PRIMORDIAL COSMOLOGY

Transcription

1 PRIMORDIAL COSMOLOGY Giovanni Montani ENEA - C.R. Frascati, ICFtANet and Dipartimento di Fisica, Universita di Roma "Sapienza", Italy Marco Valerio Battisti Dipartimento di Fisica, Universita di Roma "Sapienza", Italy and Centre de Physique Theorique, Luminy, Marseille, France Riccardo Benini ICRA and Dipartimento di Fisica," Universita di Roma "Sapienza", Italy Giovanni Imponente Queen Mary, University of London, UK World Scientific JERSEY. LONDON SINGAPORE- BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI

2 Contents Preface List of Figures vii xxiii Historical and Basic Notions 1 1. Historical Picture 3 ''' 1.1 The Concept of Universe Through the Centuries... 3 < The ancient cultures... 3 i Ancient Greek and the Mediterranean 5 0" The Hellenistic era '.' I The philosophical point of view in the Old Age.. 11 I., " The Middle Age dogmas The Renaissance revolution ; The Scientific Revolution The Enlightenment Era The XIX Century Knowledge Geometrical formalisms Difficulties for the birth of a real cosmology: 01- bers' paradox / Luigi Bianchi and the developments of differential geometry Einstein vision of space-time Birth of Scientific Cosmology. 30 ' ' Einstein proposal of a static Universe... 31

3 dv f Primordial Cosmology Galaxies and their expansion: The Hubble's dis- ^ covery 1,4 The Genesis of the Hot Big Bang Model Recent developments' ^Discovery of the acceleration Generic nature of the cosmological singularity: The Cambridge and the Landau School The inflationary paradigm The idea of non-singular cosmology: The cyclic Universe and the Big Bounce 1.5 Guidelines to the Literature ; Fundamental Taols.' Einstein Equations Matter Fields.. : Perfect fluid Scalar field Electromagnetic field Yang-Mills fields and 9-sector Hamiltonian Formulation of the Dynamics Canonical General Relativity Hamilton-Jacobi equations for gravitational field The ADM reduction of the dynamics Synchronous Reference System. 2.5 Tetradic Formalism Gauge-like Formulation of GR Lagrangian formulation..., Hamiltonian formulation On the gauge group of GR Singularity Theorems...' Definition of a space-time singularity Fluid kinematics 273 The Raychaudhuri equation.. QO Singularity Theorems 2.8 Guidelines to the Literature... 64

4 Contents xv Physical Cosmology The Structure and Dynamics of the Isotropic Universe The RW Geometry Definition of isotropy Kinematics of the isotropic Universe The particle motion The Hubble law The Hubble length and the cosmological horizon Kinetic theory and thermodynamics in the expanding Universe: The hot Big Bang The FRW Cosmology....* Field equations for the isotropic Universe Asymptotic solution toward the Big Bang The de Sitter Solution Hamiltonian dynamics of the isotropic Universe Dissipative Cosmologies Bulk viscosity Matter creation in the expanding Universe Inhomogeneous Fluctuations in the Universe The meaning of cosmological perturbations The Jeans length in a static uniform fluid The Jeans length in an expanding Universe General Relativistic Perturbation Theory Perturbed Einstein equations Scalar-vector-tensor decomposition and Fourier expansion Perturbed conservation equations Gauge modes., Evolution of scalar modes.' 147 ; Adiabatic and isocurvature perturbations Imperfect fluids Kinetic theory The Lemaitre-Tolmann-Bondi Spherical Solution Guidelines to the Literature...'...' , Features of the Observed Universe 165 '- 4.1 Current Status: The Concordance Model. 166

5 xvi f Primordial Cosmology 4.2 The Large-Scale Structure Deviations from homogeneity Dark matter. ' The power spectrum of density fluctuations The Acceleration of the Universe The Cosmic Microwave Background Sources of anisotropy The power spectrum of CMB anisotropies Acoustic oscillations Effect of the cosmological parameters Guidelines to the Literature The Theory of Inflation The Shortcomings of the Standard Cosmology The horizon and flatness paradoxes The entropy problem and the unwanted relics paradox The Inflationary Paradigm Spontaneous symmetry breaking and the Higgs phenomenon Presence of a Self-interacting Scalar Field Coupling of the scalar field with the thermal bath Inflationary Dynamics Slow-rolling phase The reheating phase Solution to the Shortcomings of the Standard Cosmology Solution to the horizon and flatness paradoxes Solution to the entropy problem and to the unwanted relics paradox General Features : Slow-rolling phase Reheating phase '...: The Coleman-Weinberg model Genesis of the seeds for structure formation Possible Explanations for the Present Acceleration of the Universe, Dark energy Modified gravity theory Guidelines to the Literature 239

