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Cosmological Perturbations and Their Effects on the Universe: from Inflation to Acceleration


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IextendmydeepestthankstoJimFryforguidanceandsoundadvicethroughoutmyventuresattheUniversityofFlorida.Forhelpfuldiscussionsandconversationsonamultitudeoftopicsovertheyears,IacknowledgeWayneBomstad,SteveDetweiler,LeanneDuy,andLisaEverett.Forsupportinvarioustimesofneed,IthankHai-PingChengandtheQuantumTheoryProject,theUniversityWomen'sClub,theUniversityofFlorida'sAlumniFellowshipProgram,theDepartmentofPhysics,andtheCollegeofLiberalArtsandSciences.IalsothankFilipposKlironomosforhisassistanceintheincarnationofthiselectronicdissertation.ExtensiveusehasbeenmadeofNASA'sAstrophysicsDataSystem'sbibliographicservices,aswellasSpires'high-energyphysicsdatabase.Onthetopicofcosmologicalinhomogeneities,IacknowledgeDanChungandUrosSeljakfortheirhelpfulinput.IwholeheartedlythankEdBertschingerformakinghisCOSMICScodeavailableforpublicuse,whichhasprovedinvaluableinunderstandingandcomputingvariousaspectsofcosmologicalperturbationtheory.JesusGallegoandJaimeZamoranoarethankedforprovidingunpublisheddatafromtheirsurveysoflocalHIIgalaxies,andRafaelGuzman,ChipKobulnicky,DavidKoo,andMarianoMolesarethankedforvaluableassistanceinmyworkandunderstandingofstarburstgalaxies.Inadditiontotheaforementionedpeople,IalsoacknowledgeEannaFlanagan,KonstantinMatchev,PierreRamondandBernardWhitingfordiscussionsongravitationalradiationinextradimensions. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTIONTOCOSMOLOGICALPERTURBATIONS ......... 10 1.1EnergyDensity ................................. 10 1.2TheoryofInation ............................... 11 1.3CosmologicalPerturbationsfromInation .................. 12 1.4CosmologicalEvolutioninaPerturbedUniverse ............... 13 1.5NonlinearEvolutionofPerturbations ..................... 17 2THEGRAVITATIONALWAVEBACKGROUND ................. 21 2.1PrimordialGravitationalWaves ........................ 21 2.2ExtraDimensions ................................ 22 2.3AThermalGravitonBackground ....................... 24 2.4DetectionofExtraDimensions ......................... 25 2.5AlternativeThermalizationMechanisms ................... 27 2.6ProblemsofExtraDimensions ......................... 28 2.7Summary .................................... 29 3STRUCTUREFORMATIONCREATESMAGNETICFIELDS ......... 30 3.1Introduction ................................... 30 3.2MagneticFields:Background ......................... 31 3.3CosmologicalPerturbations .......................... 34 3.3.1ColdDarkMatter ........................... 35 3.3.2LightNeutrinos ............................. 36 3.3.3Photons ................................. 37 3.3.4Baryons ................................. 38 3.3.5ChargeSeparations ........................... 40 3.4MagneticFields ................................. 45 3.5Discussion .................................... 47 4EFFECTSONCOSMICEXPANSION ....................... 52 4.1AcceleratedExpansion ............................. 52 4.2EectsofInhomogeneities ........................... 53 4.3EectsontheExpansionRate ......................... 56 5

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................. 56 4.5VarianceoftheEnergyinInhomogeneities .................. 60 4.6Summary .................................... 62 5FULLYEVOLVEDCOSMOLOGICALPERTURBATIONS ........... 65 5.1PrecisionCosmology .............................. 65 5.2SelectionoftheDataSample .......................... 67 5.3ConstraintsonCosmologicalParameters ................... 71 5.4ConclusionsandFutureProspects ....................... 75 6CONCLUDINGREMARKS ............................. 77 6.1CreationofPerturbations ........................... 77 6.2EarlyEvolutionofPerturbations ....................... 78 6.3FinalStateofPerturbations .......................... 80 6.4FateoftheUniverse .............................. 82 APPENDIX AERRORSINHIGH-ZGALAXIESASDISTANCEINDICATORS ........ 85 A.1UniversalityamongHIIGalaxies ....................... 85 A.2SystematicErrors ................................ 87 A.3MeasurementUncertainties ........................... 89 A.4StatisticalErrors ................................ 91 BONANELECTRICALLYCHARGEDUNIVERSE ................ 93 B.1Introduction ................................... 93 B.2GeneratingaNetElectricCharge ....................... 94 B.3NewtonianFormulation ............................ 96 B.4RelativisticFormulation ............................ 101 B.5Discussion ................................... 105 REFERENCES ....................................... 108 BIOGRAPHICALSKETCH ................................ 125 6

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Table page 5-1TableofSelectedHigh-RedshiftGalaxies ...................... 69 7

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Figure page 2-1Parameterspaceforathermalgravitationalwavebackground .......... 24 3-1qandq=Hasafunctionofredshift ........................ 44 3-2Spectralenergydensityofthemagneticeld .................... 48 4-1Spectraldensityofgravitationalpotentialenergy ................. 57 4-2Fractionalcontributionsofenergyininhomogeneities ............... 58 4-3Fractionalcontributionsofinhomogeneitiesatsecondorder ............ 61 4-4Fluctuationinpotentialenergyvs.cutoscale ................... 63 5-1Distancemodulusvs.redshift ............................ 72 5-2Constraintsonmand 74 A-1logMzvs.logLHforHIIandstarburstgalaxies ................. 88 A-2Simulationofvelocitydispersionsforstar-formingregions ............. 90 B-1Expansionfactorsforpositiveandnegativecharges ................ 99 B-2Evolutionofanetchargeasymmetry ........................ 102 8

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Theuniverseis,onthelargestscales,nearlyperfectlyisotropicandhomogeneous.Thisdegreeofsmoothnesswasaccentuatedinthepast,whendensityinhomogeneitiesdepartedfromperfectuniformitybyonlythirtypartspermillion.Thesetinyimperfectionsintheearlyuniverse,however,havehadenormousimpactincausingtheuniversetoevolveintoitspresentstate.Thisdissertationexaminestheroleofthesecosmologicalperturbationsthroughoutvariousimportanteventsduringthehistoryoftheuniverse,includingination,linearandnonlinearstructureformation,andthecurrentphaseofacceleratedexpansion.Thespectrumofperturbationsiscalculatedinthecontextofextradimensions,andshownunderwhichconditionsitcanbethermal.Theeectsofgravitationalcollapseareshowntogeneratemagneticelds,butnottosignicantlyaltertheexpansionrateorcauseacceleration.Finally,Lyman-breakgalaxiesareexaminedasapossibledistanceindicator,anditisfoundthattheymayemergeasapowerfultooltobetterunderstandtheenergycontentoftheuniverse. 9

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Onverysmall(i.e.,planetary)scales,densitycontrastscanbeashighas1030.However,aslargerandlargerscalesareexamined,thedensitycontrastofatypicalpointinspaceisfoundtobemuchsmaller.WhenscalesofO(100Mpc)orlargerareexamined,itisfoundthatdensitycontrastsarenearlyalwayssmall,suchthat1[ 1 ]. Whatcanbelearnedfromthisisthattheuniverseis,onitslargestscales,verynearlyisotropic(thesameinalldirections)andhomogeneous(thesameatallpositionsinspace).Intheframeworkofgeneralrelativity,auniversethatisbothisotropicandhomogeneousisdescribedbytheFriedmann-Robertson-Walkermetric, wherea(t)isthescalefactoroftheuniverse. Becausetheuniverseisexpandingandhasaniteage,itismanifestthatthedegreeofisotropyandhomogeneitywhichisobservedtodaywasgreaterinthepast.Thisisconrmedbyobservationsoftherelicradiationfromthebigbang,knownasthecosmicmicrowavebackground(CMB),whichshowstheuniversetohaveanamplitudeofdensityuctuationsof'3105[ 2 ].Theseuctuationsindensity,althoughinsignicantwhencomparedtothehomogeneouspartatearlytimes,playavitalroleintheuniverse's 10

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3 ]providesamechanismtoputtheseprimordialcosmologicalperturbationsinplaceatthetimeofthebigbang.Thebigbang,asacosmologicaltheory,istheonlycompellingtheoryinthecontextofEinstein'sgeneralrelativitythatprovidesanexplanationforthepresenceoftheCMBradiation,theobservedHubbleexpansionoftheuniverse,andthelightelementabundances(throughbigbangnucleosynthesis).ItisshowninPeebles1993[ 4 ]thatallreasonablealternativestothebigbangscenarioeitherfailtoreproduceoneoftheabovethreeobservationsorcannotbecompatiblewithgeneralrelativity.Thebigbang,however,isnotanoriginoftheuniverse,butisratherasetofinitialconditions.Itistheabovetheoryofinationthatnaturallyproducestheseinitialconditions. Forthebigbangtheorytoevolveintoauniversewhichiscompatiblewithobservations,theinitialconditionsmustbethattheglobalcurvatureoftheuniverseisspatiallyattoanaccuracyof2percent[ 5 ].Thetemperatureanddensityoftheuniversemustbeuniformacrossscalesfarlargerthanthehorizon,andthedensityofmagneticmonopolesintheuniversemustbeverysmall.Thesethreeproblemsareknownastheatness,horizon,andmonopoleproblem.ThemethodbywhichinationsolvesthisproblemistohaveadeSitter-likephaseofexpansionduringtheveryearlyuniverse.deSitterexpansionischaracterizedbythemetric whichissimilartoequation( 1{2 ),exceptthatthescalefactora=ep 11

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1{3 ),itismanifestthattheuniverseisexpandingatanexponentiallyfastrate.Givenenoughtime,theuniversecanexpandbyanarbitrarynumberofe-foldings.Auniverseexpandinginthiswaywillhaveitsmatterdensityreducedbyafactorof whereiandtiarethedensityandageoftheuniverseatthestartofination,andfandtfarethedensityandageattheendofination.Thisremovesanyinitialdensitiesordierencesindensity,solvingboththehorizonandmonopoleproblems.Additionally,anyinitialcurvature(departurefromatness)willbedrivenawaybyafactorofa2(ti)=a2(tf),providingasolutiontotheatnessproblemaswell. 6 { 9 ]andingravitationalradiation[ 10 11 ].Itisthesepredictionsforthedeparturesfromperfecthomogeneity,producedbyquantumuctuations,whicharethenstretchedduringinationacrossallscales,thatarethefocusofthiswork.TheperturbationsproducedbyinationareGaussianintheirstatisticalproperties,andarealsoscale-invariant. Theperturbationsingravitationalradiationareconstrainedtobeaverysmallfractionoftheenergydensityintheuniverse[ 12 ].Nevertheless,thedetectionofsuchgravitationalradiationandmeasurementofitspropertieswouldhavethecapabilitiestotellusmuchabouttheearlyuniverse,asgravitationalradiationisexpectedtobedecoupledfromtherestoftheuniversefromthetimeofitscreation.Ifascale-invariantspectrumofgravitationalwaveswereobserved,itwouldbeafurthergreatconrmationoftheinationaryparadigm,andconstraintsonthemodelofinationcouldbeinferred.However,therealsoexiststhepossibilitythatthespectrumofgravitationalwavescouldbethermal.Athermalgravitationalradiationspectrumcouldresultfrommanypossibilities, 12

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13 ].ThisisdiscussedatlengthinChapter 2 ofthiswork. Wheninationcomestoanend,theuniversereheats.Theprocessofreheatingtransferstheenergyfromthevacuum(whichwasresponsiblefortherapidexpansion)intomatterandenergy.Thisuniverseisdescribed,inthehomogeneousapproximation,bytheFriedmann-Robertson-Walkermetricofequation( 1{2 ).However,itisthedensityinhomogeneities,orprimordialcosmologicalperturbations,thatwillleadtotheformationofallstructureintheuniverse.Therefore,theequationofinterestasastartingpointformuchoftheremainderofthisworkisthatforaperturbedFriedmann-Robertson-Walkeruniverse, whereandarethescalar-modeperturbationstothegravitationalpotential.Ofcourse,itmakesnodierencewhichgaugeischosen[ 14 ],asthephysicsofcosmologicalperturbationsisthesameinallgauges.ThepreferenceoftheauthoristoworkintheconformalNewtoniangauge(alsoknownasthelongitudinalgauge),aschoseninequation( 1{5 ). 15 ]forareview).Also,intheearlyuniverse,asubstantialamountofdarkmattermustbe 13

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16 ]forareview).Theseunsolvedpuzzles,alongwiththemanyquestionssurroundingtheearlyuniversephysicsofelectroweaksymmetrybreaking,theQCDphasetransition,and(possibly)supersymmetrybreaking,areexpectedtobeunaectedbythepresenceofcosmologicalperturbations.Astheuniverseevolves,mostofthephysicsthatoccursintheearlieststagesisexpectedtooccurexactlyasitwouldinaperfectlyhomogeneousFriedmann-Robertson-Walkeruniverse. Onepossibilityintheveryearlyuniverse,however,forwhichcosmologicalperturbationsmayplayaseminalroleisiftheuniversebeginswith(orobtainsatearlytimes)anasymmetryinitsnetcharge.Ithasbeenpointedoutthatalargenetchargeintheuniversewouldberuinousatearlytimesforcosmological4Hesynthesis[ 17 ]andforthecosmicmicrowavebackground[ 18 ].Althoughaconclusivesolutionisbeyondthescopeofthisdissertation,cosmologicalperturbationsinanexpandinguniversemayhavethecapabilityofdrivinganelectricallychargeduniversetoaneutralstate.Thispossibility,andpreliminaryworkonthesubject,canbefoundinAppendix B Astheuniversecontinuestoexpandandcool,thebuildingblocksoftheuniversebegintoform.AftertheQCDphasetransition,quarksandgluonsbecomeboundintohadrons.Unstableparticlesdecayand/orco-annihilate,leavingtheuniversedevoidofexoticparticles.Neutrinosfreeze-out,anddecouplefromtherestoftheuniverse.Electronsandpositronscoannihilate,leavinganelectronasymmetrythatmatchesupnearlyperfectlywiththeprotonasymmetry.Whentheuniversecoolssubstantiallysothatstabledeuteriumcanformwithoutbeingdestroyedbythethermalphotonbath,nucleosynthesisoccurs,producingdeuterium,3He,4He,and7Li.Nucleosynthesisiscompleteroughlyfourminutesafterthebigbang.Althoughtherehasbeenworkinthepastsuggestingalternative,complexmodelsofnucleosynthesis(suchasinhomogeneous 14

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19 ]forareviewofbig-bangnucleosynthesisanditsalternatives),thestandardpictureofbigbangnucleosynthesisappearstomatchupperfectlywellwithobservationswithinthesystematicerrors[ 20 ]. Itisgenerallyassumedthatnointerestingphysicsoccursuntilthetimeofrecombination(whereelectronsandionscombinetoformneutralatoms),roughly380;000yearslater.TheonlythingsofnotewhichoccuroverthateraarethatcosmologicalperturbationsgrowaccordingtotheMeszaroseect[ 21 ],andtheuniversetransitionsfromaradiation-dominatedstatetoamatter-dominatedone.However,thereisaveryinterestingandsubtleeectthatoccursduringthistime.As(thebaryon-to-photon)ratioisverysmallandtheuniverseisstillquitehotanddenseduringthisera,everyionandelectronisconsistentlybombardedbythesehighenergyphotons.Ionshavecomparablechargestoelectrons,buttheirmassesareordersofmagnitudegreater.Thescatteringcrosssectionsofchargedparticleswithphotonsscales(fornon-relativisticscattering)as whereqisthechargeandmisthemass.Asaresultofthedierencesinmassandcrosssection,electronsareaectedbyinteractionswithphotonsinamuchmoreprofoundwaythanions.WhiletheCoulombforceskeepstheelectronsandprotonstightlycoupled,themomentumtransferfromphotonsworkstocreatechargeseparationsandcurrentsduringtheradiationera.Thetightlycoupledcomponentisdominant,andbehavesasabaryoniccomponentincosmologicalperturbationtheory(seeMaandBertschinger1995[ 22 ]foraverysophisticatedtreatment).Incontrast,thechargeseparationsandcurrentscreatedbymomentumtransferareverysmall,butnonethelessareofgreatimportforthegenerationofmagneticeldsatearlytimes[ 23 ].ThegenerationofmagneticeldsintheyounguniversebythismechanismisdetailedinChapter 3 ,whichalsodiscussesthepossibilitythattheoriginsofpresentlyobservedcosmicmagneticeldsmaylieinthismechanism. 15

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2Y(t0);(1{7) whereYisgivenby fromaninitialtimet0untilthetimeofinterest,t,wheremisthematterdensityandristheradiationdensity.Thisapproximationisvalidthroughoutthelinearregimeofgravitationalcollapseandstructureformation,andprovidesanaccuratedescriptionforthegrowthofoverdenseinhomogeneities. Duringthisepochofcompleteionization,electronsandionsareconstantlyinteractingwithoneanother,andattemptingtoformneutralatoms.Therearetwoprocessesthatimpedetheformationofneutralatoms.Therst,whichdelaystheonsetofneutralatomformation(knownasrecombination),isthatthebaryon-to-photonratio,,issolow.Eventhoughthephotontemperatureissignicantlybelowtheionizationenergyofaneutralatom(T13:6eV),thenumberofphotonsperbaryonisverygreat,andtheirenergyfollowsaPoissondistribution.Asaresult,therearestillenoughphotonsofsucientenergytokeeptheuniverse100percentionizedevenwhentheaveragetemperatureoftheuniverseissignicantlybelowthetypicalatomicionizationenergy.Thesecondprocessthatisresponsibleforimpedingtheformationofaneutral,transparentuniverseisthefactthateachLyman-seriesphoton(transitiontothegroundstateofhydrogen)emittedbyarecombiningatomwillencounterandreionizeanotherneutralatom.If,however,theemittedphotonhasenoughtimetoredshiftsucientlythatitcannotreionizeanotheratom,theuniversewillnetoneneutralatom.Also,araretwo-photonemissionprocesswillallowanatomtorecombinewithoutemittingaLyman-seriesphoton.Thisprocessofrecombiningthemajorityoftheatomsintheuniversetakesabout105yearstocomplete 16

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5 ],anddropstheionizationfraction(e)oftheuniversefrome'1toe104[ 24 ].Afterthispoint,ionsandelectronscannolongerecientlyndoneanother,andtheprocessofrecombinationfreezesout.Asthecosmicbackgroundofphotonsisnolongerconstantlyscatteringoofelectrons,itfreelystreamsfromtheepochofrecombinationuntilthepresentday,makingtheepochofrecombinationsynonomouswiththesurfaceoflastscattering. 25 ]andSchaeer1984[ 26 ],forexample. Theresultofallofthisisthataninitiallysmoothuniversewithonlyveryslightperturbationsinenergydensitybecomesacomplexwebofstructure,withsubstantialpoweronbothsmall(i.e.,galactic)andlarge(i.e.,supercluster)scales.(Foraveryinterestingcomparisonofnumericalsimulationsofstructureformationthroughthenonlinearregimeuptothepresentday,thereaderisreferredtoO'Sheaetal.2005[ 27 ].)Whilethislarge-scalestructureforms,theuniversecontinuestoexpandandcool,dropping 17

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Thevariousepochsoftheuniverse,hometotheonsetofextremelyinterestingphysics,aretrackedmosteasilybyredshift,z,denedby wherea(t)isthescalefactoroftheuniverseatagiventime,anda0isthescalefactoratpresent.Whengravitationalcollapseoccurstoasucientextentonsmallscales,themasscollectedinasmallareaofspacebecomeslargeenoughtoignitenuclearfusion.Thisistheepochatwhichtherststarsform. Exactlyatwhatepochstarformationbeginsisveryimportantforunderstandingtheevolutionofmatterandstructureinouruniverse.Asignatureoftheformationoftherststarswouldbeasureresignatureofnonlinearcollapse.Thetransitionfromasmooth,linearuniverse(suchastheuniverseatthetimeofrecombination)toahighlycomplex,nonlinearone(observedtoday)isnotyetwellunderstood.Recently,manyhavediscussedthepossibilitythatthegravitationalenergyboundinnonlinearinhomogeneitiescouldback-react,andsignicantlyimpacttheexpansionrate[ 28 { 30 ].Itappearsthattheimpactontheexpansionrateisinsignicant,however[ 31 { 34 ].ThisphysicalprocessanditseectsontheuniversearediscussedingreatdetailinChapter 4 Thedensestregionsofnonlinearstructurebecomehometotherststars,asillustratedinadaptivemeshrenementsimulations[ 35 ].ThedatafromtheWMAPsatelliteindicatethattheopticaldepthoftheuniverse,,isquitelarge[ 5 ].Fromthisinformation,itappearsthattherststarsturnedonveryearly,asthepresenceofalargenumberofstarswillreionizetheneutralgasthatformedduringrecombination.Fromtheopticaldepth,whichismeasuredtobe=0:170:04,itappearsthatreionizationoccursatroughly11.z.30.However,theobservationofaGunn-Petersontrough[ 36 ]inquasarspectraaroundz'6[ 37 ]indicatesthatreionizationisnotcompleteuntilthat 18

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38 ],havebeenproposedtoremedythissituation.Futurereleaseofdatafromsatellitesexploringthecosmicmicrowavebackgroundmayyieldlowervaluesof,whichwouldbeconsistentwithamoresimplisticexplanationofgradualreionization. Oncetherstcomplexnonlinearstructuresform,theycontinuetoevolve,withthedensestregionsattractingthemostmatterandformingthemostmassivestructures.Galaxiesgrowthroughbothmonolithiccollapseandaseriesofhierarchicalmergers,andviafurthergravitationalcollapseonlargerscales,therstclustersofgalaxieswillformaswell.Thetypesofobjectswhichcanbeobservedatearlytimesareverybrightgalaxies(intheopticalandinfrared)andquasars(primarilyintheradio),aswellasinterveningobjectsalongthelineofsight(throughabsorptionandtheLyman-forest).Foraatuniversethatcontainedacriticaldensityinmatter,theexpansionratewouldcontinuetodecreaseasthematterdensitydiluted,followingtheHubblelawofequation( 1{10 ), a2=8G wheremisagainthematterdensity(equaltothecriticaldensity)andHistheHubbleexpansionparameter.However,theuniverse'sexpansionrate,asinferredfromacombinationofmanysourcesofdata(seeChapter 5 andreferencestherein)isconsistentwithabout30percentoftheenergydensityinmatterandabout70percentinsometypeofvacuumenergy.Theexpansionlaw,then,appearstoobeyequation 1{11 a2=8G whereistheenergydensityinvacuumenergy,andthesumofmatterdensityandvacuumenergydensityisequaltothecriticaldensity. ThedatafromtypeIasupernovaehavebeenusedtoillustrateandsupportthefactthatthepictureoftheuniverseisinconsistentwithoutavacuumenergytermintheequationfortheHubblelaw[ 39 ].However,duetosystematicerrorsinherentinanysingle 19

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5 investigatesthepossibilityofusinganewmethod(rstdetailedinMelnick,TerlevichandTerlevich2000[ 40 ]andrstattemptedinSiegeletal.2005[ 41 ])tomeasurethecosmologicalparametersofmatterandvacuumenergydensityintheuniverse.Aswithanytypeofdistanceindicator,themethodofChapter 5 ,touseLyman-breakgalaxiesasadistanceindicator,issubjecttomanysourcesoferror,bothrandomandsystematic.TheseerrorsaredetailedinAppendix A Thedatasetsavailablearenowsucienttopaintacoherentpictureoftheuniverseanditsenergycontentsverywellonthelargestscales,andrelativelywellonevensmallscales[ 12 ].Therearemanyinterestingproblemsandphenomenaintheuniversethatarehithertounexplained,yetphysicsoftheanswersmaylieinsomethingassimpleasdeparturesfromtheidealmodel.Theremainderofthisworkdetailssomeinstanceswherecosmologicalinhomogeneities,whetheratearlytimesorlatetimes,onlargeorsmallscales,mayplayavitalroleinunderstandingtheuniverse.Finally,Chapter 6 willsummarizethemajorresultsofChapters 2 through 5 ,andwillpointtowardsfutureavenuesofinvestigation,suchasdeterminingthefateoftheuniverse. 20

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Inationarycosmologypredictsalow-amplitudegravitonbackgroundacrossawiderangeoffrequencies.Thischaptershowsthatifoneormoreextradimensionsexist,thegravitonbackgroundmayhaveathermalspectruminstead,dependentonthefundamentalscaleoftheextradimensions.Theenergydensityisshowntobesignicantenoughthatitcanaectnucleosynthesisinasubstantialway.Thepossibilityofdirectdetectionofathermalgravitonbackgroundusingthe21-cmhydrogenlineisdiscussed.Alternativeexplanationsforthecreationofathermalgravitonbackgroundarealsoexamined. Thesuccessoftheinationaryparadigm[ 3 ]inresolvingmanyproblemsassociatedwiththestandardbig-bangpicture[ 42 ]hasledtoitsgeneralacceptance.Inationarybigbangcosmologypredictsastochasticbackgroundofgravitationalwavesacrossallfrequencies[ 10 ],[ 11 ].Theamplitudeofthisbackgroundisdependentuponthespecic 21

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12 ]tobe gO(1010):(2{1) Ininationarycosmology,thepredictedCGWB,unliketheCMBandtheneutrinobackground,isnon-thermal.GravitationalinteractionsarenotstrongenoughtoproduceathermalCGWBattemperaturesbelowthePlanckscale(mpl1:221019GeV).Astheexistingparticlesintheuniverseleavethehorizonduringination,theonlymajorcontributionstotheenergydensitywillbethoseparticlescreatedduringorafterreheating,followingtheendofination.Unlessthereheattemperature(TRH)isgreaterthanmpl,gravitationalinteractionswillbetooweaktocreateathermalCGWB.ThemeasurementofthemagnitudeoftheprimordialanisotropiesfrommissionssuchasCOBE/DMR[ 43 ]andWMAP[ 5 ]providesanupperlimittotheenergyscaleatwhichinationoccurs[ 44 ].Fromthisandstandardcosmologicalarguments[ 45 ],anupperlimitonTRHcanbederivedtobe wheregisthenumberofrelativisticdegreesoffreedomatTRH,tplisthePlancktime,andtisthelifetimeoftheinaton.AstrongerupperlimitonTRH(1081010GeV)canbeobtainedfromnucleosynthesis[ 46 ]ifsupersymmetryisassumed.Inallreasonablecases,however,TRHmpl,indicatingthattheCGWBisnon-thermalininationarycosmology. 47 ]andKlein[ 48 ]showedthatclassicalelectromagnetismandgeneralrelativitycouldbeuniedina5-dimensionalframework.Moremodernscenariosinvolvingextradimensionsarebeingexploredinparticlephysics, 22

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49 50 ]oralargecurvature[ 51 52 ].Anyspatialdimensionswhichexistbeyondthestandardthreemustbeofasucientlysmallscalethattheydonotconictwithgravitationalexperiments.The3+1dimensionalgravitationalforcelawhasbeenverieddowntoscalesof0:22mm[ 53 ].Thus,ifextradimensionsdoexist,theymustbesmallerthanthislengthscale.Althoughthereexistmanydierenttypesofmodelscontainingextradimensions,therearesomegeneralfeaturesandsignalscommontoallofthem. Inthepresenceofextraspatialdimensions,the3++1-dimensionalactionforgravitycanbewrittenas gLm;G0N=GNm2pl wheregisthe4-dimensionalmetric,GNisNewton'sconstant,g0,G0N,andR0denotethehigher-dimensionalcounterpartsofthemetric,Newton'sconstant,andtheRicciscalar,respectively,andmDisthefundamentalscaleofthehigher-dimensionaltheory.In3+spatialdimensions,thestrengthofthegravitationalinteractionsscaleas(T=mD)(1+=2).If=0,thenmD=mpl,andstandard4-dimensionalgravityisrecovered. WhenenergiesintheuniversearehigherthanthefundamentalscalemD,thegravitationalcouplingstrengthincreasessignicantly,asthegravitationaleldspreadsoutintothefullspatialvolume.InsteadoffreezingoutatO(mpl),asin3+1dimensions,gravitationalinteractionsfreeze-outatO(mD)[ 49 ].(mDcanbemuchsmallerthanmpl,andmaybeassmallasTeV-scaleinsomemodels.)Ifthegravitationalinteractionsbecomestrongatanenergyscalebelowthereheattemperature(mD
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ParameterspaceforthecreationofathermalCGWBinthecontextofLargeExtraDimensions.Theshadedareasrepresentareasruledoutbygravitationalexperimentsandreheating,bothwithandwithouttheassumptionofsupersymmetry.Certainassumptionsaboutgravitinophysics,asdetailedinSarkar1996[ 46 ],maysignicantlylowertheboundonreheatingwithsupersymmetryinextradimensions. Othertypesofextradimensionshaveminorquantitativedierencesintheshapeoftheirparameterspaces.However,thequalitativeresult,thecreationofathermalCGWBifmD
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gg wherecisthecriticalenergydensitytoday,TCMBisthepresenttemperatureoftheCMB,andgisthenumberofrelativisticdegreesoffreedomatthescaleofmD.gisdependentontheparticlecontentoftheuniverse,i.e.whether(andatwhatscale)theuniverseissupersymmetric,hasaKKtower,etc.Otherquantities,suchasthetemperature(T),peakfrequency(),numberdensity(n),andentropydensity(s)ofthethermalCGWBcanbederivedfromtheCMBifgisknown,as Thesequantitiesarenotdependentonthenumberofextradimensions,asthelargediscrepancyinsizebetweenthethreelargespatialdimensionsandtheextradimensionssuppressesthosecorrectionsbyatleastafactorof1029.Asanexample,ifmDisjustbarelyabovethescaleofthestandardmodel,theng=106:75.ThethermalCGWBthenhasatemperatureof0.905Kelvin,apeakfrequencyof19GHz,andafractionalenergydensityg'6:1107. 55 ].WithathermalCGWBincluded,theexpansionrateoftheuniverseisslightlyincreased,causingneutron-protoninterconversiontofreeze-outslightlyearlier.AthermalCGWBcanbeeectivelyparameterizedasneutrinos, 25

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56 ].Anincreaseintheprecisionofvariousmeasurements,alongwithanimprovementinthesystematicuncertainties,mayallowfortheindirectdetectionofathermalCGWB. DirectdetectionofathermalCGWBismuchmorechallenging,butwouldprovidequitestrongevidenceforitsexistence.Conventionalgravitational-wavedetectorsincludecryogenicresonantdetectors[ 57 ],whichhaveevolvedfromthebarsofWeber[ 58 ],dopplerspacecrafttracking,andlaserinterferometers[ 59 ].ThemaximumfrequencythatthesedetectorscanprobeliesinthekHzregime,whereasathermalCGWBrequiresGHz-rangedetectors.AninterestingpossibilityfordetectionmaylieinthebroadeningofquantumemissionlinesduetoathermalCGWB.Individualphotonsexperienceafrequencyshiftduetogravitationalwaves[ 60 ].Foralargesampleofradio-frequencyphotonsinagravitationalwavebackground,theobservedlinewidth(W)willbroadenby Wh0p wheret0isthepresentageoftheuniverse,isthepeakfrequencyofthethermalCGWBandh0isthemetricperturbationtodayduetothethermalCGWB[ 61 ].AsO(1031)isaverysmallbroadening,aradiolinewithanarrownaturalwidthisthepreferredcandidatetoobservethiseect.Onepossibilityforthistypeofobservationisthe21-cmemissionlineofatomichydrogen.Solongastheemittingatomsandthedetectorsaresucientlycooled,broadeningduetothermalnoisewillbesuppressedbelowW.Becausethelifetime(1=)oftheexcitedstateofhydrogenislarge(107yr)andthefrequencyofthe 26

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Thewidthofthe21-cmlineisregrettablysevenordersofmagnitudelargerthantheexpectedbroadeningduetoathermalCGWB.Extraordinarilyaccuratemeasurementswouldneedtobetakenfordirectdetectionofthisbackground.Additionally,temperaturesoftheatomsanddetectorswouldneedtobecryogenicallycooledto1018KelvintosuppressthermalnoisebelowW.Thislastcriterionisfarbeyondthereachofcurrenttechnology,andeitheramajoradvanceorexperimentalinnovationwouldberequiredtomeasurethedesiredeectusingthistechnique. Thepredictionsofinationarenumerous[ 44 ],andmanyhavebeensuccessfullyconrmedbyWMAP[ 5 ].Themajorsuccessesofinationincludeprovidingexplanationsfortheobservedhomogeneity,isotropy,atness,absenceofmagneticmonopoles,andoriginofanisotropiesintheuniverse.Additionally,conrmedpredictionsincludeascale-invariantmatterpowerspectrum,an=1universe,andthespectrumofCMBanisotropies.ToexplainathermalCGWBbyeliminatinginationwouldrequirealternativeexplanationsforeachofthepredictionsabove.Althoughalternativetheorieshavebeenproposed,asinHollandsandWald2002[ 62 ],theyhavebeenshowntoface 27

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63 ].Thesuccessesofinationappeartosuggestthatitmaylikelyprovideanaccuratedescriptionoftheearlyuniverse. Primordialblackholeswithmasseslessthan1015gwouldhavedecayedbytoday,producingthermalphotons,gravitons,andotherformsofradiation.Densityuctuationsintheearlyuniverse,inordertoproducealargemassfractionoflow-massprimordialblackholes,andnottoproducetoolargeofamassfractionofhigh-massones,favoraspectralindexnthatislessthanorequalto2=3[ 64 ].Acceptingtheobservedscale-invariant(n'1)spectrumofdensityuctuations[ 65 ]maydisfavorprimordialblackholesasareasonablecandidateforcreatingathermalCGWB. TheDirachypothesisstatesthatthedierenceinmagnitudebetweenthegravitationalandelectromagneticcouplingstrengthsarisesduetotimeevolutionofthecouplings[ 66 ].Iftrue,gravitationalcouplingwouldhavebeenstrongerintheearlyuniverse.AttemperatureswellbelowthePlanckscale,gravitywouldhavebeenuniedwiththeotherforces,creatingathermalCGWBatthatepoch.However,thishypothesisproducesconsequencesforcosmologicalmodelsthatarediculttoreconcile[ 67 ],andanytimevariationisseverelyconstrainedbygeophysicalandastronomicalobservations[ 68 ].TheacceptablelimitsforvariationaresmallenoughthattheycannotincreasecouplingsucientlytogenerateathermalCGWBsubsequenttotheendofination.ThedicultiesfacedbyeachofthesealternativeexplanationspointstowardsextradimensionsasperhapstheleadingcandidateforthecreationofathermalCGWB. 69 ].Stringmoduliinteractionswithstandardmodeleldsarehighlysuppressed,leadingtoalonglifetimeofthestringmoduli.Stringmodulidecay,however,mustbeconsistentwithastrophysicalconstraints[ 70 ].Toaccomplishthis,stringmodulineedeitherasmallproductionamplitudeorveryspecicdecaychannels,whichbothrequire 28

