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Studies of Mixing and Relaxation in Time-Dependent Collisionless Dynamical Systems

Permanent Link: http://ufdc.ufl.edu/UFE0021720/00001

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Title: Studies of Mixing and Relaxation in Time-Dependent Collisionless Dynamical Systems
Physical Description: 1 online resource (136 p.)
Language: english
Creator: Vass, Ileana
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Astronomy -- Dissertations, Academic -- UF
Genre: Astronomy thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The current distribution of matter in galaxies tells us about the Universe we live in: how it formed, what its building blocks are, and how it evolved. Computer simulations of structure formation in the Universe allow us to investigate the processes that drive the evolution of the early Universe and the formation of galaxies. To properly interpret how observations of nearby galaxies relate to the early state of the Universe, a detailed understanding of the processes that lead galaxies to equilibrium is needed. One such process is 'mixing,' which controls how the observed distributions of galaxies relate to their initial state. This study has two main goals. The first is to investigate the processes that drive the evolution of self-gravitating systems to equilibrium and cause mixing in phase-space. The second objective is to understand mixing and other factors that result in an apparently universal power-law profile of phase-space density from cosmological N-body simulations, by performing a quantitative analysis of the evolution of phase-space density in binary $N$-body mergers and in cosmological simulations.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ileana Vass.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Gottesman, Stephen T.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-12-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021720:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021720/00001

Material Information

Title: Studies of Mixing and Relaxation in Time-Dependent Collisionless Dynamical Systems
Physical Description: 1 online resource (136 p.)
Language: english
Creator: Vass, Ileana
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Astronomy -- Dissertations, Academic -- UF
Genre: Astronomy thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The current distribution of matter in galaxies tells us about the Universe we live in: how it formed, what its building blocks are, and how it evolved. Computer simulations of structure formation in the Universe allow us to investigate the processes that drive the evolution of the early Universe and the formation of galaxies. To properly interpret how observations of nearby galaxies relate to the early state of the Universe, a detailed understanding of the processes that lead galaxies to equilibrium is needed. One such process is 'mixing,' which controls how the observed distributions of galaxies relate to their initial state. This study has two main goals. The first is to investigate the processes that drive the evolution of self-gravitating systems to equilibrium and cause mixing in phase-space. The second objective is to understand mixing and other factors that result in an apparently universal power-law profile of phase-space density from cosmological N-body simulations, by performing a quantitative analysis of the evolution of phase-space density in binary $N$-body mergers and in cosmological simulations.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ileana Vass.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Gottesman, Stephen T.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021720:00001


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Therearepeopleinmylifethatalwaysshowedabsolutetrustinme,particularlymyfamily.Thisworkwouldnothavebeenpossiblewithoutalwayshavingbymysidemymother,CeciliaVass,andmyhusband,BrentNelson.MyPhDjourneywaslongand,often,sad.Myfather,threegrandparents,andmyGodparentspassedawayduringthistime.Ithadahappyending,though,whichwasasuccessfuldefenseday.ItdenitelyhelpedthatIgrewupinascienticresearchenvironment,asbothmyparentswerechemistryPhDs.Inthislatehour,mythoughtsgoquitefarback,startingwithmyearlyyearsofschoolandtoallmyprofessorswhoselessonswerekeytomyevolution:GherasimTic,elementaryschoolandjunior-highEnglishteacher,forteachingusnottobeafraidofexpressingourselvesinaforeignlanguageandforinstillinginusagoodaccent.IoanVladescu,junior-highMathteacher,forgivingusastrongsenseof3Dgeometry.DumitruButuruga,high-schoolMathteacher,forprovidinginvaluableinsightsintohumanityandintothebeautyofcalculusandforexplainingthegreatimportanceoftheinnitesimalquantity.EugenSoosandIulianBeju,collegemechanicsprofessors,forasolidandplasticviewofthephysicsofsoliddeformablesbodies.GraciaRodrguezCaderot,collegeastronomyprofessor,foradviceduringmyrstastronomyclasses.PilarRomeroPerez,collegeastronomyprofessor,forhercounselthroughouttherststepsofastronomicalresearch.StanleyDermott,collegeastronomyprofessor,forguidancewithmyrstastronomicaldataprocessingeortsandforalotofhelpinndingnancialassistance. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 13 CHAPTER 1INTRODUCTION .................................. 14 1.1StructureFormationintheUniverse ..................... 14 1.2EvolutionofCollisionlessSystemstoDynamicalEquilibrium ........ 18 1.3ThisStudy ................................... 20 2PHASEMIXINGANDCHAOSININTEGRABLEPOTENTIALS ....... 23 2.1PhysicalExpectations ............................. 23 2.2NumericalExperiments ............................. 26 2.3ResonanceandTransientChaos ........................ 28 2.4AModelforViolentRelaxation ........................ 37 2.4.1VariablePulsationFrequencies ..................... 37 2.4.2DampedOscillations .......................... 40 2.5Summary .................................... 46 3CHAOSANDVIOLENTRELAXATIONINN-BODYMERGERS ....... 48 3.1NumericalMethods ............................... 48 3.2MacroscopicEvolutionofMergerRemnants ................. 50 3.3MicroscopicChaos ............................... 50 3.4DenitionsofMixinginN-bodySystems ................... 53 3.4.1SeparationofPairsofNearbyParticles ................ 54 3.4.2MixingofEnsemblesinPhase-Space .................. 56 3.5Results ...................................... 58 3.5.1SeparationofNearbyParticlesinPhase-Space ............ 58 3.5.2MixingofEnsemblesinPhase-Space ................. 70 3.6Summary .................................... 74 4THECONSEQUENCESOFMIXINGDURINGVIOLENTRELAXATION .. 76 4.1IncompleteRelaxationandRedistributionofParticlesinMergers ..... 76 4.2Summary .................................... 84 7

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......................... 86 5.1FineandCoarse-GrainedPhase-SpaceDensityanditsMeasurement .... 86 5.2PreviousWorkontheEvolutionofPhase-SpaceDensity .......... 88 5.3NumericalMethods ............................... 90 5.4Summary .................................... 106 6PHASE-SPACEDENSITYEVOLUTIONINCOSMOLOGICALMERGERS 110 6.1EvolutionoffandQwithredshift ...................... 114 6.2Summary .................................... 125 7CONCLUSIONS ................................... 128 7.1Summary .................................... 128 7.2FutureWork ................................... 131 REFERENCES ....................................... 132 BIOGRAPHICALSKETCH ................................ 136 8

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Table page 6-1SomeWMAPparametersoftheCDMmodel ................... 111 6-2MainPropertiesoftheHalos ............................ 114 9

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Figure page 2-1Therelativefractionfofchaoticorbitsinarepresentative1600orbitensemblewhicharesubjectedtostrictlysinusoidaloscillationswithvariablefrequency! 30 2-2Themeanvaluehiofthechaoticorbits ...................... 31 2-3Therootmeansquaredspreadinenergies ..................... 32 2-4Therelativefractionfofchaoticorbitsinarepresentative1600orbitensemblewhicharesubjectedtostrictlysinusoidaloscillations ............... 34 2-5Thepowerspectrumassociatedwiththex-componentoftheorbits ....... 36 2-6FinitetimeLyapunovexponentsforrepresentativeorbitsevolvedinanundampedPlummerpotential .................................. 38 2-7ThemeanLyapunovexponenthiandtheassociateddispersioncomputedforensemblesoforbitsevolvedinanundampedPlummerpotential ....... 39 2-8Thefourlowercurvesexhibitthemeanvaluehiasafunctionof!forchaoticorbitssubjectedtodampedoscillations ....................... 41 2-9ThexandycoordinatesofaninitiallylocalisedensembleoforbitsevolvedinaPlummerpotentialsubjectedtoanonoscillatoryperturbation .......... 42 2-10ThesameastheFigure 2-9 ,nowallowingforanoscillatoryperturbationwithm0=0:5,p=2,and!=0:035 ........................... 43 2-11ThesameasinFigure 2-10 ,for!=1:4 ....................... 44 2-12ThesameasinFigure 2-11 ,for!=3:50 ...................... 45 3-1Theseparationbetweenthemost-boundparticlesofthemergingNFWhalos .. 51 3-2Divergenceinthequantitiesln(r)inunitsoflocalcrossingtime ........ 58 3-3Thetopthreepanelsshowdivergenceinln(r),andthelowerthreepanelsshowthequantityln(v) .................................. 60 3-4Thepanelsshowdivergenceinthequantitiesln(r)withpairsofparticlesselectedatt=0:Gyrandt=3:0Gyr ............................ 62 3-5ThetopthreepanelsshowdivergenceinthequantitiesEandthelowerthreepanelsshowthequantityJ 64 3-6Thefourpanelsshowdivergenceinthefourquantitiesln(r),ln(v),EandJplottedasafunctionoftimeforpairsofparticlesforboththelowandhighresolutionsimulations ................................ 66 10

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........... 68 3-8Theevolutionofanensembleof1000nearestneighborsinphase-spaceatt=0(runHRBp)plottedatoneGyrtimeintervalsinthemerger ........... 71 3-9IsodensitycontoursforensemblesofparticlesplottedasafunctionofE;Jforshell1,shell4andshell7 .............................. 72 4-1Thefractionofparticlesthatlieineachoftheradialshellsatthenaltimestep 77 4-2Histogramsofthechangeinenergyforallparticlesinshells1,4,and7 ..... 80 4-3Histogramsofthechangeinenergyforallparticlesinthreeequalradiusbinswithinthescaleradius ................................ 81 4-4Histogramsofchangeinthezcomponentoftheangularmomentum ....... 83 4-5FractionofparticlesthatoriginatedineachoftheradialshellsinoneoftheoriginalNFWhalosinvolvedinthemergerthatendupoutsidetheducialvirialradiusoftheremnant ................................. 84 5-1Histogramsofthechangeinphase-spacedensityforallparticlesinshells1,4,7 91 5-2Twomeasuresofthephase-spacedensity:Qandf 93 5-3Q==3andfasafunctionofrfordierentepochsforrunBp1 ........ 95 5-4Q==3andfasafunctionofrforinitialandnalepochs ........... 97 5-5Isodensitycontoursforthephase-spacedensity(f)asafunctionofradiusfortheBp1run ...................................... 98 5-6ThevolumedistributionfunctionV(f)fordierentepochsintheevolutionoftheBp1run ...................................... 101 5-7ThevolumedistributionfunctionV(f),fordierentepochsandfordierentbinsintheevolutionoftheBp1run ......................... 102 5-8Isodensitycontoursforthephase-spacedensity(f)asafunctionofthetotalenergyEandtotalangularmomentumJfortheisolatedhalo .......... 104 5-9Isodensitycontoursforthephase-spacedensity(f)asafunctionofthetotalenergyEandtotalangularmomentumJfortheBp1run ............ 105 5-10TheseparationbetweenthemostboundparticlesofthetwomergingNFWhalosfordierentorbitalparameters ........................... 107 5-11Isodensitycontoursforthephase-spacedensity(f)asafunctionofthetotalenergyEandtotalangularmomentumJfordierentorbitalparameters .... 108 11

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...................... 116 6-2Q==3andmedianffordierentredshiftsfortheL25halo .......... 117 6-3Isodensitycontoursfortheparticledensityintheplane(log(f);log(r))fortheL25run ........................................ 119 6-4Tracksthoseparticlesthat,atz=2,aresituatedinthesameregionincongurationspace,butwhosephase-spacevaluesdierbythreeordersofmagnitude ..... 120 6-5Histogramsofthephase-spacedensityasafunctionofredshift .......... 122 6-6ThevolumedistributionfunctionV(f),fordierentredshiftsintheevolutionoftheL25run ..................................... 124 12

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Friedmann ( 1924 ),whoformulatedwhattodayarecalledtheFriedmann-Lema^tre-Robertson-Walker(FLRW)cosmologicalmodels.Thesemodelsleadtoapictureinwhichtheveryfabricofspace-timeisexpanding,andprovideanexplanationoftheHubbleLaw,v=Hr,whichisarelationshipbetweentherecessionvelocity(v)ofagalaxyanditsdistancefromus(r),rstrecognizedbyEdwin Hubble ( 1929 ).ThediscoverybytheCosmicBackgroundExplorer(COBE)( Fixsenetal. 1996 )satellitethattheCosmicMicrowaveBackgroundRadiation(CMBR)wasanexcellentblack-bodysupportedtheviewthattheUniversewasexpandingandhadformedinahotBigBang.Fornearly70yearsfollowingHubble'sdiscovery,thestudyofCosmologywasdominatedbythesearchfortwonumbers,theHubbleConstantH0andtheparameterdeterminingtherateofdecelerationoftheexpansionoftheUniverse.Duetolargesystematicuncertaintiesindeterminingdistancestofarawaygalaxies,theaccuratedeterminationoftheHubbleconstantwasonlyachievedrecentlyduetoadvancesininstrumentation,thelaunchoftheHubbleSpaceTelescope(HST),andtheuseofseveralcomplementarymethods.AccuratedistancestonearbygalaxiesobtainedaspartofanHSTKeyProjecthaveallowedcalibrationofvedierentmethodsfordeterminingthedistancestogalaxiesoutto500Mpc( Freedmanetal. 2001 ).Contrarytotheexpectation(basedonstandardFLRWcosmology)thattheexpansionoftheUniversewasslowingdown,observationsbytwoindependentgroupsinthelate1990sfoundthatsupernovaeathighredshiftsarefainterthanpredictedforaslowingexpansionandindicatethattheexpansionisactuallyspeedingup( Riessetal. 1998 ; Perlmutteretal. 1999 ). 14

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Spergeletal. 2007 ),theUniverseiscomposedof3%ordinarybaryonicmatter,23%darkmatter(whichdoesnotemitorabsorblight),and74%darkenergy,whichactstoaccelerateexpansion.Morethan70yearsago,Fritz Zwicky ( 1933 1937 )realizedthatclustersofgalaxiesconsistedpredominantlyofmatterinsomenonluminousform.Pioneeringworkontherotationcurvesofspiralgalaxies( Rubin&Ford 1970 ; Freeman 1970 ; Bosma&vanderKruit 1979 )showedthatthecurvesdonotdecreaseoutsidetheregionswheremostoftheluminousmassends,butareatwellbeyondtheradiusofthestellardistribution.Therotationvelocityrisestoamaximumfromthecenteroutward,butoutsideofthecentralbulge,therotationspeedisnearlyaconstantfunctionofradius.Themostwidelyacceptedexplanationforthiswasthatthereisasubstantialamountofmatterfarfromthecenterofthegalaxy,anditsgravitationalinuenceisbestseenatlargeradii.Severalcategoriesofdarkmatter(DM)havebeenpostulatedandareclassiedaseitherbaryonicdarkmatterornon-baryonicdarkmatter.Thelatterisdividedintothreedierenttypes:hotdarkmatter(HDM),warmdarkmatter(WDM),andcolddarkmatter(CDM).CandidatesforbaryonicDMarenon-luminouscoldgas( Gerhard&Silk 1996 )andMACHOs(massivecompacthaloobjects, Alcocketal. ( 1993 )).BigBangnucleosynthesisputsstringentconstraintsontheamountofbaryonsintheUniverse.Currentobservationsareconsistentwiththesetheoreticalconstraints,implyingthatalloftheDMintheUniversecannotconsistofbaryons. 15

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Navarroetal. ( 1996 )(hereafterNFW)whoarguedthatananalyticalproleintheformofadoublepower-lawprovidedagooddescriptionofhaloprolesintheirsimulations.Otherstudiesclaimdierentvaluesoftheinnerslope.However,allthesetheoreticalmodelsdivergeastheradiusapproacheszero,producingasteepcentraldensitycusp.DensityproleswithouterslopeslargerthantheNFW 16

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Taylor&Navarro 2001 ; Aradetal. 2004 ; Ascasibar&Binney 2005 ).Thesefactspointtoauniversalityintheprocessesdrivingstructureformation.Oneofthereasonswhybothparticlephysicistsandcosmologistsareinterestedinknowingaboutthephase-spacedensity(alsoknownasthedistributionfunction, Binney&Tremaine ( 1987 ))ofDMparticlesinthecurrentUniverseisthattherearecurrentlyseveralparticlephysicsexperimentssearchingforcandidateDMparticlessuchasneutralinosandaxions.IfCDMcandidatesaretheDMparticles,thentheirinteractioncross-sectionsforself-annihilationarehighestatthecentersofDMhalos,wherethedensityproleoftheDMisinacuspaspredictedbyNFW.Furthermore,ifbaryonsfallintothesesteepcuspyDMpotentialsanddissipatetheydragmoreDMinafterthemandthisprocess(generallyreferredtoasadiabaticcompression)canproduceevensteepercuspsthanpredictedinDM-onlysimulations.Inthecentersofsuchcusps-eitheratthecentersofgalaxiesliketheMWoratthecentersofthelargenumbersofpredictedDMsubhalos-WIMPparticlesshouldhaveahighcross-sectionofannihilation,andiftheydoannihilate,theywillbedetectableingamma-raysinaparticularfrequencyrangewithaparticularspectrum.Thephase-spacedensityatthepresenttimeinthecentersofgalaxycuspsdependsonboththeinitialphase-spacedensityofDMparticlesintheearlyUniverse,aswellasontheprocessesthathaveleadtochangesinthisphase-spacedensityoverthecourseofgalaxyandstructureformation.Inparticular,processessuchasmergersofcollisionlessDMhalosresultincontinuouschangesinthephase-spacedensities 17

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Chandrasekhar 1943 )wasseveralordersofmagnitudetoolongtoberelevantfortheevolutionofgalaxies( Zwicky 1939 ).\Violentrelaxation"(VR)wasrstproposedby Lynden-Bell ( 1967 )asamechanismtoexplaintheregularityandsmoothnessofthelightdistributionsinellipticalgalaxies.VR,asconsideredbyLynden-Bell,consistedof\phasemixing"-aprocessbywhichaninitiallycompactensembleofnon-interactingphase-spacepointsinatime-independentpotentialcanbestretchedintoalongribboninphase-spaceasaresultofsmalldierencesintheinitialenergies(orangularmomenta)ofparticlesintheensemble.Phasemixingconservesthenegraineddistributionfunctionbutinacoarsegrainedsense,anditproducessmooth/relaxedlookingdistributionsontimescalesthatdependonlyontherangeofvaluesoftheorbitalintegrals(oractions)intheensemble( Binney&Tremaine 1987 ).ThesecondaspectofVRstudiedbyLynden-Bellconsistsofexchangeofenergybetweentheparticlesandtheglobalpotentialwhichisundergoinguctuations.Ithasoftenbeenassumedthatthefactthatthepotentialisstronglytime-dependentiscrucialinmakingthemixingsucientlyecient.Lynden-Bellderivedthedistributionfunctionthatwouldresultfromrapiductuationsinthepotentialofthegalaxy,suchas 18

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Mahonetal. 1995 ; Merritt&Valluri 1996 ; Habibetal. 1997 ; Kandrupetal. 2000 ; Kandrup&Siopis 2003 )haveshownthatmicroscopicensemblesofiso-energetictestparticleswhichoccupyaninnitesimalvolumeofphase-spacesurroundingachaoticorbitcanevolverapidlytoanear-invariantdistributionthatuniformlycoverstheenergysurfaceavailabletotheensemble.Thisprocessofdiusiontoanear-invariantdistributionistermed\chaoticmixing"andhasbeenshowntooccurinavarietyofgalaxy-typepotentials.Thus,alocalisedensembleofinitialconditionscorrespondingtochaoticorbitswill,whenevolvedinatime-independentHamiltonian,beginbydivergingexponentiallyfromitslocalisedstateandthenconvergetowardsanequilibriumornear-equilibriumstate( Kandrup 1998 ).Likephasemixing,chaoticmixingconservesthene-grainedphase-spacedensity.Unlikephasemixing,itsratedoesnotdependontheinitialspreadintheenergydistributionofparticlesintheensemble,butonhowstronglychaotictheorbitsintheensembleare( Merritt&Valluri 1996 ).Sinceallorbitsinatime-dependentpotentialwithlong-livedoscillationscan,inprinciple,exchangeenergywiththepotential,alargefractionoftheorbitswillnotconservetheenergyintegral.Insuchsystemspreviousauthorshavefoundthatasignicantfractionoforbitscanbechaotic( Kandrupetal. 2003 ).Thisraisesthepossibilitythattheunderlyingphysicalprocessdrivingrelaxationinviolentlyuctuatingpotentialscouldbeaconsequenceofchaoticmixing.Therststepinmakingthis 19

