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Quantum Effects in Axion Dark Matter and Caustic Rings

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Title:
Quantum Effects in Axion Dark Matter and Caustic Rings
Creator:
Chakrabarty, Sankha Subhra
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (104 p.)

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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Sikivie,Pierre
Committee Co-Chair:
Matchev,Konstantin Tzvetanov
Committee Members:
Saab,Tarek
Muttalib,Khandker A
Gonzalez,Anthony Hernan

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Subjects / Keywords:
astrophysics -- darkmatter
Physics -- Dissertations, Academic -- UF
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
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Physics thesis, Ph.D.

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Abstract:
Axions or axion-like particles are highly motivated candidates for dark matter. They are often treated as classical fields in the literature. It is assumed that the classical field description is accurate because axions have large quantum degeneracy. However, it is evident that the classical and quantum descriptions are different in the context of thermalization. Certain properties such as the existence of caustic rings cannot be explained in the regime of classical field theory. On the other hand, a complete quantum description of axion dark matter is challenging from theoretical perspectives. We develop a formalism to study the departure of the quantum evolution of highly degenerate axions from its classical counterpart. First, we apply the formalism to homogeneous condensates with attractive interactions. For both contact and gravitational self-interactions, we find that the homogeneous condensate persists forever in the classical description. However, in the quantum evolution, the quanta jump out of the condensate in pairs and populate modes with wave vector less than a critical value. We calculate the time scale after which the quantum evolution differs from the classical one. We study a condensate with repulsive interactions and with small inhomogeneities in the form of a plane wave perturbation. In its classical description, the system persists in this state indefinitely. But, in the quantum description, the quasi-particles scatter among themselves and certain modes consistent with both momentum and energy conservation, become populated exponentially fast. We determine the duration of classicality in this case also. We explain why the existence of caustic rings of dark matter is consistent with their quantum description. Relative over-densities of stars near caustic rings are predicted by simulating the dynamics of half a million stars. We also determine the density profile of interstellar gas assuming the gas to be in thermal equilibrium in the gravitational potential of a caustic and the gas itself. We argue that triangular features in the recently published GAIA sky map are evidence for the fifth caustic ring. Taking the new evidence into consideration, we calculate updated values of the densities and velocities of dark matter flows near the Sun. ( en )
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In the series University of Florida Digital Collections.
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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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2019.
Local:
Adviser: Sikivie,Pierre.
Local:
Co-adviser: Matchev,Konstantin Tzvetanov.
Statement of Responsibility:
by Sankha Subhra Chakrabarty.

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QUANTUMEFFECTSINAXIONDARKMATTERANDCAUSTICRINGSBySANKHASUBHRACHAKRABARTYADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2019

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c2019SankhaSubhraChakrabarty

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Idedicatethistomyparents.

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ACKNOWLEDGMENTSIamgratefultomysupervisor,Prof.PierreSikivie,forhisprofoundguidanceandenlightenment.Iwillsurelymissourweeklydiscussionsandhisuniquewayofdeliveringdeepscienticinsightswiththesimplestexamples.IacknowledgeallthehelpsandsuggestionsIhavereceivedfromthemembersofmysupervisorycommittee:Profs.KonstantinMatchev,TarekSaab,KhandkerMuttalibandAnthonyGonzalez.IliketoextendmygratitudetoProf.Gonzalezforhisvaluablecomments.Iamthankfultomyfriendsandcolleagues:Elisa,Nilanjan,Ariel,Seishi,YaqiandIgal,foralltheilluminatingdiscussions.Igratefullyacknowledgetheunconditionallove,careandsupportIhavereceivedfrommyparents,sisterandothermembersofmyfamily.ItakethisopportunitytothankmylovingwifeAnkitawhowasasourceofcontinuousencouragementandemotionalsupportatallstagesofthisjourney.IacknowledgethesupportfromtheCollegeofLiberalArtsandSciences(CLAS)DissertationFellowshipfundedbytheMcGintyEndowmentinspring2019.IwassupportedinpartbytheInstituteofFundamentalTheory,bytheU.S.DepartmentofEnergyunderGrantNo.DE-FG02-97ER41209andbytheHeising-SimonsFoundationunderGrantNo.2015-109. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1BOSE-EINSTEINCONDENSATIONOFDARKMATTERAXIONS ...... 11 1.1QuantumMechanicsisEssential ........................ 13 1.2HowLongisaClassicalDescriptionValid? .................. 15 1.3HomogeneityisnotaNecessaryOutcomeorCriterion ............ 20 1.4WhatStatedotheParticlesCondenseinto? ................. 23 2QUANTUMANDCLASSICALDESCRIPTIONSOFAXIONS:HOMOGENEOUSCONDENSATESWITHATTRACTIVESELF-INTERACTIONS ........ 25 2.1FormalismtoCalculatetheDurationofClassicality ............. 25 2.2HomogeneousCondensatewithAttractiveContactSelf-interactions .... 31 2.2.1ClassicalDescription .......................... 31 2.2.2QuantumDescription .......................... 33 2.2.2.1Modeswithwavelengthsmallerthanthecriticalwavelength(k>kJ) ............................ 34 2.2.2.2Modeswithwavelengthlargerthanthecriticalwavelength(k
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4ASTROPHYSICALSIGNATURESOFCAUSTICRINGS:PREDICTIONS .. 64 4.1CausticRingsasManifestationsoftheQuantumBehaviorofAxions .... 64 4.1.1CausticRingsintheDarkMatterHalo ................ 64 4.1.2StructureoftheInnerCausticsandVelocityField .......... 65 4.1.3ClassicalTheoryisInadequate ..................... 66 4.1.4QuantumBehaviorofDarkMatterAxions .............. 66 4.2CausticRings .................................. 67 4.2.1FlowsNearaCausticRing ....................... 67 4.2.2Self-similarity .............................. 68 4.2.3GravitationalFieldofaCausticRing ................. 70 4.2.4GravitationalPotentialofaCaustic .................. 71 4.3EectsonStars ................................. 73 4.3.1ASingleStar .............................. 73 4.3.2DistributionofStars .......................... 76 4.3.2.1Bulkvelocities ........................ 76 4.3.2.2Overdensities ......................... 78 4.4EectsonInterstellarGas ........................... 80 5ASTROPHYSICALSIGNATURESOFCAUSTICRINGS:EVIDENCE .... 84 5.1SummaryofPreviousEvidence ........................ 84 5.1.1FlatRotationCurveatLargeRadii .................. 84 5.1.2BumpsintheRotationCurves ..................... 84 5.1.3TriangularFeatureintheIRASSkyMap ............... 85 5.1.4StellarOverdensities .......................... 85 5.2M31RotationCurve .............................. 85 5.3TriangularFeaturesintheGAIASkyMapandTheirImplications ..... 86 5.3.1LeftandRightTriangles ........................ 86 5.3.2PositionoftheSunwithrespecttotheTricusp ............ 89 5.3.3Parametersofthe5thCausticRing .................. 90 5.3.4DensitiesandVelocitiesoftheNearbyFlows ............. 91 6SUMMARY ...................................... 95 APPENDIX ADERIVATIONOFM~K0~K ............................... 98 BTHEFUNCTIONF() ............................... 99 REFERENCES ....................................... 100 BIOGRAPHICALSKETCH ................................ 104 6

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LISTOFTABLES Table page 5-1FlowsthroughtheSunfordierentvaluesof .................. 94 7