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71 72 ].Whiletheshort-wavelengthmodes(themodesinsidethehorizonwhengravitationalinteractionsfreeze-out)willthermalize,gravitationalwavesoflongerwavelengthswillbeunaected.AsthescaleofinationmustbeabovemD,theamplitudeofthesewavesisexpectedtobelarge.ThiswouldleaveanunacceptableimprintintheCMB.BothproblemsarisefromthefactthatatenergiesabovemD,macroscopicgravitybreaksdown[ 73 ].Althoughtheseproblemsmaynotberesolveduntilaquantumtheoryofgravityisrealized,theydonotchangethefactthatathermalCGWBwouldarisefromextradimensionswithmD
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Thischapterexaminesthegenerationofseedmagneticeldsonallscalesduetothegrowthofcosmologicalperturbations.Intheradiationera,localdierencesintheionandelectrondensityandvelocityeldsareinducedbymomentumtransferfromphotons.Thecurrentswhichowduetotherelativemotionoftheseuidsleadtothegenerationofmagneticelds.Magneticeldsarecreatedonallcosmologicalscales,peakingatamagnitudeofO(1023Gauss)attheepochofrecombination.Magneticeldsgeneratedinthismannerprovideapromisingcandidatefortheseedsofmagneticeldspresentlyobservedongalacticandextra-galacticscales. Inprinciple,onceaseedeldisinplace,itshouldbepossibletofollowitsevolutionandamplicationfromthecollapseofstructureandtheeectsofanyrelevantdynamos.Inthischapter,anewmechanismforthegenerationofseedeldsisputforward.Itisarguedthatcosmologicalperturbationtheoryintheradiationeraproduceschargeseparationsandcurrentsonallscales,bothofwhichcontributetomagneticelds.Theseseedeldspersistuntiltheonsetofgravitationalcollapse,atwhichpointeldamplicationanddynamoprocessescanmagnifysuchseeds,possiblytotheO(G)scalesobservedtoday. Thischapterillustratesthatthegenerationofmagneticeldsinthismannerisanecessaryconsequenceofstructureformation.Themagnitudeoftheseseedeldsiscalculated,anditisshownthattheseseedeldsmaybesucientlystrongtoaccountfor 30

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75 ].Thefourmajormethodsusedtostudyastrophysicalmagneticeldsaresynchrotronradiation,Faradayrotation,Zeemansplitting,andpolarizationofstarlight.TheseobservationaltechniquesaredetailedindepthinRuzmaikin,SokolovandShukurov1988[ 76 ],withFaradayrotationoftenprovingthemostfruitfuloftheabovemethods. Magneticeldshavebeenfoundinmanydierenttypesofgalaxies,inrichclusters,andingalaxiesathighredshifts.Spiralgalaxies,includingourown,appeartohaverelativelylargemagneticeldsofO(10G)onthescaleofthegalaxy[ 77 ],withsome(suchasM82)containinganomalouslystrongeldsupto'50G[ 78 ].Ellipticalandirregulargalaxiespossessstrongevidenceformagneticelds(oforderG)aswell[ 79 ],althoughtheyaremuchmorediculttoobserveduetothepaucityoffreeelectronsintheseclassesofgalaxies.Coherencescalesformagneticeldsinthesegalaxies,asopposedtospirals,aremuchsmallerthanthescaleofthegalaxy.Furthermore,galaxiesatmoderate(z'0:4)andhighredshifts(z&2)havebeenobservedtorequiresignicant 31

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80 81 ].Magneticeldsarealsoobservedinstructureslargerthanindividualgalaxies.Thethreemaintypesofgalaxyclustersarethosewithcoolingows,thosewithradio-halos,andthosedevoidofboth.Galaxyclusterswithcoolingowsareobservedtohaveeldsof0:2to3G[ 82 ],theComacluster(aprimeexampleofaradio-halocluster)isobservedtohaveaeldstrength2:5G[ 83 ],whileclustersselectedtohaveneithercoolingowsnorradiohalosstillexhibitindicationsofstrong(0:11G)elds[ 84 ].Thereevenexistsevidenceformagneticeldsonextraclusterscales.AnexcessofFaradayrotationisobservedforgalaxieslyingalongthelamentbetweentheComaclusterandtheclusterAbell1367,consistentwithaninterclustermagneticeldof0:20:6G[ 85 ].Onthelargestcosmologicalscales,thereexistonlyupperlimitsonmagneticelds,arisingfromobservationsofthecosmicmicrowavebackground[ 86 ]andfromnucleosynthesis[ 87 ],settinglimitsthatonscales10Mpc,eldstrengthsare108G. Observationalevidenceformagneticeldsisfoundingalaxiesofalltypesandingalaxyclusters,bothlocallyandathighredshifts,wherevertheappropriateobservationscanbemade.AreviewofobservationalresultscanbefoundinVallee1997[ 88 ].Thetheoreticalpictureofthecreationoftheseelds,however,isincomplete.FieldsofstrengthGcanbeexplainedbythemagnicationofaninitial,smallseedeldongalactic(orlarger)scalesbythedynamomechanism[ 89 { 91 ].Aprotogalaxy(orprotocluster)containingamagneticeldcanhaveitseldstrengthincreasedbymanyordersofmagnitudethroughgravitationalcollapse[ 92 93 ],andcanthenbefurtherampliedviavariousdynamos.Dynamoswhichcanamplifyasmallseedeldintothelargeeldsobservedtodayinvolvehelicalturbulence()and/ordierentialrotation(!).Varioustypesofthesedynamosincludethemean-elddynamo[ 76 94 95 ],theuctuationdynamo[ 79 96 ],andmerger-drivendynamos[ 97 ],amongothers.However,thedynamomechanismdoesnotexplaintheoriginofsuchseedelds. 32

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98 ],andtheirexistenceisnotexplainedbythedynamomechanismalone.Therearemanymechanismsthatcanproducesmall-strengthmagneticeldsonastrophysicallyinterestingscales,eitherthroughastrophysicalorexoticprocesses(seeWidrow2002[ 75 ]foradetailedreview).Exoticprocessesgenerallyrelyonnewphysicsintheearlyuniverse,suchasarst-orderQCDphasetransition[ 99 100 ],arstorderelectroweakphasetransition[ 101 102 ],brokenconformalinvarianceduringination[ 103 104 ],specicinatonpotentials[ 105 ],orthepresenceofchargedscalarsduringination[ 106 { 108 ].Astrophysicalmechanisms,incontrast,aregenerallybettergroundedinknownphysics,althoughtheyhavedicultygeneratingsucientlystrongeldsonsucientlylargescales.Thedierenceinmobilitybetweenelectronsandionscanleadtoseedmagneticeldsfromradiation-eravorticity[ 109 110 ],fromvorticityduetogas-dynamicsinionizedplasma[ 111 { 115 ],fromstars[ 116 ],orfromactivegalacticnuclei[ 117 ].Althoughtherearemanycandidatesforproducingtheseedmagneticeldsrequiredbythedynamomechanism,nonehasemergedasadenitivesolutiontothepuzzleofexplainingtheirorigins. Thenovelmechanismproposedinthischapteristhatseedmagneticeldsaregeneratedbythescatteringofphotonswithchargedparticlesduringtheradiationera.Unlikethemechanismof[ 109 110 ],whichisdisfavored[ 118 ]duetoitsrequirementofsubstantialprimordialvorticity(althoughsee[ 119 120 ]foranargumentthatsomevorticityisnecessary),theeldsofinterestherearegeneratedbytheearlieststagesofstructureformation,requiringnonewphysics.Ions(henceforthtakentobeprotons,forsimplicity)andelectronsaretreatedasseparateuids,withoppositechargesbutsignicantlydierentmasses.Themass-weightedsumsoftheirdensityandvelocityeldswilldeterminetheevolutionofbaryonsintheuniverse,andshouldagreewithprevioustreatments,suchasMaandBertschinger1995[ 22 ].Thedierenceoftheionandelectrondensityandvelocityelds,however,willprovideameasureoflocalchargeseparationand 33

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2 ],itisthesesmalldensityinhomogeneities,predictedbyinationtooccuronallscales[ 8 ],whichleadtoallofthestructureobservedintheuniversetoday.Asitistheearlyepochofstructureformationthatisofinterestforthecreationofmagneticelds,thischaptercalculatestheevolutionofinhomogeneitiesinthelinearregimeofstructureformation.ThemostsophisticatedtreatmentofcosmologicalperturbationsinthelinearregimetodateisthatofMaandBertschinger1995[ 22 ],whichprovidesevolutionequationsforaninhomogeneousuniversecontainingacosmologicalconstant,darkmatter,baryons,photons,andneutrinos.Thissectionextendstheirtreatmenttoencompassseparateprotonandelectroncomponents.Themass-weightedsumofprotonsandelectronswillrecoverthebaryoncomponent,whereasthedierenceofthedensityeldsisrepresentativeofachargeseparation,andthedierenceofthevelocityeldsisthatofanetcurrent. Thedynamicsofanycosmologicaluidcanbeobtained,ingeneral,fromthelinearEinsteinequations(seePeeblesandYu1970[ 121 ],SilkandWilkson1980[ 122 ],andWilsonandSilk1981[ 123 ]forearliertreatments).Althoughthechoiceofgaugedoesnotimpacttheresults,theConformalNewtoniangaugeleadstothemoststraightforwardcalculations.Themetricisgivenby 34

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a(_+_a a)=4Ga2T00;k2(_+_a a)=4Ga2(+P);+_a a(_+2_)+(2a a_a2 3Ga2Tii;k2()=12Ga2(+P); whereistheshearterm,whichisnegligiblefornon-relativisticmatter(butimportantforphotonsandneutrinos).Foracosmologicaluidthatiseitheruncoupledtotheotheruidsormass-averagedamonguncoupleduidsintheearlyuniverse,thefollowingevolutionequationshold: _=(1+w)(3_)3_a a(c2sw);_=_a a(13w)_w whereisdenedasthelocaldensityrelativetothespatialaverage(=),ikjvjwherevisthelocalpeculiarvelocity,andcsisthesoundspeedoftheuid. Forindividualcomponentswithinter-componentinteractions,equation( 3{3 )mustbemodiedtoincludetheseinteractions.ExamplesofsuchinteractionsincludethemomentumtransferbetweenphotonsandchargedparticlesandtheCoulombinteractionbetweenprotonsandelectrons.Forprotons,electrons,andcolddarkmatter(CDM),anequationofstatew=0isassumed,andforradiationandneutrinos,w=1 3.Themasterequationsforeachcomponentofinterestiscomputedexplicitlyinsubsections3.3.1-3.3.5. 3{3 )thattheequationswhichgovern 35

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_c=c+3_;_c=_a ac+k2: Anycold(i.e.,nonrelativistic),collisionlesscomponentwillbehaveaccordingtothedynamicsgivenbyequation( 3{4 ). inFourierspace. Theapproximationthatneutrinosaremasslessanduncoupledisverygoodfromanageoftheuniverseofapproximatelyt'1suntiltheepochofrecombination.Theevolutionequationsforlightneutrinosarethen _=4 3+4_;_=k2(1 4+);_Fl=k whereisrelatedtoFby2=F2,andtheindexlgovernsthenalequationforl2.Flisdenedbytheexpansionoftheperturbationsinthedistributionfunction,F, 36

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3{5 3{7 )arevalidforanynon-collisionalspeciesbehavingasradiation. d=3T whereTistheThomsoncross-section[ 124 ].Photonsalsoscatterwithprotons,butwithacross-sectionsuppressedbyafactorofm2e=m2p(themass-squaredratioofelectronstoprotons). 3{7 ).Photonsalsocontainanon-zerodierencebetweenthetwolinearpolarizationcomponents,denotedbyG.ThelinearizedcollisionoperatorsforThomsonscattering[ 22 125 { 127 ]yieldthesetofmasterequationsforphotons, _=4 3+4_;_=k21 4+k2+aneT(b);_F2=8 153 5kF39 5aneT+1 10aneT(G0+G2);_Fl=k 10Fm2 5Gm; 37

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3.3.1 ),butadditionallycontainimportantsound-speedtermsandtermsarisingfromThomsonscattering.Additionally,theCoulombinteractionentersthroughthecontributionoftheelectriceldtotheT0icomponentsofthestress-energytensor.ThecouplingoftheCoulombinteractiontodensityinhomogeneitiescanbecalculatedthroughacombinationoftheelectromagneticPoissonequation, whereCistheelectricchargedensity,andtheEulerequation, 1 m1 withq=masthecharge-to-massratiooftheparticleinquestionandCthecollisionoperator.TheCoulombcontributionappearsas4e(npne)qi=miintheevolutionequationfor_i,whereidenotesaspeciesofparticlewithamassmiandchargeqi.The 38

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_e=e+3_;_e=_a ae+c2sk2e+k2+e(e)4e2 forelectrons(denotedbysubscripte),and _p=p+3_;_p=_a ap+c2sk2p+k2+p(p)+4e2 forprotons,denotedbysubscriptp,wherethedampingcoecientsforelectrons(e)andprotons(p)aregivenby e4neTa Notethedierenceinthesignofthenaltermsintheequationsfor_eand_p,whichwillproveimportantintheanalysisbelow. Fromequations( 3{12 )and( 3{13 )forelectronsandprotonsthedominantgravitationalandelectromagneticcombinationscanbeconstructedseparately.Theremainderofthissubsectiondetailstheevolutionofbaryonsinthelinearregimeofaperturbeduniverse.Baryonicmattercanbetreatedasthecombinationofelectronsandprotons,thusthemassweightedsumofprotonandelectronoverdensitiesgivesrisetothebaryonicperturbations, Bysubstitutingtheexpressionsforequations( 3{12 )and( 3{13 )intoequation( 3{15 ),asetofequationsfortheevolutionofbaryonicmatterisobtained.Solongasapproximations 39

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_b=b+3_;_b=_a ab+c2sk2b+k2+b(b); wherebeme=mb.Thebaryon-photoncouplingterminequation( 3{16 )isdrivenbytheelectron-photoninteraction.Totheextentthatelectronsandprotonsmovetogether(thetight-couplingapproximation),thebaryonicuidisdraggedbytheelectron-photoninteractions,ashasbeenshownbyHarrison1970[ 109 ]andsubsequentauthors.Equation( 3{16 )isidenticaltotheequationsforbaryonevolutionderivedinMaandBertschinger1995[ 22 ]. 3{12 )and( 3{13 ),adierencecomponentaswellasasumcomponentcanbeobtained.Asthelimitsonanetelectricchargeasymmetryintheuniverseareverystrict[ 17 18 128 ],anycomponentarisingfromthedierencesindensitiesand/orvelocitiesofprotonsandelectronswillnotbestrongenoughtosignicantlyimpacttheevolutionoftheotherspeciesofparticlesintheuniverse,includingthebaryoncomponent. Thechargedierencecomponent(denotedbysubscriptq)isthedierencebetweentheprotonandelectroncomponents,suchthatq=peandq=pe.Thegravitationalpotentialoughtnottoenterintotheseequations,duetothefactthatgravityactsequivalentlyonelectronsandprotons.However,velocitiesandnumberdensitiesmaydier,npne'neq.Themasterequationsforthecharge-asymmetriccomponentareas 40

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_q=q_q=_a aq+c2sk2q+4nee2 wheretheapproximationspeandmb'mphavebeenutilizedwhereapplicable.Theterm4nee2q=meinequation( 3{17 )arisesfromtheCoulombforceactingonchargedparticles,whilethenalterm,e(b+q),arisesfromthedierenceinThomsonscatteringbetweenprotonsandelectrons.Thisnaltermisasourceofchargeseparationindependentofandinadditiontoanyinitialchargeasymmetry,andwillcreatealocalchargeasymmetryevenwhenthereisnoneinitially.Intheevaluationofequation( 3{17 ),theelectromagnetictermsdominatethecosmologicalterms,suchthatanexcellentapproximationinthepre-recombinationuniverseis _q=q;_q=4nee2 Forsomepurposes,itisusefultoexpressthesetofequationsfoundinequation( 3{17 )asasingleordinarydierentialequation.Thiscanbeaccomplishedbysetting_q=,andagainbyneglectingtheunimportantcosmologicaltermsq_a=aandc2sk2q.Manyofthecoecientsinequation( 3{17 )arefunctionsofa,butthederivativesinequation( 3{17 )arewithrespecttoconformaltime,.Achangeofvariablescanbeperformed,usingtherelationthat intheradiationera,wheret0istheageoftheuniversetoday,toexpressallderivativesasderivativeswithrespecttoa,denotedbyprimes(insteadofdots). 41

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3{20 )below, wherethesubscript0denotesthepresentvalueofaquantity.Thisissimplytheequationofadampedharmonicoscillator,withcoecientsthatchangeslowlywithtimecomparedtodampingoroscillationtimes.Thebehaviorcanbecharacterizedasoverdampedattheearliesttimes,criticallydampedwhena3:91015,andfreeatlatetimes. Ofallthetermsinequation( 3{20 ),only,b,andq(andderivatives)arefunctionsofa;allotherquantitiesareconstantcoecents.Althoughtheredoesnotexistasimpleanalyticformfor(b)ingeneral,atsucientlyearlytimesthereexiststhesimpleapproximation validwhenthefollowingconditionismet: 3{21 )isanapproximationforaatCDMcosmologywithcosmologicalparametersH0=71kms1Mpc1,m=0:27,b=0:044,andaHelium-4massfractionofY=0:248.Theseparametersareusedinallsubsequentanalysesforthecalculationofcosmologicalquantities. Theapproximationinequation( 3{21 )breaksdownatsucientlylatetimes.Whenthisoccurs,numericalmethodsmustbeusedtoobtainthequantity(b).ThesoftwarepackageCOSMICS[ 129 ]isidealforperformingthiscomputation,asitperformsnumericalevolutionofequations( 3{4 3{6 3{9 ,and 3{16 )concurrently.ComputationalresultsforthequantitiesandbaregivenbyCOSMICS,whicharevalidatalltimesinthelinear 42

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3{21 )breaksdown,thequantity(b)growsmoreslowlyinitially,andproceedstooscillateataroughlyconstantamplitudeatlatertimes.Theseoscillationsinthequantity(b)arecloselyrelatedtotheacousticoscillationsbetweenbaryonsandphotonsobservedinthecosmicmicrowavebackground[ 5 ]. Numericalintegrationofequation( 3{20 )canbeaccomplishedinvariousways,asillustratedinPressetal.1992[ 130 ].Atsucientlylatetimes(whena3:91015),numericalresultsindicatethatthequasi-equlibriumsolution obtainedbyneglectingthersttwotermsinequation( 3{20 ),isanexcellentapproximation.WithandbgivenbyCOSMICSinunitsofMpc1,theprefactorinequation( 3{22 )canbewrittenas Thequantityqthenfollowsdirectlyfromequation( 3{17 )tobe Thesolutionsinequations( 3{22 )and( 3{24 )arevaliduntilgravitationalcollapsebecomesnonlinear,whichmeansthattheyarestillvalidattheepochofrecombination(z'1089).TheresultsofnumericallyintegratingtheequationsforqandqonvariouslengthscalesupthroughrecombinationarepresentedinFigure 3-1 Itisworthpointingoutthattheresultsobtainedinthissectioncanbeappliedtoasituationwhereanetelectricchargeispresent.Inappendix B ,thepossibilityofusingtheevolutionequationsderivedforqandqisappliedtoauniversewithabrokenU(1)symmetry.Thepossibilityexiststhat,underthepropercircumstances,aninitiallychargeduniversemaybecomeneutralsimplyduetotheexpansiondynamics. 43

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3{17 ).ThisgraphusesoutputfromCOSMICS,andassuchneedstobemultipliedbytheCOBEnormalizationofBunnandWhite[ 2 ]. 44

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Anexpressionformagneticeldscanbederivedfromthecurrentsarisingfromtherelativemotionoftheprotonsandelectronsintheuniverse.MagneticeldscanbederivedfromMaxwell'sequations @t; withthecurrentdensity~Jgivenby where~vq~vp~ve,andthedisplacementcurrentisneglected. Bytakingthecurl,adirectexpressionformagneticeldsasafunctionofaandkisobtainedasaconvolution a2j~kj2Zd3k0 Whilethemagneticeldstrengthcan,inprinciple,beobtainedbysolvingequation( 3{27 ),itismorefavorabletoobtainthepowerspectrumofthemagneticeld.Thepowerspectrumisobtainedbyexaminingthesecondmomentofthemagneticeld~B(~k),which 45

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2(ij^ki^kj); whereDistheDiracdeltafunctionandPB(k)isthemagneticeldpowerspectrum.Notethatthedirectionparallelto~kdoesnotcontributetomagneticelds,andthereforethedirectionperpendicularto~kisprojectedoutinequation( 3{28 ).Thepowerspectrum,PB(k),isthengivenbytheexpression a2j~kj2!2Zd3k0 wheretheangleistheanglebetweenthevectors~kand~k0.Theexpressionforpowerinanytwoquantities,and,P(k),isgenericallydenedby Thesolutionsobtainedforqandqinequations( 3{22 )and( 3{24 )canbesubstitutedintotheequationforthepowerspectrum,equation( 3{29 ).Bynumericallyintegratingtheresultingexpression,thespectraldensitycanbeobtained.Thespectraldensityis 46

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Theresultsforthespectraldensityofmagneticeldenergyoncomovingscalesrangingfrom103Mpc1to102Mpc1attheepochofrecombinationareshowninFigure 3-2 .Thepeakofthespectraldensitycorrespondstoatypicalmagneticeldstrengthof1023Gaussoncomovingscalesof0:1Mpc1. Theresultsofthispaperareveryaccurateupthroughtheepochofrecombination.Atthisepoch,however,theuniversetransitionsfromafullyionizedstate(wheretheionizationfractione'1)toastatewheretheionizationfractionisverysmall,e104[ 24 ].Whilethephotonsaregenerallydecoupledfromthebaryonsatthispoint,thefreeelectronscontinuetointeractwiththephotons,duetotheextraordinarynumberofphotonsperfreeelectron.Intheabsenceofanyinteractionswithphotons,achargeseparationwouldevolveas q=4ne;0e2 2bH2K2q;(3{31) 47

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Spectralenergydensityofthemagneticeld(~B,ingauss)generatedbycosmologicalperturbationsonagivencomovingscale(k,inMpc1)attheepochofrecombination(z'1089).Thelineillustrates4k3PB(k)=(2)3,whichisthespectraldensityinunitsofG2;thepeakvalueisamagneticeldstrengthB1023G.Theupperlinesarethesimplepowerspectrum,PB(k). 48

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3{31 )accuratelydescribestheevolutionofchargeseparationsinthepost-recombinationuniverse. Whileeldamplicationduetogravitationalcollapseisnegligibleattheepochofrecombination,thiswillnotbethecaseatalltimes.Atrecombination,theuniversehasonlybeenmatter-dominatedforabrieftime,andthusdensityperturbationshaveonlygrownbyasmallamountinthattime,leadingtoaninsignicantamplicationoftheeldstrength.Asmagneticuxgetsfrozenin,however,nonlinearcollapsecausesj~Bjtoincreasebymanyordersofmagnitude[ 92 93 ]. Themajorsourceofamplicationofaninitialseedeld,however,comesfromdynamoeects,asdiscussedinSection2.ThekeytosolvingthepuzzleoftheoriginofcosmicmagneticeldsliesindeterminingwhethertheseedeldsproducedbyagivenmechanismcanbesuccessfullyampliedintotheO(G)eldsobservedtoday.Amajorproblemwithmanyoftheastrophysicalmechanismsthatproduceseedeldsisthattheyproducelow-magnitudeeldsatinsucientlyearlytimesfordynamoamplicationtoproduceeldsaslargeasG.TheBiermannmechanism,forinstance,canproduceseedeldsoforder1019G,butonlyataredshiftofz20.Althoughthoseinitialeldsarelargerthanthe1023Geldsproducedbythegrowthofcosmicstructure,thefactthatmagneticeldsfromstructureformationareinplaceatz'1089makesthemanextremelyattractivecandidatefortheseedsofcosmicmagneticelds.AsarguedbyDavisetal.1999[ 131 ],aseedeldassmallas1030Gatrecombinationcould 49

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Oneinterestingmechanismworthinvestigatingfurtherisforthecosmicseedeldsgeneratedbydensityperturbationstoseedsupermassiveblackholes.Itisknownthatthemagneticeldenergyinactivegalacticnucleiandquasarsiscomparabletothemagneticeldenergyinanentiregalaxy.However,thesestructurescannotgeneratetheirownmagneticeldsfromnothing;theyrequireapre-existingseedeld.Itthereforeappearstobeareasonablepossibilitythattheseedeldsgeneratedbycosmicstructureformationcouldprovidethenecessaryeldstoseedsupermassiveblackholes.Theresultantamplicationviacollapseanddynamoeectscouldexplaintheoriginoflarge-scalemagneticstructuresintheuniverse. Iflarge-scalemagneticeldsexistattheepochofrecombination,theymaybedetectablebyupcomingexperiments.TheresultsshowninFigure 3-2 provideapredictionoflarge-scalemagneticeldsattheepochofthecosmicmicrowavebackground.SucientlylargemagneticeldsonlargescalesatrecombinationmaybedetectablebyPLANCK[ 132 133 ],althoughcurrentestimatesoftheirsensitivityindicatethattheeldstrengthspredictedinthispaper(1023G)wouldbesignicantlyoutofrangeofPLANCK'scapabilities(1010G).Nonetheless,aknowledgeoftheeldstrengthsatrecombinationallowforpredictionsofCMBphotonpolarizationsandFaradayrotation,bothofwhichmaybe,atleastinprinciple,observable. Itisalsoofinteresttonotethatanyprimordialchargeasymmetryorlarge-scalecurrents(andthereforemagneticelds)createdintheveryearlyuniverse(a.3:91015)willbedrivenawaybythesedynamics.Equation( 3{20 )hasanapproximatesolutionforqwhichiscritically(exponentially)dampedata'3:91015,capableofreducinganarbitrarilylargechargeorcurrentbyasmuchasafactorofe1015.Anypre-existingqorqwillbedrivenquicklytothevaluegivenbyequations( 3{22 )and( 3{24 )attheepoch 50

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Therehasbeenotherrecentworkthatclaimstogenerateamagneticeldfromcosmologicalperturbationsvia\baryon-photonslip,"photonanisotropicstress,andasecondordervelocityvorticity[ 134 ].Theresultsofthisworkdonotrequireavelocityvorticityoranisotropicstress,nordotheyrequiresecondorderquantities.Thispaperderivesmagneticeldsfromcosmologicalperturbationsinaverystraightforwardmanner,simplybycalculatingthechargeseparationsandcurrentswhichnecessarilyarisefromthedieringinteractionsonprotonsandelectrons,andobtainingmagneticeldsdirectlyfromthosequantities.ThemotivationbehindthemethodsusedinIchikietal.2006[ 134 ]areobscureandnoteasilycomprehended,whiletheirresultsareinconsistentwiththoseobtainedinthispaper,astheirresultsformagneticeldstrengthandspectraldensityaresuspiciouslylarge.Furthermore,itisunclearhowtheirresultsforvelocityvorticityareobtained,asitiswell-knownthatthevorticityvanishesatsecondandallordersifthereisnoneinitially. Overall,thedynamicsofions,electrons,andphotonsduringtheradiationeranecessarilyleadstochargeseparationsandcurrentsonallscales,whichinturngeneratemagneticelds.Theseeldssupersedeanypre-existingeldsandareinplacepriortosubstantialgravitationalcollapse.Thus,thedynamicsofstructureformationfromcosmologicalperturbationsemergesasapromisingandwell-motivatednewcandidatetoexplaintheoriginsofcosmicmagneticelds. 51

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Weevaluatetheeectofcosmologicalinhomogeneitiesontheexpansionrateoftheuniverse.OurmethodistoexpandtoNewtonianorderinpotentialandvelocitybuttotakeintoaccountfullynonlineardensityinhomogeneities.Tolinearorderindensity,kineticandgravitationalpotentialenergycontributetothetotalenergyoftheuniversewiththesamescalingwithexpansionfactorasspatialcurvature.Inthestronglynonlinearregime,growthsaturates,andtheneteectoftheenergyininhomogeneitiesontheexpansionrateremainsnegligibleatalltimes.Inparticular,inhomogeneitycontributionsnevermimictheeectsofdarkenergyorinduceanacceleratedexpansion. 135 ]andthecosmicmicrowavebackground[ 5 ]intandemsuggestthatthecosmologicalexpansionisaccelerating.Understandingthesourceofthisacceleratedexpansionisoneofthegreatestcurrentunsolvedproblemsincosmology[ 136 ].Accelerationseemstorenderinadequateauniverseconsistingentirelyofmatter,andappearstorequireanadditional,unknowntypeofenergy(darkenergy,perhapsrealizedasacosmologicalconstant).Analternativetodarkenergyisthataccelerationarisesfromaknowncomponentoftheuniversewhoseeectsonthecosmicexpansionhavenotbeenfullyexamined.Onepossibilitycurrentlybeingexaminedisthatinhomogeneitiesinamatterdominateduniverse,oneithersub-horizon[ 29 30 ]orsuper-horizonscales[ 28 137 { 139 ],mayinuencetheexpansionrateatlatetimes.ThecentralideaisthattheenergyinducedbyinhomogeneitiesleadstoadditionalsourcetermsintheFriedmannequations,witheectsonthedynamicsthatleavenoneedforaseparatedarkenergycomponent.Intheirentirety,theseproposalspresentconictingclaimsandageneralstateofmuchconfusion:doestheenergyininhomogeneitiesproduceanacceleratedexpansion,actingineectasdarkenergy[ 138 ],ordoesitbehaveascurvature[ 31 ]?Isthemagnitudeoftheeectsmall,large,orevendivergent,oneitherlargescales[ 138 ],oronsmallscalesatlatetimes[ 30 ]? 52

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a2=8 a=4 Anymassorenergydensitythatmakesupasignicantfractionofthetotalcaninuencetheevolutionofthecosmologicalscalefactora(t).Acontributiontotheenergydensityoftheuniversewithequationofstatepi=wihasi/a3(1+w),ori=m/a3w;in 53

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3orcurvature,andacomponentwithconstantbehavesasacosmologicalconstantordarkenergy. WeintroducetheeectsofinhomogeneitiesfollowingtheformulationofSeljakandHui1996[ 140 ].IntheconformalNewtoniangauge,withmetric thetime-timeEinsteinequation(G00)yields 3_a a2(12)+(2+6)1 Where'fromthespace-spacecomponentsofG.(OurnumericalfactorsareslightlycorrectedfromthosefoundinSeljakandHui2006[ 140 ];thesefactorsmakelittledierenceintheoverallresults.)Thesourceontheright-handsideincludesadensityperturbation==inthematerialrestframe,withthetransformationtothecosmologicalframeexpandedtoleadingorderforsmallv2.Ignoringr2,(r)2,andv2,thehomogeneouspartofthisequationreproducestheusualFriedmannequation.TheinhomogeneouspartrevealsthatobeysthePoissonequationwithsource4Ga2.Thevolumeaverageoftheentireequationthenleadsto a2=8 whereWandKaretheNewtonianpotentialandkineticenergyperunitmass, 2h(1+)i;K=1 2(1+)v2:(4{5) Theseexpressionsarecorrecttorstorderinandv2,butneitheranassumptionnoranapproximationin.Weassumethathr2i=0;inallotherplacesthePoissonequationisadequatetodetermine. 54

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24Ga2Zd3k k2W(k);(4{6) anexpressioncorrectinbothlinearandnonlinearregimesifP(k)istheappropriatelinearornonlinearpowerspectrum.Thelastequalitydenesthedimensionlessspectraldensity2W(k). Inlinearperturbationtheory,validforsmallinhomogeneities,thedensitycontrastgrowsas=0(x)D(t),whereinamatterdominateduniverseD(t)/a(t)/t2=3[ 141 ].Thekineticenergyfollowsfromthelinearizedequationofcontinuity,_+rv=a=0[ 141 ], 2_a2Zd3k (theusualfactorf()'0:6=1form=1).Thekineticenergyscaleswitha(t)as_a2D2,whilethepotentialenergyscalesasa2D2;andsobothWandKgrowasD2=a/a(t),orU=(W+K)/a2.AswasnotedbyGeshnizjani,ChungandAfshordi2005[ 31 ]forsuper-horizoninhomogeneities,energyininhomogeneitieshasthesameeectontheexpansionrateasspatialcurvatureinperturbationtheory.WenotethatKlin=jWlinj=H2=4G=2 3,axedratiointhelinearregime.Thefullkineticenergyinprincipleinvolveshigherordercorrelationfunctionsandisnotasimpleintegraloverthepowerspectrum.Nonetheless,thefullkineticenergycanbeobtainedsimplyfromthepotentialenergythroughthecosmicenergyequationofIrvine1961[ 142 ]andLayzer1963[ 143 ], dt+2_a aK=d dt+_a aW;(4{8) 55

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3jWlinj.Equations( 4{6 )and( 4{8 )provideuswithexpressionssucienttocalculatenonperturbativecontributionstotheexpansionrateforboththegravitationalpotentialperturbationandkineticenergycomponents.Theresultsofthesecalculationsaregiveninthenextsection. 4{6 )and( 4{8 )determinetheenergyininhomogeneitiesoftheuniverseasafunctionofepoch,whichwecharacterizebytheexpansionfactora=a0.Fortheprimordialpowerspectrum,weusetheCDMpowerspectrumasgivenby[ 144 ],withspectralindexn=1,m=1,andCOBEnormalizedamplitudeH=1:9105[ 2 ].Toobtainthenonlinearpowerspectrumweusethelinear-nonlinearmappingofPeacockandDodds1994&1996[ 145 146 ].TheresultsofthesecalculationsareshowninFigures 4-1 and 4-2 Figure 4-1 showsthedimensionlessspectraldensityofgravitationalpotentialenergy2W(k)denedinequation( 4{6 ),evaluatedatthepresent,plottedasafunctionofwavenumberk.Thedashedcurveshowsthedensityinlinearperturbationtheory,andthesolidcurveshowsitsfullynonlinearform. Figure 4-2 showsthecontributionsofpotentialenergyandkineticenergytotheenergydensityoftheuniverse,forpastandfutureexpansionfactorsinanm=1universe.Atearlytimes,perturbationtheorygivesanaccurateresult,butata=a00:05(redshiftz20)thebehaviorstartstochange,foranintervalgrowingfasterthana1withthefastestgrowthasa1:2,andthensaturatingandgrowingsignicantlymoreslowly,eventuallyaslna. 56

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Spectraldensityofgravitationalpotentialenergy2W(k)[theintegrandofequation( 4{6 )],evaluatedatthepresent,plottedasafunctionofwavenumberk.Thedashedlineshows2Winlinearperturbationtheory;thesolidlineshowsthefullynonlinearform. 57

