AN INTRODUCTION TO MATHEMATICAL COSMOLOGY

By D. KIRAO

CONTENTS 1 SOME BASIC CONCEPTS AND AN OVERVIEW OF COSMOLOGY 2 INTRODUCTION TO GENERAL RELATIVITY 2 1 SUMMARY OF GENERAL RELATIVITY 2 2 SOME SPECIAL TOPICS IN GENERAL RELATIVITY 2 2 1 KILLING VECTORS 2 2 2 TENSOR DENSITIES 2 2 3 GAUSS AND STOKES THEOREMS 2 2 4 THE ACTION PRINCIPLE FOR GRAVITATION 2 2 5 SOME FURTHER TOPICS 3 THE ROBERTSON–WALKER METRIC 3 1 A SIMPLE DERIVATION OF THE ROBERTSON–WALKER METRIC 3 2 SOME GEOMETRIC PROPERTIES OF THE ROBERTSON– WALKER METRIC 3 3 SOME KINEMATIC PROPERTIES OF THE ROBERTSON– WALKER METRIC 3 4 THE EINSTEIN EQUATIONS FOR THE ROBERTSON–WALKER METRIC 3 5 RIGOROUS DERIVATION OF THE ROBERTSON–WALKER METRIC 4 THE FRIEDMANN MODELS 4 1 INTRODUCTION 4 2 EXACT SOLUTION FOR ZERO PRESSURE 4 3 SOLUTION FOR PURE RADIATION 4 5 EXACT SOLUTION CONNECTING RADIATION AND MATTER ERAS 4 6 THE RED-SHIFT VERSUS DISTANCE RELATION 4 7 PARTICLE AND EVENT HORIZONS 5 THE HUBBLE CONSTANT AND THE DECELERATION PARAMETER 5 1 INTRODUCTION 5 4 FURTHER REMARKS ABOUT OBSERVATIONAL COSMOLOGY APPENDIX TO CHAPTER 6 MODELS WITH A COSMOLOGICAL CONSTANT 6 1 INTRODUCTION 6 2 FURTHER REMARKS ABOUT THE COSMOLOGICAL CONSTANT 6 3 LIMITS ON THE COSMOLOGICAL CONSTANT 6 4 SOME RECENT DEVELOPMENTS REGARDING THE COSMOLOGICAL CONSTANT AND RELATED MATTERS 6 4 1 INTRODUCTION 6 4 2 AN EXACT SOLUTION WITH COSMOLOGICAL CONSTANT 6 4 3 RESTRICTION OF PARAMETER SPACE 7 SINGULARITIES IN COSMOLOGY 7 1 INTRODUCTION 7 2 HOMOGENEOUS COSMOLOGIES 7 3 SOME RESULTS OF GENERAL RELATIVISTIC HYDRODYNAMICS 7 4 DE?NITION OF SINGULARITIES 7 5 AN EXAMPLE OF A SINGULARITY THEOREM 7 6 AN ANISOTROPIC MODEL 7 7 THE OSCILLATORY APPROACH TO SINGULARITIES 7 8 A SINGULARITY-FREE UNIVERSE? 8 THE EARLY UNIVERSE 8 1 INTRODUCTION 8 2 THE VERY EARLY UNIVERSE 8 3 EQUATIONS IN THE EARLY UNIVERSE 8 4 BLACK-BODY RADIATION AND THE TEMPERATURE OF THE EARLY UNIVERSE 8 5 EVOLUTION OF THE MASS-ENERGY DENSITY 8 6 NUCLEOSYNTHESIS IN THE EARLY UNIVERSE 8 7 FURTHER REMARKS ABOUT HELIUM AND DEUTERIUM 8 8 NEUTRINO TYPES AND MASSES 9 THE VERY EARLY UNIVERSE AND IN?ATION INTRODUCTION IN?ATIONARY MODELS – QUALITATIVE DISCUSSION IN?ATIONARY MODELS – QUANTITATIVE DESCRIPTION AN EXACT IN?ATIONARY SOLUTION FURTHER REMARKS ON IN?ATION MORE IN?ATIONARY SOLUTIONS 10 QUANTUM COSMOLOGY HAMILTONIAN FORMALISM THE SCHRÖDINGER FUNCTIONAL EQUATION FOR A SCALAR ?ELD A FUNCTIONAL DI?ERENTIAL EQUATION SOLUTION FOR A SCALAR ?ELD THE FREE ELECTROMAGNETIC ?ELD THE WHEELER–DE WITT EQUATION PATH INTEGRALS CONFORMAL ?UCTUATIONS 10 10 FURTHER REMARKS ABOUT QUANTUM COSMOLOGY 11 THE DISTANT FUTURE OF THE UNIVERSE THREE WAYS FOR A STAR TO DIE GALACTIC AND SUPERGALACTIC BLACK HOLES BLACK-HOLE EVAPORATION SLOW AND SUBTLE CHANGES A COLLAPSING UNIVERSE