6 Contents xvii 6. Inhomogeneous Quasi-isotropic Cosmologies Quasi-isotropic Solution The Presence of Ultrarelativistic Matter The Role of a Massless Scalar Field The Role of an Electromagnetic Field Quasi-isotropic Inflation Geometry, matter and scalar field equations Inflationary dynamics Physical considerations Quasi-isotropic Viscous Solution Form of the energy density Comments on the adopted paradigm Solutions of the 00-Einstein and hydrodynamical equations The velocity and the three-metric Guidelines to the Literature...', '. 274 Mathematical Cosmology Homogeneous Universes Homogeneous Cosmological Models Definition of homogeneity Application to Cosmology Bianchi Classification and Line Element 286 ' ' ' 7.2 Kasner Solution 288? The role of matter 291 ' 7.3 The Dynamics of the Bianchi Models Bianchi type II: The concept of Kasner epoch.. 294! ' Bianchi type VII: The concept of Kasner era ; 7.4 Bianchi Types VIII and DC Models The oscillatory regime Stochastic properties and the Gaussian distribution Small oscillations 304 '' 7.5 Dynamical Systems Approach Equations for orthogonal Bianchi class A models. 310,". < The Bianchi I model and the Kasner circle ^ <' The Bianchi II model in vacuum. 315

7 xviii Primordial Cosmology The Bianchi IX model and the Mixmaster attractor theorem Multidimensional Homogeneous Universes On the non-diagonal cases Guidelines to the Literature Hamiltonian Formulation of the Mixmaster Hamiltonian Formulation of the Dynamics.: The Mixmaster Model in the Misner Variables Metric reparametrization Kasner solution, Lagrangian formulation Reduced ADM Hamiltonian Mixmaster dynamics Misner-Chitre Like Variables The Jacobi metric and the billiard representation Some remarks on the billiard representation The Invariant Liouville Measure Invariant Lyapunov Exponent Chaos Covariance.. ; Shortcomings of Lyapunov exponents On the occurrence of fractal basin Cosmological Chaos as a Dimensional and Matter Dependent Phenomenon 353 k The role of a scalar field The role of a vector field." Isotropization Mechanism ' Guidelines to the Literature The Generic Cosmological Solution Near the Singularity Inhomogeneous Perturbations of Bianchi IX Formulation of the Generic Cosmological Problem The Generalized Kasner solution Inhomogeneous BKL solution Rotation of the Kasner axes : The Fragmentation Process Physical meaning of the BKL conjecture The Generic Cosmological Solution in Misner Variables. 375

8 Contents xix 9.5 Hamiltonian Formulation in a General Framework Local dynamics.." Dynamics of inhomogeneities The Generic Cosmological Problem in the Iwasawa Variables Asymptotic freezing of the Iwasawa variables Cosmological billiards Multidimensional Oscillatory Regime Dilatons, p-forms and Kac-Moody algebras Properties of the BKL Map Parametrization in a generic number of dimensions Ordering properties Properties of the BKL map in the v space Guidelines to the Literature 398 Quantum Cosmology Standard Quantum Cosmology Quantum Geometrodynamics The Wheeler-DeWitt Theory Relation with the Path Integral Quantization The Problem of Time Time before quantization Time after quantization Timeless physics 419! ' 10.3 What is Quantum Cosmology? Minisuperspace models Interpretation of the theory Quantum singularity avoidance Path Integral in the Minisuperspace 427 ' 10.5 Scalar Field as Relational Time Interpretation of the Wave Function of the Universe The semiclassical approximation An example: A quantum mechanism for the isotropization of the Universe Boundary Conditions 441 * No-boundary proposal Tunneling proposal 445

9 XX Primordial Cosmology Comparison between the two approaches Quantization of the FRW Model Filled with a Scalar Field The Poincare Half Plane -J Quantum Dynamics of the Taub Universe Classical framework Quantum framework Quantization of the Mixmaster in the Misner Picture The Quantum Mixmaster in the Poincare Half Plane Continuity equation and the Liouville theorem Schrodinger dynamics Eigenfunctions and the vacuum state Properties of the spectrum Guidelines to the Literature Generalized Approaches to Quantum Mechanics The Algebraic Approach Basic elements GNS construction and Fell theorem Polymer Quantum Mechanics From Schrodinger to polymer representation Kinematics Dynamics 11 3 On the Existence of a Fundamental Scale String Theory and Generalized Uncertainty Principle Heisenberg Algebras in Non-Commutative Snyder Space- Time f a ] 11.6 Quantum Mechanics in the GUP Framework Guidelines to the Literature ~' Modern Quantum Cosmology 12.1 Loop Quantum Gravity Kinematics Implementation of the constraints Quantum constraints algebra Loop Quantum Cosmology 51i Kinematics Quantum dynamics and Big Bounce Effective classical dynamics

10 Contents. xxi 12.3 Mixmaster Universe in LQC Loop quantum Bianchi IX Effective dynamics Triangulated Loop Quantum Cosmology The triangulated model Isotropic sector: FRW Anisotropic sector: Bianchi LX Full dipole model Quantization of the model Snyder-Deformed Quantum Cosmology GUP and Polymer Quantum Cosmology: The Taub Universe Deformed classical dynamics Deformed quantum dynamics Mixmaster Universe in the GUP Approach Guidelines to the Literature 550 Bibliography 553 Index 585