DISSERTAÇÃO DE MESTRADO IFT–D.001/21 Dark Energy in String Cosmology Gabriel Lucas Andrade de Sousa Orientador Horatiu Stefan Nastase Março de 2021 Sousa, Gabriel Lucas Andrade de. S725d Dark energy in string cosmology / Gabriel Lucas Andrade de Sousa. – São Paulo, 2021 235 f. : il. Dissertação (mestrado) – Universidade Estadual Paulista (Unesp), Instituto de Física Teórica (IFT), São Paulo Orientador: Horatiu Stefan Nastase 1. Constante cosmológica 2. Energia escura (Astronomia). 3. Holografia. 4. Kaluza-Klein, teorias de. 5. Teoria de cordas 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). i Dedico esta dissertação à minha famı́lia. ii Agradecimentos Escrever esta dissertação de mestrado foi um trabalho longo e desafiador. Porém, este é somente o śımbolo de mais um ciclo que se conclui na minha vida pessoal e profissional. Sem dúvidas, sem ter tido as oportunidades que tive no passado nunca conseguiria sequer chegar ao mestrado. Deste modo, começo esta lista de agradecimentos relembrando meu peŕıodo pré-graduação. Procurarei não citar muitos nomes para evitar esquecer de pessoas importantes. Aos meus amigos de ensino médio ainda na minha cidade natal, Manaus, minha profunda gratidão. Saibam que a maior parte dos amigos com os quais pretendo contar pelo resto da vida, independente da distância, ainda vem desse peŕıodo. Certamente o colegial não teria o mesmo sentido sem nossa amizade e toda a alegria com que me recebem toda vez que visito Manaus até hoje. Em especial preciso citar meu braço direito Gabriel Akel, quem não só pude ter como melhor amigo durante o ensino médio mas também como colega de apartamento durante os quatro anos de graduação. Hoje, aos meus olhos, você faz parte da minha famı́lia. Agradeço também aos meus amigos de faculdade, com os quais pude aprender muita coisa que levarei para sempre comigo. Descobri como é importante trabalhar e estudar em grupo notando que cada um tinha seus pontos fortes e fracos que quando unidos consegúıamos aprender coisas incŕıveis sob a constante pressão de curtos peŕıodos de tempo da faculdade. Sou grato também, é claro, pelos momentos únicos de festas da faculdade que vivemos juntos. Sou especialmente grato a Carolina Yaly com quem, por mais de seis anos, compartilhei cada detalhe da minha vida e que certamente me fez uma pessoa melhor. Não imagino como teria sido minha vida na faculdade sem você. Mais recentemente, tive o privilégio de poder ter colegas de mestrado que também mar- caram minha vida. Foi um peŕıodo em que tive que me adaptar na transição entre bacharel em f́ısica médica para mestrando em f́ısica de altas energias e ao mesmo tempo aprender muita coisa nova, o que só foi posśıvel com a ajuda deles. Nosso conv́ıvio dentro e fora do IFT me deu novos amigos e posśıveis futuros colaboradores. Por tudo isso, sou muito grato. Fugindo da ordem cronológica, agradeço a todos os professores que já tive (antes, durante e depois da faculdade) e que ajudaram a me moldar como profissional, formando o adulto curioso e determinado que tenho orgulho de ser. Meus pais e minha tia também merecem agradecimentos aqui uma vez que, mesmo dando a vida para que eu pudesse estudar nas melhores escolas posśıveis dentro da nossa realidade, sempre foram muito ativos no meu iii processo de aprendizagem. Gostaria de deixar meus agradecimentos em especial ao meu orientador Horatiu Nastase, que fez muito mais que apenas me orientar. Além de ter sido meu professor em duas ocasiões, sempre esteve disposto a me dar valiosas dicas profissionais. Agradeço também a Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) pelo apoio financeiro neste projeto de mestrado, processo número 2018/25390-0. Finalmente, por último o mais importante, agradeço do fundo do coração à minha famı́lia e a Deus. Meu pai, Jerônimo, minha mãe, Clemı́lia, minha tia e segunda mãe, Marlene, e minha irmã e melhor amiga, Ana, em especial. Vocês fazem com que eu me sinta a pessoa mais sortuda do mundo. Não importa quão grandes meus sonhos possam ser, sinto que com vocês ao meu lado tudo parece fact́ıvel. Vocês me deram minha religião, meu time de futebol, me ensinaram a ter prioridades na vida, me ensinaram meus valores e caráter. Nunca seria capaz de expressar o quão privilegiado me enxergo por ser dessa famı́lia que fica mais feliz que eu pelas minhas conquistas e mais chateada que eu pelos meus insucessos. Amo vocês! iv “Todas as escolhas implicam perdas. Ninguém é digno do fundamental se não estiver disposto a perder o que é trivial.” Augusto Cury “Adversity has the effect of eliciting talents which in prosperous circumstances would have lain dormant.” Horace “We absolutely must leave room for doubt or there is no progress and there is no learn- ing. There is no learning without having to pose a question. And a question requires doubt. People search for certainty. But there is no certainty. People are terrified — how can you live and not know? It is not odd at all. You only think you know, as a matter of fact. And most of your actions are based on incomplete knowledge and you really don’t know what it is all about, or what the purpose of the world is, or know a great deal of other things. It is possible to live and not know.” Richard P. Feynman v Abstract In this master thesis, three string-inspired cosmological models are compared: string (brane) gas, holographic, and chameleon cosmologies. More precisely, dark energy features are found and the cosmological constant problem is analyzed in each scenario. Their possible solutions to this problem are quite different and common ground is hard to be found. While, within string gas cosmology, dark energy traits only appear in the primordial Hagedorn phase (analogous to inflation in standard cosmology) and have hardly anything to do with the currently observed cosmological constant, both holographic and chameleon cosmologies give us interesting insights on the dominant energy density in the universe today. On one hand, within holographic cosmology, one can map the cosmological constant problem to a renormalization group (RG) flow in the dual field theory. Then, the problem can be “holo- graphically solved”, even though the precise mechanism for the bulk solution is unknown. On the other hand, the chameleon setup encourages us to motivate a scalar potential for a modulus to create a quintessence model, hoping to explain the currently observed dark energy density. In the end, it is attempted to implement the chameleon idea in the original Kaluza-Klein theory, proposing a quintessence model from its scalar field. After fixing some problems with the initial idea, it is found that the scalar potential is still not suitable to have had relevant implications on structure formation in the universe. Key-words: Dark energy; Cosmological constant problem; String gas; Brane gas; Hologra- phy; Chameleon scalar; Kaluza-Klein theory; Quintessence models. Field of knowledge: Cosmology; Field theory; String theory. vi Resumo Nessa dissertação de mestrado, três modelos cosmológicos inspirados em teoria de cordas são comparados: cosmologias de gás de cordas (branas), holográfica e camaleônica. Mais precisamente, traços de energia escura são estudados e o problema da constante cosmológica é analisado em cada cenário. As posśıveis soluções para este problema são bem distintas e caracteŕısticas em comum são dif́ıceis de serem encontradas. Enquanto, na cosmologia de gás de cordas, elementos de energia escura só aparecem na fase primordial de Hagedorn (análoga à inflação na cosmologia padrão) e pouco tem a ver com a constante cosmological observada atualmente, tanto a cosmologia hologŕafica quanto a camaleônica nos trazem novas informações interessantes sobre a densidade de energia dominante no universo hoje em dia. Por um lado, na cosmologia holográfica, pode-se mapear o problema da constante cosmológica ao fluxo do grupo de renormalização na teoria de campos dual. Logo, o problema pode ser “holograficamente resolvido”, mesmo que o mecanismo para a solução seja desconhecido na teoria gravitacional. Por outro lado, a ideia do camaleão nos encoraja a motivar um potencial escalar para um módulo a fim de criar um modelo de quintessência, esperando explicar a densidade de energia escura observada atualmente. No final, tentamos implementar a ideia do escalar camaleão na teoria original de Kaluza-Klein, propondo um modelo de quintessência a partir do seu campo escalar. Após retificar problemas oriundos da ideia inicial, foi notado que o potencial do campo escalar continuou não tendo as caracteŕısticas necessárias para ter tido um papel importante na formação de estruturas do universo. Palavras-chaves: Energia escura; Problema da constante cosmológica; Gás de cordas; Gás de branas; Holografia; Escalar camaleão; Teoria de Kaluza-Klein; Modelos de quintessência. Áreas do conhecimento: Cosmologia; Teoria de campos; Teoria de cordas. Contents List of Figures xi Notations and Conventions xiii Introduction xv 1 The Standard Model of Cosmology 1 1.1 Initial Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Geometry of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Friedmann-Lemâıtre-Robertson-Walker (FLRW) Metric . . . . . . . . 8 1.3 Matter Content and the Friedmann Equations . . . . . . . . . . . . . . . . . 12 1.3.1 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.2 Cosmological Constant . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.4 The Inflationary Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4.1 Slow-Roll Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.2 CMBR Data: Connection to Observations . . . . . . . . . . . . . . . 36 1.5 Summary of Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2 The Cosmological Constant Problem 45 2.1 Vacuum Catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2 Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 vii viii CONTENTS 3 Relevant Aspects of String Theory 55 3.1 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1.1 Kaluza-Klein Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2 Basics of Supersymmetry and Supergravity . . . . . . . . . . . . . . . . . . . 63 3.2.1 N = 4 Super Yang-Mills (SYM) . . . . . . . . . . . . . . . . . . . . . 73 3.3 Quantization of the Bosonic String . . . . . . . . . . . . . . . . . . . . . . . 74 3.3.1 D-Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.4 Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.4.1 T-Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.5 Superstring Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.5.1 Low Energy Limit: Supergravity . . . . . . . . . . . . . . . . . . . . 97 3.5.2 The Problem of Moduli Stabilization . . . . . . . . . . . . . . . . . . 97 3.6 AdS/CFT and Gauge/Gravity Duality . . . . . . . . . . . . . . . . . . . . . 98 3.6.1 Conformal Field Theories (CFT) . . . . . . . . . . . . . . . . . . . . 99 3.6.2 The Original Realization by Maldacena . . . . . . . . . . . . . . . . . 100 3.6.3 Gauge/Gravity Duality: The General Case . . . . . . . . . . . . . . . 107 4 String Gas Cosmology 109 4.1 Basics of String Gas Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.2 Special Features of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.2.1 Avoiding the Temperature Singularity . . . . . . . . . . . . . . . . . 114 4.2.2 Three Large Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.3 Solution to the Horizon Problem . . . . . . . . . . . . . . . . . . . . 117 4.2.4 T-Duality and Moduli Stabilization . . . . . . . . . . . . . . . . . . . 121 4.3 Connection to Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.3.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.3.2 Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.4 Brane Gas Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 CONTENTS ix 5 Holographic Cosmology 133 5.1 Holographic Phenomenological Approach . . . . . . . . . . . . . . . . . . . . 135 5.1.1 Domain Wall/Cosmology Correspondence . . . . . . . . . . . . . . . 136 5.1.2 Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.1.3 Phenomenological (Pseudo-)QFT Dual . . . . . . . . . . . . . . . . . 