Instituto de F́ısica Teórica Universidade Estadual Paulista TESE DE DOUTORAMENTO IFT–T.003/19 Cosmological models from String Theory setups Heliudson de Oliveira Bernardo Orientador Prof. Horatiu Stefan Nastase Coorientador Prof. Robert Hans Brandenberger Julho de 2019 Bernardo, Heliudson de Oliveira B523c Cosmological models from string theory setups / Heliudson de Oliveira Bernardo. – São Paulo, 2019 134 f. : il. Tese (doutorado) - Universidade Estadual Paulista (Unesp), Instituto de Física Teórica (IFT), São Paulo Orientador: Horatiu Stefan Nastase Coorientador: Robert Hans Brandenberger 1. Cosmologia. 2. Modelos de corda. 3. Teoria de campos (Física). 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). To my family “I do not know what may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.” Sir Isaac Newton “Amicus Plato - amicus Aristotle - amicus Newton - amicus Einstein - magis amica veritas” Agradecimentos À toda minha famı́lia, principalmente minha querida mãe Sandra e meu irmão Adson, por sempre me apoiarem e torcerem pelo meu sucesso. Eu amo vocês. Me desculpem pela ausência. À minha parceira na vida, na f́ısica e em todo o resto, minha querida Jéssica. Obrigado pelo suporte ao longo destes anos e por sempre estar lá pra rirmos juntos de tudo, sou muito feliz ao seu lado. Aos meus amigos e amigas, que deixam o dia-dia mais leve e me fazem bem: Simão e famı́lia, que já fazem parte da minha história; Rodrigo, Nicolas, Marcella, Anderson e a galera do clã, que deixam tudo mais alegre e que trazem algumas das melhores lembranças que tenho; Márcio e famı́lia, pelas conversas e partidas de xadrez (e a Ana pela carona no dia do vestibular); aos meu amigos da UnB, principalmente a Rebeca, que me ensinou o quanto uma amizade pode crescer; Bárbara e Isabela pelas horas de conversas e ”zueiras”. Aos colegas que encontrei devido à F́ısica e que agora são também bons amigos: Priesley, Matheus, Henrique, Ana, Lúıs, Max e Carol; Daniel, pelas boas conversas sobre f́ısica, vida na academia e pelas horas de discussões sobre String Theory; Renato, por ser sempre sincero e pelos conselhos ao longo desses anos; Elisa, por me acolher tão bem na McGill e me ajudar em tudo que podia; Guilherme, por partilhar sua visão geral sobre f́ısica e filosofia. Aos meu professores, que ajudaram a moldar o que sou hoje, Jailson, Márcia, Solange e a todos do CEd 05 que sempre me apoiaram; Profs. Arsen, Olavo, Clóvis, por me fornecerem a base que precisava para fazer f́ısica; Profs. Matsas, Nathan, Rogério e Eduardo, por todas as discussões sobre f́ısica; Agradeço especialmente à Liliana, por olhar através de um simples aluno e Valéria, que já é mais que minha mestra e que acreditava em mim quando mesmo eu não o fazia: obrigado por tudo! I would like to thank so much my advisor, Horatiu Nastase, for all his patience and advises during the last 6 years; It was an honor to be your student, thank you so much for accepting me as a student, introducing me to String Cosmology and for sharing your way to do Physics. I will carry our meetings in my mind from now on. Thank you! I am grateful also to my collaborators, Horatiu Nastase, Renato Costa, Amanda Weltman, Robert Brandenberger and Guilherme Franzmann. Thank you all for the discussions and your insights. A special thank to Robert, for his patience and several conversations during my time at McGill. Agradeço, sinceramente, à CAPES pelo apoio financeiro durante o doutorado. i Resumo Nesta tese, discutimos três modelos cosmológicos que são baseados direta ou indiretamente em ideias advindas de Teoria das Cordas. Depois de uma revisão geral de Cosmologia em Teoria das Cordas, um resumo de Cosmologia e Teoria das Cordas é apresentado, com ênfase nos conceitos fundamentais e teóricos. Então descrevemos como o acoplamento camaleônico pode potencialmente afetar as predições de inflação cósmica com campo único, com trata- mento cuidadoso dos modos de perturbação cosmológica adiabáticos e de entropia. Além disso uma nova abordagem para a dualidade-T em soluções cosmológicas de supergravidade bosônica é discutida no contexto de Teoria Dupla de Campos. Por fim, propomos uma nova prescrição para o mapa holográfico em cosmologia que pode ser usado para conectar modelos fundamentais de cosmologia holográfica com outras abordagens fenomenológicas. Palavras Chaves: Cosmologia em Teoria das Cordas; Inflação Cósmica Camaleônica; Dualidade-T em Cosmologia; Cosmologia Holográfica. Áreas do conhecimento: Cosmologia Teórica; Teoria das Cordas; Teoria de Campos iii Abstract In this thesis we discuss three cosmological models that are based directly or indirectly on String Theory ideas. After a quick overview on String Cosmology a summary of both Cosmology and String Theory is presented, with emphasis on the fundamental and theoret- ical concepts. We then describe how the chameleonic coupling may potentially affect the predictions of single field cosmological inflation, with a careful treatment of adiabatic and entropy modes of cosmological perturbations. Moreover, a novel approach for T-duality of cosmological solutions of bosonic supergravity is discussed in the framework of Double Field Theory. At last, we propose a new holographic map prescription for cosmology that could be used to connect top-down setups of holographic cosmology with other phenomenological approaches. Key Words: String Cosmology; Chameleonic Cosmological Inflation; T-duality in Cosmol- ogy; Holographic Cosmology. Areas of Knowledge: Theoretical Cosmology; String Theory; Field Theory. iv Contents Preface viii 1 Introduction 1 1.1 General motivation: A crisis in fundamental Physics . . . . . . . . . . . . . . 1 1.2 Cosmology and String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Overview of String Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Inflation with Chameleon Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Cosmological Solutions in Double Field Theory . . . . . . . . . . . . . . . . . 7 1.6 Holographic Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.7 Notation and structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . 10 2 Basics of Cosmology 11 2.1 Kinematics and dynamics of an expanding Universe . . . . . . . . . . . . . . 12 2.2 Cosmological Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 The Cosmic Microwave Background Radiation . . . . . . . . . . . . . . . . . 20 2.3.1 CMB Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.2 CMB Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.3 Sound waves in the photon-baryon fluid . . . . . . . . . . . . . . . . . 22 2.4 Cosmological Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.1 The flatness and horizon problems . . . . . . . . . . . . . . . . . . . . 25 2.4.2 Inflation from scalar fields: The slow-roll inflation . . . . . . . . . . . 27 2.4.3 Primordial curvature perturbation from inflation . . . . . . . . . . . . 29 3 Basics of String Theory 35 3.1 Relativistic (or massless) quantum bosonic strings . . . . . . . . . . . . . . . 35 3.2 Relativistic quantum superstrings . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Low energy Supergravities and String Duality Web . . . . . . . . . . . . . . . 48 3.3.1 The bosonic string case . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3.2 The superstring cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.3 M-Theory, D = 11 Supergravity and String Dualities . . . . . . . . . . 52 3.4 T-duality and Double Field Theory . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.1 Toroidal Compactification of Bosonic Strings . . . . . . . . . . . . . . 53 3.4.2 Double Field Theory (DFT) . . . . . . . . . . . . . . . . . . . . . . . . 58 vi 3.4.3 T-duality in DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.5 The AdSD/CFTD−1 Correspondence . . . . . . . . . . . . . . . . . . . . . . . 63 3.5.1 AdSD Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.5.2 The N = 4 SYM as a CFT . . . . . . . . . . . . . . . . . . . . . . . . 65 3.5.3 Holographic Map and GKPW Construction . . . . . . . . . . . . . . . 66 4 Conformal Inflation with Chameleon fields 70 4.1 Motivation: String Moduli and Chameleon screening mechanism . . . . . . . 70 4.2 Conformal inflation and set-up . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.1 Conformal inflation coupled to energy density . . . . . . . . . . . . . . 71 4.2.2 Equations of motion and two cases . . . . . . . . . . . . . . . . . . . . 74 4.3 Evolution and attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3.1 The case c > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3.2 The case c < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.4 Modifications to inflationary era and CMBR observables . . . . . . . . . . . . 88 4.4.1 The case c < 0 and shortened inflation . . . . . . . . . . . . . . . . . . 88 4.4.2 The case c > 0 and modified inflation . . . . . . . . . . . . . . . . . . 89 4.4.3 Ending inflation for c > 0 . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4.4 Microscopic description . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.5 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5 T-dual Cosmological Solutions in Double Field Theory 98 5.1 Motivation: String Gas Cosmology and Stringy Symmetries . . . . . . . . . . 98 5.2 T-Dual Frames vs. T-Dual Variables . . . . . . . . . . . . . . . . . . . . . . . 99 5.3 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.4 T-duality preserving ansatz and equations for each frame . . . . . . . . . . . 102 5.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.6 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6 Holographic Cosmology from dimensional reduction of N = 4 SYM vs. AdS5 × S5 106 6.1 Motivation: Strong Gravity in the Very early universe . . . . . . . . . . . . . 106 6.2 Holographic cosmology paradigm . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.3 Top-down model from dimensional reduction of N = 4 SYM vs. AdS5 × S5 . 109 6.3.1 Transformation of dilaton and operator VEV . . . . . . . . . . . . . . 112 6.4 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7 Conclusions 116 A Holographic calculation of the scalar and tensor two-point functions 118 Bibliography 120 vii Preface The study of Cosmology raises profound questions about the origin, fate and nature of the Universe. By studying and observing the cosmological evolution we are exploring in a unique way what the laws of Physics allow or not. On the other hand, String Theory is a proposal to solve the deepest issues in Theoretical Physics and in order to make contact with what is observed, Cosmology should have a place in its framework. If String Theory really describes or not nature only experimental and observational data can tell, and in its present form it is not clear if it does or not. Regardless of that, we do not have to wait for a possible final String Theory to start exploring its cosmological applications as, similarly, we have not been waiting for a full consistent theory of Quantum Gravity to use field theory in Cosmology. The personal view of the author is that any result on String Cosmology that we could find using what we know about String Theory at present will be of importance for the final form of the theory. String Cosmology is an exciting field with increasing activity in recent years though it is a speculative area from the physical point of view. Putting the reality of String Theory aside, most of the models assume a lot of approximations, or even start from non-realistic scenarios from scratch. Generally this is due to the absence of knowledge about String Theory in many regimes. On theoretical grounds, the author has the opinion that we should understand the stringy effects of the simplest cases first. Otherwise, if we fail to do so, what is the hope of success in more complex situations? Also, the tools used to get around many issues in this field may potentially contribute to the understanding of String Theory itself or even be applied in other fields. These facts motivate us to study String Cosmology even though it is not close to give definite answers to the real Physics issues. Although the main goal of this thesis is to present the author’s line of research and contributions to it, it was written to also provide an introduction to Cosmology for string theorists and String Theory for cosmologists. So, the second and third chapters are reviews on selected topics on Cosmology and String Theory. Unfortunately, due to the limitations of the author, topics on both fields will be excluded from the discussion. If the reader is already familiar with both Cosmology and String Theory, she could skip chapters 2 and 3 after reading the Introduction. Chapters 4, 5 and 6 are strongly based on the following papers, that had significant contribution from the author, • Conformal inflation with chameleon coupling. Heliudson Bernardo et al JCAP04(2019)027. [1] viii • T-dual cosmological solutions in double field theory. II. Heliudson Bernardo, Robert Brandenberger, and Guilherme Franzmann, Phys. Rev. D 99, 063521 (2019). [2] • Holographic cosmology from ”dimensional reduction” of N = 4 SYM vs. AdS5 × S5. Heliudson Bernardo and Horatiu Nastase, ArXiv: 1812.07586 (in process of publication). [3]. ix Chapter 1 Introduction In this thesis, we will present results on three topics within the field of String Cosmology. These are cosmological models that are based on ideas of String Theory. The purpose of this chapter is twofold. Firstly we give a very broad motivation for why one should combine String Theory and Cosmology, followed by a small overview on the subject. Then we introduce the topics in which chapters 4, 5 and 6 are based. 