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LISTOFFIGURES Figure page 1-1Quantumandclassicalevolutionofagenericstate ................. 17 1-2Quantumandclassicalevolutionofhomogeneouscondensate ........... 19 1-3Longrangecorrelationvsinhomogeneity ...................... 21 3-1Plotoffortheunstablemodes ........................... 60 3-2PlotofF() ...................................... 63 4-1Trajectoriesofdarkmatterparticlesnearacausticring .............. 69 4-2Envelopeofthetrajectories ............................. 69 4-3Gravitationaleldnearacausticring ........................ 72 4-4Gravitationalpotentialnearacausticring ..................... 73 4-5TimedependenceofEandEzofastar ...................... 75 4-6Radialcoordinateofastarvstimet ....................... 77 4-7Relativeoverdensitiesofstarsnearcausticringsduetoradialdynamics ..... 80 4-8Relativeoverdensitiesofstarsnearcausticringsduetoverticaldynamics .... 81 4-9Densityproleofgasnearacausticring ...................... 83 5-1RotationcurveofM31 ................................ 86 5-2PanoramicviewoftheGAIAskymap. ....................... 87 5-3TriangularfeaturesinGAIAskymap ........................ 87 5-4Thetriangleinscribedinthetricusp ........................ 88 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyQUANTUMEFFECTSINAXIONDARKMATTERANDCAUSTICRINGSBySankhaSubhraChakrabartyAugust2019Chair:PierreSikivieMajor:PhysicsAxionsoraxion-likeparticlesarehighlymotivatedcandidatesfordarkmatter.Theyareoftentreatedasclassicaleldsintheliterature.Itisassumedthattheclassicalelddescriptionisaccuratebecauseaxionshavelargequantumdegeneracy.However,itisevidentthattheclassicalandquantumdescriptionsaredierentinthecontextofthermalization.Certainpropertiessuchastheexistenceofcausticringscannotbeexplainedintheregimeofclassicaleldtheory.Ontheotherhand,acompletequantumdescriptionofaxiondarkmatterischallengingfromtheoreticalperspectives.Wedevelopaformalismtostudythedepartureofthequantumevolutionofhighlydegenerateaxionsfromitsclassicalcounterpart.First,weapplytheformalismtohomogeneouscondensateswithattractiveinteractions.Forbothcontactandgravitationalself-interactions,wendthatthehomogeneouscondensatepersistsforeverintheclassicaldescription.However,inthequantumevolution,thequantajumpoutofthecondensateinpairsandpopulatemodeswithwavevectorlessthanacriticalvalue.Wecalculatethetimescaleafterwhichthequantumevolutiondiersfromtheclassicalone.Westudyacondensatewithrepulsiveinteractionsandwithsmallinhomogeneitiesintheformofaplanewaveperturbation.Initsclassicaldescription,thesystempersistsinthisstateindenitely.But,inthequantumdescription,thequasi-particlesscatteramongthemselvesandcertainmodesconsistentwithbothmomentumandenergyconservation,becomepopulatedexponentiallyfast.Wedeterminethedurationofclassicalityinthiscasealso. 9

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Weexplainwhytheexistenceofcausticringsofdarkmatterisconsistentwiththeirquantumdescription.Relativeover-densitiesofstarsnearcausticringsarepredictedbysimulatingthedynamicsofhalfamillionstars.Wealsodeterminethedensityproleofinterstellargasassumingthegastobeinthermalequilibriuminthegravitationalpotentialofacausticandthegasitself.WearguethattriangularfeaturesintherecentlypublishedGAIAskymapareevidenceforthefthcausticring.Takingthenewevidenceintoconsideration,wecalculateupdatedvaluesofthedensitiesandvelocitiesofdarkmatterowsneartheSun. 10

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CHAPTER1BOSE-EINSTEINCONDENSATIONOFDARKMATTERAXIONSHerewedescribethephenomenonofBose-Einsteincondensation,emphasizingthenecessaryandsucientconditionsforitsoccurrence,andthereasonwhyitoccurs.ThenwediscussfoursubtopicsthatseemtocauseconfusionsinthecontextofBECofdarkmatteraxions.ThissectionisbasedonRef.[ 1 ].ConsiderasystemofNidenticalbosonsinthermalequilibriumundertheconstraintthatthetotalnumberofparticlesisconserved.AstandardtextbookderivationyieldstheaverageoccupationnumberhNjiofparticlestatejinthelimitofahugenumberofparticles(theso-calledthermodynamiclimit): hNji=1 e1 T(j)]TJ /F4 7.97 Tf 6.59 0 Td[())]TJ /F1 11.955 Tf 11.96 0 Td[(1(1-1)whereTisthetemperature,isthechemicalpotential,andj(j=0;1;2;3;:::)theenergyofparticlestatej.Wewillassumethattheparticlestatesareorderedsothat0<1<2<:::.ThehNjimaximizethesystementropyforagiventotalenergyE=jNjjandtotalnumberofparticlesN=jNj.SinceallhNji0,itisnecessarythat<0forEq. 1-1 tomakesense.Ontheotherhand,thetotalnumberofparticlesN(T;)=jhNjiisanincreasingfunctionofforxedTsinceeachhNjiis.So,ifNisincreasedwhileTisheldxed,mustincreasebutitcannotbecomelargerthan0.Inthesystemofinteresttous,thetotalnumberofparticlesinexcited(j>0)stateshas,for=0,anitevalue Nex(T;=0)=Xj>01 e1 T(j)]TJ /F4 7.97 Tf 6.58 0 Td[())]TJ /F1 11.955 Tf 11.96 0 Td[(1:(1-2)(Inoneandtwodimensions,Nex(T;=0)maybeinnitebecauseofaninfrareddivergencebutthiscommentisnotrelevanttothesystemsofourinterestinthreespatialdimensions.)Letusconsiderthescenario,when,atxedT,NismadelargerthanNex(T;=0).TheonlypossibleresponseofthesystemisthattheextraN)]TJ /F3 11.955 Tf 12.08 0 Td[(Nex(T;= 11

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0)particlesgotothegroundstate(j=0).IndeedtheaverageoccupationnumberhN0iofthatstatebecomesarbitrarilylargeasapproaches0frombelow.Fromthediscussionabove,wededucefourconditionsforBose-Einsteincondensation:1)thesystemcomprisesalargenumberofidenticalbosons(particleshenceforth);2)thenumberofparticlesisconserved;3)theparticlesaresucientlydegenerate;and4)theparticlesareinthermalequilibrium.Thenumberofparticleshastobesucientlylarge(condition1)forthesystemtobeinthermodynamiclimit.Thenumberofparticleshastobeconserved(condition2)butonlyonthetimescaleofthermalization.Forexample,itisirrelevanttoBose-Einsteincondensationofdilutegaseswhetherbaryonsareabsolutelystable.Theonlythingthatmattersisthattheyarestableonthetimescaleofthecondensationprocess.Darkmatteraxionsarenotabsolutelystablesincetheydecaytotwophotons.However,theirlifetimeismuchlongerthantheageoftheuniverseandsoisthetimescaleofallotheraxionnumberchangingprocesses.Condition3issatisedifthedegeneracy,i.e.theaverageoccupationnumberhNjiofthemostoccupiedstate,islargerthansomecriticalnumberoforderone.Forsystemsinthermalequilibrium,thisconditionistherequirementthatthetemperatureislowerthansomecriticaltemperature.ThecriticaltemperatureissuchthattheinterparticledistanceisoforderthethermaldeBrogliewavelength.ThermalequilibriumisgenerallytakenforgrantedinthediscussionsofBose-Einsteincondensationinliquid4Heandindilutegasesbecausethesesystemsthermalizeveryquickly.Forthesesystems,condition4isreadilysatisedbutcondition3isdiculttoachievebecauseofthelowtemperaturesrequired.Thereversesituationspertainstodarkmatteraxions.Axionsthermalizeonlyveryslowly,perhapsonthetimescaleoftheageoftheuniverse,ornotatall,becauseaxionsareveryweaklyinteracting.Ontheotherhand,theirquantumdegeneracyisenormouswithN1061.Forthesereasons,westatecondition3independentlyandaheadofcondition4. 12

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Thermalizationinvolvesinteractionsandrequirestime.WedenetherelaxationtimetobethetimescaleoverwhichthedistributionfNjgoftheparticlesovertheparticlestateschangescompletely,eachNjchangingbyorder100%.Whethercondition4forBose-Einsteincondensationissatised,isanissueoftimescales.Letusassumethattherstthreeconditionsaresatisedandthatthesystemisoutofequilibrium,i.e.thesystemisinastateofentropylessthanallowed.Thesystemwillthenthermalizeonthetimescale,increasingitsentropy.ItformsaBECbecausethestateofhighestentropy,giventhattherstthreeconditionsaresatised,isoneinwhichafractionoforderoneoftheparticlesisinthelowestenergyavailablestateandtheremainingparticlesinathermaldistributioninexcitedstates.TheentropyincreaseswhenaBECforms.Theprocessisirreversible.WenowdiscussfouraspectsofBose-Einsteincondensationthatappearssometimestobesourceofconfusionintheliterature,andmustbeclariedespeciallyinthecontextofcosmicaxionBose-Einsteincondensation. 1.1QuantumMechanicsisEssentialItispossiblewithinclassicaleldtheorytoproduceaphenomenonsimilartoBose-Einsteincondensationbyintroducingacut-okUVonthewavevectorsoftheeldmodes.IndeedtheclassicalphysicsanalogofEq. 1-1 is hNji=T j)]TJ /F3 11.955 Tf 11.96 0 Td[(:(1-3)Whenapproaches0frombelow,providedT6=0,hN0idivergesasitdoesfortheBose-Einsteindistribution.However,inclassicaleldtheory,theenergygetsdistributedequallyoveralleldmodes.Ifthereisnocut-o,thespecicheatperunitvolumedivergesbecause,inanynitevolume,theeldhasaninnitenumberofmodeswithlargewavevectors~k.Inotherwords,T=0inanynitevolumecontainingniteenergyinthermalequilibrium. 13