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FractionalcontributionsofgravitationalpotentialenergyW(long-dashedline)andkineticenergyK(solidline)tothetotalenergydensityoftheuniverse,plottedasafunctionofpastandfutureexpansionfactorforanm=1universe.Theshort-dashedlineisthesumofcontributionsfrominhomogeneities.Thedottedlinesshowresultsfromlinearperturbationtheory. 58

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2(1+3w)=0:1.Since,atthistime,thetotalfractionofenergyininhomogeneitiesisU1051,thishasanegligibleeectoncosmologicalexpansiondynamics. Astheuniversefurtherevolves,sothatthemaincontributionstoWandKcomefromdeeplynonlinearscales,wecomputethepotentialenergyfromintegrationofthenonlinearpowerspectrum,andobtainkineticenergyfromthecosmicenergyequation,asdetailedinequation( 4{8 ).Inascale-invariantmodelwithpowerspectrumPknask!0,thekineticandpotentialenergiesKandWscalewiththeexpansionfactorasa(1n)=(3+n)[ 147 ](logarithmicallyinaasn!1),withratio 7+n:(4{9) NumericalsimulationsshowthatthiscontinuestoholdfortheCDMspectrumwitheectiveindexn=dlogP=dlogkatanappropriatescale,thebasisofthelinear-nonlinearmapping[ 145 146 ].FortheCDMspectrum,withn!1onlargescales,thismeansthatgrowthstops,andtheratiotendstothevirialvalueK=jWj!1 2atlatetimes.WenotethatasidefromtheintegrationoftheLayzer-Irvineequation,manyoftheseresultswereobtainedby[ 140 ]. Ourresultsshowthatthecontributionsofthepotentialandkineticenergiesofinhomogeneitieshasneverbeenstrongenoughtodominatetheexpansiondynamicsoftheuniverse.Forauniversewithm=1today,normalizedtothelargescaleuctuationsinthemicrowavebackground,theneteectofinhomogeneitiestodayisthatofaslightlyopenuniverse,withk104incurvature.Themaximumcontributioncomesfromscalesoforder1Mpc,fallingorapidlyforsmallerandlargerk,asillustratedinFigure 59

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.Thebehavioronasymptoticallysmallscales(k106hMpc1)dependsonanextrapolationthatignoressuchdetailsasstarformation,butFukugitaandPeebles2004[ 12 ]estimatethatthenetcontributionofdissipativegravitationalsettlingfrombaryon-dominatedpartsofgalaxies,includingmainsequencestarsandsubstellarobjects,whitedwarfs,neutronstars,stellarmassblackholes,andgalacticnuclei,isintotal104:9ofthecriticalenergydensity. Thesuggestionthatnonlineareectsforlargeinhomogeneitiesmaymimictheeectofdarkenergyisnotthecaseforthefullynonlineartheory.Itistruethathigherordertermsinperturbationtheorygrowfaster;thegeneraln-thordertermgrowsasDn(t).Thereindeedcomesascaleinspaceoranevolutionintimewherethebehaviorofhigherordertermsappearstodiverge.ThisisillustratedinFigure 4-3 ,whereitcanclearlybeseenthat,tosecondorderindensitycontrast,thecontributionsfrompotentialandkinetictermsappeartodiverge.Nevertheless,thefullynonlinearresultiswellbehaved.Itisonlytheperturbationexpansionthatbreaksdown,andtheactualenergysaturatesandgrowsmoreandmoreslowlyatlatetimes.AsillustratedinFigure 4-2 ,thenonlinearpotentialandkineticenergiesremainsmallcomparedtothetotalmatterdensityatalltimes,evenanexpansionfactorof103intothefuture.Inhomogeneityeectsdonotsubstantiallyaecttheexpansionrateatanyepoch. 28 ], 4h(x)(x0)i=(2Ga2)2Zd3k windowedoverthehorizonvolume(forcalculationalconvenienceweuseaGaussianrolloratherthanasharpradialedge).Forn!1ask!0,thisisindeedlogarithmically 60

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FractionalcontributionsofgravitationalpotentialenergyW(long-dashedline)andkineticenergyK(solidline)tothetotalenergydensityoftheuniverse,plottedasafunctionofpastandfutureexpansionfactorforanm=1universe.Theshort-dashedlineisthesumofcontributionsfrominhomogeneities.Thisgraphshowsthecontributionstosecondorderindensitycontrast,.Notetheapparentdivergenceisaresultofperturbationtheorybreakingdown,asthefullynonlinearresultinFigure 4-2 iswellbehaved. 61

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4-4 asafunctionoftheinfraredcutokmin. Theintegralisdominatedbythesmallestvaluesofk,whereperturbationsaredeepinthelinearregime.Forn=1theresultisveryaccuratelyW=1:45105jlnkminRHj1=2.(Wenotethatforn!1theunitsofkminareunimportant.)TheuctuationiscomparabletothemeanhWi=3:1105whenthecutoisnearthescaleofthehorizonk=H0=c,anddoesnotbecomeoforder1untilkmin10170(forn=0:95),orkmin10109(forn!1),orever(forn>1).Whilesuchanexponentiallyvastrangeofscalesmaynotbebeyondtherangeofpossibilityinaninationaryuniverse,itrequiresafearlessextrapolationwellbeyondwhatisknowndirectlyfromobservation.TheuctuationWisdominatedbycontributionsfrommodesthataredeepinthelinearperturbationregime,andscaleswithexpansionfactorasW/a2D,constantintime.ThiscontributiontotheenergywillappeardynamicallyintheFriedmannequationasanothermattercomponent.Furthermore,inthepresenceofatruedarkenergycomponent,anyeectsoncosmologicalexpansionarisingfrominhomogeneitiesquicklybecomesunimportantoncedarkenergybecomesdominant[ 140 ]. Thefactthatuctuationsinthepotentialdivergeremainstroublesome.Ithasbeenrecognizedforsometimethatpotentialuctuationsinthestandardmodelwithn!1arelogarithmicallydivergent,butsinceformostpurposesthevalueofthepotentialisunimportant,thishasnotbeenperceivedasasignicantproblem.Theeectofpotentialontheexpansiondynamicsisreal,buttheweaklogarithmicdivergenceandthefactthatitisafeedbackofagravitationalenergyongravitationaldynamicsmayleadonetohopethatthisdivergenceisalleviatedinarenormalizedquantumtheoryofgravity. 62

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Theexpecteductuationinthepotentialenergyperunitmassh(W)2i1=2evaluatedatthepresentasafunctionofinfraredcutokminforn=0:95,n=1,andn=1:05(solidlines,toptobottom).Dashedlinesareanalyticapproximationsthatasymptoticallybecomek0:025,(logk)1=2,orconstant,respectively.Thedottedlineshowstheresultforarollingspectralindexthathasn=0:95onthehorizontodaybutapproachesn=1ask!0,aspredictedbymostmodelsofslow-rollination.ThemeanvaluehWi=3:1105isshownasthehorizontaldashedline. 63

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31 { 33 ].Thepossibilitythataknowncomponentoftheuniversemayberesponsiblefortheacceleratedexpansionremainsintriguing.However,weconcludethatsub-horizonperturbationsarenotaviablecandidateforexplainingtheacceleratedexpansionoftheuniverse. 64

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Thischapterinvestigatestheuseofawell-knownempiricalcorrelationbetweenthevelocitydispersion,metallicity,andluminosityinHofnearbyHIIgalaxiestomeasurethedistancestoHII-likestarburstgalaxiesathighredshifts.Thiscorrelationisappliedtoasampleof15starburstgalaxieswithredshiftsbetweenz=2:17andz=3:39toconstrainm,usingdataavailablefromtheliterature.Abest-tvalueofm=0:21+0:300:12ina-dominateduniverseandofm=0:11+0:370:19inanopenuniverseisobtained.Adetailedanalysisofsystematicerrors,theircauses,andtheireectsonthevaluesderivedforthedistancemoduliandmiscarriedout.Adiscussionofhowfutureworkwillimproveconstraintsonmbyreducingtheerrorsisalsopresented. 5 ],typeIasupernovae[ 135 ],andgalaxysurveys[ 1 148 ].Althoughthesesourcesofdataaresucientforgeneratingconsistentvaluesforthemassdensity(m),vacuumenergydensity(),thedarkenergyequationofstateparameter(w),andthevalueofspatialcurvatureintheuniverse(k),thesevaluesmustbecheckedviaasmanyindependentmethodsaspossibleforconsistency,accuracy,andavoidingsystematicbiases.Furthermore,withoutthedatafromsupernovae,therewouldbeweakevidenceatbestforstatingthatw1,thusitisimportanttoseekanother,independentobservationsupportingtheexistenceofdarkenergy. Thecosmologicalparameterwiththegreatestnumberofobservablecross-checksism.Ithasbeenderivedusingmanytechniques,includingtheSunyaev-Zel'dovicheect[ 149 ],weakgravitationallensing[ 150 ],X-rayluminosities[ 151 ],largescaleclustering[ 152 ],peculiarvelocitiesofgalaxypairs[ 153 ],andsupernovaedata[ 39 ].Thesemethodsyieldresultsrangingfromm=0:13tom=0:35,andareallconsistentwithoneanotherat 65

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154 ].ThischapterextrapolatesalinkbetweennearbyHIIgalaxiesandHII-likestarburstgalaxiesathighredshiftstousesuchobjectsasstandardcandles.ThisisaccomplishedthroughtheapplicationoftheknowncorrelationbetweentheluminosityintheHline(LH),thevelocitydispersion(),andmetallicity(O=H)ofnearbyHIIgalaxiesdiscoveredinMelnick,TerlevichandMoles1988[ 155 ]totheHII-likestarburstgalaxiesfoundathighredshifts.Thiscorrelation,whenappliedtostarburstgalaxiesatz>2,allowsfordiscriminationbetweendierentvaluesofmasrstsuggestedinMelnick,TerlevichandTerlevich2000[ 40 ],andcandiscernwhichcosmologicalmodelismostfavoredbythedata. HIIgalaxies(andHIIregionsofgalaxies)arecharacterizedbyalargestar-formingregionsurroundedbysinglyionizedhydrogen.ThepresenceofO-andB-typestarsinanHIIregioncausesstrongBalmeremissionlinesinHandH.ThesizeofagiantHIIregionwasshowntobecorrelatedwithitsemissionlinewidthsinMelnick1978[ 156 ].ThiscorrelationwasimproveduponinTerlevichandMelnick1981[ 157 ],whoshowedthatLHofgiantHIIregionsisstronglycorrelatedwiththeir.Thisbasiccorrelation,itsextensiontoHIIgalaxies,anditsusefulnessasadistanceindicatorhavebeenexploredinthepast[ 154 155 158 ].TheempiricalcorrelationforHIIgalaxies[ 155 ]relatestheirLH,,andO=H.Therelationshipis logLH=logMz+29:60;Mz5 wheretheconstant29.60isdeterminedbyazero-pointcalibrationofnearbygiantHIIregions[ 40 ]andfromachoiceoftheHubbleparameter,H0=71kms1Mpc1[ 159 160 ]. 66

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155 ].StarburstgalaxiesobservedathighredshiftsexhibitthesamestrongBalmeremissionlinesandintensestarformationproperties[ 161 162 ]asnearbyHIIgalaxies.ThischapterfollowsthesuggestionofMelnick,TerlevichandTerlevich2000[ 40 ]thatequation 5{1 appliestotheHII-likestarburstgalaxiesfoundathighredshifts,andprovidesevidencetovalidatethatassumption. Theremainderofthischapterdiscussestheconstraintsthatcanbeplacedonmandtherestrictionsthatcanbeplacedonthechoiceofcosmologyusingstarburstgalaxies.Section5.2detailshowthedatasetwasselectedandanalyzed.Section5.3statestheresultsobtainedfromtheselecteddata.Therandomandsystematicerrorsassociatedwithanydistanceindicatorisafundamental(andoftenoverlooked)probleminherenttoobservationalcosmology.Appendix A discussestheerrorsspecictotheobservationalmethodusedinthischapter,includingadetaileddiscourseontheassumptionofuniversalitybetweenlocalHIIgalaxiesandhighredshiftstarburstgalaxies.Finally,section5.4presentstheconclusionsdrawnfromthischapter,andpointstowardsusefuldirectionsforfutureworkonthistopic. 5{1 )holdingandforwhichthedistancemodulus(DM)canbecomputedfromtheobservedquantities.Thequantitiesrequiredforanalysisofthesegalaxiesare,theuxinH(FH),O=H,theextinctioninH(AH),andtheequivalentwidthintheHline(EW). 67

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40 ],thedistancemodulusofHIIgalaxiescanbederivedfrom: wheretheconstant26:18isdeterminedbyH0andequation( 5{1 ).Thischaptermakesextensiveuseofequation( 5{2 )becauseitexpressesDMpurelyintermsofobservables.DMisinsensitivetom,,k,andwatlowredshifts(z0:1),dieringby0:1magnitudesorfewerfordrasticchangesinthechoiceofparametersabove.Athighredshifts(z>2),however,DMcanvarybyupto3magnitudesdependingonthechoiceofparameters.Ofthefourparametersaboveavailableforvariation,DMismostsensitivetochangesinm,ashasbeennotedpreviously[ 40 ].However,forvaluesofm0:3,DMissensitivetovariationsintheotherparametersby0:2to0:5magnitudes.Sinceothermeasurementsindicatethatindeedm.0:3,thischapteralsoconsidersvariationsinandk. Dataforstarburstgalaxiesatz>2arefoundinPettinietal.2001[ 161 ]andErbetal.2003[ 162 ],whichcontainmeasurementsformanyofthedesiredobservables(andrelatedquantities),alongwithredshiftdata.Partialmeasurementsexistfor36starburstgalaxies.AccordingtoMelnick,TerlevichandTerlevich2000[ 40 ],thecorrelationinequation( 5{1 )holdstrueforyoungHIIgalaxieswhosedynamicsaredominatedbyO-andB-typestarsandtheionizedhydrogensurroundingthem.AsHIIgalaxiesevolveintime,short-livedO-andB-starsburnoutquickly.AlthoughsomenewO-andB-starsareformed,eventuallythedeathrateofO-andB-starsexceedstheirbirthrate,causingagalaxytobeunder-luminousinHandHforitsmass.ThiseectcanbesubtractedoutbyexaminingtheEWofthesegalaxies,andcuttingouttheolder,moreevolvedgalaxies(thosewithsmallerequivalentwidths).Forthischapter,acutoofEW>20Aisadopted,andgalaxieswithEW20Aarenotincluded,similartothecutoof25Ausedpreviously[ 40 ].TherearealsogalaxieswithlargeEWthatdonotfollowthecorrelation 68

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High-redshiftgalaxiesselectedtobeusedasstandardcandlesonthebasisoftheirequivalentwidthsandvelocitydispersions. 5{1 )withinareasonablescatter.Itiswell-knownthatalargefractionoflocalHIIgalaxiescontainmultipleburstsofstarformation[ 158 ].Ifmultipleunresolvedstar-formingregionsarepresent,theobservedwillbeverylargeduetotherelativemotionofthevariousregions.Suchgalaxiesarenotexpectedtofollowthecorrelationofequation( 5{1 )[ 158 ].Thesimplestwaytoremovethiseectistotestfornon-gaussianityintheemissionlinesfromthiseect,butsignal-to-noiseandresolutionareinsucienttoobservethiseect.Sinceforasystemofmultiplestar-formingregionswillbemuchhigherthanforasingleHIIgalaxy,acutcanbeplacedontoremovethiseect.MonteCarlosimulations(detailedinAppendix A )indicatethatifisobservedtobegreaterthan130kms1,itislikelyduetothepresenceofmultiplestar-formingregions.Toaccountforthispresence,allgalaxieswith>130kms1arediscarded.ImposingtheabovecutsonandEWselects15ofthe36originalgalaxies,creatingthedatasampleusedfortheanalysispresentedhere.ThepropertiesofthoseselectedgalaxiesaredetailedinTable 5-1 ,withfurtherinformationaboutthegalaxiesavailableinthesourcepapers[ 41 161 162 ]. 69

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5{2 )mustbetabulatedforeachselectedgalaxy.Notallofthenecessarydataelementsareavailableintheliteratureforthesegalaxies,soassumptionshavebeenmadetoaccountforthemissinginformation.zwasmeasuredforallgalaxiesbythevacuumheliocentricredshiftsofthenebularemissionlines.wasobtainedforallgalaxiesfromthebroadeningoftheBalmeremissionlines,HforthegalaxiesfromErbetal.2003[ 162 ]andHforthegalaxiesfromPettinietal.2001[ 161 ].FHismeasureddirectlyforthegalaxiesinPettinietal.2001[ 161 ],butErbetal.2003[ 162 ]measuresFHinstead,thusFHmustbeconvertedtoFH.TheconversionforemitteduxisgiveninOsterbrock1989[ 163 ]asFH=2:75FH,butobserveduxesmustcorrectforextinction.Thus,thecompleteconversionfromFHtoFHwillbegivenbyequation( 5{3 )below, 2:75FH10(AHAH whereAHandAHaretheextinctionsinHandH,respectively.ObtainingO=Hismoredicult,asmeasurementsofmetallicityonlyexistfor5ofthe36originalstarburstgalaxies.AnaveragevalueofO=HisusedforthegalaxieswhereO=Hmeasurementsareunavailable. ValuesofO=Hareobtainedthroughmeasurementofthe[OII]emissionlineat3727Aandthe[OIII]linesat4959Aand5007AforveofthegalaxiesinPettinietal.2001[ 161 ].ThestronglineindexR23[ 164 ]isassumedtohaveitstemperature-metallicitydegeneracybrokentowardsthehighervalueofO=H,asisshowntobethecaseforluminousstarburstgalaxiesatintermediateredshiftsinKobulnickyandKoo2000[ 165 ].ThecombinationoftheoxygenlinemeasurementswiththisassumptionyieldsvaluesforO=Hforthesegalaxies.ThemeanvalueofO=Histhentakentobetheaveragemetallicityforeachoftheothergalaxieswheresuchlinemeasurementsareunavailable.Recently,measurementsofmetallicityinhighredshiftstarburstgalaxieshavebeenmade[ 166 ],usingthe[NII]/Hratioastheirmetallicityindicator.TheauthorsinShapleyet 70

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166 ]obtainanaverageO=Hof8.33forthegalaxiespreviouslyfoundinErbetal.2003[ 162 ].Thisvalueisnotedasapossibleimprovementtotheonechosenhere,andisfurtherdiscussedasasourceoferrorinthisanalysisinappendix A ofthiswork. 167 ],buthavenotbeenestablishedforstarburstgalaxiesingeneral(althoughseeCalzettietal.1994&2000[ 168 169 ]foranargumenttothecontrary).ThischapterassumesdustinHIIgalaxiestobecomparabletothatingiantHIIregions,thusAHforstarburstgalaxiesistakentobetheAHderivedinGordonetal.2003[ 167 ]fortheHIIregionsoftheLMCandSMC.AbesttappliedtothedatainGordonetal.2003[ 167 ]yields forstarburstgalaxies.Theseresultsarealsoapplicabletotheuxconversioninequation 5{3 .E(BV)isunavailableforthegalaxiesfromPettinietal.2001[ 161 ],butcanbederivedbynoticingthecorrelationbetweenE(BV)andcorrected(GR)colorsforstarburstgalaxiesinErbetal.2003[ 162 ].TheconversionadoptedisE(BV)0:481(GR).Finally,EWismeasuredforallgalaxiesinPettinietal.2001[ 161 ],butErbetal.2003[ 162 ]givesonlythespectrafortheHline.EWisestimatedfortheErbetal.2003[ 162 ]galaxiesbyestimatingthecontinuumheightfromeachspectraandtheareaundereachHpeak,calculatingtheequivalentwidthinH,andconvertingtoHusingtheBalmerdecrementsofOsterbrock1989[ 163 ].ThecompletedatasetislistedinTable 5-1 ,andisillustratedalongsidevariouscosmologiesinFigure 5-1 71

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Thedistancemodulusplottedasafunctionofredshiftsforvariouscosmologicalmodels,alongwithdatafromtheselectedgalaxies.Open-CDMuniversesand-CDMuniverseswithmof0:05,0:30,0:5,1:0,and2:0areshown.Thecrosshairsrepresentsthe1-constraintsontheDMvs.zparameterspacefromtheselecteddatasample.ThebesttstothedataareforaCDMuniversewithm=0:21and=0:79,orforanopen-CDMuniversewithm=0:11. 72

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40 ].misthereforetheparameterwhichisconstrainedmosttightlybyobservationsofstarburstgalaxies.EachgalaxyyieldsameasurementforDMandforz.Althoughtherearemultiplemodelsconsistentwitheachindividualmeasurement,observationsofmanygalaxiesatdierentredshiftswillallowtheconstructionofabest-tcurve,whichisuniquetothechoiceofcosmologicalparametersm,,k,andw.Thedatasampleof15galaxiesinthischapterisinsucienttodistinguishbetweenmodelsinthisfashion,astheuncertaintiesineachindividualmeasurementofDMaretoolarge.Themethodbywhichtheuncertaintiescanbereducedistobinthedataaccordingtoredshiftandndabest-tvalueofDMatthatpoint.Duetothesizeofthesampleinthischapter,all15pointsareaveragedintoonepointofmaximumlikelihoodtoconstrainthecosmology,witherrorsarisingfromtherandomerrorsoftheindividualpointsandfromthedistributionofpoints.TheaveragevalueobtainedisDM=47:03+0:460:56ataredshiftz=2:800:11.Thedierentcosmologicalmodels,alongwiththemostlikelypointandtherawdatapoints,aredisplayedingure 5-1 ,withH0=71kms1Mpc1. Theconstraintsplacedonmfromthisanalysisarem=0:21+0:300:12ina-dominateduniverse(m+=1;k=0)andm=0:11+0:370:19inanopenuniverse(m+k=1;=0).Figure 5-2 showsthecomparisoninmvsparameterspacebetweenthepreliminaryconstraintsofthischapterandearlyconstraintsarisingfromCMBdataandSNIadata,availableindeBernardisetal.2000[ 170 ]. CMBandSNIaconstraintsledtotherstreliableestimatesofmand.ThepreliminaryconstraintspresentedherearecomparabletoearlyconstraintsfromCMBandSNIadata,asillustratedingure 5-2 .Theaccuracyinmand,asdeterminedfromthemostrecentCMBandSNIadata[ 5 ]isnow0:04ineachparameter.Asimilar,andperhapsevensuperioraccuracycanbeachievedusingstarburstgalaxiesathighredshifts. 73

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1-constraintsinmvs.parameterspacefromstarburstgalaxies,alongwitholderconstraintsfromCMBandSNIadata,foundindeBernardisetal.2000[ 170 ]. 74

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5-1 ),alargesamplesizeisrequiredtoobtainmeaningfulconstraints.Thischaptercontainsasamplesizeofonly15galaxies,butfuturesurveysshouldbeabletoobtainhundredsofstarburstgalaxiesthatsurvivetheselectioncuts.Forasampleof500galaxies,thiswillimproveconstraintsonmtoarestrictionof0:03duetorandomerrors.Additionally,allofthesystematicsspecictothissampleduetoincompletedatawilldisappear.Appendix A discusseshowtheseerrorsmaybereduced,andhow,withsuchasample,theconcordancecosmologicalmodelcanbetestedataredshiftofz3,somethingthathasnotbeendonetodate. IftheassumptionofuniversalitybetweenlocalHIIgalaxiesandhighredshiftstarburstgalaxiesiscorrect,thismethodofmeasuringmiscapableofprovidingverytightconstraints,independentofanyconstraintsarisingfromothersources,includingCMBandSNIadata.Additionally,ifgalaxiesareobtainedatavarietyofredshiftsbetween2.z.4,dierentcosmologicalmodels(includingvacuum-energydominatedmodelswithdierentvaluesofw)canbetestedforconsistencywiththefuturedataset.Ifm0:3,thedierencesinDMatvariousredshiftsbecomequitepronounced,andmeaningfulresultsastothecompositionofthenon-mattercomponentsoftheuniversecanbeobtainedaswell.Futureworkonthistopichasthepotentialtoprovidestrongindependentevidenceeithersupportingorcontradictingtheconcordancecosmological 75

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76

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Thisdissertationhasillustratedtheinvaluableroleofcosmologicalperturbationsthroughouttheevolutionoftheuniverse.Theseperturbationsaredeparturesfromperfecthomogeneityindensityandingravitationalradiation.Theyarecreatedduringtheepochofinationbyquantumuctuations,whichareinturnstretchedacrossalllengthscalesbytheexponentialexpansionoftheuniverse.Wheninationends,thecosmologicalperturbationsbecomeoverdenseandunderdenseregionsinaradiation-dominateduniverse.Theuniversethenexpandsandcools,andthesecosmologicalperturbationsevolveundertheinuenceofgravity,radiationpressure,andalltheotherforcesoftheuniverse.Cosmologicalperturbationsgrowlinearlyatrst,andwhentheoverdenseregionshavereachedasucientdensity,theycollapsenonlinearly.Thisnonlinearcollapseleadstotherapidgrowthofcomplexstructure,formingstars,galaxies,clusters,andlaments,amongotherstructures.Thestructureexhibitedoncosmologicalscalesatthepresentisadirectresultoftheevolutionofprimordialcosmologicalperturbations.Theremainderofthischaptersummarizesthemajorndingsofthispaper,andpointstowardsfuturedirectionsforresearchonthetopicofcosmologicalperturbationsandtheireectsontheuniverse.Alsoincludedisasectiononhowcosmologicalperturbationsareexpectedtoimpacttheeventualfateoftheuniverse. 171 ]),orcanbeofpracticallyzeroamplitudecomparedtothescalarmodes(asinnewination[ 172 173 ]).Ineithercase,thespectrumofbothdensityperturbationsandgravitationalradiationarepredictedtobenearlyscaleinvariant,withapossibleslighttiltpreferringeithersmallscalesorlargescales. 77

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31 { 34 137 ]aswellasinchapter 4 .Theworkpresentedinthisdissertationindicatesthatalthoughthevarianceofthepotentialenergy(W)inthesesuperhorizonperturbationscanbecomeverylarge,itisthepotentialenergyitself(W),notW,thatcouplestotheexpansionrateoftheuniverse.Therefore,itappearsatthisjuncturethatcosmologicalperturbationsonsuperhorizonscalescannotaecttheexpansionrateoftheuniverse. Ontheotherhand,therecouldbeapreferencetowardssmallerscales,whichwouldleadtoearlynonlinearity(andfasterstructureformation)ofthesmalleststructures.Thepowerspectrumisfairlyaccuratelyknown[ 1 ]forvaluesofkupto100Mpc1,andmatchesverywellwithsimulationsofascale-invariantspectrum[ 174 ].Althoughthereisnoreasontobelievethedensityperturbationsonscalessmallerthanthisdepartsignicantlyfromann'1spectrum,theperturbationsingravitationalradiationmay.AsshowninSiegelandFry2005a[ 13 ]andinchapter 2 ,inthepresenceofextradimensionsatafundamentalscalemD,primordialgravitationalradiationwillacquireathermalspectrumandanenergydensitygivenbyequation( 2{4 )ifthereheattemperature,TRH,issucientlyhigh.Thepossibilitiesfordetectingathermalbackgroundofgravitationalradiation(andthusindirectlydetectingextradimensions)couldbeaccomplishedthroughaprecisionmeasurementoftheprimordial4Heabundanceorthroughthebroadeningofthe21cmHIline. 78

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ThenetresultofalltheinteractionsisthatoverdenseregionsintheuniversegrowaccordingtotheMeszaroseectatsucientlyearly(linear)times,asgiveninequation( 1{7 ).WhiletheMeszaroseectdoesagoodjobdescribingtheoverallevolutionoftheenergydensityinaregionofspace,itcannotgiveanyinformationabouttheevolutionofthedierenttypesofcomponentswhichcomposetheuniverse.Asanexample,atvariousepochsintheuniverse,eachofbaryons,darkmatter,photons,neutrinos,andvacuumenergycomposeatleast10percentoftheenergydensityoftheuniverse.Whiletheuniverseisradiation-dominated,photonsandneutrinosaremostimportant,whilewhenitismatter-dominated,baryonsanddarkmatterarethemostimportantcomponents. Inordertounderstandhowcosmologicalperturbationshaveevolvedintothelarge-scalestructuresobservedtoday,itisvitaltounderstandtheevolutionofthemattercomponentsoftheuniversefromveryearlytimesuntilthepresentday.Chapter 3 ,basedheavilyontheworksofMaandBertschinger1995[ 22 ]andSiegelandFry2006[ 23 ],detailstheevolutionofthemattercomponentsoftheuniverse. Onenovelideaofthisworkisthattheevolvingcosmologicalperturbationsgiverisetoseedmagneticeldsonallscales.Thiscanalsobefoundinchapter 3 andinSiegelandFry2006[ 23 ],butnotethatthereisacompetinggroupthatobtainsquitedierentresultsthroughasignicantlydierentcalculation[ 120 134 175 ].Theessentialideaisthatphotonshaveamuchlargerinteractioncrosssectionwithelectronsthanwithprotons,inducingchargeseparationsandcurrents.TheCoulombforcealsoplaysamajorrole,actingasarestoringforce.Thenetresultobtainedinchapter 3 isthatmagneticeldsonallscalesarecreated,followingthespectrumingure 3-2 .Onthemostinterestingscales(from1100Mpc),eldsofO(1023Gauss)areproducedattheepochofrecombination.Theseseedeldsmayprovidetheseedsforthemagneticeldsobservedonlargescalestoday.Thismaybeaccomplishedeitherdirectly,byhavingtheseseedelds 79

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Thesetsofequationsforadierencecomponentbetweenionsandelectronsderivedinchapter 3 aretherstoftheirkind.Problemswhichhithertocouldnotbesolvedforlackofhavingequationsthatdescribeachargedierencecannowbetackledusingthesenewtools.Appendix B presentsthepossibilitythatanetelectricchargewaspresentatsomepointintheuniverse'spast.Priortreatments(seeLyttletonandBondi1959[ 176 ]andarticleswhichciteit)havefoundthatmanyofthesescenarioswouldpresentunacceptableconsequencesforcosmology.However,Appendix B pointstowardsthepossibilitythatanetchargeintheuniverse,whichcouldarise(forinstance)fromabrokenU(1)symmetry,wouldbedrivenawaybythesimpledynamicsofcosmologicalperturbations.Ifthisprovestobethecase,manypossibilitiesforphysicsintheearlyuniverse,wherearepresentlyconsideredtohaveunacceptablecosmologicalconsequences,mayturnouttobequitevalidafterall. 1 ),butthestructureformediseventuallystabletofurthercollapseduetothevirialtheorem. Weretheuniversecompletelydevoidofangularmomentum,orrather,weretheZel'dovichapproximationexact,theuniversemightlookvastlydierentfromitspresentstate.Ifgravitationalcollapseweretooccurexactlyalongeldgradients,thennuclearreactionswouldbetheonlyinteractionsintheuniversethatpreventedallstructurefromcollapsingtosingularities.Yet,angularmomentumisafactoflifeintheuniverse,as 80

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Complexphysics,suchasshocksandheating(whichcancreatemagneticeldsviathemechanisminBiermann1950[ 111 ]),starformation,baryoniccollisions,andcollapseonmultiplescalesallplayaroleintheformationofthepresent-daystructureoftheuniverse.OnequestionthathasbeenansweredinChapter 4 ofthisworkisthequestionofwhetherthisnonlinearstructurewillbackreactsuciently,andimpacttheglobalexpansionrateoftheuniverse.Whilemanyauthors[ 28 { 30 ]contendthattheeectsofnonlinearcollapsecouldsubstantiallyimpacttheexpansionrate,ithasbeencalculated(bothinchapter 4 andinSiegelandFry2005b[ 34 ])whattheeectisexplicity.Theconclusionisthattheeectisnegligiblysmall(ofO105thenormalexpansionrate)atalltimes. Cosmologicalperturbations,inanEinstein-deSitteruniverse(m=1,nocurvatureorvacuumenergy),willgrownonlinearlyonlyoncethescaleofinterestisinsidethehorizon.Inthisscenario,structureintheuniverseisself-similar,withsmallerscalesatearliertimesbehavingidenticallytolargerscalesatlatertimes.However,observationsofstructureformation,amongotherobservables[ 177 178 ],donotsupportthispictureofanEinstein-deSitteruniverse.Thepicturewhichismostconsistentwithalltheobservationsisknowntodayastheconcordancecosmology,whichindicatesthattheuniversehasroughly30percentofthecriticalenergydensityinmatterandroughly70percentinacosmologicalconstant.Theconsequencesofthisforlarge-scalestructureintheuniverseareexaminedinsection 6.4 ofthisChapter. Oncegalaxiesform(ataredshiftofaroundz10inaCDMuniverse),theycanbeusedasdeepcosmologicalprobes.Inorderforsomethingtobeausefuldistanceindicator,theremustbearelationbetweenobservablequantitiesandacosmologicaldistance[ 179 ].Manyofthesetechniquesinvolveindividualstarsorstellarremnants,orother 81

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Thusfar,onlythesupernovae(andonlythetypeIasupernovae,atthat)havesuccessfullybeenusedasadistanceindicatoroutatthishighofaredshift.Althoughtherehasrecentlybeenanattempttousegamma-rayburstsasadistanceindicator[ 180 ],thesystematicerrorsinherenttothemethodarefarlargerthantheuncertaintlybetweendierentcosmologicalmodels.Systematicerrors,asillustratedinAppendix A ,areaconstantsourceofdicultyforanyobservationalcosmologist.Uncertaintiesintheuniversalityofadistanceindicator(i.e.thatitbehavesthesameatallredshifts),aswellasuncertaintiesintheenvironmentwherethedistanceindicatorlies,canallbiasresultsobtainedwithanyoneindicator.Itisforthesereasonsthatmanydieringdistanceindicatorsathighredshiftaredesiredforprobingcosmologicalparameters.Chapter 5 (andSiegeletal.2005[ 41 ])buildsupontheworkofMelnick,TerlevichandTerlevich2000[ 40 ],andusesstar-forminggalaxiesatz>2toconstrainthecosmologicalparameters.Althoughsystematicerrorsforthismethodarebothnumerousandworrisome,theyarequantiedandgivenadetailedtreatmentinbothAppendix A andSiegeletal.2005[ 41 ].Theresultobtainedisthat,despitesystematicerrors,ina-dominateduniverse,itcanbeconcludedthat0:09
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Theacceleratedexpansionismosteasilyandsimplyparametrizedbyacosmologicalconstantofenergydensity'6:91030gcm3.Thepresenceofanon-zerocosmologicalconstant()incorporatesabuilt-inscaletothephysicsofstructureformation.Themainconsequenceofthisnewtypeofenergydensityforstructureformationisthatscaleswhicharenotgravitationallyboundtooneanotheratthetimeofmatter-equalityneverbecomeboundtooneanother. TheHubbleexpansionparameter,H,inauniversecontainingmatter,radiation,andacosmologicalconstant,evolvesas wherer,m,andaretheenergydensitiesinradiation,matter,and,respectively.(cisthecriticaldensity.)Fromequation 6{1 ,itisfaciletodeducethatatlatetimes(whenabecomeslarge),theHubbleexpansionparameter,H,isgivenbytheconstant Therefore,thenalexpansionstateoftheuniversewillbemuchliketheinitialinationarystate,inthattherewillbeanasymptoticallyexponentialexpansion.Thescalefactoroftheuniverse,a,willevolveinthefarfutureas 83

