NUMERICAL COINCIDENCES AND TUNING IN COSMOLOGY MAR TIN - PDF document

NUMERICAL COINCIDENCES AND 'TUNING' IN COSMOLOGY MAR TIN J . R EES 1. In tr od u ction H erm ann B ondiÕ s classic book ÔC osm o logyÕ was, for m an y o f u s, an inspiring introduction t o t he science of the cosm o s. In a chapter entitled Ô M i crophysics and C o sm ologyÕ, B ondi lists the f am ous dim ensionless constants, and m entions the w ell-kno w n coincidence, Þ r st highlighted by D irac, betw een the r atio of the elec- trical and g ra vitational f orces w ithin a hydrogen atom and the r atio of the H ubble radius to the size o f an electron. H e says: ÔT hese coincidences are v ery striking, and f e w would d en y t heir possible d eep signiÞ cance, b u t t he precise nature of the conne xion the y indicate i s not understood and i s v ery m ysterious. Õ I am not sure to w h at e x tent F r ed H o yle w as inß u enced in this m atter b y B ondi, b u t h e certainly took this problem seriously too. H e also, t hrough his f am ous realisation o f t he C 12 resonance l e v elÕ s cance, m ade a celebrated addition to the list o f cosm i c coincidences. M o reo v er , in pondering t heir signiÞ cance he was l ed to conjecture t hat t he so called Ô constants o f n atureÕ m i ght not be truly uni v ersal. I n Ô G alaxies, N u clei and Q uasarsÕ H o yle w rites t hat Ô one m u st at least ha v e a m odicum o f curiosity about the strange dim ensionless num bers that appear in physics. Õ H e goes o n t o outline t wo possible attitudes t o t hem . O n e i s t hat Ôthe d im ensionless num bers are all entirely necessary t o t he logical consistenc y of physicsÕ; t he second possibility is that the num bers are not in the b roadest sense uni v b ut that Ôin other p laces their v alues w ould b e d if ferentÕ H o y le f a v oured this latter option b ecause then Ôthe curious placing of the l e v els i n C 12 and O 16 need no longer h a v e t he appearance of astonishing accidents. It could sim ply b e t hat since creatures lik e ourselv es d epend o n balance b etw een carbon and oxygen, w e can e x ist only i n t he portions of the uni v erse w here these le v els happen t o b e correctly placed. Õ W ith these te xts as m y m oti v ation, IÕ d lik e t o sum m arise b rieß y ho w t he issue looks today . I b elie v e that F r edÕ s conjecture i s n o w e v en m o re attracti v e, though the Ô portions of the uni v erseÕ betw een w h ich t he v ariation o ccurs m u st no w b e interpreted as t hem selv es v astly lar g er than the dom ain our telescopes can actually observ e Ð p erhaps e v en entire Ôuni v ersesÕ w ithin a m ulti v erse. Butbeforedwellingfurtheronthesecoincidences,itmightbeworthnotingthattheÔcoincidenceÕthatDiracandBondidiscusseddoesnotinitselfnowcausepuzzlement.Thereisreallyjustoneverylargenumberinphysics:itis/Gm(or,equivalently,thereciprocaloftheÔgravitationalÞnestructureconstantÕ whichislargerby137.TheChandrasekharmassexceedstheprotonmass.Starsaresolargebecausegravityissoweak:Dicke(1961)alsorealisedthattheyarealsolong-livedforthesamereason.TopresentDickeÕsestimateforstellarlifetimesinaslightlydifferentway,wecandeÞneacharacteristictime(cfSalpeter1964)equalto LEd=2 3e2 mec3e2 Gm2pmp ThisisthetimeitwouldtakeabodytoradiateitsrestmassenergyifhadtheÔEddingtonluminosityÕwherewhereradiationpressurebalancesgravity,andifelectronscatteringprovidedthemainopacity.Thelifetimeofanactualstarisobtainedbymultiplyingbybyvariousfactors:theefÞciencyofnuclearenergy(007);theratiooftotalpressuretoradiationpressures(1);andtheratioofactualopacitytoelectron-scatteringopacity(1).However,thekeypoint(evidentfromthesecondwayIhavewrittentheexpressionforabove)isthatstellarlifetimesarelongerthanthelighttraveltimeacrosstheelectronbythefactorinsquarebracketswhichinvolvesDiracÕslargenumber.Ifweareobservingtheuniversewhenitsageisofordertheageofastar(andthereissuchatimeinabigbangmodel)thenDiracÕsÔcoincidenceÕwouldnaturallybesatisÞed.2.Dothe‘Special’ValuesoftheConstantsNeedanExplanation?Ifweeverestablishedcontactwithintelligentaliens,howcouldwebridgetheÔculturegapÕ?Onecommonculture(inadditiontomathematics)wouldbephys-icsandastronomy.Weandthealienswouldallbemadeofatoms,andweÕdalltraceouroriginsbacktotheÔbigbangÕ13.7billionyearsago.WeÕdallsharethepotentialitiesofa(perhapsinÞnite)future.Butourexistence(andthatofthealiens,ifthereareany)dependsonouruniversebeingratherspecial.AnyuniversehospitabletolifeÐwhatwemightcallabiophilicuniverseÐhastobeÔadjustedÕinaparticularway.TheprerequisitesforanylifeofthekindweknowaboutÑlong-livedstablestars,stableatomssuchascarbon,oxygenandsilicon,abletocombineintocomplexmolecules,etcÑaresensitivetothephysicallawsandtothesize,expansionrateandcontentsoftheuniverse.Indeed,evenforthemostopenmindedscienceÞctionwriter,ÔlifeÕorÔintelligenceÕrequirestheemergenceofsomegenericcomplexstructures:itcanÕtexistinahomogeneousuniverse,notinauniversecontainingonlyafewdozenparticles.Manyrecipeswouldleadto stillbornuniverseswithnoatoms,nochemistry,andnoplanets;ortouniversestooshort-livedortooemptytoallowanythingtoevolvebeyondsterileuniformity.Consider,forexample,theroleofgravity.Starsandplanetsdependcruciallyonthisforce;however,wecouldnotexistifgravityweremuchstrongerthanitactuallyis.Alarge,long-livedandstableuniversedependsquiteessentiallyonbeingexceedinglylarge.GravityalsoampliÞesÔlinearÕdensitycontrastsinanexpandinguniverse;itthenprovidesanegativespeciÞcheatsothatdissipativeboundsystemsheatupfurtherastheyradiate.ThereÕsnothermodynamicparadoxinevolvingfromanalmoststructurelessÞreballtothepresentcosmos,withhugetemperaturedifferencesbetweenthe3degreesofthenightsky,andtheblazingsurfacesofstars.Sogravityiscrucial,buttheweakeritis,thegranderandmoreprolongedareitsconsequences.NewtonÕsconstantGneednotbeÞne-tunedÐmerelyexceedinglyweaksothatisindeedverylarge.However,thenaturalworldismuchmoresensitivetothebalancebetweenotherbasicforces.Ifnuclearforceswereslightlystrongerthantheyactuallyarerelativetoelectricforcestwoprotonscouldsticktogethersoreadilythatordinaryhydrogenwouldnotexist,andstarswouldevolvequitedifferently.Someofthedetailsarestillmoresensitive,asHoyleemphasised.Evenauniverseaslargeasourscouldbeveryboring:itcouldcontainjustblackholes,orinertdarkmatter,andnoatomsatall.Evenifithadthesameingredientsasours,itcouldbeexpandingsofastthatnostarsorgalaxieshadtimetoform;oritcouldbesoturbulentthatallthematerialformedvastblackholesratherthanstarsorgalaxies.Ðaninclementenvironmentforlife.Andouruniverseisalsospecialinhavingthreespatialdimensions.Afourdimensionalworldwouldbeunstable;intwodimensions,nothingcomplexcouldexist.Thedistinctiveandspecial-seemingrecipecharacterisingouruniverseseemstomeafundamentalmysterythatshouldnotbebrushedasidemerelyasabrutefact.Ratherthanre-addressingtheclassicÔÞnetuningÕexamples,IshallfocusontheparametersofthebigbangÐtheexpansionrate,thecurvature,theßuctuations,andthematerialcontent.Someoftheseparameters(perhapsevenall)maybeexplicableintermsofauniÞedtheory:ortheymaybesomehowderivablefromthemicrophysicalconstants.But,irrespectiveofhowthatmayturnout,itisinterestingtoexploretheextenttowhichthepropertiesofauniverseÐenvisagedhereastheaftermathofasinglebigbangÐaresensitivetothecosmologicalparameters.3.TheCosmologicalNumbersTraditionally,cosmologywasthequestforafewnumbers.TheÞrstwereH,andq.Since1965weÕvehadanother:thebaryon/photonratio.ThisisbelievedtoresultfromasmallfavouritismformatteroverantimatterintheearlyuniverseÐsomethingthatwasaddressedinthecontextofÔgranduniÞedtheoriesÕinthe1970s.(Indeed,baryonnon-conservationseemsaprerequisiteforanyplausiblein- ßationarymodel.Ourentireobservableuniverse,containingatleast10couldnothaveinßatedfromsomethingmicroscopicifbaryonnumberwerestrictlyconserved).Inthe1980snon-baryonicmatterbecamealmostanaturalexpectation,andisanotherfundamentalnumber.Wenowhavetherevivalofthecosmologicalconstantlambda(orsomekindofÔdarkenergyÕ,withnegativeassociatedpressure,whichisgenericallyequivalenttolambda).Anotherspeciallyimportantdimensionlessnumbertellsushowsmooththeuniverseis.ItÕsmeasuredbyTheSachs-Wolfeßuctuationsinthemicrowavebackgroundthegravitationalbindingenergyofclustersasafractionoftheirrestmassorbythesquareofthetypicallengthscaleofmass-clusteringasafractionoftheHubbleradius.ItÕsofcourseoversimpliÞedtorepresentthisbyasinglenumber,butinsofarasonecan,itsvalue(letÕscallitQ)ispinneddowntobe10.