1.1 General motivation: A crisis in fundamental Physics The main paradigms of fundamental Physics, General Relativity and Quantum Mechanics, are based on concepts and entities that are fundamental in the sense that they cannot be described by any further level of reductionism. In General Relativity, it is assumed that spacetime is an absolute concept that cannot be explained by some deeper structure and all that there is to know about a spacetime is encoded in its metric, which is also fundamental. For instance, gravity is a manifestation of a spacetime’s property, it’s curvature. On the other hand, all the energy residing in a spacetime is described in terms of quantum fields that are systems ruled by Quantum Mechanics. The question of what are quantum fields and what happens when they interact does not have answers within Quantum Field Theory, since these fields and their interactions are taken to be fundamental. In other words, physicists do not try to explain what are the fundamental elements of their theories but rather explain everything else in terms of it. Although there is no problem in assuming fundamental entities and explaining various phenomena in terms of them, the overall concepts and rules should be compatible with each other, that is, our fundamental description of nature should not be inconsistent. Otherwise this would indicate that the fundamental theories, concepts and entities we are using are not adequate and we should look for other elements from which we could build a consistent fundamental theory. Unfortunately, the fact that General Relativity cannot be quantized without giving non- sensical results in some extreme energy regimes, puts us in a situation where the fundamental theories that we use to describe nature are inconsistent [4–7]. This is the deepest open problem in Theoretical Physics. It is worth emphasizing that we do not have access to experimental 1 data in the regimes where our theoretical framework is conflicting. That is, from the experi- mental point of view, both General Relativity and Quantum Mechanics are excellent theories and actually they agree with each other [8,9]. At the present time, the issue is theoretical and experiments cannot directly help us in finding a framework or theory that solves this crisis, a self-consistent Quantum Gravity theory. By definition, one place where both paradigms would be important to describe nature is close to singularities. According to General Relativity, singularities of astrophysical black holes are hidden behind event horizons and so any accessible data about them are outside our reach [10,11]. Another type of singularity is of cosmological nature and, a priori, there is no no-go mechanism that prevents us from getting information on physics close to it. Moreover, modern Cosmology was constructed by combining General Relativity and Particle Physics and the agreement between all the cosmological observations and theories is astonishing [12, 13]. The use of cosmological data to constrain theories was well explored in the last few decades and in fact there is certain experimental information that could not be obtained in other ways. But from the pure theoretical point of view, the best physicists can do in constructing compatible fundamental theories is to try to recover what we already know about nature. In simple terms, it is to recover General Relativity from some theory that is compatible with Quantum Mechanics. Then, once we have a self-consistent theory we could move on and explore the physical implications of it. Due to the energy regime where we expect to get predictions, one is prone to expect that Cosmology would be useful to check if such a theoretical construction describes nature or not. 1.2 Cosmology and String Theory The final goal of Cosmology is the description of the universe on the largest scales and its time evolution as a whole. After years of theories and experiments, physicists around the world contributed to what is known as Standard Cosmological Model [13–16]. In this model, the cosmological history of the universe is described in concordance with all laws already tested in laboratory and astrophysical measurements, and it predicts that all the matter in the universe consisted of a quasi-homogeneous plasma in a state of high temperature and pressure that evolved during billions years to what we observe today. Since then, the spacetime itself is expanding and such expansion can be verified through measuring the spectrum of distant galaxies, that are receding from each other following what is known as Hubble’s law. One of the most remarkable characteristics of the observable universe is the fact that on scales bigger than 300 millions light-years, the distribution of matter is nearly homogeneous and isotropic. On such scales, we can consider the constituents of this matter distribution as galaxy clusters, each containing from a few to thousands of galaxies. In primordial times, the hot dense plasma consisted of elementary particles interacting according to the set of physical laws that we call Standard Model of Particle Physics [17,18]. Such particles collided with each other with an interaction rate that was decreasing as the universe expanded. This resulted in the fall off of the temperature and pressure of the plasma, 2 Figure 1.1: The Cosmic Microwave Background as seen from European Space Agency’s Planck satellite [16]. It is a picture of the universe around 300,000 years after the moment where our cosmological models breaks. Credits: ESA/Planck Collaboration allowing the growth of small primordial lumps on its density. Such initial irregularities got bigger under gravitational action and turned into the large scale structure we observe today, using telescopes. Amazingly, we can witness at least part of the primordial cosmological perturbations from the cosmic background radiation (CMB). The CMB is a picture of the universe at the moment of the recombination, where the temperature of the plasma was cooler enough to allow electrons and protons to make bound states and stop scattering photons, that were then free to propagate in space. The radiation corresponds almost exactly to a blackbody with temperature of ∼ 2.7K and has small anisotropies that seeded the large scales structures that we see today. Surprisingly, it seems that Quantum Mechanics plays an important role in explaining the origin of primordial inhomogeneities. The theory that describes the gravitational collapse and the formation of structures in the universe is only successful when we assume that there exists a non-luminous form of matter that interacts very weakly with other particles besides gravitationally and that such ”dark matter” dominates compared with ordinary matter, that is dubbed baryonic matter. The first evidence for dark matter came from observations of the trajectory of luminous matter in the periphery of galaxies. It was found that in order for these rotation curves to be compatible with Einstein’s General Relativity theory (in its Newtonian limit), it was necessary the existence of a distribution of matter, other than the one visible, inside each galaxy. Thus, the Standard Cosmological model assumes the existence of dark matter. At the second half of the 90’s it was discovered by supernova measurements that the uni- verse expansion is actually accelerated [19]. To describe this phase of acceleration, physicists introduced a dark energy component that enters into Einstein’s equations as a cosmological constant. It was already known that a cosmological constant drives an accelerated expansion in a homogeneous and isotropic universe. The explanation of why the universe is expanding 3 today is due to the fact that its total energy density is dominated by dark energy, though we really do not know the true nature of this component yet. So, we have a coherent picture about the cosmic history of matter. Initially, the universe was dominated by radiation or relativistic matter, that has cooled and gave origin to a phase dominated by non-relativistic matter. Then, the matter density diminished until become smaller than the dark energy density, giving the actual accelerated era. In fact, in the Standard Cosmology, the observable universe has flat spatial geometry and, today, its constitution is the following: ∼ 4% of baryonic matter, ∼ 70% of dark energy, ∼ 25% of dark matter and the radiation (photons and neutrinos) percentage is negligible. In spite of the success of the Standard Cosmological model in explaining observations, this model is commonly extended to explain why the universe has a flat geometry and how regions that appear to be causally disconnected have temperatures that are so close to each other (homogeneity). In the most popular extensions, the Universe has passed an era of accelerated expansion before being dominated by radiation. This era is called inflation, originally pro- posed in [20, 21]. During inflation, the space has expanded abruptly, going from microscopic to astronomical scales in a fraction of time of order 10−35 seconds. We do not understand completely what could generate this inflationary phase, because the energy scale involved in this period is larger than the ones tested in laboratories and particle accelerators today. Furthermore, the most striking prediction of inflation is the origin of cosmological pertur- bations [21–23]. The small inhomogeneities that have grown have no explanation within the Standard Cosmological model. A natural candidate for them is the quantum fluctuations of the degree of freedom that dominates the energy density of the Universe during inflation. Re- markably, this idea is compatible with observations, and it is the most accepted explanation for the initial conditions of the perturbations. But is worth stressing that inflation is not the only competitive candidate mechanism that explains the primordial perturbations. There are models with periods of contraction, as matter bouncing cosmologies and Ekpyrosis that are not ruled out by the observations of cosmological fluctuations. The fact that we need quantum theories to explain the Universe on large scales turns Cosmology into an area that corroborates fundamental particle physics theories. Although the Standard Model of Particle Physics is in accordance with all the tests done, including the cosmological tests, we do not know the true fundamental degrees of freedom and inter- actions at energy scales bigger than ∼ 10 TeV (that corresponds to distances of order 10−19 m). Moreover, only three of the