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Cambridge University Press 0521496500 - An Introduction to Mathematical Cosmology, Second Edition - J. N. Islam Frontmatter More information AN INTRODUCTION TO MATHEMATICAL COSMOLOGY This book provides a concise introduction to the mathematical aspects of the origin, structure and evolution of the universe. The book begins with a brief overview of observational and theoretical cosmology, along with a short introduction to general relativity. It then goes on to discuss Friedmann models, the Hubble constant and deceleration parameter, singularities, the early universe, inflation, quantum cosmology and the distant future of the universe. This new edition contains a rigorous derivation of the Robertson–Walker metric. It also discusses the limits to the parameter space through various theoretical and observational constraints, and presents a new inflationary solution for a sixth degree potential. This book is suitable as a textbook for advanced undergradu- ates and beginning graduate students. It will also be of interest to cosmologists, astrophysicists, applied mathematicians and mathematical physicists. received his PhD and ScD from the University of Cambridge. In 1984 he became Professor of Mathematics at the University of Chittagong, Bangladesh, and is currently Director of the Research Centre for Mathematical and Physical Sciences, University of Chittagong. Professor Islam has held research positions in university departments and institutes throughout the world, and has published numerous papers on quantum field theory, general relativity and cosmology. He has also written and contributed to several books. © Cambridge University Press www.cambridge.org Cambridge University Press 0521496500 - An Introduction to Mathematical Cosmology, Second Edition - J. N. Islam Frontmatter More information AN INTRODUCTION TO MATHEMATICAL COSMOLOGY Second edition J. N. ISLAM Research Centre for Mathematical and Physical Sciences, University of Chittagong, Bangladesh © Cambridge University Press www.cambridge.org Cambridge University Press 0521496500 - An Introduction to Mathematical Cosmology, Second Edition - J. N. Islam Frontmatter More information The Pitt Building, Trumpington Street, Cambridge, United Kingdom The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcón 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http: //www.cambridge.org © Cambridge University Press 1992, 2002 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1992 Second edition 2002 Printed in the United Kingdom at the University Press, Cambridge Typeface MT Times NR 10/13pt. System QuarkXPress® [] A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Islam, Jamal N. An introduction to mathematical cosmology/J. N. Islam.–2nd ed. p. cm. Includes bibliographical references and index. ISBN 0 521 49650 0 – ISBN 0 521 49973 9 (pb.) 1. Cosmology–Mathematics. I. Title. QB981 .I73 2001 523.1Ј01Ј51–dc21 2001025711 ISBN 0 521 49650 0 hardback ISBN 0 521 49973 9 paperback © Cambridge University Press www.cambridge.org Cambridge University Press 0521496500 - An Introduction to Mathematical Cosmology, Second Edition - J. N. Islam Frontmatter More information Contents Preface to the first edition page ix Preface to the second edition xi 1 Some basic concepts and an overview of cosmology 1 2 Introduction to general relativity 12 2.1 Summary of general relativity 12 2.2 Some special topics in general relativity 18 2.2.1 Killing vectors 18 2.2.2 Tensor densities 21 2.2.3 Gauss and Stokes theorems 24 2.2.4 The action principle for gravitation 28 2.2.5 Some further topics 32 3 The Robertson–Walker metric 37 3.1 A simple derivation of the Robertson–Walker metric 37 3.2 Some geometric properties of the Robertson– Walker metric 42 3.3 Some kinematic properties of the Robertson– Walker metric 45 3.4 The Einstein equations for the Robertson–Walker metric 51 3.5 Rigorous derivation of the Robertson–Walker metric 53 4 The Friedmann models 60 4.1 Introduction 60 4.2 Exact solution for zero pressure 64 4.3 Solution for pure radiation 67 4.4 Behaviour near tϭ068 4.5 Exact solution connecting radiation and matter eras 68 v © Cambridge University Press www.cambridge.org Cambridge University Press 0521496500 - An Introduction to Mathematical Cosmology, Second Edition - J. N. Islam Frontmatter More information vi Contents 4.6 The red-shift versus distance relation 71 4.7 Particle and event horizons 73 5 The Hubble constant and the deceleration parameter 76 5.1 Introduction 76 5.2 Measurement of H0 77 5.3 Measurement of q0 80 5.4 Further remarks about observational cosmology 85 Appendix to Chapter 5 90 6 Models with a cosmological constant 94 6.1 Introduction 94 6.2 Further remarks about the cosmological constant 98 6.3 Limits on the cosmological constant 100 6.4 Some recent developments regarding the cosmological constant and related matters 102 6.4.1 Introduction 102 6.4.2 An exact solution with cosmological constant 104 6.4.3 Restriction of parameter space 107 7 Singularities in cosmology 112 7.1 Introduction 112 7.2 Homogeneous cosmologies 113 7.3 Some results of general relativistic hydrodynamics 115 7.4 Definition of singularities 118 7.5 An example of a singularity theorem 120 7.6 An anisotropic model 121 7.7 The oscillatory approach to singularities 122 7.8 A singularity-free universe? 