142 5.1.4 Fitting Observations: An Alternative to Inflation . . . . . . . . . . . 146 5.2 Solutions to Pre-Inflationary Problems . . . . . . . . . . . . . . . . . . . . . 147 5.3 Insight on the Cosmological Constant Problem . . . . . . . . . . . . . . . . . 154 6 Chameleon Cosmology 157 6.1 General Chameleon Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.1.1 Conditions on the Chameleon Potential . . . . . . . . . . . . . . . . . 161 6.1.2 The Thin-Shell Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.2 Chameleon in Superstring Theory . . . . . . . . . . . . . . . . . . . . . . . . 169 6.2.1 General Remarks on the KKLT Scenario . . . . . . . . . . . . . . . . 169 6.2.2 Experimental Bounds on a KKLT Motivated Model . . . . . . . . . . 172 6.3 Chameleon in the Original Kaluza-Klein Theory . . . . . . . . . . . . . . . . 178 6.3.1 Problems with the Initial Idea . . . . . . . . . . . . . . . . . . . . . . 181 6.3.2 Attempts to Fix the Problems . . . . . . . . . . . . . . . . . . . . . . 182 6.3.3 Experimental Constraints on the Models . . . . . . . . . . . . . . . . 186 7 Conclusion 197 A Deriving the Einstein Equations 201 B Friedmann and Continuity Equations 205 C Thermodynamic Computations 213 D Kaluza-Klein Original Theory 217 x CONTENTS Bibliography 227 List of Figures 1.1 Plot of the energy densities, more precisely their ratios to the critical density, as the scale factor evolves until today. The horizontal axis is represented in log scale to highlight the early radiation-dominated era, while the y-axis is also in log scale to illustrate the flatness problem. The following current values were used to generate the plot: Ω (0) Λ = 0.69, Ω (0) m ≡ Ω (0) dm + Ω (0) b = 0.31, Ω (0) r = 10−5 and Ω (0) k = 10−3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1 Representation of different configurations fundamental strings can assume. There are open strings with ends attached to different D-branes, string (a) in the figure, open strings with both ends on the same D-brane, string (b), and closed strings detached from any D-brane, as in (c). Although D-branes 1 and 2 are both D2-branes, this is not necessarily the case, their dimensionalities were chosen for illustrational purposes. . . . . . . . . . . . . . . . . . . . . . 90 3.2 Production of Hawking radiation from black holes. From a string theory per- spective, the process is understood as the interaction of two open strings, (a) and (b), to form a closed string (c) which is understood as radiation moving away from the black hole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.1 Wound strings, on the left, interacting to form unwound strings, on the right, allowing for the dimension X to grow large. The vertical lines symbolize regions of space that are identified, X ∼ X+2πR, forming a compact dimension.117 xi xii LIST OF FIGURES 4.2 Comparison between inflationary and SGC pictures of the early universe. The figure on the left represents the SGC scenario, where the Hubble radius H−1 starts off infinite and then shrinks so that the comoving scales k1 and k2 get outside it. On the right, inflation says H−1 is almost constant in this primordial era while the scales get blown outside it. At the moment t = tr reheating happens, and its analogous within SGC, thus both behaviors agree for t > tr. In both diagrams, tout(ki) (tin(ki)) stands for the time at which the scale ki (i = 1, 2) gets outside (comes back inside) the horizon. Note that, in both cases, the earlier a scale gets outside the horizon the later such scale comes back inside it. Both x-axes represent physical distances, rather than comoving ones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.1 Representation of the shape of the KKLT potential for a superpotential W = W0 + Aeia% with a < 0. The relevant region for our analysis will be R ≤ Rmin. 170 Notations and Conventions In order to avoid confusions some notations and conventions will be listed here and those will be followed throughout the whole master’s thesis unless otherwise specified. First, indices conventions. Middle-alphabet Greek letters, e.g. µ, ν, ρ, will denote space- time indices, while middle-alphabet Latin letters, e.g. i, j, k, will stand for spatial in- dices only. We may also represent 4-dimensional (4D) spacetime vectors as, for example, pµ = (p0, ~p), where ~p is a vector in the 3-dimensional (3D) spatial subspace of the full 4D spacetime. When we consider higher dimensional spacetime, for instance in Kaluza-Klein the- ories, spacetime vectors will usually be denoted by capital Latin letters, e.g. pM = (pµ, pm) where µ still stands for the four observed dimensions and m runs through indices of the complementary subspace. If we want to write explicitly the spinorial components of a spinor, their indices will be denoted by letters from the beginning of the Greek alphabet, e.g. α, β, γ. While the beginning of the Latin alphabet, e.g. a, b, c, will be used for indices of gauge groups in some given representation. We will be adopting the mostly plus signature convention for Lorentzian spacetime met- rics, e.g. 4-dimensional Minkowski metric is ηµν = diag(−1,+1,+1,+1). As usual, spacetime indices are raised or lowered by contracting with the metric gµν or its inverse gµν . Multiplication of two objects that have the same index represents a contraction and this index is implicitly summed over: A · B ≡ AµBµ ≡ ∑ µA µBµ = ∑ µA µBνgµν . This is the Einstein summation convention. The natural units system will be used. This means ~ = c = kB = 1. Since two of the xiii xiv NOTATIONS AND CONVENTIONS three (considered to be) fundamental dimensional constants - c, ~ and G - are set to one, all quantities can be expressed, for example, in terms of powers of units of mass. Thus we have the the following equivalences in dimensions of quantities: [Mass] = [Energy] = [Length]−1 = [Time]−1 = 1 The last equality is conventional, since any quantity Q has dimensions of Massn (for any n ∈ Z) then we can represent its dimension by [Q] = n. When we need to compare against experimental values expressed in some other units we will restore dimensions of time and length via dimensional analysis. Our convention for d-dimensional Fourier transform and inverse Fourier transform will be: F (k) = ∫ ddxf(x)eik·x ; f(x) = 1 (2π)d ∫ ddkF (k)e−ik·x ; such that we have the expected integral form for the d-dimensional Dirac delta function: δd(x) = 1 (2π)d ∫ ddkeik·x The symbol log will be used to represent the Napierian logarithm, i.e. logarithm to the base e, instead of log10 as it is sometimes used. Therefore, log 10 ≡ loge 10 ' 2.3 6= 1. Introduction Cosmology has recently, i.e. in the past few decades, entered a precision era that opened windows for testing fundamental theories through cosmological observations of an early hot epoch of the universe. More precisely, there was a time in the past when the universe was too hot to allow for stable atoms. As the universe cooled down, most of the free charges permeating the cosmos were put together to form electrically neutral atoms. Since then, photons were able to travel almost freely through space. Radiation of that primordial era is constantly interacting with our detectors nowadays and it earns the name cosmic microwave background radiation (CMBR). This is one of the most important pieces of data we have to infer how the universe behaved shortly after what is called the cosmological singularity, a point in time where our standard cosmological theory breaks down. The current framework which better explains, i.e. better agrees with observations, the cosmic evolution from now back to a few minutes after the cosmological singularity is the so-called ΛCDM model or Hot Big Bang cosmology. In this model, at the largest scales, the matter content of the universe behaves like a mixture of different types of fluids whose energy densities are currently dominated by the cosmological constant Λ, hence the first let- ter in ΛCDM, followed by non-relativistic (dust) matter, which itself is mostly composed of cold dark matter (CDM) completing the model’s name, and a small amount of radiation or relativistic matter. Back in the past both dust matter and radiation were once more domi- nant than the cosmological constant. Although this model is very successful in reproducing observations, some problems arise within it. These are called pre-inflationary problems since xv xvi INTRODUCTION they are often considered to be solved by inflation, a period of rapid expansion of the fab- ric of space. Inflation plus ΛCDM cosmology are usually considered the standard model of cosmology, in analogy to the highly precise Standard Model of particle physics. However, even the inflationary paradigm comes with its caveats. At high enough energies our best physically tested theories are supposed to give place to a yet unknown (at least not completely known) more fundamental quantum theory which includes gravity. String theory is the most developed candidate for such a quantum gravity theory. More than just a quantum theory of gravity, it includes all the other known fundamental forces of nature. For this fact, it is usually classified as a theory of everything. Nevertheless, string theory is still a work in progress. It is natural then to use these cosmological observations as an opportunity to eventually be able to get experimental hints on what string theory is since our best particle experiments on Earth are still many (∼ 14) orders of magnitude away from the energy scale at which stringy effects are expected to be relevant. In this thesis, our goal is to analyze how dark energy features arise in three different string- inspired cosmological models: string/brane gas, holographic, and chameleon cosmologies. Dark energy is just a generalization of the cosmological constant as the type of energy that is responsible for the currently observed accelerated expansion of the universe. Currently, we are not aware of what dark energy is, or what it is made of, we can only feel its gravitational effects. Another related problem to be studied in the previously mentioned models is the so- called cosmological constant problem. The issue is that the theoretically expected value for the cosmological constant is off by a ∼ 10−120 factor – in the most severe case – of its observed value. This likely indicates something is fundamentally wrong with our understanding of what the cosmological constant (or dark energy) is. The first three chapters of this thesis are introductory ones on standard textbook subjects. Chapter 1 properly introduces the inflation plus ΛCDM model. The relevant issue of how CMBR data is compared to it is sketched at the end of the chapter. Chapter 2 is shorter than the rest but earns its privileged spot since it introduces dark energy and the main problem of INTRODUCTION xvii the thesis, perhaps one of the main problems in theoretical physics today, the cosmological constant problem. After that, we make a digression in chapter 3 to introduce a few string theory concepts that will be needed in the last chapters. Finally, the main focus of the thesis is in the following three chapters, one for each cosmo- logical model. Chapter 4 introduces string gas cosmology, first proposed by Brandenberger and Vafa [1], and its more general version, brane gas cosmology [2]. We start by reviewing the model’s defining properties and their consequences. Its comparison with CMBR data is also described since the main goal of any physical model must be to be compared against observations. In the end, we show how dark energy features arise in the model. Chapter 5 is on holographic cosmology, started off by Maldacena [3]. This model has increasing expecta- tions of being a reliable and robust alternative to inflation. It is argued that, within it, one can reproduce observations with fewer assumptions than within inflation, i.e. it is a broader paradigm. Most of the chapter is concerned about the gauge/gravity duality introduced in [4, 5] by McFadden and Skenderis. At the end of the chapter, we review how the cosmo- logical constant problem could potentially be “holographically