4 1 4 −1 12 −1 12 0 0 0 1 0  , (3.42) −3,−2,−2,−1,−1,−1, 0, − √ 1 − 32Rν 5 − 3 2 , √ 5 √ 5 − 32Rν − 15 10 ,−3, 1 . (3.43) 3.2.1 Zeroth order and higher order correction The solution here is written as u(λ) i where λ is the eigenvalue and i is the order correction. At zeroth order the second subscript is i = 0, at first order is i = 1, and so forth for higher order corrections. Adiabatic mode (AD) λ = 1 uAD 0 = [ −1 3 , −1 3 , −1 4 , −1 4 , −1 36 , − (23 + 4Rν) 36(15 + 4Rν) , 0, 2 3(15 + 4Rν) , 4 21(15 + 4Rν) , 1, 0 ] . (3.44) Baryon isocurvature (BI) mode: λ = −1 uBI 0 = [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0] . (3.45) Cold dark matter isocurvature (CI) mode: λ = −1 uCI 0 = [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0] . (3.46) Chapter 3. Initial conditions 61 Neutrino density isocurvature (NDI) mode: λ = −1 uNDI 0 = [ 4Rν Rν − 1 , 4, 0, 0, Rν Rν − 1 , 1, 0, 0, 0, 0, 0 ] . (3.47) Neutrino velocity isocurvature (NVI) mode: λ = −2 uNVI 0 = [ 0, 0, 0, 0, Rν Rν − 1 , 1, 0, 0, 0, 0, 0 ] . (3.48) We can use each previous normal mode (i.e. with fixed λ) to extend the solution to high-order corrections u(λ) = ( u(λ) 0 + u(λ) 1 x1 + u(λ) 2 x2 + ... ) xλ, (3.49) and we obtain the following hierarchy [(λ + 1)I − A0]u (λ) 1 = A1u(λ) 0 , (3.50) [(λ + 2)I − A0]u (λ) 2 = A1u(λ) 1 + A2u(λ) 0 , (3.51) [(λ + 3)I − A0]u (λ) 3 = A1u(λ) 2 + A2u(λ) 1 + A3u(λ) 0 , (3.52) [(λ + 4)I − A0]u (λ) 4 = A1u(λ) 3 + A2u(λ) 2 + A3u(λ) 1 + A4u(λ) 0 , (3.53) where I denotes the 11× 11 dimensional identity matrix. Let’s show Eq. (3.50) and the others follow a similar approach. The solution and the matrix up to first-order correction are u(λ) = ( u(λ) 0 + xu(λ) 1 ) xλ (3.54) A = A0 + xA1 (3.55) Taking the derivative of Eq. (3.54) with respect to ln x we have du(λ) d ln x = [ λu(λ) 0 + (λ + 1)xu(λ) 1 ] xλ. (3.56) Operating Eq. (3.55) into Eq. (3.54) and neglecting terms higher than first order in x, we have Au(λ) = [ λu(λ) 0 + xA0u(λ) 1 + xA1u(λ) 0 ] xλ. (3.57) Chapter 3. Initial conditions 62 We must have the equality du(λ) d ln x = Au(λ), (3.58) then from then rhs of Eq. (3.56) and Eq. (3.57) we have λu(λ) 0 + (λ + 1)xu(λ) 1 = λu(λ) 0 + xA0u(λ) 1 + xA1u(λ) 0 x (λ + 1 − A0)u(λ) 1 = xA1u(λ) 0 [(λ + 1) I − A0]u(λ) 1 = A1u(λ) 0 , (3.59) where Eq. (3.59) is exactly Eq. (3.50), which proves the relation. A similar proce- dure can be applied to obtain the other hierarchy relations in Eqs. (3.51), (3.52) and (3.53). Applying the Eq. (3.50) for λ corresponding to the adiabatic mode, 1,−1,−1,−1 and −2 we obtain: 3.2.2 First order correction Adiabatic mode (λ = 1): uAD 1 = [ 4 15k , 4 15k , 1 5k , 1 5k , 5Rb − Rν + 1 60k − 60kRν , 8R2 ν + 50Rν + 275 60k (8R2 ν + 90Rν + 225) , 0, 2 (4Rν − 5) 3k (8R2 ν + 90Rν + 225) , 4Rν − 5 7k (8R2 ν + 90Rν + 225) , −6 5k , −(5 + 4Rν) 12(15 + 4Rν) ] (3.60) Baryon isocurvature mode (λ = −1): uBI 1 = [ −8Rb 3k , −8Rb 3k , −2Rb k , −2Rb k , −Rb 3k , −Rb 3k , 0, 0, 0, 4Rb k , 0 ] (3.61) CDM isocurvature mode (λ = −1). uCI 1 = [ −8Rc 3k , −8Rc 3k , −2Rc k , −2Rc k , −Rc 3k , −Rc 3k , 0, 0, 0, 4Rc k , 0 ] (3.62) Chapter 3. Initial conditions 63 uNDI 1 = Neutrino isocurvature density mode (λ = −1):[ 0, 0, 0, 0, 3RbRν k (Rν − 1) 2 , 0, 0, 0, 0, 0, 0 ] (3.63) uNVI 1 = Neutrino isocurvature velocity mode (λ = −2):[ 0, 0, 0, 0, 3RbRν k (Rν − 1) 2 , 0, 0, 0, 0, 0, 0 ] (3.64) 3.2.3 Adiabatic initial conditions From the previous sections, we can combine the zeroth and first order correc- tions to obtain the adiabatic mode for the initial conditions δγ = δν = −1 3 k2τ2 +O(τ3) (3.65) δc = δb = −1 4 k2τ2 +O(τ3) (3.66) θγb = − 1 36 k4τ3 +O(τ4) (3.67) θν = − (23 + 4Rν) 36(15 + 4Rν) k4τ3 +O(τ4) (3.68) θc = 0 (3.69) σν = 2 3(15 + 4Rν) k2τ2 +O(τ3) (3.70) F(3) ν = 4 21(15 + 4Rν) k3τ3 +O(τ4) (3.71) Θ = h′ = kτ ⇒ h = 1 2 k2τ2 +O(τ3) (3.72) η = 1 − 5 + 4Rν 12(15 + 4Rν) k2τ2 +O(τ3). (3.73) Note that the results from Eq. (3.65) to Eq. (3.73) agree with results in [21, 22]. To recover the results in [19] we must perform a transformation from the synchronous gauge (used in this chapter) to the Newtonian gauge. Part II Dark Energy and Neutrinos in Cosmology 64 Chapter 4 Mass Varying Neutrinos and the Hub- ble tension The idea of neutrino-assisted early dark energy (νEDE), where a coupling between neutrinos and the scalar field that models early dark energy (EDE) is considered, was introduced with the aim of reducing some of the fine-tuning and coincidence problems that appear in usual EDE models. In order to be relevant in ameliorating the H0 tension, the contribution of EDE to the total energy density ( fEDE) should be around 10% near the redshift of matter-radiation equality. We verify under which conditions νEDE models can fulfill these requirements for a model with a quartic self-coupling of the EDE field and an exponential coupling to neutrinos. We find that in the situation where the EDE field is frozen initially, the contribution to fEDE can be significant but it is not sensitive to the neutrino- EDE coupling and does not address the EDE coincide