(Detailedmodellingoftheßuctuationsintroducesfurthernumbers:theratioofscalarandtensoramplitudes,andquantitiessuchastheÔtiltÕ,whichmeasurethedeviationfromapurescale-independentHarrison-Zeldovichspectrum.)4.AnthropicRequirementsforaUniverseWecanmakealistofwhatwouldberequiredforabigbangtoyieldanÔanthropic-allyallowedÕuniverseÐauniversewherecomplexity,whetherhumanoidormorelikeablackcloud,couldunfold.Thelistwouldincludethefollowing:Someinhomogeneities(i.e.anon-zeroQ):clearlythereisnopotentialforcomplexityifeverythingremainsinauniformultra-dilutemediumSomebaryons:complexitywouldbeprecludedinauniversesolelymadeofdarkmatter,withonlygravitationalinteractions.Atleastonestar(probably,thoughperhapssuperßuousforblack-cloud-stylecomplexity)Somesecond-generationstars:onlylater-generationstarswouldbeabletohaveorbitingplanets,unlessheavyelementswereprimordial.ItisinterestingtoengageinÔcounterfactualhistoryÕandaskwhatconstraintsthesevariousrequirementswouldimposeonhypotheticaluniverseswithdifferentcharacteristicsÐinparticular,withdifferentvaluesof:TheßuctuationamplitudeQ(whichis10inouractualuniverse)ThecosmologicalconstantThebaryon/photonratio(about10inouruniverse)Thebaryon/darkmatterratio(about0.2inouruniverse). 4.1.THEFLUCTUATIONAMPLITUDEFirst,wemightexplorewhatauniversewouldbelikewhichwasinitiallysmoother(Qsmaller)orrougher(Qlarger)thanours.WereQoforder10,therewouldbenoclustersofgalaxies;moreover,theonlygalaxieswouldbesmallandanaemic.Theywouldformmuchlaterthangalaxiesdidinouractualuniverse.Becausetheyarelooselybound,processedmaterialwouldbeexpelledfromshallowpotentialwells;theremaythereforebenosecond-generationstars,andsonoplanetarysystems.IfQwereevensmallerthan,therewouldbenostarformationatall:verysmallstructuresofdarkmatterwouldturnaroundlate,andtheirconstituentgaswouldbetoodilutetoundergotheradiativecoolingthatisaprerequisiteforstarformation.(Inalambda-dominateduniverse,isolatedclumpscouldsurviveforaninÞnitetimewithoutmergingintoalargerscaleinthehierarchy.Soeventually,forany,aÔstarÕcouldformÐbutbythattimeitmightbetheonlyboundobjectwithinthehorizon).HypotheticalastronomersinauniversewithQ=10mightÞndtheircosmicenvironmentmorevariedandinterestingthanours.Galaxiesandclusterswouldspanawiderrangeofmasses.Thebiggestclusterswouldbe30timesmoremassivethananyinouractualuniverse.TherecouldbeindividualÔgalaxiesÕÐperhapsevendiscgalaxiesÐwithmassesuptothatoftheComaclusterandinternalvelocitydispersionsupto2000km/sec.Thesewouldhavecondensedwhentheuniversewasonly3yearsold,andwhenComptoncoolingonthemicrowavebackgroundwasstilleffectiveHoweverauniversewhereQwerelargerstillÐmorethan(say)10Ðwouldbeaviolentandinhospitableplace.Hugegravitationally-boundsystemswouldcollapse,trappingtheirradiationandunabletofragment,soonaftertheepochofrecombination.(Collapseat,say,10yearswouldleadtosufÞcientpartialioniza-tion(viastrongshocks)torecouplethebaryonsandtheprimordialradiation.)Suchstructures,containingthebulkofthematerial,wouldturnintovastblackholes.Itisunlikelythatgalaxiesofanykindwouldexist;norisitobviousthatmuchbaryonicmaterialwouldevergointostars:evenifso,theywouldbeinverycompacthighlyboundsystems).(Notethat,irrespectiveoftheseanthropicconstraintsonitsvalue,Qhastobesubstantiallylessthanoneinordertomakecosmologyatractablesubject,separatefromastrophysics,ThisisbecausetheratioofthelargeststructurestotheHubbleradiusisoforderQ.NumberslikeandHareonlywell-deÞnedinsofarastheuniversepossessesÔbroadbrushÕhomogeneityÐsothatourobserva-tionalhorizonencompassesmanyindependentpatcheseachbigenoughtobeafairsample.ThiswouldnÕtbeso,andthesimpleFriedmannmodelswouldnÕtbeusefulapproximations,ifQwerenÕtmuchlessthanunity.)Accordingtomosttheoriesoftheultra-earlyuniverse,Qisimprintedbyquantumeffects:microscopicßuctuations,afterexponentialexpansion,giverisetothelarge-scaleirregularitiesobservedinthemicrowavebackgroundsky,andwhicharetheÔseedsÕforgalaxiesandclusters.ButasyetnotheoriespindownQÕsvalue. Figure1.Thedomainsinthewhichboundstructurescanform,fordifferentvaluesofQ(fromTegmarkandRees,1998).4.2.ORDARKENERGYTheoristsareevenfurtherfromunderstanding.Indeed,thenaiveguessisthatshouldbeleast120powersof10largerthanitcouldbeinouractualuniverseÐunlessthereweresomecancellationmechanism.(Indeed,inßationmodelspostu-lateaneffectivevacuumdensitythatwasindeedashighasthisforabriefinitialinterval.)Theinteresthasofcoursebeenhugelyboostedrecently,throughtheconver-genceofseverallinesofevidenceonamodelwheretheuniverseisclosetobeingÔßatÕ,butwith4percentinbaryons,about25percentindarkmatter,andtheremaining(dominant)componentinorsometime-dependentÔdarkenergyÕ.