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Cosmological Models from String Theory Setups

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Cosmological Models from String Theory Setups Instituto de F´ısicaTe´orica 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 A` toda minha fam´ılia,principalmente minha querida m~aeSandra e meu irm~aoAdson, por sempre me apoiarem e torcerem pelo meu sucesso. Eu amo voc^es. Me desculpem pela aus^encia. A` minha parceira na vida, na f´ısicae em todo o resto, minha querida J´essica.Obrigado pelo suporte ao longo destes anos e por sempre estar l´apra 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: Sim~ao e fam´ılia,que j´afazem parte da minha hist´oria;Rodrigo, Nicolas, Marcella, Anderson e a galera do cl~a,que deixam tudo mais alegre e que trazem algumas das melhores lembran¸cas que tenho; M´arcioe fam´ılia,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; B´arbarae Isabela pelas horas de conversas e "zueiras". Aos colegas que encontrei devido `aF´ısicae que agora s~aotamb´embons amigos: Priesley, Matheus, Henrique, Ana, Lu´ıs, Max e Carol; Daniel, pelas boas conversas sobre f´ısica,vida na academia e pelas horas de discuss~oessobre String Theory; Renato, por ser sempre sincero e pelos conselhos ao longo desses anos; Elisa, por me acolher t~aobem na McGill e me ajudar em tudo que podia; Guilherme, por partilhar sua vis~aogeral sobre f´ısica e filosofia. Aos meu professores, que ajudaram a moldar o que sou hoje, Jailson, M´arcia,Solange e a todos do CEd 05 que sempre me apoiaram; Profs. Arsen, Olavo, Cl´ovis,por me fornecerem a base que precisava para fazer f´ısica;Profs. Matsas, Nathan, Rog´erioe Eduardo, por todas as discuss~oes sobre f´ısica;Agrade¸coespecialmente `aLiliana, por olhar atrav´esde um simples aluno e Val´eria,que j´a´emais que minha mestra e que acreditava em mim quando mesmo eu n~aoo 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. Agrade¸co,sinceramente, `aCAPES pelo apoio financeiro durante o doutorado. i Resumo Nesta tese, discutimos tr^esmodelos cosmol´ogicosque s~aobaseados direta ou indiretamente em ideias advindas de Teoria das Cordas. Depois de uma revis~aogeral de Cosmologia em Teoria das Cordas, um resumo de Cosmologia e Teoria das Cordas ´eapresentado, com ^enfase nos conceitos fundamentais e te´oricos.Ent~aodescrevemos como o acoplamento camale^onico pode potencialmente afetar as predi¸c~oesde infla¸c~aoc´osmicacom campo ´unico, com trata- mento cuidadoso dos modos de perturba¸c~aocosmol´ogicaadiab´aticose de entropia. Al´em disso uma nova abordagem para a dualidade-T em solu¸c~oes cosmol´ogicasde supergravidade bos^onica´ediscutida no contexto de Teoria Dupla de Campos. Por fim, propomos uma nova prescri¸c~aopara o mapa hologr´aficoem cosmologia que pode ser usado para conectar modelos fundamentais de cosmologia hologr´aficacom outras abordagens fenomenol´ogicas. Palavras Chaves: Cosmologia em Teoria das Cordas; Infla¸c~aoC´osmicaCamale^onica; Dualidade-T em Cosmologia; Cosmologia Hologr´afica. Areas´ do conhecimento: Cosmologia Te´orica; 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 Introduction1 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. 5 AdS5 × S 106 6.1 Motivation: Strong Gravity in the Very early universe............. 106 6.2 Holographic cosmology paradigm......................... 107 5 6.3 Top-down model from dimensional reduction of N = 4 SYM vs. AdS5 × S . 109 6.3.1 Transformation of dilaton and operator VEV.............
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