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The internal structure of cold dark matter haloes [Elektronische Ressource] / vorgelegt von Mark Philipp Vogelsberger

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Ludwig-Maximilians-UniversitätThe internal structureof Cold Dark Matter HaloesDissertation der Fakultät für Physikder Ludwig-Maximilians-Universität Münchenfür den Grad desDoctor rerum naturaliumvorgelegt von Mark Philipp Vogelsbergeraus Bad KreuznachMünchen, 10.9.20091. Gutachter: Prof. Dr. Simon D. M. White2. Gutachter: Prof. Dr. Andreas M. BurkertTag der mündlichen Prüfung: 23.4.2010ContentsContents 1Zusammenfassung 5Abstract 9Introduction 13I General overview 191 Cosmology - a brief introduction 211.1 Theory of gravity - General Relativity . . . . . . . . . . . . . . . . . . . . 231.2 Cosmological Principle and the Metric of the Universe . . . . . . . . . . . . 241.3 The Friedmann equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.4 Cosmological vocabulary and relations . . . . . . . . . . . . . . . . . . . . 251.5 The Einstein-de Sitter Universe . . . . . . . . . . . . . . . . . . . . . . . . 272 Structure Formation 292.1 The growth of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 Collisionless medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3 Zel’dovich approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Dark Matter 393.1 Observational Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 N-body simulations . . . . . . . . . . . . . . . . . . . .

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Ludwig-Maximilians-Universität

Theinternalstructure
ofColdDarkMatterHaloes

DissertationderFakultätfürPhysik
derLudwig-Maximilians-UniversitätMünchen
fürdenGraddes
Doctorrerumnaturalium

vorgelegtvonMarkPhilippVogelsberger
ausBadKreuznach

München,10.9.2009

1.

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Gutachter:

Gutachter:

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Andreas

TagdermündlichenPrüfung:

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M.Burkert

23.4.2010

Contents
Contents1
Zusammenfassung5
Abstract9
Introduction13
IGeneraloverview19
1Cosmology-abriefintroduction21
1.1Theoryofgravity-GeneralRelativity....................23
1.2CosmologicalPrincipleandtheMetricoftheUniverse............24
1.3TheFriedmannequations...........................25
1.4Cosmologicalvocabularyandrelations....................25
1.5TheEinstein-deSitterUniverse........................27
2StructureFormation29
2.1Thegrowthofperturbations..........................31
2.2Collisionlessmedium..............................33
2.3Zel’dovichapproximation............................36
3DarkMatter39
3.1ObservationalEvidence.............................41
3.2Candidates...................................43
3.3N-bodysimulations...............................45
3.4Acriticalcommentandthesisobjectives...................47
IICoarse-graineddarkmatterdistribution51
4Phase-spacestructureinthelocaldarkmatterdistribution53
4.1Introduction...................................55
4.2TheNumericalSimulations...........................56
4.3Spatialdistributions..............................58
4.4Velocitydistributions..............................62
1

4.5Energydistributions..............................67
4.6Detectorsignals.................................73
4.7ConclusionandDiscussion...........................80

IIIFine-graineddarkmatterdistribution83
5Fine-grainedphase-spacestructureofCDMHaloes85
5.1Introduction...................................87
5.2Thegeodesicdeviationequation........................90
5.3TheDaMaFlowcode..............................97
5.4Integrablepotentials..............................99
5.5Non-integrablepotentials............................103
5.6TriaxialDarkMatterhaloes..........................110
5.7Fine-grainedphase-spaceanalysisinN-bodycodes..............117
5.8ConclusionandDiscussion...........................127

6DarkMatterCaustics129
6.1Introduction...................................131
6.2Idealisedinitialconditionsforstructureformation..............132
6.3Evolutionofthedarkmatterdistribution...................133
6.4Variationofthe3-densityalongparticletrajectories.............136
6.5ConclusionandDiscussion...........................140

7CausticsingrowingColdDarkMatterHaloes143
7.1Introduction...................................145
7.2Numericaltechniques..............................146
7.3Results......................................148
7.4ConclusionandDiscussion...........................165

8Simulatingthene-grainedphase-spaceofCDMHaloes169
8.1Introduction...................................171
8.2Initialconditions................................171
8.3Numericaltechniques..............................171
8.4Results......................................178
8.5ConclusionandDiscussion...........................184

IVConclusions
9Conclusions

VAppendices
AMeanphase-spacedensitycalculation

781981

915791

BGDEanalysisofthe1Dself-similar

CSommerfeldenhancement

Acknowledgement

IV

Bibliography

tcee

infall

199

302

702

029

Zusammenfassung
DerzeitfavorisiertekosmologischeModellezurStrukturbildungimUniversumnehmen
an,dasseingroßerAnteilderdarinenthaltenenMasse“dunkel”ist.DieseDunkleMa-
terieverrätsichdurchihrengravitativenEinuss.Beispielehierfürsinddieachen
RotationskurvenvonSpiralgalaxienunddiehohenGeschwindigkeitenvonGalaxienin
Galaxienhaufen.DerGravitationslinseneektkannheutzutagedazuverwendetwerden,
dieVerteilungDunklerMateriezuvermessen.TrotzderTatsache,dassHinweisefür
DunkleMaterieseitnunmehrals75Jahrevorhandensind,istimmernochnichtver-
standen,woraussiebesteht.DieTeilchenphysikbieteteinigeinteressanteundgutmo-
tivierteKandidaten,aberdasgesuchteDunkleMaterieTeilchenwurdebishernochnicht
entdeckt.AusdiesemGrundistdieSuchenachDunklerMaterieeinesdergemein-
samenHauptanliegenvonKosmologieundTeilchenphysik.DieeinzigeMöglichkeit,die
DunkleMaterieHypothesezubeweisen,istderdirekteNachweisvonDunkleMaterie
Teilchen.ExperimenteverwendenhierzuverschiedeneTechnikenzumNachweisdieser
Teilchen.AlldieseExperimentebenötigenjedochInformationenüberdieDichte-und
GeschwindigkeitsverteilungderDunklenMaterie.DiesekannnurmittelsderKosmolo-
gieundderTheoriederStrukturbildungbestimmtwerden.DasZieldieserDoktorarbeit
istes,diePhasenraumstrukturderDunklenMaterienahedesSonnensystemsundim
DunklenMaterieHaloderMilchstraßevorauszusagen.
EingroßerTeildieserArbeitistderAnalysedergrobkörnigenPhasenraumstrukturder
DunklenMaterienahederSonnegewidmet.DieseAnalysebasiertaufSimulationendes
AquariusProjektes,derderzeitgrößtenSimulationzumStudiumDunklerMaterieHalos
vonderArt,wiesiedieMilchstraßeumgeben.BasierendaufdiesenRechnungensagen
wirvoraus,dassdielokaleDunkleMaterieVerteilungsehrgleichförmigist:dieDichte
nahederSonnevariiertvondemMittelübereinebestangepassteellipsoidaleDichtekon-
turumwenigerals15%bei99.9%Kondenz.DielokaleGeschwindigkeitsverteilungist
ebenfallssehrgleichmäßig,abersieweichtsystematischvoneiner(multivariaten)Gauß-
Verteilungab.DieUrsachehierfüristnichtdasVorhandenseinindividuellerDunkler
MaterieKlumpenoderStrömungen,sonderndasAuftretenbreiterMerkmaleinden
VerteilungendesBetragsderGeschwindigkeitundderEnergieverteilung.DieseMerk-
malesindunveränderlichinRaumundZeitundspiegelndieEntstehungsgeschichtedes
DunkleMaterieHaloswider.WeiterhinhabendieseMerkmaleeinensignikantenEinuss
aufdiezuerwartendenSignaleinWIMPundAxionSuchexperimenten.Beispielsweise
könnenWIMP-Rückstoß-Ratenumbiszu10%vondenzuerwartetenRaten,basierend
aufeinerbestangepasstenmultivariatenGauß-Verteilung,abweichen.VonderSimulation
vorhergesagteAxion-SpektrenhabenihrMaximumtypischerweisebeiniedrigerenFre-
quenzen,alsesaufgrundeinermultivariatenGauß-Verteilungzuerwartenwäre.Auchin
diesemFallzeigendieSpektrenMerkmale,diedurchdieEntstehungsgeschichtedesHa-
loshervorgerufenwerden.Diesimpliziert,dassfallsdieDetektionvonDunklerMaterie
zurRoutinegewordenist,dieseArtvonExperimentenesermöglichenwird,Einblicke
indieEntstehungsgeschichtedesHaloszuerlangen.DannwirddasForschungsfeldder
“DunklenMaterieAstronomie”aufkeimen.
DerHauptteildieserArbeitbeschäftigtsichmitderfeinkörnigenPhasenraumstruktur
derDunklenMaterieVerteilungnahederSonne.Wirpräsentiereneineneueundallge-
meingültigeMethode,umdiefeinkörnigePhasenraumstrukturimgesamtengalaktischen
Halozuberechnen.ZieldieserTechnikistes,dieStrukturDunklerMaterieaufden