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Thecurrentbehavioroftheuniversepointstowardsthisexactscenariofortheenergycontentoftheuniverse,with70percentindarkenergyand30percentinmatter.Ifthisisthecase,andtheuniversecontinuestoevolveaccordingtotheknownlawsofphysics,thengalaxiesandtheobjectsboundtothem,thechildrenofcosmologicalperturbations,willbethelastremainingobjectsintheuniverse.Theunivesewillconsistofafewisolatedclumpsofmatterexponentiallyexpandingawayfromoneanother,withnothingbutemptyspaceinbetween.Intheend,theselonelyclumpsofmatter,havingarisenfromthegrowthandcollapseofslightlyoverdensecosmologicalperturbations,maybetheonlysubstantialthingsinacold,emptyuniverse. 84

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Therehavebeenmanyassumptionsmadealongthepathtoobtainingmviatheuseofstarburstgalaxiesasadistaceindicator,asdetailedinchapter 5 .Themajordangerineveryobservationalmethodisthateveryassumptionmadecarriesalongwithitanassociatederror.Someoftheassumptionsmadeareinherenttothemethodused,whileothersaectonlythedatasamplespecictotheoneselectedinchapter 5 .Bothwillleadtosystematicerrors,althoughthesample-specicerrorswilllargelybeeliminatedbyimprovedmeasurements,tobetakeninfutureobservingruns.Additionally,randomerrorsresultfrombothuncertaintiesinthemeasurementsandfromtheintrinsicscatterinthedistributionofpoints.Ananalysisofallthreetypesoferrorsensuesbelow. 5{1 ),asshowningure A-1 .Althoughthephysicsunderlyingstarburstgalaxieshasbeenanopenquestionforoverthirtyyears[ 183 ],itisfortunatelynotnecessarytouncoverthecompleteanswertoestablishuniversality.Itislikely(althoughunproven)thatthephysicsunderlyingthecorrelationforHIIgalaxiesissimilartothephysicsunderlyingtheTully-Fisherrelation[ 184 ]forspiralgalaxies.Specically,itisthoughtthatlinewidths(ameasureofvelocitydispersion)andtheluminosityintheHlinearebothintimatelytiedtotheamountofmassinthestar-formingregion.AtheoreticalinvestigationofexactlywhatthislinkiscouldprovequitefruitfulinunderstandingtheunderlyingphysicsofthecorrelationpresentedinMelnick,TerlevichandMoles1988[ 155 ]. ThevalidityofthecorrelationbetweenLHandMzcanbetesteddirectlytodetermineitsrangeofapplicability.Byassumingacosmology,logLHcanbewrittenpurelyintermsofluminositydistance(dL),FH,andAH,whichareeithermeasurable 85

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5{1 )and( 5{2 ).logMzcanbedeterminedthroughmeasuredvaluesforandO=H.ComparingthequantitieslogLHandlogMzthenallowsatestofthecorrelationinequation 5{1 forallgalaxiesofinterest.AllavailableHIIandHII-likestarburstgalaxieswithappropriatelymeasuredquantitiesareincludedtotestthecorrelation.LocalgalaxiesaretakenfromMelnick,TerlevichandMoles1988[ 155 ]andfromtheUniversidadComplutensedeMadrid(UCM)survey[ 185 186 ],intermediateredshiftstarburstgalaxiesaretakenfromGuzmanetal.1997[ 187 ],andhighredshiftstarburstgalaxiesarefromPettinietal.2001[ 161 ]andErbetal.2003[ 162 ].Thecosmologyassumedtotestuniversalityism=0:3,=0:7,andcutsareappliedtoallsamplessothatEW>20Aand<130kms1.Theresultsareshowningure A-1 Themajorreasonstoconcludethattheassumptionofuniversalityisvalidlieingure A-1 .ThereexistsanoverlapbetweenallfoursamplesinbothLHandMz,fromthesamplewherethecorrelationiswellestablished(nearbysamples,suchasMelnick,TerlevichandMoles1988[ 155 ]andtheUCMsurvey[ 185 186 ]),tointermediateredshiftHII-likestarburstgalaxiesGuzmanetal.1997[ 187 ],tothehighredshiftsampleusedinchapter 5 ,fromErbetal.2003[ 162 ]andPettinietal.2001[ 161 ].ThesefoursamplesallfollowthesamecorrelationbetweenLHandMzwithinthesameintrinsicscatter.(However,itisworthnotingthattheobservedscatterbroadensathighredshiftsduetomeasurementuncertainties).Byperformingastatisticalanalysisofthedatapointsingure A-1 ,itcanbeshownthatthedataselectedfromallsamplesareconsistentwiththesamechoiceofslopeandzero-pointfortheempiricalcorrelation.Forthesereasons,equation( 5{1 )appearstoholdnotjustforlocalHIIregionsandgalaxies,butforallstarburstgalaxiesregardlessofredshift. Itisimportanttonotethatthereisanuncertaintyinthezero-pointcalibrationofgure A-1 of0:08dex,whichhasnotimprovedsincethecorrelationwasrstdiscovered[ 155 ].ThiscorrespondstoanuncertaintyinDMof0:20,whichisanunacceptably 86

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155 ],andwillalsoprobetheverylowendoftherelationbetweenMzandLH,wheredataaresparse. 5{4 ),thecutonEWof20A,andthecutonof130kms1allinduceinherentsystematicerrors.MovingtheEWcutfromEW>20AuptoEW25A,assuggestedinMelnick,TerlevichandTerlevich2000[ 40 ],wouldsystematicallyraisetheDMby0:14magforallgalaxiespresentinthissample.TheEWthresholdfortheonsetofmajorevolutionaryeectsisnotyetwell-established,andnecessitatesfurtherresearch.Thecutoncomesaboutinordertoremovecontaminationfromobjectscontainingmultipleunresolvedstar-formingregions.SincethecorrelationbetweenLHandMzisonlyvalidforsingleHIIgalaxiesandHIIregions[ 158 ],acutmustbemadetoremoveobjectscontainingmultiplestar-formingregions.SingleHIIgalaxiesareobservedtohaveagaussiandistributioninpeakedat70kms1,butobjectswithmultipleunresolvedregionsareexpectedtohaveanentirelydierentdistribution.OnthebasisofMonteCarlosimulationsperformedtosimulatebothsingleandmultipleHIIgalaxies,acutonat130kms1retains95percentofthe 87

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logMzvs.logLHforlocalHIIgalaxiesandstarburstgalaxiesatintermediateandhighredshifts.Thesolidlineisthebesttofthecorrelationtothelocaldataset,ankedbythedashedlines,whichgivethe2-rmsscatter.Thelargediamondsrepresenttheselectedhighredshiftdatasample;thesmalldiamondsarethedatanotselectedonthebasisofeitherEWor.Theverticaldottedlineisthederivedcutonof130kms1.Thecrosshairsrepresentsthetypicaluncertaintyineachselecteddatapoint. 88

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A-2 below.Additionally,itcanbeshownthatthecontaminatingobjectswhicharenoteliminateddepartonlyslightlyfromtheempiricalcorrelationofequation( 5{1 ). Itisthereforeessential,foranyfuturesurvey,thatthepropercutsbedeterminedandappliedtoEWand,otherwisesubstantialuncertaintieswillarisefromtheselectionofthedatasample.Finally,thederivedextinctionlawinAHitself,fromequation( 5{4 ),hasanuncertaintyof38percent,duetothefactthattherearecompetingextinctionlawsthatgivedierentresults[ 167 { 169 ].Bothlawsarecomparablygrey,buthavedierentnormalizations.ThedierencebetweenthetwolawsleadstoasystematicuncertaintyintheDMofthehigh-redshiftgalaxiesof0:17mag. 161 ]andErbetal.2003[ 162 ].Thesystematicuncertaintiesthattheseassumptionsinducecanbeeliminatedinfuturesurveysthroughmeasurementsofallrequiredquantities.Theassumptionthatthetemperature-metallicitydegeneracyismostlikelybrokentowardstheupperbranchoftheR23vs.O=Hcurveforluminousstarburstgalaxiesathighredshiftisbasedonsoundanalysis[ 165 ],butisstilladangerousonetomake.Measurementofthe4363AoxygenlinewouldbreaktheR23vs.O=Hdegeneracy,andyieldadenitevalueformetallicityforeachgalaxy.Furthermore,O=Hhadtobeassumedfor11ofthe15galaxiesinthesample,inducingapossiblesystematicwhichcouldaectDMiftheassumedaverageO=Hdiersfromthetruevalue.IfthevalueforO=HfromShapleyetal.2004[ 166 ]isusedforthegalaxiesselectedfromErbetal.2003[ 162 ],theaverageDMisraisedby0.22mag.Thissystematiccanberemovedinfuturesurveysbyameasurementofthe[OII]lineat3727Aandthe[OIII]linesat4959Aand5007Aforeachgalaxy.Therearealsoothermetallicityindicators(seeKewleyandDopita2002[ 188 ])whichmayprovetobemorereliableat 89

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Simulationofthedistributionsofvelocitydispersions,aswouldbemeasuredforasingle,isolatedstar-formingregion(redcurve),formultiple,interactingstar-formingregions(bluecurve,asimulationofmultipleunresolvedregions),andforamixtureofbothtypes(greencurve).Theobserveddataisplottedinblack.Notethattheactualdataappearstobeasuperpositionofthesingleregiondataandthemultipleunresolvedregiondata.Multipleunresolvedregionsathighredshiftsappearintheformofenclosedgalaxymergers,andmustberemoved,astheydonotfollowthecorrelationofequation( 5{1 ).Fromthesimulations,removalofmostoftheenclosedmergerscanbeaccomplishedthroughaderivedcutonof130kms1. 90

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166 ].Intheory,manyabundanceindicatorsareavailableandmayevenbepracticallyaccessible[ 188 ],andfuturesurveysshouldallowmultiple,independenttechniquestobeused,signicantlyreducingerrors.Notealsothatitisunsettlingthatdierentmetallicityindicatorsappeartogivedierentvaluesofthemetallicityforthesamegalaxysamples;thismaybeyetanothersourceofinherenterror. Thereisalargeuncertaintyontheorderof30percentinthemeasurementofEWfortheErbetal.2003[ 162 ]sampleduetothedicultyofestablishingtheheightofthecontinuum.Somegalaxiesmayhavebeenincludedwhichshouldnothavebeen,andothersmayhavebeenexcludedwhichshouldhavebeenpresent.Theeectonthedistancemodulusisestimatedtobe0:16mag,butthiswillberemovedbymeasuringequivalentwidthinHwithahighersignal-to-noisespectraforallgalaxiesinfuturesurveys.Finally,E(BV)colors,asubstituteforAHmeasurements,areunavailableforgalaxiesfromPettinietal.2001[ 161 ],andwerederivedfromanapproximatecorrelationnoticedbetweenE(BV)andthecorrected(GR)colorsinErbetal.2003[ 162 ].ThereisanoveralluncertaintyintheextinctionduetothefactthattheaveragederivedextinctionfortheErbetal.2003[ 162 ]andthePettinietal.2001[ 161 ]samplesdierby0:34dex.Thus,thereisaninducedsystematicinDMof0:17mag,whichwillbeeliminatedwhenAHmeasurementsareexplicitytakenforallgalaxies. 91

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5{2 ),DMisdependenton5,whereasitdependsonlylinearlyontheotherquantities.Itisthereforevitaltoobtainexcellentmeasurementsofthevelocitydispersionofthesegalaxies(whichiscertainlypossible,assomeofthehigh-redshiftgalaxiesinPettinietal.2001[ 161 ]haveuncertaintiesofonly4kms1).FutureworkwillbeabletomeasuretheHandHlines,aswellasthreeoxygenlines,asthesearewherethethreewindowsintheinfraredareintheatmosphere.Forgalaxiesbetween2.z.4,theH,H,andmultipleoxygenlineswillappearattheappropriatewavelengths.Theseobservationsshouldimprovethemeasurementsof,furtherreducingtherandomuncertainties.ThedistributionofpointsmaynotimproveasstatisticsimproveduetotheintrinsicscatterontheMzvs.LHrelation,butrandomerrorsallfalloasthesamplesizeincreases.TheerrorsdecreaseasN1=2,whereNisthenumberofgalaxiesinthesample.Evenifrandomerrorsassociatedwithintrinsicproperties(suchasFH,,orO=H)remainlargeforindividualgalaxies,increasingthesamplesizewilldrivedowntheoverallrandomerrors.Hence,asampleof500galaxies,asopposedto15,willhaveitsrandomuncertaintiesreducedbyafactorof6orbetter.ThenewgenerationofNear-IRMulti-ObjectSpectrographs(suchasFLAMINGOSandEMIR)in10meterclasstelescopeswillbeidealforobtainingallnecessarymeasurementsforsuchasample. 92

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Thepossibilitythattheuniversecouldhaveanetelectricchargehasbeeninvestigatedo-and-onbymanyscientistsforthepasthalf-century.Anetelectricchargecouldariseasaconsequenceofmanydierenttypesofearly-universephysics,buttherearestrictlimitsonanetchargeintheuniversetodayfromvariouscosmologicalconstraints.Thisappendixexaminesthepossibilitythatthemathematicalformalismandequationsdevelopedinchapter 3 couldremoveaninitialnetchargefromtheuniversethroughcosmologicaldynamics.Ifthissuccessfullyoccurs,manyinterestingphysicalmechanisms,previouslythoughttobetightlyconstrainedthroughpresent-daymeasurements,mayhaveoperatedintheearlyuniverse. 176 ],whereitwaspointedoutthatasucientlylargeelectricchargeasymmetry,ontheorderof1eper1018baryons,wouldenabletherepulsiveCoulombforcetoexceedgravitationalattractiononlargescales.Theoriginalmotivationforthisproposalwastoexplaintheoriginofcosmicexpansion.Itwasexplainedthatifthemagnitudeoftherespectivechargesonelectronsandprotonsdieredby21018e,largescalerepulsionwouldfollow. Withtheadventofthebigbangtheory,whichprovedtobeanecessitytoexplaintheobservedcosmicmicrowavebackgroundradiation(CMB,discoveredinPenziasandWilson1965[ 189 ]),Hubbleexpansionwasexplainedasanecessaryconsequenceofthattheory.Itfurtherappearsthat,toamuchhigherdegreeofaccuracythan21018e,theprotonandelectronchargesareequal.Fromtheanisotropiesofcosmicrays,whichcanactasaprobeofthenetchargeintheuniverseatthepresentday,itisdeterminedthattheoverallcharge-per-baryon()isconstrainedtobejj<1029e[ 128 ].Furthermore,thedegreeofisotropyintheCMBprovidesconstraintsonthenetelectricchargeintheuniverseat 93

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18 ],whereagaintherestrictionisthatthenetcharge-per-baryonisverysmall,jj<1029e.Anetcharge-per-baryonwouldalsoimpactprimordialnucleosynthesis.Cosmologicalheliumsynthesisprovidesaprobeofthenetchargeatveryhighredshifts(z4108).Constraintsfromnucleosynthesis[ 17 ]indicatethat,intheearlyuniverse,jj.1032e,themoststringentconstraintsavailableonthenetchargeintheuniverse. Ifanetcharge-per-baryondoesexist,therearetwostraightforwardwaystoobtainit,bothofwhichwereidentiedinLyttletonandBondi[ 176 ].Either,asstatedabove,theprotoncharge(qp)diersslightlyfromtheelectroncharge(qe)inmagnitude,orthenumberdensityofprotons(np)diersfromthatofelectrons(ne).Theformerpossibilityishighlydisfavored,asterrestrialexperimentsindicatethattheelectriceldiszeroattheEarth'ssurface.AssumingequalnumbersofprotonsandelectronsonEarth(1051ofeachspecies)placesstrictconstraintsonjqpj6=jqej,asdorenedversionsoftheMillikanexperiment.Itisobservedthatchargeisquantizedinunitsofeforphysicallyobservableparticles(althoughquarksarepredictedtohavefractionalcharges),aspredictedbythestandardmodelofparticlephysics[ 53 ].Barringexoticscenarios,suchaselectricallychargedneutrinos,photons,ordarkmatter[ 190 ],itisonlyreasonabletoconsiderunequalnumberdensitiesofprotonsandelectronsasgivingrisetoanelectricchargeintheuniverse.Thecreationofanelectricchargeasymmetryinthisfashionisanalogoustothecreationofabaryonasymmetryingranduniedtheories(seeDineandKusenko2003[ 15 ]forareviewofbaryogenesis).Anelectricchargeasymmetrycanbegeneratedbyasimilarmechanismtothebaryonasymmetry,andbothareexpectedtohavethesametypesofinhomogeneities[ 191 ]. 191 { 193 ].IfthisU(1)symmetryisbrokenatsomepoint,anelectricchargeasymmetrymustbe 94

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191 ].WhentheU(1)symmetryislaterrestored,thechargeasymmetrymayremain[ 192 ].Caremustbetakentoensurethatelectricchargeremainsquantizedinunitsofe[ 193 ].Theproductionofanetchargeisanalogoustobaryogenesisingrandunicationmodels,whichhaveadecayprocessathighenergiesthatisasymmetric,preferring(slightly)toproducebaryonsoverantibaryons.Similarmechanismscouldeasilyprefertheproductionofonesignofchargeovertheother,solongasthatproductionsymmetryisrestoredtoday.Othermechanismsalsoexistwhichadmittheproductionofanetelectriccharge.ExamplesincludeKaluza-Kleinmodelswithextradimensions[ 128 194 ],cosmologieswithavaryingspeedoflight[ 195 196 ],andeectiveinteractionsallowingelectricchargenon-conservationbyunitsof2eatatime[ 197 ]. Onceanetchargehasbeenestablishedintheformofunequalprotonandelectrondensities,previoustreatmentsassumeforsimplicitythatthetotalchargewithinaspatialvolumeisconstant[ 128 191 198 ].Themajorpurposeoftheworkinthisappendixistoshowthatthisassumptionisnottrueingeneral.Electromagneticforceswillinducerelativemotionbetweenoppositelychargedspecies.Chargewillbeconservedlocally(inthattherearenochargenon-conservinginteractions),buttheexpansionratesofpositiveandnegativechargesarefoundtodier.Thisallowscurrentstoowandthenetelectricchargedensitytochangewithtime.Thisappendixexamineshowchargeasymmetries,bothlocalandglobal,evolveinanexpandinguniverse. Asaresultofthedynamicsofcosmologicalexpansion,aninitialnetchargecanbeeitherremovedcompletelyorreducedsignicantly.Solongasthecosmologicalboundsonachargeasymmetryduetocosmicrays[ 128 ],theCMB[ 18 ],andnucleosynthesis[ 17 ]aresatised,thereisnolimitonanyinitialelectriccharge.Theremainderofthisappendixfocusesonhowcosmologicaldynamicsaecttheoverallchargedensityinanexpandinguniverse.Section B.3 presentsanintuitiveNewtonianformulationofauniversewithaninitialchargeasymmetry,baseduponthegravitationalandelectromagneticforcelawsalone.Whilesection B.3 maybeusefulforgatheringanintuitivepictureofthe 95

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3 ,isfoundinsection B.4 .ThissectiontakesintoaccountnotonlytheCoulombandgravitationalforces,butalsointeractionsbetweenphotonsandchargedparticles,aswellasalloftheotherinteractionsassociatedwithstructureformation.Althoughthephysicsofsection B.4 islessintuitive,itisfarmorerigorous,andcapturesamuchgreaterdegreeoftheessentialphysicalbehavior.Adiscussionofthepreliminaryresultsdiscoveredhereandtheirimplicationsfortheearlyuniverseispresentedinsection B.5 whereidenotesatestparticleofeitheraproton(p)oranelectron(e).TheNewtoniangravitationallaw, r2;(B{2) whereGNisNewton'sconstant,andMisthetotalmassenclosedbyaspherewithradiusequaltotheuniverse'sexpansionfactor,r.Asaccelerationisdenedasar,theresultantequationfortheevolutionoftheexpansionfactorbecomes r r=4 whereistheenergydensityoftheuniverse.UsingEinstein'sequationsinsteadofNewton'sinanisotropic,homogeneousuniversewouldmodifyequation( B{3 )byreplacingwith(+3p=c2),wherepisthegeneralrelativisticpressureoftheuniverse. 96

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fortheenergydensity,wherenpandnearetheprotonandelectronnumberdensities,respectively.TheCoulombforceisgivenby 40qiQ r2;(B{5) whereqiisthechargeofaproton(qp)orelectron(qe),andQisthetotalchargeenclosedbyasphereofradiusr.Thephysicsactingonprotonsandelectronsisdierent,duetotheiroppositecharges.Therefore,whentheexpansionfactorsarederivedfromthecombinedforcelawsforelectronsandprotons,itisfoundthattheyevolvedierentlythaninequation( B{3 ).Theprotonandelectronexpansionfactors(rpandre,respectively)arefoundtobe rp and re Torewritethesetwoequationsintermsofrp,re,andtheirtimederivativesalone(i.e.toremovenpandnefromtheequations),Nisdenedtobethetotalnumberofprotonsorelectronscontainedwithinasphereofradiusrporre,respectively. Letrp;0=re;0=r01,wherethesubscript0indicatesthevalueofagivenquantityatpresent.rpandrearehenceforthwrittenasfractionsoftheirvaluetoday.TheratiooftheCoulombtothegravitationalforcesappearsinthecosmologicalevolutionequations,denedasthedimensionlessparameterK,where 97

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whereH0istheHubbleparameter(H)evaluatedtoday.Theevolutionequationsnowtaketheform rp 2H20fp fp1 re 2H20fp fe1 B{11 )and( B{12 )willonlybeimportantifthereisasignicantchargeasymmetry.Infact,ifrp=reexactly,thenbothequationsreducetoequation( B{3 ).Thecaseofanexaggeratednetchargeintheuniverseisillustratedingure B-1 TheexpansionfactorexhibitsthestandardbehaviorforanEinstein-deSitterUniverse,whichisr=r0=(t=t0)2=3.Protonsandelectronsarefoundnottoowsmoothlytogether,butrathertooscillateaboutanequilibriumwhichtheyneverreach.Thechargeasymmetryisnotaconstantoverpropervolume,butitselfoscillateswithadecreasingfrequencyanddecreasingamplitude.Theoscillationfrequencyisrapidcomparedtotherateofdecreaseofbothfrequencyandamplitude. Changingvariablestoacenter-of-massexpansionfactor(rcm)andanasymmetryparameter(r)assiststheexplorationofequations( B{11 )and( B{12 )inthelimitofasmallchargeasymmetry.Themass-weightedsumofprotonsandelectrons(rcm)andthedierencebetweenprotonsandelectrons(r)aregivenby whererisexperimentallyandtheoreticallymotivatedtobemuchlessthanrcm.Theevolutionequationsarethen rcm 2H201 2H20(K+2fpfe)r2 98

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Expansionfactorsrp(red)andre(blue),forpositiveandnegativechargedistributions(rising)andtheirtimederivatives,vpandve(falling).Theamplitudeoftheasymmetryisenhancedbyafactorof104forvisibility. 99

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r=3KH20 toleadingorderinr.Inthelimitasr!0,thestandardNewtoniancosmologicalexpansionisrecovered. SinceKislarge(K1),theasymmetry(r)behavesasaslowlydecayingharmonicoscillatorovershorttimescales.Itisusefultoparametrizer=Acos.Calculationofrshows,inconjunctionwithequation B{15 ,that _2'3KH20 Equation( B{16 )illustratesthatA_2A,since_2KH2,whereasA=AH2.(RecallthatK'1:2351036.)BecauseK1,theamplitudeoftheasymmetry,thechangeintheamplitudeoftheasymmetry,andtheexpansionrateoftheuniverseallvaryslowlywithrespecttotheoscillationfrequency. Fromequation( B{16 )andthedenitionthat!t,theoscillationfrequencyatanyepochis wheren0isthenumberdensitytoday(n01),andzistheredshiftofinterest.Thiscorrespondstoafrequencytoday(z=0)of134rads1,orapproximately21Hz.TheoscillationfrequencyatanyepochinanEinstein-deSitteruniverseistherefore21(1+z)3=2Hz. Aslightlymoresophisticatedtreatmentincludesradiationintheuniverse.Forradiation,thepressure(prad)isgivenbyprad=c2=1 3rad.Theoverallenergydensityoftheuniverseismodiedbytheadditionalterm (rad+3prad)=2hEinB wherehEiistheaverageenergyofaphoton,nBisthenumberdensityofbaryons,and'6:11010isthebaryon-to-photonratiotoday.Withradiationincluded,theequations 100

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rp 2H20fp fp1 re 2H20fp fe1 assumingphotonsfollowthesameexpansionrateasrcm(whichtheydo),andhE0iistheaveragephotonenergytoday. Equations( B{19 )and( B{20 )canbesolvednumerically,yieldingresultsforrcmandr.Thoseresultsaredisplayedingure B-2 .Althoughtheasymmetry(r)itselfincreasesslowly,therelativeamplitudeoftheasymmetrytotheexpansionfactor,r=rcm,decreases.Adiabaticdampingcausestherelativeamplitudetoevolveas rcm/r1=4cm:(B{21) Thefrequencyofoscillationinauniversecontainingbothmatterandradiationis whichreducestoequation B{17 inthelimitofnoradiation(!1).Theuniverseevolvesasmatterdominated(rcm/t2=3)atlatetimes,andasradiationdominated(rcm/t1=2)atearlytimes. Tosummarizetheresultsofthissection,whichprovidedaNewtoniantreatmentofachargeasymmetry,anexpandinguniversewithaninitialchargeasymmetryhasthatchargeasymmetryevolvewithtime.Theasymmetryoscillateswithfrequency!,asgiveninequation( B{22 ),withitsamplitudefallingasjrj=rcm/r1=4cm. 101

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Theevolutionofanetchargeasymmetryinauniversecontainingbothmatterandradiation.Thebluecurveplotstheevolutionoftheexpansionfactor,rcm,againsttime.Theredcurveistherapidlyoscillatingasymmetry,r,plottedagainsttime.Thegreencurveisananalyticttotheamplitudeoftheasymmetry,showingthatitevolvesasjrj/r3=4cm. 102

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Inchapter 3 ,withtheassistanceofMaandBertschinger1995[ 22 ]andSiegelandFry2006[ 23 ],theformalismwasdevelopedtodealwiththeevolutionofcosmologicalperturbationsofalltypesinanexpandinguniverse.Thepresenceofanetchargeintheuniverse,orachargeasymmetry,canbetreatedidenticallytoauniversecontainingachargeseparationonanarbitraryscale.Theequationsfortheevolutionofdensityperturbations(overdensitiesandunderdensities)inelectronsandprotons,asderivedinsection 3.3.4 ,are _e=e+3_;_e=_a ae+c2sk2e+k2+4 ap+c2sk2p+k2+4 whereeandparethedensityperturbationsinelectronsandprotons,andeandparethefourier-transformedvelocityperturbationsintheelectronandprotonelds. Asstatedpreviously,auniversewithanetchargewillmostlikelyachievethatstatethroughdierentnumbersofprotonsandelectronsintheuniverse.Therefore,theformalismdevelopedforalocalchargeseparationinsection 3.3.5 canbeappliedtoanetchargeasymmetryintheuniverse.Bytakingthedierencebetweenthedensityandvelocityeldsinelectronsandprotons,asetofevolutionequationsfortheevolutionofanetchargewithinagivenvolumeisobtained,identicallytoequation( 3{17 ).The 103

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_q=q_q=_a aq+c2sk2q+4e2 whereallotherquantitieshavethesamedenitionsasinchapter 3 Aninitialnetchargewillappearinequation( B{24 )asaninitialvalueofqthatisnon-zero,andpresumablylargeenoughtobeofinterest.Rewritingthisequationasasinglesecond-ordinaryordinarydierentialequation,thefollowingexpressionisobtained: q=_a a+4 Ifqisinitiallyeither0orverysmall,thenalterminequation( B{25 )willbeimportant,andthebehaviorwillscaleasitdidinchapter 3 .However,ifqisinitiallylarge,thenaltermofequation( B{25 )oughttobeunimportant. Bytransformingequation( B{25 )sothatderivativesaretakenwithrespecttoscalefactor(a)insteadofconformaltime(),equation( B{25 )becomes whereNisthenumericalfactorfromequation( 3{19 ).Byneglectingthesubdominanttermsinequation( B{26 ),andbydeningtheconstantsand!2tobe 4N;0 104

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wherethesubscript0denotesthepresentday,asimpledierentialequationcanbewrittendown.Equation( B{26 ),withthedenitionsinequations( B{27 )and( B{28 )substituted,andthesubdominanttermsneglected,becomes foraninitial,largechargeasymmetry,q. Theaboveequation,( B{29 ),willbesolvedinthefuture,mostprobablyinvolvingtheuseofintegratingfactors,followingthewell-knowntechniqueillustratedinTurnerandFry1981[ 199 ].However,averygoodapproximationforwhencritical(i.e.,exponential)dampingoccursforequation( B{29 )isgivenbysetting'!.Solvingthesubsequentequationforayieldsthatcriticaldampingoccurswhen 2431=6;0 ThiscorrespondstoatemperatureoftheuniverseofT'60GeV,whichisslightlylowerthantheelectroweakscale. Therefore,anychargeasymmetrycreatedpriortoa'3:91015willbeeliminatedbytheexpansiondynamicsoftheuniversecoupledwithscatteringandCoulombinteractions. 105

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Thisappendix,particularlysection B.4 ,illustratesthataninitialchargeasymmetrycannotnecessarilybeconstrainedbaseduponmeasurementsmadetoday[ 128 ],atrecombination[ 18 ],orevenatnucleosynthesis[ 17 ].Anychargeasymmetrywhichiscreatedbeforecriticaldampingofequation( B{29 )occurswillbeexponentiallydrivenawayaroundz'2:61014,asdeterminedinequation( B{30 ).Thisdampingissucientlystrongthatitcanremoveachargeasymmetryofanymagnitude,uptoandevenabovethecriticalenergydensityoftheuniverse. Onceanyinitialchargeiswipedout,theuniversecontinuestoevolve,andthechargeinitcontinuestoevolveaccordingtothedynamicsinequation( B{26 ).Thelate-timesolutiontothis(aftercriticaldampingbutpriortorecombination)isgivenbytheexpressionbelow, Thisnumberisalwayssmall,sothatq.1040atlatetimes,whichcertainlysatisesallobservedconstraints. Itisworthnotingthatthisanalysismaybealteredatsucientlyhighenergies,whereThomsonscatteringisapoorapproximationoftheactualelectron-photonscattering.Atenergiesofinterest(E&1GeV),non-relativisticscatteringisnotevenagoodapproximationforproton-photonscattering.AbovetheQCDscale,infact,itmaynotevenmakesensetodiscussprotons,asthoseparticleswillbedissociatedintoaquark-gluonplasma.Amoresophisticatedtreatmentmaybeneededtoextracttheexactbehavioraldetailsatthesehighenergies. Theoverallconclusionwhichcanbedrawnfromthispreliminarywork,tobereinforcedbyamorerigorouscalculationinthefuture,isthatanynetchargeintheuniversecreatedaboveatemperatureofT'60GeVisdrivenawaybycosmologicaldynamics.Thisindicatesthatanetchargeofanymagnitudecouldbegeneratedatthe 106

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PAGE 125

EthanSiegelwasbornonAugust3rd,1978,atUnionHospitalintheBronx,NewYork.HegraduatedfromtheBronxHighSchoolofSciencein1996andattendedNorthwesternUniversityfrom1996to2000,graduatingwithatripledegreeinphysics,classics,andtheIntegratedScienceProgram.HespentayearasapublicschoolteacheratKing/DrewMedicalMagnetHighSchoolinLosAngeles,California,afterwhichhemovedtoFloridatopursueaPh.D.inastrophysicsattheUniversityofFlorida.Hehasheldawidevarietyofjobsinthepast,includingemploymentasadaycampcounsellor,astockboyinalingeriestoreonBroadway,aresearchemployeeforFermilabandforNASA,abusboyinakitcheninRome,aprivatetutor,andawebpagedesigner.HecurrentlyresidesinMadison,Wisconsin,withhisroommateandtwocats.Heenjoysplayingguitar,travelling,swingdancing,outdooractivities,andswillingalcoholinitsmanyforms,aswellasscaringchildrenofallagesonHalloween. 125


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COSMOLOGICAL PERTURBATIONS AND THEIR EFFECTS ON THE UNIVERSE:
FROM INFLATION TO ACCELERATION



















By

ETHAN R. SIEGEL


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2006

















Copyright 2006

by

Ethan R. Siegel

















I dedicate this work to everybody who stood by me in my darkest hours





ACKNOWLEDGMENTS

I extend my deepest thanks to Jim Fry for guidance and sound advice throughout

my ventures at the University of Florida. For helpful discussions and conversations on

a multitude of topics over the years, I acknowledge Wayne Bomstad, Steve Detweiler,

Leanne Duffy, and Lisa Everett. For support in various times of need, I thank Hai-Ping

Cheng and the Quantum Theory Project, the University Women's Club, the University

of Florida's Alumni Fellowship Program, the Department of Physics, and the College

of Liberal Arts and Sciences. I also thank Filippos Klironomos for his assistance in the

incarnation of this electronic dissertation. Extensive use has been made of NASA's

Astrophysics Data System's bibliographic services, as well as Spires' high-energy physics

database.

On the topic of cosmological inhomogeneities, I acknowledge Dan Chung and Uros

Seljak for their helpful input. I wholeheartedly thank Ed Bertschinger for making his

COSMICS code available for public use, which has proved invaluable in understanding

and computing various aspects of cosmological perturbation theory. Jesus Gallego and

Jaime Zamorano are thanked for providing unpublished data from their surveys of local

HII galaxies, and Rafael Guzman, Chip Kobulnicky, David Koo, and Mariano Moles are

thanked for valuable assistance in my work and understanding of starburst galaxies. In

addition to the aforementioned people, I also acknowledge Eanna Flanagan, Konstantin

Matchev, Pierre Ramond and Bernard Whiting for discussions on gravitational radiation

in extra dimensions.





TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................. ............... 4

LIST OF TABLES .................................... 7

LIST OF FIGURES ...... ........... ............ ...... 8

A BST RA C T . . . . . . . . .. .. 9

CHAPTER

1 INTRODUCTION TO COSMOLOGICAL PERTURBATIONS ........ 10

1.1 Energy Density ............. .... ............. 10
1.2 Theory of Inflation ...... .......... ........... 11
1.3 Cosmological Perturbations from Inflation ......... ......... 12
1.4 Cosmological Evolution in a Perturbed Universe ....... ..... ... 13
1.5 Nonlinear Evolution of Perturbations .......... ....... ... 17

2 THE GRAVITATIONAL WAVE BACKGROUND ......... ........ 21

2.1 Primordial Gravitational Waves ................. .... .. 21
2.2 Extra Dimensions .................. ............. .. 22
2.3 A Thermal Graviton Background ................ .... .. 24
2.4 Detection of Extra Dimensions .................. ...... .. 25
2.5 Alternative Thermalization Mechanisms .................. .. 27
2.6 Problems of Extra Dimensions .................. ..... .. 28
2.7 Summary ................... .. .... .......... 29

3 STRUCTURE FORMATION CREATES MAGNETIC FIELDS ... ..... .30

3.1 Introduction ................. . . ... 30
3.2 Magnetic Fields: Background .................. .... .. 31
3.3 Cosmological Perturbations .................. ..... .. 34
3.3.1 Cold Dark Matter .................. ........ .. 35
3.3.2 Light Neutrinos .................. .......... .. 36
3.3.3 Photons .................. .............. .. 37
3.3.4 Baryons ....... ....... ........... ..... .. 38
3.3.5 Charge Separations .................. ........ 40
3.4 Magnetic Fields .................. .............. .. 45
3.5 Discussion .......... ......... ................. 47

4 EFFECTS ON COSMIC EXPANSION ................ .... .. 52

4.1 Accelerated Expansion .................. .......... .. 52
4.2 Effects of Inhomogeneities .................. ........ .. 53
4.3 Effects on the Expansion Rate .................. ..... .. 56





4.4 Contributions of Nonlinear Inhomogeneities ................ .. 56
4.5 Variance of the Energy in Inhomogeneities ..... . . ..... 60
4.6 Sum mary .................. ............... .. .. 62

5 FULLY EVOLVED COSMOLOGICAL PERTURBATIONS . . ... 65

5.1 Precision Cosmology .................. .......... .65
5.2 Selection of the Data Sample .................. ..... .. 67
5.3 Constraints on Cosmological Parameters .................. .. 71
5.4 Conclusions and Future Prospects ................ .... .. 75

6 CONCLUDING REMARKS ............... .......... .. 77

6.1 Creation of Perturbations .................. ........ .. 77
6.2 Early Evolution of Perturbations ................... . 78
6.3 Final State of Perturbations .................. ...... .. 80
6.4 Fate of the Universe .................. ......... .. .. 82

APPENDIX

A ERRORS IN HIGH-Z GALAXIES AS DISTANCE INDICATORS . ... 85

A.1 Universality among H II Galaxies ................. .. .. 85
A.2 Systematic Errors .................. ............. .. 87
A.3 Measurement Uncertainties. .................. .. .. 89
A.4 Statistical Errors .................. ............. .. 91

B ON AN ELECTRICALLY CHARGED UNIVERSE . . ...... 93

B.1 Introduction .................. ................ .. 93
B.2 Generating a Net Electric Charge ................ .... .. 94
B.3 N-',wtonian Formulation ................... ......... 96
B.4 Relativistic Formulation ................... . .. 101
B.5 Discussion .................. ................ .. 105

REFERENCES ................... ......... ....... 108

BIOGRAPHICAL SKETCH .................. ............. .. 125





LIST OF TABLES
Table page

5-1 Table of Selected High-Redshift Galaxies ......... ....... 69




LIST OF FIGURES
Figure

2-1 Parameter space for a thermal gravitational wave background


page

24


5q and Oq/H as a function of redshift .................. ..... 44

Spectral energy d-nrity of the magnetic field ....... . . 48

Spectral density of gravitational potential energy ................ ..57

Fractional contributions of energy in inhomogeneities . . 58

Fractional contributions of inhomogeneities at second order . . .... 61

Fluctuation in potential energy vs. cutoff scale . . ....... 63

Distance modulus vs. redshift .................. ......... .. 72

Constraints on Qm and QA . . . . . .. 774

log Mz vs. log LHE for H II and starburst galaxies ................ ..88

Simulation of velocity dispersions for star-forming regions . . ..... 90

Expansion factors for positive and negative charges ............... 99

Evolution of a net charge asymmetry .................. ....... 102





Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

COSMOLOGICAL PERTURBATIONS AND THEIR EFFECTS ON THE UNIVERSE:
FROM INFLATION TO ACCELERATION

By

Ethan R. Siegel

December 2006

Chair: James N. Fry
Major Department: Physics

The universe is, on the largest scales, nearly perfectly isotropic and homogeneous.

This degree of smoothness was accentuated in the past, when density inhomogeneities

departed from perfect uniformity by only thirty parts per million. These tiny imperfections

in the early universe, however, have had enormous impact in causing the universe to

evolve into its present state. This dissertation examines the role of these cosmological

perturbations throughout various important events during the history of the universe,

including inflation, linear and nonlinear structure formation, and the current phase of

accelerated expansion. The spectrum of perturbations is calculated in the context of

extra dimensions, and shown under which conditions it can be thermal. The effects of

gravitational collapse are shown to generate magnetic fields, but not to significantly alter

the expansion rate or cause acceleration. Finally, Lyman-break galaxies are examined as

a possible distance indicator, and it is found that they may emerge as a powerful tool to

better understand the energy content of the universe.





CHAPTER 1
INTRODUCTION TO COSMOLOGICAL PERTURBATIONS

1.1 Energy Density

The universe, as observed today, is filled with intricate and complex structure.

Looking at length scales on the size of an individual planet (- 10-10 pc), the solar system

(_ 10-3 pc), the galaxy (- 10-1 Mpc), or even clusters of galaxies (~ 10 Mpc), it is

evident that there are large departures from the average value of energy density, p. A

measure of the departure from homogeneity at any position is given by the quantity 6,

where

&p -f (1-1)
P P
On very small (i.e., planetary) scales, density contrasts can be as high as 6 ~ 1030.

However, as larger and larger scales are examined, the density contrast of a typical point

in space is found to be much smaller. When scales of 0(100 Mpc) or larger are examined,

it is found that density contrasts are nearly always small, such that 6 < 1 [1].

What can be learned from this is that the universe is, on its largest scales, very nearly

isotropic (the same in all directions) and homogeneous (the same at all positions in space).

In the framework of general relativity, a universe that is both isotropic and homogeneous

is described by the Friedmann-Robertson-Walker metric,


dst2dt + a2(t)(dx di), (1-2)


where a(t) is the scale factor of the universe.

Because the universe is expanding and has a finite age, it is manifest that the degree

of isotropy and homogeneity which is observed today was greater in the past. This is

confirmed by observations of the relic radiation from the big bang, known as the cosmic

microwave background (CMB), which shows the universe to have an amplitude of density

fluctuations of 6 3 x 10-5 [2]. These fluctuations in density, although insignificant

when compared to the homogeneous part at early times, play a vital role in the universe's





evolution, bringing it from a state of near-perfect homogeneity to the complex nonlinear

structures observed today.

1.2 Theory of Inflation

The theory of inflation [3] provides a mechanism to put these primordial cosmological

perturbations in place at the time of the big bang. The big bang, as a cosmological theory,

is the only compelling theory in the context of Einstein's general relativity that provides

an explanation for the presence of the CMB radiation, the observed Hubble expansion of

the universe, and the light element abundances (through big bang nucleosynthesis). It is

shown in Peebles 1993 [4] that all reasonable alternatives to the big bang scenario either

fail to reproduce one of the above three observations or cannot be compatible with general

relativity. The big bang, however, is not an origin of the universe, but is rather a set of

initial conditions. It is the above theory of inflation that naturally produces these initial

conditions.

For the big bang theory to evolve into a universe which is compatible with observations,

the initial conditions must be that the global curvature of the universe is spatially flat to

an accuracy of 2 per cent [5]. The temperature and density of the universe must be

uniform across scales far larger than the horizon, and the density of magnetic monopoles

in the universe must be very small. These three problems are known as the flatness,

horizon, and monopole problem. The method by which inflation solves this problem is to

have a de Sitter-like phase of expansion during the very early universe. de Sitter expansion

is characterized by the metric


ds2 = -dt2 + eAt(d di), (1-3)


which is similar to equation (1-2), except that the scale factor a = et, where A is a

constant. A is related to the expansion rate of the universe at that time by the equation

A = H2, where H = a/a is the Hubble parameter.




From equation (1-3), it is manifest that the universe is expanding at an exponentially

fast rate. Given enough time, the universe can expand by an arbitrary number of

e-foldings. A universe expanding in this way will have its matter density reduced by a

factor of
Pi (a(ti) = e (tj -t) (1-4)
Pf \a(tf) )
where pi and ti are the density and age of the universe at the start of inflation, and pf

and tf are the density and age at the end of inflation. This removes any initial densities or

differences in density, solving both the horizon and monopole problems. Additionally, any

initial curvature (departure from flatness) will be driven away by a factor of a2(ti)/a2(tj),

providing a solution to the flatness problem as well.

1.3 Cosmological Perturbations from Inflation

In addition to setting up the initial conditions necessary for the big bang, inflation

also predicts very slight departures from homogeneity in the universe, producing

fluctuations in both energy density [6-9] and in gravitational radiation [10, 11]. It is

these predictions for the departures from perfect homogeneity, produced by quantum

fluctuations, which are then stretched during inflation across all scales, that are the focus

of this work. The perturbations produced by inflation are Gaussian in their statistical

properties, and are also scale-invariant.

The perturbations in gravitational radiation are constrained to be a very small

fraction of the energy density in the universe [12]. Nevertheless, the detection of such

gravitational radiation and measurement of its properties would have the capabilities

to tell us much about the early universe, as gravitational radiation is expected to be

decoupled from the rest of the universe from the time of its creation. If a scale-invariant

spectrum of gravitational waves were observed, it would be a further great confirmation

of the inflationary paradigm, and constraints on the model of inflation could be inferred.

However, there also exists the possibility that the spectrum of gravitational waves could be

thermal. A thermal gravitational radiation spectrum could result from many possibilities,




one of which is as a signature of extra dimensions [13]. This is discussed at length in

Chapter 2 of this work.

When inflation comes to an end, the universe reheats. The process of reheating

transfers the energy from the vacuum (which was responsible for the rapid expansion)

into matter and energy. This universe is described, in the homogeneous approximation,

by the Friedmann-Robertson-Walker metric of equation (1-2). However, it is the density

inhomogeneities, or primordial cosmological perturbations, that will lead to the formation

of all structure in the universe. Therefore, the equation of interest as a starting point for

much of the remainder of this work is that for a perturbed Friedmann-Robertson-Walker

universe,

ds2 = a2(T)[-(1 + 20)d-2 + (1 20)dx di], (1-5)

where 4 and 4 are the scalar-mode perturbations to the gravitational potential. Of

course, it makes no difference which gauge is chosen [14], as the physics of cosmological

perturbations is the same in all gauges. The preference of the author is to work in the

conformal Newtonian gauge (also known as the longitudinal gauge), as chosen in equation

(1-5).

1.4 Cosmological Evolution in a Perturbed Universe

As the universe cools from its initial, smooth, hot, dense state, many cosmologically

interesting phenomena occur (and many others may occur, dependent upon the reheat

temperature of the universe and the physics involved at very high energy scales). A

cosmological baryon asymmetry must be generated at very early times. As the observed

baryon-to-photon ratio today is r7 6.1 x 10-10, an asymmetry in the number density

of baryons over anti-baryons of this magnitude must be produced. The process by which

this occurs is generically referred to as baryogenesis, and there are many different times

at which it may occur, including at the GUT-scale, at the electroweak scale, through

leptogenesis, or through the Affleck-Dine scenario (see Dine and Kusenko 2004 [15] for

a review). Also, in the early universe, a substantial amount of dark matter must be





generated. This can occur either through freeze-out of a stable, thermally produced relic,

through the misalignment of the vacuum, through the production of a sterile, heavy

neutrino, or through a perhaps more exotic mechanism (see Bertone, Hooper and Silk 2004

[16] for a review). These unsolved puzzles, along with the many questions surrounding

the early universe physics of electroweak symmetry breaking, the QCD phase transition,

and (possibly) supersymmetry breaking, are expected to be unaffected by the presence

of cosmological perturbations. As the universe evolves, most of the physics that occurs

in the earliest stages is expected to occur exactly as it would in a perfectly homogeneous

Friedmann-Robertson-Walker universe.

One possibility in the very early universe, however, for which cosmological perturbations

may play a seminal role is if the universe begins with (or obtains at early times) an

asymmetry in its net charge. It has been pointed out that a large net charge in the

universe would be ruinous at early times for cosmological 4He synthesis [17] and for the

cosmic microwave background [18]. Although a conclusive solution is beyond the scope

of this dissertation, cosmological perturbations in an expanding universe may have the

capability of driving an electrically charged universe to a neutral state. This possibility,

and preliminary work on the subject, can be found in Appendix B.

As the universe continues to expand and cool, the building blocks of the universe

begin to form. After the QCD phase transition, quarks and gluons become bound into

hadrons. Unstable particles decay and/or co-annihilate, leaving the universe devoid

of exotic particles. Neutrinos freeze-out, and decouple from the rest of the universe.

Electrons and positrons coannihilate, leaving an electron asymmetry that matches up

nearly perfectly with the proton asymmetry. When the universe cools substantially so

that stable deuterium can form without being destroyed by the thermal photon bath,

nucleosynthesis occurs, producing deuterium, 3He, 4He, and 7Li. Nucleosynthesis is

complete roughly four minutes after the big bang. Although there has been work in the

past suggesting alternative, complex models of nucleosynthesis (such as inhomogeneous





nucleosynthesis) to be slightly favored (see Steigman 2006 [19] for a review of big-bang

nucleosynthesis and its alternatives), the standard picture of big bang nucleosynthesis

appears to match up perfectly well with observations within the systematic errors [20].

It is generally assumed that no interesting physics occurs until the time of recombination

(where electrons and ions combine to form neutral atoms), roughly 380,000 years later.

The only things of note which occur over that era are that cosmological perturbations

grow according to the M6szdros effect [21], and the universe transitions from a radiation-dominated

state to a matter-dominated one. However, there is a very interesting and subtle effect

that occurs during this time. As tr (the baryon-to-photon) ratio is very small and the

universe is still quite hot and dense during this era, every ion and electron is consistently

bombarded by these high energy photons. Ions have comparable charges to electrons, but

their masses are orders of magnitude greater. The scattering cross sections of charged

particles with photons scales (for non-relativistic scattering) as


8r( )2 (1-6)
3 Vmc"

where q is the charge and m is the mass. As a result of the differences in mass and cross

section, electrons are affected by interactions with photons in a much more profound way

than ions. While the Coulomb forces keeps the electrons and protons tightly coupled, the

momentum transfer from photons works to create charge separations and currents during

the radiation era. The tightly coupled component is dominant, and behaves as a baryonic

component in cosmological perturbation theory (see Ma and Bertschinger 1995 [22] for a

very sophisticated treatment). In contrast, the charge separations and currents created by

momentum transfer are very small, but nonetheless are of great import for the generation

of magnetic fields at early times [23]. The generation of magnetic fields in the young

universe by this mechanism is detailed in Chapter 3, which also discusses the possibility

that the origins of presently observed cosmic magnetic fields may lie in this mechanism.





As the universe transitions from a radiation-dominated state to a matter-dominated

one, gravitational inhomogeneities begin to substantially grow for the first time. The

M6szdros effect dictates that an inhomogeneity 6 grows as


6(t) = (l+ ) 6(to), (1-7)


where Y is given by
Pm(t)
Y (1-8)
Pr(t)
from an initial time to until the time of interest, t, where Pm is the matter density and

pr is the radiation density. This approximation is valid throughout the linear regime of

gravitational collapse and structure formation, and provides an accurate description for

the growth of overdense inhomogeneities.

During this epoch of complete ionization, electrons and ions are constantly interacting

with one another, and attempting to form neutral atoms. There are two processes that

impede the formation of neutral atoms. The first, which delays the onset of neutral atom

formation (known as recombination), is that the baryon-to-photon ratio, r7, is so low. Even

though the photon temperature is significantly below the ionization energy of a neutral

atom (T, < 13.6 eV), the number of photons per baryon is very great, and their energy

follows a Poisson distribution. As a result, there are still enough photons of sufficient

energy to keep the universe 100 per cent ionized even when the average temperature of the

universe is significantly below the typical atomic ionization energy. The second process

that is responsible for impeding the formation of a neutral, transparent universe is the

fact that each Lyman-series photon (transition to the ground state of hydrogen) emitted

by a recombining atom will encounter and reionize another neutral atom. If, however, the

emitted photon has enough time to redshift sufficiently that it cannot reionize another

atom, the universe will net one neutral atom. Also, a rare two-photon emission process

will allow an atom to recombine without emitting a Lyman-series photon. This process of

recombining the majority of the atoms in the universe takes about 105 years to complete





[5], and drops the ionization fraction (Xe) of the universe from Xe 1 to Xe 10-4 [24].

After this point, ions and electrons can no longer efficiently find one another, and the

process of recombination freezes out. As the cosmic background of photons is no longer

constantly scattering off of electrons, it freely streams from the epoch of recombination

until the present day, making the epoch of recombination synonomous with the surface of

last scattering.

1.5 Nonlinear Evolution of Perturbations

Once recombination has occurred, the universe is in a matter-dominated, expanding

state, full of small density inhomogeneities on all scales. It is in this post-recombination

universe that large-scale structure formation begins to occur. The density inhomogeneities,

initially, have an amplitude of ~ 2 x 10-5 pm, with the fluctuations having a Gaussian

distribution. As the fluctuations evolve according to linear perturbation theory (with

overdensities growing according to the M6szaros effect) initially, and as gravitational

perturbations continue to grow, gravitational collapse goes nonlinear, causing a rapid

acceleration in structure formation. The overdense regions on small scales go nonlinear

first, as they enter the horizon (and thus become causally connected) first. The perturbations,

on the other hand, are across all scales, and have a roughly scale-invariant spectrum

(where the power spectrum, P(k), scales as P(k) oc k"), where the spectral index n 1.

The structure which arises from this follows scaling solutions, as described in Fry 1984 [25]

and Schaeffer 1984 [26], for example.

The result of all of this is that an initially smooth universe with only very slight

perturbations in energy density becomes a complex web of structure, with substantial

power on both small (i.e., galactic) and large (i.e., supercluster) scales. (For a very

interesting comparison of numerical simulations of structure formation through the

nonlinear regime up to the present day, the reader is referred to O'Shea et al. 2005 [27].)

While this large-scale structure forms, the universe continues to expand and cool, dropping





from a temperature of T 3000 K at recombination to a temperature of T 2.725 K at

present.

The various epochs of the universe, home to the onset of extremely interesting

physics, are tracked most easily by redshift, z, defined by

a- 1 + z, (1-9)
a(t)

where a(t) is the scale factor of the universe at a given time, and ao is the scale factor at

present. When gravitational collapse occurs to a sufficient extent on small scales, the mass

collected in a small area of space becomes large enough to ignite nuclear fusion. This is

the epoch at which the first stars form.

Exactly at what epoch star formation begins is very important for understanding the

evolution of matter and structure in our universe. A signature of the formation of the first

stars would be a surefire signature of nonlinear collapse. The transition from a smooth,

linear universe (such as the universe at the time of recombination) to a highly complex,

nonlinear one (observed today) is not yet well understood. Recently, many have discussed

the possibility that the gravitational energy bound in nonlinear inhomogeneities could

back-react, and significantly impact the expansion rate [28-30]. It appears that the impact

on the expansion rate is insignificant, however [31-34]. This physical process and its effects

on the universe are discussed in great detail in Chapter 4.

The densest regions of nonlinear structure become home to the first stars, as

illustrated in adaptive mesh refinement simulations [35]. The data from the WMAP

satellite indicate that the optical depth of the universe, r, is quite large [5]. From this

information, it appears that the first stars turned on very early, as the presence of a large

number of stars will reionize the neutral gas that formed during recombination. From

the optical depth, which is measured to be 7 = 0.17 0.04, it appears that reionization

occurs at roughly 11 < z < 30. However, the observation of a Gunn-Peterson trough [36]

in quasar spectra around z 6 [37] indicates that reionization is not complete until that





epoch. Seemingly bizarre solutions, such as a double epoch of reionization [38], have been

proposed to remedy this situation. Future release of data from satellites exploring the

cosmic microwave background may yield lower values of r, which would be consistent with

a more simplistic explanation of gradual reionization.

Once the first complex nonlinear structures form, they continue to evolve, with the

densest regions attracting the most matter and forming the most massive structures.

Galaxies grow through both monolithic collapse and a series of hierarchical mergers, and

via further gravitational collapse on larger scales, the first clusters of galaxies will form as

well. The types of objects which can be observed at early times are very bright galaxies

(in the optical and infrared) and quasars (primarily in the radio), as well as intervening

objects along the line of sight (through absorption and the Lyman-a forest). For a flat

universe that contained a critical density in matter, the expansion rate would continue to

decrease as the matter density diluted, following the Hubble law of equation (1-10),


H2- 8 p, (1-10)


where Pm is again the matter density (equal to the critical density) and H is the

Hubble expansion parameter. However, the universe's expansion rate, as inferred from

a combination of many sources of data (see Chapter 5 and references therein) is consistent

with about 30 per cent of the energy density in matter and about 70 per cent in some type

of vacuum energy. The expansion law, then, appears to obey equation 1-11


H2 2 8G- (Pm+ + P) (1-11)
a 3

where PA is the energy density in vacuum energy, and the sum of matter density and

vacuum energy density is equal to the critical density.

The data from type Ia supernovae have been used to illustrate and support the fact

that the picture of the universe is inconsistent without a vacuum energy term in the

equation for the Hubble law [39]. However, due to systematic errors inherent in any single





observational method, it is vital to collect data from a large number of methods. Chapter

5 investigates the possibility of using a new method (first detailed in Melnick, Terlevich

and Terlevich 2000 [40] and first attempted in Siegel et al. 2005 [41]) to measure the

cosmological parameters of matter and vacuum energy density in the universe. As with

any type of distance indicator, the method of Chapter 5, to use Lyman-break galaxies as

a distance indicator, is subject to many sources of error, both random and systematic.

These errors are detailed in Appendix A.

The data sets available are now sufficient to paint a coherent picture of the universe

and its energy contents very well on the largest scales, and relatively well on even small

scales [12]. There are many interesting problems and phenomena in the universe that

are hitherto unexplained, yet physics of the answers may lie in something as simple as

departures from the ideal model. The remainder of this work details some instances

where cosmological inhomogeneities, whether at early times or late times, on large or

small scales, may play a vital role in understanding the universe. Finally, Chapter 6

will summarize the major results of Chapters 2 through 5, and will point towards future

avenues of investigation, such as determining the fate of the universe.





CHAPTER 2
THE GRAVITATIONAL WAVE BACKGROUND

Inflationary cosmology predicts a low-amplitude graviton background across a wide

range of frequencies. This chapter shows that if one or more extra dimensions exist, the

graviton background may have a thermal spectrum instead, dependent on the fundamental

scale of the extra dimensions. The energy density is shown to be significant enough that

it can affect nucleosynthesis in a substantial way. The possibility of direct detection of

a thermal graviton background using the 21-cm hydrogen line is discussed. Alternative

explanations for the creation of a thermal graviton background are also examined.

2.1 Primordial Gravitational Waves

One of the most powerful windows into the early universe are backgrounds of particles

whose interactions have frozen-out. The primordial photon background, the primordial

baryon background and the primordial neutrino background are all examples of particles

that were once in thermal equilibrium. At various times during the history of the universe,

the interaction rate of the species in question dropped below the Hubble expansion rate

of the universe, causing the species in question to freeze-out. The primordial photon

background is observed as the cosmic microwave background (CMB), the baryon

background is observed as stars, galaxies, and other normal matter, and the neutrino

background, although not yet observed, is a standard component of big bang cosmology.

In addition to these backgrounds, a primordial background of gravitons (or, equivalently,

gravitational waves) is expected to exist as well, although it, too, has yet to be detected.

The frequency spectrum and amplitude of this background have the potential to convey

much information about the early universe. This chapter focuses on using the cosmic

gravitational wave background (CGWB) as a probe of extra dimensions.

The success of the inflationary paradigm [3] in resolving many problems associated

with the standard big-bang picture [42] has led to its general acceptance. Inflationary

big bang cosmology predicts a stochastic background of gravitational waves across all

frequencies [10], [11]. The amplitude of this background is dependent upon the specific





model of inflation, but the fractional energy density in a stochastic CGWB is constrained

[12] to be

g, < 0(10-10). (2-1)

In inflationary cosmology, the predicted CGWB, unlike the CMB and the neutrino

background, is non-thermal. Gravitational interactions are not strong enough to produce

a thermal CGWB at temperatures below the Planck scale (mpi & 1.22 x 1019 GeV).

As the existing particles in the universe leave the horizon during inflation, the only

major contributions to the energy density will be those particles created during or after

reheating, following the end of inflation. Unless the reheat temperature (TRH) is greater

than mpl, gravitational interactions will be too weak to create a thermal CGWB. The

measurement of the magnitude of the primordial anisotropies from missions such as

COBE/DMR [43] and WMAP [5] provides an upper limit to the energy scale at which

inflation occurs [44]. From this and standard cosmological arguments [45], an upper limit

on TRH can be derived to be


TRH ~ 6.7 x 1018 ()1/4 i GeV, (2-2)
\ )

where g* is the number of relativistic degrees of freedom at TRH, tpl is the Planck time,

and t4 is the lifetime of the inflation. A stronger upper limit on TRH (- 108 1010 GeV)

can be obtained from nucleosynthesis [46] if supersymmetry is assumed. In all reasonable

cases, however, TRH << mpl, indicating that the CGWB is non-thermal in inflationary

cosmology.

2.2 Extra Dimensions

If the universe contains extra dimensions, however, predictions about the shape

and amplitude of the CGWB may change drastically. Cosmologies involving extra

dimensions have been well-motivated since Kaluza [47] and Klein [48] showed that classical

electromagnetism and general relativity could be unified in a 5-dimensional framework.

More modern scenarios involving extra dimensions are being explored in particle physics,





with most models possessing either a large volume [49, 50] or a large curvature [51, 52].

Any spatial dimensions which exist beyond the standard three must be of a sufficiently

small scale that they do not conflict with gravitational experiments. The 3+1 dimensional

gravitational force law has been verified down to scales of 0.22 mm [53]. Thus, if extra

dimensions do exist, they must be smaller than this length scale. Although there exist

many different types of models containing extra dimensions, there are some general

features and signals common to all of them.

In the presence of 6 extra spatial dimensions, the 3+6+1-dimensional action for

gravity can be written as

S = f d4x f d l'y-1G +N /

G'N = GN 2, (2-3)


where g is the 4-dimensional metric, GN is Newton's constant, g', G', and R' denote the

higher-dimensional counterparts of the metric, Newton's constant, and the Ricci scalar,

respectively, and mD is the fundamental scale of the higher-dimensional theory. In 3+6

spatial dimensions, the strength of the gravitational interactions scale as (T'/mo)(1/2)

If 6 = 0, then mD = mpl, and standard 4-dimensional gravity is recovered.

When energies in the universe are higher than the fundamental scale mD, the

gravitational coupling strength increases significantly, as the gravitational field spreads out

into the full spatial volume. Instead of freezing out at r O(mpj), as in 3+1 dimensions,

gravitational interactions freeze-out at O(mD) [49]. (mD can be much smaller than mpl,

and may be as small as TeV-scale in some models.) If the gravitational interactions

become strong at an energy scale below the reheat temperature (mD < TRH), gravitons

will have the opportunity to thermalize, creating a thermal CGWB. Figure 2-1 illustrates

the available parameter space for the creation of a thermal CGWB in the case of large

extra dimensions, following the formalism in Giudice, Rattazzi and Wells 1999 [54].








and fractional energy density (f~) of a thermal CGWB are

7r2 (3.91 4/13
P3= (TcM)4, (2-4)

S= P 3.1 x 10-4 (9)-4/3, (2-5)
Pc
where pc is the critical energy density today, TCMB is the present temperature of the

CMB, and g, is the number of relativistic degrees of freedom at the scale of mD. 9+

is dependent on the particle content of the universe, i.e. whether (and at what scale)

the universe is supersymmetric, has a KK tower, etc. Other quantities, such as the

temperature (T), peak frequency (v), number density (n), and entropy density (s) of the

thermal CGWB can be derived from the CMB if g, is known, as

(3.91\ (3.91
1T-g = H -- Sg = S.MB

8* M 9*
g =TCMB Sg VSCMB \ g ,


These quantities are not dependent on the number of extra dimensions, as the large

discrepancy in size between the three large spatial dimensions and the 6 extra dimensions

suppresses those corrections by at least a factor of r 10-29. As an example, if mD is just

barely above the scale of the standard model, then g, = 106.75. The thermal CGWB then

has a temperature of 0.905 Kelvin, a peak frequency of 19 GHz, and a fractional energy

density Q, 6.1 x 10-7

2.4 Detection of Extra Dimensions

Although the fractional graviton energy density is expected to be small today, it may

be detectable either indirectly or directly. Nucleosynthesis provides an indirect testing

ground for a thermal CGWB. Standard big-bang nucleosynthesis predicts a helium-4

abundance of Yp = 0.2481 0.0004 [55]. With a thermal CGWB included, the expansion

rate of the universe is slightly increased, causing neutron-proton interconversion to

freeze-out slightly earlier. A thermal CGWB can be effectively parameterized as neutrinos,





as they serve the same function at that epoch in the universe (as non-collisional radiation).

The effective number of neutrino species is increased by N,-eff 27.1 (g,)-4/3, or 0.054

(for g, = 106.75). This would yield a new prediction of Yp = 0.2489 0.0004 for helium-4.

Although observations are not yet able to discriminate between these two values, the

constraints are tightening with the advent of recent data [56]. An increase in the precision

of various measurements, along with an improvement in the systematic uncertainties, may

allow for the indirect detection of a thermal CGWB.

Direct detection of a thermal CGWB is much more challenging, but would provide

quite strong evidence for its existence. Conventional gravitational-wave detectors include

cryogenic resonant detectors [57], which have evolved from the bars of Weber [58], doppler

spacecraft tracking, and laser interferometers [59]. The maximum frequency that these

detectors can probe lies in the kHz regime, whereas a thermal CGWB requires GHz-range

detectors. An interesting possibility for detection may lie in the broadening of quantum

emission lines due to a thermal CGWB. Individual photons experience a frequency

shift due to gravitational waves [60]. For a large sample of radio-frequency photons in a

gravitational wave background, the observed line width (W) will broaden by

106.75)I/3
AW ho 10-, (2-7)
vto 9

where to is the present age of the universe, v is the peak frequency of the thermal CGWB

and ho is the metric perturbation today due to the thermal CGWB [61]. As 0(10-31) is a

very small broadening, a radio line with a narrow natural width is the preferred candidate

to observe this effect. One possibility for this type of observation is the 21-cm emission

line of atomic hydrogen. So long as the emitting atoms and the detectors are sufficiently

cooled, broadening due to thermal noise will be suppressed below AW. Because the

lifetime (1/F) of the excited state of hydrogen is large (' 107 yr) and the frequency of the





emitted light (v,) is high (- 109 Hz), the natural width (W) is among the smallest known

F 2.869 x 10-15 1s-
W 2.02 x 10-24 (2-8)
v 1.42040575179 x 109 s-1

The width of the 21-cm line is regrettably seven orders of magnitude larger than the

expected broadening due to a thermal CGWB. Extraordinarily accurate measurements

would need to be taken for direct detection of this background. Additionally, temperatures

of the atoms and detectors would need to be cryogenically cooled to n 10-18 Kelvin to

suppress thermal noise below AW. This last criterion is far beyond the reach of current

technology, and either a major advance or experimental innovation would be required to

measure the desired effect using this technique.

2.5 Alternative Thermalization Mechanisms

Extra dimensions are not the only possible explanation for the existence of a thermal

CGWB. Currently, there are three known alternative explanations that would also create

a thermal CGWB. They are as follows: there was no inflation, there was a spectrum

of low-mass primordial black holes that have decayed by the present epoch, or the

gravitational constant is time-varying (the Dirac hypothesis). Each alternative is shown

below to face difficulties that may make extra dimensions an attractive explanation for the

creation of a thermal CGWB.

The predictions of inflation are numerous [44], and many have been successfully

confirmed by WMAP [5]. The major successes of inflation include providing explanations

for the observed homogeneity, isotropy, flatness, absence of magnetic monopoles, and

origin of anisotropies in the universe. Additionally, confirmed predictions include

a scale-invariant matter power spectrum, an Q = 1 universe, and the spectrum of

CMB anisotropies. To explain a thermal CGWB by eliminating inflation would require

alternative explanations for each of the predictions above. Although alternative theories

have been proposed, as in Hollands and Wald 2002 [62], they have been shown to face





significant difficulties [63]. The successes of inflation appear to suggest that it may likely

provide an accurate description of the early universe.

Primordial black holes with masses less than 1015 g would have decayed by today,

producing thermal photons, gravitons, and other forms of radiation. Density fluctuations

in the early universe, in order to produce a large mass fraction of low-mass primordial

black holes, and not to produce too large of a mass fraction of high-mass ones, favor

a spectral index n that is less than or equal to 2/3 [64]. Accepting the observed

scale-invariant (n 1) spectrum of density fluctuations [65] may disfavor primordial

black holes as a reasonable candidate for creating a thermal CGWB.

The Dirac hypothesis states that the difference in magnitude between the gravitational

and electromagnetic coupling strengths arises due to time evolution of the couplings

[66]. If true, gravitational coupling would have been stronger in the early universe.

At temperatures well below the Planck scale, gravity would have been unified with

the other forces, creating a thermal CGWB at that epoch. However, this hypothesis

produces consequences for cosmological models that are difficult to reconcile [67], and

any time variation is severely constrained by geophysical and astronomical observations

[68]. The acceptable limits for variation are small enough that they cannot increase

coupling sufficiently to generate a thermal CGWB subsequent to the end of inflation. The

difficulties faced by each of these alternative explanations points towards extra dimensions

as perhaps the leading candidate for the creation of a thermal CGWB.

