FROM COSMOS TO CHAOS This page intentionally left blank FROM COSMOS TO CHAOS The Science of Unpredictability Peter Coles AC AC GreatClarendonStreet,OxfordOX26DP OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwidein Oxford NewYork Auckland CapeTown DaresSalaam HongKong Karachi KualaLumpur Madrid Melbourne MexicoCity Nairobi NewDelhi Shanghai Taipei Toronto Withofficesin Argentina Austria Brazil Chile CzechRepublic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore SouthKorea Switzerland Thailand Turkey Ukraine Vietnam OxfordisaregisteredtrademarkofOxfordUniversityPress intheUKandincertainothercountries PublishedintheUnitedStates byOxfordUniversityPressInc.,NewYork #OxfordUniversityPress2006 Themoralrightsoftheauthorhavebeenasserted DatabaserightOxfordUniversityPress(maker) Firstpublished2006 Allrightsreserved.Nopartofthispublicationmaybereproduced, storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans, withoutthepriorpermissioninwritingofOxfordUniversityPress, orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate reprographicsrightsorganization.Enquiriesconcerningreproduction outsidethescopeoftheaboveshouldbesenttotheRightsDepartment, OxfordUniversityPress,attheaddressabove Youmustnotcirculatethisbookinanyotherbindingorcover andyoumustimposethesameconditiononanyacquirer BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressCataloginginPublicationData Coles,Peter. Fromcosmostochaos:thescienceofunpredictability/PeterColes. p.cm. ISBN-13:978–0–19–856762–2(alk.paper) ISBN-10:0–19–856762–6(alk.paper) 1. Science—Methodology. 2. Science—Forecasting. 3. Probabilities. I. Title. Q175.C61552006 5010.5192—dc22 2006003279 TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India PrintedinGreatBritain onacid-freepaperby BiddlesLtd.www.Biddles.co.uk ISBN0–19–856762–6 978–0–19–856762–2 1 3 5 7 9 10 8 6 4 2 ‘The Essence of Cosmology is Statistics’ George Mcvittie Acknowledgements IamverygratefultoAnthonyGarrettforintroducingmetoBayesian probabilityanditsdeeperramifications.Ialsothankhimforpermission to use material from a paper we wrote together in 1992. Various astronomers have commented variously on the various ideas contained in this book. I am particularly grateful to Bernard Carr and John Barrow for helping me come to terms with the Anthropic Principle and related matters. I also wish to thank the publisher for the patience over the ridiculously long time I took to produce the manuscript. Finally, I wish to thank the Newcastle United defence for helping me understand the true meaning of the word ‘random’. Contents List of Figures viii 1. Probable Nature 1 2. The Logic of Uncertainty 7 3. Lies, Damned Lies, and Astronomy 31 4. Bayesians versus Frequentists 48 5. Randomness 71 6. From Engines to Entropy 95 7. Quantum Roulette 115 8. Believing the Big Bang 138 9. Cosmos and its Discontents 161 10. Life, the Universe and Everything 180 11. Summing Up 199 Index 213 List of Figures 1 Venn diagrams and probabilities 12 2 The Normal distribution 29 3 The Hertzprung-Russell diagram 33 4 A scatter plot 34 5 Statistical correlation 40 6 Fitting a line to data 41 7 Pierre Simon, Marquis de Laplace 43 8 The likelihood for the distribution of arrival times 57 9 Inductive versus deductive logic 62 10 Correlation between height and mass for humans 74 11 Lissajous figures 82 12 The transition to chaos shown by the He´non-Heiles system 84 13 Transition from laminar to turbulent flow 85 14 The first-digit phenomenon 88 15 Randomness versus structure in point processes 91 16 A computer-generated example of a random walk 93 17 Using a piston to compress gas 97 18 The set of final microstates is never smaller than the initial set 112 19 The ultraviolet catastrophe 117 20 A classic ‘two slit’ experiment 124 21 Closed, open, and flat universes 141 22 A map of the sky in microwaves revealed by WMAP 155 23 The cosmological flatness problem 167 24 The Strong Anthropic Principle may actually be Weak 176 N O 1 Probable Nature The true logic of this world is the calculus of probabilities. James Clerk Maxwell This is a book about probability and its role in our understanding of the world around us. ‘Probability’ is used by many people in many differentsituations,oftenwithoutmuchthoughtbeinggiventowhat the word actually means. One of the reasons I wanted to write this book was to offer my own perspective on this issue, which may be peculiar because of my own background and prejudices, but which may nevertheless be of interest to a wide variety of people. My own field of scientific research is cosmology, the study of the Universeasawhole.Inrecentyearsthisfieldhasbeenrevolutionized by great advances in observational technology that have sparked a ‘data explosion’. When I started out as an ignorant young research student20yearsagotherewasvirtuallynorelevantdata,thefieldwas dominatedby theoreticalspeculation andit waswidely regardedas a branch of metaphysics. New surveys of galaxies, such as the Anglo- Australian Two-degree Field Galaxy Redshift Survey (2dFGRS) and the (American) Sloan Digital Sky Survey (SDSS), together with exquisite maps of the cosmic microwave background, have revealed the Universe to us in unprecedented detail. The era of ‘precision cosmology’hasnowarrived,andcosmologistsarenowrealizingthat sophisticatedstatisticalmethodsareneededtounderstandwhatthese new observations are telling us. Cosmologists have become glorified statisticians. Thiswasmyoriginalmotivationforthinkingaboutwritingabook, but thinking about it a bit further, I realized that it is not really correct to think that there is anything new about cosmology being a statistic subject. The quote at the start of this book, by the dis- tinguishedBritishmathematicianGeorgeMcVittieactuallydatesfrom the 1960s, long before the modern era of rapid data-driven progress.