solved” within the model [6]. Chapter 6 is on chameleon cosmology, first proposed by Khoury and Weltman [7, 8]. After the main properties of chameleon scalars are described, we review an attempt to embed this scenario in string theory and how observations constrain the model. After that, some new calculations are presented by trying to implement the chameleon idea within the Kaluza- Klein (KK) theory. We try to build a quintessence model out of the KK chameleon. Some problems are found in this attempt and solutions to them are proposed as phenomenological chameleon models which, unfortunately, end up not being suitable to have had an important role in structure formation in the past. In chapter 7 we conclude by comparing the models. The thesis includes four appendices in which long, yet relevant (but not too relevant to be placed in the main chapters), calculations are found. The most important of them is appendix D, where the Kaluza-Klein original theory is dimensionally reduced in a considerable amount of detail. This appendix is essential to the Kaluza-Klein discussion of chapter 6. Chapter 1 The Standard Model of Cosmology Cosmology is the the branch of science which studies the largest scales in our universe. Actually, this does not capture the whole picture. Since the discovery of special relativity by Albert Einstein in 1905, see original papers [9, 10], physicists realized that there is a maximum speed, c, for propagation of causal influence through space, the so-called “speed of light” for historical reasons. The existence of this threshold for causal influence implies that when we are looking at the furthest points in the sky we are actually looking back in time, since it takes a finite and considerable amount of time for light to travel from a star far away from Earth until it reaches us. This phenomenon is very much like when one throw a rock at a window. The window doesn’t break immediately after the rock is released and it takes more time between the release of the rock and the breaking of the window if one is further away from the window in the first place. Newton’s law of gravity, for example, assumes the effect of a cause to happen instantaneously. Thus Einstein’s postulate was really groundbreaking as it implies that cosmology also tells us the history of our universe. This is a rather new branch of science, compared to other subareas of physics, as one of the first observations that shaped cosmology as it is today only happened in 1929 when Edwin Hubble noticed the universe was expanding [11]. We summarize here the two main observed facts that enable us to figure out the geometry of our universe and thus, from the 1 2 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY Einstein equations, its evolution. 1.1 Initial Observations 1. Hubble law: The first such observation was the Hubble expansion cited above. To understand the consequences of this observation we first recall the definition of redshift, from the Doppler effect in special relativity: z ≡ λ− λ0 λ0 ; (1.1) where λ is the observed wavelength of the light ray emitted by the star and λ0 is the wavelength at the moment of emission of this ray. We are able to figure out λ0, even though we don’t measure it, from the atomic composition of the star. As we can see, this quantity earns the name redshift because when λ > λ0 we have z > 0 and we say that the wavelength was “redshifted” (large wavelengths on the visible spectrum correspond to the color red). On the other hand, when λ < λ0 we have z < 0 and we say the wavelength was “blueshifted” (small wavelengths on the visible sprectum correspond to the colors violet or blue). Hubble measured the distances of stars in the sky to us, r, and their respective redshifts. We will not go into much detail of how one measures the distance of astronomical objects here (for more details see chapter 3 of book [12]), but it can be done by measuring the brightness of some celestial bodies that work as standard candles, in the sense that we expect them all to have the same brightness at the same distances. He then was able to fit a straight line whose slope defines the famous Hubble constant, H. z = Hr (1.2) 1.1. INITIAL OBSERVATIONS 3 Nowadays we know that the Hubble constant is not necessarily a constant and, in general, we have H = H(t). Thus Hubble parameter is a more appropriate name and that is how we will call it from now on in this thesis. 2. Isotropy of the universe: The second observation comes from a set of data called the Cosmic Microwave Back- ground Radiation (CMBR). The CMBR is so important to cosmology that it will have a section later on better discussing its features. For now it suffices to say that this is the radiation arriving at Earth from the Big Bang that was last scattered at the last moment our universe was ionized. A long time ago the gas of photons (and by that time another types of matter coupled to it) permeating our universe was so hot that stable atoms were not able to form. Every time an atom was formed it quickly interacted with some nearby photon that would ionize it. After that period, the uni- verse cooled down allowing atoms to be stable and photons were able to travel through an electrically neutral space with low probability of interacting electromagnetically. However their gravitational interaction with massive bodies was not shut down by this cooling down process. Moreover, with the electromagnetic interaction being negligible, the gravitational interaction dominates and when we observe the CMBR nowadays we see the effect that massive bodies have on its path until it reaches us. Since we observe an isotropic distribution of this radiation coming from everywhere in the sky, up to a high precision, we conclude that photons coming from any direction were affected by objects in their paths roughly in the same manner and therefore the distribution of mass in our universe is isotropic. 3. Homogeneity of the universe: The last experimental fact that shapes our understanding of cosmology is the homo- geneity of the universe. Although it is easy to see that on scales of the Milky Way our universe is composed of small regions with a large density (stars and planets) and 4 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY basically empty space in between them, i.e. the space is inhomogeneous, at the largest scales our universe is homogeneous. On cosmological scales galaxies are just point-like objects and we can indirectly conclude, from studying peculiar velocities of galaxies (we will discuss this method in more detail when we talk about dark matter), that the distribution of mass in our universe is indeed homogeneous. Now that we listed the important facts that enable us to guess the form of the metric of our universe, we can go to a more geometrical or general relativistic discussion of how our universe looks like at the largest, i.e. cosmological, scales and how it evolves. 1.2 Geometry of the Universe We started this chapter by highlighting how groundbreaking special relativity was. Now we must briefly introduce its extension, the so-called general relativity (GR). Special relativity earns its name because it is a theory that reconciled the known fact that every inertial frame is equivalent, meaning that the laws of physics should be written in the same way in any such frame, with the peculiar fact that Maxwell equations implied a constant speed of light, later on Einstein figured out that this speed is not just that of light but it is an upper limit on the speed of any causal interaction. “Special” means that this equivalence is manifested only among inertial frames while “relativity” is what we call a theory that relates observations from one frame to another. GR, then, is the generalization of the equivalence of special relativity to any frame, including accelerated ones. Einstein realized that any accelerated frame is locally equivalent to a frame under the influence of gravity, this is the equivalence principle. The importance of this statement is incomplete until we realize that gravity is better described as the influence of the shape of space itself, spacetime to be more precise, on the objects moving on it rather than an actual force. Therefore, an object under the influence of just gravity is not actually under the influence of any force and a reference frame following its motion would be inertial. Since accelerated frames are locally equivalent to frames under 1.2. GEOMETRY OF THE UNIVERSE 5 the influence of gravity, as we said, we finally are able to relate non-inertial frames to inertial ones and can use the rules of special relativity, taking into consideration the curvature of spacetime, to any frame. In order to describe spacetime as a dynamical background on which all the physical phenomena take place one needs to understand differential geometry. Our purpose here is not to describe GR in its full geometrical fashion but to jump as soon as possible to the equations that determine the dynamics of the fabric of spacetime, i.e. the Einstein field equations, and apply them to the cosmological scenario. Therefore we will claim that spacetime, from the perspective of GR, is a 4-dimensional differential manifold M carrying a torsion-free connection compatible with a (0,2)-tensor field called metric gµν without explaining in a mathematical rigorous manner what each term means (more details can be found in classical books, for instance [13, 14]). To gain intuition, we can picture the spacetime structure (M, gµν) as just a 4D version of a 2D rubber surface, like a toy balloon without its lip, that cannot have sharp bumps on it but can change its shape smoothly and can stretch or shrink. For us the important point is that the metric is a field, roughly speaking in physicists sense a field is any quantity that may assume different values at different points on the manifold, and therefore we can try to figure out an action for it and work out the equations of motion following the usual classical field theory machinery. Fortunately, around the time Einstein found his equations [15], the mathematician David Hilbert found an action that yielded the same equations of motion as the work of Einstein [16]. The so-called Einstein-Hilbert action is: SE−H = 1 2κ ∫ d4x √ −gR ; (1.3) where κ is just a proportionality constant, g ≡ det(gµν) is the needed factor to make d4x √ −g invariant and R is the Ricci scalar which is defined as the trace of the Ricci curvature tensor R ≡ gµνRµν . In its turn the Ricci tensor is defined from the Riemann tensor as Rµν ≡ Rρ µρν . Moreover, for a torsion-free connection compatible with the metric we also can write the 6 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY Riemann tensor in terms of the Christoffel symbols: Rρ µνσ = ∂νΓ ρ µσ − ∂σΓρµν + ΓλµσΓρλν − ΓλµνΓ ρ λσ (1.4) And the Christoffel symbols in terms of the inverse metric and derivatives of the metric: Γρµν = gρσ 2 (∂µgσν + ∂νgσµ − ∂σgµν) (1.5) As we can see from the above equations, although the action looks very simple in terms of the Ricci scalar, it actually involves first and second derivative of the metric. The important point is that R is a scalar field, so every observer, i.e. frame, will agree on its value at a specific point on the manifold. We shall make a digression from the usual path of calculating the equations of motion for SE−H for a while to comment on the role of the constant κ. First, note that the metric is dimensionless, [gµν ] = 0, since this is the structure that tells us how to measure distances in a space, ds2 = gµνdx µdxν , and we can assume without loss of generality that [ds2] = 2[dxµ] = −2. Actually, not all components of dxµ need to have units of length but we can always redefine our components such that this is the case. Anyway, the important fact is [gµν ] = 0 and we will skip minor details. Thus [R] = 2, each term contains two derivatives, and since [d4x] = −4 and an action is always dimensionless – it actually has dimensions of ~ which is dimensionless in the units we are using – we have [κ] = −2. Since we must reproduce Newton’s law