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Kandrupetal. ( 2003 )and Terzic&Kandrup ( 2004 ),whodemonstratedthatlargefractionsofchaoticorbitsariseinpotentialssubjectedtolong-duration(damped)periodicoscillations.Theseauthorsfoundthatthefractionofchaoticorbitsdependedonthefrequencyaswellastheamplitudeoftheoscillationsinthepotentialandattributedthechaostoa\broadresonance."Severalprocessesaectthestructureandevolutionofcollisionlesssystems.Inparticular,VRandthestatisticaldistributionfunctionsthatresultfromthisprocesswereinvestigated( Arad&Lynden-Bell 2005 ; Arad&Johansson 2005 ; Dehnen 2005 ).Severalstudiesoftheremnantdensityproleshaveshownthattheremnantsretainasurprisinglystrongmemoryoftheirinitialdensityprolesdespitethestrengthoftherelaxationprocess( Boylan-Kolchin&Ma 2004 ; Kazantzidisetal. 2006 ). Kandrupetal. ( 2003 ).Aperiodoftime-dependencewithastrongoscillatorycomponentcangiverisetolargeamountsoftransientchaos,anditisthereforeinvestigatedwhetherchaoticphasemixingassociatedwiththistransientchaoscouldplayamajorroleinaccountingforthespeedofVR.Analysisofsimplemodelsinvolvingtime-dependentperturbationsofanintegrablePlummerpotentialindicatesthatthischaosresultsfromabroadresonancebetweenthefrequenciesoftheorbitsandthefrequenciesofthetime-dependentperturbation.Numericalcomputationsoforbitsinpotentialsexhibitingdampedoscillationssuggestthat,withinaperiodof10dynamicaltimestDorso,onecouldachievesimultaneouslyboth`near-complete'chaoticphasemixingandanearlytime-independent,integrableendstate.Asacontinuation,N-bodysimulationsofhalomergerswereanalyzedinordertoinvestigatethemechanismsresponsiblefordrivingmixinginphase-spaceandtheevolution 20

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Vallurietal. ( 2007 ).Thefocusisonmixinginenergyandangularmomentumandondetermininghowtherateofmixingofthesequantitiesscaleswithlocaldynamicaltime.ThisallowsacomparisonbetweentherateandpercentofmixingduetoVRwithotherwell-knownmixingprocessessuchasphasemixingandchaoticmixing.Itisimportanttomentionthattheterms\mixing"and\relaxation"areoftenusedinterchangeablyintheastrophysicsliteraturetomeantheapproachofagravitatingsystemtoglobalaswellassmall-scaleequilibrium.Themostrestrictivedenitionof\relaxation"isaprocesswhichleadstothecomplete\lossofmemoryofinitialconditions"orlossofcorrelationsbetweeninitiallynearbyparticlesinphase-space.Thelargechangesinintegralsofmotionrequiredforsuchrelaxationonlyoccurasaresultoftwo-bodyinteractions( Chandrasekhar 1943 )andasaresultofVR.Bothprocessescausechangesinthene-graineddistributionfunction.Incontrast,phasemixingandchaoticmixingintime-independentpotentials,undertherightconditions,canleadtolargechangesincorrelationsbutconserveintegralsofmotions,andalsoconservethene-graineddistributionfunction.ThemergerremnantsproducedduringtheVRexperimentsmentionedabovewereinvestigated,withthegoalofunderstandinghowthemixingprocessredistributesparticlesinenergy,angularmomentum,andradius.AninterestingquestionregardingVRis,\Whatgivesrisetotheuniversalpower-lawphase-spacedensityprolesseenincosmologicalmergers?"Inwhatfollowsthefocusisontheevolutionofthephase-spacedensityofcollisionless(DM)particlesfollowingmergersofsphericalpotentialswithdierentdensityproles.Thephase-spacedensityatthepresenttimeinthecentersofgalaxycuspsisexpectedtodependonboththeinitialphase-spacedensityofDMparticlesintheearlyUniverseaswellasontheprocessesthathaveleadtochangesinthisphase-spacedensityover 21

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Lichtenberg&Lieberman 1992 ),canleadtotheonsetofchaoticity.Asaparticularlysimpleexample,oneknowsthatifaregularorbitwithsubstantialpoweratsomefrequencyisperturbedbyasinusoidalperturbationwithfrequency!=,theorbitwilltypicallyexhibitadrasticresponsewhichcanmanifestsensitivedependenceoninitialconditions.Therealissueis`howclose'mustand!betotriggerstrongexponentialsensitivity?Detailedanalysisoftheeectsofresonanceoverlapsuggestsgenericallythatthewidthofsuchresonanceswillbeanincreasingfunctionofamplitude.Evenifaslightly`detuned'smallamplitudeperturbationhasnoapparenteect,alargeperturbationwithidenticalfrequencymayproduceahugeresponse.Indeed,thenumericalmodelsdescribedinSectionsx andx ofthischapterindicatethat,forlargeamplitudeperturbations{fractionalamplitudeoforder1020%ormore{theresonancecanbeverybroad.Inparticular,thesemodelsrevealthat,ifthepowerintheorbitsisconcentratedatfrequencies,onecangetanontrivialincreaseinchaos{inboththerelativenumberofchaoticorbitsandthevalueofthelargestLyapunovexponent{forsinusoidalperturbationswith0:1!30.Inotherwords,theresonancecanbemorethantwoordersofmagnitudewide! 23

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Lichtenberg&Lieberman 1992 )isoneofthemostpopularwaysofmeasuringthedegreeofchaosofanorbitinasmoothpotential.Itmeasurestherateofdivergenceoftwoinnitesimallynearbyorbitsinphase-space.Iftherateofdivergenceofphase-spacepointsislinear,thentheLyapunovexponent=0andtheorbitistermedregular.Inasmoothequilibriumpotentialwiththree-spatialdimensionssuchorbitswillconserveatleastthreeisolatingintegralsofmotion.If,ontheotherhand,therateofdivergenceofthephase-spacepointsisexponential,theorbitistermedchaotic.Thus,theLyapunovexponentdenesthetimescaleonwhichinnitesimallynearbytrajectoriesinphase-spaceundergoexponentialseparationsintheirphase-spacecoordinates.Theimportantpoint,then,isthatifresonancesofthissortarereallysobroad,onemightexpecttransientchaostobeextremelycommon,ifnotfrequentlyencountered,intime-dependentgalacticpotentials.Numericalsimulationsofgalaxyencountersandmergersandmostmodelsofgalaxyandhaloformationimplythatasystemapproachingequilibriumwillexhibitdampedoscillations.Totheextent,however,thatoneisdealingwithcollisionlessrelaxation,thereisdimensionallyonlyonenaturaltimescaleintheproblem,namelythedynamicaltD1=p Mahonetal. 1995 ; Merritt&Valluri 1996 ; Kandrup 1998 )havedemonstratedthatthepresenceofchaoscandramaticallyenhance 24

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Kandrup&Drury 1998 ).Ifthepotentialisslowlyvarying,onecanvisualiseanorbitasmovinginaphase-spacewhichisslowlychangingbutwhichisnearlyconstantoveratimescaletD.Itisthennaturaltosupposethatatsometimestheorbitndsitselfinaphase-spaceregionthatis`chaotic'inthesensethatthereisasensitivedependenceoninitialconditions,butthatatothertimesitndsitselfin`regular'regionswherethereisnosuchsensitivedependence.Inthesettingenvisionedinthischapter,thephase-spaceischangingonatimescalecomparableto{orevenshorterthan{theorbitaltimescale,sothatthispictureisnecessarilylost.Moreover,inthesettingdescribedhereonewouldexpectgenericallyasystematicincreaseintheamountanddegreeofchaos,notsimplyachangewhichcouldbeeitherpositiveornegative.Itshouldbenotedthat,toacertainextent,thetermtransientchaosisnecessarily`fuzzy.'Chaos,likeordinaryLyapunovexponents,isonlydenedinanasymptotict!1limit.Thepoint,however,isthat,evenovernitetimeintervals,itisphysicallymeaningfultoaskwhetherorbitsexhibitanexponentiallysensitivedependenceoninitialconditions.Thisis,forexample,thenotionthatmotivatesthedenitionofnitetimeLyapunovexponents( Grassbergeretal. 1988 ),whichhavebecomeacceptedtoolsinnonlineardynamics.Fromaphenomenologicalpointofview,itmakesperfectsensetoidentifyanorbitasexhibitingsignicanttransientchaosif,forsomenitetime>tD,theorbitexhibitssignicantexponentialsensitivity.Inmanycases,itisclearbyvisual 25

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(1+x2+y2+z2)1=2:(2{1)Thisformwasselectedespeciallybecauseorbitsinthispotentialareallstrictlyintegrable.Anychaosthatisobservedcanbeassociatedunambiguouslywiththeperturbationsthatwereintroduced.Thesewereassumedtotaketheform 26

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2{6 or 2{7 typicallyexhibitedcomparativelyminimalamountsoftransientchaos.Variabledrivingfrequency,assumingthat 27

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Kandrup&Novotny 2004 )typicallyexhibitanexponentialapproachtowardsanear-invariantdistributiononatimescale5tD! ,allowingfornonoscillatorydampedperturbationsoftheformgivenbyEquation 2{6 donotresultinlargemeasuresofchaos,evenforlargeamplitudesm00:5.However,allowingforoscillationsinmand/oracantriggersubstantialamountsofchaos.Thereareatleasttworeasonstobelievethatthischaosistriggeredbyaresonance.Moreobvious,perhaps,isthefactthatthelargestincreasesinchaos{asreectedbyboththenumberofchaoticorbitsandthesizeofatypicalLyapunovexponent{arisefordrivingfrequenciescomparableto,orsomewhatlargerthan,thefrequenciesforwhichtheunperturbedorbitshavemostoftheirpower.Equallysignicant,however,isthefactthatthedrivingfrequencieswhichresultinthegreatestamountofchaoscorrespondpreciselytothosefrequenciesforwhichtheorbitalenergiesare`shued'themost,i.e.,wheredierentparticleswiththesameinitialenergyendupwiththelargestspreadinnalenergies.Apriorithereneedbenodirectconnectionbetweenincreasesinchaosandlarge-scaleshuingofenergies.However,onewouldexpectresonantcouplingstoleadtosignicantchangesinenergies.Acorrelationbetweenchangesinenergyandtheamountofchaosthuscorroboratestheinterpretationthatthischaosisresonantinorigin.Atleastforthecaseoflargeamplitudeoscillations,m00:1orso,theresonanceiscomparativelybroad.OneobservessignicantincreasesinboththerelativemeasureofchaoticorbitsandthesizeofthelargestLyapunovexponentwheneverthedrivingfrequency!andthenaturalorbitalfrequenciesagreetowithinanorderofmagnitudeorso.If,forexample,oneconsidersanenergywherethenaturalfrequenciespeakform0:30:35orso,noticeableincreasesinthedegreeofchaosareobservedfor0:035!10:0,i.e.,0:1!=m30. 28

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2-1 { 2-4 ,allgeneratedfromthesamesetof1600initialconditions,evolvedforatotaltimet=512inthepresenceofastrictlysinusoidalperturbationoftheformgivenbyEquation 2{3 witha1.ThetwopanelsofFigure 2-1 exhibittherelativemeasurefofchaoticorbitsasafunctionofdrivingfrequency!forvedierentchoicesofamplitudem0.Thetoppanelexhibitsf(!)onalinearscale,thusallowingonetofocusonthebehaviouroftheresonanceathigherfrequencies;thelowerexhibitsfasafunctionoflog10!,whichallowsonetoseemoreclearlythebehaviouratlowerfrequencies.MostobviousfromFigure 2-1 isthefactthatthewidthoftheresonanceisanincreasingfunctionofm0,althoughthisincreaseappearstobecomparativelysmallform0>0:2orso.Alsoevidentisthefactthat,exceptperhapsforthelowestamplitude,m0=0:1,thefractionfappearstovarysmoothlywithfrequency.Ineachcase,fpeaksatafrequency!maxcomparabletothetypicalnaturalfrequencies0:30:35 29

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Therelativefractionfofchaoticorbitsinarepresentative1600orbitensemblewhicharesubjectedtostrictlysinusoidaloscillationswithvariablefrequency!.Thedierentcurvesrepresent,frombottomtotop,drivingamplitudesm0=0:1,0:2,0:3,0:4,and0:5.Thebottompanelshowsthesamedataplottedasafunctionoflog10!. associatedwiththeunperturbedorbits.However,!maxdoesseemtovarysomewhatasafunctionofm0.Forlargeramplitudes,near!maxessentiallyalltheorbitsarechaotic,i.e.,f(!max)1.Forlowerfrequenciesf(!max)canbesignicantlylessthanunity.Overall,thereisevidenceforsignicantamountsoftransientchaosfor0:35!10:0.ThetwopanelsofFigure 2-2 exhibithi,themeanvalueofthelargestnitetimeLyapunovexponentforthesameensembles,againplottedonbothlinearandlogarithmicscales.Herehiwasextractedbyrstidentifyingthoseorbitsintheensemblethatweredeemedchaoticandthencomputingthemeanvalueofforthoseorbits. 30

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ThemeanvaluehiofthechaoticorbitsintheensemblesusedtogenerateFigure 2-1 ,againplottedasafunctionof!.Thebottompanelshowsthesamedataplottedasafunctionoflog10!. Figure 2-3 showsE,therootmeanspreadinenergiesattimet=512,forthechaoticorbitsidentiedinFigure 2-1 andFigure 2-2 .Itisclearvisuallythat,asonewouldexpectifthetransientchaosisassociatedwitharesonantcoupling,thefrequencieswhichresultinthelargestmeasuresofchaoticorbitsandthelargesthialsoresultinthelargestshiftsinenergy.However,itisalsoevidentthat,especiallyforthehigheramplitudeperturbations,Eexhibitsamorecomplexdependenceon!thandof(!)orh(!)i.Thisdoesnotcontradicttheassertionthattransientchaosisinducedbyaresonantcoupling,butitdoessuggestthattheamountanddegreeofchaosinthebroadresonanceregionislesssensitivetothepulsationfrequencythanistheshuinginenergies.Thishas 31

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Therootmeansquaredspreadinenergies,Erms(t=512),fortheorbitensemblesusedtogenerateFigures 2-1 and 2-2 .Thebottompanelshowssamedataplottedasafunctionoflog10!. potentiallysignicantimplicationsforVR,whereoneisinterestedinboth(i)shuingtheenergiesoftheindividualmassesand(ii)randomisingtheirphase-spacelocationsonaconstantenergyregion.OneotherpointisalsoevidentfromFigure 2-3 ,namelythat,atleastforfrequencieswellintotheresonance,thespreadinenergiescanbeverylarge.Thisreectsthefactthat,becauseoftheresonantcoupling,asignicantfractionoftheorbitshaveacquiredlargeenergiesthatsetthemalongtrajectoriesinvolvingexcursionstoverylargeradii,r>10ormore.Allowingforaresonancethatisthisstrongtolastforaslongast20tD

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,whichincorporatestrongdamping,avoidthisarticialbehaviour.Inpartbecauseofthiseect,anyuniformprescriptionusedtodistinguishbetween`regular'and`chaotic'orbitsisnecessarilysomewhatimprovisedand,assuch,fraughtwithdiculties.Figures 2-1 and 2-2 reectananalysisinwhich`chaotic'wasdenedascorrespondingtoanitetimeLyapunovexponentfortheinterval010orso.Despitethis,however,itshouldberecognisedexplicitlythatthisprescriptionreallyidentiesorbitswhichare`obviously'or`strongly'chaotic,andthatitmaybeignoringaconsiderablenumberoforbitswhichareonlyveryweaklychaotic.Thisprescription,likeanyotherwhichinvolvesaxedminimumvalueandaxedintegrationtime,isopentocriticism.Forthosefrequenciesandamplitudeswheretheresonanceisespeciallystrongandmanyorbitsquicklyachievelargeenergies,itmightseemmoreappropriatetoconsiderashortertimeinterval,duringwhichtheorbitsarerestrictedtosmallerradii.However,forfrequenciesandamplitudesforwhichtheresonanceisweaker,onemightwishinsteadtointegrateformuchlongertimessoas(hopefully)toallowforcleardistinctionstobemadebetweenregularand`extremelysticky',nearlyregularorbits.Thechoiceofatimeintervalt=512representsacompromisenecessitatedbytheidealisednatureofthecomputationsdescribedinthissection.Inanyevent,whatisinevitableisthat,forchaoticorbitsinanypotential,onemustexpectcorrelationsbetweentheenergyoftheorbitandthesizeofatypicalnitetime 33

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Therelativefractionfofchaoticorbitsinarepresentative1600orbitensemblewhicharesubjectedtostrictlysinusoidaloscillationsoftheformgivenbyeq.(4)ofvariableamplitudem0with!=1:4(uppercurve),!=3:5(middlecurve),and!=7:0(lowercurve). Lyapunovexponent.Forthepotentialconsideredhere,onendstypicallythattheorbitsdeemedregulararepreciselythosewhichexhibitthesmallestchangesinenergyandhence,overall,thesmallestexcursionsfromtheorigin.However,forthoseorbitswhicharechaotic,thosespendingmoretimeatlargerradiiandwhich,forthisreason,havelargerorbitaltimescales,tendtohavesomewhatsmallervaluesof.Figure 2-4 exhibitstherelativemeasureofchaoticorbitsasafunctionofamplitudem0forthreedierentfrequencies,namely!=1:4,3:5,and7:0.For!=1:4,afrequencyinthemiddleoftheresonance,foramplitudesaslargeasm0=0:25almostalltheorbitsexhibitevidenceofchaos.For!=3:5,avaluesomewhatclosertotheedgeoftheresonance,therelativeabundanceofchaoticorbitsissomewhatsmaller.However,form0=0:25asmanyas75%oftheorbitsareclearlychaotic,afractionwhichdoesnotincreasesignicantlyifm0isincreased.For!=7:0,avalueneartheedgeofthe 34

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2-1 and 2-2 failsform00:1.Forthatreason,Figure 2-4 implementedadierent,somewhatmoresubjectivecriterionwhichhasbeenusedsuccessfullyindistinguishingbetweenregularand`verysticky'chaoticorbitsinthetriaxialgeneralisationsoftheDehnenpotentials( Siopis&Kandrup 2000 ).Whatthisentailedwasorderingthe1600computedvaluesofatvarioustimesTA,plottingtheorderedlistof(TA)'sfordierentvaluesofTA,andthenineachcasesearchingfora`kink'inthecurve.ThisprescriptionappearstopermitanaccuratedeterminationofafractionfoftheorbitsexhibitingchaoticbehaviourwhichisinsensitivetotheprecisechoiceofTA,althoughtheestimatedsizeofthenitetime(TA)doesdependonthesamplinginterval.AdditionalinsightsintothenatureoftheresonancecanbederivedfromFigure 2-5 ,whichshowsthex-componentsofthecomposite(normalised)Fouriertransformsofcoordinateandforceperunitmass,jx()jandjax()j,constructedbycombiningspectrafor1600unperturbed!=0orbitsgeneratedfromthesameinitialconditionsusedtogenerateFigures 2-1 { 2-3 .(Becauseofsphericalsymmetrythey-andz-componentsarestatisticallyidentical.)Asnotedalready,forthisparticularchoiceofenergy,jx()jpeaksatm0:30:35andiscomparativelynegligibleformuchhigherandlowerfrequencies.Bycontrast,jax()jhasmultiplepeaksassociatedwithhigherfrequencyharmonicswhicharisebecausetheforceperunitmassisanonlinearfunctionofposition.ItisnaturaltosupposethatitistheseharmonicsthatareresponsibleforthecomparativelylargeresponsesatfrequencieslargecomparedwithmthatareevidentinFigures 2-1 { 2-3 .Itisparticularlyinterestingthatthisresponsecutsofordrivingfrequencies!largecomparedwith2,ratherthan,afactthatwouldsuggestthatoneisobservingtheeectsofabroad2:1resonance. 35