four fundamental interactions can be described quantum mechanically. Gravity possesses only a classical formulation, the Einstein General relativity, and is compatible with quantum mechanics only at small energies (compared with the Planck scale). Even at such small energies, we can consider it just as an effective field theory: the quantum version of General Relativity is non-renormalizable. Actually, the search for a quan- tum theory of gravity and its unification with the other interactions is the biggest goal of the theoretical physics. In the last 60 years, through the work of various theoretical physicists, it was discovered that there exists a quantum theory that has various features that we expected from a unified theory. This is String Theory [24–27]. In this theory, the fundamental entities are quantum 4 relativistic strings and their different vibrational patterns gives different quantum particles. While this theory predicts quantum gravity, it is only consistent when the spacetime has 10 dimensions, 6 more than what is observed. But since gravity is dynamical, in the phe- nomenological models the extra dimensions are small, each with the size of the order of Planck length 10−34m and compactified in various manners, giving various different models from the 4-dimensional point of view. The goal of such models is to recover the Standard Model de- scription, unifying all the interactions in this way. Unfortunately, since the differences in such models lie at so big scales, string theory cannot be tested in laboratories today. So, recently, a new area has begun to be studied, to explore the best of Cosmology and String Theory: the applications of strings to cosmology or String Cosmology [28–34]. The goal is to explain the Standard Cosmological Model in the framework of String Theory. It is expected that in such explanation, we may obtain the first phenomenological implications of String Theory and the ultimate knowledge of cosmological questions. Among various applications, it is worth mentioning the description of initial conditions of the Universe, the implementation of inflation in String Theory, the use of AdS/CFT to cosmological models, brane universe models, description of the Standard Model vacuum, cosmological evolution of extra dimensions and the physical nature of the dark matter and dark energy. 1.3 Overview of String Cosmology Most of the models in String Cosmology are attempts to embed inflation in String Theory [29]. The most convenient link with low energy physics is done by considering classes of compactifications that gives N = 1 Supergravity in 4 dimensions (for constraints on Kähler potential and superpotential from inflation, see [35]). This procedure gives several fields in the 4 dimensional theory, organized in supersymmetric multiplets whose number depends on the details of the compactification, e.g. topology of internal manifold, fluxes and local sources. Most of them are scalar fields, called moduli fields, that should be stabilized in order for the model to be phenomenologically viable. But all fields have an explicit physical meaning, corresponding for example to positions of the local sources, volume of several different cycles of the internal manifolds and wrapping of p−forms around such cycles. The problem of embedding inflation in String Theory is then to find a consistent compactification with all fields stabilized in such a way that we get de Sitter space in 4 dimensions. Then, hopefully, it would be possible to deform this picture to get inflation, by adding new sources or displacing some moduli from their minima for instance. The problem of stabilization of all moduli is a rich subject by itself and a proper discussion of it would be outside of the scope of our analysis. Suffice to say that in type IIB compact- ifications, it is possible to stabilize all the complex structure moduli and the dilaton moduli by using fluxes (see [36, 37] and references therein). Extra Kähler moduli are stabilized by non-perturbative effects [38,39]. These results were used by KKLT to find a supersymmetric Anti-de Sitter minimum by tuning fluxes [38]. Then, they introduced an extra positive term to the potential by adding a D3-brane to ”uplift” the minimum to de Sitter space, though there are several criticisms to this last step [40–43] (see also [44]). The KKLT model is the 5 most popular model that claims to give a de Sitter vacuum in String Theory. There are other models also (see section 3.4 in [29], chapter 3 in [28] and references therein). There are also models of inflation using branes [45–52]. Generally, in these models, the inflaton is the position of a D-brane on a warped geometry or even the separation distance of two branes. For a review on the subject, see chapter 2 of [28]. The debate about the existence of de Sitter vacua in String Theory has been recently increased due to the Swampland conjecture [44, 53, 54] (see [55] for a review). In its simpler formulation, it puts bounds on the slope of the scalar potentials coming from String Theory, in particular prohibiting a stable de Sitter space. Though it was not proven, the Swampland conjecture is based on several arguments of what we should expect from a theory of Quantum Gravity based on strings. It was applied to scalar field cosmological models [56], in particular to dark energy [57, 58]. Together with the fact that inflation is not the only way to get a spectrum of initial perturbations consistent with observations, this motivates us to look also for cosmological models that are alternatives to inflation within String Theory. One of such alternative models is the Brandenberber-Vafa model, the String Gas Cosmol- ogy model (SGC) [59, 60], which is based on thermodynamics of strings and T-duality. By general arguments, based on the fact that there is a maximum effective physical temperature for a gas of strings, the model is free from singularities and has a natural mechanism to explain why only 3 spatial dimensions are decompactified. String Gas Cosmology predicts a blue-shifted spectrum of primordial gravitational waves, differing from usual inflation, in such a way that future observations could distinguish between the two. There are also other models based on the ideas of SGC, as the S-brane bouncing scenario [61–63]. It is worth emphasizing that SGC is the most ”pure” stringy model for Cosmology, in the sense that it could not be reproduced from some effective field theory based solely on usual particles and fields. Further details of the SGC model are discussed in section 1.5 and chapter 5. Another alternative model is Ekpyrosis [64]. Created using stringy ideas from D = 11 Supergravity and Horava-Witten theory [65], the ekpyrotic mechanism was rapidly detached from its origins and can now be seen as model based on scalar fields with a particular char- acteristic class of potentials (negative and highly steep), though it is difficult to explain the origin of the fields and its potentials [66, 67]. During the ekpyrotic phase, the universe con- tracts slowly but in such a way that the inhomogeneities do not increase to dominate the total energy of the Universe. A very interesting characteristic of the ekpyrotic solution is that it is an attractor solution for giving approximately scale invariant spectrum of perturbations, at the same level as inflation1 [68]. 1.4 Inflation with Chameleon Fields Chameleon fields are scalar fields with a potential energy that depends on the energy density of the environment where it is defined (hence the name ”chameleon”) [69]. Such chameleonic 1But note that, with respect to the original proposal, at least two fields are required for generating nearly scale-invariant perturbations. 6 behaviour is due to a field dependent conformal factor in the metric that other sources per- ceive. It was first presented as a proposal to explain dark energy but with a natural screening mechanism built into it from scratch [70]. The idea that a scalar can have a “chameleon” coupling to the (non-relativistic) matter density was introduced, in part, to allow for a scalar that can be very light on cosmological scales while also ”hiding” its effects from observations in the lab (on Earth), or in the Solar System [69,71]. Various laboratory searches have been initiated for such a scalar (e.g., [70,72– 88]). From a theoretical perspective, this alleviates the problem of having too many a priori light scalars in string theory (moduli): if they are chameleons, they do not contradict known experiments to date. A way to embed chameleons in string theory was suggested in [89]. On the other hand, since the chameleon is a scalar, an economical ansatz is for the same field that acts as an inflaton near the Big Bang to be the chameleon. This idea was explored in [90]. In this case, however, the two regions (inflation and chameleon) are separated by a large region of vanishing potential in field space, and the inflationary era itself is not affected by the chameleon coupling. An attempt to consider what happens if we consider inflation in the presence of a chameleon or symmetron [91] coupling was considered in [92] and [93]. In chapter 4 the issue of inflation with a chameleon coupling is considered taking into account that there can be new “attractor-like” phases due to the chameleon coupling, where various forms of matter (contributions to the energy-momentum tensor) scale in the same way with the scale factor, as seen for instance in [90] at zero potential. Since we need to consider “new inflation” type of models with a plateau, a natural starting point is the system of “conformal inflation” models (see for instance [94,95]), as analyzed in [96]. We will assume the existence of some heavy, non-relativistic matter with density ρm during the plateau phase (inflation), coupled to the inflaton via a chameleon coupling, ρmF (φ), and for the coupling the standard form F = e−cφ/MPl . We will investigate the possibility of attractor-like behaviour due to this coupling, and see that depending on the sign of c, we can have either a prolonged period of attractor behaviour before the end inflation, or an effective inflationary potential that is different, with modified values for the CMB observables, ns and r. 1.5 Cosmological Solutions in Double Field Theory Target space duality [97–101] is a key symmetry of superstring theory. Qualitatively speaking, it states that physics on small compact spaces of radius R is equivalent to physics on large compact spaces of radius 1/R (in string units). This duality is a symmetry of the mass spectrum of free strings: to each momentum mode of energy n/R (where n is an integer) there is a winding mode of energy mR, where m is an integer. Hence, the spectrum is unchanged under the symmetry transformation R → 1/R if the winding and momentum quantum numbers m and n are interchanged. The energy of the string oscillatory modes is independent of R. This symmetry is obeyed by string interactions, and it is also supposed to hold at the non-perturbative level (see e.g. [24, 25]). The exponential tower of string oscillatory modes leads to a maximal temperature for 7 a gas of strings in thermal equilibrium, the Hagedorn temperature [102]. Combining these thermodynamic considerations with the T-duality symmetry leads to the proposal of String Gas Comology [59] (see also [103]), a nonsingular cosmological model in which the Universe loiters for a long time in a thermal state of strings just below the Hagedorn temperature, a state in which both momentum and winding modes are excited. This is the ‘Hagedorn phase’. After a phase transition in which the winding modes interact to decay into loops, the T-duality symmetry of the state is spontaneously broken, the equation of state of the matter gas changes to that of radiation, and the radiation phase of Standard Big Bang expansion can begin. In addition to providing a nonsingular cosmology, String Gas Cosmology leads to an al- ternative to cosmological inflation for the origin of structure [104]: According to this picture, thermal fluctuations of strings in the Hagedorn phase