An Introduction toModern CosmologySecond Edition

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An Introduction ToModern CosmologySecond EditionAndrew LiddleUniversity ofSussex, UK WILEY

Copyright 2003John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,West Sussex POl9 8SQ, EnglandTelephone( 44) 1243779777Email (for orders and customer service enquiries): [email protected],.co.ukVisit our Home Page on W\lA\' or\\.T·\v\\'.wiley.comAll Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted 10 an\" iorm or b,'any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copnight,Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham CounRoad, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressedto the Permissions Deparonent,John \X'iley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex POIC) 85Q.England, or emailed to [email protected],·, or faxed to ( 44) 1243 "7"'05 LThis publication is designed to prO\ide accurate and authoritative information in regard to the subject matter co,·ered. It is soldon the understanding that the Publisher is not engaged in rendering professional services. If professional ad"ice or other expertassistance is required, the senices of a competent professional should be sought.Other Wiley Editorial OfficesJohn Wiley & Sons Inc. III River Street, Hoboken, NJ 0 7 030, L'SAJossey-Bass, 989 farketStreet, San Francisco, CA 94103-1741, L'SAWiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Gennan\"John Wilev & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, AustraliaJohn Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01,Jin Xing Distripark, Singapore 129809John Wiley & Sons Canada Ltd, 22 Worcester Road, Etohicoke, Omario, Canada f9\''''I L1\'('iley also publishes in books in a variety of electronic formats. Some content thatappears in print may not be ,,-ailable in electronic books.Library ofCongress Cataloging-in-Publication Data(rofollow)British Library Cataloguing in Publication DataA catalogue record for this book is "'ailable from the British LibraryISBN 0470 84834 0 Cloth0470 S4835 9 PaperProduced from author's LaTeX fIlesPrinted and bound in Great Britain by Antom' Rowe Ltd., Chippenham, WiltsThis book is printed on acid-free paper responsiblv manufactured from sustainable iorestrvin which at least two trees are planted for each nne used for paper production.

To my grandmothers

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ContentsPrefaceConstants, conversion factors and symbolsxixiv1A (Very) Brief History of Cosmological Ideas12Observational Overview2.1 In visible light. . . .2.2 In other wavebands .2.3 Homogeneity and isotropy2.4 The expansion of the Universe2.5 Particles in the Universe . . .2.5.1 What particles are there?2.5.2 Thermal distributions and the black-body spectrum3345Newtonian Gravity3.1 The Friedmann equation . . . . .3.2 On the meaning of the expansion .3.3 Things that go faster than light3.4 The fluid equation . . . . . . . . .3.5 The acceleration equation . . . . .3.6 On mass, energy and vanishing factors of c2378911111317182121222324The Geometry of the Universe4.1 Flat geometry . . . .4.2 Spherical geometry . . . .4.3 Hyperbolic geometry . . .4.4 Infinite and observable Universes.4.5 Where did the Big Bang happen? .4.6 Three values of k . . . . . . . . .30Simple Cosmological Models5.1 Hubble's law . . . . .5.2 Expansion and redshift5.3 Solving the equations .33333435252526282929

CONTENTSVlll5. Matter .5.3.2 Radiation5.3.3 MixturesParticle number densitiesEvolution including curvature36373839406Observational Parameters6.1 The expansion rate H o6.2 The density parameter no . . .6.3 The deceleration parameter qo454547487The Cosmological Constant7.1 Introducing A . . . . .7.2 Fluid description of A . . . .7.3 Cosmological models with A515152538 The Age of the Universe579636363646466676868696972The Density of the Universe and Dark Matter9.1 Weighing the Universe . . . . . . . . .9.1.1 Counting stars9.1.2 Nucleosynthesis foreshadowed.9.1.3 Galaxy rotation curves . . . .9.1.4 Galaxy cluster composition. .9.1.5 Bulk motions in the Universe9.1.6 The formation of structure . .9.1.7 The geometry of the Universe and the brightness of supernovae9.1. 8 Overview.9.2 What might the dark matter be? .9.3 Dark matter searches . . . . . .10 The Cosmic Microwave Background10.1 Properties ofthe microwave background10.2 The photon to baryon ratio . . . . . . .10.3 The origin of the microwave background.10.4 The origin of the microwave background (advanced)757577788111 The Early Universe8512 Nucleosynthesis: The Origin of the Light Elements12.1 Hydrogen and Helium. . . . . . . . . . . .12.2 Comparing with observations. . . . . . . .12.3 Contrasting decoupling and nucIeosynthesis91919496

