Naveragingofrandomvariables1 A5682IntroductiontoCosmologyCourseNotesInfacthotandcoldregionsarecorrelatedoverlargescalesandTTrmsdeclinesmuchmoreslowlythan1Themostwidelyusedstatisticalmeasure ID: 236414
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A5682:IntroductiontoCosmologyCourseNotes11.CMBAnisotropyReading:Chapter8,sections8.4and8.5GravitationalinstabilityandstructureformationToday'suniverseshowsstructureonscalesfromindividualgalaxiestogalaxygroupsandclustersuptosuperclustersofgalaxiesthatcanextendto100Mpcormore.Iftheuniverseissmoothbutnotperfectlysmooth,thengravitywillcausestructuretogrow,bydrawingmatterintoregionsthatstartslightlyabovetheaveragedensity(andoutofregionsthatstartslightlybelowtheaveragedensity).Forthisprocesstowork,theremusthavebeensmall-amplitudeinhomogeneitiespresentintheearlyuniverse,the\seeds"forthegravitationalgrowthofstructure.Theseinhomogeneitiesshouldcauseslightnon-uniformitiesintheCMB.Inthesimplestmodelswithonlybaryonicmatter,thepredicted\rucuationsintheCMBareroughlyonepartin103.Withnon-baryonicdarkmatter,itispossibletolowerthelevelofCMBanisotropiesto10 5andremainconsistentwithgravitationalgrowthoftoday'sstructure.MeasurementofCMBanisotropyThe\dipole"anisotropyoftheCMB,whichisjustthere\rexoftheearth'speculiarvelocity(whichislargelyduetothemotionoftheMilkyWay),wasdetectedinthe1970s.The rstdetectionoftheintrinsicanisotropiesoftheCMB,re\rectingstructureintheuniverseatzrecratherthanthemotionoftheearthtoday,wasobtainedbytheCOBEsatellitein1992.Thevariationsareonlyapartin105,someasuringthemrequiresextremelyhighsensitivityandextremelygoodcontrolofsystematicerrors,likecontaminationofthesignalfromothersources.SinceCOBE,therehavebeenanumberofmeasurementsofCMBanisotropywithhigherresolutionandsensitivity,fromtheground,balloons,andsatellites.TheWilkinsonMicrowaveAnisotropyProbe(WMAP)producedmuchhigherresolutionandmoresensitivemapsofCMBanisotropy,from2003-2012.Today'sstate-of-the-artCMBmapisfromtheEuropeanPlancksatellite,whichismoresensitiveandhigherresolutionthanWMAP.Ground-basedexperimentsliketheSouthPoleTelescope(SPT)andAtacamaCosmologyTelescope(ACT)cangettosmallerangularscalesbecausetheyuselargertelescopes.CharacterizingtheCMBanisotropySupposethatwesmoothaCMBmapoveranangularscale.(Anyrealmapwillautomaticallybesmoothedatsomeminimumangularscaledeterminedbythedi ractionlimitofthetelescopeusedtomakeit,andwecansubsequentlysmooththemapoverlargerscales.)WecanthenplotahistogramofthefractionaltemperaturevariationsT=Tinthesmoothedmapandmeasuretheroot-mean-squarewidthofthishistogram.Tocharacterizethestructureinthemap,wecanplot(T=T)rmsagainstthesmoothingscale.1 A5682:IntroductiontoCosmologyCourseNotesIfthemapconsistedofrandomlyplacedhotandcoldspots,wewouldexpect(T=T)rmstodeclineinproportiontothesquare-rootoftheskyarea(/1=)becauseofp Naveragingofrandomvariables.Infact,hotandcoldregionsarecorrelatedoverlargescales,and(T=T)rmsdeclinesmuchmoreslowlythan1=.ThemostwidelyusedstatisticalmeasureofstructureinCMBmapsistheangularpowerspectrum,Cl,derivedfromadecompositionofthemapintosphericalharmonics.Roughlyspeaking,l(l+1)Clisthesquareof(T=T)rmsonthescale=200=ldegrees.PhysicsofCMBanisotropyItisnotsurprisingthatifthedensityoftheuniverseisslightlyinhomogeneousatzrec,thentheCMBwillbeslightlynon-uniform.Thereareseverale ectsthatlinktheinhomogeneitiesofthematterdistributiontotheCMBanisotropies:Adiabatictemperature\ructuations.Wherethedensityishigher,thephotontemperatureishotter.Dopplershifts.Thegravitationalperturbationscausedbythedensity\ructuationsinducepeculiarvelocities,andphotonsareblueshiftedorredshiftediftheylastscatteredo electronswithpeculiarvelocitiestowardusorawayfromus,respectively.Gravitationalredshifts,asthephotonsclimboutofthedarkmatterpotentialwells,orblueshiftsastheyfallo ofpotentialhills.Thise ectmakesthephotonscomingoutofdenserregionsredder(coolertemperature).Di erentprocessesdominateondi