Главная
Cosmology: the origin and evolution of cosmic structure
Cosmology: the origin and evolution of cosmic structure
Prof Peter Coles, Francesco Lucchin, F. Lucchin, Peter Coles
0 /
0
Насколько вам понравилась эта книга?
Какого качества скаченный файл?
Скачайте книгу, чтобы оценить ее качество
Какого качества скаченные файлы?
Using an extremely accessible approach, the authors present a fascinating look at how cosmic structures, such as galaxies and clusters, began and evolved. After introducing readers to the simplest cosmological models and basic observational cosmology, they detail the Big Bang theory and the theory of structural formation by gravitational instability. The book concludes with observational techniques for testing models of largescale structure origin in the universe. Features discussions of recent spectacular breakthroughs including the COBE satellite discoveries.
Категории:
Год:
2002
Издание:
2
Издательство:
Wiley
Язык:
english
Страницы:
515
ISBN 10:
0471489093
ISBN 13:
9780470852996
Файл:
PDF, 3,38 MB
Ваши теги:
Скачать (pdf, 3,38 MB)
 Открыть в браузере
 Checking other formats...
 Конвертировать в EPUB
 Конвертировать в FB2
 Конвертировать в MOBI
 Конвертировать в TXT
 Конвертировать в RTF
 Конвертированный файл может отличаться от оригинала. При возможности лучше скачивать файл в оригинальном формате.
 Пожалуйста, сначала войдите в свой аккаунт

Нужна помощь? Пожалуйста, ознакомьтесь с инструкцией как отправить книгу на Kindle
В течение 15 минут файл будет доставлен на Ваш email.
В течение 15 минут файл будет доставлен на Ваш kindle.
Примечание: вам необходимо верифицировать каждую книгу, которую вы отправляете на Kindle. Проверьте свой почтовый ящик на наличие письма с подтверждением от Amazon Kindle Support.
Примечание: вам необходимо верифицировать каждую книгу, которую вы отправляете на Kindle. Проверьте свой почтовый ящик на наличие письма с подтверждением от Amazon Kindle Support.
Возможно Вас заинтересует Powered by Rec2Me
Ключевые слова
universe^{713}
equation^{555}
density^{544}
galaxies^{515}
cosmological^{422}
galaxy^{380}
scale^{368}
models^{360}
function^{352}
spectrum^{295}
radiation^{291}
gravitational^{284}
formation^{271}
redshift^{253}
evolution^{240}
distribution^{233}
temperature^{227}
perturbations^{223}
scales^{220}
velocity^{220}
particles^{217}
equations^{214}
cosmic^{189}
particle^{184}
horizon^{180}
cosmology^{170}
parameter^{169}
properties^{167}
expansion^{164}
principle^{164}
clusters^{149}
phase^{144}
perturbation^{142}
hubble^{141}
exp^{140}
correlation^{137}
thermal^{135}
astrophys^{134}
observations^{133}
cluster^{131}
zeq^{129}
angular^{129}
gaussian^{129}
clustering^{124}
relativistic^{115}
microwave background^{114}
physics^{113}
cmb^{109}
radius^{105}
epoch^{104}
observational^{101}
photons^{100}
einstein^{100}
primordial^{99}
metric^{99}
power spectrum^{98}
anisotropy^{94}
mpc^{93}
nonlinear^{92}
adiabatic^{92}
Связанные Подборки
1 comment
Bookhunter
Executed on the phone executed on my phone ???? this morning and evening with my friends house is on
13 October 2016 (20:49)
Вы можете оставить отзыв о книге и поделиться своим опытом. Другим читателям будет интересно узнать ваше мнение о прочитанных книгах. Независимо от того, пришлась ли вам книга по душе или нет, если вы честно и подробно расскажете об этом, люди смогут найти для себя новые книги, которые их заинтересуют.
1

2

Cosmology The Origin and Evolution of Cosmic Structure Second Edition Peter Coles School of Physics & Astronomy, University of Nottingham, UK Francesco Lucchin Dipartimento di Astronomia, Università di Padova, Italy Cosmology The Origin and Evolution of Cosmic Structure Cosmology The Origin and Evolution of Cosmic Structure Second Edition Peter Coles School of Physics & Astronomy, University of Nottingham, UK Francesco Lucchin Dipartimento di Astronomia, Università di Padova, Italy Copyright © 2002 John Wiley & Sons, Ltd Baﬃns Lane, Chichester, West Sussex PO19 1UD, England National 01243 779777 International (+44) 1243 779777 email (for orders and customer service enquiries): csbooks@wiley.co.uk Visit our Home Page on http://www.wileyeurope.com or http://www.wiley.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 Court Road, London, UK W1P 0LP, without the permission in writing of the Publisher with the exception of any material supplied speciﬁcally for the purpose of being entered and executed on a computer system for exclusive use by the purchaser of the publication. Neither the author nor John Wiley & Sons, Ltd accept any responsibility or liability for loss or damage occasioned to any person or property through using the material, instructions, methods or ideas contained herein, or acting or refraining from acting as a result of such use. The author and publisher expressly disclaim all implied warranties, including merchantability or ﬁtness for any particular purpose. There will be no duty on the author or publisher to correct any errors or defects in the software. Designations used by companies to distinguish their products are often claimed as tr; ademarks. In all instances where John Wiley & Sons, Ltd is aware of a claim, the product names appear in capital or all capital letters. Readers, however, should contact the appropriate companies for more complete information regarding trademarks and registration. Library of Congress CataloginginPublication Data (applied for) British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0 471 48909 3 Typeset in 9.5/12.5pt Lucida Bright by T&T Productions Ltd, London. Printed and bound in Great Britain by Antony Rowe Ltd., Chippenham, Wilts. This book is printed on acidfree paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. Contents Preface to First Edition Preface to Second Edition PART 1 1 First Principles 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 2 Cosmological Models The Cosmological Principle Fundamentals of General Relativity The Robertson–Walker Metric The Hubble Law Redshift The Deceleration Parameter Cosmological Distances The m–z and N–z Relations Olbers’ Paradox The Friedmann Equations A Newtonian Approach The Cosmological Constant Friedmann Models The Friedmann Models 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Perfect Fluid Models Flat Models Curved Models: General Properties 2.3.1 Open models 2.3.2 Closed models Dust Models 2.4.1 Open models 2.4.2 Closed models 2.4.3 General properties Radiative Models 2.5.1 Open models 2.5.2 Closed models 2.5.3 General properties Evolution of the Density Parameter Cosmological Horizons Models with a Cosmological Constant xi xix 1 3 3 6 9 13 15 17 18 20 22 23 24 26 29 33 33 36 38 39 40 40 41 41 42 43 43 44 44 44 45 49 vi 3 Contents Alternative Cosmologies 3.1 3.2 3.3 3.4 3.5 3.6 4 Anisotropic and Inhomogeneous Cosmologies 3.1.1 The Bianchi models 3.1.2 Inhomogeneous models The SteadyState Model The Dirac Theory Brans–Dicke Theory Variable Constants Hoyle–Narlikar (Conformal) Gravity Observational Properties of the Universe 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Introduction 4.1.1 Units 4.1.2 Galaxies 4.1.3 Active galaxies and quasars 4.1.4 Galaxy clustering The Hubble Constant The Distance Ladder The Age of the Universe 4.4.1 Theory 4.4.2 Stellar and galactic ages 4.4.3 Nucleocosmochronology The Density of the Universe 4.5.1 Contributions to the density parameter 4.5.2 Galaxies 4.5.3 Clusters of galaxies Deviations from the Hubble Expansion Classical Cosmology 4.7.1 Standard candles 4.7.2 Angular sizes 4.7.3 Numbercounts 4.7.4 Summary The Cosmic Microwave Background 67 67 69 70 72 75 79 83 83 84 84 86 86 88 89 92 94 95 97 99 100 100 Thermal History of the Hot Big Bang Model 109 The Standard Hot Big Bang Recombination and Decoupling Matter–Radiation Equivalence Thermal History of the Universe Radiation Entropy per Baryon Timescales in the Standard Model The Very Early Universe 6.1 6.2 6.3 6.4 6.5 7 67 107 5.1 5.2 5.3 5.4 5.5 5.6 6 52 52 55 57 59 61 63 64 The Hot Big Bang Model PART 2 5 51 The Big Bang Singularity The Planck Time The Planck Era Quantum Cosmology String Cosmology Phase Transitions and Inﬂation 7.1 7.2 7.3 7.4 The Hot Big Bang Fundamental Interactions Physics of Phase Transitions Cosmological Phase Transitions 109 111 112 113 115 116 119 119 122 123 126 128 131 131 133 136 138 Contents 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 8 The Lepton Era 8.1 8.2 8.3 8.4 8.5 8.6 8.7 9 Problems of the Standard Model The Monopole Problem The Cosmological Constant Problem The Cosmological Horizon Problem 7.8.1 The problem 7.8.2 The inﬂationary solution The Cosmological Flatness Problem 7.9.1 The problem 7.9.2 The inﬂationary solution The Inﬂationary Universe Types of Inﬂation 7.11.1 Old inﬂation 7.11.2 New inﬂation 7.11.3 Chaotic inﬂation 7.11.4 Stochastic inﬂation 7.11.5 Open inﬂation 7.11.6 Other models Successes and Problems of Inﬂation The Anthropic Cosmological Principle The Quark–Hadron Transition Chemical Potentials The Lepton Era Neutrino Decoupling The Cosmic Neutrino Background Cosmological Nucleosynthesis 8.6.1 General considerations 8.6.2 The standard nucleosynthesis model 8.6.3 The neutron–proton ratio 8.6.4 Nucleosynthesis of Helium 8.6.5 Other elements 8.6.6 Observations: Helium 4 8.6.7 Observations: Deuterium 8.6.8 Helium 3 8.6.9 Lithium 7 8.6.10 Observations versus theory Nonstandard Nucleosynthesis The Plasma Era 9.1 9.2 9.3 9.4 9.5 The Radiative Era The Plasma Epoch Hydrogen Recombination The Matter Era Evolution of the CMB Spectrum PART 3 Theory of Structure Formation 10 Introduction to Jeans Theory 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 Gravitational Instability Jeans Theory for Collisional Fluids Jeans Instability in Collisionless Fluids History of Jeans Theory in Cosmology The Eﬀect of Expansion: an Approximate Analysis Newtonian Theory in a Dust Universe Solutions for the Flat Dust Case The Growth Factor vii 141 143 145 147 147 149 152 152 154 156 160 160 161 161 162 162 163 163 164 167 167 168 171 172 173 176 176 177 178 179 181 182 183 184 185 185 186 191 191 192 194 195 197 203 205 205 206 210 212 213 215 218 219 viii Contents 10.9 10.10 10.11 10.12 Solution for RadiationDominated Universes The Method of Autosolution The Meszaros Eﬀect Relativistic Solutions 11 Gravitational Instability of Baryonic Matter 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 Introduction Adiabatic and Isothermal Perturbations Evolution of the Sound Speed and Jeans Mass Evolution of the Horizon Mass Dissipation of Acoustic Waves Dissipation of Adiabatic Perturbations Radiation Drag A TwoFluid Model The Kinetic Approach Summary 12 Nonbaryonic Matter 12.1 12.2 12.3 12.4 12.5 12.6 Introduction The Boltzmann Equation for Cosmic Relics Hot Thermal Relics Cold Thermal Relics The Jeans Mass Implications 12.6.1 Hot Dark Matter 12.6.2 Cold Dark Matter 12.6.3 Summary 13 Cosmological Perturbations 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 Introduction The Perturbation Spectrum The Mass Variance 13.3.1 Mass scales and ﬁltering 13.3.2 Properties of the ﬁltered ﬁeld 13.3.3 Problems with ﬁlters Types of Primordial Spectra Spectra at Horizon Crossing Fluctuations from Inﬂation Gaussian Density Perturbations Covariance Functions NonGaussian Fluctuations? 