lead to the observed inhomogeneities in the distribution of matter at late times. Making use of the holographic scaling of mat- ter correlation functions in the Hagedorn phase, one obtains a scale-invariant spectrum of cosmological perturbations with a slightly red tilt, like the spectrum which simple models of inflation predict [104]. If the string scale corresponds to that of Grand Unification, then the observed amplitude of the spectrum emerges naturally. String gas cosmology also pre- dicts a roughly scale-invariant spectrum of gravitational waves, but this time with a slightly blue tilt [105, 106], a prediction with which the scenario can be distinguished from simple inflationary models (see also [107,108] for other distinctive predictions). The phase transition at the end of the Hagedorn phase allows exactly three spatial di- mensions to expand, the others being confined forever at the string scale by the winding and momentum modes about the extra dimension (see [109–112] for detailed discussions of this point). The dilaton can be stabilized by the addition of a gaugino condensation mecha- nism [113], without disrupting the stabilization of the radii of the extra dimensions. Gaugino condensation also leads to supersymmetry breaking at a high scale [114]. For detailed reviews of the String Gas Cosmology scenario see [32,115]. However, an outstanding issue in String Gas Cosmology is to obtain a consistent descrip- tion of the background space-time. Einstein gravity is clearly not applicable since it is not consistent with the basic T-duality symmetry of string theory. Dilaton gravity, as studied in Pre-Big Bang Cosmology [116, 117] is a promising starting point, but it also does not take into account the fact, discussed in detail in [59], that to each spatial dimension there are two position operators, the first one (x) dual to momentum, the second one (x̃) dual to winding. Double Field Theory (DFT) (see [118,119] for original works and [120] for a detailed review) is a field theory model which is consistent both with the T-duality symmetry of string theory and the resulting doubling of the number of spatial coordinates (see also [121–123] for some early works). Hence, as a stepping stone towards understanding the dynamics of String Gas Cosmology it is of interest to study cosmological solutions of DFT. In an initial paper [124], point particle motion in doubled space was studied, and it was argued that, when interpreted in terms of physical clocks, geodesics can be completed arbitrarily far into the past and future. In a next paper [125], the cosmological equations of dilaton gravity were studied with a matter source which has the equation of state of a 8 gas of closed strings. Again, it was shown that the cosmological dynamics is non-singular. The full DFT equations of motion in the case of homogeneous and isotropic cosmology were then studied in [126]. The consistency of DFT with the underlying string theory leads to a constraint. In DFT, in general a stronger version of this constraint is used, namely the assumption that the fields only depend on one subset of the doubled coordinates. There are various possible frames which realize this (see also the discussion in section 3.4). In the supergravity frame it is assumed that the fields do not depend on the “doubled” coordinates x̃, while in the winding frame it is assumed that the fields only depend on x̃ and not on the x coordinates. It was shown that for solutions with constant dilaton in the supergravity frame, the consistency of the equations demands that the equation of state of matter is that of relativistic radiation, while constant dilaton in the winding frame demands that the equation of state of matter is that of a gas of winding modes. These two solutions, however, are not T-dual. In chapter 5 solutions which are T-dual are introduced, expanding on the analysis of [126] and presenting improvements in the previous non-T-dual solutions. 1.6 Holographic Cosmology The first indication that quantum gravity should have a holographic nature goes back to arguments about thermodynamics of black holes [127, 128]. In a modern view, holography in high energy physics deals with the description of quantum aspects of a spacetime with dimension D by using field theory defined on a flat background with dimension D − 1. This area of research increased a lot since the explicit construction of the correspondence between String Theory on AdS5×S5 and N =4 SYM by Maldacena [129] (see also [130,131]). Thence- forward, several gauge/gravity duality models were proposed and applied in many branches of theoretical physics, e.g. [132–135] (see [136] for applications in condensed matter). The idea of a holographic cosmology has been around for a long time. The first concrete proposal of how that would look like was put forward by Maldacena in [137] (see also [138]), stating that the wave function of the Universe, as a function of spatial 3-metrics (and scalars), ψ[hij , φ] in some gravity dual background (in his specific case, proposed for some space that asymptotes to de Sitter), equals the partition function of some (3 dimensional) field theory, with sources (for the energy-momentum tensor Tij and some scalar operator O) hij , φ, i.e., Z[hij , φ] = ψ[hij , φ]. However, at the time, there was no concrete proposal for a gravity dual pair. In [139], such a model was proposed, and a sort of phenomenological holographic cos- mology approach was born. It was first noted that, for cosmological scale factors a(t) that are both exponential (as in standard inflation, and corresponding to AdS space) or power law (as in power law inflation, and corresponding to nonconformal D-branes, for instance), a specific Wick rotation, the ”domain wall/cosmology correspondence”, turns the cosmology into a standard holographic space like a domain wall, that should have a field theory dual in 3 Euclidean dimensions. A holographic computation then relates the cosmological power spec- trum, coming from the 〈δhij(~x)δhkl(~y)〉 correlators in the bulk, with 〈Tij(~x)Tkl(~y)〉 correlators 9 in the boundary field theory. One can assume a regime where the field theory is perturbative, and the latter correlators can be calculated from Feynman diagrams. Then by comparing the cosmological power spectrum with CMBR data, we can find the best fit in a phenomenological class of field theories, with a ”generalized conformal structure”. In [140, 141] it was shown that the phenomenological fit matches the CMBR as well as the (different) standard ΛCDM with inflation, though the perturbative field theory approximation breaks down for modes with l < 30. But this holographic cosmology paradigm is more general than the specific class of phenomenological models: it includes standard inflationary cosmology, where the gravita- tional side is weakly coupled, as well as intermediate coupling field theory models, that can be treated non-perturbatively on the lattice. Another approach to holographic cosmology was considered in [142–144], where one starts with a ”top down” construction, specifically a modified version of the original N = 4 SYM vs. string theory in AdS5×S5, where an FLRW cosmology with a(t) replaces the Minkowski metric, and a nontrivial dilaton is introduced. On the field theory side, one has a time- dependent coupling now. The model has been used in [143, 144] to show how perturbations entering a Big Crunch exit after the Big Bang, one issue that has been very contentious in ekpyrotic and cyclic cosmologies. It was shown that the spectral index of perturbations exits unchanged, but there was no simple mechanism in [143,144] of calculating the power spectrum of fluctuations for CMBR. Chapter 6 is a proposal for modifying the top down construction of [142–144], to fit it into the holographic paradigm of [139], for which the common concrete realization so far is a phenomenological (bottom up) approach. We will find that we can modify the general proposal of Maldacena for Z[hij , φ] = ψ[hij , φ] to deal with this case of having both time and a radial coordinate, and then use an integration over the time coordinate, from close to zero until an arbitrary time t0 (but not to the future of it, in this way obtaining a function of t0), to argue that we have effectively a ”dimensional reduction” over the time direction. 1.7 Notation and structure of the thesis The metric signature used throughout the thesis is mostly plus (−,+, . . . ,+) and natural units are adopted otherwise stated the contrary. In chapter 3, we tried to use the same notation as references [24] and [25] and for that reason the spacetime metric is denoted as Gµν in that chapter rather than gµν as in chapter 2. The structure of the thesis is the following: in chapter 2 we review the theoretical foundations of Cosmology; in chapter 3 we discuss the main ideas of String Theory, Double Field Theory and Holography; the impact of the chameleonic coulpling to single field inflation is presented in chapter 4, for the particular case of conformal inflation; in chapter 5, T-dual cosmological solutions of DFT are presented and chapter 6 describes a novel prescription for holographic cosmology. A summary of the conclusions is presented in chapter 7 and a small digression about calculations used in chapter 6 is provided in appendix A. 10 Chapter 2 Basics of Cosmology The Standard Cosmological Model used nowadays is based on two observational facts: the universe is expanding and is very homogeneous and isotropic at large scales. The former statement refers to the space itself (at large scales) and the last is called Cosmological Prin- ciple. The expansion of the universe was discovered in 1929 by Edwin Hubble via measurements of the redshift of distant galaxies. Statistically, he discovered that the velocity v of the galaxies depended on its radial distance r by the relation v = Hr, (2.1) where H was a constant, the Hubble parameter. This is the Hubble law for an expanding Universe. Physically, it is just the statement that each galaxy is moving away from our galaxy and from each other. More remarkably in 1998 the Universe was discovered to have an accelerated expansion [19] so the Hubble constant has not a fixed value along cosmic evolution. Today1 the value of Hubble constant is given by H0 = 100h km s ·Mpc , (2.2) with h present value in the range 0.6 < h < 0.8 [145]. The isotropy and homogeneity of the energy distribution of the universe is probed by sky survey observations and the measurements of the Cosmic Microwave Background Radiation (CMBR or simply CMB). On scales larger than 300 million light years, the matter distribution acts like a ”cosmic fluid” made of clusters of galaxies containing a few dozens to hundreds of galaxies each. It is highly homogeneous at each point but not totally, since it has small inhomogeneities, such that visually the distribution has a ”cosmic-web” structure. The purpose of this chapter is to quickly review the basic aspects of Cosmology. We discuss standard cosmology and inflation, following biased topics that are thought to be a complete set needed to understand latter chapters. General references can be found in [14, 146, 147] and for the values of cosmological parameters see [16,145,148]. 1In Cosmology one works with very large spatial and temporal scales, so when one talks about the ”present” one is usually referring to a time scale small compared to billion years. 