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ToproduceaphenomenonresemblingBose-Einsteincondensationinclassicaleldtheory,acut-oisintroducedbyhandwithvaluexedsothat Tcritk2UV 2m(1-4)wheremisthebosonmassandTcritthecriticaltemperaturethatisnaturallypresentinthequantumtheory.Thenumberofmodesperunitvolumeisnitethen,oforderk3UV=(2)3.Asaresult,T6=0inthecut-oeldtheoryandN0!1whenapproaches0frombelow.However,thisdoesnotmeanthatthecut-oclassicaleldtheoryhasanyvaliditybeyondproducingsomeformofBose-Einsteincondensation.Thecut-oisnotmeanttobepresentinanyrealsense.Ingeneralthecut-oclassicaleldtheory,diersfromthequantumeldtheory,andwhenthetwomakedierentpredictions,itisthelatterthatistobebelievednottheformer.Inparticular,asdiscussedinRef.[ 2 ],theclassicaltheoryconservesvorticity,i.e.thecirculationofthevelocityeldalongaclosedpath)]TJ ET BT /F2 11.955 Tf 177.18 -346.67 Td[(C[]I)]TJ /F3 11.955 Tf 9.78 13.95 Td[(~dr~v(~r;t);(1-5)whereasthequantumtheorydoesnot.Conservationofvorticityinclassicaleldtheoryfollowsfromthecontinuityandsingle-valuednessofthewavefunction,andholdswhetherornotawavevectorcut-oisintroduced.Incontrast,vorticityisnotconservedinthequantumeldtheorybecausequantacanjumpbetweenmodesofdierentvorticity.Thecreationofvorticityisessentialtoexplainthephenomenologyofcausticringsandsolvethegalacticangularmomentumproblem[ 2 ].Itispertinent,webelieve,toremarkthatEq. 1-4 doesnotmakesenseunlessaquantitywithdimensionofaction,suchas~,isintroducedbyhand.Theclassicaleldtheorydoesnothaveanotionofparticlesnorthereforeofparticlemass,evenafterthewavevectorcut-okUVisintroduced.Ithasonlymodeswithdispersionlaw !(~k)=q !20+c2~k~k(1-6) 14

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where!0istheangularfrequencyofoscillationofthe~k=0mode.Inthequantumtheory,theparticlemassisgivenby m=!0~ c2(1-7)butintheclassicaltheory,withorwithoutcut-o,thereisnosuchthingasparticlemass.Likewise,Eq. 1-4 shouldbewritten Tcrit(~kUV)2 2m(1-8)tobedimensionallyconsistent. 1.2HowLongisaClassicalDescriptionValid?Grantedthattheoutcomeofthermalizationinadegeneratebosonicsystemisdierentfromthatofitsclassicalanalog,onemaystillaskhowlongaclassicaldescriptionofsuchasystemisaccurate.InRef.[ 3 ],itwasshownbyanalyticalargumentsandnumericalsimulationofatoymodelthatthedurationofclassicalityofadegenerateinteractingbosonicsystemisoforder,andnotlongerthan,itsthermalizationtime.WesummarizesomeresultsofRef.[ 3 ]here,andaddtoymodelsimulationsthatareanalogoustoouranalyticalcalculationspresentedinSection 2.2 .AgeneralbosonicsystemthatconservesthenumberofquantahasaHamiltonianoftheform H=Xj!jayjaj+1 4Xjklnlnjkayjaykalan(1-9)wheretheajandayjareannihilationandcreationoperatorssatisfyingcanonicalequal-timecommutationrelations.IntheHeisenbergpicture,theannihilationoperatorsaj(t)satisfytheequationsofmotion i_aj=[aj;H]=!jaj+1 2Xklnlnjkaykalan:(1-10) 15

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Theclassicaldescriptionofthesystemisobtainedbyreplacingtheoperatorsaj(t)withc-numbersAj(t).Theysatisfy i_Aj=!jAj+1 2XklnlnjkAkAlAn:(1-11)ThequantumstateoccupationnumbersNj=ayjajhavetheclassicalanalogsNj=AjAj.Onemayaskthefollowingquestion:`Giventhesameinitialvalues,howlongdotheclassicalanalogsNj(t)tracktheexpectationvalueshNj(t)iofthequantumoperators?'.Letusdenedurationofclassicalityasthetimescaleoverwhichtheclassicaldescriptionaccuratelydescribesthequantumsystemwithinsomemarginoferror,say20%.Toaddressthisissue,atoymodelofvequantumoscillatorswassimulatednumerically[ 3 ].ThetoymodelhasbeenpreviouslydiscussedandsimulatedinRef.[ 4 ]toverifynumericallythevalidityofformulaethatestimatetherateofthermalizationinthe`condensedregime'denedbythecondition)]TJ /F3 11.955 Tf 261.06 0 Td[(>where)-327(isthethermalizationrateandtheenergydispersionofthequantainthesystem.Thedarkmatteraxionuidthermalizesinthecondensedregime.TheHamiltonianofthetoymodelhastheformgiveninEq. 1-9 with!j=j!1(j=1;2;3;4;5)andlnjk=0unlessj+k=l+n.Non-zerovaluesaregivento2314;2415;3425;1322;2433;1533and3544,andtheirconjugateslnjk=jkln.TheSchrodingerequation i@tj(t)i=Hj(t)i(1-12)wassolvednumericallyforalargevarietyofinitialconditions.Inallcases,itwasfoundthatthedurationofclassicalityislessthanoratmostofordertherelaxationtime,denedasthetimescaleoverwhichthedisributionofthequantaovertheoscillatorschangescompletely.Fig. 1-1 showsinitstoppanelthequantumevolutionoftheinitialstatejN1;N2;:::;N5i=j12;25;4;12;1iasanexample.ThegureshowsthattheexpectationvalueshNjimovetowardstheirthermalaveragesontheexpectedtimescale=1=,whichisoforder0:4forthecouplingstrengthslnjkinthesimulation[ 4 ].ThethermalaveragesareshownbythedotsontherightsideofFig. 1-1 (top 16

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panel).ThebottompanelofFig. 1-1 showstheclassicalevolutionoftheinitialstate(A1;A2;:::;A5)=(p 12;p 25;p 4;p 12;p 1),inwhichtheNjandtheirtimederivatives_Njhavethesameinitialvaluesastheirquantumanaloguesinthetoppanel.Fig. 1-1 showsthattheclassicalevolutiontracksthequantumevolutiononlyforatimeoforder,andrelativelyshortcomparedto,. Figure1-1. Quantum(top)andclassical(bottom)timeevolutionoftheoccupationnumbersinthetoysystemdescribedinthetextfortheinitialstatej12;25;4;12;1i.Thedotsontherightinthetoppanelindicatethethermalaveragesinthequantumcase.Thequantumsystemapproachesthethermalaveragesontheexpectedtimescale.Theclassicalevolutiontracksthequantumevolutiononlyverybrieyanddoesnotequilibrate.ThisgureisreprintedwithpermissionfromRef.[ 3 ]. 17

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ThetoymodelcanbemadetobehaveanalogouslytothehomogeneousquantumeldcondensatesdiscussedinSection 2.2 .Thehomogeneouscondensatespersistsindenitelyintheirclassicaldescriptionbuthaveanitelifetimeintheirquantumdescription.Initialstateswiththeanalogouspropertyinthetoymodelarej0;N;0;0;0i.Intheirclassicalevolution,thesestatespersistsindenitelybecausetheRHSofEq. 1-11 vanishes.Intheir,quantumevolution,thequantainthe2ndoscillatorjumpinpairstothe1stand3rdoscillatorsandthencetothe4thand5thoscillators.Fig. 1-2 showsthehNj(t)iasafunctionoftimeforN=100inpanel(a),contrastedwiththeconstantNjinpanel(b).ForthegenericinitialstatessimulatedinRef.[ 3 ],therelaxationrateisoforder)]TJ /F2 11.955 Tf 12.86 0 Td[(p INforboththeclassicalandquantumevolutions,whereIisthenumberofrelevantinteractiontermsontheRHSofEq. 1-9 ,andandNaretypicalvaluesoftheinteractionstrengthsandofthequantumoccupationnumbers[ 4 ].Forthespecialinitialstatesj0;N;0;0;0i,therelaxationratevanishesaccordingtotheclassicalevolutionbutisoforder)]TJ /F2 11.955 Tf 90.26 0 Td[(j2213jN=log(N)accordingtothequantumevolution.Thefactorlog(N)appearsbecausetherelaxationofthesespecialstatesislimitedbytheinitialprocess2+2!1+3,whichactsasabottleneck.The2+2!1+3processcausestheoccupationnumbersofthe1stand3rdoscillatorstogrowasej2213jNt,thedierencebetweentheclassicalandquantumevolutionsbeingonlythatthegrowthisseededinthequantumevolutionwhereasitisunseededintheclassicalevolution.ApplyingthemethodspresentedinSection 2.1 ,onends hN1(t)i=hN3(t)i=sinh2(t)(1-13)with=1 2j2213jN,andtherefore hN2(t)i=N)]TJ /F1 11.955 Tf 11.96 0 Td[(2sinh2(t)(1-14)fortsucientlysmallthatthecondensatehasnotbeendepletedmuchyet.ThedottedlinesinFig. 1-2 showhN2(t)i,hN1(t)iandhN3(t)iaccordingtoEqs. 1-13 and 1-14 18