6

Skalenzubestimmen,diefürdirekteundindirekteDetektionsexperimenterelevantsind.
UnsereMethodebasiertaufderGleichungfürdiegeodätischeAbweichung,diefürjedes
individuelleSimulationsteilchengelöstwird.DieTechnikbenötigtkeineAnnahmenüber
dieSymmetrieoderStationaritätdesHalo-Entstehungsprozesses.Wirdiskutierenmitder
neuenMethodediePhasenraumstrukturallgemeinerstatischerPotentiale,dieeinekom-
pliziertereStrukturalszuvoranalysierteseparablePotentialeaufweisen.Wirzeigen,dass
füreinellipsoidaleslogarithmischesPotentialmiteinemKerndasPhasenmischenvonder
ResonanzstrukturunddamitvonderZahlunabhängigerfundamentalerFrequenzenab-
hängt.UnserMethodeerlaubtauchdieIdentikationchaotischerPhasenraumbereiche,
diesichdurcheinensehrstarkenAbfallderStrömungsdichteauszeichnen.Wirberechnen
dieStrömungsdichtenineinemellipsoidalemNFWHaloProlmitradialvariierenderPo-
tentialformundzeigen,dasseinsolchesModellungefähr105DunkleMaterieStrömean
derPositionderSonnefürdendunklenHalounsererMilchstraßevorhersagt.Derwich-
tigsteundneuesteAspektdervonunsvorgestelltenMethodebestehtdarin,dasssierelativ
einfachinkosmologischeN-KörperProgrammeeingebundenwerdenkann.Wirpräsen-
tiereneinesolcheImplementierungundzeigen,dassDiskretheitseekteinrealistischen
SituationenunterKontrollegehaltenwerdenkönnen.
DieneuentwickelteTechnikerlaubtauchdieAnalysevonKaustikeninderDun-
klenMaterieVerteilungundeinedetaillierteBerechnungderdurchsiehervorgerufe-
nenAnnihilationsstrahlung.KaustikensindeinegenerelleEigenschaftdesnichtlinearen
WachstumskalterDunklerMaterie.WäreDunkleMaterieabsolutkalt,sowürdeihre
MassendichteindenKaustikendivergieren,genausowiedieintegrierteAnnihilations-
rateindividuellerDunklerMaterieTeilcheninderKaustik.RealistischeDunkleMa-
terieKandidatenbesitzenallerdingseinenicht-verschwindendeanfänglichethermische
Geschwindigkeit,wodurchdiesesdivergenteVerhaltenregularisiertwird.Wirbeschreiben
einmathematischesVerfahrenzurAnalysedieserKaustiken.DiesesSchemakanndirekt
inN-KörperSimulationsprogrammeeingebautwerden.DadurchkönnendannKaustiken
identiziertundihreAnnihilationsstrahlungberechnetwerden.
WirverwendendieseMethoden,umdiefeinkörnigePhasenraumstrukturundKaus-
tikenvonisoliertenHalos,diesichausselbst-ähnlichenundsphärischsymmetrischenAn-
fangsbedingungenformen,zuanalysieren.WirverwendeneinmodiziertesN-KörperPro-
gramm,umdieGleichungfürdiegeodätischeAbweichungfürjedeseinzelneSimulations-
teilchenzulösen.DieradialeOrbitinstabilitäthatzurFolge,dassderHaloeinAchsen-
verhältnisvon10zu1iminnerenBereichentwickelt.DieHaloswachsenähnlichmit
derZeitundhabenähnlicheDichteprolewiediesphärischeanalytischeLösung,aber
ihredetaillierteStrukturistsehrunterschiedlich.AufgrundderhöherenDimensional-
itätderOrbitsnehmendieStrömungs-undKaustikdichtenvielschnellerab,alsinder
selbst-ähnlichenLösung.HierdurcherhöhtsichdieZahlderStrömungenanjedemPunkt
desHalos.Bei1%desUmkehrradius(wasungefährderSonnenpositioninderMilch-
straßeentspricht)besitztderHalocirca106Strömungen,imGegensatzzu102fürdie
selbst-ähnlicheLösung.DieZahlderKaustikeniminnerenHalosteigtumeinigeFak-
torenan,daeintypischerOrbitnunsechsstattnurzweiUmkehrpunktehat.Dadie
Kaustikdichtenjedochsignikantniedrigersind,reduziertsichderAnnihilationsbeitrag
vonKaustikenzurGesamtannihilationrateauf4%imVergleichzu6.5%inderselbst-
ähnlichenLösung.KaustikentragensehrvielwenigerbeikleinenRadienbei.Diesen

7

Wertenliegtein100GeVc2NeutralinomiteinerheutigenGeschwindigkeitsdispersion
von0.03cms1zugrunde.EineReduktionderDispersionumzehnGrößenordnungen
führtlediglichzurVerdopplungderKaustikhelligkeit.DarumsindKaustikenimInneren
desHalosnichtbeobachtbar.LediglichKaustikenimAußenbereichkönnenmöglicherweise
detektiertwerden.
ImletztenTeildieserArbeitbenutzenwirdieentwickeltenMethoden,umDunkleMa-
terieHalos,diesichineinemCDMUniversumentwickeln,zuanalysieren.Wirwenden
dieTechnikaufdreiverschiedeneAuösungendesAq-AHalosausdemAquariusProjekt
an.AlleSimulationenbenutzendiegleicheSofteninglänge,undwirändernnurdieAn-
zahlderSimulationsteilchen.WirerreichendamitguteKonvergenzinallenrelevanten
feinkörnigenPhasenraumeigenschaften:Kaustikdurchgänge,Strömungsdichten,Zahlder
StrömungenundAnnihilationsstrahlunginnerhalbderStrömungen.NahedesVirialradius
ndenwir107Strömungen.KaustikdichtensindsubdominantinnerhalbdesVirialradius:
nahedesVirialradiussinddieKaustikdichtenmitdermittlerenHalodichtevergleichbar,
bei10%desVirialradiussinddieKaustikdichtenallerdingsschonumeinenFaktor106
kleineralsdiemittlereHalodichte.DiesisteineFolgedesezientenPhasenmischens
imHalo.NahedesVirialradiustragendieKaustikenungefähr10%zurAnnihilations-
strahlungbei,abernur0.1%bei10%desVirialradius.