2.6 Problems of Extra Dimensions

There exist two major obstacles to the construction of a more complete phenomenological

model containing extra dimensions with mD < TRH. The first of these is the moduli

problem [6'] String moduli interactions with standard model fields are highly suppressed,

leading to a long lifetime of the string moduli. String moduli decay, however, must be

consistent with astrophysical constraints [70]. To accomplish this, string moduli need

either a small production amplitude or very specific decay channels, which both require





fine-tuning. The second problem is the overproduction of long-wavelength tensor modes

from inflation [71, 72]. While the short-wavelength modes (the modes inside the horizon

when gravitational interactions freeze-out) will thermalize, gravitational waves of longer

wavelengths will be unaffected. As the scale of inflation must be above mD, the amplitude

of these waves is expected to be large. This would leave an unacceptable imprint in the

CMB. Both problems arise from the fact that at energies above mD, macroscopic gravity

breaks down [73]. Although these problems may not be resolved until a quantum theory

of gravity is realized, they do not change the fact that a thermal CGWB would arise from

extra dimensions with mD < TRH.

Furthermore, there is a more fundamental question concerning the nature of extra

dimensions. The three observed spatial dimensions are quite large, on the order of

_ 1028 cm. On the other hand, any extra spatial dimensions must be, at most, of a length

scale less than 0.22 mm. It is very difficult to construct a compelling theory that naturally

produces three large spatial dimensions and forces the rest to be small. A possible solution

to this puzzle may lie in the work of Chodos and Detweiler 1980 [74], where it was shown

that a universe with four spatial dimensions of initially comparable size may naturally

evolve to a state with three large, expanding dimensions and one small, contracting one.

2.7 Summary

This chapter has attempted to show that extra dimensions may be responsible for the

production of a thermal gravitational wave background. A thermal CGWB, as opposed

to the stochastic CGWB of standard inflationary cosmology, is a prediction of extra

dimensions with a scale below the reheat temperature. The detection of a thermal CGWB,

although challenging at present, would provide strong evidence for the existence of extra

dimensions. The detected absence of a thermal CGWB would conversely disfavor the

existence of extra dimensions up to the energy scale of the reheat temperature.





CHAPTER 3
STRUCTURE FORMATION CREATES MAGNETIC FIELDS

This chapter examines the generation of seed magnetic fields on all scales due to the

growth of cosmological perturbations. In the radiation era, local differences in the ion

and electron density and velocity fields are induced by momentum transfer from photons.

The currents which flow due to the relative motion of these fluids lead to the generation

of magnetic fields. Magnetic fields are created on all cosmological scales, peaking at a

magnitude of 0(10-23 Gauss) at the epoch of recombination. Magnetic fields generated

in this manner provide a promising candidate for the seeds of magnetic fields presently

observed on galactic and extra-galactic scales.

3.1 Introduction

The presence of magnetic fields on galactic and extragalactic scales is a major

unsolved problem in modern astrophysics. Although the observational evidence for

magnetic fields in large-scale structures is overwhelming, there is no consensus as to their

origins. The standard paradigm for the creation of these fields is the dynamo mechanism,

in which an initial, small seed field is amplified by turbulence and/or differential rotation

to account for the fields observed today.

In principle, once a seed field is in place, it should be possible to follow its evolution

and amplification from the collapse of structure and the effects of any relevant dynamos.

In this chapter, a new mechanism for the generation of seed fields is put forward. It

is argued that cosmological perturbation theory in the radiation era produces charge

separations and currents on all scales, both of which contribute to magnetic fields.

These seed fields persist until the onset of gravitational collapse, at which point field

amplification and dynamo processes can magnify such seeds, possibly to the O(pG) scales

observed today.

This chapter illustrates that the generation of magnetic fields in this manner is

a necessary consequence of structure formation. The magnitude of these seed fields is

calculated, and it is shown that these seed fields may be sufficiently strong to account for





all of the observed magnetic fields in large-scale structures. The layout of this chapter is

as follows: the next section gives an overview of the observational evidence for magnetic

fields along with a brief theoretical picture of their generation. After that, there is an

explanation of the novel idea that the early stages of structure formation in a perturbed

universe generate magnetic fields. Subsequently, a detailed treatment of cosmological

perturbations is presented, with a specific view towards the creation and evolution of

local charge separations and currents. The magnitudes of the seed magnetic fields which

arise via this mechanism as a function of scale and epoch are then calculated. Finally,

the results of this mechanism are compared with competing theories. Also included is a

discussion of avenues for future investigation of this topic, including possible observational

signatures which would arise as predictions of this mechanism.

3.2 Magnetic Fields: Background

In all gravitationally bound or collapsing structures in which the appropriate

observations are made, magnetic fields with strength pG are seen [75]. The four

major methods used to study astrophysical magnetic fields are synchrotron radiation,

Faraday rotation, Zeeman splitting, and polarization of starlight. These observational

techniques are detailed in depth in Ruzmaikin, Sokolov and Shukurov 1988 [76], with

Faraday rotation often proving the most fruitful of the above methods.

Magnetic fields have been found in many different types of galaxies, in rich clusters,

and in galaxies at high redshifts. Spiral galaxies, including our own, appear to have

relatively large magnetic fields of 0(10 pG) on the scale of the galaxy [77], with some

(such as M82) containing anomalously strong fields up to -_ 50 pG [78]. Elliptical and

irregular galaxies possess strong evidence for magnetic fields (of order -~ pG) as well

[79], although they are much more difficult to observe due to the paucity of free electrons

in these classes of galaxies. Coherence scales for magnetic fields in these galaxies, as

opposed to spirals, are much smaller than the scale of the galaxy. Furthermore, galaxies at

moderate (z 0.4) and high redshifts (z > 2) have been observed to require significant





(- MG) magnetic fields to explain their observed Faraday rotations [80, 81]. Magnetic

fields are also observed in structures larger than individual galaxies. The three main

types of galaxy clusters are those with cooling flows, those with radio-halos, and those

devoid of both. Galaxy clusters with cooling flows are observed to have fields of 0.2to 3 G

[82], the Coma cluster (a prime example of a radio-halo cluster) is observed to have a

field strength r 2.5 pG [83], while clusters selected to have neither cooling flows nor

radio halos still exhibit indications of strong (0.1 1 pG) fields [84]. There even exists

evidence for magnetic fields on extracluster scales. An excess of Faraday rotation is

observed for galaxies lying along the filament between the Coma cluster and the cluster

Abell 1367, consistent with an intercluster magnetic field of 0.2 0.6 pG [85]. On the

largest cosmological scales, there exist only upper limits on magnetic fields, arising from

observations of the cosmic microwave background [86] and from nucleosynthesis [87],

setting limits that on scales > 10 Mpc, field strengths are < 10-s G.

Observational evidence for magnetic fields is found in galaxies of all types and in

galaxy clusters, both locally and at high redshifts, wherever the appropriate observations

can be made. A review of observational results can be found in Valle6 1997 [88]. The

theoretical picture of the creation of these fields, however, is incomplete. Fields of strength

SpiG can be explained by the magnification of an initial, small seed field on galactic

(or larger) scales by the dynamo mechanism [89-91]. A protogalaxy (or protocluster)
containing a magnetic field can have its field strength increased by many orders of

magnitude through gravitational collapse [92, 93], and can then be further amplified via

various dynamos. Dynamos which can amplify a small seed field into the large fields

observed today involve helical turbulence (a) and/or differential rotation (w). Various

types of these dynamos include the mean-field dynamo [76, 94, 95], the fluctuation

dynamo [79, 96], and merger-driven dynamos [97], among others. However, the dynamo

mechanism does not explain the origin of such seed fields.





While the initial seeds that grow into magnetic fields are anticipated to be small,

they must still come from somewhere [98], and their existence is not explained by the

dynamo mechanism alone. There are many mechanisms that can produce small-strength

magnetic fields on astrophysically interesting scales, either through astrophysical or exotic

processes (see Widrow 2002 [75] for a detailed review). Exotic processes generally rely on

new physics in the early universe, such as a first-order QCD phase transition [99, 100],

a first order electroweak phase transition [101, 102], broken conformal invariance during

inflation [103, 104], specific inflation potentials [105], or the presence of charged scalars

during inflation [106-108]. Astrophysical mechanisms, in contrast, are generally better

grounded in known physics, although they have difficulty generating sufficiently strong

fields on sufficiently large scales. The difference in mobility between electrons and ions can

lead to seed magnetic fields from radiation-era vorticity [109, 110], from vorticity due to

gas-dynamics in ionized plasma [111-115], from stars [116], or from active galactic nuclei

[117]. Although there are many candidates for producing the seed magnetic fields required

by the dynamo mechanism, none has emerged as a definitive solution to the puzzle of

explaining their origins.

The novel mechanism proposed in this chapter is that seed magnetic fields are

generated by the scattering of photons with charged particles during the radiation era.

Unlike the mechanism of [109, 110], which is disfavored [118] due to its requirement

of substantial primordial vorticity (although see [119, 120] for an argument that some

vorticity is necessary), the fields of interest here are generated by the earliest stages of

structure formation, requiring no new physics. Ions (henceforth taken to be protons,

for simplicity) and electrons are treated as separate fluids, with opposite charges but

significantly different masses. The mass-weighted sums of their density and velocity fields

will determine the evolution of baryons in the universe, and should agree with previous

treatments, such as Ma and Bertschinger 1995 [22]. The difference of the ion and electron

density and velocity fields, however, will provide a measure of local charge separation and





of local current density, both of which contribute to magnetic fields. Since cosmological

perturbations, which serve as seeds for structure formation, exist on all scales, it is

expected that seed magnetic fields will be generated on all scales by this mechanism. The

remainder of this chapter focuses on calculating the magnitude of the magnetic fields

generated by this process and discussing their cosmological ramifications.

3.3 Cosmological Perturbations

Although the early universe is isotropic and homogeneous to two parts in 10-5 [2],

it is these small density inhomogeneities, predicted by inflation to occur on all scales [8],

which lead to all of the structure observed in the universe today. As it is the early epoch

of structure formation that is of interest for the creation of magnetic fields, this chapter

calculates the evolution of inhomogeneities in the linear regime of structure formation.

The most sophisticated treatment of cosmological perturbations in the linear regime to

date is that of Ma and Bertschinger 1995 [22], which provides evolution equations for

an inhomogeneous universe containing a cosmological constant, dark matter, baryons,

photons, and neutrinos. This section extends their treatment to encompass separate

proton and electron components. The mass-weighted sum of protons and electrons will

recover the baryon component, whereas the difference of the density fields is representative

of a charge separation, and the difference of the velocity fields is that of a net current.

The dynamics of any cosmological fluid can be obtained, in general, from the linear

Einstein equations (see Peebles and Yu 1970 [121], Silk and Wilkson 1980 [122], and

Wilson and Silk 1981 [123] for earlier treatments). Although the choice of gauge does not

impact the results, the Conformal Newtonian gauge leads to the most straightforward

calculations. The metric is given by

ds2 = a2(rT)[-(1 + 2)dr2 + (1 2 i)dxidxi], (3-1)





where V) 0 when gravitational fields are weak. The linear Einstein equations are then as

follows:


k2+ 3a( a) = 47rGa2 To,


a
a a 2 k 2 4 2
+(0 + 2)+ (2 -2)+ ( ) = GGa2 6V
a a a 3 3
k2(- ) = 127rGa2(p + P)a, (3-2)


where a is the shear term, which is negligible for non-relativistic matter (but important

for photons and neutrinos). For a cosmological fluid that is either uncoupled to the

other fluids or mass-averaged among uncoupled fluids in the early universe, the following

evolution equations hold:


S= -(1+w)(- 3) -3a(C, -w)6,
a
a w c2
0 = (1- 3w)O + 1c k26 k +k24 (3-3)
a 1l+w 1+w

where 6 is defined as the local density relative to the spatial average (6 Jp/p), 0 ik'v

where v is the local peculiar velocity, and cs is the sound speed of the fluid.

For individual components with inter-component interactions, equation (3-3) must

be modified to include these interactions. Examples of such interactions include the

momentum transfer between photons and charged particles and the Coulomb interaction

between protons and electrons. For protons, electrons, and cold dark matter (CDM), an

equation of state w = 0 is assumed, and for radiation and neutrinos, w = 1. The master

equations for each component of interest is computed explicitly in subsections 3.3.1-3.3.5.

3.3.1 Cold Dark Matter

As the cold dark matter component (denoted by the subscript c) is collisionless and

pressureless, it can be simply read off from equation (3-3) that the equations which govern





its evolution are


6c = -0C + 3q,

S= -0o + k24. (3-4)
a

Any cold (i.e., nonrelativistic), collisionless component will behave according to the

dynamics given by equation (3-4).

3.3.2 Light Neutrinos

For massless (or nearly massless) particles, pressure is non-negligible. Additionally,

the shear term (a) may be important as well. The only accurate way to compute

the evolution of such a component of the universe is by integration of the Boltzmann

Equation, which is given for light neutrinos (denoted by subscript v) by


+ ik(k ), = 4[- ik(k. -)], (3-5)
87

in Fourier space.

The approximation that neutrinos are massless and uncoupled is very good from

an age of the universe of approximately t 1 s until the epoch of recombination. The

evolution equations for light neutrinos are then


4 0, + 4 ,
3

4
S= 1[l() -- (+ 1)u(l+)], (3-6)
2+ (21+16)

where cr, is related to F, by 2a, = FT2, and the index 1 governs the final equation for

I > 2. Tlv is defined by the expansion of the perturbations in the distribution function, T,,
00
=- (-i)(21 + 1),T(k,r)Pi(k h), (3-7)
l=0





where Pl(k n) are the Legendre polynomials. Equations (3-5-3-7) are valid for any

non-collisional species behaving as radiation.
3.3.3 Photons

Photons (denoted by subscript 7), although similar to light neutrinos, evolve
differently due to their large coupling to charged particles. Thomson scattering describes

the interactions of photons with electrons, where the differential cross-section is given by
the formula
du 3aT
S- (1 + Cos2 ), (3-8)
dQ 167r
where UT is the Thomson cross-section [124]. Photons also scatter with protons, but with
a cross-section suppressed by a factor of m /m2 (the mass-squared ratio of electrons to

protons).
~,, which is the polarization-summed phase-space distribution for photons, is the

same as the distribution function for neutrinos (see equation 3-7). Photons also contain a
non-zero difference between the two linear polarization components, denoted by 9,. The
linearized collision operators for Thomson scattering [22, 125-127] yield the set of master

equations for photons,

4
^ = 3 + 4,

0 k2 7 + k2V + aneT (Ob 0),
8 3 9
J2 = -15 5 k gan(eTra'

10
k
i = 21+ 1 [,(1+1) (I + 1)F,(1)] aneTJyl,

-yr = 1[m Y(m-1) (m + 1)97(m+)]
2m + 1
+anTUT -- ff g^YM (3-9)
|11U 53





where F~o = 6,, F~, = 40,/3k, JTF2 = 2-y,, the indices 1 and m are valid for 1 > 3 and

m > 0, and the subscript b denotes the baryonic component, which is the mass-weighted

sum of the electrons and protons. Electron-photon scattering is so dominant over

proton-photon scattering as to render the latter negligible, but the electron-proton

coupling (via electromagentism) is sufficiently strong that, to leading order, those two

fluids move in kinetic equilibrium.

3.3.4 Baryons

The net behavior of the baryonic component can be derived from combining the

mass-weighted contributions of the proton fluid and the electron fluid. Both protons

and electrons contain all of the terms present in the CDM equations (see section 3.3.1),

but additionally contain important sound-speed terms and terms arising from Thomson

scattering. Additionally, the Coulomb interaction enters through the contribution of

the electric field to the To components of the stress-energy tensor. The coupling of the

Coulomb interaction to density inhomogeneities can be calculated through a combination

of the electromagnetic Poisson equation,


V24 = -V = -47rpc, (3-10)


where Pc is the electric charge density, and the Euler equation,

l(a + -1(v- V) = -V q-l + C, (3-11)
a at a a ma

with q/m as the charge-to-mass ratio of the particle in question and C the collision

operator. The Coulomb contribution appears as 47re(np ne)qi/mi in the evolution

equation for Oi, where i denotes a species of particle with a mass mi and charge qi. The





evolution equations are therefore


6e = -0e + 3,

e = ae + C k ,e + k2
a
47re2
+ Fe(0 e) -- (n ne), (3-12)
me

for electrons (denoted by subscript e), and


Jp = -Op + 30,

O, = -aOP,+c k 2 +k24
a
47re2
+ FP,(, O) + (n ne), (3-13)
Tmp

for protons, denoted by subscript p, where the damping coefficients for electrons (Fe) and

protons (Fp) are given by


Fe 44p1rnrTa


S4ptnfTa ineh 2e 1.6 x 10-1i0n (3-14)
3pp \mp}

Note the difference in the sign of the final terms in the equations for 9e and Op, which will

prove important in the analysis below.

From equations (3-12) and (3-13) for electrons and protons the dominant gravitational

and electromagnetic combinations can be constructed separately. The remainder of this

subsection details the evolution of baryons in the linear regime of a perturbed universe.

Baryonic matter can be treated as the combination of electrons and protons, thus the mass

weighted sum of proton and electron overdensities gives rise to the baryonic perturbations,

m mp me mT
6b me + -- Ob =- -Oe + M Op. (3-15)
mb mb Mb Mb

By substituting the expressions for equations (3-12) and (3-13) into equation (3-15), a set

of equations for the evolution of baryonic matter is obtained. So long as approximations





such as np ne and mb mp > me hold, quantities which are obviously small compared

to the others (such as n, n,) can be neglected. The evolution equations are then


S= -b0 + 3,

Ob Ob + c k26 + k2k + r(0 b), (3-16)
a

where Fb P em/mb. The baryon-photon coupling term in equation (3-16) is driven

by the electron-photon interaction. To the extent that electrons and protons move

together (the tight-coupling approximation), the baryonic fluid is dragged by the

electron-photon interactions, as has been shown by Harrison 1970 [109] and subsequent

authors. Equation (3-16) is identical to the equations for baryon evolution derived in Ma

and Bertschinger 1995 [22].

3.3.5 Charge Separations

From equations (3-12) and (3-13), a difference component as well as a sum

component can be obtained. As the limits on a net electric charge asymmetry in

the universe are very strict [17, 18, 128], any component arising from the differences

in densities and/or velocities of protons and electrons will not be strong enough to

significantly impact the evolution of the other species of particles in the universe, including

the baryon component.

The charge difference component (denoted by subscript q) is the difference between

the proton and electron components, such that 6q = 6p Se and Oq = 0p 0e. The

gravitational potential ought not to enter into these equations, due to the fact that gravity

acts equivalently on electrons and protons. However, velocities and number densities may

differ, n, n, -_ nJq. The master equations for the charge-asymmetric component are as





follows:


6q = q
a + 47nee2
9, = -9 ^ + c k,
a me
Fre( 9b + q), (3-17)


where the approximations Fp, < Fe and mb mp have been utilized where applicable.

The term 4rrnee2jq/me in equation (3-17) arises from the Coulomb force acting on charged

particles, while the final term, F(07 Ob + Oq), arises from the difference in Thomson

scattering between protons and electrons. This final term is a source of charge separation

independent of and in addition to any initial charge asymmetry, and will create a local

charge asymmetry even when there is none initially. In the evaluation of equation (3-17),

the electromagnetic terms dominate the cosmological terms, such that an excellent

approximation in the pre-recombination universe is

4rne2
q6 = -,, ,O = e q6 Fe(0 Ob + Oq). (3-18)
me

For some purposes, it is useful to express the set of equations found in equation

(3-17) as a single ordinary differential equation. This can be accomplished by setting

Oq = -6, and again by neglecting the unimportant cosmological terms Oqa/a and c2k26,.

Many of the coefficients in equation (3-17) are functions of a, but the derivatives in

equation (3-17) are with respect to conformal time, r. A change of variables can be

performed, using the relation that

45h3c5 1/2
t = [3273G(kT)4

=Na2to,

N 72.2, (3-19)


in the radiation era, where to is the age of the universe today, to express all derivatives as

derivatives with respect to a, denoted by primes (instead of dots).





The evolution of Jq can be tracked by evolving equation (3-20) below,

S. r 1 16N2 rnr,0 e21
d6 + 2N e,0o ;g + a,
a 2 me a

= 4N2 rFo(0y Ob), (3-20)


where the subscript 0 denotes the present value of a quantity. This is simply the equation

of a damped harmonic oscillator, with coefficients that change slowly with time compared

to damping or oscillation times. The behavior can be characterized as overdamped at the

earliest times, critically damped when a z 3.9 x 10-15, and free at late times.

Of all the terms in equation (3-20), only 07, Ob, and 6q (and derivatives) are functions

of a; all other quantities are constant coefficents. Although there does not exist a simple

analytic form for (08 Ob) in general, at sufficiently early times there exists the simple

approximation

07 Ob 6.0 x 1019 k4 a5, (3-21)

valid when the following condition is met:


a < 10-5 for k < 0.1Mpc-1,

a 10-6 (1 ) for k > 0.1 Mpc-1.


Equation (3-21) is an approximation for a flat ACDM cosmology with cosmological

parameters Ho = 71 km s-1 Mpc- ,m = 0.27, Qb = 0.044, and a Helium-4 mass fraction

of Y = 0.248. These parameters are used in all subsequent analyses for the calculation of

cosmological quantities.

The approximation in equation (3-21) breaks down at sufficiently late times. When

this occurs, numerical methods must be used to obtain the quantity (0 Ob). The software

package COSMICS [129] is ideal for performing this computation, as it performs numerical

evolution of equations (3-4,3-6,3-9, and 3-16) concurrently. Computational results for

the quantities 07 and Ob are given by COSMICS, which are valid at all times in the linear





regime of structure formation. It is found that when the approximation in equation (3-21)

breaks down, the quantity (08 Ob) grows more slowly initially, and proceeds to oscillate

at a roughly constant amplitude at later times. These oscillations in the quantity (0, Ob)

are closely related to the acoustic oscillations between baryons and photons observed in

the cosmic microwave background [5].

Numerical integration of equation (3-20) can be accomplished in various ways, as

illustrated in Press et al. 1992 [130]. At sufficiently late times (when a > 3.9 x 10-15),

numerical results indicate that the quasi-equlibrium solution

S Tmb (Po' (0 Ob), (3-22)
37we2 Pb,o /

obtained by neglecting the first two terms in equation (3-20), is an excellent approximation.

With 0, and 0b given by COSMICS in units of Mpc-1, the prefactor in equation (3-22)

can be written as
(T m6b (Po) 1.64 x 10 -Mpc. (3-23)
3 7re2 \b,o/
The quantity 0q then follows directly from equation (3-17) to be

S TT b (P7,o 3-24
qo 3 7r e2 pb,o b) (3-24)

The solutions in equations (3-22) and (3-24) are valid until gravitational collapse

becomes nonlinear, which means that they are still valid at the epoch of recombination

(z 1089). The results of numerically integrating the equations for 6q and 0q on various
length scales up through recombination are presented in Figure 3-1.

It is worth pointing out that the results obtained in this section can be applied to

a situation where a net electric charge is present. In appendix B, the possibility of using

the evolution equations derived for 0q and 6q is applied to a universe with a broken U(1)

symmetry. The possibility exists that, under the proper circumstances, an initially charged

universe may become neutral simply due to the expansion dynamics.









-40
/

._ i / ,'


-45





S-50




-55





106 105 104 103
Z
z





Figure 3-1. 6q (solid lines) and Oq/H, where H is the Hubble parameter (dashed lines) as
a function of redshift (z). The lines shown are for comoving scales of (from
top to bottom) 10, 1, 0.1, and 0.01 Mpc-1. 6q rises as a a5 initially, then
ceases to grow when the scale of interest enters the horizon, and oscillates
at an amplitude which first continues to rise slowly, then falls, eventually
matching on to the equilibrium solution that 6q oc Ob. Oq can be obtained from
q6 through equation (3-17). This graph uses output from COSMICS, and as
such needs to be multiplied by the COBE normalization of Bunn and White
[2].





3.4 Magnetic Fields

With the results derived in Section 3 for Oq and 6q, values for the local current

densities and local charge separations can be obtained at any time in the pre-recombination

universe on all scales. Both q6 and Oq will contribute to magnetic fields, as currents create

magnetic fields directly, and the bulk motion of a region of net charge will also produce

a magnetic field. For each comoving distance scale (given by the value of k) and each

timeslice (determined by the scale factor a) of the universe, there will be a unique

magnetic field amplitude associated with that scale. This field may serve as the seed

for the large-scale magnetic fields observed today.

An expression for magnetic fields can be derived from the currents arising from

the relative motion of the protons and electrons in the universe. Magnetic fields can be

derived from Maxwell's equations

aE
V B = 0, V xB =47rJ+ (3-25)


with the current density J given by


J = n, e ne e v ne [6qVb (1 + b)q], (3-26)


where vq -= v ve, and the displacement current is neglected.

By taking the curl, a direct expression for magnetic fields as a function of a and k is

obtained as a convolution

S 47neo e f dk' k x k'
a2 l 2 (27)3 k [12k
+ 9q(k') b( k')]. (3-27)

While the magnetic field strength can, in principle, be obtained by solving equation

(3-27), it is more favorable to obtain the power spectrum of the magnetic field. The power

spectrum is obtained by examining the second moment of the magnetic field B(k), which




is

(Bi(k)B(2)} = (2)3D(k + k2) P PB(k),

P4 2 1 ( kS- k), (3-28)

where 6D is the Dirac delta function and PB(k) is the magnetic field power spectrum. Note
that the direction parallel to k does not contribute to magnetic fields, and therefore the
direction perpendicular to k is projected out in equation (3-28). The power spectrum,
PB(k), is then given by the expression


PB (k)= k2 sin2
a | 2 12 (27)3

S1
,x (1k Pq )(k b)P6b (k- k'I)

Ik k' 12
1V 12
; Po ( Ik) P)oqk(I k- k')
1k'

/ 2 -- -b
1 PObq(i q )P8bO, (k V')

2

PoB (k ')Pob, (i- k')J, (3-29)
\k k' 12

where the angle A is the angle between the vectors k and k'. The expression for power in
any two quantities, q and V), P(I(k), is generically defined by

((k1)(k2 )) = (27r)3JD( l + k2)PPO(k). (3-30)

The solutions obtained for Oq and 6q in equations (3-22) and (3-24) can be substituted
into the equation for the power spectrum, equation (3-29). By numerically integrating
the resulting expression, the spectral density can be obtained. The spectral density is





4lrk3PB(k)/(27r)3, and provides both a measure of the magnetic field strength on a given

scale (k) and a measure of the energy stored in magnetic fields.

The results for the spectral density of magnetic field energy on comoving scales

ranging from 10-3 Mpc-1 to 102 Mpc-1 at the epoch of recombination are shown in Figure

3-2. The peak of the spectral density corresponds to a typical magnetic field strength of

10-23 Gauss on comoving scales of 0.1 Mpc-1.

3.5 Discussion

The major result of this paper has been to illustrate that seed magnetic fields of

cosmologically interesting strengths and scales are necessarily generated by the same

processes that cause structure formation. As overdense regions in the early universe slowly

grow during the radiation era, photon interactions with both protons and electrons create

charge separations and current densities of small magnitudes, but on all scales. These

charge separations and currents grow in magnitude as the universe ages, causing magnetic

fields to grow as well. Magnetic power peaks at approximately the time of horizon

crossing, falling slowly after that. The net result is that, at the epoch of recombination

(and hence prior to significant field amplification due to gravitational collapse or dynamo

effects), seed magnetic fields of magnitude 0(10-23 G) are created by the simple dynamics

of charged particles.

The results of this paper are very accurate up through the epoch of recombination.

At this epoch, however, the universe transitions from a fully ionized state (where the

ionization fraction Xe 1) to a state where the ionization fraction is very small, Xe 10-4

[24]. While the photons are generally decoupled from the baryons at this point, the

free electrons continue to interact with the photons, due to the extraordinary number

of photons per free electron. In the absence of any interactions with photons, a charge

separation would evolve as

= 4nne2, = 3 H23 H2 K2 3q, (3-31)
me 2










10-41


10-42
7
10-43

10-44

10-45 \


0-46 \

S0-47 -\

10-4 -
10-48 y \ \ \ -

10-49

10-50 //

10-51
10-3 10-2 0.1 1 10 102
k (Mpc-1)





Figure 3-2. Spectral energy density of the magnetic field (B, in gauss) generated by
cosmological perturbations on a given comoving scale (k, in Mpc-1) at the
epoch of recombination (z 1089). The line illustrates 47rk3PB(k)/(27r)3,
which is the spectral density in units of G2; the peak value is a magnetic field
strength B ~ 10-23 G. The upper lines are the simple power spectrum, PB(k).


I I IIII"I I 111II II illi I I" '' ""I 1111111
c,





where K2 is the ratio of the electric to gravitational forces,


K -2 (4.77 x 1019)2.
Gmpme

A charge separation free of external interactions would oscillate (via plasma oscillations)

with frequency w & KH. However, as there are many complicated effects that begin

to become important after recombination, including gravitational collapse, dynamo

effects, and continued electron-photon scattering, it is unlikely that the simple equation

(3-31) accurately describes the evolution of charge separations in the post-recombination

universe.

While field amplification due to gravitational collapse is negligible at the epoch of

recombination, this will not be the case at all times. At recombination, the universe has

only been matter-dominated for a brief time, and thus density perturbations have only

grown by a small amount in that time, leading to an insignificant amplification of the

field strength. As magnetic flux gets frozen in, however, nonlinear collapse causes ABi to

increase by many orders of magnitude [92, 93].

The major source of amplification of an initial seed field, however, comes from

dynamo effects, as discussed in Section 2. The key to solving the puzzle of the origin of

cosmic magnetic fields lies in determining whether the seed fields produced by a given

mechanism can be successfully amplified into the O(uG) fields observed today. A major

problem with many of the astrophysical mechanisms that produce seed fields is that they

produce low-magnitude fields at insufficiently early times for dynamo amplification to

produce fields as large as pG. The Biermann mechanism, for instance, can produce

seed fields of order 10-19 G, but only at a redshift of z 20. Although those initial

fields are larger than the 10-23 G fields produced by the growth of cosmic structure,

the fact that magnetic fields from structure formation are in place at z 1089 makes

them an extremely attractive candidate for the seeds of cosmic magnetic fields. As

argued by Davis et al. 1999 [131], a seed field as small as 10-30 G at recombination could





possibly be amplified into a /G field today. Clearly, more work on understanding dynamo

amplification is necessary before a definitive solution to the puzzle of cosmic magnetic

fields can emerge.

One interesting mechanism worth investigating further is for the cosmic seed fields

generated by density perturbations to seed supermassive black holes. It is known that the

magnetic field energy in active galactic nuclei and quasars is comparable to the magnetic

field energy in an entire galaxy. However, these structures cannot generate their own

magnetic fields from nothing; they require a pre-existing seed field. It therefore appears

to be a reasonable possibility that the seed fields generated by cosmic structure formation

could provide the necessary fields to seed supermassive black holes. The resultant

amplification via collapse and dynamo effects could explain the origin of large-scale

magnetic structures in the universe.

If large-scale magnetic fields exist at the epoch of recombination, they may be

detectable by upcoming experiments. The results shown in Figure 3-2 provide a

prediction of large-scale magnetic fields at the epoch of the cosmic microwave background.

Sufficiently large magnetic fields on large scales at recombination may be detectable by

PLANCK [132, 133], although current estimates of their sensitivity indicate that the

field strengths predicted in this paper (r 10-23 G) would be significantly out of range of

PLANCK's capabilities (- 10-10 G). Nonetheless, a knowledge of the field strengths at

recombination allow for predictions of CMB photon polarizations and Faraday rotation,

both of which may be, at least in principle, observable.

It is also of interest to note that any primordial charge asymmetry or large-scale

currents (and therefore magnetic fields) created in the very early universe (a < 3.9 x 10-15)

will be driven away by these dynamics. Equation (3-20) has an approximate solution for

6q which is critically (exponentially) damped at a 3.9 x 10-15, capable of reducing an

arbitrarily large charge or current by as much as a factor of e-101". Any pre-existing 6q or

q8 will be driven quickly to the value given by equations (3-22) and (3-24) at the epoch





of critical damping. This ought to be applicable even to a global asymmetry, which can

be treated as a charge anisotropy (6q) on the scale of the horizon. Therefore, the results

in this paper for charge separations, currents, and magnetic fields are independent of the

initial conditions on 6q, Oq, and IB1 in the universe.

There has been other recent work that claims to generate a magnetic field from

cosmological perturbations via "baryon-photon slip," photon anisotropic stress, and a

second order velocity vorticity [134]. The results of this work do not require a velocity

vorticity or anisotropic stress, nor do they require second order quantities. This paper

derives magnetic fields from cosmological perturbations in a very straightforward manner,

simply by calculating the charge separations and currents which necessarily arise from

the differing interactions on protons and electrons, and obtaining magnetic fields directly

from those quantities. The motivation behind the methods used in Ichiki et al. 2006[134]

are obscure and not easily comprehended, while their results are inconsistent with those

obtained in this paper, as their results for magnetic field strength and spectral density are

suspiciously large. Furthermore, it is unclear how their results for velocity vorticity are

obtained, as it is well-known that the vorticity vanishes at second and all orders if there is

none initially.

Overall, the dynamics of ions, electrons, and photons during the radiation era

necessarily leads to charge separations and currents on all scales, which in turn generate

magnetic fields. These fields supersede any pre-existing fields and are in place prior

to substantial gravitational collapse. Thus, the dynamics of structure formation from

cosmological perturbations emerges as a promising and well-motivated new candidate to

explain the origins of cosmic magnetic fields.





CHAPTER 4
EFFECTS ON COSMIC EXPANSION

We evaluate the effect of cosmological inhomogeneities on the expansion rate of the

universe. Our method is to expand to Newtonian order in potential and velocity but

to take into account fully nonlinear density inhomogeneities. To linear order in density,

kinetic and gravitational potential energy contribute to the total energy of the universe

with the same scaling with expansion factor as spatial curvature. In the strongly nonlinear

regime, growth saturates, and the net effect of the energy in inhomogeneities on the

expansion rate remains negligible at all times. In particular, inhomogeneity contributions

never mimic the effects of dark energy or induce an accelerated expansion.

4.1 Accelerated Expansion

Recent observations of type-Ia supernovae [135] and the cosmic microwave background

[5] in tandem suggest that the cosmological expansion is accelerating. Understanding the

source of this accelerated expansion is one of the greatest current unsolved problems in

cosmology [136]. Acceleration seems to render inadequate a universe consisting entirely

of matter, and appears to require an additional, unknown type of energy (dark energy,

perhaps realized as a cosmological constant). An alternative to dark energy is that

acceleration arises from a known component of the universe whose effects on the cosmic

expansion have not been fully examined. One possibility currently being examined is

that inhomogeneities in a matter dominated universe, on either sub-horizon [29, 30] or

super-horizon scales [28, 137-139], may influence the expansion rate at late times. The

central idea is that the energy induced by inhomogeneities leads to additional source terms

in the Friedmann equations, with effects on the dynamics that leave no need for a separate

dark energy component. In their entirety, these proposals present conflicting claims

and a general state of much confusion: does the energy in inhomogeneities produce an

accelerated expansion, acting in effect as dark energy [138], or does it behave as curvature

[31]? Is the magnitude of the effect small, large, or even divergent, on either large scales

[138], or on small scales at late times [30]?