CONTENTSix13 The Inflationary Universe13.1 Problems with the Hot Big Bang13. I .1 The flatness problem . .13.1.2 The horizon problem .13.1.3 Relic particle abundances13.2 Inflationaryexpansion . . . . .13.3 Solving the Big Bang problems .13.3.1 The flatness problem . .13.3.2 The horizon problem .13.3.3 Relic particle abundances13.4 How much inflation? . . . .13.5 Inflation and particle physics99999910110210310410410510610610714 The Initial SingularityIII15 Overview: The Standard Cosmological ModelU5Advanced Topic 1 General Relativistic Cosmology1.1 The metric of space-time . . . .1.2 The Einstein equations . . . . .1.3 Aside: Topology of the Universe119119Advanced Topic 2 Classic Cosmology: Distances and Luminosities2.1 Light propagation and redshift2.2 The ohservable Universe .2.3 Luminosity distance. . . .2.4 Angular diameter distance2.5 Source counts.125Advanced Topic 3 Neutrino Cosmology3.1 The massless case.3.2 Massive neutrinos.3.2.1 Light neutrinos .3.2.2 Heavy neutrinos3.3 Neutrinos and structure formation137137139139140Advanced Topic 4143BaryogenesisAdvanced Topic 5 Structures in the Universe5.1 The observed structures .5.2 Gravitational instability.5.3 The clustering of galaxies5.4 Cosmic microwave background anisotropies5.4. 1 Statistical description of anisotropies5.4.2 Computing the Ct . ' . , .5.4.3 Microwave background observations.5.4.4 Spatial 2154155156

CONTENTSx5.5The origin of structure157Bibliography161Numerical answers and hints to problems163Index167

PrefaceThe development of cosmology will no doubt be seen as one of the scientific triumphs ofthe twentieth century. At its beginning, cosmology hardly existed as a scientific discipline.By its end, the Hot Big Bang cosmology stood secure as the accepted description of theUniverse as a whole. Telescopes such as the Hubble Space Telescope are capable of seeinglight from galaxies so distant that the light has been travelling towards us for most of thelifetime of the Universe. The cosmic microwave background, a fossil relic of a time whenthe Universe was both denser and hotter, is routinely detected and its properties examined.That our Universe is presently expanding is established without doubt.We are presently in an era where understanding of cosmology is shifting from thequalitative to the quantitative, as rapidly-improving observational technology drives ourknowledge forward. The tum of the millennium saw the establishment of what has cometo be known as the Standard Cosmological Model, representing an almost universal consensus amongst cosmologists as to the best description of our Universe. Nevertheless, it isa model with a major surprise - the belief that our Universe is presently experiencing accelerated expansion. Add to that ongoing mysteries such as the properties of the so-calleddark matter, which is believed to be the dominant form of matter in the Universe, and it isclear that we have some way to go before we can say that a full picture of the physics ofthe Universe is in our grasp.Such a bold endeavour as cosmology easily captures the imagination, and over recentyears there has been increasing demand for cosmology to be taught at university in anaccessible manner. Traditionally, cosmology was taught, as it was to me, as the tail end ofa general relativity course, with a derivation of the metric for an expanding Universe anda few solutions. Such a course fails to capture the flavour of modem cosmology, whichtakes classic physical sciences like thermodynamics, atomic physics and gravitation andapplies them on a grand scale.In fact, introductory modem cosmology can be tackled in a different way, by avoidinggeneral relativity altogether. By a lucky chance, and a subtle bit of cheating, the correct equations describing an expanding Universe can be obtained from Newtonian gravity.From this basis, one can study all the triumphs of the Hot Big Bang cosmology - the expansion of the Universe, the prediction of its age, the existence of the cosmic microwavebackground, and the abundances of light elements such as helium and deuterium - andeven go on to discuss more speculative ideas such as the inflationary cosmology.The origin of this book, first published in 1998, is a short lecture course at the University of Sussex, around 20 lectures, taught to students in the final year of a bachelor's