8

Abstract
CurrentlyfavouredcosmologicalmodelsforstructureformationoftheUniverseassume
thatalargefractionofthemassoftheUniverseis“dark”.Theevidencefordarkmatter
comesfromobservationsofitsgravitationalinuence.Examplessuchastheatnessof
rotationcurvesofspiralgalaxiesorthelargevelocitiesofgalaxiesingalaxyclustersare
thoughttobemanifestationsofitspresence.Gravitationallensingcannowalsobeused
tomapthedarkmatterdistributionoftheUniverse.Despitethefactthattheevidence
fordarkmatterhasexistedformorethan75years,itisstillnotclearwhatdarkmatter
ismadeof.Particlephysicsprovidessomeinterestingandwell-motivatedcandidates,
buttheelusivedarkmatterparticleshavenotyetbeendetected.Therefore,thehunt
fordarkmatterisoneofthemajorjointeortsofcosmologyandparticlephysics.The
onlywaytoprovethedarkmatterhypothesisisthedirectdetectionofdarkmatter
particlesinalaboratory.Experimentsexploitvarioustechniquestodetectdarkmatter
particles.Alloftheseexperimentsrequireasinputthephase-spacedistributionofdark
matter.Thismeanstheyrequireinformationontheconguration-spaceandvelocity-
spacedistributions.Theseinsightscanonlycomefromcosmologyandthetheoryof
structureformationintheUniverse.Thegoalofthisthesisistopredicttheexpected
darkmatterphase-spacedistributionnearthesolarsystemandinthedarkmatterhalo
oftheMilkyWay.
Alargepartofthisthesisisdedicatedtoadetailedanalysisofthecoarse-graineddark
matterdistributionneartheSunbasedontheAquariusproject,thecurrentlylargestset
ofMilkyWay-likedarkmatterhalosimulations.Basedonthesesimulationswepredict
thelocaldarkmatterdensitydistributiontoberemarkablysmooth:thedensityatthe
Sundiersfromthemeanoverabest-tellipsoidalequidensitycontourbylessthan
15%atthe99.9%condencelevel.Thelocalvelocitydistributionisalsoverysmooth,
butitdierssystematicallyfroma(multivariate)Gaussiandistribution.Thisisnotdue
tothepresenceofindividualclumpsorstreams,buttobroadfeaturesinthevelocity
modulusandenergydistributionsthatarestablebothinspaceandtimeandreectthe
detailedassemblyhistoryofeachhalo.Thesefeatureshaveasignicantimpactonthe
signalspredictedforWIMP(weaklyinteractingmassiveparticle)andaxionsearches.For
example,WIMPrecoilratescandeviateby10%fromthoseexpectedfromthebest-t
multivariateGaussianmodels.Theaxionspectrainthesimulationstypicallypeakat
lowerfrequenciesthaninthecaseofmultivariateGaussianvelocitydistributions.Alsoin
thiscase,thespectrashowsignicantimprintsoftheformationofthehalo.Thisimplies
thatoncedirectdarkmatterdetectionhasbecomeroutine,featuresinthedetectorsignal
willallowthestudyofthedarkmatterassemblyhistoryoftheMilkyWay.Aneweld,
“darkmatterastronomy”,willthenemerge.
Themainpartofthisthesisfocusesonthene-grainedphase-spacestructureofthe
darkmatterdistributionneartheSun.Anewandcompletelygeneraltechniquefor
calculatingthene-grainedphase-spacestructureofdarkmatterthroughouttheGalactic
haloispresented.Itsgoalistounderstanddarkmatterstructureonthescalesrelevantfor
directandindirectdetectionexperiments.Themethodisbasedonevaluatingthegeodesic
deviationequationalongthetrajectoriesofindividualdarkmattersimulationparticles.
Itrequiresnoassumptionsaboutthesymmetryorstationarityofthehaloformation
process.Generalstaticpotentialsthatexhibitmorecomplexbehaviourthantheseparable
potentialsstudiedpreviouslyarediscussed.Forellipsoidallogarithmicpotentialswitha
core,phasemixingissensitivetotheresonancestructure,asindicatedbythenumberof

01

independentorbitalfrequencies.Regionsofchaoticmixingcanbeidentiedbythevery
rapiddecreaseintheconguration-spacedensityoftheassociateddarkmatterstreams.
ArelevantanalysisismadeontheevolutionofthestreamdensityinellipsoidalNFW
haloeswithradiallyvaryingisopotentialshape,showingthatifsuchamodelisapplied
totheGalactichalo,atleast105streamsareexpectedneartheSun.Themostnovel
aspectofthenewapproachisthatgeneralnon-staticsystemscanbestudiedthroughits
implementationincosmologicalN-bodycodes.Thenewschemeisembeddedinacurrent
state-of-the-artN-bodycode.TestsdemonstratingthatN-bodydiscretenesseectscan
bekeptundercontrolinrealisticcongurationsarepresented.
Thenewmethodalsoallowsananalysisofcausticsinthedarkmatterdistributionand
adetailedcalculationoftheannihilationradiationassociatedwiththem.Causticsarea
genericfeatureofthenonlineargrowthofstructureinthedarkmatterdistribution.If
thedarkmatterwereabsolutelycold,itsmassdensitywoulddivergeatcaustics,andthe
integratedannihilationprobabilitywouldalsodivergeforindividualparticlesparticipating
inthem.Forrealisticdarkmattercandidates,thisbehaviourisregularisedbysmallbut
non-zeroinitialthermalvelocities.Amathematicaltreatmentofevolutionfromhot,warm
orcolddarkmatterinitialconditionsisgiven.Thisschemecanbedirectlyimplemented
incosmologicalN-bodycodes.Itallowstheidenticationofcausticsandtheestimation
oftheirannihilationradiationinfullygeneralsimulationsofstructureformation.
Themethodsdevelopedforthene-grainedphase-spaceandcausticanalysisareap-
pliedtothegrowthofisolateddarkmatterhaloesfromself-similarandsphericallysym-
metricinitialconditions.AmodiedN-bodycodeintegratesthegeodesicdeviationequa-
tioninordertotrackthestreamsandcausticsassociatedwithindividualsimulation
particles.Theradialorbitinstabilitycausesthehaloestodevelopmajor-to-minoraxis
ratiosapproaching10to1intheirinnerregions.Theygrowsimilarlyintimeandhave
similardensityprolestothesphericalsimilaritysolution,buttheirdetailedstructureis
verydierent.Thehigherdimensionalityoftheorbitscausestheirstreamandcaustic
densitiestodropmuchmorerapidlythaninthesimilaritysolution.Thisresultsina
correspondingincreaseinthenumberofstreamsateachpoint.At1%oftheturnaround
radius(correspondingroughlytotheSun’spositionintheMilkyWay)wendoforder
106streamsinoursimulations,ascomparedto102inthesimilaritysolution.Thenumber
ofcausticsintheinnerhaloincreasesbyafactorofseveral,becauseatypicalorbithassix
turningpointsratherthanone,butcausticdensitiesdropbyamuchlargerfactor.This
reducesthecausticcontributiontotheannihilationradiation.Fortheregionbetween
1%and50%oftheturnaroundradius,thisis4%ofthetotalinoursimulatedhaloes,
ascomparedto6.5%inthesimilaritysolution.Causticscontributemuchlessatsmaller
radii.Thesenumbersassumea100GeVc2neutralinowithpresent-dayvelocitydisper-
sion0.03cms1,butreducingthedispersionbytenordersofmagnitudeonlydoublesthe
causticluminosity.Therefore,causticswillbeunobservableintheinnerpartsofhaloes.
Onlytheoutermostcausticmightpotentiallybedetectable.
Finally,wepresentresultsonthene-grainedphase-spacestructureofcolddarkmatter
haloesgrowingintheconcordanceCDMcosmology.Weusethegeodesicdeviation
techniquetofollowthelocalphase-spaceevolutionofindividualsimulationparticles,and
weapplythismethodtothreedierentresolutionsoftheAq-AhalooftheAquarius
project.Weuseaxedsofteninglengthandonlychangethenumberofparticles.Good

11

convergenceisachievedforallne-grainedpropertiesofthehalo:causticpassages,stream
densities,numberofstreamsandintra-streamannihilationradiation.Atthevirialradius
weexpectabout107streams.Wendcausticdensitiestobesubdominantwithinthe
virialradius:atthevirialradiusthemaximumcausticdensityiscomparabletothemean
halodensity,whereasat10%ofthevirialradiusthecausticdensityisalreadyafactor106
smallerthanthemeandensity.Weattributethistotheveryecientphase-spacemixing.
Thecontributionofcausticstotheannihilationradiationattheturnaroundradiusis
about10%,butwellbelow0.1%at10%ofthevirialradius.