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Thepowerspectrumjx()jassociatedwiththex-componentoftheorbitsgeneratedfromthe1600initialconditionsusedtocreatedFigures 2-1 { 2-3 ,evolvedintheabsenceofanytime-dependentperturbationandplottedasafunctionof2(toppanel).Thebottompanelshowsthepowerspectrumjax()jassociatedwiththex-componentoftheforceperunitmassassociatedwiththesameorbits. MuchofthesmoothnessinplotssuchasthosegiveninFigure 2-2 reectsthefactthatthedataweregeneratedfrom`representative'ensemblesofinitialconditions.Ifthesameanalysisisrepeatedfordierentlocalisedensemblesofinitialconditions,evenensembleswiththesameenergy,oneobservessigncantvariabilityinboththesizeofatypicalnitetimeLyapunovexponentandthevalueofthefrequency!forwhichhiismaximised.Andsimilarly,plotsofquantitieslikehiasafunctionof!forsuchensemblesexhibitmorestructurethandoplotsforrepresentativeensembles.This 36

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2.4.1VariablePulsationFrequenciesAllowingforarandomlyvaryingfrequencyoftheformgivenbyEquations 2{8 and 2{9 leadstoseveralsignicantconclusions.Perhapsthemostimportantisthatallowingforacomparativelyslowfrequencydriftcanactuallyincreasetherelativemeasureofchaoticorbits.Supposethattheunperturbedfrequency!0iswellintotheresonantregionandthatisnotsucientlylargetopush!outside.Inthiscase,theintroductionofaperturbation(t)withanautocorrelationtimetclongcomparedwithtDtendstoconvertmany,ifnotall,the`regular'orbitsintoorbitswithappreciableexponentialsensitivity.Thisbehaviourcanbeunderstoodbysupposingthat,intheabsenceofthefrequencydrift,thereexistsomeregularorbitswhich,becauseoftheparticularvaluesoftheirnaturalfrequencies,barelymanagetoavoidresonatingwiththeperturbation.Allowingforadriftfacilitatesanimprovedresonancecouplingwhichcanmake(someof)theseorbitschaoticforatleastsomeofthetime.Providedthat!0and!0arewellwithintheresonanceregion,itisalsoapparantthat,astheautocorrelationtimetcdecreases,thedegreeofchaos,asprobedbythemeanhi,tendstodecrease,eventhoughtherelativefractionofchaoticorbitsmaywellremainequaltounity.Forxedautocorrelationtimetc,themeanhimanifestsonlyarelativelyweakdependenceon,atleastfor0:1=!00:8.Forsomechoicesoftcincreasingcausesanincreaseinthemeannitetime;inothercasesincreasingdecreasesthemean.However,theobservedvariationsareinvariablysmall.AllthreeofthesepointsareillustratedinFigures 2-6 and 2-7 .Thedashedandtriple-dot-dashedcurvesinFigure 2-6 correspondtoplotsoforderednitetimeLyapunovexponentsassociatedwith1600initialconditionssubjectedtoundampedpulsationswithvariablefrequency!=!0+(t)form0=0:1,!0=3:5anddierentchoicesof 37

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FinitetimeLyapunovexponentsfor1600representativeorbitsevolvedforatimet=512inanundampedPlummerpotentialpulsedwithamplitudem=0:1andafrequencyfrequency!(t)=!0+(t),with!0=3:5,autocorrelationtimetc=320=16tD,and2=0:35(dashedcurve)and2=2:8(triple-dot-dashed).Thesolidcurvecontrastsexponentsderivedforanevolutionwith0.Thesamedataisplottedfortc=160,tc=80,tc=40,tc=20,andtc=10. andtc.Thesolidcurveineachpanelcontraststheexponentsgeneratedforthesameinitialconditionsevolvedwith=0.Itisevidentthat,forthelargestvaluesoftc,thedierencesbetweenthecurveswithzeroandnonzeroarerelativelysmall,atleastforthelargestvaluesof.However,itisalsoapparentthatthetime-dependentfrequencydrifthassignicantlyreducedtherelativenumberoforbitswithverysmallnitetime's.Alsoevidentisthefactthatdecreasingtheautocorrelationtimetc,i.e.,makingthefrequencydriftmorequickly,tendsgenericallytodecreasethetypicalvalueof,althoughthe 38

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ThemeanLyapunovexponenthi(uppertwocurves)andtheassociateddispersion(lowertwocurves)computedforensemblesof1600orbitsevolvedinanundampedPlummerpotentialwithamplitudem=0:5,!0=3:5,andvariableautocorrelationtimetc.Diamondscorrespondto2=0:35,trianglesto2=2:8.(b)Thesamefordataisplottedform0=0:1andform0=0:05. 39

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2-7 2{4 or 2{5 hasacomparativelyminimaleect.Notsurprisingly,thesizeofthemeanLyapunovexponenthidecreasesastimeelapsesandtherecanbeagradualdecreaseintherelativefractionfoforbitsexhibitingexponentialsensitivity.However,thebasicphenomenonoftransientchaospersistsuntiltheperturbationhasdampedalmostcompletelyaway.Thisis,forexample,evidentfromFigure 2-8 ,whichwasagaingeneratedfromthesame1600initialconditionsusedtogenerateFigures 2-1 { 2-5 ,nowallowingforaperturbationoftheformgivenbyEquation 2{5 withm0=0:5,t0=100,andp=2,againallowingforvariable!.Herethefourlowercurves(fromtoptobottom)representmeanvaluesofthenitetimeLyapunovexponenthigeneratedseparatelyfortheintervals100
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Thefourlowercurvesexhibitthemeanvaluehiasafunctionof!forchaoticorbitssubjectedtodampedoscillationswithm0=0:5,p=2,andt0fortheintervals(fromtoptobottom)100
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ThexandycoordinatesofaninitiallylocalisedensembleoforbitsevolvedinaPlummerpotentialsubjectedtoanonoscillatoryperturbationoftheformgivenbyEquation 2{7 withm0=0:5andp=2,fort=0,t=16,t=32,t=64,t=128,andt=256. panelsexhibitthexandycoordinatesofeachoftheorbitsattimest=0,16,32,64,128,and256,thelastcorrespondingtoanintervalofapproximately12:8tD.ItisevidentvisuallyfromFigure 2-9 that,intheabsenceofoscillations,mixingiscomparativelyinecient.Indeed,theevolutioninFigure 2-9 isqualitativelyidenticaltoexamplesofregularphasemixingintime-independentpotentials(asinFigure2in Kandrup ( 1999 )).AsassertedinSectionx ,anon-oscillatoryperturbationofthe 42

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ThesameastheFigure 2-9 ,nowallowingforanoscillatoryperturbationoftheformgivenbyEquation 2{5 withm0=0:5,p=2,and!=0:035. integrablePlummerpotentialleadstolittleifanytransientchaosand,assuch,noevidenceforchaoticphasemixing.Figures 2-10 { 2-12 ,whichincorporateasystematicpulsation,allyieldphasemixingthatissubstantiallymoreecient.Figure 2-10 ,whichrepresentsorbitsthathavebeenpulsedwith!=0:035,afrequencyneartheloweredgeoftheresonance,clearlyexhibitsmorerobustmixing,althoughthemixingisstillconsiderablylessecientthanwhatisobservedforstronglychaoticowsintime-independentpotentials(asinFigure1in Kandrup ( 1999 )).Indeed,itisdicult 43

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ThesameasinFigure 2-10 ,for!=0:70. todetermineunambiguouslywhethertheorbitsusedtogenerateFigure 2-10 reallyexhibitsignicanttransientchaos.Bycontrast,Figure 2-11 ,whichwasgeneratedfor!=1:4,exhibitspreciselythesortofbehaviourwhichonehascometoassociatewithchaoticphasemixingintime-independentpotentials.Forthersttwodynamicaltimes,thelocalisedensembleoforbits,whichstartedwithaphase-spacesize<102,stillremainsconned.Byt=64,however,correspondingtoaninterval3:2tD,theorbitshavebeguntospread 44

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ThesameasinFigure 2-11 ,for!=3:50. signicantly;and,byt=128,correspondingtoroughly6:4tD,theensemblehasdispersedtollalargeportionoftheavailablephase-space,andhencerelaxationmightbeoccuringforthisvalueof!.Nevertheless,despitethisecientmixing,thepotentialisdampingrapidlytowardsanearintegrablestate.Byatimet=256,theoverallamplitudeoftheperturbationm0=(1+t=t0)2,hasdecreasedfromm0=0:5tom00:04,thiscorrespondingtoamuchmorenearlytime-independentsystem. 45

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2-12 exhibitsanexampleof`incomplete'chaoticphasemixingforthecaseofafrequency!=3:5.Hereagainoneseesclearevidenceofthedispersivebehaviourindicativeofchaoticphasemixing;but,sinceoneisclosertotheupperedgeoftheresonance,theecientmixinghasinfactturnedobeforetheensemblecouldllalltheaccessiblephase-space.Thelackoforbitsnearx=y=0isexactlywhatonewouldexpectinatime-independentsphericalpotentialfororbitswithanappreciableangularmomentum. Kandrupetal. 2003 ) 46

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47

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3{1 )(forexample, Hernquist 1990 ; Zhao 1996 ),wheredenotestheasymptoticinnerslopeoftheprole,correspondstotheouterslope,anddeterminesthesharpnessofthetransitionbetweentheinnerandouterprole. Navarroetal. ( 1996 ,hereafterNFW)prole(with(;;)=(1;3;1)),whilethesecondmodel(with(;;)=(2;3;0:2))correspondstoaprolewithashallowerinnerslope.N-bodyhalomodelsareconstructedusingtheexactphase-spacedistributionfunctionundertheassumptionsofsphericalsymmetryandanisotropicvelocitydispersiontensor( Kazantzidisetal. 2004 ).Eachoftheinitialdarkmatter(DM)haloshasavirialmassofMvir=1012Mimplyingavirialradiusofrvir'256:7kpc,andaconcentrationofc=12,resultinginascaleradiusofrs'21:4kpc.ItisworthemphasizingthattheadoptedvalueofMvirservesmerelypracticalpurposesanddoesnotimplyanythingspecialabouttheparticularchoiceofmassscale.Hence,theconclusionscanbereadilyextendedtomergersbetweenequal-masssystemsofanymassscale.Morerecently, Navarroetal. ( 2004 )proposedaproleinwhichtheslopevariescontinuouslywithradius,and Merrittetal. ( 2006 )arguedthatan Einasto ( 1969 )modelcouldbemorewidelyusedtottocharacterizethedensityproles.Oneofthemainargumentsofthelatter'sgroupisthat,unliketheNFWprole,Einasto'smodelhasa 48

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Stadel 2001 ).PKDGRAVusesasplinesofteninglength,suchthattheforceiscompletelyKeplerianattwicethequotedsofteninglength,andmulti-steppingbasedonthelocalaccelerationofparticles.TherstexperimentfollowedtheencounteroftwoNFWhalos(referredtohereafterasrunBp1),whileinthesecondexperimentanNFWhalomergedwithahalohavinganinnerslopeof=0:2(referredtohereafterasrunhBp1).TheinitialhalomodelsinthesetworunsweresampledwithN=2105particlesandforcesweresoftenedwithasplinegravitationalsofteninglengthequalto=1:5kpc.Inordertominimizeanyconcernthattheresultsmightbecompromisedbynumericalresolutioneects,weanalyzedanadditionalmergerexperimentbetweentwoNFWhalosincreasingthemassresolutionbyafactorof10andscalingdownthesofteninglengthsaccordingto/N1=3(hereafterreferredtoasrunHRBp).Initialconditionsforbinarymergersweregeneratedbybuildingpairsofhalomodelsandplacingthematadistanceequaltotwicetheirvirialradii.Inthisstudyonlymergersofsystemsonparabolicorbitswerediscussed,owingtothefactthatthisparticularorbitalcongurationisthemosttypicalofmerginghalosincosmologicalsimulations(forexample, Zentneretal. 2005 ).Thesemergersimulations(labeledBp1,hBp1,andHRBp)wereanalyzedanddescribedingreaterdetailin Kazantzidisetal. ( 2006 ).Extensiveconvergencetestscarriedoutby Kazantzidisetal. ( 2006 )indicatethatisolatedhalomodelsdidnotdeviatefromtheiroriginaldistributionfunctionsontimescalesaslongas100crossingtimesevenatsmallradii(r).InordertodistinguishthemixingthatresultsfromtheexponentialinstabilityoftheN-bodyproblemfromtheeectsofthemixingresultingduringthemerger,anN-bodysimulationoftheevolutionofasphericalisolatedNFWhalowasperformed.Allorbitsin 49

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1 2I=2T+V(3{2)whereIisthemomentofinertia,Tisthetotalkineticenergy,andVisthetotalpotentialenergyofthesystemofparticles,withallquantitiesdenedwithrespecttothecenterofmassofthesystem.Foratime-independentgravitatingsystem,I=0=2T+V.Arobustquantitativemeasureofhowfaragravitatingsystemisfromequilibriumcanbeobtainedfromthevirialratio2T=jVj,whichisunityforasysteminglobaldynamicalequilibrium.InFigure 3-1 ,thetoppanelshowstheseparationofthemostboundparticle(MBP)ofeachhalointhemerger,themiddlepanelshowstheevolutionofthevirialratio2T=jVjandthelowerpanelshowsthechangeintotalpotentialenergyofthesystem(V),asafunctionoftime.Wenotethatpericenterpassages(seenasminimaintheMBPseparation)correspondtomaximainthevirialratioandminimainpotentialenergy.Fort7Gyrthevirialratio2T=jVjandpotentialVundergostronguctuationsbutremainsclosetounitythereafter.Werefertothistimet7Gyrtobethetimewhenthesystemisgloballyrelaxed. 50

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Top:TheseparationbetweenthemostboundparticlesofthetwomergingNFWhalos(runBp1)asafunctionoftime;Middle:timeevolutionofthevirialratio2T=jVj;Bottom:timeevolutionofthetotalpotentialenergyofthesystemV. 51

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Chandrasekhar 1943 ; Binney&Tremaine 1987 )andislargerthantheageoftheUniverseingalaxy-sizedsystems.Therefore,itisgenerallyassumedthatequilibriumandnon-equilibriumevolutionofthephase-spacedensitydistributionsingalaxy-sizeobjectscanbestudiedinthemean-eldlimit,wherethecollisionlessBoltzmannequationcanbeapplied.However,aswasrstnotedby Miller ( 1964 ),theN-bodyproblemischaoticinthesensethatthetrajectoryofthe6N-dimensionalphase-spacecoordinateofthesystemexhibitsexponentialsensitivitytowardsmallchangesininitialconditions.Thisexponentialinstabilityhasbeeninvestigatedextensivelyinseveralstudiesoverthepastthreedecades(see Merritt 2005 forareview).ThesestudieshaveshownthatthelargestN-bodyLyapunovexponentdoesnotconvergetowardzeroasNincreases( Kandrup&Smith 1991 ),evenforN-bodyrealizationsofintegrablepotentials( Kandrup&Sideris 2001 ). Hemsendorf&Merritt ( 2002 )haveshownthatthedirectN-bodyproblem(withN<105)isinherentlychaoticandthatthedegreeofchaos,asmeasuredbytherateofdivergenceofnearbytrajectories(withtheLyapunovexponent),increaseswithincreasingNwithcharacteristice-foldingtimeequalto1=20ofthesystemcrossingtime(intheabsenceofsoftening).Ithasalsobeenshownthatthemeante(e-foldingtime)increaseswithincreasingparticlenumberinthecaseofsoftenedpotentials( El-Zant 2002 )sothatdiscretenesseectsarereducedasthesofteningbecomescomparabletointerparticleseparations( Kandrup&Smith 1991 ).Theexponentialseparationofparticlesinenergyoccursonthee-foldingtimescale(te)whichismuchsmallerthanthetwobodyrelaxationtime(tr)( Kandrupetal. 1994 ; Hut&Heggie 2001 ).Thedivergenceofparticlesinenergysaturatesafterafewteandthenvarieswithtimeonthestandardtwo-bodyrelaxationtimescaleindicatingthatthemicrochaosisnotanimportantprocessinchangingthene-graineddistributionfunctionsofgravitatingsystems. 52

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Valluri&Merritt 2000 ; Kandrup&Sideris 2001 ; Sideris 2004 ).Additionally,decadesofN-bodyintegrationshavedemonstratedthat,inmanyways,thebehavioroflarge-NsystemsmatchesexpectationsderivedfromthecollisionlessBoltzmannequation(forexample, Aarseth&Lecar 1975 ).ThefactthatinanN-bodysystemtheLyapunovexponentwillprimarilymeasurethemicrochaosandnotthemacroscopicchaos(arisingfromthebackgroundpotential)meansthatitishaslimitedusefulnessformeasuringchaoticbehaviorinN-bodysystems.Othermeasuresofchaos,suchasthosethatmeasurechangesinorbitalcharacteristics(forinstance,fundamentaloscillationfrequenciesoforbits;forexample, Laskar 1990 Laskaretal. 1992 Valluri&Merritt 1998 ),relyontheabilitytoidentifysuchfrequenciesfromoscillationsthatlastbetween30-100orbitalperiods. Sideris ( 2006 )developedatechniquebasedonpatternrecognitionoforbitalcharacteristicswhichisapplicableintime-dependentsystemswhoseoscillationslast10-30crossingtimes.Toourknowledge,noneofthestandardmeasuresofchaosareapplicabletoN-bodyorbitsinpotentialswithnon-periodicpotentialuctuationslastingonlyafew(<10)crossingtimes.Thismakesitnecessarytodenenewwaysofquantifyingchaosandmixing. 53

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Valluri&Merritt 2000 ; Kandrup&Sideris 2003 ).StudiesthatusedliveN-bodysimulations(forexample, Kandrupetal. 1994 )havecomparedpairsoforbitsintwodierentrealizationsofsimulationswheretheinitialconditionshadbeenperturbedbyaninnitesimalamount.HereweworkwithorbitsinagivenN-bodysimulationandarethereforelimited(bytheresolutionofthesimulations)inourabilitytochoosepairsofparticlesthatareverycloseinphase-space.Weselectalargenumber(1000)ofnearest-neighborpairsinoneofthetwomergingN-bodyhalosandfollowtheirseparationsthroughtheentiremergerprocess.Wehavecheckedthatwhentwoidenticalhalosmerge,theresultswereindependentofthehalochosentoperformtheanalysis.Topickparticlepairsweadoptthefollowingscheme:AlltheparticlesinagivenhaloaresortedintheirseparationrfromtheMBPofthathalo,andtheparticlesarebinnedintenradialshellsofequalwidth,extendingfromtheMBPtotheinitialvirialradiusofthehalo(rvir'256:7kpc).Intheanalysisthatfollowswecomparebehaviorofparticlesinthe1st,4thand7thshellfromthecenterofthepotential.Theouterradiusoftherstshellisjustoutsidethescaleradius,the4thshellliesatthehalfmassradiusand7thshellliesatthe3/4massradiusoftheinitialNFWhalo.AparticlePwithseparationrPfromtheMBPispickedatrandominagivenshell,andthe5000nearestparticlesincongurationspacetotheparticlePareselected.Thevelocityseparationsvofeachofthe5000particlesrelativetoparticleParecomputed,andthe2500particleswiththesmallestvseparationfromPareselected.ThemagnitudesoftheseparationoftotalangularmomentumJofeachofthe2500particlesaboverelativetoparticleParecomputed,andthe1250particleswiththesmallestJseparationfromPareselected.TheanglebetweentheangularmomentumJandtheangularmomentumJPiscomputedforeachofthe1250particles,andthe625particleswiththesmallestangleareselected. 54