(Incidentally,thefullresurrectionofwouldbeagreatÔcoupÕfordeSitter.Hismodel,datingforthe1920s,notonlydescribesinßation,butwouldthenalsode-scribesfutureaeonsofourcosmoswithincreasingaccuracy.Onlyforthe50-odddecadesoflogarithmictimebetweentheendofinßationandthepresentwoulditneedmodiÞcation!).Forauniversewiththeactualobservedvaluesof,itisreadilyshownthatavalueofmorethan5-10timeshigherthantheapparentÔdarkenergyÕdensitywouldhavetheÔanti-anthropicÕconsequenceofprecludinggalaxyformation.ThishappensbecausethecosmicrepulsionwouldthenbesoÞercethatitwouldtakeoverbeforeanygalaxieshadachancetoformviagravitationalinstability. 4.3.THEBARYONDARKMATTERDENSITYBaryonsareanthropicallyessential;thereareÞrmlowerlimitsontheirrequis-iteabundances,buttheyneednotbethedominantconstituent.(Indeedtheyarefarfromdominantinouractualuniverse).LowerandlowertheÔefÞcientcoolingÕdomainintheRees/Tegmark(1998)curves.reproducedinFigure1.Ifthephotonsoutnumberedthebaryonsandthedarkmatterparticlesbyastilllargerfactorthaninouractualuniverse,thentheuniversewouldremainradiation-dominatedforsolongthatthegravitationalgrowthofßuctuationswouldbeinhib-ited(Rees,1980).Ontheotherhand,ahighervalueofandal-lowsgravitationalclusteringtostartearlier.ThisreducestheminimumQrequiredforemergenceofnon-linearstructures(cf.Aguirre,2001)(Notealsothatthemechanismthatgivesrisetobaryonfavouritismmaybelinkedtothestronginteractions,andthereforecorrelatewithkeynumbersinnuc-learphysics.)4.4.DELINEATINGTHEANTHROPICALLYALLOWEDDOMAINIntheabove,Ihaveenvisagedchangingjustoneparameter,leavingtheotherswiththeiractualvalues.Butofcoursetheremaybecorrelationsbetweenthem.Forexample,supposethattherewerebigbangswithawholerangeofQ-values.Structuresformearlier(whenthematterdensityishigher)inuniverseswithlargerQ,soobviouslyahigherQisanthropically-compatiblewithahigherIfweconsideratwo-dimensionalsituationwhereQandvary,thenweÞndthatthereisananthropicallyallowedarea.Thereare(rathervaguelydeÞned)upperandlowerlimitstoQ(asalreadydiscussed)butwithintherange,thereisanupperlimittoQ(seeFigure2).Wecancarryouttheexercise,inasmanydimensionsaswewish,ofdelineatingtheanthropically-alloweddomaininparameterspace.(eventhoughtoquantifythisismoredifÞcult).Todelineatethealloweddomainsisprocedurallyuncontrover-sial,butwhataboutthemotivation?ItobviouslydependsonbelievingthatthelawsofnaturecouldhavebeenotherwiseÐunlessthereissomescientiÞcvalidityinimaginingÔcounterfactualuniversesÕthisexerciseseemsvacuous.5.Isit‘Scientic’toenquireaboutotherUniverses?IfourexistenceÐor,indeed,theexistenceofanyÔinterestingÕuniverseÐdependsonaseeminglyspecialcosmicrecipe,howshouldwereact?Thereseemtwolinestotake:wecandismissitashappenstance,orwecanconjecturethatouruniverseisaspeciallyfavoureddomaininastillvastermultiverse.Happenstance(orcoincidence) Figure2.Thisshowsinatwo-dimensionalparameterspaceandQ.TheupperandlowerlimitstoQarediscussedbyTegmarkandRees(1998).Theupperlimittostemsfromtherequirementthatgalactic-massboundsystemsshouldformbeforetheuniverseentersitsacceleratingdeSitterphase.Ouruniverse(obviously)liesintheanthropically-alloweddomain.ButwecannotsaywhetheritisatatypicallocationwithinthatdomainwithoutaspeciÞcmodelfortheprobabilitydistributionsofQandintheensemble.Maybeafundamentalsetofequations,whichsomedaywillbewrittenonT-shirts,Þxesallkeypropertiesofouruniverseuniquely.Itwouldthenbeanunassailablefactthattheseequationspermittedtheimmenselycomplexevolutionthatledtoouremergence.ButIthinktherewouldstillbesomethingtowonderabout.ItÕsnotguaranteedthatsimpleequationspermitcomplexconsequencesTotakeananalogyfrommath-ematics,considertheMandelbrotset.Thispatternisencodedbyashortalgorithm,buthasinÞnitelydeepstructure:tinypartsofitrevealnovelintricacieshowevermuchtheyaremagniÞed.Incontrast,youcanreadilywritedownotheralgorithms,superÞciallysimilar,thatyieldverydullpatterns.Whyshouldthefundamentalequationsencodesomethingwithsuchpotentialcomplexity,ratherthantheboringorsterileuniversethatmanyrecipeswouldleadto?Onehard-headedresponseisthatwecouldnÕtexistifthelawshadboringcon-sequences.Wemanifestlyarehere,sothereÕsnothingtobesurprisedabout.Ithinkwewouldneedtoknowwhytheuniquerecipeforthephysicalworldshouldpermit