11 2.1 Kinematics and dynamics of an expanding Universe From the symmetry requirements of the Cosmological Principle we get a specific form for the spacetime metric of the universe at large scales, the Friedman-Lemâıtre-Robertson-Walker (FLRW) metric. Using comoving coordinates, that is, coordinates of observers that are at rest with respect to the expansion, we have ds2 = −dt2 + a2(t) ( dr2 1− kr2 + r2 ( dθ2 + sin2 θdφ2 )) , (2.3) where k can take only three values corresponding to three possible geometries of the spatial section: for k = 1 the space is closed (a three-sphere S3), for k = 0 the space is flat (an Euclidean plane E3) and for k = −1 we have an open space (a hyperbolic space H3). The scale factor a(t) is defined up to a constant rescaling that can be chosen such that a(t0) = 1 and can be increasing or decreasing, describing expanding or contracting universes and its time dependence is related with the energy content of the Universe via Einstein’s equations. The Hubble law can be obtained from the FLRW metric, regardless of the functional form of the scale factor. Suppose that the coordinates are such that our galaxy is located at r = 0 and there is another galaxy at r = rg from which we want to check the Hubble law. At a proper time t, the proper distance to the far away galaxy is d(t) = a(t) ∫ rg 0 dr (1− kr2)1/2 =  a(t) sin−1(r) (for k = 1); a(t)r (for k = 0); a(t) sinh−1(r) (for k = −1), (2.4) and in an expanding universe, the velocity of the galaxy recession as seen by us is v = ḋ(t) = ȧ(t) a(t) d(t), (2.5) which is the Hubble law with Hubble parameter given by H(t) = ȧ(t) a(t) . (2.6) We see that the Hubble law is a kinematic feature independent of the cosmological solution for a(t). Another kinematic feature is the redshift of distant sources due to the expansion of space. Consider two pulses of light emitted from a distant source located at r = rg at times t = tg and t = tg+δtg and received at r = 0 at times t0 and t0+δt0, respectively. Since the comoving distance covered by the pulses is the same, we have∫ t0 tg dt a(t) = ∫ t0+δt0 tg+δtg dt a(t) . (2.7) Assuming a(t) to be constant over small time intervals δtg and δt0, we get δtg a(tg) = δt0 a(t0) , (2.8) 12 so there is a shift z between the wavelength of the emitted and received light, λg and λ0 respectively, given by z = λ0 − λg λg = a(t0) a(tg) − 1. (2.9) It is a redshift for an expanding universe and blueshift for a contracting one. This cosmological shift is due to the expansion of the space between the emitter and receiver and so is different from the Doppler shift, where the source of light and the receiver have a relative motion in the same inertial frame. The last kinematic feature worth stressing is the existence of horizons in an expanding universe. Intuitively, from Hubble’s law, two sufficiently distant points will be moving away from each other faster than the speed of light and then be causally disconnected by the expansion of the universe. To formalize this idea one defines the cosmological event horizon, as the maximum distance that light will be able to travel if emitted at time t: deh(t) = ∫ ∞ t dt′ a(t′) , (2.10) then by definition, at a given time, an event cannot affect any other event if their separation is greater than deh. There is also the particle horizon at time t, which is the distance that light could have travelled since t = 0: dph(t) = ∫ t 0 dt′ a(t′) . (2.11) Finding some correlation between two points with comoving spatial separation greater than dph is very unlikely, as these points could never be in causal contact. In section 2.4 we will see that this situation happens to be true for different points in the CMB. Another characteristical distance is the Hubble horizon, distance or radius, given by H−1. The equations describing perturbations in an expanding universe are such that super-Hubble effects are negligible to the dynamics on scales well inside the Hubble radius. In fact, at such small scales and at time intervals much smaller than the Hubble distance, one could neglect the expansion and the physics would be the same as in Minkowski spacetime. Until now we have focused on the kinematic features of an expanding universe. Turning to the dynamics, the equations ruling the evolution of a(t) are given by Einstein’s field equations, sourced by a perfect fluid energy-momentum tensor Rµν − 1 2 gµνR = 8πGTµν , (2.12) where, on the grounds of homogeneity and isotropy, Tµν = p(t)gµν + (ρ(t) + p(t))uµuν , (2.13) where p(t) and ρ(t) are the time-dependent pressure and energy densities of the fluid, respec- tively, and uµ is the velocity vector field of the fluid. Then, equation (2.12) relates a(t), k, ρ(t) and p(t). Its time-time component 00 gives the first Friedmann equation, ȧ2(t) + k = 8πG 3 ρ(t)a2(t), (2.14) 13 and each diagonal spatial components ii gives 2a(t)ä(t) + ȧ2(t) + k = −8πGp(t)a2(t). (2.15) Combining these equations by derivating the first with respect to time, we get ρ̇(t) + 3H(t)(ρ(t) + p(t)) = 0, (2.16) which is exactly the continuity equation for the energy-momentum tensor, ∇µTµν = 0. It is also possible to combine equations (2.14) and (2.15) to obtain the second Friedmann equation ä(t) a(t) = −4πG 3 (ρ(t) + 3p(t)), (2.17) from which we can see that the sign of ρ+3p determines whether the expansion is accelerating or decelerating. From the first Friedmann equation (2.14), for ρ(t) > 0 and k = 0 or k = −1 (flat and open universes), ȧ2(t) will not vanish and so the universe expands forever. While for k = 1 (closed universe), the expansion stops at some time tc defined by ρ(tc)a 2(tc) = 3/8πG, after which the universe start to contract. Defining the critical density ρc = 3H2 8πG , (2.18) we can recast the condition of open, flat and closed universes as ρ < ρc, ρ = ρc and ρ > ρc, respectively. Measuring the total energy density in the universe today and at CMB redshift, we know that the Universe is very close to flat [145]. So, from now on we focus on the case k = 0. In order to find the time evolution of a(t), ρ(t) and p(t), we need an equation of state relating energy and pressure. Assuming a constant relation, we have p(t) = wρ(t), (2.19) where the constant equation of state parameter w has different values depending on the nature of the cosmic fluid. Then we can find the solutions, for w 6= −1 a(t) ∝ t 2 3(1+w) , ρ(t) ∝ a3(1+w)(t) ∝ t2 (2.20) There are 3 simple but useful cases for w: • Radiation, w = 1/3. In this case the energy-momentum tensor can be obtained from the electromagnetic action, which gives a traceless Tµν . This implies w = 1/3 for an homogeneous and isotropic distribution of photons. We then have ρ ∝ a−4 and a(t) ∝ t1/2. • Non-relativistic matter or dust, w = 0. In this case the rest energy dominates of the kinetic energy of the components of the fluid and the pressure is negligible. The energy density scales as ρ ∝ a−3 and the scale factor evolves as a ∝ t2/3. 14 • Cosmological constant or dark energy, w = −1. Related to the vacuum energy density, this case corresponds to an energy momentum tensor proportional to the metric, Tµν = M2 PlΛgµν . The continuity equation gives a constant energy density, which implies an exponential expansion, a ∝ eHt with H = √ Λ/3. Then, the FRW metric covers a patch of the de Sitter spacetime. In Standard Cosmology the Universe is assumed to be initially in thermal equilibrium in a radiation phase. Then, after two phase transitions (hadronic and electroweak), it be- comes dominated by non-relativistic matter. Finally, recent observations strongly support an accelerated phase that can be modelled by dark energy. All the discussion in this section applies under the assumption of exact homogeneity and isotropy, an unperturbed universe. But in reality we have cosmological structures and so our Universe is not exactly unperturbed for sure. In the next section we discuss cosmological perturbations of an homogeneous and isotropic universe and their observational implications. 2.2 Cosmological Perturbations At each fixed time, the comoving coordinates used to write (2.3) define a time slice of the spacetime that has uniform energy density and is orthogonal to comoving worldlines. Con- sidering the k = 0 case it will also be flat. These slices are labelled with the time coordinate t, but there is another possible coordinate choice, the conformal time τ defined as dτ = adt. Once we perturb the background, it is impossible to find a coordinate system that preserves all these features of the unperturbed universe. We need to write the metric in a modified way such that (2.3) is retrieved for zero perturbations. There are various different choices or gauges. Each gauge could still have some properties of the unperturbed metric, but not all. Then, one may choose to fix the gauge and work on the perturbation or to construct gauge invariant variables and write all equations in terms of them. In the following, we will consider the gauge fixing approach. The perturbation of the metric corresponds to 10 degrees of freedom that can be decom- posed as scalars, vectors and tensors with respect to 3-rotations. In fact, consider the most general form of the perturbed metric in conformal time, ds2 = a2(τ) { −(1 + 2φ)dτ2 − 2Sidτdx i + [(1 + 2ψ)δij + Eij ] dx idxj } . (2.21) The spacetime functions φ(xµ), ψ(xµ) are two of the 4 scalar degrees of freedom. The other two, B(xµ) and E(xµ), lie within the decomposition of Si(x µ) and Eij(x µ): Si = ∂iB +Bi, Eij = 2 ( ∂i∂j − 1 3 δij∇2 ) E + 2∂(iEj) + hij , (2.22) where Bi, Ei and hij are divergenceless, with hij traceless, giving 4 vector and 2 tensor degrees of freedom. 15 Note that by perturbing the metric, we are also perturbing the velocity vector field of any fluid on the spacetime. Indeed from uµuν = −1, we get 2uµδu µ = −δgµνuµuν . To first order in perturbations and in (τ, xi) coordinates, the perturbed 4-velocity is uµ(xµ) = a−1(τ) ( 1− φ(xµ), vi(xµ) ) , (2.23) where the coordinate 3-velocity vi = vi is first order in perturbation. Then using (2.13), the perturbed energy momentum tensor components in (τ, xi) coordinates are T 0 0 = −(ρ+ δρ), T 0 i = (ρ+ p)(vi −Bi), T i 0 = −(ρ+ p)vi, T i j = (p+ δp)δ ij + Π i j , (2.24) where the traceless Π i j is an anisotropic stress perturbation. At linear order, the Einstein’s equations does not mix scalar, vector and tensor pertur- bations. We will not discuss vector perturbations, since they decay quickly in our expanding Universe and do not play a big role in large scale structure formation. Tensor perturbations may be interpreted as gravitational waves and they are discussed in section 2.4, in the context of inflation. In the rest of this thesis we shall focus on scalar perturbations, as they have more impact on the CMB anisotropies. The perturbations of the metric and energy-momentum tensor are not invariant under coordinate transformations, xµ → x̃µ = xµ + ξµ(xν). (2.25) The infinitesimal parameters ξµ have 4 degrees of freedom, that give rise to ”fake” metric fluctuations. Decomposing ξi = ∂iξ+ ξi⊥ with ∂iξ i ⊥ = 0, the components ξ0 and ξ correspond to scalar fluctuations, while ξi⊥ is related to vector mode perturbations. As tensor fluctuations are gauge invariant, coordinate transformations cannot produce fake tensor perturbations. Using the expression for the transformation of the components of any rank-2 tensor Aµν , õν(x̃) = ∂xα ∂x̃µ ∂xβ ∂x̃ν Aαβ(x), (2.26) it is possible to find the transformations for scalar mode perturbations in the metric, φ→ φ̃ = φ− ξ0′ − aHξ0, (2.27) ψ → ψ̃ = ψ − aHξ0 − 1 3 ∇2ξ, (2.28) B → B̃ = B + ξ0 − ξ′, (2.29) E → Ẽ = E − ξ0, (2.30) and in the energy-momentum tensor, δρ→ δ̃ρ = δρ− ξ0ρ′, (2.31) δp→ δ̃p = δp− ξ0p′, (2.32) v → ṽ = v + ξ′, (2.33) Π→ Π̃ = Π, (2.34) 16 where Π and v are the scalar parts of Πij and vi, respectively. The prime in the equations above denotes derivative with respect to conformal time. There are special combinations of the metric and energy-momentum tensor perturbations that are gauge invariant. For the metric, we have the Bardeen variables, Φ = φ+B′ − E′′ + aH(B − E′), (2.35) Ψ = −ψ − aH(B − E′) + 1 3 ∇2E, (2.36) and for the energy-momentum tensor, we have the comoving density perturbation, ∆ = δρ ρ + ρ′ ρ (v +B). (2.37) Using coordinate transformations, we can set two of the 8 scalar perturbations in the metric and in the cosmic fluid to zero. Each choice defines a different gauge2. The most common gauges in cosmology are the following: • Newtonian or longitudinal gauge: B = 0 = E. In this gauge, the time slices are isotropic and orthogonal to the wordlines of comoving observers. In terms of Bardeen variables, Φ = φ and Ψ = −ψ, and the form of the metric ds2 = a2(τ) [ −(1 + 2Φ)dτ2 + (1− 2Ψ)δijdx idxj ] , (2.38) is similar to the one valid in regions of spacetime containing weak gravitational fields (hence the gauge’s name), the difference being that for the latter metric Φ = Ψ. We will see that this is exactly the case for zero anisotropic stress. • Spatially-flat or uniform curvature gauge: ψ = 0 = E. As the name indicates, in this gauge the constant time hypersurfaces are flat. This choice simplifies curvature perturbation computations. • Synchronous gauge: φ = 0 = B. In this case the wordlines of constant xi are geodesics and the time slices are orthongonal to them. Since there is no time lapse, the clocks of observers within a Hubble radius are synchronized, giving the name for this gauge. One also defines the uniform density gauge, with δρ = 0, and comoving gauge, with v = 0, though they do not fix the gauge redundancy entirely since only the choice of ξ0 is used. There are different versions of these gauges for different uses of the ξ choice (most commonly, B = 0 is assumed). Now we discuss the dynamical equations for perturbations. They are obtained from the conservation of the energy momentum tensor and Einstein’s equations at first order in perturbation theory (which are not totally independent due to the Bianchi identity). To get 2Some gauge choices do not fix all the gauge invariance, as there could still be residual coordinate trans- formations that preserves the gauge fixing. For us, it is a important fact that for the longitudinal gauge, if φ and ψ vanishes at infinity, the gauge is completely fixed. 