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Kandrup&Sideris ( 2003 )).Todeterminethebehaviorofa\typicalpair"ofparticlesinagivenshell,werstpick1000pairsofparticlesineachshell.Wethenreportthemedianvalueoftheseparationofthatphase-spacequantityasafunctionoftimeforthe1000particlepairs.Wend(asiswellknown)thatthemedianisamorereliableestimatorthanthemeanseparationofparticlesbecauseeachparticlehasasmallbutniteprobabilityofbeingejectedfromthesystemasaresultofthemerger,andsuchejectionscauseasmallfractionofparticlestoformalongtailinthehistogramofseparationsateachtimestep.Thus,forallquantitiesdenedbelowitshouldbeimplicitlyunderstoodthatwereportthemedianvalueoftheparameterfor1000pairsofparticles.Specically,wemonitorthefollowingquantities:\ln(r)"{thenaturallogarithmoftheseparationinthespatialcoordinaterofthemedianpairofparticles.\ln(jvj)"{thenaturallogarithmoftheabsolutevalueoftherelativethreedimensionalvelocityvofthemedianpairofparticles.\E"{theseparationintotalenergyofthemedianpairofparticles.\J"{theseparationinthetotalamplitudeoftheangularmomentumofthemedianpairofparticles.(Wepickpairswhosevectordirectionshaveangularseparationsoflessthan90deg.Astheparticlesseparate,wetracktheseparationintheamplitudeofJbutnotthedirection.)Thequantitiesdenedabovedierinoneimportantaspectfrompreviousstudiesinwhichsimilarquantitieshavebeenused:allparticlesinthepairsareself-gravitating(andnottestparticles)andarepickedfromaninitialdistributionfunctionthatisself-consistentwiththesphericalNFWpotential.Thisimpliesthatthetypicalinitialpairseparationln(r)ofparticlesdependsontheirlocationinthepotentialsuchthat 55

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. Mahonetal. 1995 ; Merritt&Valluri 1996 ; Kandrupetal. 2000 ; Kandrup&Siopis 2003 ).Theseauthorspickedensemblesoforbitswithinitialconditionsthatuniformlysampledaninnitesimallysmallregionofphase-spaceandmeasuredtherateatwhichtheensemblespreadsandllsaregionofcongurationspace.Ensemblesassociatedwithstronglychaoticorbitswerefoundtoevolvetoanear-invariantdistributionwithin5-10orbitalcrossingtimes,whileensemblesassociatedwithweaklychaoticor\sticky"orbits(orbitsthatlieclosetoresonanceinphase-space)evolvedonmuchlongertimescalesandoftendidnotreachaninvariantdistribution.Ensemblesassociatedwithregularorbitsonlyspreadslightly,duetophasemixingthatresultedfromthesmalldierencesintheirinitialintegralsofmotion. 56

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)toobtainthe1000nearestparticlestoP.Conguration-spacecoordinateshavetraditionallybeenusedtostudychaoticmixingofmicroscopicensemblesinsmoothpotentials.Insuchexperimentsmicroscopicensemblesofregularorbitsdonotmixatallincongurationspacesoalltheobservedmixingcanbeattributedtochaos.HowevertestscarriedoutwithensemblesselectedbytheaboveschemeintheN-bodyisolatedsphericalNFWhaloshowedquitesignicantspreadinginconguration-space.Theevolutionoftheensemblesincongurationspacecoordinatesfollowedthefollowingpattern:ensemblesspreadrapidlyincongurationspaceonashorttimescale(duetotheMillerinstability,seeSectionx ),andthencontinuedtospreadmoreslowlyataratethatscaledwithlocalorbitalperiod.Atagivenradiusthemixingratewasfoundtobesensitivetotheparticularensembleinquestion-thenatureoftheparentorbit(whetheritisbox-likeortube-like),thelocationoftheorbitrelativetotheorientationofthemergeretc.SincetheirmixingincongurationspaceisdominatedbytheMillerinstabilityandphasemixing,ourensemblesaretoolargetobeusefullycomparedwiththemicroscopicensemblesusedtomeasurechaoticmixingintheworksofpreviousauthors.However,wefoundthatensemblesselectedbytheaboveschemeintheisolatedhaloshowednegligiblemixinginthecoordinatesE;J(inagreementwiththepreviousndingsof Hut&Heggie 2001 ).Consequentlywefocusourattentiononmixinginthesetwovariables.ThemainobjectiveofcarryingouttheseexperimentsistoobservemixinginE;Jthatresultsfromthemerger. 57

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Divergenceinthequantitiesln(r)=ln(r(t=0))asafunctionoftime(inunitsoflocalcrossingtime)forparticlesinthe1st,4thand7thradialshells.ThesolidlinesareforthemergeroftwoNFWhalos(runBp1).Thedot-dashedlinesareforthecontrolsimulationofanisolatedNFWhalo.Thedashedstraightlinesarebest-tstoboththesolidlinesandthedot-dashedlinesintheregiont=tc=[0:050:35]. Wedenethequantities(E;J)foreachparticleintheensembleasfollows: (3{3) (3{4) whereE(P);J(P)aretheenergy,magnitudeofthetotalangularmomentumoftheparticlePrespectively.ThequantitiesE;J;arecomputedforeachofthe1000particlesintheensemble.TheresultsoftheseexperimentsaredescribedinSectionx 3.5.1SeparationofNearbyParticlesinPhase-SpaceWebeginbydemonstratingtheabilityofln(r)toidentifychaoticbehavior.Thisquantity(andalltheothersthatweuse)arecomparedtoequivalentquantitiesmeasuredinanisolatedequilibriumNFWhalo.ThethreepanelsofFigure 3-2 showtheinitialseparationinspatialcoordinateln(r)(relativetoitsvalueatt=0Gyr)inthree 58

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).Thisphaseofexponentialdivergenceisshortlived(<35%ofthelocalcrossingtime).Thee-foldingtimeoftheMillerinstabilityte=0:39;1:06;2:73Gyrforshells1,4,7respectively(roughlyhalfthelocalcrossingtimeineachshell).Inboththemergersimulationandintheisolatedhalo,theinitialrapidchangeinln(r)saturatesatthesamevalue-avaluedeterminedbythelengthscaleoverwhichthesmoothpotentialdominatesoverthegraininess( Kandrup&Smith 1991 ).Atsmallerradii,thehigherparticledensitiesresultinsaturationvaluesofln(r)thatarehigherthatatlargerradii.Figure 3-2 isclearevidencethatmicro-chaosisdetectabledespiteourinability(imposedbyouruseofaN-bodydistributionfunction)tochooseparticlesarbitrarilyclosetoeachotherinphase-space.Wenowfocusonthebehaviorofthefourquantities(ln(r),ln(v),EandJ)overlongertime-scales.AsbeforeallquantitiesarecomparedwiththeequivalentquantitiesmeasuredinanisolatedequilibriumNFWhalotocontrolforexponentialMillerinstability.Figure 3-3 showstheevolutionofln(r)andln(v)asafunctionoftimeinthreedierentradialshells(1st,4thand7thfromthecenter).Thequantitiesplottedarescaledusingtheinitialvaluesofthequantitiesatt=0Gyr;i.e.,thetoppanelsshow[ln(r)=ln(r(0))]andthelowerpanelsshow[ln(v)=ln(v(0))](wherer(0);v(0)aretheseparationsatt=0).Thesolidlinesshowtheevolutionofthequantitiesinthelow-resolutionmergersimulations(runBp1:twoNFWhaloswith2105particlesperhalo),andthedot-dashedcurvesshowtheevolutionofthesequantitiesintheisolated 59

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Thetopthreepanelsshowdivergenceinthequantitiesln(r)=ln(r(t=0))asafunctionoftimeforthreedierentshellsinthehalo(fromlefttoright:therst,fourth,andseventhshellsrespectively).Thelowerthreepanelsshowthequantityln(v)=ln(v(t=0)),forthesamethreeshells.ThesolidlinesareforthemergeroftwoNFWhalos(runBp1).Thedot-dashedlinesareforthecontrolsimulationofanisolatedNFWhalo. equilibriumN-bodyNFWhalo.InwhatfollowswepointoutsomecharacteristicfeaturesofFigure 3-3 thatareseenelsewhereinthissection.ThelowerpanelsofFigure 3-3 showtheseparationinln(v)inboththeisolatedhaloandthemerginghalos.Thereisanoticeableinitialdecreaseinln(v)(whichoccurssimultaneouslywiththelinearincreasein[ln(r)=ln(r(0))]),whichweattributetothemutualgravitationalattractionbetweenthepairofparticlesthatcausesadecelerationoftheparticleswhiletheirradialseparationincreases.Theinitialdecreaseinln(v)isseeninboththemerginghalosandtheisolatedhalo,conrmingthatthisisnotduetothemergerbutisacharacteristicoftheN-bodysimulation. 60

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3-1 .Itistherstandsecondpericenterpassagesthatcausethemostsignicantchangesintheparticleseparations(ln(r),ln(v)).Itisquiteremarkablehowlittlesystematicchangeinseparationofeitherrorvisseeninbetweenpericenterpassages,suggestingthatverylittlechangeinthemacroscopicseparationsofparticlesoccurringatanytimeduringthemergerexceptduringthepericenterpassages.Thiscanbeunderstoodasfollows:duringpericenterpassagesoftheMBPsofthetwohalos,thereissignicantoverlapinthetwopotentialsleadingtoasuddendeepeninginthepotentialexperiencedbytheparticlesleadingtoatransientcompressivetidaleldthatsimultaneouslydecreasestheseparationsinparticles(seenbythedipsinln(r))andincreasestheirkineticenergies(seeninthesharpincreasesinln(v)).Dynamicalfrictionbetweenthetwohalosisalsostrongestatpericenterandconsequentlythemajorityoftheirorbitalenergyandangularmomentumarelostimpulsivelyatthesetimes.Whenthetwohalosseparateenoughthattheirinnermostshellsnolongeroverlaptheyexperiencelittleornochangeintheirpotentials,sothereislittlefurtherchangeinseparationoftrajectories.Thus,theglobalpotentialuctuationsthatoccurbetweenpericenterpassagesdonotcauseanyfurtherincreaseinparticleseparations.Behaviorsimilartothatinshell1isseeninshells2and3butisnotshownhere.Theeectoftherstpericenterpassageisseentooccursimultaneouslyatallradii.Inshells4,7weobserveadipinln(r)andpeakinln(v)attherstpericenterpassagewhichisbothbroaderandlessintensethaninshell1.Atlargerradiitherstpericenterpassageisfollowedbyacontinuouschangeinln(r)andln(v)butsubsequentpericenter 61

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Thepanelsshowdivergenceinthequantitiesln(r)=ln(r(t=0))asafunctionoftimefor1st,4th,7thshells.ThesolidcurvesareforthemergeroftwoNFWhalos(runBp1)withpairsofparticlesselectedatt=0:Gyr.Thedashedcurvesareforpairsofparticlesselectedatt=3:0Gyrintheisolatedhaloandthedot-dashedcurvesareforpairsofparticlesselectedatt=3:0Gyr.Thecurvesforthepairsstartingatt=3:0Gyrhavebeentranslatedintimetot=0Gyr.Thedot-dashedcurvesandthedashedcurvesindicatethattheinitialexponentialinstabilityofnearbyorbitsduringthemergerisnotverydierentfromthatintheisolatedhalo,independentofthetimeatwhichtheparticlepairsareselected. passagesarelessclearlyvisible.Thisindicatesthatwhilethereisnopropagationdelayintheimpulsivetidalshockwithradius,theresponseislongerlivedatlargerradii.Thisisbecause,thecentersofthetwomerginghalosdonotseparatebeyondtheradiusofthe4thand7thshellsaftertherstpericenter.Consequentlythechangeintheexternalpotentialisnotassuddenasfortheinnershells.AsdiscussedinSectionx itisextremelydiculttomakeareliablequantitativemeasurementofanexponentialinstabilityonashorttimescale.However,boththequalitativeandquantitativebehaviorofinitiallynearbyparticlesseeninFigure 3-2 indicatesthatweareinfactdetectingtheMillerinstability.Iftheoscillatingexternalpotentialisalsocontributingtoanexponentialinstabilityoforbitsonemightexpectthatifpairsofnearbyparticleswerepickedduringthephasewhenthepotentialischangingveryrapidly,therateofchangeofseparationinsomequantity(sayr)wouldbegreater 62

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3-4 comparestheseparationsofparticlesln(r)whenpairswerepickedatt=0Gyrwithpairspickedatt=0Gyrintheisolatedhaloandseparationofpairsofparticlespickedatt=3Gyr.Forthepairsofparticlespickedatt=3Gyrinthemerger,thetimeaxisist3sothatinitialslopesofthethreecurvescanbemorereadilycompared.Wechoseastarttimet=3GyrtocompareagainstsinceitcorrespondstoatimeatwhichthevirialratioischangingveryrapidlyandcorrespondstotherstapocenterseparationoftheMBPsofthetwohalos(identicalbehaviorwasfoundwhenparticleswerepickedatt=4Gyr).Sincethepotentialischangingrapidlyatthistime,wemightexpecttoobserveanenhancedrateofseparationofnearbypairsoforbitsduetochaosinducedbythepotentialuctuationsprovidedthee-foldingtimeforthisinstabilityiscomparabletoorgreaterthanthatduetotheMillerinstability.Figure 3-4 shows,onthecontrary,thattheinitialexponentialseparationofparticlesinratt=3Gyrdoesnothaveasystematicallygreaterslopethantherateofseparationofparticlesatt=0Gyr.Inshell1theslopeofthedot-dashcurveissmallerthanthatofthesolidcurve(whichweattributetothefactthatheatingfromtherstpericenterpassagecausednearbyparticlestobefurtherapartinphase-space);inshell4theslopesofallthreecurvesareessentiallyidentical;onlyinshell7istheslopeofthedot-dashcurveslightlysteeperthanthatoftheblackcurvesindicatingthatchaoticitymaybeplayingabiggerroleatlargeradii.Whilethisisclearlynotconclusiveevidence,itissuggestiveofthehypothesisthatthedegreeofchaosduetothetime-dependentpotential,isnotsignicantcomparedtothechaoticityduetotheMillerinstability.Itmayplayasmallroleintheevolutionofthesystematlargerradii.Howeverthereislittleevidencetosuggestthatchaoticmixingfromorbitsinthetime-dependentpotentialisdrivingtherelaxation.Tounderstandbetterthemechanismdrivingtherelaxationwenowlookatseparationofparticlesinenergyandangularmomentumspace.Inintegrabletime-independent 63

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ThetopthreepanelsshowdivergenceinthequantitiesE=E(t=0)asafunctionoftimeforthreedierentshellsinthehalo(fromlefttoright:therst,fourthandseventhshellsrespectively).ThelowerthreepanelsshowthequantityJ=J(t=0),forthesamethreeshells.ThesolidlinesareforthemergeroftwoNFWhalos(runBp1).Thedot-dashedlinesarefortheisolatedNFWhalo. potentials,energyandsomequantityakintoangularmomentumaregenerallyintegralsofmotion.Itisthereforeofinteresttoquantifyhowthesequantitieschangeduringamerger.Figure 3-5 showstheseparationsinenergies(toppanels)andangularmomentum 3-3 .Asbefore,thedot-dashedlinesareforpairsofparticlesintheisolatedhalo,whilethesolidlinesareforpairsofparticlesinoneofthemerginghalos.ThequantitiesE;Jarescaledrelativetotheirinitialvaluesatt=0Gyr.EandJ(asexpected)areveryclosetozerointheisolated 64

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3-6 comparesseparationsinallfourquantitiesln(r),ln(v),EandJfortwomergersimulationswithdierentnumericalresolutions.WerecallthattherunHRBp 65

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Thefourpanels(toptobottom)showdivergenceinthefourquantitiesln(r),ln(v),EandJplottedasafunctionoftimeforpairsofparticlesintheinnermostshellofoneofthetwomergingNFWhalos.Thesolidlinesareforthemergeroftwohaloswith2105particlesineachhalo(runBp1);dot-dashedlinesareforthemergeroftwohaloswith2106particlesineachhalo(runHRBp).TheinsertsinthetoptwopanelsshowdierencesintheinitialseparationduetotheMillerinstability.Thedot-dashlinestartsatlowervaluesinbothln(r),ln(v)whichisaresultofpickingparticlescloserinphase-spaceinrunHRBp.ForrunHRBpln(r)hasashortere-foldingtime,duetothehigherparticlenumberandsmallersofteninglength.Apartfromthebehaviorofinitialseparationsinallfourquantities,thelowandhighresolutionmergersarealmostidentical. 66

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3-7 showsseparationsinln(r)andln(v)forparticlesindierentradialshellsinrunBp1andinthemergerofanNFWhalowithashallowcusp.Thebehaviorisremarkablysimilarinbothcasesexceptintheinnermostshell.Thisisexpected,sincetheinnermostshellistheonlyoneinwhichthedensityproles(andconsequentlythephase-spacedensitydistributions)diersignicantly.Itiswellknownfrompreviousstudies( Boylan-Kolchin&Ma 2004 ; Kazantzidisetal. 2006 )thatcuspsinshallowpotentialsarelessrobustthancuspsinNFWhalos.Particlesintheinnermostshellof 67

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Solidlinesshowseparationsinln(r)andln(v)forthemergeroftwoNFWhalosandthedot-dashlinesshowtheseparationsofthesequantitiesforthemergerbetweenanNFWhaloandahalowithaninnershallowcuspof=0:2.Thebehaviorissimilarinbothcasesexceptintheinnermostshellwherethedierencesinthedensityprolesaresignicant. theshallowcuspexperiencealargerincreaseinenergyduetoeachpericenterpassagethanparticlesintheNFWhalobecausetheyexperienceagreaterrelativeincreaseinthedepthoftheexternalpotentialarisingfromtheoverlapofthetwohalos,andconsequentlythephase-spacevolumeaccessibletothemislarger.Beyondthescaleradiusthetwohaloshaveessentiallyidenticaldensityproles,andseparationsinln(r)andln(v)arealmostindistinguishable.Theresultsofourinvestigationofparticleseparationsinln(r),ln(v),EandJcanbesummarizedasfollows:TheinitialseparationofpairsofnearbyparticlesinN-bodysystemssuchasthosestudiedhereisaresultoftheexponentialinstabilityoftheN-bodyproblem,thesocalled\Millerinstability."Thequalitativeandquantitativebehavioroftheseparation 68

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3-3 ),theirdependenceonsofteningandforceresolution(Figure 3-6 )arecompletelyconsistentwithpreviousstudiesofthisinstability.ItisimportanttopointoutthattheveryfactthatweareabletodetecttheexponentialdivergenceoforbitsduetotheMillerinstabilityindicatesthatdespitethefactthatpairsofnearbyparticlesintheN-bodysimulationsareseparatedbymacroscopicdistances(i.e.,arenotinnitesimallycloseby),ourmethodforidentifyinganexponentialinstabilitycanbeappliedsuccessfullytoanN-bodysystem.SubsequenttothesaturationoftheseparationsduetotheMillerinstability,themostsignicantincreaseinseparationsofnearbyparticles(inr,v,E,orJ)occurduringpericenterpassagesoftheMBPofthetwohalosandisaconsequenceofthecompressivetidalshocking(duetotheoverlapofthetwohalos)anddynamicalfrictionbetweenthetwohalosthatoccursduringthepericenterpassages.Intheinnermostshellsthereislittleornoseparationinphase-spacequantitiesoccurringbetweenpericenterpassages,despitethelargeglobaluctuationsinpotentialoccurringduringthesetimes.Atlargerradiiseparationinr,vcontinuefollowingpericenterpassages.Atlargerradii,thereisalsoasmall(butshortlived)increaseinseparationinEandJbetweentherstandsecondpericenterpassages.Ifweconsidertheincreaseinthemacroscopicseparationofparticlesin(r;v)tobeameasureofthemixingofparticlesinthesequantities,theseplotswouldleadustoconcludethatthemajorityofmixingoccursduetotheMillerinstabilityandseveralphasesofmixingthatoccurduringpericenterpassagesofthetwohalos.Inallradialshellsofmergersimulationweobservethatthemacroscopicseparationofinitiallynearbyparticlesincreasesuntilitsaturates.Inthemerginghalos,saturationoccursatlargervaluesofln(r),ln(v)respectively,thanintheisolatedhalo.Thisisaconsequenceofthefactthattheaccessiblephase-spacevolumehasincreasedduringthemergerprocess.WesawfromFigure 3-5 thatEandJ(whichrepresentincreasesintheavailablephase-spacestatesaccessibletoparticles)increasedinstep-wisefashionprimarilyduringpericenterpassagesoftheMPBs,aswaspredictedby Spergel&Hernquist ( 1992 ).Byt=7Gyrthevirialratio2T=jVj1,althoughthepotentialcontinuestoexperiencelong-livedbutlow-amplitudeoscillationsuntil13Gyr.Thelow-levelpotentialuctuationscauselittlechangeintheseparationofeitherrorvbeyond7Gyr.InLynden-Bell's(1967)modelfor\violentrelaxation",particlesexperiencechangesintheirphase-spacecoordinatesduetotheirinteractionwithatime-varyingbackgroundpotential.Ithasbeenarguedbyvariousauthorsthatatime-dependentpotentialaloneisinadequatetocauserelaxation,sincenearbyparticlesinphase-spacewillmerelyberelabeledinenergyandwillnotseparate(ormix).Theresultsfromthissectionindicatethatduringthemergeroftwohalos,nearestneighborparticlesinphase-spacedomixin 69