14 Nonlinear Evolution 14.1 14.2 14.3 14.4 14.5 14.6 The Spherical ‘TopHat’ Collapse The Zel’dovich Approximation The Adhesion Model Selfsimilar Evolution 14.4.1 A simple model 14.4.2 Stable clustering 14.4.3 Scaling of the power spectrum 14.4.4 Comments The Mass Function NBody Simulations 14.6.1 Direct summation 14.6.2 Particle–mesh techniques 14.6.3 Tree codes 14.6.4 Initial conditions and boundary eﬀects 221 223 225 227 229 229 230 231 233 234 237 240 241 244 248 251 251 252 253 255 256 259 260 261 262 263 263 264 266 266 268 270 271 275 276 279 281 284 287 287 290 294 296 296 299 300 301 301 304 305 306 309 309 Contents 14.7 Gas Physics 14.7.1 Cooling 14.7.2 Numerical hydrodynamics 14.8 Biased Galaxy Formation 14.9 Galaxy Formation 14.10 Comments 15 Models of Structure Formation 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 Introduction Historical Prelude Gravitational Instability in Brief Primordial Density Fluctuations The Transfer Function Beyond Linear Theory Recipes for Structure Formation Comments ix 310 310 312 314 318 321 323 323 324 326 327 328 330 331 334 Observational Tests 335 16 Statistics of Galaxy Clustering 337 PART 4 16.1 16.2 16.3 16.4 Introduction Correlation Functions The Limber Equation Correlation Functions: Results 16.4.1 Twopoint correlations 16.5 The Hierarchical Model 16.5.1 Comments 16.6 Cluster Correlations and Biasing 16.7 Counts in Cells 16.8 The Power Spectrum 16.9 Polyspectra 16.10 Percolation Analysis 16.11 Topology 16.12 Comments 17 The Cosmic Microwave Background 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 Introduction The Angular Power Spectrum The CMB Dipole Large Angular Scales 17.4.1 The Sachs–Wolfe eﬀect 17.4.2 The COBE DMR experiment 17.4.3 Interpretation of the COBE results Intermediate Scales Smaller Scales: Extrinsic Eﬀects The Sunyaev–Zel’dovich Eﬀect Current Status 18 Peculiar Motions of Galaxies 18.1 18.2 18.3 18.4 18.5 18.6 Velocity Perturbations Velocity Correlations Bulk Flows Velocity–Density Reconstruction RedshiftSpace Distortions Implications for Ω0 337 339 342 344 344 346 348 350 352 355 356 359 361 365 367 367 368 371 374 374 377 379 380 385 389 391 393 393 396 398 400 402 405 x Contents 19 Gravitational Lensing 19.1 19.2 19.3 19.4 19.5 Historical Prelude Basic Gravitational Optics More Complicated Systems Applications 19.4.1 Microlensing 19.4.2 Multiple images 19.4.3 Arcs, arclets and cluster masses 19.4.4 Weak lensing by largescale structure 19.4.5 The Hubble constant Comments 20 The HighRedshift Universe 20.1 20.2 20.3 20.4 20.5 20.6 20.7 Introduction Quasars The Intergalactic Medium (IGM) 20.3.1 Quasar spectra 20.3.2 The Gunn–Peterson test 20.3.3 Absorption line systems 20.3.4 Xray gas in clusters 20.3.5 Spectral distortions of the CMB 20.3.6 The Xray background The Infrared Background and Dust Numbercounts Revisited Star and Galaxy Formation Concluding Remarks 21 A Forward Look 21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 21.9 21.10 21.11 21.12 Introduction General Observations Xrays and the Hot Universe The Apotheosis of Astrometry: GAIA The Next Generation Space Telescope: NGST Extremely Large Telescopes FarIR and Submillimetre Views of the Early Universe The Cosmic Microwave Background The Square Kilometre Array Gravitational Waves Sociology, Politics and Economics Conclusions 409 409 412 415 418 418 419 420 421 422 423 425 425 426 428 428 428 430 432 432 433 434 437 438 444 447 447 448 449 450 452 453 454 456 456 458 460 461 Appendix A. Physical Constants 463 Appendix B. Useful Astronomical Quantities 465 Appendix C. Particle Properties 467 References 469 Index 485 Preface to First Edition This is a book about modern cosmology. Because this is a big subject – as big as the Universe – we have had to choose one particular theme upon which to focus our treatment. Current research in cosmology ranges over ﬁelds as diverse as quantum gravity, general relativity, particle physics, statistical mechanics, nonlinear hydrodynamics and observational astronomy in all wavelength regions, from radio to gamma rays. We could not possibly do justice to all these areas in one volume, especially in a book such as this which is intended for advanced undergraduates or beginning postgraduates. Because we both have a strong research interest in theories for the origin and evolution of cosmic structure – galaxies, clusters and the like – and, in many respects, this is indeed the central problem in this ﬁeld, we decided to concentrate on those elements of modern cosmology that pertain to this topic. We shall touch on many of the areas mentioned above, but only insofar as an understanding of them is necessary background for our analysis of structure formation. Cosmology in general, and the ﬁeld of structure formation in particular, has been a ‘hot’ research topic for many years. Recent spectacular observational breakthroughs, like the discovery by the COBE satellite in 1992 of ﬂuctuations in the temperature of the cosmic microwave background, have made newspaper headlines all around the world. Both observational and theoretical sides of the subject continue to engross not only the best undergraduate and postgraduate students and more senior professional scientists, but also the general public. Part of the fascination is that cosmology lies at the crossroads of many disciplines. An introduction to this subject therefore involves an initiation into many seemingly disparate branches of physics and astrophysics; this alone makes it an ideal area in which to encourage young scientists to work. Nevertheless, cosmology is a peculiar science. The Universe is, by deﬁnition, unique. We cannot prepare an ensemble of universes with slightly diﬀerent parameter values and look for diﬀerences or correlations in their behaviour. In many branches of physical science such experimentation often leads to the formulation of empirical laws which give rise to models and subsequently theories. Cosmology is diﬀerent. We have only one Universe, and this must provide the empirical laws we try to explain by theory, as well as the experimental evidence we use to test the theories we have formulated. Though the distinction between them is, of course, not completely sharp, it is fair to say that physics is predominantly characterised by experiment and theory, and cosmology by observation and paradigm. xii Preface to First Edition (We take the word ‘paradigm’ to mean a theoretical framework, not all of whose elements have been formalised in the sense of being directly related to observational phenomena.) Subtle inﬂuences of personal philosophy, cultural and, in some cases, religious background lead to very diﬀerent choices of paradigm in many branches of science, but this tendency is particularly noticeable in cosmology. For example, one’s choice to include or exclude the cosmological constant term in Einstein’s ﬁeld equations of general relativity can have very little empirical motivation but must be made on the basis of philosophical, and perhaps aesthetic, considerations. Perhaps a better example is the fact that the expansion of the Universe could have been anticipated using Newtonian physics as early as the 17th century. The Cosmological Principle, according to which the Universe is homogeneous and isotropic on large scales, is suﬃcient to ensure that a Newtonian universe cannot be static, but must be either expanding or contracting. A philosophical predisposition in western societies towards an unchanging, regular cosmos apparently prevented scientists from drawing this conclusion until it was forced upon them by 20th century observations. Incidentally, a notable exception to this prevailing paradigm was the writer Edgar Allan Poe, who expounded a picture of a dynamic, cyclical cosmos in his celebrated prose poem Eureka. We make these points to persuade the reader that cosmology requires not only a good knowledge of interdisciplinary physics, but also an open mind and a certain amount of selfknowledge. One can learn much about what cosmology actually means from its history. Since prehistoric times, man has sought to make sense of his existence and that of the world around him in some kind of theoretical framework. The ﬁrst such theories, not recognisable as ‘science’ in the modern sense of the word, were mythological. In western cultures, the Ptolemaic cosmology was a step towards the modern approach, but was clearly informed by Greek cultural values. The Copernican Principle, the notion that we do not inhabit a special place in the Universe and a kind of forerunner of the Cosmological Principle, was to some extent a product of the philosophical and religious changes taking place in Renaissance times. The mechanistic view of the Universe initiated by Newton and championed by Descartes, in which one views the natural world as a kind of clockwork device, was inﬂuenced not only by the beginnings of mathematical physics but also by the ﬁrst stirrings of technological development. In the era of the Industrial Revolution, man’s perception of the natural world was framed in terms of heat engines and thermodynamics, and involved such concepts as the ‘Heat Death of the Universe’. With hindsight we can say that cosmology did not really come of age as a science until the 20th century. In 1915 Einstein advanced his theory of general relativity. His ﬁeld equations told him the Universe should be evolving; Einstein thought he must have made a mistake and promptly modiﬁed the equations to give a static cosmological solution, thus perpetuating the fallacy we discussed. It was not until 1929 that Hubble convinced the astronomical community that the Universe was actually expanding after all. (To put this aﬀair into historical perspective, remember that it was only in the mid1920s that it was demonstrated – by Hubble and Preface to First Edition xiii others – that faint nebulae, now known to be galaxies like our own Milky Way, were actually outside our Galaxy.) The next few decades saw considerable theoretical and observational developments. The Big Bang and steadystate cosmologies were proposed and their respective advocates began a long and acrimonious debate about which was correct, the legacy of which lingers still. For many workers this debate was resolved by the discovery in 1965 of the cosmic microwave background radiation, which was immediately seen to be good evidence in favour of an evolving Universe which was hotter and denser in the past. It is reasonable to regard this discovery as marking the beginning of ‘Physical Cosmology’. Counts of distant galaxies were also showing evidence of evolution in the properties of these objects at this time, and the ﬁrst calculations had already been made, notably by Alpher and Herman in the late 1940s, of the elemental abundances expected to be produced by nuclear reactions in the early stages of the Big Bang. These, and other, considerations left the Big Bang model as the clear victor over the steadystate picture. By the 1970s, attention was being turned to the question that forms the main focus of this book: where did the structure we observe in the Universe around us actually come from? The fact that the microwave background appeared remarkably uniform in temperature across the sky was taken as evidence that the early Universe (when it was less than a few hundred thousand years old) was very smooth. But the Universe now is clearly very clumpy, with large ﬂuctuations in its density from place to place. How could these two observations be reconciled? A ‘standard’ picture soon emerged, based on the known physics of gravitational instability. Gravity is an attractive force, so that a region of the Universe which is slightly denser than average will gradually accrete material from its surroundings. In so doing the original, slightly denser region gets denser still and therefore accretes even more material. Eventually this region becomes a strongly bound ‘lump’ of matter surrounded by a region of comparatively low density. After two decades, gravitational instability continues to form the basis of the standard theory for structure formation. The details of how it operates to produce structures of the form we actually observe today are, however, still far from completely understood. To resume our historical thread, the 1970s saw the emergence of two competing scenarios (a terrible word, but sadly commonplace in the cosmological literature) for structure formation. Roughly speaking, one of these was a ‘bottomup’, or hierarchical, model, in which structure formation was thought to begin with the collapse of small objects which then progressively clustered together and merged under the action of their mutual gravitational attraction to form larger objects. This model, called the isothermal model, was advocated mainly by American researchers. On the other hand, many Soviet astrophysicists of the time, led by Yakov B. Zel’dovich, favoured a model, the adiabatic model, in which the ﬁrst structures to condense out of the expanding plasma were huge agglomerations of mass on the scale of giant superclusters of galaxies; smaller structures like individual galaxies were assumed to be formed by fragmentation processes within the larger structures, which are usually called ‘pancakes’. The debate xiv Preface to First Edition between the isothermal and adiabatic schools never reached the level of animosity of the Big Bang versus steadystate controversy but was nevertheless healthily animated. By the 1980s it was realised that neither of these models could be correct. The reasons for this conclusion are not important at this stage; we shall discuss them in detail during Part 3 of the book. Soon, however, alternative models were proposed which avoided many of the problems which led to the rejection of the 1970s models. The new ingredient added in the 1980s was nonbaryonic matter; in other words, matter in the form of some exotic type of particle other than protons and neutrons. This matter is not directly observable because it is not luminous, but it does feel the action of gravity and can thus assist the gravitational instability process. Nonbaryonic matter was thought to be one of two possible types: hot or cold. As had happened in the 1970s, the cosmological world again split into two camps, one favouring cold dark matter (CDM) and the other hot dark matter (HDM). Indeed, there are considerable similarities between the two schisms of the 1970s and 1980s, for the CDM model is a ‘bottomup’ model like the old baryon isothermal picture, while the HDM model is a ‘topdown’ scenario like the adiabatic model. Even the geographical division was the same; Zel’dovich’s great Soviet school were the most powerful advocates of the HDM picture. The 1980s also saw another important theoretical development: the idea that the Universe may have undergone a period of inﬂation, during which its expansion rate accelerated and any initial inhomogeneities were smoothed out. Inﬂation provides a model which can, at least in principle, explain how such homogeneity might have arisen and which does not require the introduction of the Cosmological Principle ab initio. While creating an observable patch of the Universe which is predominantly smooth and isotropic, inﬂation also guarantees the existence of small ﬂuctuations in the cosmological density which may be the initial perturbations needed to feed the gravitational instability thought to be the origin of galaxies and other structures. The history of cosmology in the 20th century is marked by an interesting interplay of opposites. For example, in the development of structureformation theories one can see a strong tendency towards change (such as from baryonic to nonbaryonic models), but also a strong element of continuity (the persistence of the hierarchical and pancake scenarios). The standard cosmological models have an expansion rate which is decelerating because of the attractive nature of gravity. In models involving inﬂation (or those with a cosmological constant) the expansion is accelerated by virtue of the fact that gravity eﬀectively becomes repulsive for some period. The Cosmological Principle asserts a kind of largescale order, while inﬂation allows this to be achieved locally within a Universe characterised by largescale disorder. The confrontation between steadystate and Big Bang models highlights the distinction between stationarity and evolution. Some variants of the Big Bang model involving inﬂation do, however, involve a large ‘metauniverse’ within which ‘miniuniverses’ of the size of our observable patch are continually being formed. The appearance of miniuniverses also emphasises Preface to First Edition xv the contrast between whole and part : is our observable Universe all there is, or even representative of all there is? Or is it just an atypical ‘bubble’ which just happens to have the properties required for life to evolve within it? This brings into play the idea of an Anthropic Cosmological Principle which emphasises the special nature of the conditions necessary to create observers, compared with the general homogeneity implied by the Cosmological Principle in its traditional form. Another interesting characteristic of cosmology is the distinction, which is often blurred, between what one might call cosmology and metacosmology. We take cosmology to mean the scientiﬁc study of the cosmos as a whole, an essential part of which is the testing of theoretical constructions against observations, as described above. On the other hand, metacosmology is a term which describes elements of a theoretical construction, or paradigm, which are not amenable to observational test. As the subject has developed, various aspects of cosmology have moved from the realm of metacosmology into that of cosmology proper. The cosmic microwave background, whose existence was postulated as early as the 1940s, but which was not observable by means of technology available at that time, became part of cosmology proper in 1965. It has been argued by some that the inﬂationary metacosmology has now become part of scientiﬁc cosmology because of the COBE discovery of ﬂuctuations in the temperature of the microwave background across the sky. We think this claim is premature, although things are clearly moving in the right direction for this to take place some time in the future. Some metacosmological ideas may, however, remain so forever, either because of the technical diﬃculty of observing their consequences or because they are not testable even in principle. An example of the latter diﬃculty may be furnished by Linde’s chaotic inﬂationary picture of eternally creating miniuniverses which lie beyond the radius of our observable Universe. Despite these complexities and idiosyncrasies, modern cosmology presents us with clear challenges. On the purely theoretical side, we require a full integration of particle physics into the Big Bang model, and a theory which treats gravitational physics at the quantum level. We also need a theoretical understanding of various phenomena which are probably based on wellestablished physical processes: nonlinearity in gravitational clustering, hydrodynamical processes, stellar formation and evolution, chemical evolution of galaxies. Many observational targets have also been set: the detection of candidate darkmatter particles in the galactic halo; gravitational waves; more detailed observations of the temperature ﬂuctuations in the cosmic microwave background; larger samples of galaxy redshifts and peculiar motions; elucidation of the evolutionary properties of galaxies with cosmic time. Above all, we want to stress that cosmology is a ﬁeld in which many fundamental questions remain unanswered and where there is plenty of scope for new ideas. The next decade promises to be at least as exciting as the last, with ongoing experiments already probing the microwave background in ﬁner detail and powerful optical telescopes mapping the distribution of galaxies out to greater and greater distances. Who can say what theoretical ideas will be advanced in light of these new observations? Will the theoretical ideas described in this book xvi Preface to First Edition turn out to be correct, or will we have to throw them all away and go back to the drawing board? This book is intended to be an uptodate introduction to this fascinating yet complex subject. It is intended to be accessible to advanced undergraduate and beginning postgraduate students, but contains much material which will be of interest to more established researchers in the ﬁeld, and even nonspecialists should ﬁnd it a useful introduction to many of the important ideas in modern cosmology. Our book does not require a high level of specialisation on behalf of the reader. Only a modest use is made of general relativity. We use some concepts from statistical mechanics and particle physics, but our treatment of them is as selfcontained as possible. We cover the basic material, such as the Friedmann models, one ﬁnds in all elementary cosmology texts, but we also take the reader through more advanced material normally available only in technical review articles or in the research literature. Although many cosmology books are on the market at the moment thanks, no doubt, to the high level of public and media interest in this subject, very few tackle the material we cover at this kind of ‘bridging’ level between elementary textbook and research monograph. We have also covered some material which one might regard as slightly oldfashioned. Our treatment of the adiabatic baryon picture of structure formation in Chapter 12 is an example. We have included such material primarily for pedagogical reasons, but also for the valuable historical lessons it provides. The fact that models come and go so rapidly in this ﬁeld is explained partly by the vigorous interplay between observation and theory and partly by virtue of the fact that cosmology, in common with other aspects of life, is sometimes a victim of changes in fashion. We have also included more recent theory and observation alongside this pedagogical material in order to provide the reader with a ﬁrm basis for an understanding of future developments in this ﬁeld. Obviously, because ours is such an exciting ﬁeld, with advances being made at a rapid rate, we cannot claim to be deﬁnitive in all areas of contemporary interest. At the end of each chapter we give lists of references – which are not intended to be exhaustive but which should provide further reading on the fundamental issues – as well as more detailed technical articles for the advanced student. We have not cited articles in the body of each chapter, mainly to avoid interrupting the ﬂow of the presentation. By doing this, it is certainly not our intention to claim that we have not leaned upon other works for much of this material; we implicitly acknowledge this for any work we list in the references. We believe that our presentation of this material is the most comprehensive and accessible available at this level amongst the published works belonging to the literature of this subject; a list of relevant general books on cosmology is given after this preface. The book is organised into four parts. The ﬁrst, Chapters 1–4, covers the basics of general relativity, the simplest cosmological models, alternative theories and introductory observational cosmology. This part can be skipped by students who have already taken introductory courses in cosmology. Part 2, Chapters 5–9, deals with physical cosmology and the thermal history of the universe in Big Bang models, including a discussion of phase transitions and inﬂation. Part 3, Chap Preface to First Edition xvii ters 10–15, contains a detailed treatment of the theory of gravitational instability in both the linear and nonlinear regimes with comments on darkmatter theories and hydrodynamical eﬀects in the context of galaxy formation. The ﬁnal part, Chapters 16–19, deals with methods for testing theories of structure formation using statistical properties of galaxy clustering, the ﬂuctuations of the cosmic microwave background, galaxypeculiar motions and observations of galaxy evolution and the extragalactic radiation backgrounds. The last part of the book is at a rather higher level than the preceding ones and is intended to be closer to the ongoing research in this ﬁeld. Some of the text is based upon an English adaptation of Introduzione alla Cosmologia (Zanichelli, Bologna, 1990), a cosmology textbook written in Italian by Francesco Lucchin, which contains material given in his lectures on cosmology to ﬁnalyear undergraduates at the University of Padova over the past 15 years or so. We are very grateful to the publishers for permission to draw upon this source. We have, however, added a large amount of new material for the present book in order to cover as many of the latest developments in this ﬁeld as possible. Much of this new material relates to the lecture notes given by Peter Coles for the Master of Science course on cosmology at Queen Mary and Westﬁeld College beginning in 1992. These sources reinforce our intention that the book should be suitable for advanced undergraduates and/or beginning postgraduates. Francesco Lucchin thanks the Astronomy Unit at Queen Mary & Westﬁeld College for hospitality during visits when this book was in preparation. Likewise, Peter Coles thanks the Dipartimento di Astronomia of the University of Padova for hospitality during his visits there. Many colleagues and friends have helped us enormously during the preparation of this book. In particular, we thank Sabino Matarrese, Lauro Moscardini and Bepi Tormen for their careful reading of the manuscript and for many discussions on other matters related to the book. We also thank Varun Sahni and George Ellis for allowing us to draw on material cowritten by them and Peter Coles. Many sources are also to be thanked for their willingness to allow us to use various ﬁgures; appropriate acknowledgments are given in the corresponding ﬁgure captions. Peter Coles and Francesco Lucchin London, October 1994 Contents 14.7 Gas Physics 14.7.1 Cooling 14.7.2 Numerical hydrodynamics 14.8 Biased Galaxy Formation 14.9 Galaxy Formation 14.10 Comments 15 Models of Structure Formation 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 Introduction Historical Prelude Gravitational Instability in Brief Primordial Density Fluctuations The Transfer Function Beyond Linear Theory Recipes for Structure Formation Comments ix 310 310 312 314 318 321 323 323 324 326 327 328 330 331 334 Observational Tests 335 16 Statistics of Galaxy Clustering 337 PART 4 16.1 16.2 16.3 16.4 Introduction Correlation Functions The Limber Equation Correlation Functions: Results 16.4.1 Twopoint correlations 16.5 The Hierarchical Model 16.5.1 Comments 16.6 Cluster Correlations and Biasing 16.7 Counts in Cells 16.8 The Power Spectrum 16.9 Polyspectra 16.10 Percolation Analysis 16.11 Topology 16.12 Comments 17 The Cosmic Microwave Background 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 Introduction The Angular Power Spectrum The CMB Dipole Large Angular Scales 17.4.1 The Sachs–Wolfe eﬀect 17.4.2 The COBE DMR experiment 17.4.3 Interpretation of the COBE results Intermediate Scales Smaller Scales: Extrinsic Eﬀects The Sunyaev–Zel’dovich Eﬀect Current Status 18 Peculiar Motions of Galaxies 18.1 18.2 18.3 18.4 18.5 18.6 Velocity Perturbations Velocity Correlations Bulk Flows Velocity–Density Reconstruction RedshiftSpace Distortions Implications for Ω0 337 339 342 344 344 346 348 350 352 355 356 359 361 365 367 367 368 371 374 374 377 379 380 385 389 391 393 393 396 398 400 402 405 xx Preface to Second Edition new chapter on gravitational lensing, another ‘hot’ topic for today’s generation of cosmologists. We also