consequencesasinterestingasthoseweseearoundus(andwhich,asabyproduct,allowedustoexist)Aspecialuniversedrawnfromanensemble,ormultiverseButthereisanotherperspectiveÐahighlyspeculativeone,however.TheremaybemanyÔuniversesÕofwhichoursisjustone.Intheothers,somelawsandphysicalconstantswouldbedifferent.ButouruniversewouldnÕtbejustarandomone.Itwouldbelongtotheunusualsubsetthatofferedahabitatconducivetotheemergenceofcomplexityandconsciousness.Ifouruniverseisselectedfromamultiverse,itsseeminglydesignedorÞnetunedfeatureswouldnÕtbesurprising.Thismightseemarcanestuff,disjointfromÔtraditionalÕcosmologyÐorevenfromseriousscience.Butmyprejudiceistobeopenmindedaboutensemblesofuniverseandsuchlike,andeventosuspectthatwemaynotbeabletoaccountforsomefeaturesofourownuniversewithoutinvokingthem.First,asemanticdigression:thewordÔuniverseÕtraditionallydenotesÔeverythingthereisÕ.Thereforeifwearetoconsiderotherdomainsofspacetime(originatinginotherbigbangs)weshouldreallydeÞnethewholeensembleasÔtheuniverseÕ,andintroduceanewwordÐÔmetagalaxyÕforinstanceÐtodenotewhatobservationalcosmologiststraditionallystudy.However,solongasthiswholeidearemainsspec-ulative,itisprobablybesttocontinuetodenotewhatcosmologistsobserveasÔtheuniverseÕ,andtointroduceanewterm,ÔmultiverseÕ,forthewholehypotheticalensemble.SomemightregardotheruniversesÑregionsofspaceandtimethatwecan-notobserve(perhapseveninprincipleandnotjustinpractice)Ðasbeingintheprovinceofmetaphysicsratherthanphysics.Scienceisanexperimentalorob-servationalenterprise,anditÕsnaturaltobetroubledbyinvocationsofsomethingunobservable.ButIthinkÔotheruniversesÕ(inthissense)alreadyliewithintheproperpurviewofscience.ItisnotabsurdormeaninglesstoaskÔDounobservableuniversesexist?Õ,eventhoughnoquickanswerislikelytobeforthcoming.ThequestionplainlycanÕtbesettledbydirectobservation,butrelevantevidencesought,whichcouldleadtoananswer.Thereisactuallyablurredtransitionbetweenthereadilyobservableandtheabsolutelyunobservable,withaverybroadgreyareainbetween(seeFigure3).Toillustratethis,onecanenvisageasuccessionofhorizons,eachtakingusfurtherthanthelastfromourdirectexperience:Limitofpresent-daytelescopesThereisalimittohowfaroutintospaceourpresent-dayinstrumentscanprobe.Obviouslythereisnothingfundamentalaboutthislimit:itisconstrainedbycur-renttechnology.Manymoregalaxieswillundoubtedlyberevealedinthecomingdecadesbybiggertelescopesnowbeingplanned.WewouldobviouslynotdemotesuchgalaxiesfromtherealmofproperscientiÞcdiscoursesimplybecausetheyhavenÕtbeenseenyet.Limitinprincipleatpresentera Figure3.Extendinghorizonsbeyondthedirectly-observable.Eveniftherewereabsolutelynotechnicallimitstothepoweroftelescopes,ourobservationsarestillboundedbytheparticlehorizon,whichdemarcatesthesphericalshellaroundusatwhichtheredshiftwouldbeinÞnite.Ifouruniverseweredecelerating,thenthehorizonofourremotedescendantswouldencompassextragalaxiesthatarebeyondourhorizontoday.Itis,tobesure,apracticalimpedimentifwehavetoawaitacosmicchangetakingbillionsofyears,ratherthanjustafewdecades(maybe)oftechnicaladvance,beforeapredictionaboutaparticulardistantgalaxycanbeputtothetest.Butdoesthatintroduceadifferenceofprinciple?Surelythelongerwaiting-timeisamerelyquantitativedifference,notonethatchangestheepistemologicalstatusofthesefarawaygalaxies?Never-observablegalaxiesfromÔourÕBigBangButwhataboutgalaxiesthatwecanneversee,howeverlongwewait?ItÕsnowbelievedthatweinhabitanacceleratinguniverse.Asinadeceleratinguniverse,therewouldbegalaxiessofarawaythatnosignalsfromthemhaveyetreachedus;butifthecosmicexpansionisaccelerating,wearenowrecedingfromtheseremotegalaxiesatanever-increasingrate,soiftheirlighthasnÕtyetreachedus,itneverwill.SuchgalaxiesarenÕtmerelyunobservableinprinciplenowÐtheywillbebeyondourhorizonforever.Butifagalaxyisnowunobservable,ithardlyseemstomatterwhetheritremainsunobservableforever,orwhetheritwouldcomeintoviewifwewaitedatrillionyears.(AndIhaveargued,under(ii)above,thatthelattercategoryshouldcertainlycountasÔrealÕ.)