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Gnedinetal. 1999 ; Valluri 1993 ).Theparticlesrespondtosuddenincrementsintheirorbitalenergyandangularmomentumbyjumpingtoneworbits,resultinginmixinginphase-spaceandevolutiontoanewdistribution.Thislatterphaseofevolutioncouldindeedbetheresultofsomeweakerformofchaoticmixing,coupledwithphasemixing,bothofwhichoccuronalongertimescale. weconneouranalysistomixinginthescaledenergyandangular-momentumvariablesE;J.Figure 3-8 showstheevolutionofanensembleof1000self-gravitatingparticlesinthe4thshellinrunHRBp.Thedensitycontoursareobtainedusingakernelsmoothingalgorithm.Att=2andt=5Gyrwhichcorrespondtotherstandsecondpericenterpassage,respectively,theensembleundergoesasuddenspreadinginenergy(E)whichisaconsequenceofthedeepeningofthepotentialduringtheoverlapofthetwohalos.Byt=6GyrtheparticledistributionllsatriangularregioninE;Jspace.AgreaterspreadinginEatthesmallestvaluesofangularmomentumisaconsequenceofthefactthattheseorbitsareonthemostradialorbitsandconsequentlyexperiencegreaterchangesintheirpotentialenergies.Particleswiththesmallest(mostnegative)valuesofEexperiencethegreatestspreadinangularmomentumlargelyasaconsequenceoftheirbeingejectedduringthetidalshockstolargerradii.Wecarriedoutsimilarmixingexperimentsforover25dierentensemblesineachshell.ThebehavioroftheensembleinFigure 3-8 wasfoundtobequiterepresentativeofallthedierentensembles. 70

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Theevolutionofanensembleof1000nearestneighborsinphase-spaceinshell4att=0(runHRBp)plottedatoneGyrtimeintervalsinthemerger.Theplotshowsisodensitycontoursfortheparticledistribution.Theredcontourscorrespondtothehighestdensityregionsandblacktothelowestdensityregions.Thecontoursarespacedatlogarithmicdensityintervalsrelativetothemaximumdensitycontourtoenhancethevisibilityoflowdensityregions.Pericenterpassages(att=2andt=5Gyr)areseentocauseasuddenspreadingoftheentireensembleinE(aresultofthetidalshockingatthesetimes).Twoadditionalpericenterpassagesoccurbetweent=5andt=6(seeFigure 3-1 ).Byt=6Gyrtheensemblehasexperienced4pericenterpassagesandhassettledtollatriangularregionintheE;Jspace. 71

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IsodensitycontoursforensemblesofparticlesplottedasafunctionofE;Jforshell1(lefthandcolumn),shell4(middlecolumn)andshell7(right-handcolumn)plottedfordierenttimesintheirevolution(seetextfordetails). 72

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3-9 showscontoursofprojecteddensityofparticlesasafunctionofthequantities(E;J)inrunHRBp,forensemblesin3dierentradialshellsat4dierenttimesintheevolution.Theevolutionoftheensemblesinshells1,4and7areverysimilar.Att=2Gyr(rstpericenterpassage)theensemblespreadsinE.Similarincreasesinspreadandareseenduringthe2ndpericenterpassages(butarenotshownhere).Aftertherstpericenterpassagethemajorityoftheparticlesintheensembleatt=4Gyrreturntoamorecompactdistributionin(E;J)butonewithalargerspreadinJatsmallE.Thebottompanelscorrespondtot=6Gyr,aftertherstthreepericenterpassageshaveoccurred.AtthistimealltheensemblesllaroughlytriangularregioninE;Jspace.Thischaracteristictriangularnaldistributionisseeninallshellsandforessentiallyallofoverahundreddierentensemblesweexamined.Thisnaldistributionisnotuniformindensity,butthedensitycontrastacrosstheensembleismuchsmallerthanintheinitialdistribution.Wenotethatwhentheseensembleswerere-observedat15Gyrtherewasslightlygreateruniformityofdensityinthelower-densitytailsbutotherwisetheyhadchangedverylittlefromtheirdistributionsat8Gyr.Therearetwoimportantobservationsthatcanbemade:(a)changesinE;Joccurprimarilyduringpericenterpassages(Figure 3-8 ),and(b)thenaldistributioninE;Jspaceoftheensemblesatallradiiaresimilar,andtherateatwhichthisnaldistributionisreachedisindependentofradius(Figure 3-9 ).Themediancrossingtimeintheinnermostshell(r25kpc)istc5108yrs,whileinthe7thshellistc5Gyr.ItisobviousfromFigure 3-9 thatalthoughthecrossingtimeinshell7isafactorof10longerthaninshell1,thediusionoftheensemblesin(E;J)-spaceappearstooccurprimarilyduetothepericenterpassagesoftheMBPsandthereisonlyaslightlyincreasedrateofspreadingatlargerradii.Phasemixingandchaoticmixinginisolatedpotentialsoccuratratesthatscalewiththelocaldynamicaltimesoonewouldexpectthatmixingeectsthatdependonthelocaldynamicaltime(suchaschaoticmixing)wouldoccurfasteratsmallradiithanatlargeradii.The 73

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3-9 providesthestrongestevidencesofarthatmixinginenergyandangularmomentumisdrivenbycompressiveshockinganddynamicalfrictionthatoccuratpericenterpassages.Whilechaoticmixing,asdenedforstaticsmoothpotentials,mightbeoccurringduringthemergeroftwohalos,itplaysaminorroleindrivingtheremnanttoequilibrium. 74

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75

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Kazantzidisetal. 2006 ).Twoquestionsthatarisefromthesendingsare:1)howdothenalradialdistributions,energiesandangularmomentaofparticlesintheremnantdependontheiroriginallocationinthemerginghalosand,2)Fromwheredotheparticlesthatendupoutsidethevirialradiusinthemergerremnantoriginate?Addressingtherstquestionisimportantsincethemergercausestheredistributionofparticlestooccurinawaythatthedensityprolesandphase-spacedistributionfunctionspreservehomology.Thismaybeinterpretedasanindicationthatparticlesatallradiiarebeingheateduniformlyatallradii.However,itisalsoreasonabletoassumethatlessboundparticlesintheouterpartofthehaloarepreferentiallyheated.Itisalsoofinteresttounderstandhowtherelativechangeinenergyorangularmomentumofparticlesdependsontheirinitiallocationinthehalo.Inthissectionweexaminetheredistributionofparticlesinradius,energyandangularmomentumandthedependenceoftheirnallocationsontheirinitiallocationinthemerginghalos.Wenoteinpassingthatwhiletheisolatedhalosandtheinitialmerginghalosarespherical,justifyingtheuseofsphericalradialshells,themergerremnantissignicantlyprolate-triaxialwithminor-to-majorprincipalaxisratioc=avaryingfrom0:5atthecenterto0:7atthevirialradius.Althoughthemergerremnantsexhibitsignicantdeparturesfromsphericalsymmetry,inwhatfollowsweignorethetriaxialityofthenaldistributionwhenbinningparticlesinradius.EachpanelinFigure 4-1 showshowparticlesthatwereoriginallyinagivenshell(indicatedbythecaption)areredistributedinthenalradialshellsatt=9Gyrwhen 76

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Thefractionofparticles(N=Nmax)thatlieineachoftheradialshellsatthenaltimestep.Theverticaldashedlineandthecaptionineachpanelindicatetheshellinwhichparticleswereatt=0Gyr.ParticlesfromoneofthetwoNFWhalos(inrunBp1)isshownasasolidlinewhileparticlesinthesameshellsoftheisolatedNFWhaloareshownbydot-dashedline. 77

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Boylan-Kolchin&Ma 2004 ; Kazantzidisetal. 2006 ).Figure 4-2 showsdistributionsofthechangesinthekinetic,potential,andtotalenergiesofallparticlesindierentshellsattheendofthemerger(E=EnalEinitial).They-axisgivesthefractionalchangeinnumberofparticles(relativetothetotalnumberofparticlesinagivenradialshell).Particlesintheinnermostshellexperienceanetdecreaseintotalenergythatisaresultoftheoveralldecreaseinthepotentialenergyoftheparticlespointingtoanenergysegregationphenomenonduringmergers( Funatoetal. 1992 ).Thisisexpected,sincethereisanoverallincreaseof60%inthemassofthecuspcomparedtothemassintheinitialcusp( Kazantzidisetal. 2006 ).Atallotherradii,thereisanetincreaseintotalenergythatismostsignicantatlargerradii.Thisisaresultofsignicantfractionsofparticlesgainingpotentialenergyandbeingredistributedtolargerradiiasseenfromthemultiplepeaksinthepotentialenergydistributionsarisingfromparticlesheatedduringeachpericenterpassage.ItisstrikingtonotethatinallshellsthekineticenergydistributionsremainpeakedaboutE=0andaremorepeakedthanGaussian,indicatingthatthemajorityofparticlesexperienceonlyasmallchangeintheirkineticenergiesdespiteexperiencinglargechangesintheirpotentialenergies.Thisisadditionalconrmationthattheredistributioninenergythatwesawinthemixingexperiments(Sectionx )isprimarilyduetochangesinpotentialenergiesofparticlesduetointeractionswiththebackgroundpotentialduringpericenterpassages.Theinnermostshellincludesallparticleswithin25kpc,whichisslightlylargerthantheinitialscaleradius(rs=21kpc)ofthemergingNFWhalos.Figure 4-3 issimilartoFigure 4-2 butshowshowparticlesinthreeequalradialshellswithinrsare 79

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Histogramsofthechangeinenergy(E=EnalEinitial)forallparticlesinshells1(top),4(middle),and7(bottom).Thesolidlineshowsthechangeintotalenergy,thedot-dashedshowsthechangeinkineticenergy,andthedashedlineshowsthechangeinpotentialenergy. 80

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Histogramsofthechangeinenergy(E=EnalEinitial)forallparticlesinthreeequalradiusbinswithinthescaleradius(rs=21kpc).Thesolidlinerepresentstotalenergy,thedashed-dotrepresentskineticenergyandthedashedrepresentspotentialenergy. 81

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4-4 showshistogramsofchangeinJz(thecomponentofJperpendiculartotheorbitalplaneofthemerger)forallparticlesinradialshells1,4and7.ThehighlypeakeddistributionsintheinnermostshellindicatesthatthereisvirtuallynonetchangeinJzfortheparticleshere.Intheoutertwoshells,Jzhasanincreasinglywiderdistribution.Althoughitisnotplottedhere,wenotethatintheoutermostshells(shells9,10)themedianofthedistributionofJzshiftstoslightlypositivevalues,indicatingasmallgaininnetangularmomentumfortheoutermostparticles.Theobservedincreaseisaresultoftransferoftheorbitalangularmomentumofthemerginghalosintotheangularmomentumofindividualparticles.Ourresultsindicatethattheangularmomentumofthemergerisabsorbedprimarilybyparticlesatlargeradii.Aswasnotedpreviously,about10%ofthemassoftheinitialDMhalosliesoutsidetheducialvirialradius,butithasbeenfoundthatnearly40%ofthemassofthenalremnantliesoutsidethevirialradiusoftheremnanthalo( Kazantzidisetal. 2006 ).WenowexaminethedistributionofparticlesinthemergerremnantofrunBp1todeterminewheretheseparticlesoriginatefrom.Figure 4-5 showsthefraction(Fvir)ofthetotalnumberofparticleslyingbeyondthevirialradiusoftheremnantthatoriginatedfromeachofthe10shellsoftheoriginalhalos.Allparticlesthatlayoutsidethevirialradiusintheoriginalhalosareassignedtoshell11(whichextendsfromthevirialradiustotheouteredgeofthesimulationvolume).Thehighestfraction(about25%)oftheparticlesoutsidethevirialradiusoftheremnantwerealreadyoutsidethevirialradiusoftheinitialsystems.Theinnermostshellisthemostrobustwithfewerthan(<1%)beingejectedbeyondrvir,Interestingly,allshellsfromthehalfmassradiusonward(shell3andbeyond)contributeroughlyequally(811%)totheparticlesthatlieoutsidervirinthenalremnant.Figure 4-5 andFigure 4-1 togethershowthattheradialredistribution 82

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HistogramsofchangeinJz(componentofJperpendiculartotheorbitalplaneofthemerger)forallparticlesintheselectedshells. 83

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Fractionofparticlesthatoriginatedineachoftheradialshells(1-11)inoneoftheoriginalNFWhalosinvolvedinthemergerthatendupoutsidetheducialvirialradiusoftheremnant.(Allparticlesbeyondthevirialradiusareassignedtoshell11.) ofparticlesoccursessentiallyindependentlyoftheoriginalradiusoftheparticle,providedthattheparticleslieoutsidethecentral3shells. 3 liesoutsideitsformalvirialradiusandthatthismatterisejectedroughlyuniformlyfromallradiioutsidetheinnerregions.Thishighlightsthefactthatmass,initsstandardvirialdenition,isnotadditiveinmergers.WeconrmthendingsofseveralotherauthorsthatcuspsofDMhalosareremarkablyrobust.Therobustnessofcuspsisaconsequenceofthefactthat80%oftheparticlesintherstshellhaveapocentersthatliewithintherstshell.Duringthemergeronlyasmallfraction(20%)ofparticleswithapocenterswithintheoriginalcuspareejectedtolargerradii.Themajorityoftheejectedparticlesdonotgetbeyondshell4.Amuchlargerfraction(45%)ofparticlesinthecentralregionsofcore-likeprolesisejectedtoradialshellsdirectlyoutsidethecuspduringamerger. 84

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@t+vrfr@f @v=0; whereisthegravitationalpotential.TheCBEstatesthatdf=dt=0inacollisionlesssystem,whichimpliesthatf(x;v;t)isaconstant.Inpractice,itisimpossibletomeasurethene-graineddistributionfunctionf,andwhatismeasuredisanaverageoffoversomevolumeofphase-space.Thus,thecoarse-graineddistributionfunctionfatanyphase-spacepoint(x;v)istheaveragevalueoffinsomesmallvolumeelementinphase-spacecenteredon(x;v).WhilefdoesnotobeytheCBE,itis,infact,theonlyquantitythatcanactuallybemeasuredforadynamicalsystem.Sincethecoarse-graineddistributionfunctionfistheonlyquantitythatcanbemeasured,itisinterestingtodeterminehowitvarieswithtime.Inthischapter,wewillfocusonthetimeevolutionofthecoarse-grainedphase-spacedensityduringthemergerandrelaxationoftwoDMhalos.Theactualvalueofthecomputedfunctiondependsonthespecicdetailsofthecoarse-grainingemployed.However,theMixingTheorem( Binney&Tremaine 1987 ; Tremaineetal. 1986 ; Mathur 1988 )statesthatprocessesthatoperateduringtherelaxationofcollisionlesssystems(e.g.phasemixing,chaoticmixing,andthemixingofenergyandangularmomentumthataccompaniesviolentrelaxation)resultina 86

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Binney&Tremaine ( 1987 ); Tremaineetal. ( 1986 ),especiallyfollowingsucheventsasmergersandviolentrelaxation.Oneofthesimplestmethodsforcomputingcoarse-grainedphase-spacedensityoftheDMhalosthatresultfromcosmologicalsimulationswasrstusedby Taylor&Navarro ( 2001 ),whodenedaspherically-averagedspacedensity where(r)istheconguration-spacedensityaveragedinsphericalshells,and(r)isthesphericallyaveragedvelocitydispersion.TheyfoundthatQ(r)iswellapproximatedbyasinglepowerlaw,Q/r1:87,from102:5ofthevirialradiustojustbeyondthevirialradius(i.e.overmorethan2.5ordersofmagnitudeinradius),factthatwaspredictedby Bertschinger ( 1985 )forasecondaryinfallmodel.Inrecentyears,severalgroupshavedevelopednumericaltechniquesforcomputingthecoarse-grainedphase-spacedensityinN-bodysimulations.Allofthesetechniquesrelyuponwaystodivide6-dimensionalphase-spaceintobinsinwhichparticlescanbecounted.These\binning"techniquesarealsoreferredtoastesselationschemes: ( 2004 )usesa\Delaunaytesselationeldestimator,"whichcomputesthedensitiesofasetofpointsfromthevolumeoftheDelaunaycellstowhichtheybelong.Inaspaceofddimensions,a\Delaunaycell"isdenedasad-dimensionalpolyhedronmadebyconnectingeverysetofd+1points,suchthata(d{1)-dimensionalspherepassesthroughallofthembutdoesnotencompassanyotherpointfromthesample.Thecodethattheywrotetoimplementtheschemeisreferredtoas\Sheshdel."Theseauthorsusetheircodetoshowthatthehighestphase-spacedensityregionsintheUniverseatz=0areatthecentersofDMhalosandsubhalos.ThemainproblemwiththeDelaunaytesselationschemeisthatthephase-spacevolumes,andthus,thedensityestimatesderivedarenotmetric-free. ( 2005 )useamethod(encodedinthealgorithm\FiEstAS")thatismetric-free,andveryfastcomparedwithothertesselationschemes,e.g.,Sheshdel. 87

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Sharma&Steinmetz 2006 )isanalgorithmwhichemploysamethodwhichisverysimilartothemethodusedby Ascasibar&Binney ( 2005 ),butwiththedierencethat,ateachstep,thespaceistesselatedalongthedimensionhavingminimumentropy(andmaximuminformation).Thisschemeoptimizesthenumberofdivisionstobemadeinaparticulardimension(asopposedtoFiEstAS)andextractsmaximuminformationfromthedata.ThemajordierencebetweenEnBiDandFiEstASisthat,becauseitemploysaminimumentropycriterion,EnBiDismoreaccurateatmeasuringthephase-spacedensitywhenitislow.Sincewearemainlyinterestedintheevolutionoftheoveralldensityproleand,inparticular,therobustnessoftheregionswiththehighestphase-spacedensity,wedonotexpectourresultstobesignicantlyalteredbythechoiceofcode.Furthermore,wecomparedtheoutputsoftheSheshdelcodeandtheFiEstAScodesonsomeofoursimulationsandfoundtheresultstobequalitativelyindependentofcodeemployed. Taylor&Navarro ( 2001 ).TheyfoundthatQ(r)iswellapproximatedbyasinglepowerlaw,Q/r1:87,overmorethan2.5ordersofmagnitudeinradius.Sincethispower-lawprolewasdiscovered,severalstudieshavefocusedonunderstandingitsoriginand,inparticular,whethertheproleis:(a)aresultofthehierarchicalgrowthduringstructureformation,(b)theresultoftherelaxationprocessessuchasviolentrelaxation,(c)theresultofinitialconditions,orsomecombinationofthethree. Austinetal. ( 2005 )studiedtheissueoftheoriginofthepower-lawprolesinQbyusingsemi-analyticalextendedsecondaryinfallmodels.Thesemodelsfollowtheevolutionofcollisionlesssphericalshellsofmatterthatareinitiallysettobeoutofdynamical 88