21

Introduction

Mankindhasalwaysaskedthequestion“Wheredowecomefrom?”.Bothreligionand
philosophycameupwithideasonhowtoanswerthisquestion.Butnoneofthesewere
scienticinthesensetheywerebasedonconstructionsofideasratherthanempirical
facts.Astronomy,beingtheoldestscienticdiscipline,hascontributedtothesequestions
signicantlyfromtheverybeginning,withhumancuriosityasthemaindriverinthat
eld.Butitwasalongwayuntilcosmologyasabranchofastronomicalresearchbecame
aprecisionscienceinitsmodernsense.Earlydiscussionsofcosmologicalideashadmore
theavourofphilosophicalandmetaphysicaldebatesontheoriginofourUniverse.It
wasduringthelastcenturywhenthischangedforthersttimeanddramatically.Weare
nowadaysinapositionwheremodernsciencecanattempttogoveryclosetothecreation
oftheUniverseandtounderstandhowitworksindetail.
ItisthepassionforunderstandingourUniverseandourexistencethathasdrivenall
majorbreakthroughsincosmology.Themainenginesforthisprogress,especiallyover
thelastdecades,havebeenthedevelopmentofpath-breakingobservationaltechniques
thatcould,forthersttime,conrmideasaboutourUniverseandthereforeremovethe
purephilosophicalcharacterfromcosmologyandturnitintooneofthemostexciting
scienticadventuresofthecurrentepoch.
Itwasbackinthe60softhelastcenturywhentheradiationofthecosmicmicrowave
background(CMB)wasaccidentallyfoundbyPenziasandWilson.Thisradiationwas
predictedbyGamowinthe40sastheafterglowoftheBigBang.Itwastherstpieceof
evidenceforthehotbigbangmodel,whichistodaythestandardmodelforthebeginning
andevolutionofourUniverse.CMBobservationsbecamemorepreciseoverthelast
decadesanditistodaypossibletoseewhattheUniverselookedlikejustabout380.000
yearsaftertheBigBanghappened.Atthebeginningofthe21stcenturytheWMAP
satellitehasseentheseedsofstructureformationinthisradiationastinytemperature
variations.Thereforewearetodayinapositionwhereweareclosertothebirthof
ourUniversethaneverbefore.Itisanenormoussuccessofmankindunderstandingto
investigatethefullUniverseanditshistorybasedonobservationsfromatinyplanetcalled
Earth.
Itisnotonlythekindofquestionscosmologyasksthatmakesitdierentfrommany
otherscienticdisciplines,itisalsothespeedofdevelopmentoftheeld.Agoodexample
ofdramaticchangesincosmologyisthediscoveryofthecosmologicalconstant.Thisis
ahundredyearsoldstorywhichbeganwhenEinsteinformulatedhistheoryofgeneral
relativityasanecessaryextensionofNewton’stheoryofgravity.Thistheorywasamajor
breakthroughinourinterpretationofthespace-time,postulatingthatmatterdeforms
it.WhenEinsteinwrotedownhisequations,hefoundthattheydonotallowastatic

41

Universe.ButatthistimeitwasgenerallybelievedthattheUniversewasstatic,neither
expandingnorcontracting.Thushisequationsseemedtobewrongwhenappliedto
thewholeUniverse.Torescuehistheoryandgetastaticsolution,headdedanow
famoustermtotheseequations:theso-calledcosmologicalconstant.Thismadethem
compatiblewithastaticUniverse.SomeyearslaterHubblefoundthattheUniverseis
notstatic,butexpanding,sotherewasnoneedforanymore.Later,Einsteinthought
thattheintroductionofthecosmologicalconstantwasthebiggestmistakeheevermade
inhislife.Infollowingdecadesmostpeopleassumedthatwaszeroandwasnot
requiredtodescribeourUniverse.Thecosmologicalconstantdied,sotospeak,with
Hubble’sobservationoftheexpandingUniverse.Butinthelate90softhelastcentury
twosupernovaeobservationgroupsrealisedthatourUniverseseemstoexpandfasterthan
inthepast:itshowsanacceleratedexpansion.Thisbehaviourcouldonlybeexplained
bya-liketerminEinstein’sequations.Thistermcontributesabout70%ofthetotal
energydensityofourUniversetoday!Fromonedaytothenext,cosmologistsrealised
thattheyhadmissed70%oftheUniverseinalltheirequations.Oneshouldremember
thatthisdramaticchangehappenedjust10yearsago!Thisshowshowfasttheeldis
evolving.Todaythephenomenaisgiventhemoregeneralname“darkenergy”.
Althoughcosmologyisevolvingsofastitalsohaslongstandingproblemsjustlikeother
eldsofscience.Oneofthebiggestmysteriesistheso-called“darkmatter”problem.This
datesbacktoobservationsbyZwickyin1933.HeobservedgalaxiesintheComacluster,
measuringhowfasttheymovethroughthiscluster.Tohissurprise,hefoundthattheyare
movingextremelyfast,sofastthattheyshouldyrightoutofthecluster,becausethere
appearedtobetoolittlemasstogravitationallybindthemtothecluster.Sincethecluster
seemsstableandlonglived,Zwickyinferredthatitmustbealotheavierthanexpected
basedontheestimatedmassesofallthegalaxies.Someunseenmattercomponentwhat
gluetheclustertogetherthroughitsgravitationalattraction.Thiswasthebirthofthe
“unseen”or“dark”matterproblem.Thisnamesimplyreectsthefactthatthereappears
tobematterpresent,butitdoesnotemitanyelectromagneticradiation.Laterinthe70s
Rubinfoundasimilarproblemwhensheobservedspiralgalaxies.Sherealisedthatthe
outerregionsofthesegalaxiesarerotatingtoofasttobeheldinbythevisiblestars.The
onlyexplanationwastoassumesomesortofdarkmatterhalosurroundingthegalaxythat
keepsthestarsandgastogether.Nowadayswehaveevenmoreevidencefordarkmatter
comingfromdetailedobservationsoftheCMBandfromgravitationallensingstudies.
LensingstudiesaresensitivetothebendingoflightbymatteraspredictedbyEinstein’s
theoryofgeneralrelativity.Thistracesallkindofmatter,includingdarkmatter.All
thesedierentobservationsconsistentlyindicatethatabout25%oftheUniverseconsists
ofdarkmatter.
Thebottomlineofcurrentstate-of-the-artcosmologyistherefore,thatwedonotknow
what(70+25)%=95%oftheUniverseismadeof!Weonlyknow5%oftheUniverse.
Thistinyamountaccountsforallthestars,galaxies,planets,gas,andanylife-formslike
thereaderofthisthesis.
Thisfactshowsthatalthoughcosmologyisevolvingfast,itisstillfacingsomeof
themostfundamentalunresolvedquestionsofNature;ithasnotgivenananswertothe
question:“WhatistheUniversemadeof?”Thisisaveryseriousproblem,butalsoan
excitingone.

51

Thedarkenergyanddarkmatterproblemsarequalitativelydierent.Thedarkenergy
componentaccountsfor70%oftheUniversetodaybutatearliertimesitwassubdominant
andtheUniversewasdominatedbydarkmatteror(atveryearlytimes)byradiation.To
explainthegrowthofstructureandtheevolutionoftheUniversefromearlytimesuntil
today,itismorecriticaltounderstandthedarkmatterproblem.
Darkmatterhasanotherbigadvantage.Mostofthedarkmattercandidatesoer
apossibilityofdirectdetection.Moderncosmologicaltheoriesassumethatgalaxieslike
theMilkyWayareembeddedinalargehaloofdarkmatterparticles.Ifthisistrue,
theEarthisyingthroughaseaofdarkmatterparticlesandeverythingonEarthis
penetratedbyaconstantuxofdarkmatterparticles.Manytheoriespredictthatdark
matterparticlesshouldinteractwithordinarymatterviaweakinteractions.Therefore,
itshouldatleastinprinciplebepossibletodetectthesemysteriousparticlesusinga
sensitiveenoughdetector.Unfortunately,detectionisnotstraightforward.Themain
complicationisthatthecross-sectionforreactionsbetweenthedetectormaterialand
theelusivedarkmatterparticleisextremelysmall,evensmallerthanthatforneutrino
interactions.Therefore,itisextremelyhardtodetecttheseparticles.Anotherproblem
isthatthedarkmatterparticlepropertiesarenotstronglyconstrained.Itistherefore,
astrangesituation:Weassumethatthereisaconstantuxofdarkmatterparticles
throughthesolarsystem,butneverthelessthedetectionisdicultbecauseofthelow
reactioncross-sections.Currentlymorethan20experimentsallovertheworldsearchfor
theelusivedarkmatterparticle.
Anotherwayofdarkmatterdetectionexploitsthefactthatsomedarkmattercandi-
datesareabletoself-annihilate:iftwoofthemcollide,theycanproduce-radiationand
standardmodelparticles.Thisradiationshouldbestrongestinthedensestregionsofthe
darkmatterdistribution.Thesearethecentresofhaloesorsubhaloesofdarkmatter,
where,forexample,theMilkyWayisembedded.Manyexperimentsarealsosearching
forthesesignals.Recently,PAMELAandotherexperimentsreportedanomaliesinthe
cosmicrayspectrathatcouldbeduetodarkmatterannihilation.Butuntilnowordinary
astrophysicalexplanationscannotbeexcluded.
Fordarkmattersearchexperimentsitisusefultoknowhowthedarkmatterparticles
aredistributedintheGalactichalo.Especially,fordirectsearchesitisimportanttoknow
howdarkmatterisdistributednearthesolarsystem,sincethedetectionratesdependon
thedarkmatterphase-spacedistribution.
Howcanwepredictthedistributionofsomethingwhenwedonotknowwhatitis?It
istruethatwecurrentlydonotknowwhatdarkmatterismadeof,butweknowhowit
shouldbehave.Itinteractsalmostexclusivelygravitationallywithitselfandothermatter
andsoevolveslikeacollisionlessself-gravitatinguid.Duetothegreatprogressspendin
CMBobservations,wenowknowtheinitialconditionsfornonlinearstructureformation
intheUniverse.Basedontheseinitialconditionswecanfollowtheevolutionofthedark
matterparticles.Theequationsofmotionneedtobemodiedonlyslightlytoaccount
fordarkenergythatisnegligibleatthebeginningofstructureformationbutdominant
today.Moderncosmologyuseslargesupercomputerstosolvetheseequationsandcreate
avirtualUniverseinthecomputer.Onecanthenusethesemodelstogetanideaabout
thedarkmatterdistributionofourUniverse.
Thegoalofthisthesisistousesuchcomputersimulationstopredictthedarkmatter