of gravity in the non-relativistic limit, we get κ = 8πG, where G is Newton’s gravitational constant in 4 dimensions (see [14]). Furthermore, we can rewrite it in terms of the 4D reduced Planck mass M2 Pl = 1 8πG = κ−1. Note that [κ] = −2⇒ [MPl] = 1 ; [G] = −2 as expected. This discussion on the role of κ will be important when we talk about higher dimensions of spacetime. Now we end our digression and comeback to the equations of motion. We get the Einstein field equations without presence of matter by requiring that δSE−H = 1.2. GEOMETRY OF THE UNIVERSE 7 0, from equation (1.3), when we vary the metric (or its inverse). If we want to include matter fields to this scenario we need to consider an action Sm = ∫ d4x √ −gLm that is added to the Einstein-Hilbert one. Moreover, another simple addition we can make is to include a cosmological constant term SΛ = − 1 κ ∫ d4x √ −gΛ to it, where Λ is just a spacetime constant. This cosmological constant term turned out to be very relevant to describe cosmology as we will see later on. All in all, we end up with S = SE−H +SΛ +Sm and we require that δS = 0 when the metric is varied. The resulting equations of motion are, see appendix A: Gµν + Λgµν ≡ Rµν − 1 2 gµνR + Λgµν = 8πGTµν ; (1.6) where Tµν ≡ −2√ −g δSm δgµν is the energy-momentum (or stress-energy) tensor that represents the content of energy and momentum of matter fields in our system, i.e. for cosmology in the whole universe. Equation (1.6) indicates how matter, through the energy-momentum tensor in the right hand side of the equation, influences the shape of the fabric of spacetime, i.e. shape of the universe in our cosmological analysis since our physical system is the whole universe. Futhermore, since Sm[ψi] contains a factor of the metric in it – at least through √ −g but generally not just this factor – if we calculate the equations of motion with respect to the matter fields ψi they will depend on how the universe is curved. In conclusion, GR tells us that gravity is not a force in the usual classical sense. It is actually geometry, i.e. the effects we think of being caused by a force are a reflect of the curvature of spacetime in our region. Both the matter content of the universe influence its shape and this shape influence the path of particles traveling through spacetime. We end this section by noticing that the term Λgµν coming from the action SΛ can either be interpreted as a piece of the gravitational action and therefore lying in the gravitational side of the last paragraph’s analysis or it can be put in the right hand side (RHS) of the equation (1.6) and compose another type of matter content of the universe, such that the new energy momentum tensor would be T̃µν = Tµν − Λ 8πG gµν . Both interpretations, of course, 8 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY generate the same experimental predictions thus it is a matter of convenience interpreting the cosmological constant as a component of the matter content or not. The observations made in this paragraph will be important later on when we discuss the types of matter composing our model of the universe. 1.2.1 Friedmann-Lemâıtre-Robertson-Walker (FLRW) Metric Now that we briefly summarized the main features of GR needed for us, we will look for a specif realization of the Einstein equations to our system or interest, the whole universe. The first thing to be noted is that the Ricci scalar and tensor in equation (1.6) have up to second derivatives of the metric in all sorts of complicated ways. Therefore, the Einstein equations are a set of highly nonlinear equations that are nearly impossible to be solved analytically without symmetry restrictions to the metric. That is where the observational facts discussed in the first section of this chapter comes in handy. Spatial homogeneity and isotropy at the largest scales mean that the spacetime is composed of spatial slices that lie on top of each other in a time ordered manner. The metric needs to have the form: ds2 = −dt2 + dl2phys ; (1.7) where dl2phys represents the spatial subspace that we are going to analyze in detail now. We must notice that isotropy forbids g0i components to assume a non-zero value and that, in principle, we could have written g00dt̃ 2 instead of just dt2, however we could always redefine dt = √ g00dt̃ in order to write the metric in the form of equation (1.7). The observation of the Hubble expansion implies that the physical spatial slices change in side with time, therefore we can define comoving coordinates dl, i.e. coordinates that follow the expansion of the universe, such that dlphys = a(t)dl. The function dictating the expansion of the universe a(t) is called the scale factor. If we want dl to have units of length the scale factor is dimensionless. Nevertheless, for now, it is useful to attribute the length dimension to a(t) and work with a dimensionless dl. 1.2. GEOMETRY OF THE UNIVERSE 9 We must now figure out the possible forms of dl2. Again, homogeneity and isotropy play an essential role. We must only consider maximally symmetric 3D spaces. There are three possibilities: 1. a spatially flat universe: dl2 is the line element of a 3-dimensional Euclidean space. dl2 = δijdx idxj = d~x2 (1.8) 2. a positively curved spatial subspace: a sphere S3. The 3D sphere’s line element can be defined from its embedding in a 4-dimensional Euclidean space as: dl2 = δijdx idxj + dy2 = d~x2 + dy2 ; where: δijx ixj + y2 = ~x2 + y2 = R2 (1.9) Note that by a redefinition of xi and y we can rewrite the constraint as ~x2 + y2 = 1 without loss of generality. Therefore, by taking differentials of the new constraint we get: dy2 = (~x · d~x)2 1− ~x2 ⇒ dl2 = d~x2 + (~x · d~x)2 1− ~x2 (1.10) 3. a negatively curved spatial subspace: a 3D Lobachevsky space or a hyperboloid H3. This space’s line element can be defined from its embedding in a 4D Minkowski space as: dl2 = δijdx idxj − dy2 = d~x2 − dy2 ; where: δijx ixj − y2 = ~x2 − y2 = −R2 (1.11) Again we can get rid of the radius R for the same argument as in the previous case and work with the constraint ~x2 − y2 = −1. Again we can calculate the differential dy in terms of the other coordinates and plug it back into equation (1.11). dy2 = (~x · d~x)2 1 + ~x2 ⇒ dl2 = d~x2 − (~x · d~x)2 1 + ~x2 (1.12) 10 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY It is easy to see that we can write all three cases into one equation for the spacetime line element by introducing a constant k. ds2 = −dt2 + a2(t) [ d~x2 + k (~x · d~x)2 1− k~x2 ] ; where: k ≡  0 flat space; +1 positively curved; −1 negatively curved. (1.13) That is the famous Friedmann-Lemâıtre-Robertson-Walker (FLRW) metric. Some books may call it just Robertson-Walker (RW) or Friedmann-Robertson-Walker (FRW) metric. Since the name does not change the physics, it is just important to notice they are talking about the same metric. Now, we may want to rewrite equation (1.13) such that the scale factor becomes dimen- sionless and dl recovers dimension of length. We are able to do that since the FLRW metric has a rescaling symmetry, meaning that if we change a→ a/λ, xi → λxi and k → k/λ2 the metric remains the same. Thus, we can not only choose λ such that [λ] = −1 leading to a dimensionless a(t) but also set a(t0) = 1 where t0 is the cosmological time (we will shortly define precisely what it means) of today. From now on in this thesis, we will consider a dimensionless scale factor such that its value is equal to one at the present time. A perhaps more commonly found, for instance [12, 17], form of this metric is the realization of equation (1.13) in spherical coordinates. In these coordinates we have d~x2 = dr2 + r2dθ2 + r2 sin2 θdφ2 = dr2 + r2dΩ2, ~x2 = r2 and ~x · d~x = rdr, leading to: ds2 = −dt2 + a2(t) ( dr2 1− kr2 + r2dΩ2 ) (1.14) One may also define a conformal time η such that dη = dt/a(t). Then, equation (1.14) becomes: ds2 = a2(η) ( −dη2 + dr2 1− kr2 + r2dΩ2 ) ; (1.15) 1.2. GEOMETRY OF THE UNIVERSE 11 where a(η) is the same a(t) when we plug in the expression t = t(η) that we can calculate by integrating dη = dt/a(t). Moreover, notice that the reason for the name “conformal” time is that in equation (1.15) the scale factor works as a conformal factor of a conformal transformation, i.e. g̃µν = a2(η)gµν . Since we introduced the scale factor, we are now able to define the Hubble parameter from it. As we discussed in the beginning of the chapter, the Hubble parameter, first thought to be a constant, quantified the rate of expansion of the universe, therefore it is natural to express it in terms of the change (derivative) of a(t). More precisely, from equation (1.1) and the fact that the speed of light is constant in the vacuum (c = 1): z ≡ λ− λ0 λ0 = 1/ν − 1/ν0 1/ν0 = ν0 − ν ν ; (1.16) where ν0 is the emitted frequency and ν is the observed one. From the expression of the relativistic Doppler effect we can rewrite ν0 in terms of ν as: ν0 = ν √ 1 + v 1− v = ν(1 + v) +O(v2); (1.17) where v is the velocity of the source of light (in our case, a bright star in the sky) with respect to the detector (us on Earth). In our convention v is positive if the source is moving away from the detector. Thus, up to first order, z = v and Hubble actually found a relation between the velocity of a star moving away from us with the distance to us. Now, coming back to the implications the scale factor has on cosmological scales, recall that any physical distance can be written as the scale factor times a comoving distance, ~rphys = a(t)~r, and notice that we can calculate the physical velocity of any object moving with respect to us as: ~vphys ≡ d~rphys dt = da dt ~r + a d~r dt ≡ ȧ a ~rphys + ~vpec ; (1.18) where we defined the peculiar velocity ~vpec as the velocity that is not due to the change of 12 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY a(t) and the other component of ~vphys, (ȧ/a)~rphys, is called the Hubble flow. If an observer follows the Hubble flow, i.e. if it is a comoving observer, then the scale factor would be just a constant from his or her perspective, therefore the observer would only measure the peculiar velocity. Equation (1.18) shows us that the angular coefficient of the linear relation z = v = Hr measured by Hubble is actually equal to: H ≡ 1 a da dt ≡ ȧ a ; (1.19) and we arrive at our expression for the Hubble parameter in terms of the scale factor as promised. One last comment is in order. In the beginning of the chapter, we skipped the fact that the universe is essentially isotropic in a special frame, the CMBR (or cosmic) frame, not in any frame. The CMBR frame is defined as the one in which this fluid that is composed of photons traveling around from the Big Bang is homogeneous and isotropic. Observations made on Earth point out that the CMBR has a dipole (see [12]). However, it is actually associated to the motion of Earth with respect to the CMBR frame. Therefore, since we constructed the FLRW metric from observations valid in the cosmic frame, the cosmological time which appears in equation (1.13) is defined in this special frame. It is with respect to this time that we compute the age of the universe, for example. At first sight, on special relativistic grounds, it seems weird that we can compute the age of the universe and usually do not say with respect to which observer. Nonetheless, we can do such a thing because we have a special cosmic frame in this case and it is perfectly fine according to GR. 1.3 Matter Content and the Friedmann Equations We are almost ready to apply the Einstein equations, equation (1.6), to the universe as a whole. We already know the form of the metric which is going to be used, however 1.3. MATTER CONTENT AND THE FRIEDMANN EQUATIONS 13 we still have to describe the energy-momentum tensor of our system. First, we must recall the discussion of the first section, where we pointed out that, at the largest scales, galaxies and even clusters of galaxies are point-like objects. Therefore, if we could zoom out our perspective to these scales we would see a bunch of points potentially interacting with each other which could be heated