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Barnesetal. ( 2006 )demonstratedthatthedynamicalequilibriumofhaloswithvelocityanisotropyisnotenoughtoproducethepowerlawfor=3,but Navarroetal. ( 2004 )andSersicdensitydistributionsindynamicalequilibriumdoproducenearlyscale-freeQproles. Barnesetal. ( 2007 )extendedtheirpreviousworkandincludedconstantanisotropiesandanisotropydistributions,andtheyfoundthatthereisadistinctsimilaritybetweentheshapesofradialdensityandanisotropyproles. MacMillanetal. ( 2006 )arguedthattheradialorbitinstabilityisthemissinglinkthatproducesthepower-lawstructure,andthatitmightalsodriveisolatedsystemstoacommonnalstateindependentofinitialperturbations. Peiranietal. ( 2006 )studiedtheevolutionofQinthecoreofDMhalosandshowedthattheQrelationdependsonthemechanismbywhichhalosacquiremass,andthathalosthathaveundergonemajormergereventshavelowerphase-spacedensitiesthanthosethathaveaccretedmassquietly.TheirresultswereindependentofthechoiceofthespecicDMmodelchosen:colddarkmattermodelsresultedinthesameprolesaswarmdarkmatterscenarios. Peirani&deFreitasPacheco ( 2007 )denedaglobalvalueforthephase-spacedensity,Q,whichwasfoundtodecreaserapidlywithtime.Thisisconsideredtobeaconsequenceoftherandomizationprocessofinitialbulkmotionsafterredistributionofenergyfollowingeachsuccessivemerger.Inotherwords,ashalosmerge,theirorbitalkineticenergyis 89

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3 .InadditiontothesimulationsdescribedinChapter 3 ,wealsobrieydescribetheresultsoffourothersimulationsinwhichwestudytheeectsofvaryingorbitalparametersandtheslopeofthecentralcusp. ,theCBEdeterminestheevolutionofne-grainedphase-spacedensitywithtime,andthisequationimpliesthatf(t)isconstant.Thecoarse-grainedphase-spacedensityistheonlyquantitythatisactuallymeasurableviarealisticexperiments,andthiscanbecomputedinanumberofdierentways.Onewaytocomputethecoarse-grainedphase-spacedensityproleistoaveragespherically,asinthecaseofthequantityQ,usedbyseveralotherauthors.FortheresultsdescribedinthischapterandinChapter 6 ,weusetheFiEstASalgorithmwhichwaskindlyprovidedbyYagoAscasibar,whichisdescribedingreaterdetailin( Ascasibar&Binney 2005 ).Sincewecanonlyestimatethecoarse-grainedphase-spacedensityfinarealdynamicalsystemorinN-bodysimulations,andwecannevermeasurethene-grainedphase-spacedensityf,wesimplifyournotationanduseftorepresentthecoarse-grainedphase-spacedensityobtainedfromFiEstAS.TheFiEstASalgorithmtakesasinputthephase-spacecoordinates(x;v)ofeachparticleintheN-body 90

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Histogramsofthechangeinphase-spacedensity(f=(fnalfinitial)=finitial)forallparticlesinshells1(top),4(middle),7(bottom).ParticlesfromoneofthetwoNFWhalos(inrunBp1)areshownasasolidline,whileparticlesinthesameshellsfromtheisolatedNFWhaloareshowninadashedline. simulationandgivesasoutputtheconguration-spacedensity,(x),andphase-spacedensity,f(x;v),atthelocationofeachparticle.Inordertounderstandhowtointerprettheinformationprovidedbyf(x;v)asitevolveswithtime,wecomparehistogramsofthechangeinf(x;v)fromthebeginningtotheendofasimulation.Figure 5-1 showsthedistributionsofthechangesinf(f=(fnalfinitial)=finitial).ThedashedcurvesareforparticlesinanisolatedsphericalNFWhalowhichisevolvedinisolationfor15Gyr,whilethesolidcurvesareforparticlesintheparabolicmergeroftwo 91

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3 andshowallparticlesinradialshells1,4,and7.They-axisgivesthefractionofthetotalnumberofparticlesinagivenshell.Thesolidcurvesclearlyshowthat,inthecaseofthemergeroftwohalos,thepeakofthedistributionisbelowzero,indicatingthatthemedianphase-spacedensityhasdecreasedasexpectedduringthemerger.Itisimportanttonotethat,inthecaseoftheisolatedhaloinequilibrium,f=fphasemixingisexpectedtoresultinaslightoveralldecreaseincoarse-grainedphase-spacedensitywhenaveragedovertheentiresystem.Whenwefollowthechangeinphase-spacedensityaroundaspecicparticle(aswedowithFiEstAS)weseethatthedistributionoff=fisalmostGaussian,sinceeachparticleintheequilibriumhalocantraveltoeitherahigherorlowerregionofphase-space.Wenote,therefore,thatitisnecessarytoobtaineitherameanoramediandistributionoftheFiEstASoutputftoproperlyinterpretthechangesweobserveinthecontextofourstandardexpectationthatcoarse-grainedphase-spacedensitydecreaseswithevolutionduetomixing.Inordertocomparewithresultsfrompreviouswork,wealsocomputethesphericallyaveragedphase-spacedensityQ(r),employedbyotherauthors.Throughoutthisstudy,Q==3wasestimatedasfollows:allparticlesinagivenhaloweresortedintheirseparationrfromthemostboundparticle(MBP)ofthathaloandthenwerebinnedin100radialshellsofequalwidth.Themassdensitywasevaluatedforeachradialbin,andthetotalvelocitydispersionwascalculatedas=q 5-2 showsacomparisonofQ(r)fortheisolatedNFWhaloaswellasthemedian(dottedline)andthemeanvalue(thinsolidline)offforthishaloproducedbyFiEstAS.Themedianandmeancurveswerecomputedforalltheparticlesineachofthe100sphericalshellsmentionedabove.Thisgureillustratesthat,inthecaseofNFWhalos,Q(r)isapowerlawoverseveralordersofmagnitude(uptothevirialradius).It 92

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Twomeasuresofthephase-spacedensityasafunctionofr=rvir:Q==3(dot-dashedline)andf(dottedlinecorrespondstothemedianvaluesoff,whilethethinsolidlinerepresentsthemean-seetextforexplanations),forthecontrolsimulationofanisolatedNFWhaloatt=0Gyr(toppanel).Apower-lawtQ/r1:87(redline)isoverplotted.ResidualsforthetinQareinthebottompanel. deviatesfrompowerlawatlargeradiiwithabumpinthedistribution.ThisdeviationfrompowerlawarisesbecauseofthenecessityofsettingupaninitialdistributionfunctionthatisinequilibriumandcouldalsobebecausetheNFWhalosusedhereweredesignedtohaveisotropicvelocitydispersionprolesatallradii,whilecosmologicalNFWhaloshavelargeradialvelocityanisotropyespeciallyatlargeradii.ThedensityproleoftheisolatedNFWprolefallsomorerapidlythaninthecaseofasimpleextrapolation 93

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5-3 showsQ(dashedline)andthemedianoff(dottedline)fordierentepochsduringthemergeroftwosphericalNFWhalosonaparabolicorbit(Bp1merger),asafunctionofradius(inunitsofthevirialradiusatt=0Gyr).BothQandfspreadorextendtolargerradiiasthemergerprogresses.Thisisbecause(aswediscussedinChapter 5 )materialisejectedtolargeradiiduringthemerger.FortbetweentwoandthreeGyr(threeGyrisnotshowninthegure),weseeatemporaryshrinkagewithradiusandalargeassociateductuationinbothQandf.Thesetimesaresignicant,astheycorrespondtotherstpericenterpassageandtherstapocenterpassage,respectively.Forr<0:9rvir,Qremainsroughlypowerlawthroughouttheevolution.ParticlesbeyondthatradiushavehighervaluesofQthanwouldbeexpectedfromasimpleextrapolationoftheinnerpowerlaw.Thisissignicantbecause,asweshallnoteinChapter 6 ,thisdeviationabovethepower-lawdistributionisnotseenincosmologicalproles(seeFigure 6-2 ). 94

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5-2 )isalsoanexcellenttatt=0GyrinthiscaseanditcontinuestoprovideagoodttoQuptofourGyr.Figure 5-4 showsQ(bluedot-dashedline)andf(bluedottedline)asafunctionofradiusfortheinitialepoch(includingparticlesfromonlyonehalo)andforthenalone(blacklines,includingparticlesfrombothhalos)forrunBp1(toppanel),fortheBp2run(middlepanel),andforthehBp2run(bottompanel).Theorangelinesinthebottompanelrepresenttheparticlesthat,att=0Gyr,formthehalowithashallowercusp(=0.2),andtheblueonesareforthehalowithasteepercusp(=1.7)inthehBp2run.ItisremarkablethatbothQandfprolesareextremelysimilarfortheintialcongurationandfortheremnantforradiilessthan0.8rvirintheBp1(`NFW-NFW')andBp2(`shallow-shallow')mergers.ForthehBp2case(`shallow-steep'),theremnantprolesareverymuchidenticalwiththoseofthesteepercase,whichisinagreementwithresultsof Boylan-Kolchin&Ma ( 2004 )and Kazantzidisetal. ( 2006 ).Inadditiontoknowingthemeanorthemedianvalueoffatagivenradius,itisusefultoknowhowthevaluesoffaredistributedateachradius.InFigure 5-5 ,weplotisodensitycontourstorepresentthenumberdensityofparticlesinthe(f;r)plane.Inaddition,themedianvalueoffateachradiusisoverplottedasathickwhiteline.Theorangeregionsoftheplotcorrespondtothehighestnumberdensityofparticles,andtheblackregionscorrespondtothelowestnumberdensityofparticles.Contoursarespaced 96

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Isodensitycontoursforthephase-spacedensity(f)asafunctionofradiusfortheBp1run,plottedfordierenttimesinitsevolution.Theorangecontourscorrespondtothehighestdensityregionsandtheblacktothelowestdensityregions.Contoursarespacedatlogarithmicdensityintervalsrelativetothemaximumdensitycontour.Themedianvalueoff(whiteline)isevaluatedinequallyspacedlogarithmicbinsinradius.Thepresenceofislandsofhigherparticledensityatdierentradiiisclearlynoticeable,suggestingthatthesystemisnotyetcompletelymixed. 98

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Kazantzidisetal. 2006 ).Ascanbeseen,someoftheseejectedshellshavehighphase-spacedensity(orangeregions).Similarwiggleswereseenby Aradetal. ( 2004 )(theirFigure7)andwereattributedtosub-halos,butoursimulationsdonothavesubhalos,andhence,weconcludethatthefeaturestheyobservedcouldalsobeshells.Asdiscussedearlierinthissection,thecollisionlessBoltzmannequationstatesthatthene-grainedphase-spacedensityfaroundthephase-spacepointofagivenparticlealwaysremainsconstant.So,V(f)df,thevolumeofphase-spaceoccupiedbyphase-spaceelementswhosedensityliesbetween(f;f+df),isalsoconserved.However,itisimportanttonotethatthecoarse-grainedphase-spacedensityfanditsassociatedphase-spacevolumeV(f)arenotconserved.Ontheotherhand,therearesimplerelationshipsthatexistbetweentheinitialV(f)andV(f)atalatertimethatarisefromtheMixingTheorem( Tremaineetal. 1986 ; Mathur 1988 ).TheMixingTheoremstatesthatmixingreducesfsuchthat,atanytime,f
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Sharma&Steinmetz ( 2006 )): wherefbin=(fi+1+fi)=2TheplotinFigure 5-6 showsV(f)prolesfortheBp1runateachtimeintheevolutionofthemerger.Inaddition,abest-tpowerlaw(V(f)[f]2:450:012),overtherange1010<[f]<107:5(where[f]isfbindenedinEquation 5{5 )isplottedatallepochs.Wenotethat,sincethereisinsignicantmixingintheisolatedhalo,V(f)fortheisolatedhalo(notplotted)doesnotchangewithtime.Thevolumeofphase-spacedensityV(f)associatedwithlowphase-spacedensity[f]increasesafterthemergeriscomplete(sixGyr).However,wenotethatthevolumeofphase-spacedensityV(f)associatedwithhigherphase-spacedensityvaluesexhibitsverylittlechangeduringtheevolution.Theincreaseinvolumeofphase-spaceatlowphase-spacedensitiesispredictedtobeaconsequenceoftheMixingTheorem.However,thepreservationofthepower-lawdistributionofV(f)over2.5ordersofmagnitudein[f]duringtheevolutionissurprising.IstheconservationoftheV(f)proleshapetheresultofconservationofthedensityprole(whichremainsaNFWprole)atvariousradii,oristheresomethingmorefundamentalthatresultsinthisevolution?Weinvestigatetherobustnessofthepower-lawproleinV(f)byover-plottingtheV(f)prolesseparatelyforparticlesinfourdierentradialbins(Figure 5-7 ).V(f)ofparticlesintherstbin(inner10%byradius)isplottedwithasolidline.Theparticlesinthenextthreeradialbinsarecombinedandrepresentedbythedottedline.Theparticlesinbinsve,six,andsevenarerepresentedbyadashed-dottedline,andV(f)forall 100

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ThevolumedistributionfunctionV(f)(seeEquation 5{3 ),fordierentepochsintheevolutionoftheBp1run.Apower-lawtV(f)/f2:45isshownontopofeachcurveforthevaluesof[f]intherangeof[1010;107:5]. 101

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ThevolumedistributionfunctionV(f),fordierentepochsandfordierentbinsintheevolutionoftheBp1run.Thesolidlinerepresentstheparticlesthat,att=0Gyr,werelocatedwithin0.1rvir;thedottedlineisfor[0.1-0.4]rvir;dashed-dotfor[0.4-0.7]rvir;longdashesforallparticlesbeyond0.7rvir. 102

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4-1 ,toppanel,whichshowswhereparticlesthatwereoriginallyinthecuspareredistributedattheendofthemerger.About57%remainintheinnermostshell,andtheremaining43%spreadtoalltheothershells.Itisthis43%whichleavethecuspduringthemergerthatcausethespreadoverfourordersofmagnitudein[f].Itisclearthatthelargelowphase-spacedensityregionsobservedinFigure 5-6 originatefromtheparticlessituatedoutsidetheseventhbin(long-dashedline).Figure 5-8 plotsfasafunctionofthetotalenergy(E)andtotalangularmomentum(J)oftheparticlesatt=0Gyrfortheisolatedrun.Itshowscontoursofequalfforparticlesasafunctionoftheir(E;J).Weseethatthehighestphase-space-densityparticleshavethemostnegativeenergies.Ateachenergythereisaspreadofangularmomenta.ThisinitialdistributionarisesbecausetheoriginalNFWhalosweresetuptohavedistributionfunctionsdependentsolelyupontheenergy,f=f(E).So,ateachenergy,particlesweredistributeduniformlywithangularmomentum.Thus,thelowerthephase-spacedensityoftheparticlesis,thelargertheirenergyEandthegreatertheirspreadinJ.InFigure 5-9 weplotfasafunctionofthetotalenergy(E)andtotalangularmomentum(J)ofalltheparticlesatdierentepochsfortheBp1run.Figure 5-9 showscontoursofequalfforparticlesasafunctionoftheir(E;J).Attheinitialepochweseethatthehighestphase-space-densityparticleshavethemostnegativeenergies.Ateachenergythereisaspreadofangularmomenta.ThisinitialdistributionarisesbecausetheoriginalNFWhalosweresetuptohavedistributionfunctionsdependentsolelyuponthe 103

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Isodensitycontoursforthephase-spacedensity(f)asafunctionofthetotalenergyEandtotalangularmomentumJfortheisolatedrun,plottedatt=0Gyr.Thepurplecontourscorrespondtothehighestdensityregionsandtheblacktothelowestdensityregions.Contoursarespacedatlogarithmicdensityintervalsrelativetothemaximumdensitycontour. energy,f=f(E).So,ateachenergy,particlesweredistributeduniformlywithangularmomentum.Thus,thelowerthephase-spacedensityoftheparticlesis,thelargertheirenergyEandthegreatertheirspreadinJ.Asthemergerprogresses,thereissignicantspreadingandmixinginenergyandangularmomentumaswellasanincreaseinthevolumeofphase-spaceoccupiedbyparticles(asrepresentedbythegreentoblackcolorsinthecontourlevels,indicatinglowphase-spacedensity).However,inthenalepoch,theparticledistributionregainsitstriangulardistributionwiththephase-spacedensityof 104

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Isodensitycontoursforthephase-spacedensity(f)asafunctionofthetotalenergyEandtotalangularmomentumJfortheBp1run,plottedfordierenttimesinitsevolution.Thepurplecontourscorrespondtothehighestdensityregionsandtheblacktothelowestdensityregions.Contoursarespacedatlogarithmicdensityintervalsrelativetothemaximumdensitycontour. 105

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5-10 showstheseparationbetweenthemostboundparticle(MBP)ofeachhalointhemergers,fordierentorbitalparameters.Thetoppanelshowsthecaseoftheradial(Br1)case,themiddlepanelisfortheparabolic(Bp1)case,andthebottompanelisforthecircular(Bc1)run.Wenotethatthepericenterpassagesintheradialcasearenotasfrequentasintheparabolicmerger,andforthecircularsituation,theyareabsent.Figure 5-11 showsf(E;J)atthenalepochformergerswithdierentorbitalparameters-i.e.wemonitortheeectofvaryingtheorbitfrompurelyradialtopurelycircularonthenalf(E;J)distribution.Thenalepochisdened,ineachcase,tobethetimewheneachsystemisgloballyrelaxed,whichisadierenttimeineachsimulation,asitcanbeseenfromFigure 5-10 .Thetimesforeachcasearet6,7,and25Gyr,respectively.Theparaboliccasehasthehighestincidenceoflowphase-spaceregionsathighenergies,whichareindicativeofunboundparticles.Thelargestspreadinangularmomentumathighenergiesoccursforthecircularcase. 5-1 showsthatinanequilibriumhaloparticlescanmovefromregionsoflowphase-spacedensitytohighphase-spacedensitywithaprobabilitydistributionthatisaGaussiancenteredonzero-indicatingthatthecoarse-grainedphase-spacedensityaroundthemedian(ormean)particledoesnotchange.Duringamergerthecoarse-grainedphasespace-densityaroundthemedianparticledecreases.InFigure 5-2 weshowedthatQ(r)forsphericalisotropicNFWhalosisapower-lawwithaslopeof1:87over2.5ordersofmagnitudeinradius.f(r)isnotapower-lawbuthasaslopethatisslowlychangingwithradiusonalogarithmicscalewitha\knee"-likefeatureat0:8rvir.BothQ(r)andf(r)haveabumpatrrvirwhichisnotseenincosmologicalNFWhalos.WeattributethisfeaturetothefactthatourNFWhalosaredesignedtohaveisotropicvelocitydispersionproleswithandexponentialcutoindensityandaredesignedtobestableinisolationwhileNFW 106

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TheseparationbetweenthemostboundparticlesofthetwomergingNFWhalosfortheBr1run(toppanel),theBp1run(middlepanel),andfortheBc1run(bottompanel). 107

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Isodensitycontoursforthephase-spacedensity(f)asafunctionofthetotalenergyEandtotalangularmomentumJfortheBr1run(toprow),fortheBp1run(middlerow),andfortheBc1run(bottomrow). 108