changed the structure of the ﬁrst part of the book to make a gentler introduction to the subject instead of diving straight into general relativity. We also added problems sections at the end of each chapter and reorganised the references. We decided to keep our account of the basic physics of perturbation growth (Chapters 10–12) while other books concentrate more on modelbuilding. Our reason for this is that we intended the book to be an introduction for physics students. Models come and models go, but physics remains the same. To make the book a bit more accessible we added a sort of ‘digest’ of the main ideas and summary of modelbuilding in Chapter 15 for readers wishing to bypass the details. Other bits, such as those covering theories with variable constants and inhomogeneous cosmologies, were added for no better reason than that they are fun. On the other hand, we missed the boat in a signiﬁcant way by minimising the role of the cosmological constant in the ﬁrst edition. Who knows, maybe we will strike it lucky with one of these additions! Because of the dominance that observation has assumed over the last few years, we decided to add a chapter at the end of the book exploring some of the planned developments in observation technology (gravitational wave detectors, new satellites, groundbased facilities, and so on). Experience has shown us that it is hard to predict the future, but this ﬁnal chapter will at least point out some of the possibilities. We are grateful to everyone who helped us with this second edition and to those who provided constructive criticism on the ﬁrst. In particular, we thank (in alphabetical order) George Ellis, Richard Ellis, Carlos Frenk, Andrew Liddle, Sabino Matarrese, Lauro Moscardini and Bepi Tormen for their comments and advice. We also acknowledge the help of many students who helped us correct some of the (regrettably numerous) errors in the original book. Peter Coles and Francesco Lucchin Padua, January 2002 PART 1 Cosmological Models 1 First Principles In this chapter, our aim is to provide an introduction to the basic mathematical structure of modern cosmological models based on Einstein’s theory of gravity, the General Theory of Relativity or general relativity for short. This theory is mathematically challenging, but fortunately we do not really need to use its fully general form. Throughout this chapter we will therefore illustrate the key results with Newtonian analogies. We begin our study with a discussion of the Cosmological Principle, the ingredient that makes relativistic cosmology rather more palatable than it might otherwise be. 1.1 The Cosmological Principle Whenever science enters a new ﬁeld and is faced with a dearth of observational or experimental data some guiding principle is usually needed to assist during the ﬁrst tentative steps towards a theoretical understanding. Such principles are often based on ideas of symmetry which reduce the number of degrees of freedom one has to consider. This general rule proved to be the case in the early years of the 20th century when the ﬁrst steps were taken, by Einstein and others, towards a scientiﬁc theory of the Universe. Little was then known empirically about the distribution of matter in the Universe and Einstein’s theory of gravity was found to be too diﬃcult to solve for an arbitrary distribution of matter. In order to make progress the early cosmologists therefore had to content themselves with the construction of simpliﬁed models which they hoped might describe some aspects of the Universe in a broadbrush sense. These models were based on an idea called the Cosmological Principle. Although the name ‘principle’ sounds grand, principles are generally introduced into physics when one has no data to go on, and cosmology was no exception to this rule. The Cosmological Principle is the assertion that, on suﬃciently large scales (beyond those traced by the largescale structure of the distribution of galaxies), 4 First Principles the Universe is both homogeneous and isotropic. Homogeneity is the property of being identical everywhere in space, while isotropy is the property of looking the same in every direction. The Universe is clearly not exactly homogeneous, so cosmologists deﬁne homogeneity in an average sense: the Universe is taken to be identical in diﬀerent places when one looks at suﬃciently large pieces. A good analogy is that of a patterned carpet which is made of repeating units of some basic design. On the scale of the individual design the structure is clearly inhomogeneous but on scales larger than each unit it is homogeneous. There is quite good observational evidence that the Universe does have these properties, although this evidence is not completely watertight. One piece of evidence is the observed nearisotropy of the cosmic microwave background radiation. Isotropy, however, does not necessarily imply homogeneity without the additional assumption that the observer is not in a special place: the socalled Copernican Principle. One would observe isotropy in any spherically symmetric distribution of matter, but only if one were in the middle of the pattern. A circular carpet bearing a design consisting of a series of concentric rings would look isotropic only to an observer standing in the centre of the pattern. Observed isotropy, together with the Copernican Principle, therefore implies the Cosmological Principle. The Cosmological Principle was introduced by Einstein and subsequent relativistic cosmologists without any observational justiﬁcation whatsoever. Indeed, it was not known until the 1920s that the spiral nebulae (now known to be galaxies like our own) were outside our own galaxy, the Milky Way. A term frequently used to describe the entire Universe in those days was metagalaxy, indicating that it was thought that the Milky Way was essentially the entire cosmos. The Galaxy certainly does not look the same in all directions: it presents itself as a prominent band across the night sky. In advocating the Cosmological Principle, Einstein was particularly motivated by ideas associated with Ernst Mach. Mach’s Principle, roughly speaking, is that the laws of physics are determined by the distribution of matter on large scales. For example, the value of the gravitational constant G was thought perhaps to be related to the amount of mass in the Universe. Einstein thought that the only way to put theoretical cosmology on a ﬁrm footing was to assume that there was a basic simplicity to the global structure of the Universe enabling a similar simplicity in the local behaviour of matter. The Cosmological Principle achieves this and leads to relatively simple cosmological models, as we shall see shortly. There are various approaches one can take to this principle. One is philosophical, and is characterised by the work of Milne in the 1930s and later by Bondi, Gold and Hoyle in the 1940s. This line of reasoning is based, to a large extent, on the aesthetic appeal of the Cosmological Principle. Ultimately this appeal stems from the fact that it would indeed be very diﬃcult for us to understand the Universe if physical conditions, or even the laws of physics themselves, were to vary dramatically from place to place. These thoughts have been taken further, leading to the Perfect Cosmological Principle, in which the Universe is the same not only The Cosmological Principle 5 in all places and in all directions, but also at all times. This stronger version of the Cosmological Principle was formulated by Bondi and Gold (1948) and it subsequently led Hoyle (1948) and Hoyle and Narlikar (1963, 1964) to develop the steadystate cosmology. This theory implies, amongst other things, the continuous creation of matter to keep the density of the expanding Universe constant. The steadystate universe was abandoned in the 1960s because of the properties of the cosmic microwave background, radio sources and the cosmological helium abundance which are more readily explained in a Big Bang model than in a steady state. Nowadays the latter is only of historical interest (see Chapter 3 later). Attempts have also been made to justify the Cosmological Principle on more direct physical grounds. As we shall see, homogeneous and isotropic universes described by the theory of general relativity possess what is known as a ‘cosmological horizon’: regions suﬃciently distant from each other cannot have been in causal contact (‘have never been inside each other’s horizon’) at any stage since the Big Bang. The size of the regions whose parts are in causal contact with each other at a given time grows with cosmological epoch; the calculation of the horizon scale is performed in Section 2.7. The problem then arises as to how one explains the observation that the Universe appears homogeneous on scales much larger than the scale one expects to have been in causal contact up to the present time. The mystery is this: if two regions of the Universe have never been able to communicate with each other by means of light signals, how can they even know the physical conditions (density, temperature, etc.) pertaining to each other? If they cannot know this, how is it that they evolve in such a way that these conditions are the same in each of the regions? One either has to suppose that causal physics is not responsible for this homogeneity, or that the calculation of the horizon is not correct. This conundrum is usually called the Cosmological Horizon Problem and we shall discuss it in some detail in Chapter 7. Various attempts have been made to avoid this problem. For example, particular models of the Universe, such as some that are homogeneous but not isotropic, do not possess the required particle horizon. These models can become isotropic in the course of their evolution. A famous example is the ‘mixmaster’ universe of Misner (1968) in which isotropisation is eﬀected by viscous dissipation involving neutrinos in the early universe. Another way to isotropise an initially anisotropic universe is by creating particles at the earliest stage of all, the Planck era (Chapter 6). More recently still, Guth (1981) proposed an idea which could resolve the horizon problem: the inﬂationary universe, which is of great contemporary interest in cosmology, and which we discuss in Chapter 7. In any case, the most appropriate approach to this problem is an empirical one. We accept the Cosmological Principle because it agrees with observations. We shall describe the observational evidence for this in Chapter 4; data concerning radiogalaxies, clusters of galaxies, quasars and the microwave background all demonstrate that the level of anisotropy of the Universe on large scales is about one part in 105 . 6 1.2 First Principles Fundamentals of General Relativity The strongest force of nature on large scales is gravity, so the most important part of a physical description of the Universe is a theory of gravity. The best candidate we have for this is Einstein’s General Theory of Relativity. We therefore begin this chapter with a brief introduction to the basics of this theory. Readers familiar with this material can skip Section 1.2 and resume reading at Section 1.3. In fact, about 90% of this book does not require the use of general relativity at all so readers only interested in a Newtonian treatment may turn directly to Section 1.11. In Special Relativity, the invariant interval between two events at coordinates (t, x, y, z) and (t + dt, x + dx, y + dy, z + dz) is deﬁned by ds 2 = c 2 dt 2 − (dx 2 + dy 2 + dz2 ), (1.2.1) where ds is invariant under a change of coordinate system and the path of a light ray is given by ds = 0. The paths of material particles between any two events are such as to give stationary values of path ds; this corresponds to the shortest distance between any two points being a straight line. This all applies to the motion of particles under no external forces; actual forces such as gravitation and electromagnetism cause particle tracks to deviate from the straight line. Gravitation exerts the same force per unit mass on all bodies and the essence of Einstein’s theory is to transform it from being a force to being a property of space– time. In his theory, the space–time is not necessarily ﬂat as it is in Minkowski space–time (1.2.1) but may be curved. The interval between two events can be written as ds 2 = gij dx i dx j , (1.2.2) where repeated suﬃxes imply summation and i, j both run from 0 to 3; x 0 = ct is the time coordinate and x 1 , x 2 , x 3 are space coordinates. The tensor gij is the metric tensor describing the space–time geometry; we discuss this in much more detail in Section 1.3. As we mentioned above, particle moves in such a way that the integral along its path is stationary: δ ds = 0, (1.2.3) path but such tracks are no longer straight because of the eﬀects of gravitation contained in gij . From Equation (1.2.3), the path of a free particle, which is called a geodesic, can be shown to be described by k l d2 x i i dx dx = 