61

distributionneartheSunandthereforesupplydarkmattersearchexperimentswiththe
requireddarkmatterphase-spacedistribution.Thisisachallengingproblemandithas
onlybecomepossiblewithinthelastyearstorunsimulationswithenoughresolutionfor
thistask.Inthisthesiswewillpresentresultsbasedonthebiggestcomputersimulation
projectfollowingthestructureformationofMilkyWay-likedarkmatterhaloes.Weuse
asetofsimulationstoanalysethephase-spacestructureneartheSunandpredictdark
matterdetectorsignalsforthersttime.Althoughthesimulationsofthisprojectarethe
biggestdonesofar,theyarestillquitelimitedifoneisinterestedinthesmallestlength
scalesrelevantfordarkmattersearches.Wethereforepresentacompletelynewtechnique
toanalysethedarkmatterdistributiononscalesfarbelowwhatispossibletodaywith
standarddarkmattersimulations.Thisnewtechniqueallowsustomakepredictionsfor
thene-grainedphase-spacestructureofdarkmatterhaloes.Inthisthesiswepresentthe
mostdetailedpredictionsforthedarkmatterdistributiontodate.
Therstthreechaptersofthisthesispresentanintroductiontotheeldofcosmology
anddarkmatter.Intherstchapterwedescribethebasicsofgeneralrelativity.Basedon
this,wederivetheFriedmannequationsthatdescribetheevolutionofahomogeneousand
isotropicUniverse.Inthesecondchapterwefocusonthequestionhowstructurecangrow
inanexpandingUniverse.Wedemonstratethatthiscaneasilybedoneanalyticallyin
thelinearregime,wherestructureformationisdrivenbysmalldensityuctuationsabout
auniformdensitydistribution.Inthethirdchapterwewillpresentsomeobservational
factsthatpointtowardstheexistenceofdarkmatter.Weclosethatchapterwithexplain-
ingthemaintechniquesusedtoruncomputersimulationsofstructureformation.We
startourdetailedstudyofthedarkmatterdistributioninthesecondpartofthisthesis.
Inchapter4weuseasetofveryhighresolutiondarkmattersimulationstopredictthe
phase-spacestructureofdarkmatterneartheSun.Therewealsopresentexpecteddirect
detectionresultsbasedonthesesimulations.Inchapter5wepresentanewmethodthat
canbeusedtoresolvethene-grainedphase-spacestructureofcolddarkmatterhaloes.
Wedemonstratethisusingstaticpotentialsandpresentaworkingimplementationina
cosmologicalsimulationcode.Thismethodallowstogofarbeyondthecoarse-grained
phase-spaceanalysisofchapter4.Chapter6isbasedonchapter5anddescribeshowto
handlephase-spacecatastrophes,so-calledcaustics,toaccountcorrectlyfortheirannihi-
lationradiation.Wealsoshowtherehowthiscanbeimplementedinsimulationcodesto
trackcausticsandtheirradiationcorrectly.Inchapter7weapplythemethodsdeveloped
intheprevioustwochapterstoasimpliedhaloformationmodel.Thismodelwasused
bymanyotherauthorstoanalysethene-grainedstructureofcolddarkmatterhaloes.
Weshowthatallthesestudiesareincorrectbecausetheydonotdescribecorrectlythe
physicsoftheformationprocess.Mostnotably,theyallmissinstabilitiesthatcompletely
alterthene-grainedphase-spacestructureofthedevelopingdarkmatterhalo.Thishas
directimplicationsfordirectandindirectdarkmattersearches.Inchapter8weapply
themethodsofthefthandsixthchaptertothehaloesstudiedinchapter4.Thisallows
usnallyandforthersttimetoanalysethene-grainedphase-spacestructureofhaloes
forminginthefullcosmologicalframework.Inthelastpartofthisthesiswegiveour
conclusions.

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General

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overview

1Cosmology-abriefintroduction

Inthischapterwepresentthemathematicalfoundationsofmoderncosmology.
Westartbygivingabriefoverviewofourcurrentunderstandingofgravity
asthemainactoroncosmologicalscales.WedescribegravitybyEinstein’s
generalrelativitythatrelatesthegeometryofspace-timewithmatterand
energyintermsofEinstein’seldequations.Basedonthecosmologicalprin-
ciplewecanderivetheFriedmannequationsthatdescribetheevolutionofa
homogeneousandisotropicUniverse.Attheendofthischapterweintroduce
somecosmologicalandastrophysicalvocabularythatwillbeusedintherest
ofthisthesis.

22

Cosmology

-

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1.1Theoryofgravity-GeneralRelativity

32

Theresultsofthisandthenextchaptercanbefoundinstandardtextbookslike
Weinberg(1972);Misneretal.(1973);ColesandLucchin(1995);Peacock(1999).

1.1Theoryofgravity-GeneralRelativity
ThedominatingforceonthelargescalesoftheUniverseisgravity.Currently,thebest
theoryforgravityisEinstein’sgeneralrelativitydevelopedatthebeginningofthelast
century(Einstein,1916,1917).Wewillheredescribebrieythekeyresultsneededto
understandthebasicequationsofcosmology.
Inspecialrelativitytheinvariantintervaldsbetweentwoeventsatspace-timecoor-
dinates(t,x,y,z)and(t+dt,x+dx,y+dy,z+dz)isgivenby
ds2=c2dt2(dx2+dy2+dz2).(1.1)
ThisisinvariantunderLorentzcoordinatetransformationsanddescribesdistancesin
Minkowskispace-time.Oneofthefundamentalideasofgeneralrelativityisthatmatter
canchangethemetricofspace-timefromtheatMinkowskispacetoacurvedspace-time,
wherewewrite(usingEinstein’ssumconvention)
ds2=gdxdx,=0,...,3,(1.2)
wherex0=ctisthetimecoordinate,x1,x2,x3arespacecoordinatesandgisthe
metric.Theequationsofmotionforparticlesinsuchaspace-timecanbederivedfrom
thestationarityoftheirpathZ
ds=0.(1.3)
htapFromthisequationwecandeducethegeodesicequationofmotionforatestparticle
d2xdxdx
ds2+dsds=0,(1.4)
wherearetheChristoelsymbols
1@g@g@g
#"=2g@x+@x@x.(1.5)
Letusnextdenetheenergy-momentumtensorTforaperfectuidwithpressureP
andenergydensity,
T=(P+2c)UUPg,(1.6)
whereU=gdx/dsdenotesthefour-velocity.Wecanwritedownacovariantconser-
vationlawas
T;=0,(1.7)
where;standsforthecovariantderivativeofatensor
A@A;=+A.(1.8)
x@

1()8.

