A873CosmologyCourseNotesThecosmologicalmodelssubsequentlydevelopedbyFriedmannandLeMaitreretainsomeideasfromtheEinsteincosmologyGRlargescalehomogeneitylargescalecurvaturebuttheydroptheassumptiont ID: 332073
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A873:CosmologyCourseNotesIII.HomogeneousCosmologicalModelsReadingsmaterialinthissectioniscoveredinchapters3-7ofRyden,andinchapter3ofPeacock.IverymuchlikethediscussioninJimGunn's(1978)articleTheFriedmannModelsandOpticalObservationsinCosmology,fromtheSAAS-FEEProceedingsObservationalCos-mology,editedbyA.Maederetal.Thisarticleishardtotrackdown,andIwilldistributethe rst40pagesorso(whicharethemostuseful).Fordistancemeasures,IrecommendthecompactsummarybyDavidHogg,DistanceMea-suresinCosmology,availableasastro-ph/9905116.Itgenerallydoesnotgivederivations,butitsummarizesthestandardresultsinadmirablyclearform.Notationdi ersfromonetreatmenttoanother.IwilltrytomostlyfollowRyden'snota-tion.EinsteincosmologicalmodelCosmologicalConsiderationsontheGeneralTheoryofRelativity(Einstein,1917,reprintedinthePrincipleofRelativitybook)Einsteinconstructsthe rstmoderncosmologicalmodel,drawingonnewconceptsofrelativity.Givesarguments,notparticularlypersuasive,againstanin niteuniverse.Introduceshypothesis:universeishomogeneousonlargescales.Howcanuniversebe niteandhomogeneous?Whataboutboundary?GRallowsasolution:spaceispositivelycurved,likethesurfaceofasphere.Finite,butnoboundary.Seeksstaticsolutionwithconstant,positivecurvature.Provesthatnosuchsolutionexists.Inresponse,Einsteinmodi esthe eldequationfromG=8TtoG g=8T;addingthe\cosmologicalterm."Ofthe\principles"thatleadtothe eldequation,theonehedropsistherequirementthatspacetimeis\ratwhenT=0everywhere.Foraspeci crelationbetweenthecosmologicalconstantandthematterdensity,thisallowsastaticsolution.11 A873:CosmologyCourseNotesThecosmologicalmodelssubsequentlydevelopedbyFriedmannandLeMa^itreretainsomeideasfromtheEinsteincosmology|GR,largescalehomogeneity,largescalecurvature|buttheydroptheassumptionthattheuniverseisstatic.Theythereforedonotrequireacosmologicalconstant.Einsteinabandonedthecosmologicaltermforgoodwhenthecosmicexpansionwasdis-covered(in1929).Heisreputedtohavecalledit\thegreatestblunderofmylife."TheCosmologicalConstantThecosmologicalconstantideahasnevercompletelygoneaway.Ithasbeenespeciallyprominentinthelast20years,butitisnowviewedaspartofT.InsteadofG g=8T;G=8(Tmatter+TVAC);whereTVACg=8:TVACisthestress-energytensorofa\falsevacuum"or\scalar eld"withequationofstatep= .Thebasice ectcanbeseenfromourNewtonianlimitresult:r2=4(+3p):Forp =3,gravitypushesinsteadofpulls.Withtherightchoiceof,canhavestaticmodelinwhichpushofvacuumenergybalancespullofmatter(butunstable).Withlarger,getacceleration.TheFriedmann-Robertson-WalkermetricThespatialmetricoftheEinsteincosmologicalmodelisthatofa3-sphere:dl2=dr2+R2sin2(r=R)d\r2;whered\r2d2+sin2d 2istheangularseparation.HereRisthecurvatureradiusofthe3-dimensionalspace,andrisdistancefromtheorigin.Inthecoordinateframeofafreelyfallingobserver,timeisjustpropertimeasmeasuredbytheobserver,andthespacetimemetricisds2= c2dt2+dl2:AnaturalgeneralizationoftheEinsteinmodelistoallowthecurvatureradiusR(t)tobeafunctionoftime.12 