0, + Γ kl ds 2 ds ds (1.2.4) where the Γ s are called Christoﬀel symbols, i Γkl = 12 g im ∂gml ∂gkl ∂gmk + − , ∂x l ∂x k ∂x m (1.2.5) Fundamentals of General Relativity 7 and g im gmk = δik (1.2.6) is the Kronecker delta, which is unity when i = k and zero otherwise. Free particles move on geodesics but the metric gij is itself determined by the matter. The key factor in Einstein’s equations is the relationship between the distribution of matter and the metric describing the space–time geometry. In general relativity all equations are tensor equations. A general tensor is a quantity which transforms as follows when coordinates are changed from x i to x i : ∂x k ∂x l ∂x r ∂x s · · · · · · Amn... (1.2.7) Akl... pq... = r s... , ∂x m ∂x n ∂x p ∂x q where the upper indices are contravariant and the lower are covariant. The difference between these types of index can be illustrated by considering a tensor of rank 1 which is simply a vector (the rank of a tensor is the number of indices it carries). A vector will undergo a transformation according to some rules when the coordinate system in which it is expressed is changed. Suppose we have an original coordinate system x i and we transform it to a new system x k . If the vector A transforms in such a way that A = ∂x k /∂x i A, then the vector A is a contravariant vector and it is written with an upper index, i.e. A = Ai . On the other hand, if the vector transforms according to A = ∂x i /∂x k A, then it is covariant and is written A = Ai . The tangent vector to a curve is an example of a contravariant vector; the normal to a surface is a covariant vector. The rule (1.2.7) is a generalisation of these concepts to tensors of arbitrary rank and to tensors of mixed character. In Newtonian and specialrelativistic physics a key role is played by conservation laws of mass, energy and momentum. Our task is now to obtain similar laws for general relativity. With the equivalence of mass and energy brought about by Special Relativity, these laws can be written ∂Tik = 0. ∂x k (1.2.8) The energy–momentum tensor Tik describes the matter distribution: for a perfect ﬂuid, with pressure p and energy density ρ, it is Tik = (p + ρc 2 )Ui Uk − pgik ; (1.2.9) the vector Ui is the ﬂuid fourvelocity Ui = gik U k = gik dx k , ds (1.2.10) where x k (s) is the world line of a ﬂuid element, i.e. the trajectory in space–time followed by the particle. Equation (1.2.10) is a special case of the general rule for raising or lowering suﬃxes using the metric tensor. 8 First Principles It is easy to see that the Equation (1.2.8) cannot be correct in general relativity since ∂T ik /∂x k and ∂Tik /∂x k are not tensors. Since Tmn = ∂x i ∂x k Tik , ∂x m ∂x n it is evident that ∂Tmn /∂x n involves terms such as ∂ 2 x i /∂x m ∂x n , so it will not be a tensor. However, although the ordinary derivative of a tensor is not a tensor, a quantity called the covariant derivative can be shown to be one. The covariant derivative of a tensor A is deﬁned by Akl... pq...;j = ∂Akl... pq... ∂x j k l s r kn... kl... kl... + Γmj Aml... pq... + Γnj Apq... + · · · − Γpj Ar q... − Γqj Aps... − · · · (1.2.11) in an obvious notation. The conservation law can therefore be written in a fully covariant form: Ti k;k = 0. (1.2.12) A covariant derivative is usually written as a ‘;’ in the subscript; ordinary derivatives are usually written as a ‘,’ so that Equation (1.2.8) can be written Tik,k = 0. Einstein wished to ﬁnd a relation between matter and metric and to equate Tik to a tensor obtained from gik , which contains only the ﬁrst two derivatives of gik and has zero covariant derivative. Because, in the appropriate limit, Equation (1.2.12) must reduce to Poisson’s equation describing Newtonian gravity ∇2 ϕ = 4π Gρ, (1.2.13) it should be linear in the second derivative of the metric. The properties of curved spaces were wellknown when Einstein was working on this theory. For example, it was known that the Riemann–Christoﬀel tensor, i Rklm = i i ∂Γkm ∂Γkl i n n i − + Γnl Γkm − Γnm Γkl , ∂x l ∂x m (1.2.14) could be used to determine whether a given space is curved or ﬂat. (Incidentally, i is not a tensor so it is by no means obvious, though it is actually true, that Γkm i Rklm is a tensor.) From the Riemann–Christoﬀel tensor one can form the Ricci tensor : Rik = R lilk . (1.2.15) Finally, one can form a scalar curvature, the Ricci scalar : R = g ik Rik . (1.2.16) Now we are in a position to deﬁne the Einstein tensor 1 Gik ≡ Rik − 2 gik R. (1.2.17) The Robertson–Walker Metric 9 Einstein showed that Gi k ;k = 0. (1.2.18) The tensor Gik contains second derivatives of gik , so Einstein proposed as his fundamental equation Gik ≡ Rik − 12 gik R = 8π G Tik , c4 (1.2.19) where the quantity 8π G/c 4 (G is Newton’s gravitational constant) ensures that Poisson’s equation in its standard form (1.2.13) results in the limit of a weak gravitational ﬁeld. He subsequently proposed the alternative form 1 Gik ≡ Rik − 2 gik R − Λgik = 8π G Tik , c4 (1.2.20) where Λ is called the cosmological constant ; as gi k ;k = 0, we still have Ti k;k = 0. He actually did this in order to ensure that static cosmological solutions could be obtained. We shall return to be the issue of Λ later, in Section 1.12. 1.3 The Robertson–Walker Metric Having established the idea of the Cosmological Principle, our task is to see if we can construct models of the Universe in which this principle holds. Because general relativity is a geometrical theory, we must begin by investigating the geometrical properties of homogeneous and isotropic spaces. Let us suppose we can regard the Universe as a continuous ﬂuid and assign to each ﬂuid element the three spatial coordinates x α (α = 1, 2, 3). Thus, any point in space–time can be labelled by the coordinates x α , corresponding to the ﬂuid element which is passing through the point, and a time parameter which we take to be the proper time t measured by a clock moving with the ﬂuid element. The coordinates x α are called comoving coordinates. The geometrical properties of space–time are described by a metric; the meaning of the metric will be divulged just a little later. One can show from simple geometrical considerations only (i.e. without making use of any ﬁeld equations) that the most general space–time metric describing a universe in which the Cosmological Principle is obeyed is of the form ds 2 = (c dt)2 − a(t)2 dr 2 2 2 2 2 + r (dϑ + sin ϑ dϕ ) , 1 − Kr 2 (1.3.1) where we have used spherical polar coordinates: r , ϑ and ϕ are the comoving coordinates (r is by convention dimensionless); t is the proper time; a(t) is a function to be determined which has the dimensions of a length and is called the cosmic scale factor or the expansion parameter ; the curvature parameter K is a constant which can be scaled in such a way that it takes only the values 1, 0 or −1. The metric (1.3.1) is called the Robertson–Walker metric. 10 First Principles The signiﬁcance of the metric of a space–time, or more speciﬁcally the metric tensor gik , which we introduced brieﬂy in Equation (1.2.2), ds 2 = gik (x) dx i dx k (i, k = 0, 1, 2, 3) (1.3.2) (as usual, repeated indices imply a summation), is such that, in Equation (1.3.2), ds 2 represents the space–time interval between two points labelled by x j and x j + dx j . Equation (1.3.1) merely represents a special case of this type of relation. The metric tensor determines all the geometrical properties of the space–time described by the system of coordinates x j . It may help to think of Equation (1.3.2) as a generalisation of Pythagoras’s theorem. If ds 2 > 0, then the interval is timelike and ds/c would be the time interval measured by a clock which moves freely between x j and x j + dx j . If ds 2 < 0, then the interval is spacelike and ds 2 1/2 represents the length of a ruler with ends at x j and x j + dx j measured by an observer at rest with respect to the ruler. If ds 2 = 0, then the interval is lightlike or null; this type of interval is important because it means that the two points x j and x j + dx j can be connected by a light ray. If the distribution of matter is uniform, then the space is uniform and isotropic. This, in turn, means that one can deﬁne a universal time (or proper time) such that at any instant the threedimensional spatial metric dl2 = γαβ dx α dx β (α, β = 1, 2, 3), (1.3.3) where the interval is now just the spatial distance, is identical in all places and in all directions. Thus, the space–time metric must be of the form ds 2 = (c dt)2 − dl2 = (c dt)2 − γαβ dx α dx β . (1.3.4) This coordinate system is called the synchronous gauge and is the most commonly used way of slicing the fourdimensional space–time into three space dimensions and one time dimension. To ﬁnd the threedimensional (spatial) metric tensor γαβ let us consider ﬁrst the simpler case of an isotropic and homogeneous space of only two dimensions. Such a space can be either (i) the usual Cartesian plane (ﬂat Euclidean space with inﬁnite curvature radius), (ii) a spherical surface of radius R (a curved space with positive Gaussian curvature 1/R 2 ), or (iii) the surface of a hyperboloid (a curved space with negative Gaussian curvature). In the ﬁrst case the metric, in polar coordinates ρ (0 ρ < ∞) and ϕ (0 ϕ < 2π ), is of the form dl2 = a2 (dr 2 + r 2 dϕ2 ); (1.3.5 a) we have introduced the dimensionless coordinate r = ρ/a, which lies in the range 0 r < ∞, and the arbitrary constant a, which has the dimensions of a length. On the surface of a sphere of radius R the metric in coordinates ϑ (0 ϑ π ) and ϕ (0 ϕ < 2π ) is just dr 2 2 2 dl2 = a2 (dϑ2 + sin2 ϑ dϕ2 ) = a2 , (1.3.5 b) + r dϕ 1 − r2 The Robertson–Walker Metric 11 where a = R and the dimensionless variable r = sin ϑ lies in the interval 0 r 1 (r = 0 at the poles and r = 1 at the equator). In the hyperboloidal case the metric is given by dl2 = a2 (dϑ2 + sinh2 ϑ dϕ2 ) = a2 dr 2 2 2 + r dϕ , 1 + r2 (1.3.5 c) where the dimensionless variable r = sinh ϑ lies in the range 0 r < ∞. The Robertson–Walker metric is obtained from (1.3.4), where the spatial part is simply the threedimensional generalisation of (1.3.5). One ﬁnds that for the threedimensional ﬂat, positively curved and negatively curved spaces one has, respectively, dl2 = a2 (dr 2 + r 2 dΩ 2 ), (1.3.6 a) dr 2 + r 2 dΩ 2 , 1 − r2 dr 2 2 2 dl2 = a2 (dχ 2 + sinh2 χ dΩ 2 ) = a2 + r dΩ , 1 + r2 dl2 = a2 (dχ 2 + sin2 χ dΩ 2 ) = a2 (1.3.6 b) (1.3.6 c) where dΩ 2 = dϑ2 + sin2 ϑ dϕ2 ; 0 χ π in (1.3.6 b) and 0 χ < ∞ in (1.3.6 c). The values of K = 1, 0, −1 in (1.3.1) correspond, respectively, to the hypersphere, Euclidean space and space of constant negative curvature. The geometrical properties of Euclidean space (K = 0) are well known. On the other hand, the properties of the hypersphere (K = 1) are complex. This space is closed, i.e. it has ﬁnite volume, but has no boundaries. This property is clear by analogy with the twodimensional case of a sphere: beginning from a coordinate origin at the pole, the surface inside a radius rc (ϑ) = aϑ has an area S(ϑ) = 2π a2 (1 − cos ϑ), which increases with rc and has a maximum value Smax = 4π a2 at ϑ = π . The perimeter of this region is L(ϑ) = 2π a sin ϑ = 2π ar , which is 1 maximum at the ‘equator’ (ϑ = 2 π ), where it takes the value 2π a, and is zero at the ‘antipole’ (ϑ = π ): the sphere is therefore a closed surface, with ﬁnite area and no boundary. In the threedimensional case the volume of the region contained inside a radius rc (χ) = aχ = a sin−1 r (1.3.7) has volume V (χ) = 2π a3 (χ − 1 2 sin 2χ), (1.3.8) which increases and has a maximum value for χ = π , Vmax = 2π 2 a3 , (1.3.9) and area S(χ) = 4π a2 sin2 χ, (1.3.10) 12 First Principles Figure 1.1 Examples of curved spaces in two dimensions: in a space with negative curvature (open), for example, the sum of the internal angles of a triangle is less than 180◦ , while for a positively curved space (closed) it is greater. maximum at the ‘equator’ (χ = 12 π ), where it takes the value 4π a2 , and is zero at the ‘antipole’ (χ = π ). In such a space the value of S(χ) is more than in Euclidean space, and the sum of the internal angles of a triangle is more than π . The properties of a space of constant negative curvature (K = −1) are more similar to those of Euclidean space: the hyperbolic space is open, i.e. inﬁnite. All the relevant formulae for this space can be obtained from those describing the hypersphere by replacing trigonometric functions by hyperbolic functions. One can show, for example, that S(χ) is less than the Euclidean case, and the sum of the internal angles of a triangle is less than π . In cases with K ≠ 0, the parameter a, which appears in (1.3.1), is related to the curvature of space. In fact, the Gaussian curvature is given by CG = K/a2 ; as expected it is positive for the closed space and √ negative for the open space. −1/2 = a/ K is, respectively, positive or The Gaussian curvature radius RG = CG imaginary in these two cases. In cosmology one uses the term radius of curvature to describe the modulus of RG ; with this convention a always represents the radius of spatial curvature. Of course, in a ﬂat universe the parameter a does not have any geometrical signiﬁcance. As we shall see later in this chapter, the Einstein equations of general relativity relate the geometrical properties of space–time with the energy–momentum tensor describing the contents of the Universe. In particular, for a homogeneous and isotropic perfect ﬂuid with restmass energy density ρc 2 and pressure p, the The Hubble Law 13 solutions of the Einstein equations are the Friedmann cosmological equations: p 4 (1.3.11 a) ä = − 3 π G ρ + 3 2 a, c ȧ2 + Kc 2 = 83 π Gρa2 (1.3.11 b) (the dot represents a derivative with respect to cosmological proper time t); the time evolution of the expansion parameter a which appears in the Robertson– Walker metric (1.3.1) can be derived from (1.3.11) if one has an equation of state relating p to ρ. From Equation (1.3.11 b) one can derive the curvature 1 ȧ 2 ρ K = 2 −1 , (1.3.12) a2 c a ρc where ρc = 2 3 ȧ 8π G a (1.3.13) is called the critical density. The space is closed (K = 1), ﬂat (K = 0) or open (K = −1) according to whether the density parameter Ω(t) = ρ ρc (1.3.14) is greater than, equal to, or less than unity. It will sometimes be useful to change the time variable we use from proper time to conformal time: dt ; (1.3.15) τ= a(t) with such a time variable the Robertson–Walker metric becomes dr 2 2 2 ds 2 = a(τ)2 (c dτ)2 − + r dΩ . 