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5-3 and 5-5 weshowedthatbothQ(r)andf(r)areremarkablyrobustandarepreservedduringthemergeroftwoNFWhalos.ThevolumedistributionfunctionV(f)isfoundtoincreaseatlowphase-spacedensitiesduetotheejectionofmatterduringthemerger.However,overtwoordersofmagnitude,thisfunctionisfoundtobesurprisinglystableduringthemergeroftwoNFWhalos.TherobustnessofV(f)isnotaconsequenceofsimilarlevelsofmixingatdierentradii-infact,V(f)intheinnermostshellisfoundtobeessentiallyconstant,demonstrating,onceagain,therobustnessofthecusp.MuchoftheincreaseinV(f)atlowvaluesoffistheresultofmatterejectedfromtheouter30%ofthehalosduringthemerger.Bydesign,theinitialdistributionfunctionsusedinoursimulationshaveaformthatisapproximatelyf(E)(Figure 5-8 ).DuringthemajormergeroftwoNFWhalosonaparabolicoraradialorbitthenaldistributionisfoundtohaveadistributionthatisremarkablysimilartotheinitialdistributionfunction-atagivenenergyEparticlesarealmostuniformlydistributedinangularmomentumJexceptfortheoutermostregionsofthehalo(lowestphase-spacedensityregions). 109

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6-1 includesalistofsomeoftheCDMmodelparameters.Theerrorsquotedare1,andtheyhavebeenderivedusingaMarkovchainMonteCarloanalysisbytheWMAPcollaboration( Spergeletal. 2007 ). 110

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SomeWMAPparametersoftheCDMmodel ParameterDescriptionValue 5 ,oneoftheinterestingndingsofstructureformationsimulations(CDM,standardCDM),aswellasmoresimplecollapsemodelssuchasthesphericalinfallmodels( Bertschinger 1985 ),hasbeenthediscoverythatthecoarse-grainedphase-spacedensityQ(r)(seeEquation 5{2 )followsauniversalpower-lawproleover2.5ordersofmagnitudeinradius( Taylor&Navarro 2001 )andthevolumedensityofphase-spaceV(f)alsohasapower-lawproleovernearlyfourordersofmagnitudeinf( Aradetal. 2004 ).AswesawinFigures 5-3 and 5-6 inChapter 5 ,majormergers(thosewhicharethemostviolentandthereforelikelytoresultinthegreatestamountofmixinginphase-space)preservebothprolesoverasignicantrangeinbothrandfbutdeviatefromthecosmologicalprolesatlowphase-spacedensitiesf(i.e.,larger).WhilemajormergerssuchasthosestudiedinChapter 5 areinstructivesincetheyrepresentthemostextremeformofmixing,theyarenotthemostcommonmodeofmassaccretionintheUniverse.Infact,onthescalesofgalaxy-sizedDMhalos,mosthalosareexpectedtohavehadtheirlastmajormergerbeforez2.Thus,mostofthemasscurrentlyingalaxy-sizedDMhaloswasnotacquiredinamajormerger.IfwearetounderstandbettertheoriginoftheuniversalprolesinQ(r)andV(f),itisvitaltofollowtheevolutionofthesequantitiesincosmologicalstructureformationsimulations. 111

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( 2007 )studiedtheevolutionofphase-spacedensityprolesinDMhalosthroughconstrainedsimulationsdesignedtocontrolthemerginghistoryofagivenhalo.Theyshowedthathalosevolvethroughaseriesofquiescentphasesofslowmassaccretioninterruptedbyoccasionalmajormergers.Duringthequiescentphasesofslowaccretion,thedensityprolecloselyfollowsthatofaNFWhaloandthephase-spacedensityQ(r)iswellrepresentedbyapower-lawrwith=1:90:1overaHubbletime.Theyshowedthattheprolesdeviatefrompower-lawmoststronglyduringmajormergersbutrecovertheirpower-lawprolethereafter.Theyconcludethat,whilethephysicaloriginoftheNFWdensityproleandthepower-lawphase-spacedensityproleisstillunknown,virialequilibriuminthequiescentphasesimpliesthataDMhaloevolvesalongasequenceofNFWproleswithconstantenergyperunitvolumewithinthescaleradiusrs.AsdiscussedinChapter 5 ,Q(r)isreallynotatruemeasureofphase-spacedensitydespitethefactthatithasthedimensionsofphase-spacedensity.Itisthereforeworthwhiletoinvestigatetheevolutionofthetruephase-spacedensityf(x;v;t),thequantitythatappearsinthecollisionlessBoltzmannequation(CBE),aswellasprojectionsofthisquantity,suchasf(r)andf(E;J),asafunctionoftime.Asecondmotivationforstudyingtheevolutionofphase-spacedensityincosmologicalN-bodysimulationsistotestsomeoftheassumptionsthathavebeenusedtoputconstraintsonthepropertiesofDMparticles. Tremaine&Gunn ( 1979 )werethersttousephase-spacedensityargumentsbasedontheobservedcentralmassdensitiesandcentralvelocitydispersionsofkinematicaltracersinobjectsonvariousscales:massivegalactichalos,binarygalaxypairssuchastheMilkyWayandAndromeda,groupsofgalaxies,andclusters.Theyusedkinematicalestimatesofmaximumcentralphase-spacedensityinthesesystemstoputconstraintsonthephase-spacedensityoftheDMparticle.Theyarguedthatifstructureformationishierarchical,andiftheDMthatdominatestheformationofstructureiscollisionless,thentheCBErequiresthatthemaximumne-grainedphase-spacedensityisconserved.Sincethemaximumcoarse-grained 112

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Hogan&Dalcanton 2000 )aswellasonupdatedobservationalconstraintsonthedistributionofDMindwarfgalaxies,low-surfacebrightnessgalaxies,andonmoremassiveobjects( Dalcanton&Hogan 2001 ).Thecentralpremiseoftheseargumentsistheassumptionthat,whilecollisionlessparticlesdecreasetheircoarse-grainedphase-spacedensityduringtheprocessofhierarchicalassemblyofstructure,thecentralcuspsthatforminCDMsimulationsaretheresultoflow-entropymaterialthatsinkstothecenteroftheclusterhaloandpreservestheprimordialentropyaswellasprimordialphase-spacedensityofDM.Thisassumptionhasnotyetbeentestedviadetailedcosmologicalsimulations.Inthischapter,ourmaingoalistoinvestigatetheevolutionofphase-spacedensityintheformationandevolutionoffourMilky-Way-sizedhalosinaCDMcosmology((m;;h;8)=(0:3;0:7;0:7;0:9)),andinparticular,wefocusonthedistributionofthematterwiththehighestphase-spacedensityanditsrelationshiptoitsprimordialvalues.Thisisaworkinprogressandwepresentpreliminaryresults.Thesimulationsanalyzedbelowweredescribedingreaterdetailintheworksof Kravtsovetal. ( 2004 ); Gnedin&Kravtsov ( 2006 ).ThesimulationswerecarriedoutusingtheAdaptiveRenementTreeN-bodycode(ART, Kravtsovetal. ( 1997 )).Thesimulationstartswithauniform2563gridcoveringtheentirecomputationalbox.Thisgriddenesthelowest(zeroth)levelofresolution.Higherforceresolutionisachievedintheregionscorrespondingtocollapsingstructuresbyrecursivereningofallsuchregionsbyusinganadaptiverenementalgorithm.Eachcellcanberenedorde-renedindividually.The 113

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MainPropertiesoftheHalos HaloMvir(h1M)rvir(h1kpc) G11:661012298G21:241012278G31:191012281L201:41012231 cellsarerenediftheparticlemasscontainedwithinthemexceedsacertainspeciedvalue.Thegridisthusrenedtofollowthecollapsingobjectsinaquasi-Lagrangianfashion.Threeofthegalactichalosweresimulatedinacomovingboxof25h1Mpc(hereafterL25);theywereselectedtoresideinawell-denedlamentatz=0.Twohalosareneighbors,located425h1kpcfromeachother.Thethirdhaloisisolatedandislocated2Mpcawayfromthepair.Hereafter,werefertotheisolatedhaloasG1andthehalosinthepairasG2andG3.ThevirialmassesandvirialradiiforthehalosstudiedaregiveninTable 6-2 .Thevirialradius(andthecorrespondingvirialmass)waschosenastheradiusencompassingthedensityof180timesthemeandensityoftheUniverse.ThemassesoftheDMhalosarewellwithintherangeofpossibilitiesallowedbymodelsforthehalooftheMilkyWaygalaxy( Klypinetal. 2002 ).Thefourthgalactichalowassimulatedinacomovingboxof20h1Mpcbox(hereafterL20)anditwasusedtofollowtheevolutionoftheMilkyWay-sizedhalowithhighresolution. 114

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6-1 showsthesphericallyaveragedQ(dashed-dottedline)andboththemedianandmeanvaluesoff(thickandthinsolidlines,respectively)in100concentricradialbinsatz=0.ThecomputationalmethodsusedthroughoutthischapterwerethesameasinChapter 5 .Thelargeuctuationsinthemeanvalueoff(thinline)aretheresultofthepresenceofsubstructure,which(asweshallseelaterinthissection)canhaveextremelyhighvaluesoff,whichdominatethemeandistributionoffatradiibeyond0:1rvir.Themedianvalueoffinconcentricradialbinsismuchsmoother.Inwhatfollows,wewillusethemedianvalueoffinconcentricradialbinstofollowtheevolutionoffinthemainhalo,sinceitislesssensitivetosubstructure.Wenotethat,asforthecaseofthespherical,isotropicNFWhalomodelsweanalyzedinthepreviouschapter,Q(r)resultingfromthecosmologicalsimulationisalsoapower-lawover2:5ordersofmagnitudeinr.Apower-lawtisalsoplottedforf/r1:460:01forarangeof[0.004{0.562]inrvir.Figure 6-2 showsthesphericallyaveragedQ(r)(dashed-dottedline)andmedianoffasafunctionofredshift.Thepower-lawtQ/r1:840:01atz=0providesagoodtonlyuptoz=2.Forhighervaluesofz,Q(r)deviatessignicantlyfromapower-law.Thisbehaviorshouldbecontrastedwiththendingsof Homanetal. ( 2007 ),whofoundthatQwasttedbyapower-lawasfarbackintimeasz=5:3,althoughwenotethattheirplotsshowsignicantdeviationsfromthepower-lawtbeyondonescaleradius(approximately0:1rvir)atthisredshift.AsweshowedinFigure 6-1 ,f(r)deviatesfrompower-lawatz=0.Thisdeviationisevenstrongeratotherredshifts.Inparticular,theimportanceofhighphase-spacedensity 115

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6-2 becomingatterwithradius.Atz=9(theearliestredshiftatwhichweareabletoreliablytracktheparticlesbackintime),Qandfareessentiallyconstantatallradiibeyondtheinner0:01rvirandhavemeanvaluesashighasorhigherthantheirmaximumvaluesatz=0.Smoothprolesdeveloponlyforz2.Figure 6-3 showscontoursofconstantparticlenumberperunitareaontheplane(log(f);log(r)),whilethesolidwhitecurveshowsthemedianvalueoffinlogarithmicallyspacedbinsinr.Asexpected,themedianvalueofftracestheregionsoflargestparticlenumbers.Atradiir>0:05rvir,thehighfvaluesappearasspikesthatarewellconcentratedinradius.Atz3andtowardsz=0,itisclearthatthesespikesinf(whichhaveorangecentralcores)alsohavelargeparticlenumbersandconsequentlycorrespondtoDMsub-halos.Thesub-halosmergecontinuouslyandareheatedandmix.Astheinfallofsubhalosproceeds,thereisanoverallloweringofthemedianfcurveaswellasadeclineinthelowerenvelopeofthedistributiontosmallerfvalues.Thisisademonstrationofthedecreaseofcoarse-grainedphase-spacedensityexpectedtoresultfromthemergers.ThemostimportantfeatureofthedistributionsinFigure 6-3 isthatthepeaksinfseeninthesubhalosatallredshiftsremainhighatvaluesoff105fromz=9allthewaytoz=0.However,duringthattimethemedianvalueoffatthesmallestradiusdropsfrom107f106atz=9downto1010f107atz=0.Furthermore,thehighestvaluesoffatthecenterofthehaloalsodropbyalmostanorderofmagnitude.ThecentralvalueoffinaMilky-Way-sizedDMhaloisthereforelessrepresentativeoftheprimordialphase-spacedensitythanthecentralvalueoffinthehighphase-spacedensitysubhalos.Anotherstrikingfeatureofthedistributionoffvaluesatlargerradiiisthat,atseveralredshifts(e.g.atz=3),thereareparticlessituatedinthesameradiuswhich 118

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Isodensitycontoursfortheparticledensityintheplane(log(f);log(r))fortheL25run,plottedfordierentredshiftsinitsevolution.Theorangecontourscorrespondtotheregionswiththehighestparticledensitiesandtheblacktothelowestdensityregions.Contoursarespacedatlogarithmicdensityintervalsrelativetothemaximumdensitycontour.Themedianvalueoff(whiteline)isevaluatedinequallyspacedlogarithmicbinsinradius. 119

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Tracksthoseparticlesthat,atz=2,aresituatedinthesameregionincongurationspace,butwhosephase-spacevaluesdierbythreeordersofmagnitude.Theparticleswithhighvaluesoffareshowninorange,whilethosewithlowvaluesoffareinblue. producebothahighfspike(orangearrow)correspondingtoasubhaloandalowfspike(bluearrow).Thephase-spacevaluesofthesespikesdierby5-7ordersofmagnitude.InFigure 6-4 ,weidentifytheparticlesassociatedwiththeseextremelylowvaluesoffanddeterminetheirrelationshiptotheparticleswiththehighestvaluesoff.Inparticular,wetrackallparticleswithradius0:4r=rvir<0:5atz=3.Theparticleswithhighphase-spacedensities(f106)areshowninorange,andparticleswithlowphase-spacedensities(f1010)areshowninblue.Theparticlesareidentiedatz=3(middleleft-handpanel)andthentrackedstartingatanearliertimez=3:3(toprow)throughtoalatertimez=2:6(bottomrow).Themiddlecolumnshowstheevolutionoftheparticlesintwospatialcoordinates(xandy),andtherightmostcolumnshows 120

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6-4 givesagraphicalillustrationofhowthesubhalosinteractwitheachotheraswellasDMparticlesinthegeneralhaloandcausemixinginphase-spaceandleadtoasmoothdistributioninf.Itisstillunclearwhatgivesrisetotheobservednaldistributionoff,butthemicroscopicprocessofheatingviamergersofsmallsubhalosisapparentlynotverydierentfromthetypesofheatingandmixingweobservedinmajormergersinChapter 5 .WesawinFigure 6-3 thatthehighestphase-spacedensityregions,bothatthecenterofthemainhaloandwithinsubhalos,havevaluesoff107.InFigure 6-5 wedetermineinamorequantitativemannerhowrepresentativeoftheprimordialvalueof 121

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Histogramsofthephase-spacedensityasafunctionofredshift.Thethickreddashedlinefollowsthoseparticleswhich,atz=0,aredeviatedfromtheGaussianthatapproximatesthedistributiontillz=1.Thethinreddashedlinefollowsthesameparticlesfromz=0,butthecorrespondingy-axisvaluesaretentimeshigherforclarity. 122

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6-3 ,mostofthesehigh-ftailparticlesareinsubhalosatz=0,whilesomearealsointhehighestphase-spacedensityparticlesinthecentralcuspofthehaloatz=0.Ineachoftheplots,thedistributionofthispopulationofparticlesisplottedasadashedcurve.Weseethatthissub-populationhasaGaussiandistributionatz=9withameanlog(f)=6:40comparedwiththemeanlog(f)=6:53foralltheparticles(inthesolidcurve)atz=9.ThisimpliesthatthematerialinthecentersofDMsubhalosatz=0hasphase-spacedensitiesthatarerepresentativeofthemeanphase-spacedensityofmaterialevenatz=9.Oursimulationsdoproceedbackintimetohigherredshifts,howeverwedonotanalyzethemherebecausethemassresolutiondoesnotallowustoresolvetheevolutionofmostoftheobjectsbeyondthisepoch,butweanticipatethat,athigherz,thedistributionsbecomemuchmoresharplypeakedandprobablymovetoevenhighervaluesoff.AswediscussedinChapter 5 ,whilethene-grainedversionofthevolumeV(f)ofphase-spaceassociatedwithmaterialofphase-spacedensityfisconservedbytheCBE,thecoarse-grainedversionofthisquantityalwaysincreasesduetomixingbytheMixing

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ThevolumedistributionfunctionV(f),fordierentredshiftsintheevolutionoftheL25run.Apower-lawtV(f)/f2:51isshownatz=0(redsolidline). 124

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Mathur 1988 ).WenowconsidertheevolutionofV(f),thecoarse-grainedvolumedensityofphase-space.ThisquantitywasdenedinChapter 5 ,Equation 5{3 ,tobethevolumeofphase-spaceoccupiedbyphase-spaceelementswhosedensityliesbetween(f;f+df).Figure 6-6 showsthetime-evolutionofV(f).Atz=0, Aradetal. ( 2004 )showedthatV(f)exhibitsapower-lawproleoverfourordersofmagnitudeinlog(f).Athinsolidlinewithproley/f2:510:07isoverplottedontheV(f)curveatz=0.AsforthecaseofthemajormergerswesawinChapter 5 (e.g.theBp1mergerinFig. 5-6 ),thereisanoticeabledeviationfromthepower-lawproleatlowvaluesoff.However,wenotethat,whereasinFigure 5-6 thevolumeofphase-spaceV(f)associatedwithlowvaluesforf<1010continuedtoincreasewithdecreasingfattheendofthemajormerger,theoppositebehaviorisseeninthecosmologicalsimulationatz=0.InthebottomrightpanelofFigure 6-6 ,wendthat,whileupturnsinV(f)atlowvaluesoffareseenfromz=2tillz0:5,thevolumeofphase-spaceassociatedwiththislowphase-spacedensitymaterialdecreasesthereafterandapparently\disappears"entirelybyz=0.ThisisaresultofthefactthatthehaloiscontinuouslyaccretingmassandgrowinginthesimulationL25.TheupturnsinV(f)inthesimulationBp1aswellasinthecosmologicalsimulationarisefollowingasignicantaccretionofmatter(inthecaseoftheL25simulation,thiswasprobablyduetothesimultaneousaccretionofseveralsubhalosandtheejectionofmaterialseeninthelowphase-spacedensityspikesinFigure 6-3 ratherthanamajormerger).Fromz=0:7toz=0,thislowphase-spacedensitymatterispulledbackintothehalofollowinganincreaseinthedepthofitspotentialwellbythecontinuedaccretionofmatter. 125

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6-1 ).Fromz=5onward,materialwithinrvir(z)atthatredshiftfollowsapower-lawinQandanapproximatepower-lawinf.Beyondthisradius,Qdecreasesbutfisalmostconstantwithradius.Thisisbecausefisameasureoftruephase-spacedensity,anditsconstancyshowsthatmaterialoutsidethevirialradiusstillhasnotmixed.Qdecreasesbecausethevolume,r3,isincreasingrapidlywithradius,whilelocaldensityofmaterialisapproximatelyconstant(Figure 6-2 ).Thehighestvaluesoffatthecenterofthehaloislessrepresentativeoftheinitialphase-spacedensitythanthecentralvalueoffinthehighphase-spacedensitysubhalos.Regionsofhighestf(f107)areinthesubhalosatallz5.Subhaloscontributeminimallytothemedianfatlargeradii(Fig6.3).Inthecenter,themedianoffdropsbyalmostoneorderofmagnitude,fromf106tof107.Atrvir,themedianoffdropsfromf106tof1010,nearlyfourordersofmagnitude(Figure 6-3 ).Atallredshifts,weseeregionsofverylowfco-spatialwithsubhalosatlargeradii.Thesearefoundtoberegionswheremultiplesubhalosareundergoinggravitationalinteractions.Unboundparticlesaretransientlyheatedastheyaregravitationallyacceleratedbytheinteractingsubhalos.Thisproducestransientlowfspikes(Figure 6-4 ).Histogramsoflog(f)areapproximatelyGaussianatallredshifts.Atz=9themeanandmedianvaluesoffareapproximately106:46.Thedistributionisslightlyskewedtowardslowervaluesoff.Astimeprogresses,themeanandmedianoflog(f)shifttoprogressivelylowervalues.Atz=1thedistributiondevelopsadistinctiveskewnessduetotheformationofahighftail.Thehighphase-spacedensityparticlesatz=0havephase-spacedensitiesthatarerepresentativeofthemeanphase-spacedensityofmaterialatz9,andhencewelearnaboutthenatureofDM,sincemeasuringfnowisthesameasmeasuringitatz9(Figure 6-5 ).Thematterwiththehighestfatz=0(f107),whentracedbacktoz=9,isfoundtohavethesamemeanastheoveralldistribution.WhilewendthatthevolumedistributionfunctionV(f)isfollowingthepower-lawprolethatwasrstobservedby Aradetal. ( 2004 ),itwasalsofoundthattheitsvaluesassociatedwithlowphase-spacedensityregionsincreaseatearlierredshifts, 126

