An Introduction to Mathematical Cosmology, Second Edition - J

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.

AN INTRODUCTION TO MATHEMATICAL COSMOLOGY Second edition

J. N. ISLAM

Research Centre for Mathematical and Physical Sciences, University of Chittagong, Bangladesh

          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

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

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

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

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

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

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. For more than three decades she has been friend, philosopher and mentor for me and my wife and in recent years a very affectionate godmother (‘Goddy’) to my daughters. This book is fondly dedicated to this remarkable person.

Jamal Nazrul Islam Chittagong, 1991

Preface to the second edition

The material in the earlier edition, to which there appears to have been a favourable response, has been kept intact as far as possible in this new edition except for minor changes. A number of new additions have been made. Some standard topics have been added to the introduction to general relativity, such as Killing vectors. Not all these topics are used later in the book, but some may be of use to the beginning student for mathematical aspects of cosmological studies. Observational aspects have been brought up to date in an extended chapter on the cosmological constant. As this is a book on mathematical cosmology, the treatment of observations is not deﬁnitive or exhaustive by any means, but hopefully it is adequate. To clarify the role of the cosmological constant, much discussed in recent years, an exact, somewhat unusual solution with cosmological constant is included. Whether the solution is new is not clear: it is meant to provide a ‘comprehension exercise’. One reviewer of the earlier edition wondered why the Hubble constant and the deceleration parameter were chosen for a separate chapter. I believe these two parameters are among the most important in cosmology; adequate understanding of these helps to assess observations generally. Within the last year or two, through anal- yses of supernovae in distant galaxies, evidence seems to be emerging that the universe may be accelerating, or at least the deceleration may be not as much as was supposed earlier. If indeed the universe is accelerating, the nomenclature ‘deceleration parameter’ may be called into question. In any case, much more work has to be done, both observational and theoretical, to clarify the situation and it is probably better to retain the term, and refer to a possible acceleration as due to a ‘negative deceleration parameter’ (in case one has to revert back to ‘deceleration’!). I believe it makes sense, in most if not all subjects, constantly to refer back to earlier work, observational, experimental or practical, as well as theoretical aspects, for

xii Preface to the second edition this helps to point to new directions and to assess new developments. Some of the material retained from the first edition could be viewed in this way. A new exact inflationary solution for a sixth degree potential has been added to the chapter on the very early universe. The chapter on quantum cosmology is extended to include a discussion on functional differential equations, material which is not readily available. This topic is relevant for an understanding of the Wheeler–De Witt equation. Some additional topics and comments are considered in the Appendix at the end of the book. Needless to say, in the limited size and scope of the book an exhaustive treatment of any topic is not possible, but we hope enough ground has been covered for the serious student of cosmology to benefit from it. As this book was going to press, Fred Hoyle passed away. Notwith- standing the controversies he was involved in, I believe Hoyle was one of the greatest contributors to cosmology in the twentieth century. The controversies, more often than not, led to important advances. Hoyle’s predic- tion of a certain energy level of the carbon nucleus, revealed through his studies of nucleosynthesis, confirmed later in the laboratory, was an out- standing scientific achievement. A significant part of my knowledge of cosmology, for what it is worth, was acquired through my association with the then Institute of Theoretical Astronomy at Cambridge, of which the Founder-Director was Hoyle, who was kind enough to give me an appoint- ment for some years. I shall always remember this with gratitude. I am grateful to Clare Hall, Cambridge, for providing facilities where the manuscript and proofs were completed. I am grateful for helpful comments by various CUP readers and refer- ees, although it has not been possible to incorporate all their suggestions. I thank the various reviewers of the earlier edition for useful comments. I am grateful to Simon Mitton, Rufus Neal, Adam Black and Tamsin van Essen for cooperation and help at various stages in the preparation of this edition. I thank ‘the three women in my life’ (Suraiya, Sadaf and Nargis) and my son-in-law Kamel for support and encouragement.

Jamal Nazrul Islam Chittagong, November 2000

IN MEMORIAM Mary Wraith (1908–1995) in aﬀection, admiration and gratitude