1 − Kr 2 (1.3.16) 1.4 The Hubble Law The proper distance, dP , of a point P from another point P0 , which we take to deﬁne the origin of a set of polar coordinates r , ϑ and ϕ, is the distance measured by a chain of rulers held by observers which connect P to P0 at time t. From the Robertson–Walker metric (1.3.1) with dt = 0 this can be seen to be r a dr dP = = af (r ), (1.4.1) 2 1/2 0 (1 − Kr ) where the function f (r ) is, respectively, f (r ) = sin−1 r (K = 1), (1.4.2 a) f (r ) = r (K = 0), (1.4.2 b) f (r ) = sinh−1 r (K = −1). (1.4.2 c) 14 First Principles Of course this proper distance is of little operational signiﬁcance because one can never measure simultaneously all the distance elements separating P from P0 . The proper distance at time t is related to that at the present time t0 by dP (t0 ) = a0 f (r ) = a0 dP (t), a (1.4.3) where a0 is the value of a(t) at t = t0 . Instead of the comoving coordinate r one could also deﬁne a radial comoving coordinate of P by the quantity dc = a0 f (r ). (1.4.4) In this case the relation between comoving coordinates and proper coordinates is just a0 dP . dc = (1.4.5) a The proper distance dP of a source may change with time because of the timedependence of the expansion parameter a. In this case a source at P has a radial velocity with respect to the origin P0 given by vr = ȧf (r ) = ȧ dP . a (1.4.6) Equation (1.4.6) is called the Hubble law and the quantity H(t) = ȧ/a (1.4.7) is called the Hubble constant or, more accurately, the Hubble parameter (because it is not constant in time). As we shall see, the value of this parameter evaluated at the present time for our Universe, H(t0 ) = H0 , is not known to any great accuracy. It is believed, however, to have a value around H0 65 km s−1 Mpc−1 . (1.4.8) The unit ‘Mpc’ is deﬁned later on in Section 4.1. It is conventional to take account of the uncertainty in H0 by deﬁning the dimensionless parameter h to be H0 /100 km s−1 Mpc−1 (see Section 4.2). The law (1.4.6) can, in fact, be derived directly from the Cosmological Principle if v c. Consider a triangle deﬁned by the three spatial points O, O and P. Let the velocity of P and O with respect to O be, respectively, v(r) and v(d). The velocity of P with respect to O is v (r ) = v(r) − v(d). (1.4.9) From the Cosmological Principle the functions v and v must be the same. Therefore v(r − d) = v (r − d) = v(r) − v(d). (1.4.10) 15 Redshift Equation (1.4.10) implies a linear relationship between v and r: β vα = Hα xβ (α, β = 1, 2, 3). (1.4.11) If we impose the condition that the velocity ﬁeld is irrotational, ∇ × v = 0, (1.4.12) β which comes from the condition of isotropy, one can deduce that the matrix Hα is symmetric and can therefore be diagonalised by an appropriate coordinate transformation. From isotropy, the velocity ﬁeld must therefore be of the form vi = Hxi , (1.4.13) where H is only a function of time. Equation (1.4.13) is simply the Hubble law (1.4.6). Another, simpler, way to derive Equation (1.4.6) is the following. The points O, O and P are assumed to be suﬃciently close to each other that relativistic space– time curvature eﬀects are negligible. If the universe evolves in a homogeneous and isotropic manner, the triangle OO P must always be similar to the original triangle. This means that the length of all the sides must be multiplied by the same factor a/a0 . Consequently, the distance between any two points must also be multiplied by the same factor. We therefore have l= a l0 , a0 (1.4.14) where l0 and l are the lengths of a line segment joining two points at times t0 and t, respectively. From (1.4.14) we recover immediately the Hubble law (1.4.6). One property of the Hubble law, which is implicit in the previous reasoning, is that we can treat any spatial position as the origin of a coordinate system. In fact, referring again to the triangle OO P, we have vP = vO + vP = Hd + vP = Hr (1.4.15) vP = H(r − d) = Hr , (1.4.16) and, therefore, which again is just the Hubble law, this time expressed about the point O . 1.5 Redshift It is useful to introduce a new variable related to the expansion parameter a which is more directly observable. We call this variable the redshift z and we shall use it extensively from now on in describing the evolution of the Universe because many of the relevant formulae are very simple when expressed in terms of this variable. 16 First Principles We deﬁne the redshift of a luminous source, such as a distant galaxy, by the quantity λ0 − λe , (1.5.1) z= λe where λ0 is the wavelength of radiation from the source observed at O (which we take to be the origin of our coordinate system) at time t0 and emitted by the source at some (earlier) time te ; the source is moving with the expansion of the universe and is at a comoving coordinate r . The wavelength of radiation emitted by the source is λe . The radiation travels along a light ray (null geodesic) from the source to the observer so that ds 2 = 0 and, therefore, t0 te c dt = a(t) r 0 dr = f (r ). (1 − Kr 2 )1/2 (1.5.2) Light emitted from the source at te = te + δte reaches the observer at t0 = t0 + δt0 . Given that f (r ) does not change, because r is a comoving coordinate and both the source and the observer are moving with the cosmological expansion, we can write t0 c dt = f (r ). (1.5.3) t a(t) If δt and, therefore, δt0 are small, Equations (1.5.2) and (1.5.3) imply that δt0 δt . = a0 a (1.5.4) If, in particular, δt = 1/νe and δt0 = 1/ν0 (νe and ν0 are the frequencies of the emitted and observed light, respectively), we will have νe a = ν0 a0 (1.5.5) a0 a = , λe λ0 (1.5.6) or, equivalently, from which 1+z = a0 . a (1.5.7) A line of reasoning similar to the previous one can be made to recover the evolution of the velocity vp (t) of a test particle with respect to a comoving observer. At time t + dt the particle has travelled a distance dl = vp (t) dt and thus ﬁnds itself moving with respect to a new reference frame which, because of the expansion of the universe, has an expansion velocity dv = (ȧ/a) dl. The velocity of the particle with respect to the new comoving observer is therefore vp (t + dt) = vp (t) − ȧ ȧ dl = vp (t) − vp (t) dt, a a (1.5.8) The Deceleration Parameter 17 which, integrated, gives vp ∝ a−1 . (1.5.9) The results expressed by Equations (1.5.5) and (1.5.11) are a particular example of the fact that, in a universe described by the Robertson–Walker metric, the momentum q of a free particle (whether relativistic or not) evolves according to q ∝ a−1 . There is also a simply way to recover Equation (1.5.7), which does not require any knowledge of the metric. Consider two nearby points P and P , participating in the expansion of the Universe. From the Hubble law we have dvP = H dl = ȧ dl, a (1.5.10) where dvP is the relative velocity of P with respect to P and dl is the (inﬁnitesimal) distance between P and P . The point P sends a light signal at time t and frequency ν which arrives at P with frequency ν at time t + dt = t + (dl/c). Since dl is inﬁnitesimal, as is dvP , we can apply the approximate formula describing the Doppler eﬀect : ν − ν dν dvP da ȧ = − = − dt = − . (1.5.11) ν ν c a a The Equation (1.5.11) integrates immediately to give (1.5.5) and therefore (1.5.7). 1.6 The Deceleration Parameter The Hubble parameter H(t) measures the expansion rate at any particular time t for any model obeying the Cosmological Principle. It does, however, vary with time in a way that depends upon the contents of the Universe. One can express this by expanding the cosmic scale factor for times t close to t0 in a power series: 1 a(t) = a0 [1 + H0 (t − t0 ) − 2 q0 H02 (t − t0 )2 + · · · ], (1.6.1) where q0 = − ä(t0 )a0 ȧ(t0 )2 (1.6.2) is called the deceleration parameter ; the suﬃx ‘0’, as always, refers to the fact that q0 = q(t0 ). Note that while the Hubble parameter has the dimensions of inverse time, q is actually dimensionless. Putting the redshift, deﬁned by Equation (1.5.7), into Equation (1.6.1) we ﬁnd that z = H0 (t0 − t) + (1 + 12 q0 )H02 (t0 − t)2 + · · · , (1.6.3) which can be inverted to yield t0 − t = 1 1 [z − (1 + 2 q0 )z2 + · · · ]. H0 (1.6.4) 18 First Principles To ﬁnd r as a function of z one needs to recall that, for a light ray, t0 t c dt = a r 0 dr , (1 − Kr 2 )1/2 (1.6.5) which becomes, using Equations (1.5.7) and (1.6.3), c a0 t0 t [1 + H0 (t0 − t) + (1 + 12 q0 )H02 (t0 − t)2 + · · · ] dt = r + O(r 3 ), (1.6.6) and therefore r = c [(t0 − t) + 12 H0 (t0 − t)2 + · · · ]. a0 (1.6.7) Substituting Equation (1.6.4) into (1.6.7) we have, ﬁnally, r = c [z − 12 (1 + q0 )z2 + · · · ]. a0 H 0 (1.6.8) Expressions of this type are useful because they do not require full solutions of the Einstein equations for a(t); the quantity q0 is used to parametrise a family of approximate solutions for t close to t0 . 1.7 Cosmological Distances We have shown how the comoving coordinate system we have adopted relates to proper distance (i.e. distances measured in a hypersurface of constant proper time) in spaces described by the Robertson–Walker metric. Obviously, however, we cannot measure proper distances to astronomical objects in any direct way. Distant objects are observed only through the light they emit which takes a ﬁnite time to travel to us; we cannot therefore make measurements along a surface of constant proper time, but only along the set of light paths travelling to us from the past – our past light cone. One can, however, deﬁne operationally other kinds of distance which are, at least in principle, directly measurable. One such distance is the luminosity distance dL . This is deﬁned in such a way as to preserve the Euclidean inversesquare law for the diminution of light with distance from a point source. Let L denote the power emitted by a source at a point P, which is at a coordinate distance r at time t. Let l be the power received per unit area (i.e. the ﬂux) at time t0 by an observer placed at P0 . We then deﬁne dL = L 4π l 1/2 . (1.7.1) The area of a spherical surface centred on P and passing through P0 at time t0 is just 4π a20 r 2 . The photons emitted by the source arrive at this surface having been redshifted by the expansion of the universe by a factor a/a0 . Also, as we Cosmological Distances 19 have seen, photons emitted by the source in a small interval δt arrive at P0 in an interval δt0 = (a0 /a)δt due to a timedilation eﬀect. We therefore ﬁnd 2 a L , l= 4π a20 r 2 a0 (1.7.2) from which dL = a20 r . a (1.7.3) Following the same procedure as in Section 1.6, one can show that dL = c [z + 12 (1 − q0 )z2 + · · · ], H0 (1.7.4) in contrast with the proper distance, dP , deﬁned by Equation (1.4.1), which has the form dP = a0 r , with f (r ) given by Equations (1.4.2). Next we deﬁne the angulardiameter distance dA . Again, this is constructed in such a way as to preserve a geometrical property of Euclidean space, namely the variation of the angular size of an object with its distance from an observer. Let DP (t) be the (proper) diameter of a source placed at coordinate r at time t. If the angle subtended by DP is denoted ∆ϑ, then Equation (1.2.1) implies DP = ar ∆ϑ. (1.7.5) We deﬁne dA to be the distance dA = DP = ar ; ∆ϑ (1.7.6) it should be noted that a decreases as r increases for the same DP and, in some models, the angular size of a source can actually increase with its luminosity distance. Other measures of distance, less often used, are the parallax distance d µ = a0 r , (1 − Kr 2 )1/2 (1.7.7) and the proper motion distance dM = a0 r . (1.7.8) Evidently, for r → 0, and therefore for t → t0 , we have dp dL dA dµ dM dc , so that at small distances we recover the Euclidean behaviour. (1.7.9) 20 First Principles 1.8 The m–z and N–z Relations The general relationship we have established between redshift and distance allows us to establish some interesting properties of the Universe which could, in principle, be used to probe its spatial geometry and, in particular, to test the Cosmological Principle. In fact, there are severe complications with the implementation of this idea, as we discuss in Section 4.7. If celestial objects (such as galaxy clusters, galaxies, radio sources, quasars, etc.) are distributed homogeneously and isotropically on large scales, it is interesting to consider two relationships: the m–z relationship between the apparent magnitude of a source and its redshift and the N(> l)–z relationship between the number of sources of a given type with apparent luminosity greater than some limit l and redshift less than z. These relations are also important because, in principle, they provide a way of determining the deceleration parameter q0 . As we have seen previously, dL = c [z + 12 (1 − q0 )z2 + · · · ], H0 (1.8.1) from which l= LH02 L = [1 + (q0 − 1)z + · · · ]. 