A873:CosmologyCourseNotesTheuniverseisstillhomogeneousandisotropiconasurfaceofconstantt,butitisnolongerstatic.Inthe1930s,RobertsonandWalker(independently)showedthatthereareonlythreepossiblespacetimemetricsforauniversethatishomogeneousandisotropic.Theycanbewrittends2= c2dt2+a2(t)dr2+S2k(r)d\r2;whereSk(r)=R0sin(r=R0);k=+1;=r;k=0;=R0sinh(r=R0);k= 1:Inthisnotation,a(t)isdimensionless.Itisde nedsothata(t0)=1atthetimet0(usuallytakentobethepresent)whenthecurvatureradiusisR0.Atothertimesthecurvatureradiusisa(t)R0.TheradialcoordinaterandtheradiusofcurvatureR0haveunitsoflength(e.g.,Mpc).IhavefollowedRyden'snotationingivingSk(r)unitsoflength.InGunn'snotation,Sk(r)isdimensionless,anda(t)isreplacedbyR( ),whereR( )hasunitsoflength.Fork=+1,thespacegeometryatconstanttimeisthatofa3-sphere.Fork=0,thespacegeometryatconstanttimeisEuclidean,a.k.a.\\ratspace."Fork= 1,thespacegeometryatconstanttimeisthatofanegativelycurved,3-dimensional\pseudo-sphere."FriedmannandLeMa^itreusedthismetricintheircosmologicalmodelsofthe1920s.RobertsonandWalkerprovedthattheyaretheonlyformsconsistentwiththeCosmolog-icalPrinciple(homogeneityandisotropy).ItiscommonlycalledtheFriedmann-Robertson-Walkermetric,orsometimestheRobertson-Walkermetric.TheFRWmetric,spacecurvature,andspacetimecurvaturek=+1=)positivecurvature,sphericalgeometry, nitespacek=0=)nocurvature,Euclideangeometry,in nitespacek= 1=)negativecurvature,pseudo-spheregeometry,in nitespaceNote:thesearedescriptionsofspaceatconstantt.Formanyformsofa(t),spacetimeispositivelycurvedevenifspaceisnot,(thisisalwaysthecaseunlessacosmologicalconstantorsomeotherformofenergywithnegativepressureisimportant).Inthespecialrelativistic,Milnecosmology,spacetimeis\rat,butsurfacesofconstanttimearenegativelycurved.13 A873:CosmologyCourseNotesPositivecurvature=)geodesics\accelerate"(in2ndderivativesense)towardseachother.Initially\parallel"geodesicsconverge.Example:greatcirclesonasphere.Zerocurvature=)nogeodesic\acceleration."Initiallyparallelgeodesicsstayparallel.Euclideangeometry.Example:straightlinesonaplane.Negativecurvature=)geodesics\accelerate"awayfromeachother.Initiallyparallelgeodesicsdiverge.Example:geodesicsonasaddle.Thesubstitutionx=Sk(r)allowstheFRWmetrictobewritteninanotherfrequentlyusedform:ds2= c2dt2+a2(t)dx21 kx2=R20+x2d\r2:Demonstrationisleftasa(simple)exerciseforthereader.Withrasradialcoordinate,radialdistancesare\Euclidean"butangulardistancesarenot(unlessk=0).Withxasradialcoordinate,thereverseistrue.ComovingObserversThemetricdependsonthecoordinateframeoftheobserver.Evenahomogeneousandisotropicuniverseonlyappearssotoaspecialsetoffreelyfallingobservers,called\FundamentalObservers(FOs)"or\ComovingObservers."Theseobserversare\goingwiththe\row"oftheexpandinguniverse,andtheproperdistancebetweenthemincreasesinproportiontoa(t).Inthecoordinateframeoftheseobservers,theFRWmetricapplies,andthetimecoordinateoftheFRWmetricisjustpropertimeasmeasuredbytheseobservers.ComovingspatialcoordinatestrackthepositionsoftheseFOs,i.e.,thecomovingseparationbetweenanypairofFOsremainsconstantintime.AnobservermovingrelativetothelocalFOshasa\peculiarvelocity,"wherepeculiarisusedinthesenseof\speci