(iv)Galaxiesindisjointuniverses Thenever-observablegalaxiesin(iii)wouldhaveemergedfromthesameBigBangaswedid.Butsupposethat,insteadofcausally-disjointregionsemergingfromasingleBigBang(viaanepisodeofinßation)weimagineseparateBigBangs.Arespace-timescompletelydisjointfromoursanylessrealthanregionsthatnevercomewithinourhorizoninwhatweÕdtraditionallycallourownuni-verse?SurelynotÐsotheseotheruniversestooshouldcountasrealpartsofourcosmos,tooWhetherotheruniversesexistornotisascientiÞcquestion.Thosewhoareprejudicedagainsttheconceptshouldregardtheabovestep-by-stepargumentasanexerciseinÔaversiontherapyÕ.Fromareluctancetodenythatgalaxieswithredshift10areproperobjectsofscientiÞcenquiry,youareledtowardstakingseriouslyquiteseparatespace-times,perhapsgovernedbyquitedifferentÔlawsÕ.Linde,VilenkinandothershaveperformedcomputersimulationsdepictinganÔeternalÕinßationaryphasewheremanyuniversessproutfromseparatebigbangsintodisjointregionsofspacetimesÐeachsuchregionitselfvastlylargerthanourobservationalhorizon.Guth,HarrisonandSmolinhave,fromdifferentviewpoints,suggestedthatanewuniversecouldsproutinsideablackhole,expandingintoanewdomainofspaceandtimeinaccessibletous.AndRandallandSundrumsuggestthatotheruniversescouldexist,separatedfromusinanextraspatialdi-mension;thesedisjointuniversesmayinteractgravitationally,ortheymayhavenoeffectwhatsoeveroneachother.Noneofthesescenarioshasbeensimplydreamedupoutoftheair:eachhasaserious,albeitspeculative,theoreticalmotivation.However,oneofthem,atmost,canbecorrect.Quitepossiblynoneis:therearealternativetheoriesthatwouldleadjusttooneuniverse.Firmingupanyoftheseideaswillrequireatheorythatconsistentlydescribestheextremephysicsofultra-highdensities,howstructuresonextradimensionsareconÞgured,etc.Perhaps,inthe21st-centurytheory,physi-cistswilldevelopatheorythatyieldsinsightinto(forinstance)whytherearethreekindsofneutrinos,andthenatureofthenuclearandelectricforces.Suchatheorywouldtherebyacquirecredibility.Ifthesametheory,appliedtotheverybeginningofouruniverse,weretopredictmanybigbangs,thenwewouldhaveasmuchreasontobelieveinseparateuniversesaswenowhaveforbelievinginferencesfromprimordialnucleosynthesisabouttheÞrstfewminutesofcosmichistory.6.UniversalLaws,orMereBylaws?Sometheorists,FrankWilczekforinstance,regardÔarethelawsofphysicsunique?ÕasakeyscientiÞcchallengeforthenewcentury.TheanswerdetermineshowmuchvarietytheotheruniversesÐiftheyexistÐmightdisplay.Ifthereweresomethinguniquelyself-consistentabouttheactualrecipeforouruniverse,thentheaftermathofanybigbangwouldbeare-runofourownuniverse.Butafarmoreinterest-ingpossibility(whichiscertainlytenableinourpresentstateofignoranceofthe Figure4.ÔDecisiontreeÕ.Progressin21-stcenturyphysicsshouldallowustodecidewhetheranthropicexplanationsareirrelevantor,ontheotherhand,thebestwecaneverhopefor.underlyinglaws)isthattheunderlyinglawsgoverningtheentiremultiversemayallowvarietyamongtheuniverses.SomeofwhatwecallÔlawsofnatureÕmayinthisgranderperspectivebelocalbylaws,consistentwithsomeoverarchingtheorygoverningtheensemble,butnotuniquelyÞxedbythattheory.ManythingsinourcosmicenvironmentÐforinstance,theexactlayoutoftheplanetsandasteroidsinourSolarSystemÐareaccidentsofhistory.Likewise,therecipeforanentireuniversemaybearbitrary.MorespeciÞcally,someaspectsmaybearbitraryandothersnot.Asananalogy(whichIowetoPaulDavies)considertheformofsnowßakes.Theirubiquitoussix-foldsymmetryisadirectconsequenceofthepropertiesandshapeofwatermolecules.Butsnowßakesdisplayanimmensevarietyofpatternsbecauseeachismouldedbyitsmicro-environments:howeachßakegrowsissensitivetothefortuitoustemperatureandhumiditychangesduringitsgrowth.Ifphysicistsachievedafundamentaltheory,itwouldtelluswhichaspectsofnatureweredirectconsequencesofthebedrocktheory(justasthesymmetricaltemplateofsnowßakesisduetothebasicstructureofawatermolecule)andwhichare(likethedistinctivepatternofaparticularsnowßake)theoutcomeofaccidents.Thecosmologicalnumbersinouruniverse,andperhapssomeoftheso-calledconstantsoflaboratoryphysicsaswell,couldbeÔenvironmentalaccidentsÕ,ratherthanuniquelyÞxedthroughoutthemultiversebysomeÞnaltheory.Someseem-inglyÔÞnetunedÕfeaturesofouruniversecouldthenonlybeexplainedbyÔan-thropicÕarguments[seeFigure4].Althoughthisstyleofexplanationraiseshackles