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Figure1-2. Quantum(top)andclassical(bottom)evolutionoftheinitialstatej0;100;0;0;0i.Initsclassicalevolutionthisstatepersistsindenitely.Initsquantumevolution,thestatethermalizes.ThedottedlinesshowthepredictionsofEqs. 1-13 and 1-14 .Afteratimeoforder0.1,theseequationsareinaccuratebecausequantajumpfromthe1stand3rdoscillatorsbacktothe2ndoscillatorandfromthe3rdtothe4thand5thoscillators. 19

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Insummary,thedurationofclassicalityoftheinitialstatej0;N;0;0;0i,whichpersistsindenitelyaccordingtoitsclassicalevolution,isafactorlog(N)longerthanthedurationofclassicalityofgenericstatesjN1;N2;N3;N4;N5i.Thej0;N;0;0;0iisthetoymodelanalogofthehomogeneouscondensatesdiscussedinSection 2.2 .Thosehomogeneouscondensatesalsopersistforeveraccordingtotheirclassicalevolution,buthaveanitedurationofclassicalityaccordingtotheirquantumevolution.Weexpectthedurationofclassicalityofinhomogeneouscondensatestobeshorterthanthatofhomogeneouscondensatesforthesamereasonthatthedurationofclassicalityofgenerictoymodelinitialstatesisshorterthanthatofthej0;N;0;0;0iinitialstate,thereasonbeingtheabsenceinthecaseofgenericstatesofthethermalizationbottleneckthatispresentfortheinitialstatej0;N;0;0;0i. 1.3HomogeneityisnotaNecessaryOutcomeorCriterionContrarytostatementsappearingoccasionallyintheliterature,thecondensedstateneednotbeastateofmomentum~p=0.Generally,itisnot.Thestate~p=0ishomogeneousandminimizesthekineticenergy!~p=~p~p 2mofaparticleinemptyspace.But,ingeneral,spaceisnotemptyandtheparticleenergiesjthatappearinEq. 1-1 dierfromthefrequencies!jthatappearinEq. 1-9 becauseofinteractions.Eveninemptyspace,thelowestenergyavailablestateneednotbethezeromomentumstate.LackofhomogeneityisnotimpedimenttoBose-Einsteincondensation.Fig. 1-3 showsaBECthatishighlyinhomogeneousonsomelengthscaledbutextendsoveramuchlargerlengthscaleL.Itmayberealizedbyplacingsuperuid4Heinalongtubewithvariousobstructionsonthelengthscaledinsidethetube.Althoughinhomogeneousonthelengthscaled,thecondensatehaslongrangecorrelationsonthelengthscaleL,aswenowshowexplicitly.ForageneralsystemundergoingBose-Einsteincondensation,letuj(~x;t)bethewavefunctionoftheparticlestatewithenergyj.Thewavefunctionsformacomplete 20

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Figure1-3. Cartoonofsuperuid4HellingatubeoflengthLwithvariousobstructionsinsidethatmaketheuidinhomogeneousonthelengthscaled.AlthoughinhomogeneousonscaledtheuidiscorrelatedonscaleL,whichmaybearbitrarilylargecomparedtod.LikewisetheBose-Einsteincondensateofdarkmatteraxionsmaybecorrelatedonthescaleofthehorizonalthoughinhomogeneousonthescaleofgalaxies. orthonormalset: ZVd3xuj(~x;t)uk(~x;t)=kjXjuj(~x;t)uj(~y;t)=3(~x)]TJ /F3 11.955 Tf 11.83 0 Td[(~y): (1-15) Thequantumscalareld(~x;t)describingtheparticlesundergoingBose-Einsteincondensationanditscanonicallyconjugateeld(~x;t)maybeexpandedintermsofthosewavefunctions: (~x;t)=Xj1 p 2muj(~x;t)bj(t)+uj(~x;t)bj(t)y(~x;t)=Xj1 ir m 2uj(~x;t)bj(t))]TJ /F3 11.955 Tf 11.95 0 Td[(uj(~x;t)bj(t)y (1-16) wherebj(t)andbj(t)yareannihilationandcreationoperatorssatisfyingcanonicalequaltimecommutationrelations.Notethatbj(t)andbj(t)yinEq. 1-16 aredierentfromtheaj(t)andaj(t)yintheprevioussubsectionsincethelatterannihilateandcreateparticlesineigenstatesofthefreeHamiltonian,whereasbj(t)andbj(t)yannihilateandcreateparticlesineigenstatesoftheone-particleHamiltonianinwhichtheinteractionsoftheone 21

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particlewithalltheotherparticlesarederivedfromthefullHamiltonianusingmeaneldtheory.Ageneralsystemmaybewritten ji=XfNjgc(fNjg)jfNjgit(1-17)wherefNjgisanarbitrarydistributionoftheoccupationnumbersovertheparticlestates, jfNjgit=Yj1 p Nj!)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(bj(t)yNjj0i(1-18)wherej0iistheemptystate.Sincethetotalnumberofparticlesisconserved,wemaytakejitobeaneigenstateofthetotalnumberoperator Xjbj(t)ybj(t)ji=Nji;(1-19)inwhichcase,c(fNjg)=0unlessPjNj=N.Inthestateji,theeld(~x;t)hasequal-timecorrelationfunction h(~x;t)(~y;t)i=XfNjgXfN0jgc(fNjg)c(fN0jg)Xk;l1 2m[uk(~x;t)ul(~y;t)tfNjgjbk(t)ybl(t)jfN0jgt+uk(~x;t)ul(~y;t)tfNjgjbk(t)bl(t)yjfN0jgt]: (1-20) IfaBEChasformed,thelowestenergyavailablestatehasoccupationnumberN0oforderN.Inthatcase,c(fNjg)=0unlessN0=N0andtherefore h(~x;t)(~y;t)i=N0 2m[u0(~x;t)u0(~y;t)+u0(~x;t)u0(~y;t)]+:::(1-21)wherethedotsarecontributionsfromparticlestatesotherthanthecondensedstate,thatfalloexponentiallyorasapowerlawwithdistancej~x)]TJ /F3 11.955 Tf 12.18 0 Td[(~yjr.Thecontributionfromthecondensedstatedoesnotfallwithdistancer.Instead,forgiven~y,ithassupport 22

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whereveru0(~x;t)hassupport.ThusanyBECiscorrelatedoverthewholeextentofthecondensate. 1.4WhatStatedotheParticlesCondenseinto?SincethermalizationisaconditionforBose-Einsteincondensation,itfollowsthatthestatetheparticlescondenseintoisthelowestenergystatethatisavailabletothemthroughthethermalizinginteractions.Ingeneral,itisnotthelowestenergystateinanabsolutesense.Forexample,ifabeakerofsuperuid4Hesitsonatable,amacroscopicallylargenumberofatomsareinacondensedstate.Thecondensedstateiscertainlynotthelowestenergystatesinceitsenergycanbeloweredbyplacingthebeakerontheoor.Itis,however,thelowestenergystateavailabletothe4Heatomsthroughthethermalizinginteractions.Asalreadymentioned,thecondensedstateneednotbestable.Ideally,itshouldbestableonthethermalizingtimescale.Acomplicatingfactoristhatthermalizationisrarelycomplete.Fortunately,Bose-Einsteincondensationoccursimmediatelyandexplosivelyonthethermalizationtimescale.Therateatwhichparticlesmovetothecondensedstateisproportionaltothenumberofparticlesalreadyinthecondensedstate[ 5 ].ThetimescaleoverwhichacompleteBose-EinsteindistributionisestablishedisgenerallymuchlongerthanthetimescaleoverwhichtheBECforms[ 6 ].Nonetheless,inthecaseofBose-Einsteincondensationofdarkmatteraxions,wehavetodealwiththecomplicationthattheaxionuidismadeunstablebytheveryinteractionthatthermalizesit.AfterBose-Einsteincondensationhasoccurred,furtherthermalizationisrequiredbecausetheinstabilitycausesthelowestenergyavailablestatetochangewithtime.Ourworkismotivatedbythequestionwhattheoutcomeofthermalizationiswhiledensityperturbationsgrowandhowthisoutcomediersfromthepredictionsofcosmologicalperturbationtheorywithordinarcolddarkmatter(CDM).InSection 2.1 ,weconstructaformalismthatallowsonetodiscussmoreclearlythethermalizationofauidmadeunstableandthereforeinhomogeneous,bythevery 23