Part of the confusion arises from the fully relativistic perturbation theory formulation

of many of these calculations. Although this is undeniably a valid approach, the number

of terms in a perturbation theory calculation can be large and can mask the underlying

physics. In this Chapter, taking advantage of phenomenological results that have been

derived from a combination of quasilinear perturbation theory, nonlinear theory, and

numerical simulations, we compute the potential and kinetic inhomogeneity energies

within the horizon to Newtonian order in potential and velocity for fully nonlinear

density contrasts. We find these energies to be small at present, and their projected

values remain small, even far into the future. The following section considers the

effect of inhomogeneities, for weak gravity and slow motions but for arbitrary density

perturbations, characterized in terms of the density power spectrum. After that, the

results for the kinetic and potential energies in both the linear and the fully nonlinear

regimes are presented, as a function of the cosmological expansion factor. Finally, the

implications of these results for the present and future expansion history of the universe

are discussed.

4.2 Effects of Inhomogeneities

The purpose of this chapter is to investigate whether energy in inhomogeneities can

mimic the effects of dark energy for a universe containing only matter. To this end, we

work in an = 1 Einstein-de Sitter universe, with no curvature or cosmological constant,

and compute the effects of inhomogeneities on the cosmic expansion rate. The dynamics of

cosmological expansion are governed by the Friedmann equations,


)= Gp, G(p + 3p). (4-1)
a 3 a 3

Any mass or energy density that makes up a significant fraction of the total can influence

the evolution of the cosmological scale factor a(t). A contribution to the energy density

of the universe with equation of state pi = wpi has pi oc a-3(1+w), or pi/pm oc a-3W; in





particular, a component with p oc a-2 behaves as w = -1 or curvature, and a component

with constant p behaves as a cosmological constant or dark energy.

We introduce the effects of inhomogeneities following the formulation of Seljak and

Hui 1996 [140]. In the conformal Newtonian gauge, with metric

ds2 = a2() [_(1 + 2)dr2 +2 (1 2)dX2], (4-2)

the time-time Einstein equation (G'O) yields


3()2(1- 2)+(2 + 6)a V20 + ,(VO)2

= 87rGp(1 + )(1 + v2), (4-3)

Where V'- from the space-space components of G. (Our numerical factors are

slightly corrected from those found in Seljak and Hui 2006 [140]; these factors make

little difference in the overall results.) The source on the right-hand side includes a

density perturbation 6 = 6p/p in the material rest frame, with the transformation to the

cosmological frame expanded to leading order for small v2. Ignoring 0 V20, (VI)2, and

v2, the homogeneous part of this equation reproduces the usual Friedmann equation. The
inhomogeneous part reveals that q obeys the Poisson equation with source 47Gpa2j. The

volume average of the entire equation then leads to

(a2 -Gp8 (1- 5W+ 2K), (4-4)
a 3

where W and K are the Newtonian potential and kinetic energy per unit mass,

1 1
W = 1 ((1+6) ), K = 1 ((1 + 6)v2) (4-5)

These expressions are correct to first order in q and v2, but neither an assumption nor an

approximation in 6. We assume that (V20) = 0; in all other places the Poisson equation is

adequate to determine 0.





The Newtonian potential and kinetic energies thus can influence cosmological

expansion. We can compute both W and also K completely and exactly from knowledge
only of the density power spectrum. The potential is related to the density inhomogeneity
by the Poisson equation, V24 = 47rGpa2 an expression which holds even for nonlinear

inhomogeneities. From this, we obtain

W 47rGpa2 f dk P(k) d A2(k), (4-6)

an expression correct in both linear and nonlinear regimes if P(k) is the appropriate linear
or nonlinear power spectrum. The last equality defines the dimensionless spectral density

A4k).
In linear perturbation theory, valid for small inhomogeneities, the density contrast

grows as 6 = Jo(x)D(t), where in a matter dominated universe D(t) oc a(t) oc t2/3 [141].
The kinetic energy follows from the linearized equation of continuity, 6 + V v/a = 0 [141],


Kn =2 j P(k) (4-7)
2 (27)3 k2

(the usual factor f(Q) Q .6 = 1 for ,m = 1). The kinetic energy scales with a(t)

as &2D2, while the potential energy scales as pa2D2; and so both W and K grow as
D2/a oc a(t), or Pu = p(W + K) oc a-2. As was noted by Geshnizjani, Chung and

Afshordi 2005 [31] for super-horizon inhomogeneities, energy in inhomogeneities has the
same effect on the expansion rate as spatial curvature in perturbation theory. We note
that Kimn/ljWin = H2/47Gp = 2, a fixed ratio in the linear regime. The full kinetic energy

in principle involves higher order correlation functions and is not a simple integral over
the power spectrum. Nonetheless, the full kinetic energy can be obtained simply from the

potential energy through the cosmic energy equation of Irvine 1961 [142] and Layzer 1963

[143],
(d 2dd=a W, (4-8)
dt a dt a
+- dt





with initial conditions set in the linear regime, Kjm = j|Wi Equations (4-6) and

(4-8) provide us with expressions sufficient to calculate nonperturbative contributions to

the expansion rate for both the gravitational potential perturbation and kinetic energy

components. The results of these calculations are given in the next section.

4.3 Effects on the Expansion Rate

Equations (4-6) and (4-8) determine the energy in inhomogeneities of the universe

as a function of epoch, which we characterize by the expansion factor a/ao. For the

primordial power spectrum, we use the CDM power spectrum as given by [144], with

spectral index n = 1, ~, = 1, and COBE normalized amplitude JH = 1.9 x 10-5 [2]. To

obtain the nonlinear power spectrum we use the linear-nonlinear mapping of Peacock and

Dodds 1994 & 1996 [145, 146]. The results of these calculations are shown in Figures 4-1

and 4-2.

Figure 4-1 shows the dimensionless spectral density of gravitational potential energy

A2(k) defined in equation (4-6), evaluated at the present, plotted as a function of
wavenumber k. The dashed curve shows the density in linear perturbation theory, and the

solid curve shows its fully nonlinear form.

Figure 4-2 shows the contributions of potential energy and kinetic energy to the

energy density of the universe, for past and future expansion factors in an Qm = 1

universe. At early times, perturbation theory gives an accurate result, but at a/ao w 0.05

(redshift z f 20) the behavior starts to change, for an interval growing faster than a1

with the fastest growth as a1'2, and then saturating and growing significantly more slowly,

eventually as In a.

4.4 Contributions of Nonlinear Inhomogeneities

In this chapter we have evaluated the size and the time evolution of the contribution

of inhomogeneities to the expansion dynamics of a matter-dominated universe, including

the effects of fully nonlinear density inhomogeneities. When density fluctuations are

in the linear regime, the ratio of the inhomogeneity contribution to the matter density


















10-6





10-7





10-8

1'


Figure 4-1.


Spectral density of gravitational potential energy A2v(k) [the integrand of
equation (4-6)], evaluated at the present, plotted as a function of wavenumber
k. The dashed line shows A' in linear perturbation theory; the solid line
shows the fully nonlinear form.


10-5


0-3 10-2 0.1 1 10 102
k (h Mpc-1)










10-2


I 11111"'I 11"1 1I 11"1 1I 11"11I 11"1 1I ,". I-"I


10-3



10-4



10-5



10- -



10-7



10-7
10-3 10-2 0.1 1 10 102 103
a /a





Figure 4-2. Fractional contributions of gravitational potential energy W (long-dashed
line) and kinetic energy K (solid line) to the total energy density of the
universe, plotted as a function of past and future expansion factor for an
,m = 1 universe. The short-dashed line is the sum of contributions from
inhomogeneities. The dotted lines show results from linear perturbation
theory.





grows linearly with expansion factor, as does curvature in an open universe, making

only a very small contribution to the expansion rate. As density fluctuations begin to

go nonlinear, the energy in inhomogeneities grows at a slightly faster rate, at most as

a1'2 oc a-3w, or w = -0.4. This by itself, even if the dominant energy component, would

be only temporarily and only very slightly accelerating, with deceleration parameter

q0 = (1 + 3w) = -0.1. Since, at this time, the total fraction of energy in inhomogeneities

is Qvu 10-5 < 1, this has a negligible effect on cosmological expansion dynamics.

As the universe further evolves, so that the main contributions to W and K come

from deeply nonlinear scales, we compute the potential energy from integration of the

nonlinear power spectrum, and obtain kinetic energy from the cosmic energy equation,

as detailed in equation (4-8). In a scale-invariant model with power spectrum P k"

as k --> 0, the kinetic and potential energies K and W scale with the expansion factor as

a(-n)/(3+n) [147] (logarithmically in a as n -* 1), with ratio

K 4
(4-9)
\W\ 7+n

Numerical simulations show that this continues to hold for the CDM spectrum with

effective index n = d log P/d log k at an appropriate scale, the basis of the linear-nonlinear

mapping [145, 146]. For the CDM spectrum, with n -- 1 on large scales, this means that

growth stops, and the ratio tends to the virial value Ki/WI | at late times. We note

that aside from the integration of the Layzer-Irvine equation, many of these results were

obtained by [140].

Our results show that the contributions of the potential and kinetic energies of

inhomogeneities has never been strong enough to dominate the expansion dynamics of the

universe. For a universe with Qm = 1 today, normalized to the large scale fluctuations

in the microwave background, the net effect of inhomogeneities today is that of a slightly

open universe, with Qfk Z 10-4 in curvature. The maximum contribution comes from

scales of order 1 Mpc, falling off rapidly for smaller and larger k, as illustrated in Figure





4-1. The behavior on asymptotically small scales (k > 106 h Mpc-1) depends on an

extrapolation that ignores such details as star formation, but Fukugita and Peebles

2004 [12] estimate that the net contribution of dissipative gravitational settling from

baryon-dominated parts of galaxies, including main sequence stars and substellar objects,

white dwarfs, neutron stars, stellar mass black holes, and galactic nuclei, is in total 10-4.9

of the critical energy density.

The suggestion that nonlinear effects for large inhomogeneities may mimic the effect

of dark energy is not the case for the fully nonlinear theory. It is true that higher order

terms in perturbation theory grow faster; the general n-th order term grows as D"(t).

There indeed comes a scale in space or an evolution in time where the behavior of higher

order terms appears to diverge. This is illustrated in Figure 4-3, where it can clearly be

seen that, to second order in density contrast, the contributions from potential and kinetic

terms appear to diverge. Nevertheless, the fully nonlinear result is well behaved. It is only

the perturbation expansion that breaks down, and the actual energy saturates and grows

more and more slowly at late times. As illustrated in Figure 4-2, the nonlinear potential

and kinetic energies remain small compared to the total matter density at all times, even

an expansion factor of 103 into the future. Inhomogeneity effects do not substantially

affect the expansion rate at any epoch.

4.5 Variance of the Energy in Inhomogeneities

It has been pointed out that although the average inhomogeneity energy is small, its

variance has a logarithmically divergent contribution from the variance of the potential on

super-horizon scales [28],


((AW)2) = f3 3x ) )

= (2Gpa P(k)W2(kR), (4-10)
2 2 (2 )3 k4(

windowed over the horizon volume (for calculational convenience we use a Gaussian rolloff

rather than a sharp radial edge). For n -- 1 as k -- 0, this is indeed logarithmically









10-2


A /ao
10-3 / c -

10-4 / n' K -


10-5.""-



10 1


10-7
10-6



10-3 10-2 0.1 1 10 102 103
a /a




Figure 4-3. Fractional contributions of gravitational potential energy W (long-dashed
line) and kinetic energy K (solid line) to the total energy density of the
universe, plotted as a function of past and future expansion factor for an
,m = 1 universe. The short-dashed line is the sum of contributions from
inhomogeneities. This graph shows the contributions to second order in
density contrast, 6. Note the apparent divergence is a result of perturbation
theory breaking down, as the fully nonlinear result in Figure 4-2 is well
behaved.


/





dependent on the low-k cutoff (and if n < 1 the divergence is worse), but the rest of the

integral is finite for the CDM spectrum. The fluctuation in potential energy, ((AW)2) 1',

is shown in Figure 4-4 as a function of the infrared cutoff knn.

The integral is dominated by the smallest values of k, where perturbations are deep in

the linear regime. For n = 1 the result is very accurately AW = 1.45 x 10-5 1 In kminRH 1/2

(We note that for n -> 1 the units of kn, are unimportant.) The fluctuation is comparable

to the mean (W) = 3.1 x 10-5 when the cutoff is near the scale of the horizon k = Ho/c,

and does not become of order 1 until kmin 10-170 (for n = 0.95), or knin ~ 10-10`

(for n -+ 1), or ever (for n > 1). While such an exponentially vast range of scales may

not be beyond the range of possibility in an inflationary universe, it requires a fearless

extrapolation well beyond what is known directly from observation. The fluctuation AW

is dominated by contributions from modes that are deep in the linear perturbation regime,

and scales with expansion factor as AW oc pa2D, constant in time. This contribution

to the energy will appear dynamically in the Friedmann equation as another matter

component. Furthermore, in the presence of a true dark energy component, any effects on

cosmological expansion arising from inhomogeneities quickly becomes unimportant once

dark energy becomes dominant [140].

The fact that fluctuations in the potential diverge remains troublesome. It has been

recognized for some time that potential fluctuations in the standard model with n -- 1

are logarithmically divergent, but since for most purposes the value of the potential is

unimportant, this has not been perceived as a significant problem. The effect of potential

on the expansion dynamics is real, but the weak logarithmic divergence and the fact that

it is a feedback of a gravitational energy on gravitational dynamics may lead one to hope

that this divergence is alleviated in a renormalized quantum theory of gravity.

4.6 Summary

We have found that, to leading order in 4 and v2 but with fully nonlinear density

fluctuations, inhomogeneities on sub-horizon scales have only a minimal effect on the









10-3


< 10-4









10-5

-50 -45 -40 -35 -30 -25 -20 -15 -10 -5
log1 kmin




Figure 4-4. The expected fluctuation in the potential energy per unit mass ((AW)2)1/2
evaluated at the present as a function of infrared cutoff kmn for n = 0.95,
n = 1, and n = 1.05 (solid lines, top to bottom). Dashed lines are
analytic approximations that asymptotically become k-0.025, (log k)1/2, or
constant, respectively. The dotted line shows the result for a rolling spectral
index that has n = 0.95 on the horizon today but approaches n = 1 as
k -- 0, as predicted by most models of slow-roll inflation. The mean value
(W) = 3.1 x 10-5 is shown as the horizontal dashed line.





cosmological expansion dynamics, even far into the future, and in particular never result in

an accelerated expansion. Other authors have also shown that recent attempts to explain

an accelerated expansion through super-horizon perturbations face significant difficulties

[31-33]. The possibility that a known component of the universe may be responsible for

the accelerated expansion remains intriguing. However, we conclude that sub-horizon

perturbations are not a viable candidate for explaining the accelerated expansion of the

universe.





CHAPTER 5
FULLY EVOLVED COSMOLOGICAL PERTURBATIONS

This chapter investigates the use of a well-known empirical correlation between the

velocity dispersion, metallicity, and luminosity in H/ of nearby H II galaxies to measure

the distances to H II-like starburst galaxies at high redshifts. This correlation is applied to

a sample of 15 starburst galaxies with redshifts between z = 2.17 and z = 3.39 to constrain

,m, using data available from the literature. A best-fit value of ,m = 0.2 1+0 in a

A-dominated universe and of m, = 0.110 7 in an open universe is obtained. A detailed

analysis of systematic errors, their causes, and their effects on the values derived for the

distance moduli and Qm is carried out. A discussion of how future work will improve

constraints on ,m by reducing the errors is also presented.

5.1 Precision Cosmology

Precision cosmology, or accurately constraining the parameters describing the

universe, has recently become an active field of research due to the precision of available

data sets. Stringent contraints have recently been placed on cosmological parameters from

measurements of the microwave background [5], type Ia supernovae [135], and galaxy

surveys [1, 148]. Although these sources of data are sufficient for generating consistent

values for the mass density (2,m), vacuum energy density (QA), the dark energy equation

of state parameter (w), and the value of spatial curvature in the universe (2k), these

values must be checked via as many independent methods as possible for consistency,

accuracy, and avoiding systematic biases. Furthermore, without the data from supernovae,

there would be weak evidence at best for stating that w ~ -1, thus it is important to seek

another, independent observation supporting the existence of dark energy.

The cosmological parameter with the greatest number of observable cross-checks is

0,. It has been derived using many techniques, including the Sunyaev-Zel'dovich effect

[149], weak gravitational lensing [150], X-ray luminosities [151], large scale clustering [152],

peculiar velocities of galaxy pairs [153], and supernovae data [39]. These methods yield

results ranging from ,m = 0.13 to ,m = 0.35, and are all consistent with one another at





the 2-a7 level. However, they all face difficulties when attempting to differentiate between

cosmological models, as they are only weakly dependent on QA, 0k, and w. If a reliable

standard candle were found at high redshifts, cosmological models could be discriminated

between by precise and accurate observations, as the distance modulus becomes sensitive

to fA, fk, and w at higher redshifts. It is known that local H II galaxies and giant H II

regions in local galaxies are physically similar systems [154]. This chapter extrapolates

a link between nearby H II galaxies and H II-like starburst galaxies at high redshifts to

use such objects as standard candles. This is accomplished through the application of the

known correlation between the luminosity in the H/3 line (LHQ), the velocity dispersion

(oa), and metallicity (O/H) of nearby H II galaxies discovered in Melnick, Terlevich

and Moles 1988 [155] to the H II-like starburst galaxies found at high redshifts. This

correlation, when applied to starburst galaxies at z > 2, allows for discrimination between

different values of ,m as first suggested in Melnick, Terlevich and Terlevich 2000 [40], and

can discern which cosmological model is most favored by the data.

H II galaxies (and H II regions of galaxies) are characterized by a large star-forming

region surrounded by singly ionized hydrogen. The presence of 0- and B-type stars in

an H II region causes strong Balmer emission lines in Ha and H/3. The size of a giant

H II region was shown to be correlated with its emission line widths inMelnick 1978 [156].

This correlation was improved upon in Terlevich and Melnick 1981 [157], who showed that

LHp of giant H II regions is strongly correlated with their a. This basic correlation, its

extension to H II galaxies, and its usefulness as a distance indicator have been explored in

the past [154, 155, 158]. The empirical correlation for H II galaxies [155] relates their LHg,

(7, and O/H. The relationship is

a5
log LHf = log Mz + 29.60, Mz (5-1)


where the constant 29.60 is determined by a zero-point calibration of nearby giant H II

regions [40] and from a choice of the Hubble parameter, Ho = 71 kms-1 Mpc-1 [159, 160].





The 1-a rms scatter about this correlation is 0.33 dex on log LHp from the local sample

of H II galaxies found in Melnick, Terlevich and Terlevich 1988 [155]. Starburst galaxies

observed at high redshifts exhibit the same strong Balmer emission lines and intense

star formation properties [161, 162] as nearby H II galaxies. This chapter follows the

suggestion of Melnick, Terlevich and Terlevich 2000 [40] that equation 5-1 applies to the

H II-like starburst galaxies found at high redshifts, and provides evidence to validate that

assumption.

The remainder of this chapter discusses the constraints that can be placed on f, and

the restrictions that can be placed on the choice of cosmology using starburst galaxies.

Section 5.2 details how the data set was selected and analyzed. Section 5.3 states the

results obtained from the selected data. The random and systematic errors associated

with any distance indicator is a fundamental (and often overlooked) problem inherent to

observational cosmology. Appendix A discusses the errors specific to the observational

method used in this chapter, including a detailed discourse on the assumption of

universality between local H II galaxies and high redshift starburst galaxies. Finally,

section 5.4 presents the conclusions drawn from this chapter, and points towards useful

directions for future work on this topic.

5.2 Selection of the Data Sample

The goal of the analysis presented here is to obtain distances for each H II-like

starburst galaxy at high redshift. H II galaxies must first be detected at high redshift. A

sample is then selected on the basis of the correlation in equation (5-1) holding and for

which the distance modulus (DM) can be computed from the observed quantities. The

quantities required for analysis of these galaxies are a, the flux in H/3 (FHf), O/H, the

extinction in H/3 (AHg), and the equivalent width in the H/3 line (EW).





Following the analysis in Melnick, Terlevich and Terlevich 2000 [40], the distance

modulus of H II galaxies can be derived from:


DM = 2.5 log( ) 2.5 log(O/H) A 26.18, (5-2)
FHu

where the constant 26.18 is determined by Ho and equation (5-1). This chapter makes

extensive use of equation (5-2) because it expresses DM purely in terms of observables.

DM is insensitive to f,, QA, 0k, and w at low redshifts (z < 0.1), differing by 0.1

magnitudes or fewer for drastic changes in the choice of parameters above. At high

redshifts (z > 2), however, DM can vary by up to 3 magnitudes depending on the choice

of parameters. Of the four parameters above available for variation, DM is most sensitive

to changes in Qm, as has been noted previously [40]. However, for values of ,m < 0.3, DM

is sensitive to variations in the other parameters by 0.2 to 0.5 magnitudes. Since other

measurements indicate that indeed Q, < 0.3, this chapter also considers variations in QA

and fk.

Data for starburst galaxies at z > 2 are found in Pettini et al. 2001 [161] and Erb

et al. 2003 [162], which contain measurements for many of the desired observables (and

related quantities), along with redshift data. Partial measurements exist for 36 starburst

galaxies. According to Melnick, Terlevich and Terlevich 2000 [40], the correlation in

equation (5-1) holds true for young H II galaxies whose dynamics are dominated by O-

and B-type stars and the ionized hydrogen surrounding them. As H II galaxies evolve in

time, short-lived O- and B-stars burn out quickly. Although some new O- and B-stars are

formed, eventually the death rate of O- and B-stars exceeds their birth rate, causing a

galaxy to be under-luminous in Ha and H/3 for its mass. This effect can be subtracted

out by examining the EW of these galaxies, and cutting out the older, more evolved

galaxies (those with smaller equivalent widths). For this chapter, a cutoff of EW > 20 A is

adopted, and galaxies with EW < 20 A are not included, similar to the cutoff of 25 A used

previously [40]. There are also galaxies with large EW that do not follow the correlation





Table 5-1. High-redshift galaxies selected to be used as standard candles on the basis of
their equivalent widths and velocity dispersions.

z 7 1 FHf 2 12+log (O/H) AH_ 3 EW 4 DM 5
2.17 62 29 0.9 0.2 8.55 0.013 23 47.49+2_1
2.18 51 22 1.9 0.5 8.55 0.157 72 45.45 +1
2.54 < 42 1.3 0.3 8.55 0.141 21 44.82+0 31
2.44 < 60 1.3 0.3 8.55 0.285 25 46.64+0 31
2.32 75 21 2.4 0.6 8.55 0.735 47 46.72+1_8
2.17 107 15 2.6 0.7 8.55 0.214 31 48.96+.01
3.11 < 63 3.4 1.0 8.55 0.505 28 45.77+0.31
3.23 69 4 < 1.7 8.55 0.237 < 27 47.12+0.44
-0.32
3.39 87 + 12 2.7 0.3 8.70 0.08 0.773 37 46.96 +1
3.10 116 + 8 4.1 0.4 8.62 0.07 0.237 43 48.81+.31
3.09 67 + 6 < 2.3 8.55 0.110 < 31 46.76+0.56
3.07 113 + 7 1.3 0.3 8.39 0.16 1.01 25 49.711+43
3.32 100 + 4 3.5 0.4 8.55 0.852 25 47.73'+.
3.09 55 15 3.0 1.0 8.55 0.284 40 45.22 +1.
2.73 81 1.35 0.2 8.49 0.10 1.14 26 47.49+1-


of equation (5-1) within a reasonable scatter. It is well-known that a large fraction of

local H II galaxies contain multiple bursts of star formation [158]. If multiple unresolved

star-forming regions are present, the observed a will be very large due to the relative

motion of the various regions. Such galaxies are not expected to follow the correlation of

equation (5-1) [158]. The simplest way to remove this effect is to test for non-gaussianity

in the emission lines from this effect, but signal-to-noise and resolution are insufficient

to observe this effect. Since a for a system of multiple star-forming regions will be much

higher than for a single H II galaxy, a cut can be placed on a to remove this effect. Monte

Carlo simulations (detailed in Appendix A) indicate that if a is observed to be greater

than 130 km s-1, it is likely due to the presence of multiple star-forming regions. To

account for this presence, all galaxies with a > 130 km s-1 are discarded. Imposing the

above cuts on a and EW selects 15 of the 36 original galaxies, creating the data sample

used for the analysis presented here. The properties of those selected galaxies are detailed

in Table 5-1, with further information about the galaxies available in the source papers

[41, 161, 162].





Once the sample has been selected, the quantities required to calculate DM using

equation (5-2) must be tabulated for each selected galaxy. Not all of the necessary data

elements are available in the literature for these galaxies, so assumptions have been made

to account for the missing information, z was measured for all galaxies by the vacuum

heliocentric redshifts of the nebular emission lines, a was obtained for all galaxies from the

broadening of the Balmer emission lines, Ha for the galaxies from Erb et al. 2003 [162]

and H3 for the galaxies from Pettini et al. 2001 [161]. FHf is measured directly for the

galaxies in Pettini et al. 2001 [161], but Erb et al. 2003 [162] measures FHa instead, thus

FHa must be converted to FHP. The conversion for emitted flux is given in Osterbrock

1989 [163] as FHu = 2.75 FHf, but observed fluxes must correct for extinction. Thus, the

complete conversion from FHa to FHP will be given by equation (5-3) below,

1 AH--AH,3
FH = FHa 1 0 2( ) (5-3)
2.75

where AHa and AHP are the extinctions in Ha and H/3, respectively. Obtaining O/H is

more difficult, as measurements of metallicity only exist for 5 of the 36 original starburst

galaxies. An average value of O/H is used for the galaxies where O/H measurements are

unavailable.

Values of O/H are obtained through measurement of the [O II] emission line at 3727 A

and the [0 III] lines at 4959 A and 5007 A for five of the galaxies in Pettini et al. 2001

[161]. The strong line index R23 [164] is assumed to have its temperature-metallicity

degeneracy broken towards the higher value of O/H, as is shown to be the case for

luminous starburst galaxies at intermediate redshifts in Kobulnicky and Koo 2000 [165].

The combination of the oxygen line measurements with this assumption yields values

for O/H for these galaxies. The mean value of O/H is then taken to be the average

metallicity for each of the other galaxies where such line measurements are unavailable.

Recently, measurements of metallicity in high redshift starburst galaxies have been made

[166], using the [N II]/Ha ratio as their metallicity indicator. The authors in Shapley et





al. 2004 [166] obtain an average O/H of 8.33 for the galaxies previously found in Erb et

al. 2003 [162]. This value is noted as a possible improvement to the one chosen here, and

is further discussed as a source of error in this analysis in appendix A of this work.

AH, is derived from the E(B V) color of the galaxy in question. Extinction laws are

known and established for the Milky Way, the Large and Small Magellanic Clouds (LMC

and SMC, respectively), and the H II regions of the LMC and SMC [167], but have not

been established for starburst galaxies in general (although see Calzetti et al. 1994 & 2000

[168, 169] for an argument to the contrary). This chapter assumes dust in H II galaxies to

be comparable to that in giant H II regions, thus AHp for starburst galaxies is taken to be

the AHP derived in Gordon et al. 2003 [167] for the H II regions of the LMC and SMC. A

best fit applied to the data in Gordon et al. 2003 [167] yields


AH~ = (3.28 0.24) E(B V),

AHa = (2.14 0.17) E(B V), (5-4)

for starburst galaxies. These results are also applicable to the flux conversion in equation

5-3. E(B V) is unavailable for the galaxies from Pettini et al. 2001 [161], but can be

derived by noticing the correlation between E(B V) and corrected (G R) colors

for starburst galaxies in Erb et al. 2003 [162]. The conversion adopted is E(B V) ,

0.481 (G R). Finally, EW is measured for all galaxies in Pettini et al. 2001 [161], but

Erb et al. 2003 [162] gives only the spectra for the Ha line. EW is estimated for the Erb

et al. 2003 [162] galaxies by estimating the continuum height from each spectra and the

area under each Ha peak, calculating the equivalent width in Ha, and converting to H/3

using the Balmer decrements of Osterbrock 1989 [163]. The complete data set is listed in

Table 5-1, and is illustrated alongside various cosmologies in Figure 5-1.

5.3 Constraints on Cosmological Parameters

In the previous section, the distance modulus was calculated for each galaxy in the

selected sample. The comparison of these values of DM and the predicted values of
































1.5 2 2.5 3 3.5 4
Redshift


Figure 5-1. The distance modulus plotted as a function of redshifts for various
cosmological models, along with data from the selected galaxies. Open-CDM
universes and A-CDM universes with f, of 0.05, 0.30, 0.5, 1.0, and 2.0
are shown. The crosshairs represents the 1-a constraints on the DM vs. z
parameter space from the selected data sample. The best fits to the data are
for a ACDM universe with Q, = 0.21 and QA = 0.79, or for an open-CDM
universe with = 0.11.





DM at a given redshift for different cosmological models provides a constraint on the

cosmological parameters. DM is most sensitive at high redshifts to the variation of the

cosmological parameter Qm, as pointed out by Melnick, Terlevich and Terlevich 2000

[40]. ~, is therefore the parameter which is constrained most tightly by observations

of starburst galaxies. Each galaxy yields a measurement for DM and for z. Although

there are multiple models consistent with each individual measurement, observations of

many galaxies at different redshifts will allow the construction of a best-fit curve, which

is unique to the choice of cosmological parameters f,, QA, Qk, and w. The data sample

of 15 galaxies in this chapter is insufficient to distinguish between models in this fashion,

as the uncertainties in each individual measurement of DM are too large. The method by

which the uncertainties can be reduced is to bin the data according to redshift and find a

best-fit value of DM at that point. Due to the size of the sample in this chapter, all 15

points are averaged into one point of maximum likelihood to constrain the cosmology, with

errors arising from the random errors of the individual points and from the distribution of

points. The average value obtained is DM = 47.03+.4 at a redshift z = 2.80 0.11. The

different cosmological models, along with the most likely point and the raw data points,

are displayed in figure 5-1, with Ho = 71 km s-1 Mpc-1

The constraints placed on f, from this analysis are = 0.2 10.30 in a A-dominated

universe (,m + 2A = 1; 2k = 0) and Q, = 0.110. 3 in an open universe (Qm + 2k = 1;

QA = 0). Figure 5-2 shows the comparison in ,m vs fA parameter space between the

preliminary constraints of this chapter and early constraints arising from CMB data and

SNIa data, available in de Bernardis et al. 2000 [170].

CMB and SNIa constraints led to the first reliable estimates of ,m and OA. The

preliminary constraints presented here are comparable to early constraints from CMB and

SNIa data, as illustrated in figure 5-2. The accuracy in ,m and QA, as determined from

the most recent CMB and SNIa data [5] is now 0.04 in each parameter. A similar, and

perhaps even superior accuracy can be achieved using starburst galaxies at high redshifts.
















V^M~- "-^.7





The final section below discusses this possibility, and the systematic and statistical errors

which must be overcome are discussed in Appendix A.

5.4 Conclusions and Future Prospects

This chapter has demonstrated that using H II-like starburst galaxies at high redshifts

as a standard candle is a promising and well-motivated avenue to explore for precision

cosmology. A future survey of high redshift starburst galaxies with measurements of z, a,

AH1, FHQ, O/H, and EW will reduce both random and systematic errors dramatically.

Since the inherent scatter of the method is large (as can be seen in figure 5-1), a large

sample size is required to obtain meaningful constraints. This chapter contains a sample

size of only 15 galaxies, but future surveys should be able to obtain hundreds of starburst

galaxies that survive the selection cuts. For a sample of 500 galaxies, this will improve

constraints on Qm to a restriction of 0.03 due to random errors. Additionally, all of the

systematics specific to this sample due to incomplete data will disappear. Appendix A

discusses how these errors may be reduced, and how, with such a sample, the concordance

cosmological model can be tested at a redshift of z r 3, something that has not been done

to date.

If the assumption of universality between local H II galaxies and high redshift

starburst galaxies is correct, this method of measuring Qm is capable of providing very

tight constraints, independent of any constraints arising from other sources, including

CMB and SNIa data. Additionally, if galaxies are obtained at a variety of redshifts

between 2 < z < 4, different cosmological models (including vacuum-energy dominated

models with different values of w) can be tested for consistency with the future data set.

If 1m < 0.3, the differences in DM at various redshifts become quite pronounced, and

meaningful results as to the composition of the non-matter components of the universe

can be obtained as well. Future work on this topic has the potential to provide strong

independent evidence either supporting or contradicting the concordance cosmological





model of Q~ + QA = 1, w = -1, in addition to providing a very stringent constraint on the

Qm + QA parameter space.




CHAPTER 6
CONCLUDING REMARKS

This dissertation has illustrated the invaluable role of cosmological perturbations

throughout the evolution of the universe. These perturbations are departures from perfect

homogeneity in density and in gravitational radiation. They are created during the epoch

of inflation by quantum fluctuations, which are in turn stretched across all length scales

by the exponential expansion of the universe. When inflation ends, the cosmological

perturbations become overdense and underdense regions in a radiation-dominated

universe. The universe then expands and cools, and these cosmological perturbations

evolve under the influence of gravity, radiation pressure, and all the other forces of the

universe. Cosmological perturbations grow linearly at first, and when the overdense

regions have reached a sufficient density, they collapse nonlinearly. This nonlinear collapse

leads to the rapid growth of complex structure, forming stars, galaxies, clusters, and

filaments, among other structures. The structure exhibited on cosmological scales at the

present is a direct result of the evolution of primordial cosmological perturbations. The

remainder of this chapter summarizes the major findings of this paper, and points towards

future directions for research on the topic of cosmological perturbations and their effects

on the universe. Also included is a section on how cosmological perturbations are expected

to impact the eventual fate of the universe.

6.1 Creation of Perturbations

The perturbations produced by inflation are capable of imprinting both primordial

scalar modes (density perturbations) and tensor modes (gravitational radiation). Vector

modes may also be produced, but these decay over time, and are unimportant for

cosmology. The tensor modes that are produced can either be of a comparable amplitude

to the scalar modes (as in chaotic inflation [171]), or can be of practically zero amplitude

compared to the scalar modes (as in new inflation [172, 173]). In either case, the spectrum

of both density perturbations and gravitational radiation are predicted to be nearly scale

invariant, with a possible slight tilt preferring either small scales or large scales.





If there is a preference towards larger scales, it is possible that, on vastly superhorizon

scales, the increased power can affect either the expansion rate or the global spatial

curvature of the universe. Differing opinions and many discussions can be found in the

literature [31-34, 137] as well as in chapter 4. The work presented in this dissertation

indicates that although the variance of the potential energy (AW) in these superhorizon

perturbations can become very large, it is the potential energy itself (W), not AW, that

couples to the expansion rate of the universe. Therefore, it appears at this juncture that

cosmological perturbations on superhorizon scales cannot affect the expansion rate of the

universe.