17 gauge invariant equations, we work in longitudinal gauge and then use Φ = φ and Ψ = −ψ to write the final equations in terms of gauge invariant metric perturbations.3 From the 0-component of the conservation equation ∇µTµν = 0, we get the continuity equation for the perturbations, δ′ + (1 + w) ( ∂iv i − 3Φ′ ) − 3aHw ( δ − δp p ) = 0, (2.39) where δ ≡ δρ/ρ. The divergence of the perturbed Euler equation, coming from the i- components, gives (∂iv i )′ + aH(1− 3w)∂iv i + w′ 1 + w ∂iv i + 1 ρ+ p ∇2δp− 2 3 w 1 + w ∇2Π = 0. (2.40) The 00-component of Einstein equations gives, ∇2Ψ− 3aH(Ψ′ + aHΦ) = 4πGa2ρδ, (2.41) while the scalar part of the 0i-component yields Ψ′ + aHΦ = −4πGa2(ρ+ p)v. (2.42) The ij-component of Einstein’s equations have a trace and trace-free scalar parts. From the former we obtain Ψ′′ + 1 3 ∇2(Φ−Ψ) + ( 2(aH)′ + (aH)2 ) Φ + aH(Φ′ + 2Ψ′) = 4πGa2δp, (2.43) and from the latter we get( ∂i∂j − 1 3 δij∇2 ) (Ψ− Φ) = 8πGa2Πij . (2.44) In the absence of anisotropic stress tensor, Πij , equation (2.44) gives Φ = Ψ, and we can write all the equations in terms of the gravitational potential Φ. At first order, we can assume that to be the case. Combining equations (2.41) and (2.42) we get the gauge invariant version of the Poisson equation, ∇2Φ = 4πGa2ρ∆, (2.45) from which we have that the gauge invariant function ∆ defined in (2.37) can be interpreted as the overdensity of energy in comoving gauge (v = 0 = B). The gauge invariant comoving curvature perturbation, R(xµ), defined as R = ψ − 1 3 ∇2E + aH(B + v), (2.46) 3Another possibility is to find the gauge transformation for the perturbation in Einstein and energy- momentum tensor and construct combinations of these perturbations to get gauge invariant equations, as done in [149] 18 is the curvature perturbation in the slices of constant τ in the comoving gauge. Indeed, using the perturbed metric (2.21), the intrisic curvature of the constant time hypersurfaces is R(3) = − 4 a2 ∇2 ( ψ − 1 3 ∇2E ) , (2.47) and we see that R reduces to the scalar curvature perturbation ψ − 1 3 ∇2E, (2.48) in the comoving gauge. There is also the uniform density curvature perturbation, ζ(xµ), defined as ζ = ψ − 1 3 ∇2E − aH δρ ρ′ , (2.49) that is also gauge invariant and reduces to the scalar curvature perturbation for δρ = 0. It is related with R by R = ζ − 2 9(1 + w) ∇2Φ (aH)2 , (2.50) as one can show using equation (2.45). So, on super-Hubble scales, we have R = ζ. A crucial property of the comoving and uniform density perturbations is the fact that it does not evolve outside the Hubble radius under some assumptions on the energy-momentum tensor perturbations. For vanishing anisotropic stress, we have 3 2 (aH)2(1 + w)R′ = −4πGa2aH ( δp− p′ ρ′ δρ ) + aH p′ ρ′ ∇2Φ, (2.51) and since the last term is negligible for superhorizon modes, a barotropic equation of state, p = p(ρ), (2.52) implies thatR and ζ are frozen outside the Hubble radius. The condition (2.52) is an adiabatic condition, that implies that the perturbations in the pressure can be written as perturbations in the energy density. In presence of different fluid components ρi with equation of state wi, from the adiabatic condition for energy densities ρi = ρi(ρ), (2.53) where ρ is the sum of all ρi, we have that the individual perturbations δρi are zero in the uniform density energy gauge, on which δρ = 0. For other slicings, δρi are obtained by a common local shift in time, δρi = −ρ′iδτ, (2.54) that together with the continuity equation (and in absence of energy transfer between the fluid components), gives δi 1 + wi = δj 1 + wj . (2.55) In cosmological evolution, the fluid components that will be relevant correspond to matter δm and radiation δr. Then (2.55) gives δr = 4δm/3. So, all adiabatic perturbations are related 19 and can be written in terms of the total energy density perturbation δρ. There are also isocurvature or entropy perturbations, that correspond to uncorrelated perturbations between different components, but all observations are consistent with purely adiabatic perturbations. The fact that the gauge invariant curvature perturbations are conserved outside the hori- zon is a very important result. It is used to explain how microphysics may generate primordial perturbations, as if we have a mechanism to make the perturbation modes to get out of the Hubble horizon and then get back inside it, then the initial conditions for the CMB perturba- tions have a casual cause. All proposals for explaining structure formation explore this idea, the most promising and famous being inflation, to be discussed latter in this chapter. 2.3 The Cosmic Microwave Background Radiation 2.3.1 CMB Anisotropies The Cosmic Microwave Background Radiation (CMBR or simply CMB) is the radiation permeating all observable universe and corresponds to photons travelling freely since recom- bination, a moment in time when the electrons and protons first combined to form H atoms and Thomson scattering (e+ γ → e+ γ) was not enough to keep the baryons and photons in equilibrium. The CMB has an approximate thermal blackbody spectrum with temperature T̄ ' 2.7K. At the level of background cosmology, the CMB defines a frame where it is isotropic, due to homogeneity of the Universe at the time of recombination. When we observe CMB today, the motion of the Solar System with respect to this frame produces a dipolar effect in the observed temperature. The relation between the momentum of an observed photon coming from direction n̂ in the sky, po(n̂), and its momentum p in the CMB rest frame is given by po(n̂) = p γ(v)(1− n̂ · ~v) ≈ p(1 + n̂ · ~v), (2.56) where ~v is the Solar System velocity and γ(v) is its relativistic gamma factor. Since CMB has a blackbody spectrum and T ∝ 1/a, we can write this change in momenta as a change in the observed temperature of the CMB: δT (n̂) T̄ ≡ To(n̂)− T̄ T̄ = po(n̂)− p p = n̂ · ~v = v cos θ. (2.57) From CMB observations we find v ≈ 368km/s. To find how the photons are affected by cosmological perturbations let us consider their geodesics in a perturbed background (in longitudinal gauge), ds2 = a2(τ)[−(1 + 2Φ)dτ2 + (1− 2Ψ)δijdx idxj ]. (2.58) From the 0th component of the geodesic equation, we get d dτ ln(ap) = −dΦ dτ + ∂ ∂τ (Φ + Ψ), (2.59) 20 where p is the photon momentum. Integrating this equation along a line-of-sight, gives ln(ap)|0 = ln(ap)|∗ + (Φ∗ − Φ0) + ∫ τ0 τ∗ dτ∂τ (Φ + Ψ), (2.60) where we assume that all photons were emitted at a fixed time τ∗, i.e., an instantaneous recombination. We call τ∗ the moment of last scattering. To relate this result with temperature anisotropies, we use ap ∝ aT̄ ( 1 + δT T̄ ) (2.61) to get δT T̄ ∣∣∣∣ 0 = δT T̄ ∣∣∣∣ ∗ + (Φ∗ − Φ0) + ∫ τ0 τ∗ dτ∂τ (Φ + Ψ) (2.62) The term Φ0 only affects the monopole perturbation, and so is unobservable and we’ll drop it from the equation. Since ργ ∝ T̄ 4 =⇒ δργ ∝ 4T̄ 3δT = 4ργδT/T̄ , we have δT T̄ ∣∣∣∣ 0 = ( 1 4 δγ + Φ ) ∗ + ∫ τ0 τ∗ dτ∂τ (Φ + Ψ). (2.63) The last term is called Integrated Sachs-Wolfe (ISW) term and it vanishes during matter domination, when Φ̇ ≈ Ψ̇ = 0. The combination 1/4δγ + Φ is called Sachs-Wolfe term (SW). Including the motion of electrons at the surface of last scattering, leads to an extra term for δT/T̄ : δT T̄ (n̂) = ( 1 4 δγ + Φ + n̂ · ~ve ) ∗ + ∫ τ0 τ∗ dτ(Φ̇ + Ψ̇) (2.64) For adiabatic initial conditions, we can relate all the fluid perturbations with the metric ones. Working in longitudinal gauge, we have ζ = −Ψ− aH δρ ρ′ = −Ψ− δ 3(1 + w) , (2.65) and so, at super Hubble scales we get ζ = −Ψ− 2 3 Φ + (aH)−1Φ̇ 1 + w , (2.66) and since ζ is constant, the non-decaying solution for the gravitational potential before horizon entry is (for Π = 0) is Φ = Ψ = −3 + 3w 5 + 3w ζ. (2.67) Together with equation (2.65), this implies that, during radiation domination, δ = −2Φ = 4 3 ζ. (2.68) This result will be used as initial condition for the oscillations in the baryon-photon fluid. Also, at super-Hubble scales the SW term is proportional to Φ∗, since at τ∗ we have matter domination and so δγ = −4δm/3 = −8Φ/3, giving (δγ/4 + Φ)∗ = Φ∗/3. Thus, an overdense region (Φ < 0) appears as a cold spot in the sky today. 21 2.3.2 CMB Power Spectrum The anisotropies on the CMB temperature are of the order δT/T̄ ∼ 10−5. The primordial perturbations in the early universe are imprinted in the observed angular statistic of these fluctuations. In this section we briefly comment on the CMB power spectrum. Assuming isotropic initial conditions, we have〈 δT (n̂) T̄ δT (n̂′) T̄ 〉 = ∑ l 2l + 1 4π ClPl(cos θ), (2.69) since the correlation will only depend on the relative orientations of n̂ and n̂′, n̂ · n̂′ = cos θ. The Pl(cos θ) functions are the Legendre polynomials and the expansion coefficients Cl are the angular power spectrum. The observed spectrum of CMB anisotropies results from sub-horizon evolution of per- turbations in the photon density and metric, that were sourced by initial super-horizon per- turbations. Such evolution corresponds to sound waves in the photon-baryon fluid. We can write Cl = 4π (2l + 1)2 ∫ d ln kT 2 l (k)Pζ(k), (2.70) where Pζ(k) is the power spectrum of the uniform density curvature perturbation ζ and Tl(k), called the transfer function, comes from the solution of the line-of-sight equation, as we shall see in the next subsection. Ignoring the ISW term, we have Tl = TSW (k)jl(kr∗) + TD(k)j′l(kr∗) TSW ≡ (1/4δγ + Φ)∗ ζ(k) , TD(k) ≡ −(ve)∗ ζ(k) . (2.71) In equations above, jl and j′l are Bessel funcions and its derivatives. They act like delta functions mapping Fourier modes k to the harmonics moments l ∼ kr∗, with r∗ = τ0− τ∗ the distance from last scattering surface. So, for a scale invariant Pζ , Cl ∼ 4π (2l + 1)2 [T 2 SW (k) + T 2 D(k)]k∼l/r∗ (2.72) and the cross term TSW (k)TD(k) is negligible. 2.3.3 Sound waves in the photon-baryon fluid From the continuity and Euler equations for perturbations, we get: δ′′γ + R′ 1 +R δ′γ − c2 s∇2δγ = 4Ψ′′ + 4 3 ∇2Φ + R′ 1 +R Ψ′ (2.73) where R ≡ 3ρ̄b/4ρ̄γ and c2 s ≡ 1/3(1 + R). The metric potentials, Φ and Ψ are determined by Einstein’s equations as in the previous section. To gain some intuition, let us obtain approximate analytic results from this equation. 