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interactionsthatthermalizeit.InSections 2.2 and 2.3 ,weapplytheformalismtoahomogeneouscondensatewithattractivecontactandgravitationalself-interactions,respectively.InChapter 3 ,weprovidethequantumdescriptionofacondensatewithrepulsivecontactinteractionsandsmallinhomogeneities.InChapter 4 ,westartwithabriefreviewonthepropertiesofacausticringandexplainwhyandhowquantumeectsplayakeyroleintheformationofdarkmattercaustics.InSection 4.3 ,westudytheeectsofthecausticringsonthedynamicsofasinglestarandofadistributionofstars.InSection 4.4 ,wedeterminethedensityproleoftheinterstellargasnearacausticringassumingthegastobeinthermalequilibriuminthepotentialofthecausticandthegasitself.InChapter 5 ,wediscusssomerecentevidenceofthecausticringsandtheirimplicationsonthedensitiesandvelocitiesofthedarkmatterowsatthelocationoftheSun. 24

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CHAPTER2QUANTUMANDCLASSICALDESCRIPTIONSOFAXIONS:HOMOGENEOUSCONDENSATESWITHATTRACTIVESELF-INTERACTIONSInthischapter,wedevelopaformalismtostudythedepartureofthequantumevolutionofahighlydegeneratescalareldfromitsclassicalcounterpart(Section 2.1 ).Thenweapplytheformalismtoclassicalhomogeneouscondensateswithbothattractivecontactself-interactions(Section 2.2 )andgravitationalself-interactions(Section 2.3 ). 2.1FormalismtoCalculatetheDurationofClassicalityInthissection,weintroduceaformalismtoanalyzetheevolutionofahighlydegenerateBosonicuidwithcontactinteractions.Wedemonstratetheformalismusing4theorywiththefollowingHamiltonian H=Zd3x1 22+1 2(~r)2+1 2m22+ 4!:4:(2-1)where(~x;t)and(~x;t)areconjugateHermitianscalareldssatisfyingcanonicalequaltimecommutationrelations.Thedoublecolon:...:symbolinthelasttermofEq. 2-1 indicatesnormalordering.Thattermdescribescontactinteractionswhicharerepulsiveif>0andattractiveif<0.Theandeldssatisfythefollowingequationsofmotion @t=;@t)-222(r2+m2+ 6:3:=0:(2-2)Weareconcernedonlywiththenon-relativisticregimeofthetheory.Thenon-relativisticlimitisobtainedbysetting (~x;t)=1 p 2m (~x;t)e)]TJ /F4 7.97 Tf 6.59 0 Td[(imt+ (~x;t)yeimt(~x;t)=r m 2()]TJ /F3 11.955 Tf 9.3 0 Td[(i) (~x;t)e)]TJ /F4 7.97 Tf 6.59 0 Td[(imt)]TJ /F3 11.955 Tf 11.96 0 Td[( (~x;t)yeimt; (2-3) basedonRef.[ 1 ]. 25

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byneglectingtermsoforder@t versustermsoforderm ,andbyignoringtermsproportionaltoe2imtande)]TJ /F9 7.97 Tf 6.58 0 Td[(2imtthatoscillatesofastintimethattheyeectivelyaveragestozero. (~x;t)isanon-Hermitianscalareldsatisfyingtheequaltimecommutationrelations [ (~x;t); (~y;t)]=0;[ (~x;t); (~y;t)y]=3(~x)]TJ /F3 11.955 Tf 11.83 0 Td[(~y);(2-4)andtheequationofmotion i@t =)]TJ /F1 11.955 Tf 15.61 8.08 Td[(1 2mr2 + 8m2 y :(2-5)Thenumberofparticlesisconservedinthenon-relativisticlimiteventhoughthenumberofparticlesisnotconservedintheoriginaltheory,Eq. 2-1 .Weexpandthe eldinanorthonormalandcompletesetofwavefunctionsu~k(~x;t)labeledby~k: (~x;t)=X~ku~k(~x;t)a~k(t):(2-6)Thewavefunctionsu~k(~x;t)satisfy ZVd3xu~k(~x;t)u~k0(~x;t)=~k0~kandX~ku~k(~x;t)u~k(~y;t)=3(~x)]TJ /F3 11.955 Tf 11.83 0 Td[(~y) (2-7) whereVisthevolumeofthespaceinwhichthetheoryisdened.Thea~k(t)anda~k(t)yareannihilationandcreationoperatorssatisfyingequaltimecommutationrelations [a~k(t);a~k0(t)]=0;[a~k(t);a~k0(t)y]=~k0~k;(2-8)andtheequationofmotion i@ta~k(t)=X~k0M~k0~k(t)a~k0(t)+1 2X~k2;~k3;~k4~k3~k4~k~k2(t)a~k2(t)ya~k3(t)a~k4(t)(2-9) 26

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with M~k0~k(t)=ZVd3xu~k(~x;t)()]TJ /F3 11.955 Tf 9.3 0 Td[(i@t)]TJ /F1 11.955 Tf 18.27 8.09 Td[(1 2mr2)u~k0(~x;t)~k3~k4~k~k2(t)= 4m2ZVd3xu~k(~x;t)u~k2(~x;t)u~k3(~x;t)u~k4(~x;t): (2-10) Inthenon-relativisticlimit,theaboveequationsareexactforanyorthonormalcompletesetofstatesu~k(~x;t).Herewewilldevelopamethodtostudythedepartureofthequantumevolutionfromtheclassicaloneandtocalculatethedurationofclassicality.Theclassicaldescriptionisobtainedbyreplacingthequantumeld (~x;t)byawavefunction(~x;t)whichsatisesthec-numberversionofEq. 2-5 i@t=)]TJ /F1 11.955 Tf 15.61 8.09 Td[(1 2mr2+ 8m2jj2(2-11)calledtheSchrodinger-Gross-Pitaevskii(SGP)equation.Althoughwavefunctionsarehistoricallyassociatedwithaquantummechanicaldescription,fromthepointofviewofquantumeldtheory,awavefunctionismerelyasolutionoftheclassicaleldequationsinthenon-relativisticlimit.Thewavefunctionmaybewrittenas (~x;t)=A(~x;t)ei(~x;t);(2-12)whereA(~x;t)and(~x;t)arereal.Thewavefunction(~x;t)describesauidofnumberdensityn(~x;t)andvelocityeld~v(~x;t): n(~x;t)=A(~x;t)2 (2-13) ~v(~x;t)=1 m~r(~x;t): (2-14) Eq. 2-11 impliesthecontinuityequation @tn+~r(n~v)=0(2-15) 27

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andtheEuler-likeequation @t~v+(~v~r)~v=)]TJ /F1 11.955 Tf 12.68 8.09 Td[(1 m~rV)]TJ /F3 11.955 Tf 13.09 3.02 Td[(~rq(2-16)where V(~x;t)= 8m2n(~x;t)(2-17)and q(~x;t)=)]TJ /F1 11.955 Tf 17.98 8.09 Td[(1 2m2r2p n p n:(2-18)q(~x;t)issometimescalled\quantumpressure".Exceptforthe)]TJ /F3 11.955 Tf 10.44 3.02 Td[(~rqterm,Eq. 2-16 istheEulerequationforauidofclassicalparticlesmovinginthepotentialV(~x;t).The)]TJ /F3 11.955 Tf 10.43 3.02 Td[(~rqtermisaconsequenceoftheunderlyingwavenatureoftheuidandaccounts,forexample,forthetendencyofawavepackettospread.Letusconsideraparticularsolutionoftheclassicalequationofmotion.Weaskthefollowingquestion:`Forhowlongdoesitprovideanaccuratedescriptionofthequantumsystem?'.Asshownbelow,wetreatthesolution(~x;t)oftheSGPequation 2-11 asonemodeofthequantumeld.Whenthis,andonlythismode,ishighlyoccupied,theclassicaldescriptionisveryaccurate,quantumcorrectionsbeingoforder1=NwhereNistheoccupationnumberofthemode(~x;t).Thequestioniswhetherthequantastayinthismode,andiftheydonot,atwhichrate,theyleavethemode(~x;t).Toaddresstheseissues,weintroduceasetofmodesthataresimilarto(~x;t)butdierfromitbylongwavelengthmodulations: u~k(~x;t)=1 p N(~x;t)ei~k~(~x;t):(2-19)The~(~x;t)areco-movingcoordinateschosensothatthedensityin~-spaceisconstantinbothspaceandtime: d3N d3=n(~x;t) J(~x;t)=n0(2-20) 28