amongsomephysicistsitisanalogoustowhatanyobserverorexperimenterdoeswhentheyallowforselectioneffectsintheirmeasurements:iftherearemanyuniverses,mostofwhicharenothabitable,weshouldnotbesurprisedtoÞndourselvesinoneofthehabitableones!TheentirehistoryofouruniversecouldjustbeanepisodeoftheinÞnitemul-tiverse;whatwecallthelawsofnature(orsomeofthem)maybejustparochialbylawsinourcosmicpatch.Suchspeculationsdramaticallyenlargeourconceptofreality.PuttingthemonaÞrmfootingmustawaitasuccessfulfundamentaltheorythattellsuswhethertherecouldhavebeenmanyÔbigbangsÕratherthanjustone,and(ifso)howmuchvarietytheymightdisplay.Untilthisfundamentalissueissettledonewayortheother,wewonÕtknowwhetheranthropicargumentsareirrelevantorunavoidable.7.TestingMultiverseTheoriesHereandNowWemayonedayhaveaconvincingtheorythataccountsfortheverybeginningofouruniverse,tellsuswhetheramultiverseexists,and(ifso)whethersomesocalledlawsofnaturearejustparochialby-lawsinourcosmicpatch.ButwhileweÕrewaitingforthattheoryÐanditcouldbealongwaitÐwecancheckwhetheranthropicselectionoffersatenableexplanationfortheapparentÞnetuning.Suchahypothesiscouldevenberefuted:thiswouldhappenifouruniverseturnedouttoevenmorespeciallytunedthanourpresencerequires.Wecouldapplythisstyleofreasoningtotheimportantnumbersofphysics(for)totestwhetherouruniverseistypicalofthesubsetthatthatcouldharbourcomplexlife.MostphysicistswouldconsidertheÔnaturalÕvalueofbelarge,becauseitisaconsequenceofaverycomplicatedmicrostructureofspace.Perhapsthereisonlyararesubsetofuniverseswhereisbelowthethresholdthatallowsgalaxiesandstarstoform.universeobviouslyhadtobebelowthatthreshold,Butifouruniverseweredrawnfromanensembleinwhichequallylikelytotakeanyvalue,wewouldnÕtexpectittobetoofarbelowitCurrentevidencesuggeststhatifconstitutedtheÔdarkenergyÕ,itsactualvalueis5-10timesbelowthatthreshold.Thatwouldputouruniversebetweenthe10thor20thpercentileofuniversesinwhichgalaxiescouldform.Inotherwords,ouruniverseisnÕtsigniÞcantlymorespecial,withrespectto,thanouremergencedemanded.Butsupposethat(contrarytocurrentindications)observationsshowedmadenodiscerniblecontributiontotheexpansionrate,andwasoftimesbelowthethreshold,notjust5Ð10times.ThisÔoverkillprecisionÕwouldraisedoubtsaboutthehypothesisthatwasequallylikelytohaveanyvalue,andsuggestthatitwaszeroforsomefundamentalreason(orthatithadadiscretesetofpossiblevalues,andalltheotherswerewellaboutthethreshold).IÕvetakenjustasanexample.Wecouldanalyseotherimportantnumbersofphysicsinthesameway,totestwhetherouruniverseistypicalofthehabitablesubsetthatthatcouldharbourcomplexlife.Themethodologyrequiresustodecide whatvaluesarecompatiblewithouremergence.ItalsorequiresaspeciÞctheorythatgivestheprobabilityofanyparticularvalue.Withthisinformation,onecanthenaskifouractualuniverseisÔtypicalÕofthesubsetinwhichwecouldhaveemerged.Ifitisanatypicalmemberevenofthissub-set(notmerelyoftheentiremultiverse)thenourhypothesiswouldbedisproved.OtherparameterscouldbeanalysedsimilarlyÐtestinginamulti-parameterspacewhetherouruniverseisatypicalmemberwithintheanthropicallyalloweddomain.Asatwo-dimensionalexample,considerthejointconstraintsonandQinFigure2.WecannotdecidewhetherouruniverseistypicalwithoutatheorythattellsuswhatÔmeasureÕtoputoneachpartofthe2-dimensionalparameterspace.Ifhigh-Quniversesweremoreprobable,andtheprobabilitydensityofwereuniform,thenweshouldbesurprisednottoÞndourselvesinauniversewithhigherandhigherQ.Theseexamplesshowthatsomeclaimsaboutotheruniversesmayberefutable,asanygoodhypothesisinscienceshouldbe.WecannotconÞdentlyassertthatthereweremanybigbangsÐwejustdonÕtknowenoughabouttheultra-earlyphasesofourownuniverse.NordoweknowwhethertheunderlyinglawsareÔpermissiveÕ:settlingthisissueisachallengeto21stcenturyphysicists.Butiftheyare,thenso-calledanthropicexplanationswouldbecomelegitimateÐindeedtheyÕdbetheonlytypeofexplanationweÕlleverhaveforsomeimportantfeaturesofouruniverse.Modelswithlowomega,non-zerolambdatwokindsofdarkmatter,andtherestmayugly.Sometheoristsareupsetbythesedevelopments,becauseitfrustratestheircravingformaximalsimplicity.Ithinkwecanlearnalessonfromcosmologicaldebatesinthe17thcentury.GalileoandKeplerwereupsetthatplan-etsmovedinellipticalorbits,notinperfectcircles.Newtonlatershowed,however,thatallellipticalorbitscouldbeunderstoodbyasingleuniÞedtheoryofgravity.Likewiseouruniversemaybejustoneofanensembleofallpossibleuniverses,constrainedonlybytherequirementthatitallowsouremergence.ButtoregardthisoutcomeasuglymaybeasmyopicasKeplerÕsinfatuationwithcircles:NewtonwasperhapsthegreatestscientiÞcintellectofthesecondmillennium.Perhapshisthird-millenniumcounterpartwilluncoveramathematicalsystemthatgovernstheentiremultiverse.ReferencesAguirre,A.:2001,Phys.RevD,3508.Dicke,R.:1961,Nature,446.Rees,M.J.:1980,PhysicaScripta,614.Salpeter,E.E.:1964,Astrophys.J.,796.Tegmark,M.andRees,M.J.:1998,Astrophys.J.,526.Tegmark,M.:2003,in:J.D.Barrowetal.