42

Cosmology-abriefintroduction

Theenergy-momentumtensordescribesthedistributionofmatterandsothesourcesof
gravity.Themainpointofgeneralrelativityisthelinkbetweenmatterandmetric.Let
usthereforerstdenetheRiemann-Christoeltensor
@@
R=+,(1.9)
x@x@thatcanbeusedtomeasurecurvatureofspace-time.Fromthistensorwecanconstruct
theRiccitensorR=RandtheRicciscalarR=gR.Finallywecandenethe
Einsteintensor
1G=RgR.(1.10)
2WiththistensorwecanformulateEinstein’seldequations
G8G=4T,(1.11)
cwheretheprefactor8G/4censuresthatPoisson’sequationresultsinthelimitofaweak
gravitationaleld.Aswewillshowbelowtheseeldequationsdonotallowastatic
Universe.AtthetimewhenEinsteindevelopedhistheorytherewerenohintsforanon-
staticUniverse,soheintroducedacosmologicalconstanttoallowstaticsolutions.The
modiedEinsteintensorhasthefollowingform
G81G=RgRg=4T,(1.12)
c2whereistheconstantintroduced.

1.2CosmologicalPrincipleandtheMetricoftheUni-
esrev

ThecosmologicalprinciplestatesthatonsucientlylargescalestheUniverseisbothho-
mogeneousandisotropic.Onthesescalesitspropertiesarethereforeidenticaleverywhere
inspace(homogeneous)anditlooksthesameinalldirections(isotropic).Itcanbeshown
thatthemostgeneralspace-timemetricdescribingaUniverseinwhichtheCosmological
Principleholdsisgivenby
2rd#"ds2=(cdt)2a(t)2+r2(d2+sin2d2),(1.13)
2rK1wherer,,arecomovingsphericalcoordinates,tisthepropertime,a(t)isthescale
factordescribingthetimeevolutionofthehomogeneousUniverseandKcharacterises
itsspatialcurvaturetakingthevalues1(closed),0(at)or1(open).Thismetricis
calledtheRobertson-Walkermetric.Wenotethatthisresultwasderivedwithoutusing
Einstein’seldequations.ThemetricisapureconsequenceoftheCosmologicalPrinciple.

1.3TheFriedmannequations25
1.3TheFriedmannequations
ToderivethetimeevolutionoftheUniverseasgivenbythetimedependenceofthe
scalefactor,weneedtorelatea(t)totheenergycontent.Forthis,weuseEinstein’s
eldequations.IfweputtheRobertson-Walkermetricintotheseeldequationsand
assumeaperfectuidwecanderivethefollowingtwoequations(knownastheFriedmann
equations)forthescalefactorbymatchingthetime-timeandspace-spacecomponents
(space-timecomponentsmerelygive0=0)
a¨4GP
=+32,(1.14)
a232c
a_=8GKc.(1.15)
2a3aTherstFriedmannequationshowsthatastaticUniverseisonlypossibleif=3P/c2,
soeithertheenergydensityorthepressuremustbenegative.Addingthecosmological
constanttotheEinsteintensoryieldsthefollowingslightlymodiedFriedmannequations
2a¨=4G+3P+c,
2a_28GKc2c2
a3c3
=2+.(1.16)
3a3a1.4Cosmologicalvocabularyandrelations
Hubblefunction:AveryusefulquantityistheHubblefunctionorHubbleparameter
denedas
a_H=.(1.17)
aThecurrentvalueH(t0),wheret0standsforthepresenttime,isdenotedasH0and
usuallyparametrisedinthefollowingway
a_kms1
at0Mpc
H0==100h,(1.18)
whereweintroducedhtoparametriseH0.
Redshift:Theredshiftzhasafundamentalimportanceincosmology.Itisdenedas
z=obsem=obs1=em1,(1.19)
ememobs
whereobs(obs)andem(em)arethewavelengths(frequencies)observedatEarthand
emittedbyadistantsource,respectively.Ifweneglectpeculiarmotionoftheemitting
objectandsolvethegeodesicequationofthelightraywendtherelation
aaz+1=obs=0,(1.20)
aemaem

26Cosmology-abriefintroduction
wherea0=a(t0)denotesthescalefactortoday.Wenotethatthisred-shiftingeectis
dierentfromtheordinaryDoppler-eect.Thecosmologicalredshiftarisesduetothe
expansionofspace.Itiscommontousetheredshiftasaconvenienttimevariabledue
toitsrelationtotemmentionedabove.Thepresentepochisthengivenbyz=0andthe
pastz>0.
Criticaldensity:Tocharacterisedensitiesitisusefultointroduceareferencedensity.
AnalysingtheFriedmannequationsinmoredetailitturnsoutthatanaturalchoicefor
thisdensityisthecriticaldensitydenedas
3H2(t)
c(t)=.(1.21)
G8Inthecase=0theUniverseisalwaysat(K=0)ifitsdensityequalsc(t).Ifitis
larger,theUniverseisclosed(K=1),ifitissmalleritisopen(K=1).Thisjusties
thenamecriticaldensity.Itisconvenienttoexpressthecontributionsofthedierent
componentsoftheUniverseintermsofthecriticaldensitybydening
)t(
l(t)=l,(1.22)
)t(cwhereldenotesthecomponent(e.g.
=c2/(3H2)).Ifwerefertothepresentday
valuesatt0wewillwrite
l0(e.g.
0=c2/(3H02)).Thedensityofcomponentlevolves
thenasafunctionofredshift
l(z)=
l(z)c(z)=
l0c0(1+z)3(1+w).(1.23)
UsingthedenitionsabovewecanrewritethetwoFriedmannequationsinthefollowing
yawa¨=aH_+H2,(1.24)
H2=H02E2(z),(1.25)
wherewedenedthefunction
ihE2(z)=(1
t0)(1+z)2+
m0(1+z)3+
r0(1+z)4+
0,(1.26)
where
t0,
m0,
r0arethepresent-daytotal,matterandradiationdensityparameters,
respectively.Bydenitionthecriticaldensityevolveslike
c(z)=c0E2(z),(1.27)
andtherefore
l0(1+z)3(1+w)(1+z)3(1+w)

l(z)=2=2.(1.28)
c0E(z)E(z)
Time-redshiftrelation:Inthefollowingwewilloftendenotetimesbytheircorrespond-
ingredshiftvalues.Itisthereforedesirabletohavetheexplicitrelationbetweentimeand

1.5TheEinstein-deSitterUniverse

72

redshifttotranslatebetweenthem.Takingthetimederivativeoftheazrelation,we
obtainz_a_
1+z=a=H,(1.29)
dnadz1dzdz
dt=(1+z)H(z)=H0(1+z)E(z)=tH(1+z)E(z),(1.30)
wherewehavedenedtheHubbletimetHas
tH=1=9.7958h1Gyr.(1.31)
H0Thetimeelapsedbetweenz!1(BigBang)andarbitraryredshiftzcanthenbe
calculatedviaintegrationas
+Z1dz0+Z1dz0
t(z)=(1+z)0H(z)0=tH(1+z)0E(z)0.(1.32)
zzThecurrentageoftheUniversecanbederivedfromthisequationbyevaluatingt(z=0).

1.5TheEinstein-deSitterUniverse
AsanexampleforasolutionoftheFriedmannequations,wewillherebrieydescribethe
so-calledEinstein-deSitter(EdS)Universe.ThisisadustUniversewithcriticalmatter
density.Althoughthediscoveryofdarkenergyrepresentingacosmologicalconstantby
observationsofsupernovaSNIain1998clearlyrulesoutthismodel,itisstillaverygood
approximationatearlytimes,wherethecontributionofthetotheenergydensityof
theUniverseisnegligible.
ThesolutionoftheFriedmannequationsisstraightforwardintheEdSmodelandwe
simplystatetheresults,assuminga0=a(t0)=1

:egA

ScaleFactor:

HubbleFunction:

t0=3H20=32tH=6.5h1Gyr.

t2/3
a(t)=t0

H(t)=32t

(1.33)

(1.34)

(1.35)

82

Cosmology-abriefintroduction

WenotethatthereisalsoananalyticsolutionforaatUniversewithcosmological
constant,
0=1
m0
0m22a3/2 a3!1/23
!1/3
mm0
H0t=3(1
m0)1/2ln4a+1+a5,am=
.
(1.36)

H0

=t

m

2/133!,5

3/10m!0

0

(1.36)

2StructureFormation

IfwelookaroundinourUniversewecanclearlyseethatitisnothomogeneous
andisotropic.Onthescalesofclustersofgalaxiesorgalaxiesweclearlysee
dramaticstructures.Thedevelopmentofthisstructurecannotbedescribed
withthehomogeneousandisotropicmodelspresentedinthelastchapter.
Herewedescribehowstructureevolves.Wewillfocusonthegrowthof
smalldensityperturbationsatearlytimes.Wewillshowhowthesecanbe
describedinlineartheory.Attheendofthischapterwewillalsointroducethe
Zel’dovichapproximationthatisusedtosetupinitialconditionsforN-body
simulations.