22 At early times, during radiation domination R� 1 and so let us consider R = 0. Ignoring time dilation terms, we get Θ̈− c2 s∇2Θ = 0, (2.74) where c2 s ≈ 1/3 and Θ ≡ δγ/4 + Φ. We have Θ(k, τ) = Ak cos(cskτ) +Bk sin(cskτ), (2.75) with Ak and Bk fixed by initial conditions. For adiabatic initial conditions, all perturbations in the limit τ → 0 are analytic function of k2, so4 Bk = 0. The coefficient Ak is obtained by matching with super-horizon data at the last scattering surface. Thus, Θ(k, τ∗) = ζ(k) 3 cos(cskτ∗) (2.76) This solution for the SW term already indicates the origin of the oscillations in the CMB power spectrum Cl, since TSW (k) ∼ cos(cskτ∗). Modes at the extrema of their oscillations, kn = nπ/(csτ∗), will produce enhanced fluc- tuations. These will correspond to peaks at multipoles of the fundamental scale k∗ ≡ π/s∗, where s∗ = csτ∗ ≈ τ∗/ √ 3 is the sound horizon at recombination. The mode k∗ corresponds to a characteristic angular scale: θ∗ = λ∗ DA , l∗ = k∗DA ≈ τ∗ τ0 , (2.77) where DA is the angular diameter distance from the last scattering. In a flat universe, DA = τ0 − τ∗ ≈ τ0. Assuming a purely matter dominated universe after recombination, τ ∝ a1/2, we have θ∗ ≈ ( 1 1100 )1/2 ≈ 2◦, l∗ ≈ 200. (2.78) Comparing with the actual CMB temperature power spectrum in Figure 2.1, we see that the first peak is indeed around l∗ ≈ 200. Observations of θ∗ are consistent with a flat universe. Let us consider the effects of baryons in the fluid. We have R ∝ a increasing with time until R ∼ O(1) at recombination. We now have: d2 dτ2 (Θ +RΦ)− 1 3 ∇2(Θ +RΦ) = 0, (2.79) where we have ignored the time variation of R. We see that the equilibrium point of oscillation shifts to Θequi = −RΦ. This leads to odd and even peaks in the CMB having unequal heights, because the Cl’s depends of the square of the solution. During the radiation era, the gravitational potential Φ decays inside horizon. So, for a mode that enters the sound horizon during radiation domination, the gravitational potential decays after horizon crossing and drives the acoustic amplitude higher. 4This is basically due to the fact that super-horizon modes enter the Hubble radius with vanishing velocity. 23 Figure 2.1: The cosmic microwave background radiation temperature power spectrum [16]. The combination l(l + 1)Cl/2π is plotted in the vertical axis. There is another effect we must consider: the photon-baryon fluid is not a perfect fluid. The coupling between photons and electrons is not perfect and the photons have a finite mean free path λC = 1 neσTa , (2.80) where ne is the electron density and σT is the Thomson cross section. This leads to a damping of small-scale fluctuations, known as Silk damping. The random walk of the photons through baryons will mix cold and hot regions and so fluctuations will be erased below the diffusion length, λD ≡ √ NλC = √ τ/λCλC = √ τλC . (2.81) This effectively generates viscosity in the fluid: Θ̈ + µc2 sk 2Θ̇ + c2 sk 2Θ = 0, µ ≡ λC ( 16 15 + R2 1 +R ) , (2.82) and so there will be exponential suppression for the modes with k > kD ≡ 2π/λD. Indeed, the photon transfer function receives the following correction: T (k)→ D(k)T (k), where D(k) = e−k 2/k2 D . (2.83) The approach taken on this subsection was a simplified description of what is actually done, the real world is more complex. In practice, there are codes to calculate the transfer function by solving a Boltzmann equation for the radiation distribution functions (for details, see chapter 11 in [14]). In the following section, we will see how inflation provides the initial condition for CMB anisotropies, ζ(k). 24 2.4 Cosmological Inflation 2.4.1 The flatness and horizon problems Despite its successes, the Big Bang cosmology does not explain why the Universe is so close to the flat geometry and what is the origin of the primordial inhomogeneity of the cosmic fluid, that stayed imprinted in the cosmic microwave background as observed on its anisotropy. There is still another problem related to causality due to the existence of horizons. Let us focus on the flatness problem first. One can write the Friedmann equations in terms of the energy density parameter Ω, defined as the ratio of ρ by the critical density (2.18), Ω = ρ ρc . (2.84) Then we have dΩ d ln a = (1 + 3w)Ω (Ω− 1) (2.85) Although Ω = 1 for flat geometry, in general Ω is not constant. For matter or radiation we have (1 + 3w) > 0 which leads to the conclusion that flat space is an unstable fixed point: if Ω begins smaller than 1 then it evolves for even smaller values; if Ω begins greater than 1 then it tends to increase even more. That is, we have d|Ω− 1| d ln a > 0, for (1 + 3w) > 0. (2.86) So, having an Universe close to the flat geometry today is a highly fine-tuned state. To give a quantitative idea, from the CMB measurements we know that at present time, |Ω0−1| < 0.02, at least. So, taking Ω0 = 1±0.05 implies that at the recombination we have Ωrec = 1±0.0004, and at the time of primordial nucleosynthesis, Ωnuc = 1± 10−12. This fine-tuned situation is called the flatness problem. Moving to the issue of causal structure of an expanding universe, let us work with confor- mal time τ . Using such coordinates, the FLRW metric is conformally related to the Minkowski metric and since a conformal factor does not affect the condition for having null geodesics, we can analyze causal structure as in Minkowski spacetime. Note that, by the definition of the particle horizon (2.11), we have dph(t) = ∫ t 0 dt′ a(t′) = ∫ τ(t) 0 dτ ′ = τ(t), (2.87) and thus the comoving particle horizon size is the age of the universe in conformal time. The expanding Universe has finite age, and therefore unlike the Minkowski spacetime we cannot prolong the light cone of an observer to infinite past: the light cone ”stops” at τ = 0. In this way, two points on the CMB at 180◦ apart in the sky will have past light cones that do not overlap. They ”stop” at the spatially infinite surface at τ = 0. Thus, these points are causally disconnected and there is no reason why such points reach the thermal equilibrium observed. 25 In quantitative terms, the particle horizon size at the time of the recombination trec is much smaller than the comoving distance that the radiation travels after decoupling:∫ trec 0 dt a(t) � ∫ t0 trec dt a(t) . (2.88) Therefore, there is no way to explain why the temperature of CMB is so isotropic within the standard cosmological evolution. This is the horizon problem. Let us again use the first Friedmann equation, but now in the form |Ω− 1| = |k| a2H2 . (2.89) In the Standard Cosmological model aH is always decreasing, and then Ω evolves away from flatness. But if we require that d dt ( H−1 a ) < 0, (2.90) which means that the Hubble length H−1 in comoving coordinates decreases with time, then Ω evolves towards flatness. This condition is equivalent to ä > 0, which is the definition of inflation, an primordial epoch of accelerated expansion. From equation (2.85) inflation is achieved if (1 + 3w) < 0, d|Ω− 1| d ln a < 0, for (1 + 3w) < 0. (2.91) and we see that this result implies inflation by looking at (2.17). Inflation solves the flatness problem automatically, but any model has to describe enough inflation in order not to lose the almost flat character after the end of inflation. From (2.90) it is possible to conclude that inflation solves also the horizon problem. Since the comoving Hubble length is shrinking during inflation, distances that can be seen before inflation are much larger than distances that can be seen after inflation. This is true for scales compared to the horizon too, that is, distances that are smaller than the horizon before inflation are ”red-shifted” to scales larger than the horizon after inflation. In other words, inflation is a mechanism to turn sub-horizon scales into super-horizon ones. As an example let us consider the de Sitter case, for which we have p = −ρ and d ln Ω d ln a = 2(1− Ω). (2.92) In this case, the conformal time is dτ = dt a(t) = exp (−Ht) dt, (2.93) and so, τ = − 1 aH . (2.94) Then, during inflation, the conformal time is negative and in this case the value τ = 0 represents the transition of inflationary expansion to the radiation dominated era. The two opposite points in the sky cited before can now share a causal past in the negative domain of the conformal time. 26 2.4.2 Inflation from scalar fields: The slow-roll inflation The de Sitter example of last section is an explicit picture of inflation. But the physics re- sponsible for accelerated expansion at early times cannot be Einstein’s cosmological constant, because it dominates over matter and radiation contributions at late times. We need a tran- sition between the inflation era and the radiation dominated phase, i.e., the ”vacuum” energy that drives inflation must be time dependent. For this purpose, inflation is implemented with fields, most simply with scalar fields. We are interested in models with single real scalar field, that in this case is called the inflaton. Thus, it is natural to begin with the action for the inflaton minimally coupled to gravity, S = ∫ d4x √ −g [ 1 16πG R− 1 2 gµν∂µφ∂νφ− V (φ) ] , (2.95) where V (φ) is the potential energy of the scalar field. Assuming flat geometry, the equation of motion for φ in a FLRW background is φ̈+ 3Hφ̇−∇2φ+ δV δφ = 0 (2.96) We are interested in an homogeneous field for which ∇φ = 0. The equation of motion simplifies to φ̈+ 3Hφ̇+ V ′(φ) = 0, (2.97) where the prime represents derivative with respect to the field. The energy-momentum tensor of the scalar field is Tµν = ∂µφ∂νφ− gµν ( 1 2 gρσ∂ρφ∂σφ+ V (φ) ) , (2.98) and for the homogeneous case it has the same form as the energy-momentum tensor for a perfect fluid, with energy and pressure densities given by ρ = 1 2 φ̇2 + V (φ), p = 1 2 φ̇2 − V (φ). (2.99) Then we see that the de Sitter limit p ' −ρ is the regime in which the potential energy dominates the kinetic energy, φ̇2 � V (φ). In this case the universe expands almost exponen- tially. The implementation with scalar fields and the last approximation frequently leads to a quasi-de Sitter universe. For this reason, it is convenient to define the number of e-folds N as dN ≡ −Hdt, (2.100) note that N decreases as time (and also the scale factor) increases. Inserting (2.99) into the Friedmann equations (2.14) and (2.15) gives H2 = 8πG 3 ( 1 2 φ̇2 + V (φ) ) , (2.101) 27 ( ä a ) = −4πG 3 (ρ+ 3p) ≡ H2(1− ε), (2.102) where we have defined the dimensionless parameter ε = 3 2 ( p ρ + 1 ) = 4πG ( φ̇ H )2 = − Ḣ H2 . (2.103) In the de Sitter limit, ε→ 0, the potential energy dominates the kinetic energy, and we have H2 ' 8πG 3 V (φ). (2.104) We will assume that the frictional term in the equation (2.97) dominates the ”acceleration” term φ̈: φ̈� 3Hφ̇ (2.105) and then the equation of motion for the scalar field is, approximately, 3Hφ̇+ V ′(φ) ' 0. (2.106) The assumption (2.105) can be written in terms of another dimensionless parameter |η| � 1, where we defined η ≡ − φ̈ Hφ̇ (2.107) Using equations (2.104) and (2.105) we can write the parameters ε and η approximately as ε = 4πG ( φ̇ H )2 ' 1 16πG ( V ′(φ) V (φ) )2 , η = − φ̈ Hφ̇ ' 1 8πG ( V ′′(φ) V (φ) ) . (2.108) The approximations (2.104) and (2.105) are called the slow-roll approximation and they are valid as long as the slow-roll parameters parameters ε and η, satisfy ε� 1, |η| � 1. (2.109) The first of these conditions tells us that the field has a very flat potential, V ′(φ) � V (φ) and the second slow roll condition informs that the curvature V ′′(φ) of the potential must be small. Using the definition of ε we can write the number of e-folds of inflation as an integral in the field space: N∗ = − ∫ Hdt = − ∫ H φ̇ dφ = 2 √ πG ∫ dφ√ ε ' 8πG ∫ φN∗ φe V (φ) V ′(φ) dφ, (2.110) where φe is the field value at the end of inflation, that is obtained by breaking the slow-roll approximation, ε(φe) = 1, and φN∗ is just an adjusted value that leads to a certain total 28 e-folding number N∗. The number of e-folds of inflation needed to solve the horizon and flatness problem comes from the condition of having present cosmological scales inside the horizon before inflation. It turns out that we need N∗ & 60. We now can put all this together and construct a qualitative picture of slow-roll inflation. At very early times the energy density of the Universe is dominated by the potential energy of a scalar field, the inflaton. The potential is nearly constant and inflation stands as long as the inflaton rolls down this region slowly, while the space expands almost exponentially. Inflation ends when the field enters a steeper and more curved region and