up (or cooled down) if we find a mechanism to supply the system with (or extract from it) some energy. In conclusion, the components of the universe behave as a fluid. It is usual to model them as a perfect fluid - in this background cosmology description, on which we will consider perturbations later on - and we will restrict ourselves to this standard analysis in this thesis. A fluid is described by its pressure P , its energy density ρ and its comoving 4-velocity uµ (or, equivalently, its dual covetor uµ ≡ gµνu ν). The stress-energy tensor, i.e. the energy-momentum tensor, of a perfect fluid assumes the form: Tµν = ρuµuν + P (gµν + uµuν) (1.20) It is important to notice that P and ρ are not two independent variables. In fact, we call the equation of state of an energy content in our universe the relation between its pressure and its energy density, P = P (ρ). We will shortly see that we can classify all energy content in our universe basically in three categories. Heavy matter, with respect to the average temperature of the CMBR, is called non-relativistic matter, baryonic matter or just matter. Matter which is light with respect to the CMBR is called relativistic matter or just radiation. Finally, we also have an exotic type of matter called dark energy. Even though the relation between P and ρ for any type of energy doesn’t have to be linear, the equation of state of these three energy contents can be written as P = wρ, where w is a different constant for each type of matter. We will describe exactly what is the value of w for each energy content but first, for the sake of simplicity, let us assume only one energy content described by equation (1.20). We finally are able to work out the equations of motion of cosmology, the Friedmann equations, from the Einstein equations, equation (1.6), see appendix B for the derivation. The first Friedmann equation is: 14 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY ( ȧ a )2 = 8πG 3 ρ− k a2 (1.21) And the second Friedmann equation is: ä a = −4πG 3 (ρ+ 3P ) (1.22) Since equations (1.21) and (1.22) tell us the evolution of the scale factor, they actually dictate how our universe behaves. Another important equation which will be analyzed in more detail now is the continuity equation, see appendix B equation (B.26). Each matter component obey this equation. ρ̇ = −3 ȧ a (ρ+ P ) (1.23) We will consider a equation of state P = wρ for a generic parameter w 6= −1. The specific, and relevant, case in which w = −1 will be described in the cosmological constant section. Soon we will derive the value of this parameter for each of the three main components of the universe and we will come back to the following general relations. From the continuity equation: dρ dt = −3ρ(1 + w) 1 a da dt ⇒ ∫ ρ(a) ρ(0) dρ̃ ρ̃ = −3(1 + w) ∫ a a0 dã ã (1.24) The subscript (or superscript) 0 indicate the value of quantities nowadays. Recall that we set a0 = 1, then: ρ(a) = ρ(0)a−3(1+w) (1.25) In equation (1.21), we can define a curvature energy density ρk ≡ − 3k 8πGa2 such that the equation becomes H2 = (8πG/3)(ρ + ρk). However, we will shortly discuss an experimental observation [18] which leads to ρ(0) � ρ (0) k , i.e. an almost spatially flat universe, today. We 1.3. MATTER CONTENT AND THE FRIEDMANN EQUATIONS 15 will therefore drop the curvature contribution to the first Friedmann equation. In fact, we haven’t showed that ρ� ρk at all times and we could only drop this term if we integrate the expression between times close to today. Nevertheless we will drop this term and integrate to any time and just later we will argue that we could do that since ρ is indeed much greater than ρk at all times. Then, we can plug the result of equation (1.25) into equation (1.21) neglecting the curvature term to obtain: ȧ a = √ 8πG 3 ρ(0)a− 3 2 (1+w) ⇒ ∫ a(t) 0 da′a′ 1 2 (1+3w) = √ 8πG 3 ρ(0) ∫ t 0 dt′ ⇒ a(t) ∝ t 2 3(1+w) (1.26) Note that in order to get to equation (1.26) we considered that a → 0 as t → 0. This is called the cosmological singularity. It is a natural expectation from the expansion of the universe observation. Furthermore, it would be hard to come up with a natural explanation, within this simple fluid model for matter components, for a universe that would have started off huge, then shrunk to a minimum and would be now expanding to break our assumption of a cosmological singularity. From equation (1.26), the Hubble parameter can be related to time as: a(t) = Ct 2 3(1+w) ⇒ ȧ(t) = Ct 2 3(1+w) 2 3(1 + w)t ⇒ H ≡ ȧ a = 2 3(1 + w)t ∝ a− 3(1+w) 2 (1.27) From equation (1.27) it is easy to see that if 2 3(1+w) ∼ 1, which we will see happens for most of the history of our universe therefore is a good approximation, then t ∼ H−1 which means that the age of the universe is t0 ∼ H−1 0 and the Hubble constant, i.e. the Hubble parameter evaluated at today’s time, is H0 ' 70 (km/s)/Mpc. Therefore, H0 ' 70 × 3.2 × 10−20s−1 ' 7× 10−11 years−1 ⇒ t0 ∼ 10 billions years. We must point out that nowadays there are two independent ways of measuring H0 which are not compatible, i.e. they differ from each other 16 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY with a high confidence interval, see for example [18, 19, 20, 21, 22]. This incompatibility is the so-called Hubble tension and explaining what is going on is a current challenge for cosmologists. Nevertheless, for our order of magnitude calculation H0 ∼ 100 (km/s)/Mpc was precise enough. Finally, from equations (1.27) and (1.21) neglecting the curvature term, we find how the energy density of a type of matter whose equation of state parameter is w 6= −1 depends on time: ρ(t) = 3 8πG H2 = 3 8πG [ 2 3(1 + w)t ]2 = 1 6πG(1 + w)2t2 (1.28) Notice that, although we were considering just one type of matter in our universe in the above analysis for simplicity, it is easy to generalize to many types of matter, each with its own w value. First, all types of matter have their own continuity equation (1.23). In the Friedmann equations (1.21) and (1.22), we just have to replace ρ → ∑ i ρi and P → ∑ i Pi. Equation (1.25) is derived entirely from the continuity equation, thus is valid for each type of matter, taking into consideration their different values of w, of course. And equations (1.26), (1.27) and (1.28) are all valid if we have an energy density which dominate over the others such that ∑ i ρi ' ρdom as long as we replace w → wdom. A form of the first Friedmann equation that will be useful when we compare against observations is the following. First, let us define the critical density as: ρc ≡ 3H2 8πG (1.29) And the ratios of the energy densities to the critical one as: Ωi ≡ ρi ρc ; (1.30) where i stands for the different types of matter in the universe. Considering ρk as we introduced before and therefore interpreting the curvature as a type 1.3. MATTER CONTENT AND THE FRIEDMANN EQUATIONS 17 of energy content, equation (1.21) can be rewritten as: H2 H2 0 = 8πG 3H2 0 ∑ i ρi = ∑ i ρ (0) i ρ (0) c a−3(1+wi) = ∑ i Ω (0) i a−3(1+wi) ; (1.31) where we have used the definitions (1.29) and (1.30) and the expression for ρ = ρ(a) found in equation (1.25). The subscript 0 and the superscript (0) stand for quantities evaluated at today’s time, as usual. Equation (1.31) is useful since CMBR data [18] are expressed in terms of Ω (0) i and H0. Now we should derive the equation of state parameter for relativistic, i.e. radiation, and non-relativistic matter, i.e. dust matter. First, it is important to point out that the matter contents of our universe are believed to have begun as a hot fluid in an equilibrium state, see book [12] chapter 5. This quickly reached equilibrium state allow us to describe the fluids composing the universe as gases within equilibrium thermodynamics. Remember that, from statistical mechanics, we get the Fermi-Dirac and Bose-Einstein distributions calculating the average number of particles with energy E for gases of fermions and bosons, respectively. f(E) = 1 eβ(E−µ) ± 1 ; (1.32) where β = 1/T , T is the temperature of the gas and µ is its chemical potential. The plus sign stands for Fermi-Dirac while the minus sign indicates a Bose-Einstein distribution. For T � E − µ, we get the Maxwell-Boltzmann distribution, fMB: f(E) = 1 eβ(E−µ)[1± e−β(E−µ)] = e−β(E−µ)[1∓ e−β(E−µ) +O ( (e−β(E−µ))2 ) ] = e−β(E−µ) +O ( (e−β(E−µ))2 ) ⇒ fMB(E) = e−β(E−µ) (1.33) For a general distribution f(E), i.e. either Fermi-Dirac, Bose-Einstein or Maxwell- 18 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY Boltzmann, integral expressions for the number density (n), energy density (ρ) and pressure (P ) of the gas can be written. Here the expressions are given below. For their derivation see appendix C. n = g 2π2 ∫ ∞ m dE √ E2 −m2Ef(E) (1.34) ρ = g 2π2 ∫ ∞ m dE √ E2 −m2E2f(E) (1.35) P = g 6π2 ∫ ∞ m dE(E2 −m2)3/2f(E) (1.36) We are ready to use equations (1.32) and (1.33) to calculate w for the two relevant cases. (i). Radiation (wr = 1/3). First, let us present a heuristic derivation which will add intuition to the obtained result. Imagine a box full of relativistic particles, e.g. photons. If we cannot add nor remove energy from this box, with the expansion of the universe, its energy density will drop because ρr = E/V and the volume is increasing as V = Vcoma 3, where the subscript stands for comoving. However, while that happens, the wavelengths of the photons also expand and since E ∝ λ−1 = λ−1 coma −1. Adding these two contributions we have ρr ∝ a−4. If we compare this result with equation (1.25) it is easy to figure out that −3(1 + wr) = −4 ∴ wr = 1/3. Now, back to our thermodynamic calculations, if we take the ultrarelativistic limit, E � m, of equations (1.35) and (1.36) it is easy to see that: P = ρ 3 = g 6π2 ∫ ∞ m dEE3f(E)⇒ wr = 1 3 ; (1.37) which confirms our heuristic derivation. 1.3. MATTER CONTENT AND THE FRIEDMANN EQUATIONS 19 (ii). Non-relativistic/dust matter (wm = 0). Again, first we give a heuristic argument to the derivation. The only difference in our thought experiment in comparison to the previous relativistic case is that the particles in the box need not to be thought of as photons, which had an associated wavelength, therefore the only decreasing component to the energy density comes from the expanding volume of the box. Thus, ρm ∝ a−3 which after comparing against equation (1.25) gives −3(1 + wm) = −3 ∴ wm = 0. Back to the thermodynamic analysis, we will have to compute n, ρ and P this time. The non-relativistic limit to be considered is T � µ and also m2 � p2 such that: E2 = m2 + p2 ∴ E = m √ 1 + p2 m2 = m [ 1 + 1 2 p2 m2 +O ( p4 m4 )] ' m+ p2 2m (1.38) Note that in the non-relativistic limit we are dealing with a Maxwell-Boltzmann gas. From equation (C.2), we have: n = g (2π)3 ∫ ∞ 0 dp4πp2e−β(E(p)−µ) ' g 2π2 ∫ ∞ 0 dpp2e−β(m−µ)e− βp2 2m ; (1.39) where in the second equality we used the approximation of equation (1.38). Recall the definition of the gamma function: Γ(z) = ∫ ∞ 0 xz−1e−xdx ; for Re(z) > 0 (1.40) Then we can rewrite the integral of equation (1.39) as: n ' g 2π2 e− m−µ T (2mT )3/2 ∫ ∞ 0 dxe−x 2 x2 = g 4π2 e− m−µ T (2mT )3/2 ∫ ∞ 0 dxe−xx1/2 ' g 4π2 e− m−µ T (2mT )3/2Γ(3/2) = g ( mT 2π )3/2 e− m−µ T (1.41) where we used the fact Γ(3/2) = √ π/2. 20 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY For the energy density, from equation (C.4), we can see that if we consider only the 0th order in the expansion for the energy, namely E(p) ' m, we have: ρ = g (2π)3 ∫ d3pE(p)f(E(p)) ' m [ g (2π)3 ∫ d3pf(E(p)) ] = mn (1.42) Finally, if we keep only E ' m in the non-exponential piece of the integral expression for the pressure of the gas, equation (C.6), becomes: P = g (2π)3 ∫ d3p p2 3E f(E(p)) ' g 2π2 1 3m ∫ ∞ 0 dpp4e− m−µ T e− p2 2mT ' g 2π2 1 3m e− m−µ T (2mT )5/2 ∫ ∞ 0 dxe−x 2 x4 = g 4π2 1 3m e− m−µ T (2mT )5/2 ∫ ∞ 0 dxe−xx3/2 ' g 4π2 1 3m e− m−µ T (2mT )5/2Γ(5/2) = [ g ( mT 2π )3/2 e− m−µ T ] T ' nT (1.43) We conclude that, from equations (1.42) and (1.43), P ' nT � nm ' ρ thus P/ρ→ 0 and we indeed get wm = 0 as anticipated. Now that wr and wm are known, note that, since ρr ∝ a−4 and ρm ∝ a−3, the total energy density, i.e. sum of all components, tends to be dominated by radiation in the early history of the universe but then, after a while, dust matter energy density overcomes the former radiation dominance. We will soon discuss dark energy and add the last relevant component in order to tell the history of the universe. 