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6-6 )atearliertimescontinuetocollapseontothemainprogenitor,andsotheyendupinregionsofhigherdensityandtheirfincreases.ThehighV(f)regionswereseeninChapter 5 tobetheresultofejectionofmaterialduetomajormergers.Thismaterialispulledbackintothemainhalobyz=0,duetothecontinuedgrowthofthehaloandduetothecosmicexpansion. 127

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2 discussedsome`natural'physicalexpectationregardingthepossibilityoftransientchaosinducedbytime-dependentperturbations.Firstweconsideredwhatcanhappenwhenanintegrablesystemissubjectedtoanundampedoscillatoryperturbationwithasinglefrequency,andthenweextrapolatedfromthisexampletoconsidermorerealisticcomputationsthatdealwithtwoimportantaspectassociatedwithviolentrelaxation,i.e.allowingfordampedoscillationsandallowingforvariablepulsationfrequency.AnextensionoftheworkpresentedinChapter 2 involvedsearchingforevidenceoftransientchaosinthecontextoffullyself-consistentnumericalsimulationsforself-gravitatingsystems.Theobviousissuewaswhetherthedegreeofchaoticmixingobservedinasimulationofviolentrelaxationinaself-gravitatingsystemcorrelateswiththedegreetowhichthebulkpotentialadmitsasignicanttime-dependentoscillatorycomponent.ThemodelsusedinChapter 2 (Plummerspheres)donotconstituteespeciallyrealisticmodelsofearly-typegalaxies;asidefromtheidealizationofsphericalsymmetry,realgalaxiesoftenhaveacentralcusp,andtheirdensitiesfallomuchmoreslowlyatlargeradii.Also,Plummersphereshaveconstant-densitycores,wherealltheorbitshave 128

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3 and 4 focusonsimulationsofthemergeroftwoDMhaloswithNFWpotentials,mostoftheresultsaregenerictomergersofcollisionlessgravitatingsystemsandarethereforeapplicabletodissipationless(\dry")mergersofellipticalgalaxies.Inparticular,theabsenceoflargeamountsofmixinginradiusoverandabovethatexpectedfromphasemixinginanisolatedhalosupportspreviousworkthatindicatesthatradialgradientsinstellarproperties(suchasmetallicitygradients)arelikelytosurviveintacteveninmajormergers( Boylan-Kolchin&Ma 2004 ).Theseresultsarealsoexpectedtobegenericallyapplicabletocosmologicalmergers.Theworkof Faltenbacheretal. ( 2006 )-astudyoftherelaxationprocessesthatoperateinhierarchicalmergersofDMhalosincosmologicalN-bodysimulations-independentlyarrivesatthesameconclusions,namelythattheprimarydriverformixingandrelaxationincosmologicalmergersistidalshocking.Wendstrongevidencethattheprocessesthatdrivethemixinginphase-spaceandevolutiontoadynamicalequilibriumdonotoccurcontinuouslyduetothetime-dependentpotentialofthesystem.Infact,strongmixingoccursprimarilyfollowingepisodicinjectionofenergyandangularmomentumintotheinternalenergiesofparticlesduringpericenterpassages.Theinjectionofenergyandangularmomentumcausesnearestparticlesinphase-spacetoseparateinE;J.Previousstudiesoforbitalevolutionintime-dependentandtime-independentpotentialshavearguedthatthepresenceoflargefractionsofchaoticorbitscouldbetheprincipaldriverofthemixingandevolutiontoanequilibriumdistribution.Whilechaoticmixingcouldwellbeoccurring,wedonotndevidencethatsuchchaoticmixingisdrivingtherelaxation.Thisisprobablyaconsequenceofthefactthatthetimescales 129

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2 occurswhenthedrivingfrequencyoftheoscillationisclosetotheorbitalfrequencyforalargefractionoforbits.InNFWhalos,thepotentialissteeplycusped,andconsequently,thereisamuchwiderrangeoforbitalfrequencies.TheresultspresentedinChapter 4 furtherhighlightthefactthatmass,initsstandardvirialdenition,isnotadditiveinmergers.Modelsinwhichthemergersdoublethemasswithinthevirialradiusoftheremnantgreatlyoverestimateboththemassanddensityofthemergerproduct.OneofthemainndingsofouranalysisinChapter 4 isthatparticlesinthecentraldensitycuspsofsteeplycuspeddarkmatter(DM)proles(liketheNFWprole)retainastrongmemoryoftheirinitialconditionsandthatmixingoccursprimarilyduetoredistributionofparticlesatlargerradii.AsmentionedintheIntroduction,oneofthereasonswhybothparticlephysicistsandcosmologistsareinterestedinknowingaboutthephase-spacedensityofDMparticlesinthecurrentUniverseisthattherearecurrentlyseveralparticlephysicsexperimentssearchingforcandidateDMparticles.InChapter 5 ,wefocusedontheevolutionofthephase-spacedensityofcollisionless(DM)particlesfollowingmergersofsphericalpotentialswithdierentdensityproles.WeshowedthatbothQ(r),the\poor-man's"phase-spacedensity'sproxy,andf(r),thedistributionfunctionprovidedbytheeldestimatordevelopedby Ascasibar&Binney ( 2005 ),aresurprisinglystableandarepreservedduringthemergeroftwoNFWhalos.Thevolumeofphase-spaceoccupiedbyphase-spaceelementswhosedensityliesintherange(f;f+df),V(f)(thevolumedistributionfunction)ispower-lawoverseveralordersofmagnitude,conrmingtheresultsof Aradetal. ( 2004 ); Ascasibar&Binney ( 2005 ); Sharma&Steinmetz ( 2006 ).Thedeparturesfrompower-lawaretheresultofmatterejectedfromtheouter30%ofthehalosduringthemerger. 130

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6 illustratedthat,atpresenttime,thehighphase-spacedensityparticleshavephase-spacedensitiesthatarerepresentativeofthemeanphase-spacedensityofmaterialatz9;thisprovidessomeinsightintothenatureofDM,asmeasuringfnowisthesameasmeasuringitatmuchearlierredshifts. 3 ,whereouranalysisofthemergersstoppedaround10-15Gyr,slowmixingduetochaoticprocessesarelikelytobecomeimportantatthisphase.Thiscanbestudiedwiththetraditionaltoolsofnon-lineardynamicssincethepotentialisonlyslowlychangingafterthistime.ThemajormergersstudiedinChapter 3 arenotthemosttypicalmodeofmassaccretionintheUniverse.Itisthereforenecessarytostudymultipleminormergersinthegrowthofhalostounderstandtheroleofdynamicalfrictionandtidalcompressioninhierarchicalmergers.Thetoolsdevelopedcanstillbeappliedtothisfuturestudy.Newsimulationsofmultipleminormergerswillneedtoberunandanalysed.Thiswillgiveusbetterinsightsintosuchquestionsaswhethermixingisaseectivewhenthemergerislessviolent,andonhowdynamicalfrictionandtidalcompressionandtidalstripping(whichactincomplementaryways)workinstructureformation.Chapter 4 showedthat40%ofthemassofthenalremnantliesoutsidetheformalvirialradiusinthecaseofamajormerger.Itisimportanttoknowifthisisalsotrueforminormergersandsucessivemergers,orformergersconsideringdierentorbitalparametersordierentinnerdensityproles,whichwerepartiallystudiedinChapter 5 .ThecompletionofChapter 6 requiresanecessaryanalysisofthedistributionfunction'sdependenceonthetotalenergyandtotalangularmomentum,f(E;J),similartotheonedescribedattheendofChapter 5 ,inordertomonitoritschangesasafunctionofhalomassonbothsmallandlargescales. 131

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Aarseth,S.J.,&Lecar,M.1975,ARA&A,13,1 Alcock,C.,Akerlo,C.W.,Allsman,R.A.,Axelrod,T.S.,Bennett,D.P.,Chan,S.,Cook,C.H.,Freeman,K.C.,Griest,K.,Marshall,S.L.,Park,H.S.,Perlmutter,S.,Peterson,B.A.,Pratt,M.R.,Quinn,P.J.,Rodgers,A.W.,Stubbs,C.W.,&Sutherland,W.1993,Nature,365,621 Arad,I.,Dekel,A.,&Klypin,A.2004,MNRAS,353,15 Arad,I.,&Johansson,P.H.2005,MNRAS,362,252 Arad,I.,&Lynden-Bell,D.2005,MNRAS,361,385 Ascasibar,Y.,&Binney,J.2005,MNRAS,356,872 Austin,C.G.,Williams,L.L.R.,Barnes,E.I.,Babul,A.,&Dalcanton,J.J.2005,ApJ,634,756 Barnes,E.I.,Williams,L.L.R.,Babul,A.,&Dalcanton,J.J.2006,ApJ,643,797 Barnes,E.I.,Williams,L.L.R.,Babul,A.,&Dalcanton,J.J.2007,ApJ,654,814 Bertschinger,E.1985,ApJS,58,39 Binney,J.,&Tremaine,S.1987,Galacticdynamics(Princeton,NJ,PrincetonUniversityPress,1987,747p.) Bosma,A.,&vanderKruit,P.C.1979,A&A,79,281 Boylan-Kolchin,M.,&Ma,C.-P.2004,MNRAS,349,1117 Chandrasekhar,S.1943,ReviewsofModernPhysics,15,1 Dalcanton,J.J.,&Hogan,C.J.2001,ApJ,561,35 Dehnen,W.2005,MNRAS,360,892 Einasto,J.1969,AstronomischeNachrichten,291,97 El-Zant,A.A.2002,MNRAS,331,23 Faltenbacher,A.,Gottloeber,S.,&Mathews,W.G.2006,ArXivAstrophysicse-prints Fixsen,D.J.,Cheng,E.S.,Gales,J.M.,Mather,J.C.,Shafer,R.A.,&Wright,E.L.1996,ApJ,473,576 Freedman,W.L.,Madore,B.F.,Gibson,B.K.,Ferrarese,L.,Kelson,D.D.,Sakai,S.,Mould,J.R.,Kennicutt,R.C.,Jr.,Ford,H.C.,Graham,J.A.,Huchra,J.P.,Hughes,S.M.G.,Illingworth,G.D.,Macri,L.M.,&Stetson,P.B.2001,ApJ,553,47 132

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Friedmann,A.1924,ZeitschriftfurPhysik,21,326,EnglishtranslationinCosmologicalConstants,editedbyJ.BernsteinandG.Feinberg(ColumbiaUniversity,NewYork,1986,59p.) Funato,Y.,Makino,J.,&Ebisuzaki,T.1992,PASJ,44,291 Gerhard,O.,&Silk,J.1996,ApJ,472,34 Gnedin,N.Y.,&Kravtsov,A.V.2006,ApJ,645,1054 Gnedin,O.Y.,Lee,H.M.,&Ostriker,J.P.1999,ApJ,522,935 Grassberger,P.,Badii,R.,&Politi,,A.1988,JournalofStatisticalPhysics,51,135 Habib,S.,Kandrup,H.E.,&Mahon,M.E.1997,ApJ,480,155 Hemsendorf,M.,&Merritt,D.2002,ApJ,580,606 Hernquist,L.1990,ApJ,356,359 Homan,Y.,Romano-Diaz,E.,Shlosman,I.,&Heller,C.2007,ArXive-prints,706 Hogan,C.J.,&Dalcanton,J.J.2000,Phys.Rev.D,62,063511 Hubble,E.1929,ProceedingsoftheNationalAcademyofScience,15,168 Hut,P.,&Heggie,D.C.2001,ArXivAstrophysicse-prints Kandrup,H.E.1998,MNRAS,301,960 Kandrup,H.E.1999,inAstronomicalSocietyofthePacicConferenceSeries,Vol.182,GalaxyDynamics-ARutgersSymposium,ed.D.R.Merritt,M.Valluri,&J.A.Sellwood,197 Kandrup,H.E.,&Drury,J.1998,inNonlinearDynamicsandChaosinAstrophysics:AFestschriftinHonorofGeorgeContopoulos.,ed.J.R.Buchler,S.T.Gottesman,&H.E.Kandrup,306 Kandrup,H.E.,Mahon,M.E.,&Smith,H.J.1994,ApJ,428,458 Kandrup,H.E.,&Novotny,S.J.2004,CelestialMechanicsandDynamicalAstronomy,88,1 Kandrup,H.E.,Pogorelov,I.V.,&Sideris,I.V.2000,MNRAS,311,719 Kandrup,H.E.,&Sideris,I.V.2001,Phys.Rev.E,64,056209 Kandrup,H.E.,&Sideris,I.V.2003,ApJ,585,244 Kandrup,H.E.,Sideris,I.V.,Terzic,B.,&Bohn,C.L.2003,ApJ,597,111 133

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Kandrup,H.E.,&Smith,H.J.1991,ApJ,374,255 Kandrup,H.E.,Vass,I.M.,&Sideris,I.V.2003,MNRAS,341,927 Kazantzidis,S.,Magorrian,J.,&Moore,B.2004,ApJ,601,37 Kazantzidis,S.,Zentner,A.R.,&Kravtsov,A.V.2006,ApJ,641,647 Klypin,A.,Zhao,H.,&Somerville,R.S.2002,ApJ,573,597 Kravtsov,A.V.,Gnedin,O.Y.,&Klypin,A.A.2004,ApJ,609,482 Kravtsov,A.V.,Klypin,A.A.,&Khokhlov,A.M.1997,ApJS,111,73 Laskar,J.1990,Icarus,88,266 Laskar,J.,Froeschle,C.,&Celletti,A.1992,PhysicaDNonlinearPhenomena,56,253 Lichtenberg,A.J.,&Lieberman,M.A.1992,RegularandChaoticDynamics(AppliedMathematicalSciences,NewYork:Springer,1992) Lynden-Bell,D.1967,MNRAS,136,101 MacMillan,J.D.,Widrow,L.M.,&Henriksen,R.N.2006,ApJ,653,43 Mahon,M.E.,Abernathy,R.A.,Bradley,B.O.,&Kandrup,H.E.1995,MNRAS,275,443 Mathur,S.D.1988,MNRAS,231,367 Merritt,D.2005,NewYorkAcademySciencesAnnals,1045,3 Merritt,D.,Graham,A.W.,Moore,B.,Diemand,J.,&Terzic,B.2006,AJ,132,2685 Merritt,D.,&Valluri,M.1996,ApJ,471,82 Miller,R.H.1964,ApJ,140,250 Navarro,J.F.,Frenk,C.S.,&White,S.D.M.1996,ApJ,462,563 Navarro,J.F.,Hayashi,E.,Power,C.,Jenkins,A.R.,Frenk,C.S.,White,S.D.M.,Springel,V.,Stadel,J.,&Quinn,T.R.2004,MNRAS,349,1039 Peirani,S.,&deFreitasPacheco,J.A.2007,ArXivAstrophysicse-prints Peirani,S.,Durier,F.,&deFreitasPacheco,J.A.2006,MNRAS,367,1011 Perlmutter,S.,Turner,M.S.,&White,M.1999,PhysicalReviewLetters,83,670 Riess,A.G.,Filippenko,A.V.,Challis,P.,Clocchiatti,A.,Diercks,A.,Garnavich,P.M.,Gilliland,R.L.,Hogan,C.J.,Jha,S.,Kirshner,R.P.,Leibundgut,B.,Phillips,M.M., 134

PAGE 135

Rubin,V.C.,&Ford,W.K.J.1970,ApJ,159,379 Sharma,S.,&Steinmetz,M.2006,MNRAS,373,1293 Sideris,I.V.2004,CelestialMechanicsandDynamicalAstronomy,90,147 Sideris,I.V.2006,PhysicalReviewE(inpress),73,1 Siopis,C.,&Kandrup,H.E.2000,MNRAS,319,43 Spergel,D.N.,Bean,R.,Dore,O.,Nolta,M.R.,Bennett,C.L.,Dunkley,J.,Hinshaw,G.,Jarosik,N.,Komatsu,E.,Page,L.,Peiris,H.V.,Verde,L.,Halpern,M.,Hill,R.S.,Kogut,A.,Limon,M.,Meyer,S.S.,Odegard,N.,Tucker,G.S.,Weiland,J.L.,Wollack,E.,&Wright,E.L.2007,ApJS,170,377 Spergel,D.N.,&Hernquist,L.1992,ApJ,397,L75 Stadel,J.G.2001,Ph.D.Thesis Taylor,J.E.,&Navarro,J.F.2001,ApJ,563,483 Terzic,B.,&Kandrup,H.E.2004,MNRAS,347,957 Tremaine,S.,&Gunn,J.E.1979,PhysicalReviewLetters,42,407 Tremaine,S.,Henon,M.,&Lynden-Bell,D.1986,MNRAS,219,285 Valluri,M.1993,ApJ,408,57 Valluri,M.,&Merritt,D.1998,ApJ,506,686 Valluri,M.,&Merritt,D.2000,inTheChaoticUniverse,ProceedingsoftheSecondICRANetworkWorkshop,AdvancedSeriesinAstrophysicsandCosmology,vol.10,EditedbyV.G.GurzadyanandR.Runi,WorldScientic,2000,p.229,ed.V.G.Gurzadyan&R.Runi,229 Valluri,M.,Vass,I.M.,Kazantzidis,S.,Kravtsov,A.V.,&Bohn,C.L.2007,ApJ,658,731 Zentner,A.R.,Berlind,A.A.,Bullock,J.S.,Kravtsov,A.V.,&Wechsler,R.H.2005,ApJ,624,505 Zhao,H.1996,MNRAS,278,488 Zwicky,F.1933,HelveticaPhysicaActa,6,110 Zwicky,F.1937,ApJ,86,217 Zwicky,F.1939,Proc.Natl.Acad.Sci.,25,604 135

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IwasborninBucharest,Romania,whereIwenttovariousschoolsofvariouseducationallevels.Iclearlyremembermyrstlessoninastronomy,whenIwasthreeyearsold.Themoonhadjustrisenfrombehindtheapartmentbuildinginfrontofours,anditwasred,andfull,andincrediblyimmense.Iaskedmyfather,howwasitpossibleforthemoontochangecolors.Heexplainedtomethatitwasn'tactuallyred,butthatwasthewaywewereseeingitduetotheEarth'satmosphere.IfIwaitedlongenough,Iwouldseeitchangetoyellow,andthenwhenitgotnearertothetopoftheskyitwouldturnalmostwhite.Justthensomebodycalledandaskedtotalktomyfather,andhesaidthathewasdoingastronomywithhisdaughter.ThecalleraskedhowoldIwas,andmyfatherreplied,\She'sthree."ThatishowIremembermyagebackthen.Istayedawaketoseethemoonchangingfromredtoyellow,justasmyfatherhadsaiditwould.Laterthatnight,Igotupfrommybedandaskedmymothertotakemetothewindow,andIsawthemoonturningwhitewhenitreachedthetopofthesky.Sincethen,lifeformewasquitefullofexperiencesthatproducedequallywondrous,strong,ever-lastingimpressionsastheoneIdescribed.IdidgraduateresearchinRomania,Spain,andintheUS.IntheUS,IdidgraduateresearchattheUniversityofFlorida,NorthernIllinoisUniversity,andUniversityofChicago.Iwouldliketothinkthat,whereverthepathoflifewilltakemefromnowon,themoonwillstillchangecolorsjustforme. 136