ctoitself"ratherthan\odd."Anobserverwithanon-zeropeculiarvelocitydoesnotseeanisotropicuniverse{e.g.,dipoleanisotropyofthecosmicmicrowavebackgroundcausedbyre\rexofthepeculiarvelocity.Examplesofapplicationofmetric(1)Anyfreelyfallingparticlefollowsgeodesicsinspacetime,whosesolutionincomovingcoor-dinatescouldbefoundfromthegeodesicequation.Forcomovingparticles(FO's),solutionistrivial:r;; =constant.14 A873:CosmologyCourseNotes(2)Lightraystravelalongnullgeodesics:ds2=0.Therefore,alongaradialray(d\r=0),dr=cdt=a(t)=)ro re=Rtotecdt=a(t):(3)Inasurfaceofconstantt,metricdistancealongaradialpathofconstant; isl=Zds=Zr2r1a(t)dr=a(t)(r2 r1):(4)Inasurfaceofconstantt,metricdistancealongapathofconstantr;betweentwopointsofdi erent isl=Zds=Z 2 1a(t)Sk(r)d\r=a(t)Sk(r)sin( 2 1):Notethatthisisnotagreatcircle(andhenceshortest)pathunless==2.(5)Inasurfaceofconstantt,metricvolumeofashellofradiusrandwidthr risV=Zd3s=Zr+rra(t)drZ4a2(t)S2k(r)d\r2=4S2k(r)a3(t)r:Fork=0thisisjust4r2ra3(t).Notethatthe\metric"distancesandvolumesin(3)-(5)are\proper,"physicaldistancemeasuresandthatristhecomovingradialcoordinate.RedshiftofphotonsAphotonemittedbyanearbycomovingsourceatdistancedisDopplershifted:d= vc= Hdc= Hdt;whereHistheHubbleparameterandthelastequalityfollowsbecaused=cdt.TheHubbleparameterisH=_d=d=_(ar)=(ar)=_a=a:Thus,d= _adta= daa=)dln= dlna:Letthephotonbeemittedwithfrequencyeattimeteandobservedwithfrequencyoattimeto.Integratetogeteo=aoae=oe(1+z);z=redshift:Constantofintegration xedbydemandingo !easao !ae.Photonwavelengthproportionaltoa(t).15 A873:CosmologyCourseNotesFrequencyshift=)timedilation.Reale ectobservedin,e.g.,supernovalightcurves.Couldalsoderiveby(a)consideringsuccessivecreststravelingonnullgeodesics,or(b)usingequationforevolutionof4-momentumalongnullgeodesic.KinematicredshiftGiveaparticlea\peculiar"velocitywithrespecttothecomovingframe.Thepeculiarvelocitydecaysasitcatchesupwithrecedingparticles.Thisisapurelykinematice ect,thoughitlookslikea ctitious\friction."Inthenon-relativisticlimit,aparticlewithspeedvgoesdistancevdtandchangesitspeculiarvelocitybydv= H(vdt)=)dvv= _adta= daa=)pepo=mvemvo=aoae:Momentumredshiftslikethefrequencyofaphoton.Gunnshowsthatthiscontinuestoholdintherelativisticcase(dvlarge).OnecanalsoshowthatthisimpliesthatthedeBrogliewavelengthofaparticleredshiftsjustlikephotonwavelengths.Kinematicredshiftprofoundlya ectsthedynamicsofinstabilities:inanexpandinguni-verse(oranyexpandingmedium),undrivendisturbancesdecayinsteadofcoast.Flux,diameter,andsurfacebrightnessvs.redshiftds2= c2dt2+a2(t)[dr2+S2k(r)d\r2]:Frommetricapplication(2)above,wehavethecomovingdistancetoanobjectthatemittedlightattimeteasDc=Zt0tecdta(t):Froma(1+z) 1wehaveda= dz(1+z) 2= a2dz;andfromH_a=awehavedt=da=(aH)=)dt=a(t)=da=(a2H)= dz=H:PuttingtheseresultstogetheryieldsDc=Zz0cdz0H(z0)=cH0Zz0dz0H0H(z0)forthecomovingdistancetoanobjectatredshiftz.UsingtherelationforH0=H(z0)thatwewillderivelaterfromtheFriedmannequationthenreproducesequation(15)ofHogg(1999).16 