1.3.1 Dark Matter If we want to observe the amount of energy density for each type of matter in the universe today ρ (0) i , or their ratios to the critical density Ω (0) i , we would realize there is a discrepancy in the amount of non-relativistic matter which is measured to exist via gravitational observations and which can be seen, i.e. which interacts electromagnetically and therefore we can see their 1.3. MATTER CONTENT AND THE FRIEDMANN EQUATIONS 21 brightness through telescopes. There is much more matter than what we can see according to gravitational observations. The conclusion is that there is an exotic type of matter, dark matter, in the universe which adds to the usual baryonic matter, i.e. any matter composed of Standard Model particles and which interacts with light, increasing the expected gravitational pull in distant galaxies. The name “dark matter” was given since we cannot directly observe them, i.e. they don’t shine, thus looking to the sky we would see just darkness instead of them. Yet, from their gravitational effects, they are known to be there. Actually, ignoring gravitational lensing effects, we would see through them, since they don’t interact with light, and therefore perhaps “transparent matter” would be a better name. Nevertheless the name got stuck and we will keep calling it dark matter. Here, I will quickly discuss a few observations that led to conclusion which was pointed out last paragraph in order to properly define the ΛCDM model later. The main goal of this thesis is not to study dark matter, even though its composition is a relevant open question, therefore to find a more comprehensive list of observations see chapter 4 of book [12]. The first evidence for dark matter was due to Zwicky and dates back to 1933 [23]. Zwicky was analyzing the Coma Cluster and realized that the visible mass was much less than it was expected to be. Since in this cluster galaxies can already be approximated as points, he mod- eled the cluster as a gas with gravitational interaction among components. He noticed, using the virial theorem, that the visible mass was way less than the expected to counterbalance the kinetic term. Something seemed off, nowadays we know that is because mdark � mvisible and he was not taking dark matter into consideration since he could not see it. Another evidence for dark matter is the unusual feature of light being distorted in the sky by no visible object. This phenomenon is explained by the presence of an invisible matter in the path between Earth and the light ray coming to us. In GR massive objects bend the path of light and that is why we observe such a distortion of light for apparently no good reason. The last observation we are going to discuss here which led to the conclusion that there 22 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY is such an exotic type of matter in our universe is the CMBR data. The Planck satellite collected data of the radiation coming to us from the primordial times of the universe. The latest published paper with its analyzed data is [18]. We will come back to it later but for now it is enough to say they are able to figure out both Ω (0) dm and Ω (0) b – in our notation dm stands for dark matter and b for baryonic matter – from the fitting of the six parameters of the ΛCDM model to observations. The best fitting generates the values: Ω (0) dm ' 0.26 ; and Ω (0) b ' 0.05 ; (1.44) which agrees with the previous observation by Zwicky that there is much more dark matter than visible matter in the universe. 1.3.2 Cosmological Constant As anticipated, there is yet another relevant type of matter due to the existence of a non-zero cosmological constant in equation (1.6) which dictates the behaviour of the scale factor with time through the Friedmann equations. The cosmological constant term in the Einstein equations is being interpreted here as another type of energy content and, as we have already discussed, a stress-energy tensor of the form T (Λ) µν = − Λ 8πG gµν is associated to it. It is easy to figure out the equation of state parameter for the cosmological constant as: T (Λ)µ ν = − Λ 8πG δµν = diag(−ρΛ, PΛ, PΛ, PΛ)⇒ PΛ = −ρΛ = − Λ 8πG ⇒ wΛ = −1 (1.45) Therefore our analysis of equations (1.24) through (1.28) is not valid for the cosmological constant parameter. Some weird features of the cosmological constant will be pointed out now. First, from the continuity equation (1.23), we get ρ̇Λ = 0 which means the energy density is constant with time, even with the expansion of the universe. This fact seems strange when we try to think of a box containing this type of matter and expanding according to the 1.3. MATTER CONTENT AND THE FRIEDMANN EQUATIONS 23 Hubble flow, as we did for radiation and matter. This exotic type of energy is not diluted as the volume expands, its energy has to grow ∝ a3 to compensate the increasing factor in the volume due to the Hubble flow. However, as we will discuss in detail in chapter 2, a w = −1 equation of state parameter can be associated with the vacuum energy of quantum fields, thus since when the box is expanded by the Hubble flow there is automatically an increasing volume of vacuum – the “amount of vacuum” increases – inside the box justifying the weird fact of ρ̇Λ = 0. The association of the cosmological constant with a vacuum energy is problematic and is further discussed in chapter 2. This problem is so central to this thesis that it deserves a separate chapter. From equation (1.21), again dropping the curvature term for the reasons already men- tioned, we realize that H is constant since ρΛ in an universe dominated by the cosmological constant energy density – like ours as we will shortly see. This implies an exponential be- haviour for a = a(t). ∫ a(t) 1 da′ a′ = ∫ t t0 Hdt′ ∴ a(t) = eH(t−t0) (1.46) Finally, we must note that, from the Planck data [18], Ω (0) Λ ' 0.69 for the cosmological constant. Therefore our universe is indeed dominated by this type of energy today. This leads to the conclusion Λ > 0 which in turns means we live in a de Sitter space, named after Willem de Sitter. For our purposes, a de Sitter or dS (anti-de Sitter or AdS) space is a space whose size grows (shrinks) exponentially with time, as it is our case from equation (1.46). More generally, a dS (AdS) space is a maximally symmetric Lorentzian manifold with constant positive (negative) Ricci curvature. The relation between the curvature and the cosmological constant is easily obtained by taking the trace of the Einstein equations (1.6) without matter, i.e. pure gravity: gµνRµν − 1 2 gµνgµνR + gµνgµνΛ = 0 ∴ R = 4Λ (1.47) 24 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY 1.4 The Inflationary Paradigm Up until now, we have constructed a model which is successful in describing the early history of our universe and can be used also to predict the its future. Nonetheless, even the pre-inflationary ΛCDM model, also called Hot Big Bang cosmology, has its problems when we try to extend its predictive power close to the cosmological singularities. Three problems will be presented here to indicate where exactly this model fails. See book [12] chapter 8 for a more comprehensive list. 1. The flatness problem We have mentioned that the curvature energy density is negligible today compared with other types of energy density, ρ (0) k � ∑ i ρ (0) i . This is not a problem per se, however within our framework we are able to realize that something strange happens at the beginning of the universe. First note that we can rewrite the first Friedmann equation (1.21) as: H2 = 8πG 3 ρ− k a2 ∴ 1 = ρ ρc − k a2H2 ∴ Ω− 1 = k a2H2 (1.48) where ρ ≡ ∑ i ρi and similarly Ω ≡ ∑ i Ωi = ∑ i ρi/ρc with i running through all types of energy excluding the curvature. In the first ∴ we have just divided the equation by H2 and used the definition of critical density, ρc ≡ 3H2 8πG . Notice from equation (1.48) that Ω(t)→ 1 means the universe is nearly flat at time t. From equation (1.26) and (1.27), valid for w 6= 1, it can be concluded that Ω − 1 ∝ t 2 3 1+3wdom 1+wdom . Therefore Ω−1 increases with time (decreases going back in time) in periods of domination by a type of energy whose equation of state parameter is either wdom < −1 or wdom > −1/3. Although we are living in a cosmological-constant-dominated period, this dominance started just recently – compared with the age of the universe – and most of our history was dominated by either matter or radiation, both with w > −1/3. 1.4. THE INFLATIONARY PARADIGM 25 Therefore the weird fact is that not only |Ω(0)−1| � 1 today but the flatness tended to be even more severe in the beginning of the universe. We must seek for an explanation of this fine-tuning, |Ω(t)− 1|≪ 1 for t < t0, in the early universe. Inflation [24, 25, 26, 27, 28] solves the problem assuming a period of rapid accelerated expansion in the beginning of the universe. This paradigm started as what we call today “old inflation”, but we will be concerned only with the “new inflation” here. In order to get an accelerated expansion ä > 0, from equation (1.22), we need wdom < −1/3. It is usually considered an exponential growth of space, obtained for wdom = −1, i.e. a cosmological-constant-dominated period. Despite being favoured by experiments [18], this is not a requirement though. One usually considers, for simplicity, a scalar field model obeying slow-roll conditions, which will be described in the next section. The important facts needed in this discussion to solve the problems are the, at least nearly, exponential growth of the scale factor and the definition of the number of e-folds Ne, which quantifies the amount of inflation that happens, as: Ne ≡ log [ a(te) a(tb) ] = HI(te − tb) ≡ HI∆t ; (1.49) where in the second equality we have used the fact a(t) ∝ eHI t ∴ a(te) = a(tb)e HI(te−tb) with te being the moment in which inflation ends and tb its beginning. Recall that we are considering w ' −1 thus HI is constant. Back to the flatness problem, we saw in equation (1.48) that Ω(t)− 1 decreases in the past for wdom > −1/3, but during inflation Ω(t) − 1 ∝ a−2 ∝ e−2HI t which increases going back in time. Therefore, we get rid of the fine-tuning since Ω − 1 could have assumed any value at the beginning of the universe and inflation is the mechanism which forces our universe to be almost spatially flat at the beginning of the radiation- dominated era. It is as if our universe was the surface of a deflated balloon and inflation was a person who blew it up stretching its surface and flattening it out. 26 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY Quantitatively, if we assume Ω(tb)− 1 ∼ O(1), then: Ω(0) − 1 = k a2 0H 2 0 = k H2 0 = k a2 eH 2 I a2 eH 2 I H2 0 = k a2 bH 2 I e−2Ne a2 eH 2 I H2 0 = [Ω(tb)− 1]e−2Ne a2 eH 2 I H2 0 ∼ e−2Ne a2 eH 2 I H2 0 (1.50) Since Ω(0) − 1� 1 we have the constraint: eNe > aeHI H0 (1.51) Based on some expectations of the energy scale in which inflation should have occurred, the usual number of e-folds needed is ∼ 60, see book [12] chapter 8.3 for details. 