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wheren0isaconstant,n(~x;t)isthephysicalspacedensityimpliedby(~x;t)(seeEq. 2-12 and 2-13 )and J(~x;t)=det@~ @~x(2-21)istheJacobianofthemap.The~(~x;t)canbeconstructedasfollows.Thewavefunction(~x;t)impliesavelocityeld~v(~x;t)givenbyEq. 2-14 ,andhenceamap~x(~;t); @~x @t~=~v(~x(~;t);t):(2-22)If~v(~x;t)werethevelocityeldofaowofparticles,~wouldlabelindividualparticlesintheow.Forexample,~maybethepositionoftheparticleatsomeinitialtimet.Themap~(~x;t)istheinverseof~x(~;t).Thisconstructionensuresthatthedensityin~-spaceistime-independent.Furthermore,itisalwayspossibletochangevariables~!~0suchthatthedensityin~0-spaceis~0-independentaswell.Wechoosetheregionin~-spacewherethetheoryisdened,tobeacubeofvolumeV0=L30withperiodicboundaryconditionsatitssurface.ThusthewavevectorsappearinginEq. 2-19 are~k=2 L0(n1;n2;n3)wherethenj=0;1;2;:::(j=1;2;3).Wehavethen ZVd3xu~k(~x;t)u~k0(~x;t)=1 V0ZV0d3ei(~k0)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k)~=~k0~kandX~ku~k(~x;t)u~k(~y;t)=V0 Nn(~x;t) J(~x;t)3(~x)]TJ /F3 11.955 Tf 11.83 0 Td[(~y)=3(~x)]TJ /F3 11.955 Tf 11.83 0 Td[(~y); (2-23) i.e.theu~k(~x;t)formacompleteorthonormalset.SubstitutingEq. 2-19 inEqs. 2-10 ,weget ~k3~k4~k~k2(t)= 4m2N2ZVd3xn(~x;t)2ei(~k3+~k4)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k2)~(~x;t)= 4m2N~n(~k+~k2)]TJ /F3 11.955 Tf 11.35 3.15 Td[(~k3)]TJ /F3 11.955 Tf 11.36 3.15 Td[(~k4;t) (2-24) where ~n(~q;t)=1 V0ZV0d3n(~x(~;t);t)e)]TJ /F4 7.97 Tf 6.59 0 Td[(i~q~;(2-25) 29

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and M~k0~k(t)=)]TJ /F3 11.955 Tf 17.49 8.09 Td[( 8m2~n(~k)]TJ /F3 11.955 Tf 11.36 3.16 Td[(~k0;t)+1 2mNZVd3xn(~x;t)~r(~k~(~x;t))~r(~k0~(~x;t))ei(~k0)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k)~(~x;t): (2-26) ThederivationofEq. 2-26 isgiveninAppendix A .Theannihilationoperatorsa~k(t)satisfyEq. 2-9 .Thecorrespondingclassicalequationsofmotionareobtainedbyreplacingtheoperatorsa~k(t)bycomplexnumbersA~k(t) i@tA~k(t)=X~k0M~k0~k(t)A~k0(t)+1 2X~k2;~k3;~k4~k3~k4~k~k2(t)A~k2(t)A~k3(t)A~k4(t):(2-27)Theclassicalsolutionwithwhichwestartedis(~x;t)=p Nu~0(~x;t).Therefore,A~k(t)=p N~0~kmustsolveEq. 2-27 .Onecanverifythatthisisindeedthecasesince M~0~k(t)=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(N 2~0~0~k~0(t):(2-28)TheEq. 2-28 providesaconsistencycheckforourformalism.OnceEq. 2-9 issolved,theaverageoccupationnumberofmode~kintheHeisenbergpictureisgivenby N~k(t)=inja~k(t)ya~k(t)jin(2-29)wherejiniistheinitialstateofthesystem.Tondoutthedurationofclassicality,weassumealltheparticlesareinitiallyinthemodecorrespondingtotheclassicalsolution(~k=0mode).Thetotalnumberofparticlesthathaveleftthe~k=0modeaftertimetis Nev(t)=X~k(6=0)N~k(t):(2-30)ThedurationofclassicalityisthetimescaletclforwhichNev(tcl)N,i.e.thetimescalewhenalmostalltheparticleshaveleftthemodecorrespondingtotheclassicalsolution. 30

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2.2HomogeneousCondensatewithAttractiveContactSelf-interactions 2.2.1ClassicalDescriptionTheclassicaldescriptionofabosonicuidwithattractivecontactself-interactionsisobtainedbysolvingtheSGPEq. 2-11 whichadmitsthehomogeneoussolution 0=p n0e)]TJ /F4 7.97 Tf 6.59 0 Td[(i!t(2-31)where !=n0 8m2:(2-32)Considerasmallperturbationaboutthehomogeneoussolution (~x;t)=0(t)+1(~x;t):(2-33)Tothelowestorder,theperturbationsatises i@t1=)]TJ /F1 11.955 Tf 15.62 8.08 Td[(1 2mr21+!(21+e)]TJ /F9 7.97 Tf 6.59 0 Td[(2i!t1):(2-34)WedecomposetheperturbationinFouriermodesasfollows 1(~x;t)=e)]TJ /F4 7.97 Tf 6.58 0 Td[(i!tX~kC~k(t)ei~k~x:(2-35)TheFouriercoecientsC~k(t)satisfy i@tC~k(t)=k2 2m+!C~k(t)+!C)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k(t):(2-36)TheFouriercoecientsC~k(t)canbedecomposedas C~k(t)=s~k(t)+r~k(t)(2-37)where s~k(t)=s)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k(t)andr~k(t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(r)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k(t):(2-38) 31

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Eq. 2-36 impliesthat r~k(t)=2im k2@ts~k(t)(2-39)andthats~k(t)isasolutionof @2ts~k(t)=)]TJ /F3 11.955 Tf 12.93 8.09 Td[(k2 2mk2 2m+2!s~k(t):(2-40)Thecaseofrepulsiveinteractions(>0)willbediscussedinChapter 3 .Intheattractivecase(=jj<0),thereisacriticalwavelength2=kJ,similartotheJeanslengthforgravitationalinteractions,with kJ=r jjn0 2m:(2-41)Formodeswithk>kJ,i.e.withwavelengthsmallerthanthecriticalwavelength,theperturbationsarestableandoscillatewithangularfrequency !(~k)=s k2 2mk2 2m)]TJ 13.15 8.08 Td[(jjn0 4m2:(2-42)ThemostgeneralsolutiontoEq. 2-36 is C~k(t)=c~kk2 2m+!(~k)e)]TJ /F4 7.97 Tf 6.58 0 Td[(i!(~k)t+c)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~kk2 2m)]TJ /F3 11.955 Tf 11.95 0 Td[(!(~k)ei!(~k)t(2-43)wherethec~karethecomplexnumberswhichcanbedeterminedintermsoftheFouriercoecientsoftheinitialperturbation1(~x;0).Formodeswithk
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wherec~k;arecomplexnumberssubjecttotheconstraintsc~k;=c)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k;.Theinstabilityoccursbecausetheattractivecontactforcesproduceatendencyofquantatomovetowardsregionsofhighdensity,hencecrowdplacesthatarecrowdedalready.Thistendencyovercomestheeectofquantumpressureonlengthscaleslargerthank)]TJ /F9 7.97 Tf 6.59 0 Td[(1J. 2.2.2QuantumDescriptionNowwewillapplytheformalismdevelopedinSection 2.1 tothehomogeneouscondensatewithattractiveinteractionswhichisdescribedbyEqs. 2-31 and 2-32 .Since,theclassicalvelocityeld~v=0inthisstate,wechoose~=~xandhence,followingEqs. 2-19 2-26 and 2-24 u~k(~x;t)=1 p Ve)]TJ /F4 7.97 Tf 6.58 0 Td[(i!t+i~k~xM~k0~k=k2 2m)]TJ /F3 11.955 Tf 11.96 0 Td[(!~k0~k~k3~k4~k~k2=2! N~k3+~k4~k+~k2: (2-46) Theequationsofmotion(Eq. 2-9 )fortheannihilationoperatorsa~k(t)are i@ta~k(t)=k2 2m)]TJ /F3 11.955 Tf 11.96 0 Td[(!a~k(t)+! NX~k1;~k2a~k1+~k2)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k(t)ya~k1(t)a~k2(t):(2-47)ToanalyzethebehaviorofthesystemwhenthehomogeneousparticlestateisoccupiedbyalargenumberNofquanta,wefollowBogoliubov'sapproach[ 7 ]andsubstitute a~k(t)=p N~0~k+b~k(t):(2-48)Theb~k(t)operatorssatisfycanonicalcommutationrelations.For~k6=0,theequationsofmotionare i@tb~k=k2 2m+!b~k+!by)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+! p NX~k0(b~k)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k0b~k0+2by~k0)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~kb~k0)+! NX~k1;~k2by~k1+~k2)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~kb~k1b~k2: (2-49) 33