A873:CosmologyCourseNotesAnobjectofangularsized\rattimetehasatransversephysicalsizedl=a(te)Sk(r)d\r=(1+z) 1Sk(r)d\r:TheangulardiameterdistanceisDA=dld\r=(1+z) 1Sk(r);whereristhecomovingdistanceDcasgivenabove.Wewilllater ndfromtheFriedmannequationthatthecurvatureradiusisR0=cH0j\nkj 1=2;where\ntot=1 \nkistheratioofthetotalenergydensityoftheuniverse(mass,radiation,darkenergy,...)tothecriticaldensity.For\nk!0,R0!1andthecurvature1=R20!0.Togetherwiththede nitionDH=c=H0,thisyieldstheresultofequations(16)and(18)ofHogg(1999):DA=(1+z) 1\n 1=2ksinh(\n1=2kDc=DH)k= 1;=(1+z) 1Dck=0;=(1+z) 1j\nkj 1=2sin(j\nkj1=2Dc=DH)k=+1:Thephotonsfromasourceatredshiftzaredistributedoveranarea4S2k(r)atthepresentday,sincea(t0)1:Photonsemittedinatimedtearereceivedoveranintervaldt0=(1+z)dte,andtheyareshifteddownwardinenergyby(1+z).Thebolometric\ruxFisthereforereducedbyanadditionalfactor(1+z)2:F=L4S2k(r)(1+z)2L4D2L;whereListhesourcebolometricluminosityandDL=(1+z)2DA(Hogg1999,eq.21).Inc.g.s.units,[F]=ergs 1cm 2;[L]=ergs 1:ThesolidanglesubtendedbyasourceofprojectedareaAis\n=A=D2A,makingthesurfacebrightnessI0F\n=L4D2LD2AA=L4A1(1+z)4=Ie(1+z)4:17 A873:CosmologyCourseNotesThisisthefamous(1+z)4surfacebrightnessdimmingofcosmologicalsources,whichcanmakehighredshiftgalaxiesverydiculttodetect.Note,however,thattheserelationsfor\ruxandsurfacebrightnessarebolometric,inte-gratedoverallwavelengths.Therelationformonochromatic\ruxescanbewritten0S0=eLe4D2L:Themonochromaticorpassband\ruxofanastronomicalobjectisa ectedbytheredshiftingofthebandpassfrom0toe=0(1+z).Thise ectisreferredtoastheK-correction(seeHoggetal.2002,astro-ph/0210394).Mostimportantfactaboutredshift:Ifwemeasuretheredshiftofasource(e.g.,fromthefrequencyofaspectralline),weknowao=ae.Givenamodelofa(t),wealsoknowtheradialdistance.Forangulardiameterandlumi-nositydistances,wealsoneedtoknowthespacecurvature(R0andk,or\nk).Nearby(i.e.,to rstorderinz),cz=co ee=H0d;H0_a(t0)a(t0);independentofothercosmologicalparameters.Alternatively,ifwecaninferdistancefromobserved\rux(ofasourceofknownluminosity)orobservedangularsize(ofasourceofknownphysicalsize),wecanreconstructa(t)fromobservations,andconstraincosmologicalparameters.18 A873:CosmologyCourseNotesDynamics:theFriedmannequationGravityhasn'tenteredthepictureyet.Buttogoanyfurther,weneeda(t).AssumeGRiscorrect.Wecouldgetequationsfora(t)bypluggingtheFRWmetricintothe eldequation.Thisyieldstwonon-trivialequations,oneofwhichistheintegraloftheother.offollowingthisderivation,we'llusetheNewtonianlimit,r2=4G(+3p),whichwillgetusalmostalltheway.WeappealtoBirkho 'stheorem,whichimpliesthatwecanthinkaboutasmallsphericalvolumeinisolation,ignoringthegravitationale ectsoftherestoftheuniverse(whichcanceloutinsphericalsymmetry).ConsiderashellofphysicalradiusRcomovingwiththeHubble\row:R= 43G(+3p)R31R2:ButR=arwithrconstant,soR=ar.Thus,a= 43G(+3p)a= 43G[3(+p)a 2a]:Thisisan\acceleration"equationforthecosmicexpansion.Weseealreadythata0if+3p�0,gravityslowsexpansion.Wewouldliketohavean\energy"equationfor_a,whichwecangetbyintegratingifweknowhowandpchangewitha.Usethe