the slow-roll conditions are not satisfied anymore. To obtain the standard Big Bang cosmology, the energy of the inflaton must decay into the Standard Model particles in a process that is called reheating. The theory of reheating is far from complete, as it depends on the extension of the Standard Model to high energies. This process is model dependent, the inflaton being or not a fundamental field. It is remarkable that only the potential energy of the inflaton is important for the physcis of slow roll inflation. Thus, the potential specifies the model. This single view of the dynamics of inflation driven by scalar fields is an effective rep- resentation of some underlying theory that is not yet established. That is one reason why trying to get inflaton potentials from String Theory has dominated the work of the String Cosmology community in these past years. In the next subsection, we discuss how inflation generates the primordial density pertur- bation, R or ζ. Though inflation is not the only model that explains the origin of ζ, it is the most popular one, since it solves other puzzles (horizon and flatness problems) at the same time. 2.4.3 Primordial curvature perturbation from inflation In addition to its success in solving the flatness and horizon problems, it is remarkable that the primordial density perturbation of the cosmic fluid can be obtained from inflation. This is done by considering the quantum nature of the inflaton. In essence, the primordial inhomo- geneity comes from quantum fluctuations of the inflaton. It is worthwhile to remember that it is this very primordial inhomogeneity that leads to the formation of the large structures of our Universe. It is this density perturbation that leads to the initial conditions for the inhomogeneities of cosmic radiation background, see e.g. (2.76). Indeed, from observations of the CMB at large scales, one can construct the power spectrum of this perturbation on the cosmic fluid. We begin by solving the Klein-Gordon equation for a free scalar field ϕ, that is not the inflaton and does not affect the scale factor. It is just an arbitrary scalar field in a fixed background. The equation of motion is (2.97), but with V = 0 and in conformal time coordinate, we have ϕ′′ + 2 ( a′ a ) ϕ′ −∇2ϕ = 0, (2.111) where now the prime indicates derivative with respect to the conformal time, ′ = d/dτ . 29 Consider its Fourier expansion in comoving wave vectors k: ϕ(τ,x) = ∫ d3k (2π)3/2 ( ϕk(τ)bke ik·x + ϕ̄k(τ)b̄ke −ik·x ) . (2.112) Then, the equation of motion for the mode ϕk is ϕ′′k + 2 ( a′ a ) ϕ′k + k2ϕk = 0. (2.113) Using the field redefinition uk ≡ a(τ)ϕk(τ) gives u′′k + ( k2 − a′′ a ) uk = 0. (2.114) We shall analyze this equation in two regimes: • Long wavelength limit, k � a′′/a. In this limit, we have a′′uk = au′′k (2.115) with solution uk ∝ a which implies that ϕk = const.. Then, the field modes ϕk are not dynamical, but have a non-zero amplitude. This is an example of mode freezing with the mode asymptoting to a constant; • Short wavelength limit, k � a′′/a. In this case, equation (2.114) reduces to the Klein- Gordon equation in Minkowski space, but in conformal coordinates, u′′k + k2uk = 0, (2.116) that has the solution uk(τ) = 1√ 2k ( Ake −ikτ +Bke ikτ ) . (2.117) This can be identified with the exact Minkowskian solution in the ultraviolet limit. Hence, all we need to do is set the boundary conditions on field perturbations in the ultraviolet limit (short wavelength limit). But in such limit we have a Minkowskian description of the fields, equation (2.116), and we know how to quantize fields in Minkowski spacetime. In fact, one of the conditions comes from quantization and the other comes from the vacuum selection, which is the same as in the Minkowski case, the Bunch-Davies vacuum. Together these conditions specify Ak and Bk. Under canonical quantization, the coefficients bk and b̄k of the expansion (2.112) are promoted to annihilation and creation operators with the commutation relation[ b̂k, b̂ † k′ ] = δ3(k − k′) (2.118) The quantum field ϕ̂(τ,x) is written in terms of these operators and has a canonical conjugate momentum Π̂(τ,x) = a2(t) ∂ϕ̂ ∂τ . (2.119) 30 The commutation relation [ ϕ̂(τ,x), Π̂(τ,x′) ] = iδ3(x− x′), (2.120) leads to uk ∂ūk ∂τ − ūk ∂uk ∂τ = i, (2.121) which gives, for the ultraviolet mode solution (2.117), |Ak|2 − |Bk|2 = 1. (2.122) The second condition on these constants comes from the vacuum selection. We choose the vacuum which gives the usual Minkowski vacuum in the ultraviolet limit: Ak = 1, Bk = 0→ uk(τ) ∝ e−ikτ . Now we can apply this analysis to perturbations of the inflaton field. We split the inflaton as the background value plus fluctuation parts φ(t, x) = φ(t) + ϕ(t, x), (2.123) where in the general case of a perturbed background we will assume a flat slicing in order to define ϕ. We need to take into account the evolution of the background spacetime that is encoded in a(τ). A good approximation is to consider the slow parameter ε as constant, but in general this is not the case. For an arbitrary equation of state parameter, we get τ = − ( 1 aH )( 1 1− ε ) , (2.124) and from (2.14) and (2.17) with flat geometry, we have a′′ a = a2H2(2− ε). (2.125) Then equation (2.114) is u′′k + [ k2 − a2H2(2− ε) ] uk = 0, (2.126) and using (2.124) to write aH in terms of the conformal time, we have τ2(1− ε)2u′′k + [ (kτ)2(1− ε)2 − (2− ε) ] uk = 0. (2.127) This is a Bessel equation and its solutions depends on the first and second Bessel functions, Jν and Yν , uk ∝ √ −kτ (Jν(−kτ)± iYν(−kτ)) , (2.128) where ν = 3− ε 2(1− ε) . (2.129) For the de Sitter case, ε = 0, the Bessel index is ν = 3/2 and the solution is uk ∝ ( kτ − i kτ ) e±ikτ . (2.130) 31 The short wavelength limit is kτ →∞. This can be seen from the equation (2.124), (−kτ)(1− ε) = k aH , (2.131) since the quantity (k/aH) represents the wavenumber k in units of the comoving Hubble size rH = (aH)−1, then the limit kτ → 0 corresponds to (k/aH)� 1, the long wavelength limit. Taking the short wavelength limit of the de Sitter solution and selecting the Bunch-Davies vacuum, uk ∝ eikτ , with the normalization fixed by the canonical quantization, we have uk = 1√ 2k e−ikτ . (2.132) Thus, the exact solution to the mode function of the inflaton in the de Sitter case is uk = 1√ 2k ( kτ − i kτ ) e−ikτ . (2.133) This solution is valid for all wavelengths, we have used the short limit just to find the proper normalization. Let us confirm the long wavelength limit, −kτ −→ 0: uk −→ 1√ 2k ( −i kτ ) = −i√ 2k ( aH k ) ∝ a. (2.134) Comparing with the solution of (2.115), we see the consistency with the vacuum choice. Therefore, the field amplitude ϕk is, in the long wavelength limit, |ϕk| = ∣∣∣uk a ∣∣∣ = H√ 2k3 = const. (2.135) The amplitude of the quantum fluctuations can be obtained from the two-point correlation function of the quantum field ϕ̂, 〈0|ϕ̂(τ,x)ϕ̂(τ,x′)|0〉 = ∫ d3k (2π)3 |ϕ|2eik·(x−x′) ≡ ∫ dk k Pϕ(k)eik·(x−x ′), (2.136) where we have defined the power spectrum for the quantum field fluctuations Pϕ(k) as Pϕ(k) ≡ ( k3 2π2 ) ∣∣∣uk a ∣∣∣2 = ( H 2π )2 . (2.137) Note that this power spectrum is scale independent, as H is constant. For a more general model, the spacetime is not exactly de Sitter and the power spectrum of the field fluctuations will only be approximately scale invariant. Since the initial density perturbations for CMB will be given by super-horizon modes, it is conventional to evaluate the power spectrum at aH = k, i.e., at horizon crossing. As mentioned before, the primordial density perturbation, that is responsible for struc- ture formation, is generated by the fluctuations of the inflaton. Using the general relativistic perturbation theory reviewed in section 2.2, it is possible to show that the power spectrum of 32 the primordial curvature perturbations ζ is related to the power spectrum of the inflaton fluc- tuations Pϕ(k). Indeed, using equation (2.49), on the flat slice where the inflaton fluctuations were defined, we have ζ = −aH ϕ φ′ , (2.138) and so the power spectrum of the density curvature perturbation ζ is Pζ = ( H φ̇ )2(H 2π )2 , (2.139) where the right hand side is evaluated at the horizon exit, k = aH. Using the slow-roll approximation (2.109), we have Pζ ' (8πG)2V 24π2ε . (2.140) The spectral index ns is a parameter that measures the departure from scale invariance, ns − 1 ≡ d lnPζ d ln k , (2.141) that is, the power spectrum can be parametrized as Pζ = Aζ ( k k0 )(ns−1) , (2.142) where k0 is some reference scale and Aζ is an amplitude. For ns = 1 we have scale invariance. It is possible to use the slow roll conditions to find ns − 1 = −6ε+ 2η. (2.143) Since in most models η � ε, the spectral index is smaller than 1. Thus Pζ receives a bigger contribution from modes with k < 1, and hence we say that the spectrum has a red tilt. The tensor perturbations hij are related to the production of gravitational waves. Since the transverse and longitudinal polarization states of the gravitational waves evolves as inde- pendent scalars, we can use the power spectrum of the inflaton fluctuations to calculate the power spectrum of the tensor perturbations as the sum of the 2 correlated functions for the separate polarizations, Ph = 2× (8πG) ( H 2π )2 , (2.144) where the factor (8πG) comes from the Einstein-Hilbert action normalization. This power spectrum has a spectral index nt = −2ε, and so it is also red-tilted. Then we have a consistent condition between the amplitudes of the power spectra Ph and Pζ , r = Ah Aζ ' 16ε. (2.145) As inflation is not the only mechanism to generate the scalar and tensor primordial per- turbations, their measurement alone are not a proof of inflation. Unfortunately, we have not detected primordial tensor perturbations yet, but it would be a very important check 33 of inflation as the consistency condition for r could then be verified. It could be however that the tensor perturbations spectrum is not red-tilted or even if it is, it does not satisfy (2.145). In this case we will have to look for alternatives models for generating the primordial perturbations. Therefore, for each model specified by the inflaton potential, we have three independent parameters to calculate: the amplitude of the scalar power spectra Aζ , the scalar spectral index ns and the tensor spectral index nt. Since tensor perturbation were not measured yet, we only have observational values for Aζ and ns and a constraint for r. From Planck 2018 cosmological parameter results [148], we have ns = 0.9649± 0.0042, r < 0.11, (2.146) Aζ ' 2.01× 10−9. 34 Chapter 3 Basics of String Theory The purpose of this chapter is to quickly review the foundations of String Theory. We will state several results without proof, but special attention will be devoted on the assumptions they stand on. Historically, String Theory was created as an attempt to explain phenomeno- logical results of hadronic physics in the late 60’s, that led to several amplitudes with nice analytical properties. But this goal was never completed and Quantum Chromodynamics turned up to be the right theory ruling strong interactions. Then in early 70’s, a new inter- pretation of the energy scale of the stringy process was taken and String Theory was first recognized as a potential theory of quantum gravity. After that, research in String Theory was conducted with the hope of getting a final theory that would describe all the interactions of nature, a Theory of Everything. In the following sections, we show how to get the string spectra and how to find low energy actions describing the behaviour of the massless fields that appear in the various kinds of superstrings. Type II theories have a special treatment, as solution to them are used to define the AdS/CFT correspondence that is used in chapter 6. General references for the present chapter are [24–27]. Double Field Theory was initially introduced in [119, 150] (for reviews see [120, 151]) and the discussion presented in section 3.4 was inspired by [152]. For further details on AdS/CFT see [153] and references therein. 3.1 Relativistic (or massless) quantum bosonic strings All modern