XIICONTENTSdegree or the penultimate year of a master's degree. The prerequisites are all very standardphysics, and the emphasis is aimed at physical intuition rather than mathematical rigour.Since the book's publication cosmology has moved on apace, and I have also becomeaware of the need for a somewhat more extensive range of material, hence this second edition. To summarize the differences from the first edition, there is more stuff than before.and the stuff that was already there is now less out-of-date.Cosmology is an interesting course to teach, as it is not like most of the other subjectstaught in undergraduate physics courses. There is no perceived wisdom, built up over acentury or more, which provides an unquestionable foundation, as in thermodynamics.electromagnetism, and even quantum mechanics and general relativity. Within our broadbrush picture the details often remain rather blurred, changing as we learn more about theUniverse in which we live. Opportunities crop up during the course to discuss new resultswhich impact on cosmologists' views of the Universe, and for the lecturer to impose theirown prejudices on the interpretation of the ever-changing observational situation. UnlessI've changed jobs (in which case I'm sure www. google. corn will hunt me down), youcan follow my own current prejudices by checking out this book's WWW Home Page rnlThere you can find some updates on observations, and also a list of any errors in the bookthat I am aware of. If you are confident you've found one yourself, and it's not on the list.I'd be very pleased to hear of it.The structure of the book is a central 'spine', the main chapters from one to fifteen,which provide a self-contained introduction to modem cosmology more or less reproducing the coverage of my Sussex course. In addition there are five Advanced Topic chapters,each with prerequisites, which can be added to extend the course as desired. Ordinarilythe best time to tackle those Advanced Topics is immediately after their prerequisites havebeen attained, though they could also be included at any later stage.I'm extremely grateful to the reviewers of the original draft manuscript, namely SteveEales, Coel Hellier and Linda Smith, for numerous detailed comments which led to thefirst edition being much better than it would have otherwise been. Thanks also to thosewho sent me useful comments on the first edition, in particular Paddy Leahy and MichaelRowan-Robinson, and of course to all the Wiley staff who contributed. Matthew Colless.Brian Schmidt and Michael Turner provided three of the figures, and Martin Hendry, Martin Kunz and Franz Schunck helped with three others, while two figures were generatedfrom NASA's SkyVtew facility (http://skyview.gsfc located at theNASA Goddard Space Flight Center. A library of images, including full-colour versionsof several images reproduced here in black and white to keep production costs down, canbe found via the book's Home Page as given above.Andrew R LiddleBrigbtonFebruary 2003

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xivSome fundamental constantsNewton's constantSpeed of lightGcReduced Planck constantBoltzmann constantIiRadiation constantElectron mass--energyProton mass--energyNeutron mass--energyThomson cross-sectionFree neutron half-lifeo 7[2 k /15li3c3m e c2m p c2m n c2or h/27[kBorUethalf6.672 X 10- 11 m 3 kg- 1 sec- 22.998 x 108 msec- 13.076 x 10- 7 Mpcyr- 11.055 x 10- 34 m 2 kg sec- 11.381 x 10- 23 J K- 18.619 x 1O- 5 eVK- 17.565 x 10- 16 J m- 3 K- 40.511 MeV938.3 MeV939.6 MeV6.652 x 10- 29 m 2614 secSome conversion factors1 pc 3.261 light years 3.086 x 10 16 m1 yr 3.156 x 10 7 sec1 eV 1.602 x 10- 19 J1 M 0 1.989 X 1030 kgIJ lkgm 2 sec- 21 Hz 1 sec- 1

XyCommonly-used symbolsI ::"TIT, CeredshiftHubble constantphysical distancevelocityfrequencytemperatureBoltzmann constantenergy densityradiation constantNewton's gravitational constantmass densityscale factorcomoving distancecurvaturepressure(or occasionally momentumHubble parameternumber densityHubble constant(or Planck's constantpresent density parametercritical densitydensity parametercurvature 'density parameter'deceleration parametercosmological constantcosmological constant density parametertimepresent agebaryon density parameterhelium abundanceluminosity distanceangular diameter distancecosmic microwave background anisotropiesdefined on page 9,359,4599121313151517181919202211 )34394612)4747