4π c 2 z2 4π d2L (1.8.2) Astronomers do not usually work with the absolute luminosity L and apparent ﬂux l. Instead they work with quantities related to these: the absolute magnitude M and the apparent magnitude m (for more details see Section 4.1). The magnitude scale is deﬁned logarithmically by taking a factor of 100 in received ﬂux to be a diﬀerence of 5 magnitudes. The zeropoint can be ﬁxed in various ways; for historical reasons it is conventional to take Polaris to have an apparent magnitude of 2.12 in visible light but diﬀerent choices can and have been made. The absolute magnitude is deﬁned to be the apparent magnitude the source would have if it were placed at a distance of 10 parsec. The relationship between the luminosity distance of a source, its apparent magnitude m and its absolute magnitude M is, therefore, just dL = 101+(m−M)/5 pc. (1.8.3) m − M = −5 + 5 log dL (pc) (1.8.4) The quantity is called the distance modulus. Using Equation (1.8.2) we ﬁnd m−M 25 − 5 log10 H0 + 5 log cz + 1.086(1 − q0 )z + · · · , (1.8.5) with H0 in km s−1 Mpc−1 and c in km s−1 . Here one should remember that 1 Mpc = 106 pc and the logarithms are always deﬁned to the base 10. The behaviour of m(z) is sensitive to the value of q0 only for z > 0.1. In reality, as we shall see, there are many other factors which intervene in this type of analysis with the The m–z and N–z Relations 21 result that we can say very little about q0 , or even its sign. In the regime where it is accurate, that is for z < zmax 0.2, Equation (1.8.5) can provide an estimate of H0 , together with a strong conﬁrmation of the validity of the Hubble law and, therefore, of the Cosmological Principle. Another test of this principle is the socalled Hubble test, which relates the number N(> l) of sources of a particular type with apparent luminosity greater than l as a function of l. If the Universe were Euclidean and galaxies all had the same absolute luminosity L, and were distributed uniformly with mean numberdensity n0 , we would have 4 N(l) = 3 π n0 d3l , with dl given by dl = L 4π l (1.8.6) 1/2 , (1.8.7) from which N(l) ∝ l−3/2 (1.8.8) and, therefore, introducing the apparent magnitude in the form m = 2.5 log10 l + const., log N(l) = 0.6m + const. (1.8.9) Equation (1.8.9) is also true if the sources have an arbitrary distribution of luminosities around L; in this case all that changes is the value of the constant. In the nonEuclidean case we have r n[t(r )]a[t(r )]3 r 2 dr , (1.8.10) N(l) = 4π (1 − Kr 2 )1/2 0 where t(r ) is the time at which a source at r emitted a light signal which arrives now at the observer. If the galaxies are neither created nor destroyed in the interval t(r ) < t < t0 , so that na3 = n0 a30 , we see that, upon expanding as a power series, Equation (1.8.10) leads to N(l) = 4π n0 a30 ( 13 r 3 + 1 5 10 Kr + · · · ). (1.8.11) Recalling that r = c [z − 12 (1 + q0 )z2 + · · · ], a0 H0 (1.8.12) Equation (1.8.11) becomes log N(l) = 3 log z − 0.651(1 + q0 )z + const., (1.8.13) from which one can, in principle, recover q0 . In practice, however, there are many eﬀects (the most important being various evolutionary phenomena) which eﬀectively mean that the constant terms in the above equations all actually depend on z. Nevertheless, Equation (1.8.13) works well for z < 0.2, where the term in q0 is negligible and the constant is, eﬀectively, constant. 22 First Principles 1.9 Olbers’ Paradox Having established the behaviour of light in the expanding relativistic cosmology, it is worth revisiting an idea from the prerelativistic era. Before the development of relativity, astronomers generally believed the Universe to be inﬁnite, homogeneous, Euclidean and static. This picture was of course shattered by the discovery of the Hubble expansion in 1929, which we discuss in Chapter 4. It is nevertheless interesting to point out that this model, which we might call the Eighteenth Century Universe, gave rise to an interesting puzzle now known as Olbers’ Paradox (Olbers 1826). As a matter of fact, Olbers’ Paradox had previously been analysed by a number of others, including (incorrectly) Halley (1720) and (correctly) Loys de Chéseaux (1744). The argument proceeds from the simple observation that the night sky is quite dark. In an Eighteenth Century Universe, the apparent luminosity l of a star of absolute luminosity L placed at a distance r from an observer is just L (1.9.1) l= 4π r 2 if one neglects absorption. This is the same as Equation (1.7.1). Let us assume, for simplicity, that all stars have the same absolute luminosity and the (constant) number density of stars per unit volume is n. The radiant energy arriving at the observer from the whole Universe is then ∞ ∞ L 2 4π r dr = nL dr , (1.9.2) ltot = 2 0 4π r 0 which is inﬁnite. This is the Olbers Paradox. It was thought in the past that this paradox could be resolved by postulating the presence of interstellar absorption, perhaps by dust; such an explanation was actually advanced by Lord Kelvin in the 19th century. What would happen if this were the case would be that, after a sufﬁcient time, the absorbing material would be brought into thermodynamic equilibrium with the radiation and would then emit as much radiation as it absorbed, though perhaps in a diﬀerent region of the electromagnetic spectrum. To be fair to Kelvin, however, one should mention that at that time it was not known that light and heat were actually diﬀerent aspects of the same phenomenon, so the argument was reasonable given what was then known about the nature of radiation. In the modern version of the expanding Universe the conditions necessary for an Olbers Paradox to arise are violated in a number of ways we shall discuss later: the light from a distant star would be redshifted; the spatial geometry is not necessarily ﬂat; the Universe may not be inﬁnite in spatial or temporal extent. In fact, the basic reason why an Olbers Paradox does not arise in modern cosmological theories is much simpler than any of these possibilities. The key fact is that no star can burn for an inﬁnite time: a star of mass M can at most radiate only so long as it takes to radiate away its rest energy Mc 2 . As one looks further and further out into space, one must see stars which are older and older. In order for them all, out to inﬁnite distance, to be shining light that we observe now, they must The Friedmann Equations 23 have switched on at diﬀerent times depending on their distance from us. Such a coordination is not only unnatural, it also requires us to be in a special place. So an Olbers Paradox would only really be expected to happen if the Universe were actually inhomogeneous on large scales and the Copernican Principle were violated. The other eﬀects mentioned above are important in determining the exact amount of radiation received by an observer from the cosmological background, but any cosmology that respects the relativistic notion that E = mc 2 (and the Cosmological Principle) is not expected to have an inﬁnitely bright night sky. Exactly how much background light there is in the Universe is an observation that can in principle be used to test cosmological models in much the same way as the numbercounts discussed in Section 1.8. 1.10 The Friedmann Equations So far we have developed much of the language of modern relativistic cosmology without actually using the ﬁeld Equations (1.2.20). We have managed to discuss many important properties of the universe in terms of geometry or using simple kinematics. To go further we must use general relativity to relate the geometry of space–time, expressed by the metric tensor gij (xk ), to the matter content of the universe, expressed by the energy–momentum tensor Tij (xk ). The Einstein equations (without the cosmological constant; see Section 1.12) are Rij − 12 gij R = 8π G Tij , c4 (1.10.1) where Rij and R are the Ricci tensor and Ricci scalar, respectively. A test particle moves along a space–time geodesic, that is a trajectory in a fourdimensional space whose ‘length’ is stationary with respect to small variations in the trajectory. In cosmology, the energy–momentum tensor which is of greatest relevance is that of a perfect ﬂuid: Tij = (p + ρc 2 )Ui Uj − pgij , (1.10.2) where p is the pressure, ρc 2 is the energy density (which includes the restmass energy), and Uk is the ﬂuid fourvelocity, deﬁned by Equation (1.2.10). If the metric is of Robertson–Walker type, the Einstein equations then yield p 4π G ρ + 3 2 a, (1.10.3) ä = − 3 c for the time–time component, and p aä + 2ȧ2 + 2Kc 2 = 4π G ρ − 2 a2 , c (1.10.4) for the space–space components. The space–time components merely give 0 = 0. Eliminating ä from (1.10.3) and (1.10.4) we obtain 8 ȧ2 + Kc 2 = 3 π Gρa2 . (1.10.5) 24 First Principles In reality, as we shall see, Equations (1.10.3) and (1.10.5) – the Friedmann equations – are not independent: the second can be recovered from the ﬁrst if one takes the adiabatic expansion of the universe into account, i.e. d(ρc 2 a3 ) = −p da3 . (1.10.6 a) The last equation can also be expressed as ṗa3 = or 1.11 d [a3 (ρc 2 + p)] dt p ȧ = 0. ρ̇ + 3 ρ + 2 c a (1.10.6 b) (1.10.6 c) A Newtonian Approach Before proceeding further, it is worth demonstrating how one can actually get most of the way towards the Friedmann equations using only Newtonian arguments. Birkhoﬀ’s theorem (1923) proves that a spherically symmetric gravitational ﬁeld in an empty space is static and is always described by the Schwarzschild exterior metric (i.e. the metric generated in empty space by a point mass). This property is very similar to a result proved by Newton and usually known as Newton’s spherical theorem which is based on the application of Gauss’s theorem to the gravitational ﬁeld. In the Newtonian version the gravitational ﬁeld outside a spherically symmetric body is the same as if the body had all its mass concentrated at its centre. Birkhoﬀ’s theorem can also be applied to the ﬁeld inside an empty spherical cavity at the centre of a homogeneous spherical distribution of mass–energy, even if the distribution is not static. In this case the metric inside the cavity is the Minkowski (ﬂatspace) metric: gij = ηij (ηij = −1 for i = j = 1, 2, 3; ηij = 1 for i = j = 0; ηij = 0 for i ≠ j). This corollary of Birkhoﬀ’s theorem also has a Newtonian analogue: the gravitational ﬁeld inside a homogeneous spherical shell of matter is always zero. This corollary can also be applied if the space outside the cavity is inﬁnite: the only condition that must be obeyed is that the distribution of mass–energy must be spherically symmetric. A proof of Birkhoﬀ’s theorem is beyond the scope of this book, but we will use its existence to justify a Newtonian approach to the timeevolution of a homogeneous and isotropic distribution of material. Let us consider the evolution of the mass m contained inside a sphere of radius l centred at the point O in such a universe. By Birkhoﬀ’s theorem the space inside the sphere is ﬂat. If the radius l is such that Gm 1, (1.11.1) lc 2 one can use Newtonian mechanics to describe the behaviour of the particle. Equation (1.11.1) means in eﬀect that the freefall time for the sphere, τﬀ (Gρ)−1/2 , is A Newtonian Approach 25 much greater than the lightcrossing time τ l/c. Alternatively, Equation (1.11.1) means that the radius of the sphere is much larger than the Schwarzschild radius corresponding to the mass m, rS = 2mG/c 2 . As we have seen in Section 1.4, the Cosmological Principle requires that l = dc a , a0 (1.11.2) where a is the expansion parameter of the universe which, according to our conventions, has the dimensions of a length, while the comoving coordinate dc is dimensionless. One can always pick dc small enough so that at any instant the inequality (1.11.1) is satisﬁed. We shall see, however, that this quantity actually