126 8 The early universe 128 8.1 Introduction 128 8.2 The very early universe 135 8.3 Equations in the early universe 142 8.4 Black-body radiation and the temperature of the early universe 143 8.5 Evolution of the mass-energy density 148 8.6 Nucleosynthesis in the early universe 153 8.7 Further remarks about helium and deuterium 159 8.8 Neutrino types and masses 164 © Cambridge University Press www.cambridge.org Cambridge University Press 0521496500 - An Introduction to Mathematical Cosmology, Second Edition - J. N. Islam Frontmatter More information Contents vii 9 The very early universe and inflation 166 9.1 Introduction 166 9.2 Inflationary models – qualitative discussion 167 9.3 Inflationary models – quantitative description 174 9.4 An exact inflationary solution 178 9.5 Further remarks on inflation 180 9.6 More inflationary solutions 183 Appendix to Chapter 9 186 10 Quantum cosmology 189 10.1 Introduction 189 10.2 Hamiltonian formalism 191 10.3 The Schrödinger functional equation for a scalar field 195 10.4 A functional differential equation 197 10.5 Solution for a scalar field 199 10.6 The free electromagnetic field 199 10.7 The Wheeler–De Witt equation 201 10.8 Path integrals 202 10.9 Conformal fluctuations 206 10.10 Further remarks about quantum cosmology 209 11 The distant future of the universe 211 11.1 Introduction 211 11.2 Three ways for a star to die 211 11.3 Galactic and supergalactic black holes 213 11.4 Black-hole evaporation 215 11.5 Slow and subtle changes 216 11.6 A collapsing universe 218 Appendix 220 Bibliography 238 Index 247 © Cambridge University Press www.cambridge.org Cambridge University Press 0521496500 - An Introduction to Mathematical Cosmology, Second Edition - J. N. Islam Frontmatter More information Preface to the first edition Ever since I wrote my semi-popular book The Ultimate Fate of the Universe I have been meaning to write a technical version of it. There are of course many good books on cosmology and it seemed doubtful to me whether the inclusion of a chapter on the distant future of the universe would itself justify another book. However, in recent years there have been two interesting developments in cosmology, namely inflationary models and quantum cosmology, with their connection with particle physics and quantum mechanics, and I believe the time is ripe for a book containing these topics. Accordingly, this book has a chapter each on inflationary models, quantum cosmology and the distant future of the universe (as well as a chapter on singularities not usually contained in the standard texts). This is essentially an introductory book. None of the topics dealt with have been treated exhaustively. However, I have tried to include enough introductory material and references so that the reader can pursue the topic of his interest further. A knowledge of general relativity is helpful; I have included a brief exposition of it in Chapter 2 for those who are not familiar with it. This material is very standard; the form given here is taken essentially from my book Rotating Fields in General Relativity. In the process of writing this book, I discovered two exact cosmological solutions, one connecting radiation and matter dominated eras and the other representing an inflationary model for a sixth degree potential. These have been included in Sections 4.5 and 9.4 respectively as I believe they are new and have some physical relevance. I am grateful to J. V. Narlikar and M. J. Rees for providing some useful references. I am indebted to a Cambridge University Press reader for helpful comments; the portion on observational cosmology has I believe improved considerably as a result of these comments. I am grateful to ix © Cambridge University Press www.cambridge.org Cambridge University Press 0521496500 - An Introduction to Mathematical Cosmology, Second Edition - J. N. Islam Frontmatter More information x Preface to the first edition F. J. Dyson for his ideas included in the last chapter. I thank Maureen Storey of Cambridge University Press for her efficient and constructive subediting. I am grateful to my wife Suraiya and daughters Nargis and Sadaf and my son-in-law Kamel for support and encouragement during the period this book was written. I have discussed plans for my books with Mrs Mary Wraith, who kindly typed the manuscript for my first book.