On the other hand, there could be a preference towards smaller scales, which would

lead to early nonlinearity (and faster structure formation) of the smallest structures.

The power spectrum is fairly accurately known [1] for values of k up to ~ 100 Mpc-1,

and matches very well with simulations of a scale-invariant spectrum [174]. Although

there is no reason to believe the density perturbations on scales smaller than this depart

significantly from an n 1 spectrum, the perturbations in gravitational radiation may. As

shown in Siegel and Fry 2005a [13] and in chapter 2, in the presence of extra dimensions

at a fundamental scale mD, primordial gravitational radiation will acquire a thermal

spectrum and an energy density given by equation (2-4) if the reheat temperature, TRH,

is sufficiently high. The possibilities for detecting a thermal background of gravitational

radiation (and thus indirectly detecting extra dimensions) could be accomplished through

a precision measurement of the primordial 4He abundance or through the broadening of

the 21-cm HI line.

6.2 Early Evolution of Perturbations

Once the initial cosmological perturbations of the universe are in place amidst the

other initial conditions of the big bang (expanding, dense, radiation-dominated universe),

their densities evolve in accordance with all of the physical effects acting on them.

These include the gravitational force, which provides a gradient towards the overdense





perturbations and away from underdense ones, radiation pressure, the Coulomb force, the

nuclear (strong and weak) forces, and scattering from particle-particle interactions.

The net result of all the interactions is that overdense regions in the universe grow

according to the Miszaros effect at sufficiently early (linear) times, as given in equation

(1-7). While the M6sziros effect does a good job describing the overall evolution of the

energy density in a region of space, it cannot give any information about the evolution of

the different types of components which compose the universe. As an example, at various

epochs in the universe, each of baryons, dark matter, photons, neutrinos, and vacuum

energy compose at least 10 per cent of the energy density of the universe. While the

universe is radiation-dominated, photons and neutrinos are most important, while when it

is matter-dominated, baryons and dark matter are the most important components.

In order to understand how cosmological perturbations have evolved into the

large-scale structures observed today, it is vital to understand the evolution of the matter

components of the universe from very early times until the present day. Chapter 3, based

heavily on the works of Ma and Bertschinger 1995 [22] and Siegel and Fry 2006 [23],

details the evolution of the matter components of the universe.

One novel idea of this work is that the evolving cosmological perturbations give rise

to seed magnetic fields on all scales. This can also be found in chapter 3 and in Siegel and

Fry 2006 [23], but note that there is a competing group that obtains quite different results

through a significantly different calculation [120, 134, 175]. The essential idea is that

photons have a much larger interaction cross section with electrons than with protons,

inducing charge separations and currents. The Coulomb force also plays a major role,

acting as a restoring force. The net result obtained in chapter 3 is that magnetic fields

on all scales are created, following the spectrum in figure 3-2. On the most interesting

scales (from 1 100 Mpc), fields of 0(10-23 Gauss) are produced at the epoch of

recombination. These seed fields may provide the seeds for the magnetic fields observed on

large scales today. This may be accomplished either directly, by having these seed fields





directly grow into the fields observed today, or by these fields seeding the supermassive

black holes/AGNs which then amplify the field, and populate the universe with the

resultant magnetic field energy.

The sets of equations for a difference component between ions and electrons derived in

chapter 3 are the first of their kind. Problems which hitherto could not be solved for lack

of having equations that describe a charge difference can now be tackled using these new

tools. Appendix B presents the possibility that a net electric charge was present at some

point in the universe's past. Prior treatments (see Lyttleton and Bondi 1959 [176] and

articles which cite it) have found that many of these scenarios would present unacceptable

consequences for cosmology. However, Appendix B points towards the possibility that a

net charge in the universe, which could arise (for instance) from a broken U(1) symmetry,

would be driven away by the simple dynamics of cosmological perturbations. If this proves

to be the case, many possibilities for physics in the early universe, where are presently

considered to have unacceptable cosmological consequences, may turn out to be quite valid

after all.

6.3 Final State of Perturbations

Cosmological perturbations continue to grow linearly, with the most substantial

growth occurring once the universe has become matter-dominated, until a critical density

is reached. Once this occurs, density perturbations enter the nonlinear regime, and grow

very rapidly. The nonlinear structure formed in this manner collapses to often very large

density contrasts (see chapter 1), but the structure formed is eventually stable to further

collapse due to the virial theorem.

Were the universe completely devoid of angular momentum, or rather, were the

Zel'dovich approximation exact, the universe might look vastly different from its present

state. If gravitational collapse were to occur exactly along field gradients, then nuclear

reactions would be the only interactions in the universe that prevented all structure from

collapsing to singularities. Yet, angular momentum is a fact of life in the universe, as





evidenced by the rotations of many astronomical systems. As a result, when structure

formation goes nonlinear and gravitational collapse becomes a major effect, angular

momentum conservation also becomes a major factor in forming a stable, final-state

structure.

Complex physics, such as shocks and heating (which can create magnetic fields via the

mechanism in Biermann 1950 [111]), star formation, baryonic collisions, and collapse on

multiple scales all play a role in the formation of the present-day structure of the universe.

One question that has been answered in Chapter 4 of this work is the question of whether

this nonlinear structure will backreact sufficiently, and impact the global expansion rate

of the universe. While many authors [28-30] contend that the effects of nonlinear collapse

could substantially impact the expansion rate, it has been calculated (both in chapter 4

and in Siegel and Fry 2005b [34]) what the effect is explicit. The conclusion is that the

effect is negligibly small (of 0 ~ 10-5 the normal expansion rate) at all times.

Cosmological perturbations, in an Einstein-de Sitter universe (fQ = 1, no curvature

or vacuum energy), will grow nonlinearly only once the scale of interest is inside the

horizon. In this scenario, structure in the universe is self-similar, with smaller scales at

earlier times behaving identically to larger scales at later times. However, observations of

structure formation, among other observables [177, 178], do not support this picture of an

Einstein-de Sitter universe. The picture which is most consistent with all the observations

is known today as the concordance cosmology, which indicates that the universe has

roughly 30 per cent of the critical energy density in matter and roughly 70 per cent in a

cosmological constant. The consequences of this for large-scale structure in the universe

are examined in section 6.4 of this Chapter.

Once galaxies form (at a redshift of around z ~ 10 in a ACDM universe), they can be

used as deep cosmological probes. In order for something to be a useful distance indicator,

there must be a relation between observable quantities and a cosmological distance

[179]. Many of these techniques involve individual stars or stellar remnants, or other





low-luminosity objects. To be used at the largest distance scales, a distance indicator must

first be visible at such large distances. There are only a few classes of objects which are

visible out at redshifts z > 2, such as the most luminous galaxies, supernovae, gamma-ray

bursts, and quasars. It is only at z > 2 that it becomes reasonable to constrain the various

cosmological parameters (such as Qm, QA, and the dark energy equation of state, w).

Thus far, only the supernovae (and only the type Ia supernovae, at that) have

successfully been used as a distance indicator out at this high of a redshift. Although

there has recently been an attempt to use gamma-ray bursts as a distance indicator

[180], the systematic errors inherent to the method are far larger than the uncertainty

between different cosmological models. Systematic errors, as illustrated in Appendix A,

are a constant source of difficulty for any observational cosmologist. Uncertainties in

the universality of a distance indicator (i.e. that it behaves the same at all redshifts), as

well as uncertainties in the environment where the distance indicator lies, can all bias

results obtained with any one indicator. It is for these reasons that many differing distance

indicators at high redshift are desired for probing cosmological parameters. Chapter 5

(and Siegel et al. 2005 [41]) builds upon the work of Melnick, Terlevich and Terlevich 2000

[40], and uses star-forming galaxies at z > 2 to constrain the cosmological parameters.

Although systematic errors for this method are both numerous and worrisome, they are

quantified and given a detailed treatment in both Appendix A and Siegel et al. 2005 [41].

The result obtained is that, despite systematic errors, in a A-dominated universe, it can

be concluded that 0.09 < f < 0.51, with a best-fit value of fm = 0.21. This is the only

known independent estimate for cosmological parameters from distance indicators at such

high redshifts other than supernovae or the cosmic microwave background.

6.4 Fate of the Universe

One of the most puzzling cosmological discoveries of the last decade has been the

discovery that the expansion rate of the universe is accelerating (discovered by Riess et

al. 1998 [181] and Perlmutter et al. 1999 [182], independently). These observations have





been confirmed and appear very convincing, yet a compelling theoretical mechanism

for explaining the existence of this so-called dark energy has not yet been discovered.

Understanding the nature of this accelerated expansion and the physics behind it is one

of the great challenges for modern cosmology. An interesting avenue for future research

would be to predict the future history and evolution of structure in the universe based on

the presence of this accelerated expansion.

The accelerated expansion is most easily and simply parametrized by a cosmological

constant of energy density PA 6.9 x 10-30 g cm-3. The presence of a non-zero

cosmological constant (A) incorporates a built-in scale to the physics of structure

formation. The main consequence of this new type of energy density for structure

formation is that scales which are not gravitationally bound to one another at the time of

matter-A equality never become bound to one another.

The Hubble expansion parameter, H, in a universe containing matter, radiation, and

a cosmological constant, evolves as

H2= H2 Pa-4 + Pa-3 +PA) (6-1)
(Pc Pc Pc

where p,, pm, and PA are the energy densities in radiation, matter, and A, respectively. (pc

is the critical density.) From equation 6-1, it is facile to deduce that at late times (when a

becomes large), the Hubble expansion parameter, H, is given by the constant


H = Ho QA PA (6-2)
Pc

Therefore, the final expansion state of the universe will be much like the initial inflationary

state, in that there will be an asymptotically exponential expansion. The scale factor of

the universe, a, will evolve in the far future as

a eo Vt. (6-3)





The net result is that the structure that is bound at the time of matter-A equality

remains bound, whereas structure that is not yet bound never becomes bound, and will

exponentially recede from one another. The local group contains the Andromeda galaxy,

the large and small Magellanic clouds, and a few other, small structures. At a time

- 5 x 1010 years in the future, the local group, gravitationally bound to our galaxy, will be

the only matter in our universe within 500 Mpc of the Milky Way! In fact, in twice that

time (~ 1011 years), everything outside of our local group will "red out," meaning that

objects in our local group will be the only objects causally in contact with our galaxy in

the universe.

The current behavior of the universe points towards this exact scenario for the energy

content of the universe, with 70 per cent in dark energy and 30 per cent in matter.

If this is the case, and the universe continues to evolve according to the known laws

of physics, then galaxies and the objects bound to them, the children of cosmological

perturbations, will be the last remaining objects in the universe. The universe will consist

of a few isolated clumps of matter exponentially expanding away from one another, with

nothing but empty space in between. In the end, these lonely clumps of matter, having

arisen from the growth and collapse of slightly overdense cosmological perturbations, may

be the only substantial things in a cold, empty universe.





APPENDIX A
ERRORS IN HIGH-Z GALAXIES AS DISTANCE INDICATORS

There have been many assumptions made along the path to obtaining m via the use

of starburst galaxies as a distance indicator, as detailed in chapter 5. The major danger

in every observational method is that every assumption made carries along with it an

associated error. Some of the assumptions made are inherent to the method used, while

others affect only the data sample specific to the one selected in chapter 5. Both will lead

to systematic errors, although the sample-specific errors will largely be eliminated by

improved measurements, to be taken in future observing runs. Additionally, random errors

result from both uncertainties in the measurements and from the intrinsic scatter in the

distribution of points. An analysis of all three types of errors ensues below.

A.1 Universality among HII Galaxies

The most important assumption made was the assumption of universality of the

distance indicator used for both local H II galaxies and H II-like starburst galaxies.

Support for this assumption is provided by the fact that both galaxy types follow the

empirical correlation of equation (5-1), as shown in figure A-1. Although the physics

underlying starburst galaxies has been an open question for over thirty years [183], it

is fortunately not necessary to uncover the complete answer to establish universality.

It is likely (although unproven) that the physics underlying the correlation for H II

galaxies is similar to the physics underlying the Tully-Fisher relation [184] for spiral

galaxies. Specifically, it is thought that line widths (a measure of velocity dispersion)

and the luminosity in the Ho line are both intimately tied to the amount of mass in the

star-forming region. A theoretical investigation of exactly what this link is could prove

quite fruitful in understanding the underlying physics of the correlation presented in

Melnick, Terlevich and Moles 1988 [155].

The validity of the correlation between Lfl and Mz can be tested directly to

determine its range of applicability. By assuming a cosmology, log LHp can be written

purely in terms of luminosity distance (dL), FHp, and AHfl, which are either measurable





or computable from observables for each galaxy, as shown in equations (5-1) and (5-2).

log Mz can be determined through measured values for a and O/H. Comparing the

quantities log LHf and log M, then allows a test of the correlation in equation 5-1 for all

galaxies of interest. All available H II and H II-like starburst galaxies with appropriately

measured quantities are included to test the correlation. Local galaxies are taken from

Melnick, Terlevich and Moles 1988 [155] and from the Universidad Complutense de Madrid

(UCM) survey [185, 186], intermediate redshift starburst galaxies are taken from Guzman

et al. 1997 [187], and high redshift starburst galaxies are from Pettini et al. 2001 [161] and

Erb et al. 2003 [162]. The cosmology assumed to test universality is Qm = 0.3, QA = 0.7,

and cuts are applied to all samples so that EW > 20 A and a < 130 km s-1. The results

are shown in figure A-1.

The major reasons to conclude that the assumption of universality is valid lie in

figure A-1. There exists an overlap between all four samples in both LHO and M,, from

the sample where the correlation is well established (nearby samples, such as Melnick,

Terlevich and Moles 1988 [155] and the UCM survey [185, 186]), to intermediate redshift

H II-like starburst galaxies Guzman et al. 1997 [187], to the high redshift sample used in

chapter 5, from Erb et al. 2003 [162] and Pettini et al. 2001 [161]. These four samples

all follow the same correlation between LHp and M, within the same intrinsic scatter.

(However, it is worth noting that the observed scatter broadens at high redshifts due to

measurement uncertainties). By performing a statistical analysis of the data points in

figure A-i, it can be shown that the data selected from all samples are consistent with

the same choice of slope and zero-point for the empirical correlation. For these reasons,

equation (5-1) appears to hold not just for local H II regions and galaxies, but for all

starburst galaxies regardless of redshift.

It is important to note that there is an uncertainty in the zero-point calibration of

figure A-1 of 0.08 dex, which has not improved since the correlation was first discovered

[155]. This corresponds to an uncertainty in DM of 0.20, which is an unacceptably





large error for the accuracy desired. If starburst galaxies are to be taken seriously as a

distance indicator for precision cosmology, it is essential that the zero-point be determined

to significantly greater accuracy. This can be accomplished via a twofold approach: a

comprehensive survey of the nearby (z < 0.1) H II galaxy population, and a survey of the

nearby, very low-luminosity, low velocity-dispersion H II regions. Accomplishing both of

these goals will allow a marked reduction in the zero-point error by significantly increasing

the sample size from the sample used in Melnick, Terlevich and Moles 1988 [155], and will

also probe the very low end of the relation between M, and LHf, where data are sparse.

A.2 Systematic Errors

The other assumptions which are inherent to this method are the choices of where

to cut on EW and on a, and the assumption that AHp is the same for starburst galaxies

as it is for local H II regions. These two sources of uncertainty (how to select the sample

and what the correct extinction law is for high redshift starburst galaxies) are systematics

that cannot be removed by improved observations. Each assumption that is made has an

associated error. The assumed extinction laws of equations (5-4), the cut on EW of 20 A,

and the cut on a of 130 km s-1 all induce inherent systematic errors. Moving the EW cut

from EW > 20 A up to EW > 25 A, as suggested in Melnick, Terlevich and Terlevich

2000 [40], would systematically raise the DM by 0.14 mag for all galaxies present in

this sample. The EW threshold for the onset of major evolutionary effects is not yet

well-established, and necessitates further research. The cut on a comes about in order to

remove contamination from objects containing multiple unresolved star-forming regions.

Since the correlation between LHp and Mz is only valid for single H II galaxies and H II

regions [158], a cut must be made to remove objects containing multiple star-forming

regions. Single H II galaxies are observed to have a gaussian distribution in a peaked at

- 70 km s-1, but objects with multiple unresolved regions are expected to have an entirely

different distribution. On the basis of Monte Carlo simulations performed to simulate

both single and multiple H II galaxies, a cut on a at 130 km s-1 retains 95 per cent of the








44






Q


0
"-


40-





38 1
8


Figure A-1.


log Mz vs. log LHfP for local H II galaxies and starburst galaxies at
intermediate and high redshifts. The solid line is the best fit of the correlation
to the local data set, flanked by the dashed lines, which give the 2-a rms
scatter. The large diamonds represent the selected high redshift data sample;
the small diamonds are the data not selected on the basis of either EW or a.
The vertical dotted line is the derived cut on a of 130 km s-1. The crosshairs
represents the typical uncertainty in each selected data point.


10 12 14
log MZ





valid, single H II galaxies, while eliminating 75 per cent of the contaminating objects. The

results of the Monte Carlo simulations can be seen in figure A-2 below. Additionally, it

can be shown that the contaminating objects which are not eliminated depart only slightly

from the empirical correlation of equation (5-1).

It is therefore essential, for any future survey, that the proper cuts be determined and

applied to EW and o, otherwise substantial uncertainties will arise from the selection of

the data sample. Finally, the derived extinction law in AHp itself, from equation (5-4), has

an uncertainty of 38 per cent, due to the fact that there are competing extinction laws

that give different results [167-169]. Both laws are comparably grey, but have different

normalizations. The difference between the two laws leads to a systematic uncertainty in

the DM of the high-redshift galaxies of 0.17 mag.

A.3 Measurement Uncertainties

There have also been assumptions made specifically to compensate for incomplete

data in the data sets of Pettini et al. 2001 [161] and Erb et al. 2003 [162]. The systematic

uncertainties that these assumptions induce can be eliminated in future surveys through

measurements of all required quantities. The assumption that the temperature-metallicity

degeneracy is most likely broken towards the upper branch of the R23 vs. O/H curve for

luminous starburst galaxies at high redshift is based on sound analysis [165], but is still a

dangerous one to make. Measurement of the 4363 A oxygen line would break the R23 vs.

O/H degeneracy, and yield a definite value for metallicity for each galaxy. Furthermore,

O/H had to be assumed for 11 of the 15 galaxies in the sample, inducing a possible

systematic which could affect DM if the assumed average O/H differs from the true

value. If the value for O/H from Shapley et al. 2004 [166] is used for the galaxies selected

from Erb et al. 2003 [162], the average DM is raised by 0.22 mag. This systematic

can be removed in future surveys by a measurement of the [O II] line at 3727 A and

the [O III] lines at 4959 A and 5007 A for each galaxy. There are also other metallicity

indicators (see Kewley and Dopita 2002 [188]) which may prove to be more reliable at






































100 200 300
sigma (km / s)


Figure A-2.


Simulation of the distributions of velocity dispersions, as would be measured
for a single, isolated star-forming region (red curve), for multiple, interacting
star-forming regions (blue curve, a simulation of multiple unresolved regions),
and for a mixture of both types (green curve). The observed data is plotted
in black. Note that the actual data appears to be a superposition of the single
region data and the multiple unresolved region data. Multiple unresolved
regions at high redshifts appear in the form of enclosed galaxy mergers,
and must be removed, as they do not follow the correlation of equation
(5-1). From the simulations, removal of most of the enclosed mergers can
be accomplished through a derived cut on ar of 130 km s-.


250


200





150


100





50





0





high redshifts. Measurement of the [N II] line at 6584 A, along with Ha, can provide

another measurement of metallicity [166]. In theory, many abundance indicators are

available and may even be practically accessible [188], and future surveys should allow

multiple, independent techniques to be used, significantly reducing errors. Note also that

it is unsettling that different metallicity indicators appear to give different values of the

metallicity for the same galaxy samples; this may be yet another source of inherent error.

There is a large uncertainty on the order of 30 per cent in the measurement of

EW for the Erb et al. 2003 [162] sample due to the difficulty of establishing the height

of the continuum. Some galaxies may have been included which should not have been,

and others may have been excluded which should have been present. The effect on the

distance modulus is estimated to be 0.16 mag, but this will be removed by measuring

equivalent width in H/3 with a higher signal-to-noise spectra for all galaxies in future

surveys. Finally, E(B V) colors, a substitute for AHp measurements, are unavailable for

galaxies from Pettini et al. 2001 [161], and were derived from an approximate correlation

noticed between E(B V) and the corrected (G R) colors in Erb et al. 2003 [162].

There is an overall uncertainty in the extinction due to the fact that the average derived

extinction for the Erb et al. 2003 [162] and the Pettini et al. 2001 [161] samples differ

by 0.34 dex. Thus, there is an induced systematic in DM of 0.17 mag, which will be

eliminated when AHP measurements are explicit taken for all galaxies.

A.4 Statistical Errors

Random errors, due to both uncertainties in measurement and to the large scatter

in the distribution of points, are perhaps the best understood of the sources of error.

Measurements of AHf are uncertain by 0.04 to 0.11 dex, depending on the galaxy's

brightness. Improved measurements, which rely on the Ha/H/3 ratio instead of solely

on E(B V) colors, may reduce the uncertainty significantly. Measurements of FHp are

uncertain by roughly 20 to 25 per cent on average, and random uncertainties in O/H are

of order 0.10 dex. The largest measurement uncertainty comes from measurements of a,





which is obtained by the broadening of the Balmer emission lines. Even relatively small

uncertainties in a of order 15 per cent can induce uncertainties in DM of 0.8 mag per

galaxy. The induced uncertainty is so large because, as seen in equation (5-2), DM is

dependent on a5, whereas it depends only linearly on the other quantities. It is therefore

vital to obtain excellent measurements of the velocity dispersion of these galaxies (which

is certainly possible, as some of the high-redshift galaxies in Pettini et al. 2001 [161] have

uncertainties of only 4 km s-1). Future work will be able to measure the Ha and HP

lines, as well as three oxygen lines, as these are where the three windows in the infrared

are in the atmosphere. For galaxies between 2 < z < 4, the Ha, H3, and multiple oxygen

lines will appear at the appropriate wavelengths. These observations should improve the

measurements of ao, further reducing the random uncertainties. The distribution of points

may not improve as statistics improve due to the intrinsic scatter on the Mz vs. LHf

relation, but random errors all fall off as the sample size increases. The errors decrease as

N-1/2, where N is the number of galaxies in the sample. Even if random errors associated

with intrinsic properties (such as FHg, a, or O/H) remain large for individual galaxies,

increasing the sample size will drive down the overall random errors. Hence, a sample

of 500 galaxies, as opposed to 15, will have its random uncertainties reduced by a factor

of 6 or better. The new generation of Near-IR Multi-Object Spectrographs (such as

FLAMINGOS and EMIR) in 10 meter class telescopes will be ideal for obtaining all

necessary measurements for such a sample.





APPENDIX B
ON AN ELECTRICALLY CHARGED UNIVERSE

The possibility that the universe could have a net electric charge has been investigated

off-and-on by many scientists for the past half-century. A net electric charge could arise

as a consequence of many different types of early-universe physics, but there are strict

limits on a net charge in the universe today from various cosmological constraints.

This appendix examines the possibility that the mathematical formalism and equations

developed in chapter 3 could remove an initial net charge from the universe through

cosmological dynamics. If this successfully occurs, many interesting physical mechanisms,

previously thought to be tightly constrained through present-day measurements, may have

operated in the early universe.

B.1 Introduction

This appendix explores the consequences of a cosmological charge asymmetry in

cosmology. One interesting effect of this was first presented in Lyttleton and Bondi 1959

[176], where it was pointed out that a sufficiently large electric charge asymmetry, on

the order of r 1 e per 1018 baryons, would enable the repulsive Coulomb force to exceed

gravitational attraction on large scales. The original motivation for this proposal was

to explain the origin of cosmic expansion. It was explained that if the magnitude of the

respective charges on electrons and protons differed by r 2 x 10-s e, large scale repulsion

would follow.

With the advent of the big bang theory, which proved to be a necessity to explain the

observed cosmic microwave background radiation (CMB, discovered in Penzias and Wilson

1965 [189]), Hubble expansion was explained as a necessary consequence of that theory.

It further appears that, to a much higher degree of accuracy than 2 x 10-18 e, the proton

and electron charges are equal. From the anisotropies of cosmic rays, which can act as a

probe of the net charge in the universe at the present day, it is determined that the overall

charge-per-baryon (A) is constrained to be |A| < 10-29 e [128]. Furthermore, the degree

of isotropy in the CMB provides constraints on the net electric charge in the universe at





a redshift of z 1089 [18], where again the restriction is that the net charge-per-baryon

is very small, |A| < 10-29 e. A net charge-per-baryon would also impact primordial

nucleosynthesis. Cosmological helium synthesis provides a probe of the net charge at very

high redshifts (z 4 x 108). Constraints from nucleosynthesis [17] indicate that, in the

early universe, |A| < 10-32 e, the most stringent constraints available on the net charge in

the universe.

If a net charge-per-baryon does exist, there are two straightforward ways to obtain

it, both of which were identified in Lyttleton and Bondi [176]. Either, as stated above,

the proton charge (q,) differs slightly from the electron charge (q%) in magnitude, or the

number density of protons (np) differs from that of electrons (ne). The former possibility

is highly disfavored, as terrestrial experiments indicate that the electric field is zero at the

Earth's surface. Assuming equal numbers of protons and electrons on Earth (- 1051 of

each species) places strict constraints on qp| -4 \qel, as do refined versions of the Millikan

experiment. It is observed that charge is quantized in units of e for physically observable

particles (although quarks are predicted to have fractional charges), as predicted by the

standard model of particle physics [53]. Barring exotic scenarios, such as electrically

charged neutrinos, photons, or dark matter [190], it is only reasonable to consider unequal

number densities of protons and electrons as giving rise to an electric charge in the

universe. The creation of an electric charge asymmetry in this fashion is analogous to the

creation of a baryon asymmetry in grand unified theories (see Dine and Kusenko 2003 [15]

for a review of baryogenesis). An electric charge asymmetry can be generated by a similar

mechanism to the baryon asymmetry, and both are expected to have the same types of

inhomogeneities [191].

B.2 Generating a Net Electric Charge

A global charge asymmetry can be generated via many channels. The most intuitive

method is to temporarily break the U(1) electromagnetic gauge symmetry [191-193].

If this U(1) symmetry is broken at some point, an electric charge asymmetry must be





produced [191]. When the U(1) symmetry is later restored, the charge asymmetry may

remain [192]. Care must be taken to ensure that electric charge remains quantized in

units of e [193]. The production of a net charge is analogous to baryogenesis in grand

unification models, which have a decay process at high energies that is asymmetric,

preferring (slightly) to produce baryons over antibaryons. Similar mechanisms could easily

prefer the production of one sign of charge over the other, so long as that production

symmetry is restored today. Other mechanisms also exist which admit the production

of a net electric charge. Examples include Kaluza-Klein models with extra dimensions

[128, 194], cosmologies with a varying speed of light [195, 196], and effective interactions

allowing electric charge non-conservation by units of 2e at a time [197].

Once a net charge has been established in the form of unequal proton and electron

densities, previous treatments assume for simplicity that the total charge within a spatial

volume is constant [128, 191, 198]. The major purpose of the work in this appendix is

to show that this assumption is not true in general. Electromagnetic forces will induce

relative motion between oppositely charged species. Charge will be conserved locally (in

that there are no charge non-conserving interactions), but the expansion rates of positive

and negative charges are found to differ. This allows currents to flow and the net electric

charge density to change with time. This appendix examines how charge asymmetries,

both local and global, evolve in an expanding universe.

As a result of the dynamics of cosmological expansion, an initial net charge can be

either removed completely or reduced significantly. So long as the cosmological bounds

on a charge asymmetry due to cosmic rays [128], the CMB [18], and nucleosynthesis

[17] are satisfied, there is no limit on any initial electric charge. The remainder of this

appendix focuses on how cosmological dynamics affect the overall charge density in an

expanding universe. Section B.3 presents an intuitive Newtonian formulation of a universe

with an initial charge asymmetry, based upon the gravitational and electromagnetic force

laws alone. While section B.3 may be useful for gathering an intuitive picture of the





cosmological dynamics, it does not capture all of the correct physical behavior. A fully

relativistic formulation, based upon the evolution equations derived in chapter 3, is found

in section B.4. This section takes into account not only the Coulomb and gravitational

forces, but also interactions between photons and charged particles, as well as all of the

other interactions associated with structure formation. Although the liv-Y i-, of section

B.4 is less intuitive, it is far more rigorous, and captures a much greater degree of the

essential physical behavior. A discussion of the preliminary results discovered here and

their implications for the early universe is presented in section B.5.

B.3 Newtonian Formulation

The simplest scenario that can be written down to explore a universe with a global

charge asymmetry is a Newtonian cosmology containing an additional Coulomb term. The

standard Newtonian cosmology is derivable from the Newtonian force law


Fi = mi ai, (B-l)


where i denotes a test particle of either a proton (p) or an electron (e). The Newtonian

gravitational law,
GN m I M
Fi = -- M (B-2)

where GN is Newton's constant, and M is the total mass enclosed by a sphere with radius

equal to the universe's expansion factor, r. As acceleration is defined as a f r, the

resultant equation for the evolution of the expansion factor becomes

T 47r
S Gp, (B-3)
r 3

where p is the energy density of the universe. Using Einstein's equations instead of

Newton's in an isotropic, homogeneous universe would modify equation (B-3) by replacing

p with (p + 3p/c2), where p is the general relativistic pressure of the universe.





A toy model of interest is an Einstein-de Sitter universe composed solely of protons

and electrons. In this model,

p mpn, + mn, (B-4)

for the energy density, where n, and nr are the proton and electron number densities,

respectively. The Coulomb force is given by

1 qiQ
Fc ,0 r (B-5)
47c0 r2

where qj is the charge of a proton (qp) or electron (q,), and Q is the total charge enclosed

by a sphere of radius r. The physics acting on protons and electrons is different, due

to their opposite charges. Therefore, when the expansion factors are derived from the

combined force laws for electrons and protons, it is found that they evolve differently than

in equation (B-3). The proton and electron expansion factors (rp and r6, respectively) are

found to be
r 47 e2
G(mn, + mene) + (np n,) (B-6)
rp 3 3com,
and
-e 4 G(mpn + mene) (n, n). (B-7)
r, 3 3Eom6
To rewrite these two equations in terms of r,, r6, and their time derivatives alone (i.e.

to remove n, and n, from the equations), N is defined to be the total number of protons

or electrons contained within a sphere of radius rp or re, respectively.

4xr 4xr
N 3npr, 3 ner 3. (B-8)

Let rp,o = re,o = ro = 1, where the subscript 0 indicates the value of a given quantity at

present. r, and re are henceforth written as fractions of their value today. The ratio of

the Coulomb to the gravitational forces appears in the cosmological evolution equations,

defined as the dimensionless parameter K, where


K -- 1.235 x 1036. (B-9)
47eo G(m, + me)2





Finally, define quantities f,, fe, and Ho by


Sm me 8Hr2
fp = fe H2 = o, (B-10)
mp + me mp + me 3

where H0 is the Hubble parameter (H) evaluated today. The evolution equations now take

the form
r? 1 (fp fe K 1 1)
HO -- -Hr3 (B- 11)
rP 2 r r f rP r
f,- 2 H 1) + l 1- r -12)
-2 + r3 +fe rP3 r3y (B }

K is large, but rp is almost equal to re, therefore the final terms in equations (B-11)

and (B-12) will only be important if there is a significant charge asymmetry. In fact, if

r, = re exactly, then both equations reduce to equation (B-3). The case of an exaggerated

net charge in the universe is illustrated in figure B-1.

The expansion factor exhibits the standard behavior for an Einstein-de Sitter

Universe, which is r/ro = (t/to)2/3. Protons and electrons are found not to flow smoothly

together, but rather to oscillate about an equilibrium which they never reach. The charge

asymmetry is not a constant over proper volume, but itself oscillates with a decreasing

frequency and decreasing amplitude. The oscillation frequency is rapid compared to the

rate of decrease of both frequency and amplitude.

Changing variables to a center-of-mass expansion factor (re) and an asymmetry

parameter (6r) assists the exploration of equations (B-11) and (B-12) in the limit of a

small charge asymmetry. The mass-weighted sum of protons and electrons (rem) and the

difference between protons and electrons (Jr) are given by

rem frp + fere, r =_ rp re, (B-13)

where 6r is experimentally and theoretically motivated to be much less than rem. The

evolution equations are then

S 12 1 3 jr2
m O2 r (K + 2pfe) -' (B-14)
rem 2 ,m 2 rem











102


10



1-



0.1 -



10-2



10 -



10-4 1 11
10-6 10- 10-4 10-3 10-2 0.1 1
t





Figure B-1. Expansion factors rp (red) and re (blue), for positive and negative charge
distributions (rising) and their time derivatives, v, and Ve (falling). The
amplitude of the asymmetry is enhanced by a factor of 104 for visibility.





r 3KH 1 (B 5)
2(B-15)
Jr 2f~fe r32'
to leading order in Jr. In the limit as Jr > 0, the standard Newtonian cosmological

expansion is recovered.

Since K is large (K >> 1), the asymmetry (Jr) behaves as a slowly decaying harmonic

oscillator over short timescales. It is useful to parametrize Jr = A cos q. Calculation of Sr

shows, in conjunction with equation B-15, that

3KH2 1
2 A2 constant. (B-16)

Equation (B-16) illustrates that A C< 02A, since 02 KH2, whereas A/A H2.

(Recall that K -- 1.235 x 1036.) Because K > 1, the amplitude of the asymmetry, the

change in the amplitude of the asymmetry, and the expansion rate of the universe all vary

slowly with respect to the oscillation frequency.

From equation (B-16) and the definition that 4 = wt, the oscillation frequency at any

epoch is

w = s eno Ho(1 + z)3/2 (B-17)
8VEoGmpme
where no is the number density today (no = 1), and z is the redshift of interest. This

corresponds to a frequency today (z = 0) of 134 rad s-1, or approximately 21 Hz. The

oscillation frequency at any epoch in an Einstein-de Sitter universe is therefore 21(1 + z)3/2

Hz.

A slightly more sophisticated treatment includes radiation in the universe. For

radiation, the pressure (Prad) is given by prad/C2 = jPrd. The overall energy density of the

universe is modified by the additional term


(Prad + 3Prad) = 2 ((B-18)

where (E-) is the average energy of a photon, nB is the number density of baryons, and

rq 6.1 x 10-10 is the baryon-to-photon ratio today. With radiation included, the equations