rstlawofthermodynamics(energyconservation),assumingthattheexpansionisadiabatic: pdV=dU=d(V)=dV+Vd=)d= (+p)dVV= 3(+p)daa=)_a= a3(+p)_:(Theadiabaticassumptionisvalidduringmostofthecosmicexpansion,butitisviolatedatsomespecialepochswhenthenumberofparticleschangessubstantially.)Multiplybothsidesoftheaccelerationequationby_atoget_aa= 43G a2_ 2a_a=43Ga2_+d(a2)d :19 A873:CosmologyCourseNotesRecognizethat_aa=d(_a2=2)=dtandthatthetermin[]isd(a2)=dt.Integratewithrespecttottoget_a2 8G3a2=constant:Unfortunately,derivingtheintegrationconstantreallydoesrequiretheGR eldequation.Wecanguessthatifthedensityishigh,spacewillbepositivelycurved,andtheuniversewillbegravitationallybound,makingtheconstant(whichplaystheroleofapotentialenergy)negative.Conversely,ifislow,spacewillbenegativelycurved,andtheuniversewillbeunbound,withapositiveconstant.GRleadstotheconclusionthattheintegrationconstantis kc2=R20.Indimensionallycorrectform,theFriedmannequationcanbewritten_aa2 8G3+kc2a2R20=0:Wewillsometimesrefertothe rsttermasthe\kinetic"term,thesecondasthe\gravi-tational"term,andthethirdasthe\curvature"term.ThedensityparameterNotethat_a=a=H,soifk=0theFriedmannequation=)=3H2=(8G).De nethe\criticaldensity"c=3H28G=densityofak=0Friedmannuniverse.Wecande neadimensionless\cosmologicaldensityparameter"\n=c=)=\n3H28G:TheFriedmannequationcanalsobewrittenH2(1 \n)= kc2a2R20:Matchingsignsimplies\n�1 !k=+1;closeduniverse\n=1 !k=0;\ratuniverse\n1 !k= 1;openuniverse20 A873:CosmologyCourseNotesNotethat\n=\n(t),butbecausekdoesn'tchange,\nalwaysremainswithinwhicheverofthese3regimesitstartsin.Ifwede ne\nk=1 \n,where\nisthesumofallotherenergydensitiesrelativetoc,thentheaboveequationimpliesR0=cH0j\nkj 1=2:EvolutionofenergydensityConsideranenergycomponentwithequationofstatep=w(c=1units).Howdoesitsenergydensitychangewithexpansionfactora(t)?dU= pdVd(V)=dV+Vd= wdVVd= (1+w)dVdln= (1+w)dlnV=3(1+w)dlna(V/a3):Integratingyields/a 3(1+w):Pressurelessmatter:w=0,/a 3(dilution)Radiation:w=1=3,/a 4(dilutionplusredshift)Cosmologicalconstant:w= 1,=const.Ifthesearetheenergycomponentsintheuniverse,thentheFriedmannequationbecomes_aa2 8G3m;0a0a3+r;0a0a4+;0= kc2a2R20:Herethesubscript0canrepresentany ducialtimet0.Ifitrepresentsthepresentday,thena0=a=(1+z).Notethatevenifthecurvaturetermiscomparabletothegravitationaltermtoday,itwillbenegligibleatsucientlyhighredshiftbecausethemandrtermsgrowmorerapidlywith(1+z).Thus,\ratuniverse(k=0)solutionsarealwaysaccurateathighz.SolutionsoftheFriedmannequation:singlecomponentuniverseEmptyuniverse:=0,k= 1_aa2=c2a2R20:21 A873:CosmologyCourseNotesSolutiona=ct=R0,R0=ct0.MetricandexpansionrateoftheMilnecosmology.Flatuniverse:k=0,=0 a0an._aa2=8G30a0an:Solutiona/t2=n.Pressurelessmatter:n=3,a=a0(t=t0)2=3.Radiation:n=4,a=a0(t=t0)1=2.(Ourstandardnotationhasa0=1.)