2. The horizon problem The CMBR observed today is basically photons coming to us from a special moment of the universe’s evolution which goes by the name of last scattering surface, more details will be given shortly on CMBR’s own subsection. The important fact for our discussion now is that the photons come to us from this specific and early time tls, where the sub- script stands for last scattering. This radiation is isotropic to a high degree of precision, as anticipated, therefore if we look for opposite directions in the sky we would observe approximately the same distribution of energies for the photons. The problem is, within pre-inflationary cosmology, those regions in the skies are causally disconnected. Why would we observe almost the same things from causally disconnected regions? It would be only expected from regions that were once in causal contact and had the time to thermalize. To be more concrete, we must calculate the distance dls traveled by light from the beginning of the universe, at t = 0, to the surface of last scattering, tls, and the 1.4. THE INFLATIONARY PARADIGM 27 distance d0 traveled by it from tls to us today, at t = t0, in order to compare the two of them. From ds2 = −dt2 + a2dl2 plus the fact that light travels on lightlike trajectories, ds2 = 0, the desired (comoving) distances are calculated as: dls = ∫ tls 0 dt a(t) ; and d0 = ∫ t0 tls dt a(t) (1.52) Some additional information are in order here. Although we skipped a description of the early history of the universe, it is known that before and after the last scattering surface the universe was dominated by matter for most of the time, therefore we will use a(t) ∝ t2/3 ∴ a(t) = a0(t/t0)2/3 = (t/t0)2/3 in both integrals. dls = ∫ tls 0 dt ( t t0 )−2/3 = 3t 2/3 0 t 1/3 ls ; d0 = ∫ t0 tls dt ( t t0 )−2/3 = 3t 2/3 0 (t 1/3 0 − t1/3ls ) (1.53) If we want to explain how two photons from opposite regions of the sky are strongly correlated, one of these photons must have passed through the point in this comoving space where the second photon was sent to us at the last scattering surface at some point before tls. In other words, one of the photons should have travelled at least the comoving diameter of nowadays observable universe from t = 0 to t = tls before traveling the radius of the universe until it gets to us now, dls ≥ 2d0, which means: N ≡ 2d0 dls ≤ 1 ; for causal connection (1.54) However, its known that the surface of last scattering happened at redshift zls ' 1300. Thus: N = 2 t 1/3 0 − t1/3ls t 1/3 ls ' 2 ( t0 tls )1/3 = 2a −1/2 ls = 2 √ 1 + zls ' 72 ; (1.55) 28 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY where in first equality the results of equation (1.53) were used, in the second the fact t0 � tls was used, in the third we used a(t) = a0(t/t0)2/3 and, finally, the last equality comes from equation (1.16) plus the fact that wavelengths are stretched by the Hubble flow as any physical length, λ ∝ a−1. z ≡ λ− λ0 λ0 = 1 a − 1 ∴ 1 a = 1 + z (1.56) Thus, equation (1.55) shows us we need a mechanism to increase dls (or decrease d0) by, at least, a factor of ∼ 70 if we aim to explain the similarity between two distant regions in the sky. Inflation solves this problem since the two opposite points in our universe today would have been once close together before inflation begins. After the last scattering surface, inflation is supposed to have ended thus the calculation of d0 remains unchanged, we will just rewrite it. However, inflation plays a significant role in the determination of dls so we must recalculate it taking inflation into consideration now. From equation (1.52), considering the integral is dominated by the inflationary period that happens before tls: dls ' ∫ te tb dt eHI(te−t) ae = − e HI te aeHI (e−HI te − e−HI tb) = 1 aeHI (eNe − 1) ' 1 aeHI eNe (1.57) Recalling that we are considering a matter-dominated era before and after the last scattering surface, since it is the case for most of the time, then H0 = 2/3t0. Therefore, d0 is rewritten as: d0 = 3t 2/3 0 (t 1/3 0 − t1/3ls ) = 3t0 [ 1− ( tls t0 )1/3 ] ' 2 H0 (1.58) 1.4. THE INFLATIONARY PARADIGM 29 From equation (1.54), in order to the regions to be causally connected in the past it is needed that: N = 4 H0 aeHIe −Ne ≤ 1 ∴ eNe > aeHI H0 ; (1.59) where we have just plugged the results of equations (1.57) and (1.58) into the definition of N . The obtained constraint on Ne is the same as the derived one for the flatness problem. 3. The monopole/relic problem As the universe cooled down from its initial hot state it is believed that it could have passed through some phase transitions, for instance the period when a grand unified theory (GUT) symmetry is broken, when topological defects are expected to be created. Different kinds of relics or topological defects are expected to be created, however magnetic monopoles will be our focus here and at the end we will argue that this is the most relevant case indeed. By the Kibble mechanism [29, 30] we would expect one monopole per horizon as well as one nucleon per horizon created during a phase transition. Without inflation we would expect this 1 : 1 proportion to be maintained up until today. However, it is observed less than 10−30 monopoles per nucleon on Earth materials, see [31] chapter 4.1, so we must have some mechanism to dilute the concentration of monopoles away but don’t do the same to nucleons. Inflation does exactly this job. Actually the rapid expansion of inflation dilutes away both concentrations, nevertheless inflation is followed by a transition period named reheating before the standard Hot Big Bang scenario and during reheating the scalar field responsible for inflation, namely the inflaton, decay into mostly radiation which will later create more nucleons. All in all, the density of nucleons is expected to remain the same as if there wasn’t inflation, thus the overall effect of inflation is to dilute away only the monopoles. Therefore a 10−30 reduction 30 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY on the ratio of monopoles to nucleons implies an 1030 expansion of the volume of the universe which, in turns, implies a growth of 1010 in the scale factor. That is the least amount of inflation we must have to avoid incompatibility with experiments. Ne ≥ log ( 1010 ) ' 23 (1.60) It should be pointed out that, in order for the inflation mechanism to solve the problem, the phase transition must have happened before it, or at least in the middle of it. If it happens in the middle, Ne & 23 should be the least amount of inflation which happened after the phase transition. Other relics give us less stringent bounds on the number of e-folds and are therefore redundant. The minimum amount of dilution needed for other relics is a 10−11 reduction in their number density, see [17] chapter 7.5. It should be pointed out, though, that within holographic cosmology, an alternative scenario to inflation, the origin of the monopole problem is different from other types of relics and they will be discussed separately in chapter 5. It is clear that the above problems are basically solved by only one feature of inflation, the rapid and long expansion of space before the start of the Hot Big Bang period. A reheating consideration was needed to solve the monopole problem though. Other problems, such as why we have large entropy nowadays or how perturbations which formed galaxies today were generated, are solved in a similar fashion, see [12]. 1.4.1 Slow-Roll Inflation As anticipated, the current most accepted paradigm for inflation is the “new inflation”. As it is usual practice, we will be concerned with scalar field models for inflation with an almost exponential growth of the scale factor, i.e. an approximate cosmological constant. The scalar field goes by the name of inflaton, φ, and it obeys some conditions, known as slow-roll 1.4. THE INFLATIONARY PARADIGM 31 conditions, in order to give us the features want for inflation. To find the slow-roll conditions we must define the two slow-roll parameters first. Notice that the horizon problem was solved because the rapid expansion of the universe made physical scales, which were once in causal contact, go outside the horizon where they got frozen, i.e. there wasn’t any new quantum fluctuation to destroy the correlation, and in the radiation and matter-dominated eras, which came after inflation, these scales came back inside it. The horizon, which is understood as the region in causal contact, is quantified here by the Hubble radius rH ≡ H−1. This is the simplest type of horizon, but not the only, and it is approximately constant during inflation, H−1 I . Therefore, we need the comoving Hubble radius, defined as the physical one divided by the scale factor as usual, to shrink during inflation. d dt (aH)−1 = − ȧH + aḢ (aH)2 = 1 a ( − Ḣ H2 − 1 ) < 0⇒ εH ≡ − Ḣ H2 � 1 (1.61) The first slow-roll parameter is εH and its role, as we saw, is to ensure the shrinking of the comoving Hubble radius, among other things. Since we are considering approximate cosmological constant models Ḣ → 0 ∴ εH � 1 not only εH < 1. The shrinking of the above radius also implies an accelerated expansion: d dt (aH)−1 = d dt ( 1 ȧ ) = − ä ȧ2 < 0 ∴ ä > 0 (1.62) The second requirement, which defines the second slow-roll parameter ηH , is that we need εH to be small for a sufficiently large amount of time or, equivalently, for a large number of e-folds Ne. Therefore we define: ηH ≡ d log εH dNe = 1 εHH dεH dt = ε̇H HεH ; (1.63) where it was used the fact Ne ≡ log(ae/ai) = log eHt ∴ dNe = Hdt, since H ' HI = constant, in the second equality. The second slow-roll condition is |ηH | � 1. Now, it is time to associate εH and ηH , whose definitions are valid in general, with 32 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY characteristics of the scalar field in the scalar field model for inflation. Some comments on the inflaton model are in order. Initially, the potential V (φ) was thought to have to be composed of a plateau, nearly flat and large, followed by a drop. Nowadays, it is known that many types of potential can be used in inflationary models, as long as the slow-roll conditions are satisfied. From the action: Sφ = ∫ d4x √ −g [ −1 2 gµν∂µφ∂νφ− V (φ) ] (1.64) The stress-energy tensor is found by varying the action with respect to the inverse metric: δSφ = ∫ d4x { δ( √ −g) [ −1 2 gµν∂µφ∂νφ− V (φ) ] − √ −gδgµν 1 2 ∂µφ∂νφ } (1.65) Recall from equation (A.5) that δ( √ −g) = −(1/2) √ −ggµνδgµν . Thus: δSφ = ∫ d4x √ −gδgµν [ −1 2 ∂µφ∂νφ+ 1 4 gµνg ρσ∂ρφ∂σφ+ 1 2 gµνV (φ) ] ⇒ Tµν ≡ − 2√ −g δSφ δgµν = ∂µφ∂νφ− 1 2 gµνg ρσ∂ρφ∂σφ− gµνV (φ) (1.66) The background inflaton is expected to preserve homogeneity and isotropy we observe nowadays in the universe, therefore φ = φ(t), leading to: T µν = gµρ∂ρφ∂νφ− δµν [ −1 2 φ̇2 + V (φ) ] ⇒  T 0 0 = −ρφ = − φ̇ 2 − V (φ) ; T ij = δijPφ = δij [ φ̇2 2 − V (φ) ] . (1.67) where φ̇ ≡ dφ/dt as usual. From equation (1.67) it is easy to see that, in general: wφ ≡ Pφ ρφ = φ̇2/2− V (φ) φ̇2/2 + V (φ) (1.68) 1.4. THE INFLATIONARY PARADIGM 33 The first Friedmann equation (1.21) for a universe composed solely by the inflaton be- comes: H2 = 8πG 3 ρφ − k a2 = 1 3M2 Pl [ φ̇2 2 + V (φ) ] − k a2 ; (1.69) where we will ignore the term k/a2 from now on, since it goes to zero quickly because of the exponential behaviour of the scale factor. Recall that the Friedmann equations are the equations of motion for the metric, i.e. obtained by varying the action with respect to the metric. The equation of motion for the inflaton is the Klein-Gordon equation, in FLRW spacetime though. δ(SE−H + Sφ) = δSφ = ∫ d4x √ −g[−gµν∂µ(δφ)∂νφ− V ′(φ)δφ] = ∫ d4x[∂µ( √ −ggµν∂νφ)δφ− √ −gV ′(φ)δφ] = ∫ d4x √ −g[∇µ(gµν∂νφ)δφ− V ′(φ)δφ] = ∫ d4x √ −gδφ[gµν∇µ(∂νφ)− V ′(φ)] ; (1.70) where in the third equality we just partially integrated the first term, considering a bound- ary condition that vanishes the boundary term. In the fourth equality we used the identity ∂µ( √ −gXµ) = √ −g∇µX µ valid for any tensor field Xµ, as proved in equation (A.11). Fi- nally, in the last equality, ∇µg µν = 0 was used. Thus, the equation of motion is: δ δφ (SE−H + Sφ) = 0 ∴ �φ− V ′(φ) = gµν(∂µ∂νφ− Γρµν∂ρφ)− V ′(φ) = 0 −φ̈− giiΓ0 iiφ̇− V ′(φ) = 0 φ̈+ [ 1− kr2 a2 ( aȧ 1− kr2 ) + 1 a2r2 (aȧr2) + 1 a2r2 sin2 θ (aȧr2 sin2 θ) ] φ̇+ V ′(φ) = 0 φ̈+ 3Hφ̇+ V ′(φ) = 0 ; (1.71) where Γs and the inverse metric were taken from appendix B. 34 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY Our goal now is to relate εH and ηH to the potential V (φ) seeking to understand how the slow-roll conditions constrain the potential. Taking the time derivative of equation (1.69) and applying the result of equation (1.71) we get: 2HḢ = 1 3M2 Pl [φ̇φ̈+ V ′(φ)φ̇] = φ̇ 3M2 Pl (−3Hφ̇) ∴ Ḣ = − φ̇2 2M2 Pl (1.72) Now, from the first slow-roll condition, it is clear why the name “slow-roll” inflation from the following: εH � 1 ∴ |Ḣ| � H2 ∴ φ̇2 2M2 Pl � φ̇2 6M2 Pl + V (φ) 3M2 Pl ∴ φ̇2 � V (φ) (1.73) The potential term dominates over the kinetic one, which means a slow roll – compared to the potential – due to the Hubble friction term in equation (1.71)! From equation (1.73), note that Pφ ' −ρφ ' −V (φ) ∴ wφ ' −1, an approximate cosmological constant as anticipated. Moreover, from the first Friedmann equation: H2 ' V (φ) 3M2 Pl (1.74) Now, note that, from the second slow-roll condition: ηH = ε̇H HεH = 1 H ( −H 2 Ḣ )( − Ḧ H2 + 2 Ḣ2 H3 ) = Ḧ HḢ + 2εH ' Ḧ HḢ � 1 ; (1.75) where we considered εH � 1 in the last equality. Plugging in the result of equation (1.72) into equation (1.75) we get: Ḧ = − φ̇φ̈ M2 Pl ⇒ ηH ' ( − φ̇φ̈ M2 Pl )( −2M2 Pl φ̇2 ) 1 H � 1 ∴ φ̈ φ̇ 1 H � 1 (1.76) Therefore, the above result tells us the Klein-Gordon equation may be approximated by 3Hφ̇+V ′(φ) ' 0, which leads to φ̇ ' −V ′(φ)/3H ∴ φ̇2 ' [V ′(φ)]2/9H2 ' [V ′(φ)]2M2 Pl/3V (φ). 