disappears from the formulae. Applying a Newtonian approximation to describe the motion of a unit mass at a point P on the surface of the sphere yields d2 l Gm 4π Gρl, =− 2 =− dt 2 l 3 (1.11.3) d l̇2 Gm d Gm = − 2 l̇ = , dt 2 l dt l (1.11.4) or, multiplying by l̇, and, integrating, l̇2 = 2Gm + C, l (1.11.5) which is nothing more than the law of conservation of energy per unit mass: the constant of integration C is proportional to the total energy. From Equations (1.11.2) and (1.11.5) it is easy to obtain the Equation (1.10.4) in the form ȧ2 + Kc 2 = 83 π Gρa2 by putting C = −K dc c a0 (1.11.6) 2 . (1.11.7) It is clear that, with an appropriate redeﬁnition of dc , one can scale K so as to take the values 1, 0 or −1. The case K = 1 corresponds to C < 0 (negative total energy). In this case the expansion eventually ceases and collapse ensues. In the case K = −1 the total energy is positive, so the expansion never ends. The case K = 0 corresponds to total energy of exactly zero: this represents the ‘escape velocity’ situation where the expansion ceases at t → ∞. Equation (1.11.3) implies that there are no forces due to pressure gradients, which is in accord with our assumption of homogeneity and isotropy. Equation (1.11.6) was obtained under the assumption that the sphere contains only nonrelativistic matter (p ρc 2 ). A result from general relativity shows that, in 26 First Principles the presence of relativistic particles, one should replace the density of matter in Equation (1.11.3) by p ρeﬀ = ρ + 3 2 , (1.11.8) c where ρ now means the energy density (including the restmass energy) divided by c 2 . In this way, Equation (1.11.3) becomes p 4π G ρ + 3 2 a. (1.11.9) ä = − 3 c It is important to note that, from Equation (1.10.6 a), d(ρc 2 a3 r03 ) = −p d(a3 r03 ); (1.11.10) from (1.11.9) one obtains (1.11.6) in both the nonrelativistic (p 0, ρ = ρm ) and ultrarelativistic (p ρc 2 ) cases. In fact Equation (1.11.9), after multiplying by ȧ, gives p 1 d 2 4π ȧ = − G ρaȧ + 3 2 aȧ . (1.11.11) 2 dt 3 c From (1.11.10) we have 3 p aȧ = −3ρaȧ − ρ̇a2 , c2 which, substituted in Equation (1.11.11), yields d 4π 1 d 2 ȧ = Gρa2 . 2 dt dt 3 (1.11.12) (1.11.13) From Equation (1.11.13), by integration, one obtains Equation (1.11.6). What this shows is that, with Birkhoﬀ’s theorem and a reinterpretation of the quantity ρ to take account of intrinsically relativistic eﬀects, we can derive the Friedmann equations using an essentially Newtonian approach. 1.12 The Cosmological Constant Einstein formulated his theory of general relativity without a cosmological constant in 1916; at this time it was generally accepted that the Universe was static. We outlined the development of this theory in Section 1.2, and the ﬁeld equations themselves appear as Equation (1.10.1). A glance at the equation p 4π G ρ+3 2 a ä = − (1.12.1) 3 c shows one that universes evolving according to this theory cannot be static, unless ρ = −3 p ; c2 (1.12.2) The Cosmological Constant 27 in other words, either the energy density or the pressure must be negative. Given that this type of ﬂuid does not seem to be physically reasonable, Einstein (1917) modiﬁed the Equation (1.10.1) by introducing the cosmological constant term Λ: 1 Rij − 2 gij R − Λgij = 8π G Tij ; c4 (1.12.3) as we shall see, with an appropriate choice of Λ, one can obtain a static cosmological model. Equation (1.12.3) represents the most general possible modiﬁcation of the Einstein equations that still satisﬁes the condition that Tij is equal to a tensor constructed from the metric gij and its ﬁrst and second derivatives, and is linear in the second derivative. This modiﬁcation does not change the covariant character of the equations, and does not alter the continuity condition (1.2.12). The strongest constraint one can place on Λ from observations is that it should be suﬃciently small so as not to change the laws of planetary motion, which are known to be well described by (1.10.1). The Equation (1.12.3) can be written in a form similar to (1.10.1) by modifying the energy–momentum tensor: 8π G T̃ij , c4 1 Rij − 2 gij R = (1.12.4) with T̃ij formally given by T̃ij = Tij + Λc 4 gij = −p̃gij + (p̃ + ρ̃c 2 )Ui Uj , 8π G (1.12.5) where the eﬀective pressure p̃ and the eﬀective density ρ̃ are related to the corresponding quantities for a perfect ﬂuid by p̃ = p − Λc 4 , 8π G ρ̃ = ρ + Λc 2 ; 8π G (1.12.6) these relations show that Λ−1/2 has the dimensions of a length. One can then show that, for a universe described by the Robertson–Walker metric, we can get equations which are analogous to (1.10.3) and (1.10.5), respectively: ä = − p̃ 4π G ρ̃ + 3 2 a 3 c (1.12.7) and ȧ2 + Kc 2 = 8π G ρ̃a2 . 3 (1.12.8) These equations admit a static solution for ρ̃ = −3 3Kc 2 p̃ = . 2 c 8π Ga2 (1.12.9) 28 First Principles For a ‘dust’ universe (p = 0), which is a good approximation to our Universe at the present time, Equations (1.12.9) and (1.12.6) give Λ= K , a2 ρ= Kc 2 . 4π Ga2 (1.12.10) Since ρ > 0, we must have K = 1 and therefore Λ > 0. The value of Λ which makes the universe static is just 4π Gρ ΛE = . (1.12.11) c2 The model we have just described is called the Einstein universe. This universe is static (but unfortunately unstable, as one can show), has positive curvature and a curvature radius c −1/2 aE = ΛE = . (1.12.12) (4π Gρ)1/2 After the discovery of the expansion of the Universe in the late 1920s there was no longer any reason to seek static solutions to the ﬁeld equations. The motivation which had led Einstein to introduce his cosmological constant term therefore subsided. Einstein subsequently regarded the Λterm as the biggest mistake he had made in his life. Since then, however, Λ has not died but has been the subject of much interest and serious study on both conceptual and observational grounds. The situation here is reminiscent of Aladdin and the genie: after he released the genie from the lamp, it took on a life of its own. For more than 60 years the genie lingered, providing neither compelling observational evidence of its existence nor strong theoretical reasons for it to be taken seriously. However, observations do now suggest that it may have been there all along. We shall return to this resurgence of Λ in the next chapter and also in Chapter 7, but in the meantime we shall restrict ourselves to brief comments on two particularly important models involving the cosmological constant, because we shall encounter them again when we discuss inﬂation. The de Sitter universe (de Sitter 1917) is a cosmological model in which the universe is empty (p = 0; ρ = 0) and ﬂat (K = 0). From Equations (1.12.6) we get p̃ = −ρ̃c 2 = − Λc 4 , 8π G (1.12.13) which, on substitution in (1.12.8), gives 1 ȧ2 = 3 Λc 2 a2 ; (1.12.14) this equation implies that Λ is positive. Equation (1.12.14) has a solution of the form 1 a = A exp[( 3 Λ)1/2 ct], (1.12.15) corresponding to a Hubble parameter H = ȧ/a = c(Λ/3)1/2 , which is actually constant in time. In the de Sitter vacuum universe, test particles move away from Friedmann Models 29 each other because of the repulsive gravitational eﬀect of the positive cosmological constant. The de Sitter model was only of marginal historical interest until the last 20 years or so. In recent years, however, it has been a major component of inﬂationary universe models in which, for a certain interval of time, the expansion assumes an exponential character of the type (1.12.15). In such a universe the equation of state of the ﬂuid is of the form p −ρc 2 due to quantum eﬀects which we discuss in Chapter 7. In the Lemaître model (1927) the universe has positive spatial curvature (K = 1). One can demonstrate that the expansion parameter in this case is always increasing, but there is a period in which it remains practically constant. This model was invoked around 1970 to explain the apparent concentration of quasars at a redshift of z 2. Subsequent data have, however, shown that this is not the explanation for the redshift evolution of quasars, so this model is again of only marginal historical interest. 1.13 Friedmann Models Having dealt with a few special cases, we now introduce the standard cosmological models described by the solutions (1.10.3) and (1.10.5). Their name derives from A. Friedmann, who derived their properties in 1922. His work was not well known at that time partly because his models were not static, and the discovery of the Hubble expansion was still some way in the future. His work was in any case not widely circulated in the western scientiﬁc literature. Independently, and somewhat later, the Belgian priest George Lemaître obtained essentially the same results and his work achieved more immediate attention, especially in England where he was championed by Eddington. When the work of Lemaître (1927) was published, Hubble’s observations were just becoming known, so in the West Lemaître is often credited with being the father of the Big Bang cosmology, although that honour should probably be conferred on Friedmann. The Friedmann models are so important that we shall devote the next chapter to their behaviour. Here we shall just whet the readers appetite with some basic properties. First, we assume a perfect ﬂuid with some density ρ and pressure p. The form of equation of state giving p as a function of ρ does not concern us for now; we discuss it in Section 2.1. For the moment we also ignore the cosmological constant. The equations we need to solve are (1.10.3) and (1.10.5), which we rewrite here for completeness: p 4π G ρ+3 2 a (1.13.1) ä = − 3 c and ȧ2 + Kc 2 = 8π G ρa2 , 3 (1.13.2) 30 First Principles as well as the Equation (1.10.6) d(ρa3 ) = −3 p 2 a da. c2 (1.13.3) The Equations (1.13.1)–(1.13.3) allow one, at least in principle, to calculate the time evolution of a(t) as well as ρ(t) and p(t) if we know the equation of state. Let us focus for now on Equation (1.13.3), which can be rewritten in a convenient form for a = a0 : 2 2 a 8π Kc 2 ρ0 ȧ Gρ − = H02 1 − (1.13.4) = H02 (1 − Ω0 ) = − 2 , a0 3 a0 ρ0c a0 where H0 = ȧ0 /a0 , Ω0 is the (present) density parameter and ρ0c = 3H02 . 8π G (1.13.5) The suﬃx ‘0’ refers here to a generic reference time t0 which is also used in the particular case where t is the present time. Equation (1.13.5) is a reminder of the importance of ρ0c : if ρ0 < ρ0c , then K = −1, while if ρ0 > ρ0c , K = 1; K = 0 corresponds to the ‘critical’ case when ρ0 = ρ0c . Let us now include the cosmological constant term Λ. In Section 1.12 we showed how one can treat the cosmological constant as a form of ﬂuid with a strange equation of state, as well as a modiﬁcation of the law of gravity. In that sense, Λ can be thought of as belonging either on the lefthand or righthand side of the Einstein equations. Either way, the upshot is that Equations (1.13.1) and (1.13.2) become p Λc 2 a 4π G ρ+3 2 a+ (1.13.6) ä = − 3 c 3 and ȧ2 + Kc 2 = 8π G Λc 2 a2 ρa2 + , 3 3 (1.13.7) respectively. If we ignore the original terms in p and ρ we can see that Equation (1.13.7) can be written in a form similar to Equation (1.13.4): 2 ȧ Λc 2 Kc 2 Λ = H02 1 − − (1.13.8) = H02 (1 − Ω0Λ ) = − 2 . a0 3 Λc a0 In this case the ‘critical’ value is Λc = 3H02 , c2 (1.13.9) so that Ω0Λ = Λc 2 /3H02 . If we now reinstate the ‘ordinary’ matter we began with, we can see that the curvature is zero as long as Ω0Λ + Ω0 = 1. The cosmological constant therefore breaks the relationship between the matter density and curvature. Even if Ω0 < 1, a suitably chosen value of Ω0Λ = 1 − Ω0 can be invoked to ensure ﬂat space sections. Friedmann Models 31 Bibliographic Notes on Chapter 1 The classic papers of Einstein (1917), de Sitter (1917), Friedmann (1922) and Lemaître (1927) are all well worth reading for historical insights. A particularly erudite overview of the role of observation in expanding world models is given by Sandage (1988). More detailed discussions of the basic background, including the role of general relativity in cosmology, can be found in Berry (1989), Harrison (1981), Kenyon (1990), Landau and Lifshitz (1975), Milne (1935), Misner et al. (1972), Narlikar (1993), Peebles (1993), Peacock (1999), Raychaudhuri (1979), Roos (1994), Wald (1984), Weinberg (1972) and Zel’dovich and Novikov (1983). Problems 1. Suppose that it is discovered that Newton’s law of gravitation is incorrect, and that the force F on a test particle of mass m due to a body of mass M has an additional term that does not depend on M and is proportional to the separation r : F =− GMm Amr + . r2 3 Assuming that Newton’s sphere theorem continues to hold, derive the appropriate form of the Friedmann equation in this case and comment on your result. 2. The most general form of a space–time fourmetric in the synchronous gauge is ds 2 = c 2 dt 2 − gαβ dx α dx β = c 2 dt 2 − dl2 , where gαβ is the threemetric of the spatial hypersurfaces. By writing the equation of the threespace as that of a constrained surface in four dimensions, show that the most general form of the threemetric compatible with homogeneity and isotropy is given by