-dominated\ratuniverse:k=0,=_aa2=8G3:Since_a/a,solutionisexponentialgrowth:a=a0et=tH;tH=8G3 1=2:SolutionsoftheFriedmannequation:twocomponentuniverseMatter+curvatureWehavepreviouslywrittentheFriedmannequationintheforms_aa2 8G3+kc2a2R20=0andH2(1 \n)= kc2a2R20:Evaluatingthesecondequationata=a0=1gives(1 \n0)= kc2R20H20;whichwecanusetochangetheR0dependenceinthe rstformtoan\n0dependence.22 A873:CosmologyCourseNotesForamatter-dominateduniverse(withorwithoutcurvature),=0a 3=\n03H208Ga 3;allowingtheFriedmannequationtobewrittenH2 \n0H20a 3+H20(\n0 1)a 2=0;orH2H20=\n0a 3+(1 \n0)a 2:Weseethatfor\n0�1,Hbecomeszeroatthe\turnaround"epochamax=\n0\n0 1:For\n01,Hdoesnotreachzero,soitcannotchangesign;anexpandingsub-criticaluniverseexpandsforever.For\n0�1,thesolution(whichcanbeveri edbydirectsubstitutionandabitofalgebra)canbewrittenintheparametricforma()=amax2(1 cos);t()=amax21H0(\n0 1)1=2( sin)=R0camax2( sin):Maximumexpansionisreachedat=,andtheuniversecollapsesina\bigcrunch"at=2.For\n01,de nea=\n01 \n0;andtheparametricsolutionisa()=a2(cosh 1);t()=a21H0(1 \n0)1=2(sinh )=R0ca2(sinh ):whererunsfromzerotoin nity.Atlatetimes( 1),thesolutionapproachesa/tasuniverseenters\freeexpansion."Atearlytimes( 1, 1),bothsolutionsapproacha/t2=3,asfork=0.23 A873:CosmologyCourseNotesMatter+Fora\ratuniversewithacosmologicalconstant,\n;0=1 \nm;0,andtheFriedmannequationcanbewrittenH2H20=\nm;0a 3+(1 \nm;0):For\nm;01,\n;0�0,andthematterdensityandcosmologicalconstantareequalatanexpansionfactoram=\nm;0\n;01=3:TherelationbetweentimeandexpansionfactorcanbewritteninthecumbersomebutexplicitformH0t=23(1 \nm;0)1=2ln2aam3=2+ 1+aam3!1=23:Atearlytimesa(t)32p\nm;0H0t2=3;likea\rat,matter-dominateduniverse,whileatlatetimesa(t)amexp(p\n;0H0t);givingtheexponentiallyexpandingsolutionfora-dominateduniverse.Curvature,Destiny,TopologyAstheabovesolutionshows,amatterdominatedk=+1universeeventuallycollapses,whileamatterdominatedk=0ork= 1universeexpandsforever.Thisequationofclosedgeometrywithabounduniverseand\rat/opengeometrywithanunbounduniversecontinuestoholdifradiationisadded.Butvacuumenergycanchangethepicture.FollowingtheargumentsinProblemSet2,theFriedmannequationcanbewrittenintheformH2(a)=H20\n ;0 (a) ;0+\nk;0a0a2+\nm;0a0a3+\nr;0a0a4;where (a)isthevacuumenergydensityatexpansionfactoraand\nk;0=1 \nm;0 \nr;0 \n ;0:24 A873:CosmologyCourseNotesNotethatk=+1correspondstonegative\nkandviceversa.Fortheuniversetorecollapse,wemusthaveH(a)=0atsometimeinthefuture(a�a0).For\n ;0=0,thismusthappenif\nk0cannothappenif\nk�0.For\n ;0�0,recollapsecanbeavoidedif (a)= ;0fallsslowerthana 2.Bestguesscurrentparametersare\n ;00:7,j\nk;0j 1, (a)const:,implyingthattheuniversecouldbeopen,\rat,orclosed,butthatexpansionforeverislikely.Futurerecollapseispossibleif\nk;00andvacuumenergychangesitsequationofstateandstartstofallfasterthana 2inthefuture.If\n ;00(anegativevacuumenergyisnotfavoredbyobservations,butitisnotobviouslyimpossibleinprinciple),thenonecouldhavean\nk�1(open)universethatrecollapses.GRdoesnotprohibittheuniversefromhavingacomplextopology,e.g.atoroidaltopologyinwhichheadingo inonedirectioneventuallybringsyoubacktowhereyoustarted.Thus,inprinciple,theuniversecouldbenegativelycurvedor\ratandstillbespatially nite.havebeensome(unconvincing)claimsforperiodicredshiftsthatcouldbeinterpretedasevidenceforcomplextopology.PeopleareseriouslysearchingforsignsofcomplextopologyinthepatternofCMBanisotropies.25