1.4. THE INFLATIONARY PARADIGM 35 This was the relation we were looking for in order to be able to rewrite the first slow-roll parameter in terms of V (φ) as: εH ≡ − Ḣ H2 ' φ̇2 2M2 Pl 3M2 Pl V (φ) ' 3 2V (φ) [V ′(φ)]2M2 Pl 3V (φ) = M2 Pl 2 [ V ′(φ) V (φ) ]2 (1.77) We can then define a new slow-roll parameter in terms of V (φ) and its slow-roll condition, which is equivalent to εH � 1, as: ε ≡ M2 Pl 2 [ V ′(φ) V (φ) ]2 � 1 (1.78) It would be nice to also redefine ηH in terms of V (φ). From equations (1.74) and (1.77) plus the condition εH ' ε� 1 and the approximate Klein-Gordon equation already derived φ̇ ' −V ′(φ)/3H, after a bit of algebra we realize that: ηH = ε̇H HεH ' ε̇ Hε = M2 Pl ( V ′ V ) φ̇ [ V ′′ V − ( V ′ V )2 ] 1 H 2 M2 Pl ( V V ′ )2 ' 2 H ( V V ′ )( − V ′ 3H )[ V ′′ V − ( V ′ V )2 ] ' −2V 3 ( 3M2 Pl V )[ V ′′ V − ( V ′ V )2 ] ' −2M2 Pl V ′′ V + 4ε ' −2M2 Pl V ′′(φ) V (φ) (1.79) Therefore, from equation (1.79) it is clear we can define η in terms of the potential such that, from |ηH | � 1, we get: η ≡M2 Pl V ′′(φ) V (φ) ; with slow-roll condition: |η| � 1 (1.80) The importance of redefining εH and ηH as ε and η depending of the potential lies in the fact that, from a V (φ) of any proposed model, we can check directly if this is an appropriate model to describe slow-roll inflation and in which regions of φ the potential obey the slow-roll conditions. 36 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY At last, it is important to connect the previously analyzed cosmological epochs with the inflationary one. The transition period from inflation to the pre-inflationary scenario is called reheating. As was said, at first V (φ) was thought to necessarily be composed of a plateau followed by a drop. After falling off the cliff, the potential would oscillate around a minimum where the inflaton would decay into the modes of fields we observe nowadays, starting the standard pre-inflationary eras. As mentioned, this is crucial to solve the relic problem. 1.4.2 CMBR Data: Connection to Observations So far we have been considering a perfectly homogeneous and isotropic universe. Although this is a good approximation, some inhomogeneities of order 10−5 of its average value are actually measured in the temperature of the CMBR. As it can be seen, this is consistent with our first approximation of an homogeneous and isotropic universe since fluctuations are orders of magnitude smaller than the average. However, in order to form the large scale structures we observe on telescopes, those fluctuations are essential. In the usual terminology, we have been studying the background dynamics of the universe so far. Now, perturbations to the background will be considered. Perturbations both in the metric, affecting the left-hand side of the Einstein equations (1.6), and in the energy momentum tensor, right-hand side of the equations, must be con- sidered. At the end, a connection between observations of the CMBR spectrum and these perturbations has to be constructed. If it was intended to self-consistently connect, at any level of detail, those two ends, at least another separate chapter would be needed. Here, we will just point out the main ideas behind cosmological perturbation theory and describe which parameters CMBR experiments, more precisely the Planck satellite data [18], con- strains. These ideas will be important to compare inflation against other models to solve pre-inflationary problems discussed in the last chapters of this thesis. A more detailed de- scription can be found on books [12] (chapters 7 and 11) and [32], besides the lecture notes [33]. 1.4. THE INFLATIONARY PARADIGM 37 We should start by giving an intuitive picture of how fluctuations during inflation generate the large scale structures we observe today. First consider a perturbation δφ(t, ~x) to the inflaton background field φ̄(t), which was called just φ(t) until last subsection avoiding a notation overkill. Note that the bar is now important to distinguish the background field from the total field φ(t, ~x) = φ̄(t) + δφ(t, ~x). If inflation ends at φ = φe then, regions where δφ > 0 will reach the end of inflation before regions where δφ < 0. This will eventually lead to a perturbation δρ(t, ~x) in the background energy density of these regions, ρ̄. This simple example illustrate why we need to consider perturbations both in the metric and in the energy-momentum tensor. An useful procedure to make calculations easier is the decomposition of the metric and the stress-energy perturbations into scalar, vector and tensor components, the so-called SVT decomposition. This decomposition is usually done in momentum space via Fourier transform Qk(t,~k) = ∫ d3xQ(t, ~x)ei ~k·~x, where Q stands for any metric or stress-energy quantity, e.g. δφ. To first order, each type of perturbation evolves independently in a SVT decomposition, thus they can be treated individually. That is the usefulness of this procedure. Let us start by considering perturbations on the line element, or equivalently on the metric, as: ds2 = a2(η){−(1 + 2A)dη2 − 2Bidx idη + (δij + hij)dx idxj} ; (1.81) where a is the scale factor as usual and A, Bi and hij are the perturbations to be SVT decomposed, in the physical space, below. Note that the conformal time η is being used instead of t. Bi = ∂iB − B̂i ; where: ∂iB̂i = 0 (1.82) hij = 2Cδij + 2∂〈i∂j〉E + 2∂(iÊj) + 2Êij ; (1.83) where:  ∂〈i∂j〉E ≡ ( ∂i∂j − 1 3 δij∇2 ) E ; ∂(iÊj) ≡ 1 2 (∂iÊj + ∂jÊi) ; ∂iÊi = 0 ; ∂iÊij = 0 and Êi i = 0 38 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY After the SVT decomposition, we still have 10 degrees of freedom, as expected for a 4- dimensional metric. There are four scalars (A, B, C and E) two divergenceless vectors (B̂i and Êi) and one traceless and divergenceless 3-dimensional symmetric tensor (Êij), summing up to 4 + 4 + 2 = 10 degrees of freedom. It should be pointed out that vector perturbations quickly decay with the Hubble expansion, therefore the relevant quantities to us will be only the scalars and the tensor Êij. Here comes the subtle issue that the perturbations defined above are not gauge invariant. It can be shown that a simple time coordinate transformation in the FLRW metric can lead to unphysical perturbations. Therefore we will need to search for gauge-independent quantities. Choosing the so-called Newton (or Newtonian) gauge, the line element of equation (1.81) becomes: ds2 = a2(η)[−(1 + 2Ψ)dη2 + (1− 2Φ)δijdx idxj] ; (1.84) where Ψ and Φ are two of the Bardeen variables, which are invariant under general coordinate transformations. We may then define a gauge-independent variable from a combination of metric, Φ, and energy-momentum tensor, δρ, perturbations: ζ ≡ −Φ− H ˙̄ρ δρ (1.85) Again, we refer to the notes [33] appendix A to a motivation to this definition. The power spectrum of scalar perturbations, which is going to be constrained by CMBR data, is defined from the quantity defined in equation (1.85). The goal is to link Gaussian perturbations of the CMBR to the same perturbations of the gauge-invariant scalar ζ. The variance of any quantity q = q(~x) can be defined as: σ2 q (~x) ≡ 〈q2(~x)〉 − 〈q(~x)〉2 ⇒ σ2 q (~x) = 〈q2(~x)〉 ; for 〈q(~x)〉 = 0 ; (1.86) where 〈〉 means average. In the case of the CMBR quantity to be analyzed shortly Θ ≡ δT/T 1.4. THE INFLATIONARY PARADIGM 39 it stands for an ensemble average, i.e. average over the sky. Note that only perturbations, i.e. quantities which by definition average to zero, will be concerning us in the following analysis, therefore σ2 q = 〈q2〉 will be considered. The probability density function at q ≡ q(~x0), often introduced as the probability of measuring q(~x) between q and q + dq, is generically ∝ e − q2 2σ2q , i.e. it follows a Gaussian distribution. Therefore, Gaussian perturbations are obtained by two-point functions 〈q2(~x)〉. The two-point correlation function of scalar perturbations in real, i.e. physical, space is: ξ(r) ≡ 〈ζ(~x)ζ(~x+ ~r)〉 ; (1.87) where we make the assumption ξ only depends on r ≡ |~r|. The two-point function in momentum space is found by taking the Fourier transform on both ~x and ~x+ ~r. 〈ζkζk′〉 = ∫ d3x ∫ d3rξ(r)ei ~k′·~xei ~k·(~x+~r) = ∫ d3xei( ~k+~k′)·~x ∫ d3rξ(r)ei ~k·~r (1.88) Finally, we define the power spectrum Pζ(k) as the Fourier transform of the two-point function ξ(r): Pζ(k) ≡ ∫ d3rξ(r)ei ~k·~r ⇒ 〈ζkζk′〉 = ∫ d3xei( ~k+~k′)·~xPζ(k) = (2π)3δ3(~k + ~k′)Pζ(k) ; (1.89) where we assumed Pζ only depends on k ≡ |~k| and after the implication arrow we plugged the definition into equation (1.88). Moreover, its usual to define a dimensionless power spectrum of scalar perturbations, ∆2 s(k), as: σ2 ζ (~x) = 〈ζ2(~x)〉 = ξ(0) = 1 (2π)3 ∫ d3kPζ(k) = 1 (2π)3 ∫ dk4πk2Pζ(k) ≡ ∫ d log k∆2 s(k) = ∫ dk k ∆2 s(k) ; (1.90) such that ∆2 s(k) ≡ k3 2π2Pζ(k). 40 CHAPTER 1. THE STANDARD MODEL OF COSMOLOGY The parameter which quantifies the scale-dependence of ∆2 s is called scalar spectral index, or equivalently scalar tilt, ns. Furthermore, there is another parameter which tells us how the tilt varies with k and is called the running of the spectral index, αs. They are defined as: ns − 1 ≡ d log ∆2 s d log k ; and αs ≡ dns d log k (1.91) Solving the differential equations of equation (1.91) assuming a constant αs gives us: ∆2 s(k) = ∆2 s(k∗) ( k k∗ )ns(k∗)−1+(1/2)αs log(k/k∗) ; (1.92) where k∗ is a pivot scale, i.e. an arbitrary scale of reference. The connection between CMBR spectrum and the scalar perturbation power spectrum is done by the transfer function Tl(k). The anisotropy Θ ≡ δT/T is what is measured by CMBR observations (actually we observe intensities of waves which are proportional to Θ). This quantity depends not only on spacetime coordinates η, the conformal time of today, and ~x, the position of the Earth or the detector, but also on the angular direction from which the photons come to us from the sky ~e. Therefore, one may perform a multipole expansion of Θ: Θ(η, ~x,~e) = ∑ lm Θlm(η, ~x)Ylm(~e) ; (1.93) in order to isolate the ~e-dependence. All in all, the precise previously mentioned connection is done by Θl(k) = Tl(k)ζk. Therefore, if we know the transfer function and the scalar perturbations power spectrum, the power spectrum of Θl can be obtained. The same argument we made to define ∆2 s can be made to analogously define a power spectrum of tensor perturbations. These perturbations were not yet observed in the CMBR, however one may expect future observations to measure them. This detection would be relevant since it would tell us the scale in which inflation happened, which nowadays can only be estimated. Tensor perturbations arise from the tensor hij in equation (1.81). They are described by two propagating degrees of freedom, i.e. the graviton polarizations, h+ and 1.4. THE INFLATIONARY PARADIGM 41 h×. The power spectra (dimensional and dimensionless) of tensor perturbations are: 〈hkhk′〉 = (2π)3δ3(~k + ~k′)Ph(k) ; and ∆2 h ≡ k3 2π2 Ph(k) ; (1.94) where h stands for any of the two polarizations. The total tensor power spectrum is defined to be the sum of the power spectrum of each degree of freedom. Assuming the power spectra of h+ and h× are equal, we end up with ∆2 t (k) ≡ ∆2 h