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Oneshouldnotethatthelasttwotermsaresuppressedrelativetothersttwolineartermsbyoneortwofactorsof1=p N.Ignoringthequadraticandcubicterms,wehave i@t0B@b~kby)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k1CA=0B@k2 2m+!!)]TJ /F3 11.955 Tf 9.3 0 Td[(!)]TJ /F4 7.97 Tf 11.97 4.71 Td[(k2 2m)]TJ /F3 11.955 Tf 11.96 0 Td[(!1CA0B@b~kby)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k1CA:(2-50)TodiagonalizethematrixappearinginEq. 2-50 ,weperformacanonicalBogoliubovtransformation: 0B@b~kby)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k1CA=0B@u(~k)v(~k)v(~k)u(~k)1CA0B@~ky)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k1CA(2-51)whereu(~k)andv(~k)arerealandu(~k)2)]TJ /F3 11.955 Tf 12.16 0 Td[(v(~k)2=1.Onemaywriteu(~k)=cosh(~k)andv(~k)=sinh(~k).Theequationsofmotionofthenewoperators~k(t)are i@t0B@~ky)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k1CA=0B@acosh(2)+bsinh(2)asinh(2)+bcosh(2))]TJ /F3 11.955 Tf 9.3 0 Td[(asinh(2))]TJ /F3 11.955 Tf 11.96 0 Td[(bcosh(2))]TJ /F3 11.955 Tf 9.29 0 Td[(acosh(2))]TJ /F3 11.955 Tf 11.95 0 Td[(bsinh(2)1CA0B@~ky)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k1CA(2-52)where a=k2 2m+!andb=!:(2-53) 2.2.2.1Modeswithwavelengthsmallerthanthecriticalwavelength(k>kJ)ThematrixthatappearsinEq. 2-52 canbemadediagonalbychoosing(~k)suchthat tanh(2)=)]TJ /F3 11.955 Tf 11.08 8.09 Td[(b a;(2-54)withthemagnitudeofthediagonalelementsequalto acosh(2)+bsinh(2)=s k2 2mk2 2m)]TJ 13.15 8.09 Td[(jjn0 4m2=!(~k)(2-55)where!(~k)istheangularfrequencythatappearsintheclassicaldescription.Eq. 2-54 isvalidonlywhenk>kJ=q jjn0 2m.Forthesemodesandwiththischoiceof(~k), i@t~k=!(~k)~k:(2-56) 34

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TheHamiltonianfortheoperators~kbecomes H=X~k!(~k)y~k~k(2-57)wherewehavenotshownhigherorderinteractiontermswhicharesuppressedbyfactorsof1=p N.They~kand~kcreateandannihilatethe`quasi-particles'whicharethequantaofexcitationofthesystemwhenthehomogeneousparticlestateishighlyoccupied. 2.2.2.2Modeswithwavelengthlargerthanthecriticalwavelength(kjaj.Forthesemodes,wesetthediagonalelementsofthematrixinEq. 2-52 equaltozerobychoosing(~k)suchthat tanh(2)=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(a b;(2-58)withtheo-diagonalelementsgivenby asinh(2)+bcosh(2)=)]TJ /F10 11.955 Tf 9.3 21.56 Td[(s k2 2mjjn0 4m2)]TJ /F3 11.955 Tf 15.58 8.08 Td[(k2 2m=)]TJ /F3 11.955 Tf 15.3 8.08 Td[(k 2mq k2J)]TJ /F3 11.955 Tf 11.95 0 Td[(k2=)]TJ /F3 11.955 Tf 9.3 0 Td[((~k)(2-59)where(~k)istherateofinstabilitythatappearsintheclassicaldescription.Withthischoiceof(~k),wehave i@t~k=)]TJ /F3 11.955 Tf 9.3 0 Td[((~k)y)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k:(2-60)TheHamiltonianfortheoperators~kbecomes H=X~kk>kj!(~k)y~k~k+X~kk0()]TJ /F3 11.955 Tf 9.3 0 Td[((~k))(~k)]TJ /F4 7.97 Tf 6.16 2.11 Td[(~k+y~ky)]TJ /F4 7.97 Tf 6.16 2.11 Td[(~k)(2-61)wherewehavenotshownhigherorderinteractionterms.Wemayrewritethekinetictermsforthek
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Wenowshowthatthissystemexhibitsparametricresonance.ConsidertheHamiltonianforanyofthek
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Thenormalizationcondition 1Xp=0jc2p(t)j2=1(2-73)yields c0(t)=1 p cosh(t):(2-74)Theprobabilitythatthesystemisfoundinthe(2p)thexcitedstateisthus jc2p(t)j2=(2p)]TJ /F1 11.955 Tf 11.96 0 Td[(1)!! 2pp!(tanh(t))2p cosh(t):(2-75)TheaverageoccupationnumbercanbeobtaineddirectlyfromEq. 2-66 : hN(t)i=0j(t)y(t)j0=sinh2(t):(2-76)Likewise,theaverageoccupationnumbersquared N(t)2=0j(t)y(t)(t)y(t)j0=sinh4(t)+2cosh2(t)sinh2(t):(2-77)Therootmeansquaredeviationfromtheaverageoccupationnumberistherefore N(t)=q hN(t)2i)-222(hN(t)i2=1 p 2jsinh(2t)j:(2-78)Boththeaverageoccupationnumberanditsrootmeansquaredeviationgrowase2t. 2.2.3DurationofClassicalityWefoundthattheoccupationnumbersofall~kand0~kmodesinthewavevectorrange0
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~k(0)and0~k(0)by b~k(t)=u p 2)]TJ /F3 11.955 Tf 5.47 -9.69 Td[(~k(t)+0~k(t)+v p 2y~k(t))]TJ /F3 11.955 Tf 11.95 0 Td[(0y~k(t)=1 p 2)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(~k(0)+0~k(0)hucosh((~k)t))]TJ /F3 11.955 Tf 11.96 0 Td[(ivsinh((~k)t)i+1 p 2y~k(0))]TJ /F3 11.955 Tf 11.96 0 Td[(0y~k(0)hvcosh((~k)t)+iusinh((~k)t)i (2-79) whereweusedEqs. 2-63 and 2-66 with=(~k).Weconsidertheinitialstatejisuchthat ~k(0)ji=0~k(0)ji=0(2-80)forall~k6=0.UsingEq. 2-79 ,wendthattheaveragenumberofquantathathavejumped,intimet,fromthecondensateintomodeb~kwith0
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Theaveragenumberofquantathathaveevaporatedfromthe~k=0condensateis Nev(t)=X~kDjby~k(t)b~k(t)jE=VZk
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where'(~r;t)istheoperator '(~r;t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(GmZVd3r0 (~r0;t)y (~r0;t) j~r)]TJ /F3 11.955 Tf 11.56 0 Td[(~r0j(2-88)whoseclassicalanalogisthegravitationalpotential.Wemayexpand (~r;t)inanyorthonormalandcompletesetofwavefunctionsu~k(~r;t),aswedidinSection 2.1 .Eqs. 2-6 2-10 remainunchangedexceptthat ~k3~k4~k~k2(t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(Gm2ZVd3rZVd3r01 j~r)]TJ /F3 11.955 Tf 11.56 0 Td[(~r0ju~k(~r;t)u~k2(~r0;t)u~k3(~r;t)u~k4(~r0;t)+u~k3(~r0;t)u~k4(~r;t) (2-89) issubstitutedforthesecondequationofEq. 2-10 2.3.1ClassicalDescriptionIntheclassicaldescription,theoperator (~r;t)isreplacedbyac-numberwavefunction(~r;t).ThewavefunctionsatisestheclassicalanalogofEq. 2-87 : i@t(~r;t)=)]TJ /F1 11.955 Tf 15.62 8.09 Td[(1 2mr2(~r;t)+m(~r;t)(~r;t)(2-90)with (~r;t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(GmZVd3r0j(~r0;t)j2 j~r)]TJ /F3 11.955 Tf 11.56 0 Td[(~r0j:(2-91)Thegravitationalpotential(~r;t)satisesthePoissonequation r2(~r;t)=4Gmj(~r0;t)j2:(2-92)LetusremarkhoweverthatEq. 2-92 impliesEq. 2-91 onlyuptoasolutionoftheLaplaceequation.TheSchrodinger-Poissonequations,Eqs. 2-90 and 2-92 arecommonlyusedtodescribeself-gravitatingdegenerateaxionsoraxion-likeparticles[ 8 ][ 9 ][ 10 ][ 11 ][ 12 ][ 13 ][ 14 ][ 15 ][ 16 ].TheywereusedinRef.[ 17 ]todescribethehomogeneousexpandinguniverseandtheevolutionofdensityperturbationstherein.WesummarizetheresultsofRef.[ 17 ]astheyarethestartingpointforouranalysisofthesystem'squantumevolutioninthenextsubsection. 40