erentscales,soifwecanmeasureCMBanisotropyoverawiderangeofscaleswecanseparatethemtosomedegree.AcousticoscillationsAslongastheuniverseisionized,sothatphotonsaretightlycoupledtotheelectronsandbaryons,thephotonsandbaryonsbehavelikeasingle,high-pressure\ruidwitha\soundspeed"ofapproxi-matelyc=p 3:Anoverdenseregionmaystarttocollapseunderitsowngravity,buteventuallythebuildupofpressurewillhaltandreversetheexpansion.Ifthereisenoughtime,aperturbationcancontractandre-expandseveraltimes,acycleof\acousticoscillations."Becauseofthise ect,thereisapreferredscaleonwhichthetemperature\ructuationsarelargest.Thisisthescaleforwhichanoverdenseregionhashadjustenoughtimetocollapsetomaximumcompressionwhentheuniverserecombines.Onverylargescalestherehasbeeninsucienttimeforcontractontooccur,andgravitationalredshiftsproduceapproximately\scale-invariant"\ructuations.Thephysicalsizeoftheacousticscalecanbecomputedfrom rstprinciples,oncethedensitiesofdarkmatter,baryons,andradiationarespeci ed.Theacousticscaleisa\standardruler."Theangularsizeoftheacousticscaledependsonthecomovingdistancetothelastscatteringsurfaceandon2 A5682:IntroductiontoCosmologyCourseNotes Plancksatellitemeasurementsoftheangularpowerspectrumoftemperature\ructuationsintheCMB. Plancksatellitemeasurementsoftheangularpowerspectrumofpolarization\ructuationsintheCMB.3 A5682:IntroductiontoCosmologyCourseNotesgeometry(curvature).MeasuringcosmologicalparameterswiththeCMBIfwespecifythestatisticalpropertiesofthedensity\ructuationspresentatzrec,andthematterandenergycontentsoftheuniverse(\nr;0,\nm;0,\n;0,H0,etc.),thenwecanpredictthefullpatternofCMBanisotropy.Modelpredictionscanbetestedagainstmeasurements,andthemeasurementscanbeusedtoinferthepropertiesoftheprimordial\ructuationsandthematterandenergycontentsoftheuniverse.Changingparameterschangesthepredictions.Thereissomedegeneracyamongtheparameters(i.e.,changesinonecanbetradedo againstchangesinanother),butwithgoodmeasurementsoverawiderangeofscalesthesedegeneraciescanmostlybebroken.Thereisaniceillustrationofthesee ectsathttp://space.mit.edu/home/tegmark/movies.htmlPlotsonthenextpageconveythebasicidea.4 A5682:IntroductiontoCosmologyCourseNotes Theoreticalpredictionsofthetemperaturepowerspectrumfora ducialcosmologicalmodel(blacksolidline,parametersasindicated)andmodelsinwhichoneparameterisvariedatatime.Topandbottompanelsshowlogarithmicandlinearlaxes,respectively.Dottedred:Highbaryondensity.Dashedgreen:Changeofinitialconditions,fromdi erentin\rationmodel.Long-dashedblue:LowerHubbleconstant.Dot-dashedgreen:Nocosmologicalconstant,\nm=0:3,openuniverse(k= 1)Dot-dashedred:Nocosmologicalconstant,\nm=1,\ratuniverse(k=0)Theheightsofthepeaksdependsmoststronglyonthematterdensityandthebaryondensity.Theangularlocationofthepeaksdependsmoststronglyoncurvature.Theoverall\tilt"dependsonin\rationaryinitialconditions.5 A5682:IntroductiontoCosmologyCourseNotesCMBPolarizationCMBanisotropiesarepolarized,i.e.,theyhaveslightlydi erentamplitudeswhenmeasuredinorthogonalpolarizations.Thepolarizedsignalisonlyabout10%ofthefullsignal(whichisonly10 5inthe rstplace),andthepolarizationfromforegroundcontaminantsishardtoremove,sothesemeasurementsaretough.Firstdetectionsin2005,consistentwithpredictionsfromin\ration.Roughlyspeaking,polarizationdoubles(oreventriples)theinformationcontentofCMBanisotropies andbetterpinsdownwhatiscausinganisotropyonagivenscale.Thesee ectscanproduceonlyparticularkindsofstatisticalpatterns,knownas\E-mode"polar-ization.\B-mode"polarizationcanbeproducedbygravitationallensingoftheE-modepatternbyclusteredmatteratlowredshifts.B-modepolarizationcanalsobeproducedbygravitywavescreatedintheearlyuniverse,a.k.a.\tensor\ructuations."DetectionofB-modepolarization,especiallyfromgravitywaves,isthecurrentholygrailofexper-imentalCMBresearch.Thechallengeispartlyoneofrawsensitivity,butalsocontrollinginstrumentale ectsandastro-physicalforegrounds(e.g.,emissionfromdustintheMilkyWay)atpart-in-a-millionlevels.6