com An Introduction to Modern Cosmology Second Edition www.com This page intentionally left blank www.com An Introduction To Modern Cosmology Second Edition Andrew Liddle University of Sussex, UK WILEY www.com Copyright © 2003 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.uk Visit our Home Page on www.com or www.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Cour Road, London WIT 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ. England, or emailed to permreq@wiley.uk, or faxed to (+44) 1243 770571.
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12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 dementi Loop #02-01, |in Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes in books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Cataloging-in-Publication Data (to follow) British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0470 84834 0 Cloth 047084835 9 Paper Produced from author's LaTeX files Printed and bound in Great Britain by Antony Rowe Ltd., Chippenham, Wilts This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.com To my grandmothers www.com This page intentionally left blank www.com Contents Preface xi Constants, conversion factors and symbols xiv 1 A (Very) Brief History of Cosmological Ideas 1 2 Observational Overview 3 2.1 In visible light 3 2.2 In other wavebands 7 2.3 Homogeneity and isotropy 8 2.4 The expansion of the Universe 9 2.5 Particles in the Universe 11 2.1 What particles are there? 11 2.2 Thermal distributions and the black-body spectrum 13 3 Newtonian Gravity 17 3.1 The Friedmann equation 18 3.2 On the meaning of the expansion 21 3.3 Things that go faster than light 21 3.4 The fluid equation 22 3.5 The acceleration equation 23 3.6 On mass, energy and vanishing factors of c2 24 4 The Geometry of the Universe 25 4.4 Infinite and observable Universes 29 4.5 Where did the Big Bang happen? 29 4.6 Three values of k 30 5 Simple Cosmological Models 33 5.2 Expansion and redshift 34 5.3 Solving the equations 35 www.com viii CONTENTS 5.4 Particle number densities 39 5.5 Evolution including curvature 40 6 Observational Parameters 45 6.1 The expansion rate HO 45 6.2 The density parameter Q0 47 6.3 The deceleration parameter QQ 48 7 The Cosmological Constant 51 7.2 Fluid description of A 52 7.3 Cosmological models with A 53 8 The Age of the Universe 57 9 The Density of the Universe and Dark Matter 63 9.1 Weighing the Universe 63 9.3 Galaxy rotation curves 64 9.4 Galaxy cluster composition 66 9.5 Bulk motions in the Universe 67 9.6 The formation of structure 68 9.7 The geometry of the Universe and the brightness of supernovae .2 What might the dark matter be? 69 9.3 Dark matter searches 72 10 The Cosmic Microwave Background 75 10.1 Properties of the microwave background 75 10.2 The photon to baryon ratio 77 10.3 The origin of the microwave background 78 10.4 The origin of the microwave background (advanced) 81 11 The Early Universe 85 12 Nucleosynthesis: The Origin of the Light Elements 91 12.1 Hydrogen and Helium 91 12.2 Comparing with observations 94 12.3 Contrasting decoupling and nucleosynthesis 96 www.com CONTENTS ix 13 The Inflationary Universe 99 13.1 Problems with the Hot Big Bang 99 13.1 The flatness problem 99 13.2 The horizon problem 101 13.3 Relic particle abundances 102 13.3 Solving the Big Bang problems 104 13.1 The flatness problem 104 13.2 The horizon problem 105 13.3 Relic particle abundances 106 13.4 How much inflation? 106 13.5 Inflation and particle physics 107 14 The Initial Singularity 111 15 Overview: The Standard Cosmological Model 115 Advanced Topic 1 General Relativistic Cosmology 119 1.1 The metric of space-time 119 1.2 The Einstein equations 120 1.3 Aside: Topology of the Universe 122 Advanced Topic 2 Classic Cosmology: Distances and Luminosities 125 2.1 Light propagation and redshift 125 2.2 The observable Universe 128 2.4 Angular diameter distance 132 2.5 Source counts 134 Advanced Topic 3 Neutrino Cosmology 137 3.1 The massless case 137 3.3 Neutrinos and structure formation 140 Advanced Topic 4 Baryogenesis 143 Advanced Topic 5 Structures in the Universe 147 5.1 The observed structures 147 5.3 The clustering of galaxies 150 5.4 Cosmic microwave background anisotropies 152 5.1 Statistical description of anisotropies 152 5.2 Computing the Ct 154 5.3 Microwave background observations 155 5.4 Spatial geometry 156 www.5 The origin of structure 157 Bibliography 161 Numerical answers and hints to problems 163 Index 167 www.com Preface The development of cosmology will no doubt be seen as one of the scientific triumphs of the twentieth century. At its beginning, cosmology hardly existed as a scientific discipline.