03

tS

tcur

uer

Format

noi

2.1Thegrowthofperturbations31
2.1Thegrowthofperturbations
InthelastchapterwedescribedtheevolutionofahomogeneousandisotropicUniverse
andwefoundthattheFriedmannequationsfullycharacteriseit.Onscalesuptothose
ofgalaxyclustersweseelotsofstructureintheUniverse.Thisstructureisbelievedto
formfromweakdensityandenergyinhomogeneitiesimposedatearlytimesandobserved
directlyintheCMB.Theseinhomogeneitieswereprobablyseededbyquantumuctua-
tionsatveryearlytimesandblownupbyanexponentialgrowthoftheUniverseduring
itsinationaryphase.Gravityasthemainactorinstructureformationthenleadsto
theevolutionoftheseinitialseedsthatnallygiverisetothestructuresweobserveto-
day.MosttheoriesassumethattheinitialseedeldcanbewelldescribedasaGaussian
randomeldinthedensitycontrast
(x,t)=(x,t)b(t),(2.1)
)t(bwhere(x,t)isthedensityatpositionxandtimetandb(t)isthebackgroundmean
densityoftheUniverseatthattime.AtearlytimesfollowsaGaussiandistribution
withameanhi=0andavariance2=h2.iThisrandomeldwillthenevolveunder
gravity.Usuallytheevolutionisdividedintotworegimes:||1iscalledthelinear
regime,whereas||1correspondstothenonlinearregimeofstructureformation.Our
galaxy,forexample,clearlyrepresentsahighlynonlinearregionoftheUniverseduetoits
highdensitycontrastandinstructureslikethesewetypicallyhave350fortheMilky
Wayhalo,and105withintheorbitoftheSun.
Lineargrowth
Theevolutionofthedensityeldcanbefollowedanalyticallyinthelinearregime.
Theideaisthattheequationsgoverningtheevolutionofthesystemcanbeexpandedina
seriesandfor||1allhighordertermscanbedropped.Ifweforgetforashortmoment
abouttheexpansionoftheUniverse,wecanwritedowntheequationsthatgovernthe
evolutionofauidinagravitationaleld.Wedescribetheuidintermsofitspressure
P,densityandvelocityeldv
Continuityequation:
@+r(v)=0,(2.2)
t@Eulerequation:
Pv@r+(vr)v=r,(2.3)
t@Poisson’sequation:
r2=4G.(2.4)
Letusnextsupposethatwehaveasolutiontotheseequations:=b,P=Pb,=b
andv=0.Wenotethatsuchasolutiondoesactuallynotexist,butthefollowing
calculationneverthelessyieldsthecorrectresults.Wefollowheretheoriginalderivation
ofJeans,wherehealsoassumedtheexistenceofthestaticsolution.Letusslightlyperturb

32StructureFormation
thissolutionandaskhowtheperturbationswillevolveintime.Sinceweareinterested
inlinearordereects,wewritetheperturbedquantitiesinthefollowingway
=b+,|b/|1,
P=Pb+P,|P/bP|1,
=b+,|/b|1,
.v=vIfwesubstitutetheseexpressionsbackintotheevolutionequationsoftheuidandwork
onlyuptolinearorder,wearriveatthefollowingsetoflinearisedequations
Continuityequation:
@+br(v)=0,(2.5)
t@Eulerequation:
2@v=csrr,(2.6)
t@bPoissonequation:
r2=4G,(2.7)
wherewedenedthespeedofsoundcsatconstantentropy
PP@! cs2==.(2.8)
@SSolutionsofthissetofequationscanbefoundinFourierspaceifthedispersionrelation
!2cs2k2+4Gb=0,(2.9)
isfullled.Growingandexponentiallyunstablemodesonlyexistif!isimaginary,which
impliesp
Gk<2b=kJ,(2.10)
cswhereweintroducedtheJeanswave-numberkJ.Thetypicallengthscaleassociatedwith
thiswave-numberiscalledtheJeanslength
cJ=ps.(2.11)
GbThismeansthatanobjectwithatypicalsizewith>Jwillbegravitationally
unstableandcollapse,whichimpliesthatthedensitycontrastisgrowing.Thetypical
timethatittakesforthecollapseisgivenbythefree-falltime
1tcollp.(2.12)
GbSofarweworkedouttheequationsfortheinstabilitiesinastaticUniverse.Foran
expandingUniverse,theequationsmustbeslightlymodied.Thishastheconsequence

2.2Collisionlessmedium

33

thattheexponentialgrowthwejustfoundwillbedilutedbytheexpansion,sothatin
theendonlyapower-lawgrowthsurvives.Thedensitycontrastinamatterdominated
Universeevolvesaccordingto
a_¨+2_+cs2k24Gb=0,(2.13)
awheretheequationiswritteninFourierspace.
Onecanshowthatthedensitycontrastequationhasagrowinganddecayingsolution,
whichareusuallycalledD+andD,respectively,andbotharepowerlawsintime.The
generalsolutionisalinearcombinationofD+andD.Thisimpliesthatafteracertain
periodoftimethegrowingmodewilldominateandwehave/D+.
Asanexamplewebrieystatethegrowthfactorforthematterdominatedcase
D+(a)/a,D+(t)/t2/3.(2.14)
ForsimplicitywewillsetD+=Dinthefollowingsinceweareonlyinterestedinthe
growingmode.Wenotethatthederivationherewasbasedonacollisionaluid.Further
downwewillalsodiscussthesituationforacollisionlessuid.Thisisveryimportant,
becausethereisstrongevidence(seenextchapter)thatthebulkofmattercomesfroma
collisionlessandcoldcomponent,namelycolddarkmatter.

Nonlineargrowth
Whilethelinearanalysispresentedaboveisstraightforward,theanalysisofthenon-
linearregime1isadicultproblem.Insightsintothisstageofevolutioncomemainly
fromcomputersimulations(calledN-bodysimulations)thatfollowthegravitationalin-
teractionofaMonte-CarlosampleofthematterdistributionintheUniverse.Wewill
comebacktothispointinthenextchapter.

2.2Collisionlessmedium
InthenextchapterwewillseethatalargefractionoftheUniverseseemstoconsistof
ayetunknowndarkcomponent,calleddarkmatter.Darkmatterisassumedtobehave
asacollisionlessuid.Furthermore,thenumberofdarkmatterparticlesisverylarge
sothattwo-bodyscatteringeventsoccuronlyataverylowrate,andthesystemcanbe
describedintermsofasmoothdistributionfunctionf(x,v,t)inphase-space.
Sincetherearenocollisionsbetweenparticles,thephase-spaceevolutionofthedark
matteruidisgivenbythecollisionlessBoltzmannequation(alsocalledVlasovequation).
Itsimplystatestheconservationofphase-spacedensityalongaparticletrajectory
@f+v@f@@f=0.(2.15)
@t@x@x@v
Thestatementthatthephase-spacedensityisconservedalongtheparticletrajectoryis
alsoknownasLiouville’stheoreminHamiltoniandynamics.Thegravitationalpotential
isgeneratedinaself-consistentwaybytheparticlesthemselvesandrelatedtothe

43

StructureFormation

densitydistributionbyPoisson’sequation
r2(x,t)=4G(x,t),(2.16)
wherethedensityisgivenbythevelocityintegralofthephase-spacedensity
(x,t)=dvf(x,v,t).(2.17)
ZInthesamewaywecanalsodeneameanvelocityeldas
1v¯(x,t)=dvvf(x,v,t).(2.18)
ZTakinghighervelocitymomentsleadstoanequationforthemeanstreamingvelocitythat
issimilartotheEulerequationforthecollisionaluidsdiscussedabove
@v¯1
+(v¯r)v¯=rPr.(2.19)
t@ThisequationlooksremarkablysimilartotheEulerequationforuids.Theonlydier-
enceisthatthepressureisnotscalaranymore.Thelasttermonther.h.s.describesan
anisotropickineticpressurearisingfromrandommotionsoftheparticles.Thisso-called
stresstensorPisdenedas
Pij=(vhivjivihivjh)i,(2.20)
erehw

Pij=(vhivjivihivjh)i,

(2.20)