(eds.),ScienceandUltimateReality:fromQuantumtoCosmos,CambridgeUniversityPress,inpress. N M T J . R E 1 I t o u c H e a B o s c b Ô o o l w f m a y o f u s a i i t o t h s o t c o s I a c e Ô M i c a C o s o B o l t f a o d e c a m e t w e w n c Þ r s h b D i b e t r a o t e t a g r v f o w i a h a a t r a o t H u r t t s o f a e H e s Ô h c a v e s a f e w w d e y t h p d e s c b u t t h p n o t c x t y i i s n u a i s v e m y Õ I a n s t w h a e x t F r e H o y w a i u e i t m a b y B o b u t h e c t t p s t H e a t h h f a o r o f t h C 1 r l e v e s s c m a a c a t t l o f c i c c M o r v e , i p t h s c h w l e t c t h t h s c Ô c o f n a m i g n b t u v e I n Ô G a N u c a Q u H o y w r t h Ô o m u s a l h v e a m o o f c a t s d e n b t a i p Õ H e g o n t o o t w p a t o t h . O n e i s t h Ô d i e n b a a e n t o t h l c y o p t h s p i t t n b a n i t b r s u v e b u t Ô o p l t v a w o b e d i f H o y l f a v o t l o b e t Ô c p o t l e v e i n C 1 a O 1 n n l h a v e t h a o a a I c s p b e t h s c l e o e d e o n a b b e e c a o w e c e x i o i n t h p o t u v e w h t l v e h t o b e c p Õ W i t t x a m y m o v a I d l e t o s m a b r y h w t h i l t . I b e v e t F r e s c i s n o w e v e m o r a v e t t Ô p o t u v e b e w h i t h v a o c m u s n w b e i a t h s e v a l g e t t d a o t c a o e Ð p e e v e e Ô v e w i a m u v e A e o NUMERICAL COINCIDENCES AND 'TUNING' IN COSMOLOGY MAR TIN J . R EES 1. In tr od u ction H erm ann B ondiÕ s classic book ÔC osm o logyÕ was, for m an y o f u s, an inspiring introduction t o t he science of the cosm o s. In a chapter entitled Ô M i crophysics and C o sm ologyÕ, B ondi lists the f am ous dim ensionless constants, and m entions the w ell-kno w n coincidence, Þ r st highlighted by D irac, betw een the r atio of the elec- trical and g ra vitational f orces w ithin a hydrogen atom and the r atio of the H ubble radius to the size o f an electron. H e says: ÔT hese coincidences are v ery striking, and f e w would d en y t heir possible d eep signiÞ cance, b u t t he precise nature of the conne xion the y indicate i s not understood and i s v ery m ysterious. Õ I am not sure to w h at e x tent F r ed H o yle w as inß u enced in this m atter b y B ondi, b u t h e certainly took this problem seriously too. H e also, t hrough his f am ous realisation o f t he C 12 resonance l e v elÕ s signiÞ cance, m ade a celebrated addition to the list o f cosm i c coincidences. M o reo v er , in pondering t heir signiÞ cance he was l ed to conjecture t hat t he so called Ô constants o f n atureÕ m i ght not be truly uni v ersal. I n Ô G alaxies, N u clei and Q uasarsÕ H o yle w rites t hat Ô one m u st at least ha v e a m odicum o f curiosity about the strange dim ensionless num bers that appear in physics. Õ H e goes o n t o outline t wo possible attitudes t o t hem . O n e i s t hat Ôthe d im ensionless num bers are all entirely necessary t o t he logical consistenc y of physicsÕ; t he second possibility is that the num bers are not in the b roadest sense uni v ersal, b ut that Ôin other p laces their v alues w ould b e d if ferentÕ H o y le f a v oured this latter option b ecause then Ôthe curious placing of the l e v els i n C 12 and O 16 need no longer h a v e t he appearance of astonishing accidents. It could sim ply b e t hat since creatures lik e ourselv es d epend o n a balance b etw een carbon and oxygen, w e can e x ist only i n t he portions of the uni v erse w here these le v els happen t o b e correctly placed. Õ W ith these te xts as m y m oti v ation, IÕ d lik e t o sum m arise b rieß y ho w t he issue looks today . I b elie v e that F r edÕ s conjecture i s n o w e v en m o re attracti v e, though the Ô portions of the uni v erseÕ betw een w h ich t he v ariation o ccurs m u st no w b e interpreted as t hem selv es v astly lar g er than the dom ain our telescopes can actually observ e Ð p erhaps e v en entire Ôuni v ersesÕ w ithin a m ulti v erse. Abstract. Fred Hoyle famously drew attention to the significance of apparent coincidences in the energy levels of the carbon and oxygen nucleus. This paper addresses the possible implications of other coincidences in cosmology.