By its end, the Hot Big Bang cosmology stood secure as the accepted description of the Universe as a whole. Telescopes such as the Hubble Space Telescope are capable of seeing light from galaxies so distant that the light has been travelling towards us for most of the lifetime of the Universe. The cosmic microwave background, a fossil relic of a time when the Universe was both denser and hotter, is routinely detected and its properties examined. That our Universe is presently expanding is established without doubt.
We are presently in an era where understanding of cosmology is shifting from the qualitative to the quantitative, as rapidly-improving observational technology drives our knowledge forward. The turn of the millennium saw the establishment of what has come to be known as the Standard Cosmological Model, representing an almost universal con- sensus amongst cosmologists as to the best description of our Universe. Nevertheless, it is a model with a major surprise — the belief that our Universe is presently experiencing ac- celerated expansion. Add to that ongoing mysteries such as the properties of the so-called dark matter, which is believed to be the dominant form of matter in the Universe, and it is clear that we have some way to go before we can say that a full picture of the physics of the Universe is in our grasp.
Such a bold endeavour as cosmology easily captures the imagination, and over recent years there has been increasing demand for cosmology to be taught at university in an accessible manner. Traditionally, cosmology was taught, as it was to me, as the tail end of a general relativity course, with a derivation of the metric for an expanding Universe and a few solutions. Such a course fails to capture the flavour of modern cosmology, which takes classic physical sciences like thermodynamics, atomic physics and gravitation and applies them on a grand scale. In fact, introductory modern cosmology can be tackled in a different way, by avoiding general relativity altogether.
By a lucky chance, and a subtle bit of cheating, the cor- rect equations describing an expanding Universe can be obtained from Newtonian gravity. From this basis, one can study all the triumphs of the Hot Big Bang cosmology — the ex- pansion of the Universe, the prediction of its age, the existence of the cosmic microwave background, and the abundances of light elements such as helium and deuterium — and even go on to discuss more speculative ideas such as the inflationary cosmology. The origin of this book, first published in 1998, is a short lecture course at the Uni- versity of Sussex, around 20 lectures, taught to students in the final year of a bachelor's www.com xii CONTENTS degree or the penultimate year of a master's degree. The prerequisites are all very standard physics, and the emphasis is aimed at physical intuition rather than mathematical rigour.
Since the book's publication cosmology has moved on apace, and I have also become aware of the need for a somewhat more extensive range of material, hence this second edi- tion. To summarize the differences from the first edition, there is more stuff than before, and the stuff that was already there is now less out-of-date. Cosmology is an interesting course to teach, as it is not like most of the other subjects taught in undergraduate physics courses. There is no perceived wisdom, built up over a century or more, which provides an unquestionable foundation, as in thermodynamics, electromagnetism, and even quantum mechanics and general relativity.
Within our broad- brush picture the details often remain rather blurred, changing as we learn more about the Universe in which we live. Opportunities crop up during the course to discuss new results which impact on cosmologists' views of the Universe, and for the lecturer to impose their own prejudices on the interpretation of the ever-changing observational situation. Unless I've changed jobs (in which case I'm sure www. com will hunt me down), you can follow my own current prejudices by checking out this book's WWW Home Page at http://astronomy.uk/~andrewl/cosbook.html There you can find some updates on observations, and also a list of any errors in the book that I am aware of.
If you are confident you've found one yourself, and it's not on the list. I'd be very pleased to hear of it. The structure of the book is a central 'spine', the main chapters from one to fifteen, which provide a self-contained introduction to modern cosmology more or less reproduc- ing the coverage of my Sussex course. In addition there are five Advanced Topic chapters, each with prerequisites, which can be added to extend the course as desired.
Ordinarily the best time to tackle those Advanced Topics is immediately after their prerequisites have been attained, though they could also be included at any later stage. I'm extremely grateful to the reviewers of the original draft manuscript, namely Steve Eales, Coel Hellier and Linda Smith, for numerous detailed comments which led to the first edition being much better than it would have otherwise been. Thanks also to those who sent me useful comments on the first edition, in particular Paddy Leahy and Michael Rowan-Robinson, and of course to all the Wiley staff who contributed. Brian Schmidt and Michael Turner provided three of the figures, and Martin Hendry, Mar- tin Kunz and Franz Schunck helped with three others, while two figures were generated from NASA's SkyView facility (http: / /skyview.
gov) located at the NASA Goddard Space Flight Center. A library of images, including full-colour versions of several images reproduced here in black and white to keep production costs down, can be found via the book's Home Page as given above. Andrew R Liddle Brighton February 2003 www.com This page intentionally left blank www.com XIV Some fundamental constants Newton's constant G 6.672 x 10- 1 1 m3 kg 1 sec 2 Speed of light c 2.076 x 10" 7 Mpcyr" 1 Reduced Planck constant h = h/2-rr 1.