1Zvhii=dvvif(x,v,t),(2.21)
1Zvhivji=dvvivjf(x,v,t).(2.22)
Wecanalsointroduceacosmologicalconstantasapressureofthevacuum.Asa
simplicationwewillassumeanisotropicvelocitypressureintheformofP=2,where
wecallthevelocitydispersion.Withthesemodicationsandassumptionswearriveat
thefollowingEulerequationforacollisionlessuid(x_=v¯)
@x_rxP
+(x_rx)x_=rx+
H02x.(2.23)
t@whererxdenotesaderivativew.r.t.x.TodiscussevolutioninanexpandingUniverse
itisconvenienttoworkincomovingcoordinatesx0whichwedenebythefollowingtwo
relations
x=ax,0v=a_x0+ax_0=Hx+av.0(2.24)
Ifwealsoassumeacoldmediumwithaverysmallvelocitydispersionwecanneglect
thepressurePintheEulerequation.NextwerewritetheEulerequationincomoving
coordinates.Therefore,wenotethatx¨=ax¨0+2a_x_0+a¨x0andrx=1/arx0.Thea¨term

2.2Collisionlessmedium

53

canbeeliminatedusingtheFriedmannequation.Takingtheserelationsintoaccountwe
canrewritetheEulerequationinthefollowingformthatisvalidinthelinearregime
a_14G
x¨0+2x_0=rx+bx0.(2.25)
3aaInthisequationthe
termdoesnotexplicitlyshowup.Butwenotethatthea(t)
evolutionofcoursedependson
asdescribedbytheFriedmannequations.Wecan
furthersimplifythisequationifweintroducethepeculiargravitationalpotential
2000(x)0=Gdx˜0(x˜)b=a(x)0+aa¨x02,(2.26)
Z|x0x˜|02
whereweintroducedthecomovingdensity0=a3.Ifwesubstitutethepeculiarpotential
wearriveatthefollowingequationincomovingcoordinates
x¨0+2a_x_0=1rx0.(2.27)
3aaThisallowsusnallytowritedownasetofequationsthatisthecounterpartofthethree
equationswewrotedownwhenanalysingthecollisionaluidabove.
Continuityequation:

Eulerequation:

Poisson’sequation:

_0x+(1+)r0x_=0,

1a_00xx¨+2x_=r0,
3aa

(2.28)

(2.29)

rx20=4G(0b0)=4Gb0.(2.30)
Ifwenowassumethattheuctuationeldissmallandneglecthigherordertermsin,
wecancombinethesethreeequationsandobtain
0a_¨+2_=4G3b=4Gb.(2.31)
aaThisequationlooksexactlylikethecorrespondingequationfortheuiddiscussedabove
ifweneglectthespeedofsoundthere,equivalenttoneglectingthepressure.Wealready

63

StructureFormation

wrotedownthegrowingmodeforamatterdominatedUniverse.Forageneralcosmology,
thegrowingsolutionisgivenby
Ztdt0aZ(t)da
D(t)=H(t)a2(t)0H2(t)0=H(t)a_3.(2.32)
00ThecosmologyentersinthisequationbytheHubblefunctionHandthescalefactora.

2.3Zel’dovichapproximation
Wewillnowbrieydescribeanotherapproachforperturbationtheorythatisequiva-
lenttothetreatmentabove,butisespeciallyusefulforsettingupinitialconditionsfor
cosmologicalsimulations.Wehaveseenthatthedensitycontrastevolvesaccordingto
(x,0t)=D(t)0(x)0,(2.33)
ifweneglectthesubdominantdecaymode.Thusthedensityeldgrowsself-similarly
withtimeinthelinearregime.Thisisalsotrueforthegravitationalaccelerationandthe
peculiarvelocity.Theself-similardensitygrowthimpliesforthepeculiargravitational
potential
(x,0t)=D(t)0(x)0withrx200=4Gb00=4Gba30.(2.34)
WecanrewritetheEulerequationinconformaltime(d=dt/a)
0a_01d2x0dadx0ddx0
! x¨+2ax_=a3rx0)ad2+dd=rx0)dad=rx0,(2.35)
andthismanipulationallowsastraightforwardintegrationoftheequation
dx01Z
=dDrx00.(2.36)
adIfweintegrateanothertimeweget
ZdZ!
x0=q0dDrx00.(2.37)
aBydenitionDsatisestheuctuationgrowthequationanditcanbeshownthatthe
doubleintegralonther.h.s.isproportionaltoD.Wethereforearriveatthefollowing
equations
00D(t)0D_(t)
x=q4Ga3rx00,v=4Ga3rx00,(2.38)
bbwherewewentbacktotimetinsteadofconformaltime.Thisformulationoflinear
theoryisduetoZel’dovich.ItisaLagrangiandescriptionwherethegrowthofstructure
isspeciedbygivingthedisplacementx0q0andthepeculiarvelocityv0ofeachmass

2.3Zel’dovichapproximation

73

elementasafunctionofitsinitial(Lagrangian)positionq.0Inthisso-calledZel’dovich
approximationtrajectoriesarestraightlineswiththedistancetravelledproportionalto
.D

83

tS

tcur

eru

Format

noi

3DarkMatter

Inthischapterwemotivateandintroducetheideaofdarkmatterasthemain
actorinstructureformation.Afterdiscussingevidencefordarkmatteron
variousscales,webrieydescribepossibledarkmattercandidates.Although
itisnotfullyclearwhatdarkmatterismadeof,wewillshowthatcomputer
simulationscanneverthelessbeusedtostudyitsevolutionandfollowthe
structureformationintheUniverse.Weclosethischapterbymakingsome
criticalcommentsonthedarkmatterhypothesisandmotivatetheobjectives
ofthisthesis.

04

kraD

Matter

3.1ObservationalEvidence

14

3.1ObservationalEvidence
TherstevidenceoftheneedforadarkmattercomponentcamefromZwicky’sComa
clusterobservationin1933.Sincethentheevidencefordarkmatterhasbecomevery
strongandweheregiveashortoverviewoftheobservationalevidencefordarkmatter
onvariousscales.ArecentreviewonthissubjectcanbefoundinBertoneetal.(2005).
Evidenceongalacticscales
Evidencefordarkmatterongalacticscalescomesfromtheobservationofrotation
curvesofspiralgalaxies(e.g.RubinandFord,1970).Thesecurvesshowthecircular
velocityasafunctionofradiusandareusuallymeasuredbyopticalsurfacephotometry
andbyHI21cmlineemissioninregionswherenootheropticaltracerisavailable.Based
ontheDopplereectcircularvelocitiescanthenbemeasuredandarotationcurvecan
beconstructed.Nowadaysmanyofthesecurveshavebeenmeasuredandtheyshowa
veryinterestingbehaviour:theyareatouttoradiifarbeyondtheedgeofthevisible
isexpectedtobev(r)=GM(r)/rforasphericalsystem,whereM(r)istheenclosed
disk.ThisisquiteunexpeqctedsinceNewtoniandynamicssaysthatthecircularvelocity
masswithinradiusr.SinceM(r)isexpectedtobpeconstantbeyondthevisibledisk,
onewouldexpectatypicalKeplerfallov(r)/1/rforthecircularvelocity.Thefact
thattherotationcurveisatbeyondtheopticaldiskimpliesthattheremustbeahalo
ofmatterwithM(r)/r.Therefore,rotationcurvesrequiretheexistenceofaextended
darkmatterhalotoexplaintheiratnessfarbeyondthevisibledisk.
Evidenceongalaxyclusterscales
Historically,clustersofgalaxiesgavetherstevidenceforsome’missingmass’com-
ponent.In1933,Zwickyinferredfrommeasurementsofvelocitydispersionsofgalaxies
intheComaclusteramass-to-lightratioofabout400M/L.Thisexceedstheratioin
thesolarneighbourhoodbytwoordersofmagnitude.ItwasZwicky’sobservationthat
markedthebeginningofthe’darkmatterproblem’.
Adetailedmeasurementoftheamountofdarkmatterinagalaxyclusterrequires
methodstomeasureits(total)mass.AcommonmethodistheobservationofthehotX-
rayemittinggasinthecluster.Assuminghydrostaticequilibriumandsphericalsymmetry
yieldsthefollowingrelationbetweenthetemperatureTofthehotgasandtheenclosed
massM(r) ! !
M(r)1Mpc
kBT=(1.31.8)keV14.(3.1)
rM01