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Thewavefunctionthatdescribesthehomogeneousexpandinguniverseis 0(~r;t)=p n0(t)ei 2mH(t)r2(2-93)whereH(t)istheHubbleexpansionrate.IndeedthevelocityeldimpliedbyEq. 2-93 is ~v(~r;t)=H(t)~r:(2-94)Furthermore,Eqs. 2-90 and 2-92 implythecontinuityequation @tn0+3Hn0=0(2-95)andtheFriedmannequation H(t)2+K a(t)2=8G 3mn0(t):(2-96)K=+1;0;)]TJ /F1 11.955 Tf 9.3 0 Td[(1dependingonwhethertheuniverseisclosed,criticaloropen,respectively,anda(t)isthescalefactordenedbyH(t)=_a=a.Equations 2-94 2-95 and 2-96 arethestandardequationsthatdescribethehomogeneousmatter-dominatedexpandinguniverse.Densityperturbationsareintroducedbywriting (~r;t)=0(~r;t)+1(~r;t):(2-97)TheperturbationinthewavefunctionisFouriertransformedintermsofco-movingwavevectors~kasfollows: 1(~r;t)=0(~r;t)Zd3k1(~k;t)ei~k~r a(t):(2-98)TheSchrodinger-Poissonequationsaresatisedtolinearorderprovided 1(~k;t)=1 2(~k;t)+ima(t)2 k2@t(~k;t)(2-99)and @2t(~k;t)+2H(t)@t(~k;t))]TJ /F1 11.955 Tf 11.95 0 Td[(4Gmn0(~k;t)+k4 4m2a(t)4(~k;t)=0:(2-100) 41

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The(~k;t)aretheFouriercomponentsofthedensitycontrast (~r;t)=n1(~r;t) n0(~r;t)=Zd3k(~k;t)ei~k~r a(t);(2-101)wheren1(~r;t)isthedensityperturbation.TheFouriercomponentsofthevelocityperturbationsaregivenby ~v1(~k;t)=ia(t)~k ~k~k@t(~k;t):(2-102)Equations 2-100 2-102 arethestandardequationsdescribingtheevolutionofdensityperturbationsinanexpandingmatterdominateduniverseexceptforthelastterminEq. 2-100 .Thattermisabsentifthematterisnon-degeneratecoldcollisionlessparticles,suchasWIMPsorsterileneutrinos.Itisduetotheeectofthe`quantumpressure'qinEqs. 2-16 and 2-18 whenthematterisawave.ItimpliesaJeanslength[ 18 ][ 19 ][ 20 ][ 21 ] lJ=(16Gm3n0))]TJ /F7 5.978 Tf 7.78 3.26 Td[(1 4=1:011014cm10)]TJ /F9 7.97 Tf 6.59 0 Td[(5eV m1 2 10)]TJ /F9 7.97 Tf 6.58 0 Td[(29g/cm3 mn0!1 4:(2-103)Fork>a(t) lJ=kJ,thedensityperturbationsoscillateintime.Forka(t) lJ,themostgeneralsolutionofEq. 2-100 is (~k;t)=A(~k)t t02 3+B(~k)t0 t;(2-104)inthecriticaluniversecase(K=0)wherea(t)/t2 3.Beforewediscussthequantumevolutionoftheinitiallyhomogeneousexpandinguniverse,letuspointoutthatthewavefunction0inEq. 2-93 satisesEq. 2-90 withthegravitationalpotential 0(~r;t)=2 3Gmn0(t)r2(2-105)whichisindeedanappropriatesolutionofthePoissonequation,Eq. 2-92 butwhichdiersfromEq. 2-91 byaconstantthatdivergesintheinnitevolumelimit.TheclassicaldescriptionaboveusestheSchrodinger-Poissonequations,Eqs. 2-90 and 2-92 .However,toobtainthequantumevolution,wewillnditmoreconvenienttostartwithasolution 42

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ofEqs. 2-90 and 2-91 .ThewavefunctionandgravitationalpotentialthatdescribethehomogeneousexpandinguniverseandsolveEqs. 2-90 and 2-91 are 0(~r;t)=p n0(t)ei 2mH(t)r2)]TJ /F4 7.97 Tf 6.58 0 Td[(imRtdt00(0;t0)0(~r;t)=2 3Gmn0(t)r2+0(0;t): (2-106) ThewavefunctiongiveninEqs. 2-106 isthestartingpointforourdiscussioninthenextsubsection. 2.3.2QuantumDescriptionInthissubsection,wederivethequantumevolutionofauniversethatstartsobeingdescribedbythehomogeneousexpandinguniversesolution0oftheclassicalequationsofmotion,Eqs. 2-106 .Againweareinterestedtoseehowlongtheclassicalsolutiongivesadescriptionconsistentwiththequantumevolution.WeusethegeneralmethodpresentedinSection 2.1 .Forageneralsolution(~r;t)oftheclassicaleldequation 2-90 ,weexpandthequantumeld (~r;t)aswedidfor4theory.Theinteractioncoecientsare ~k3~k4~k1~k2(t)=)]TJ /F3 11.955 Tf 10.49 8.08 Td[(Gm2 N2ZVd3rZVd3r0n(~r;t)n(~r0;t) j~r)]TJ /F3 11.955 Tf 11.56 0 Td[(~r0jhei(~k3)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k1)~(~r;t)+i(~k4)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k2)~(~r0;t)+ei(~k4)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k1)~(~r;t)+i(~k3)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k2)~(~r0;t)i (2-107) andthekineticcoecientsare M~k0~k(t)=)]TJ /F3 11.955 Tf 20.65 8.08 Td[(m NZVd3r(~r;t)n(~r;t)ei(~k0)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k)~(~r;t)+1 2mNZVd3rn(~r;t)~r(~k~(~r;t))~r(~k0~(~r;t))ei(~k0)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k)~(~r;t): (2-108) Equation 2-108 isobtainedbyfollowingthesamestepsasinAppendix A butforthegravitationalcase.Notethattheself-consistencyconditionEq. 2-28 isalwayssatised. 43

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Forthespecialsolution0(~r;t)describingahomogeneousexpandinguniverse, (~r;t)isexpandedintotheorthonormalcompletesetofwavefunctions u~k(~r;t)=1 p N0(~r;t)ei~k~r a(t)=r n0(t) Nei 2mH(t)r2)]TJ /F4 7.97 Tf 6.59 0 Td[(imRtdt00(0;t0)+i~k~r a(t):(2-109)Theu~k(~r;t)aresimilarto0butdierfromitbylongwavelengthmodulations.TheyhavethepropertiesdescribedbyEqs. 2-19 2-23 ,with~(~r;t)=~r=a(t).Wespecializehenceforthtothecriticaluniverse(K=0)forwhich a(t)=t t2 3andn(t)=nt t2(2-110)wheretisanarbitrarilychoseninitialtime.TheinteractioncoecientsinthebasisofEq. 2-109 areinthatcase ~k3~k4~k1~k2(t)=)]TJ /F1 11.955 Tf 10.49 8.09 Td[(4Gm2 Vt t2 3~k3+~k4~k1+~k2 1 (~k4)]TJ /F3 11.955 Tf 11.35 3.15 Td[(~k1)2+2+1 (~k3)]TJ /F3 11.955 Tf 11.35 3.15 Td[(~k1)2+2!(2-111)whereV=N=nisthevolumeoccupiedbythesystemattheinitialtimet,andisaninfraredcuto.Thekineticcoecientsare M~k0~k(t)= ~k~k0 2mt t4 3+2m 3t2t t2 31 (~k)]TJ /F3 11.955 Tf 11.35 3.15 Td[(~k0)2+2!~k0~k:(2-112)Theequationsofmotionforthea~k(t)operatorsandtheirclassicalanalogsA~k(t)arethesameasintheprevioussection,Eqs. 2-9 and 2-27 ,butwiththe~k3~k4~k1~k2(t)andM~k0~k(t)givenbytheaboveexpressions.TheconsistencyconditionEq. 2-28 issatisedsincetheFriedmannequationimplies 4Gmn=2 3t2:(2-113)Theconsistencyconditionensuresthat A~k(t)=p N~0~k;(2-114)whichdescribesthehomogeneousexpandinguniverse,isasolutionoftheclassicalequationsofmotion. 44

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Toanalyzethebehaviorofthequantumsystemwhenthehomogeneousparticlestate(~k=0)isoccupiedbyahugenumberNofquanta,wesubstitute a~k(t)=p N~0~k+b~k(t):(2-115)Theb~k(t)operatorssatisfycanonicalcommutationrelations,andtheequationsofmotion i@tb~k=M~k0~k+N~k0~0~k~0b~k0+1 2N~0~0~k~k0by~k0+:::= ~k~k 2mt t4 3)]TJ /F1 11.955 Tf 13.15 8.09 Td[(2m 3t2t t2 31 ~k~k!b~k)]TJ /F1 11.955 Tf 13.15 8.09 Td[(2m 3t2t t2 31 ~k~kby)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+::: (2-116) wherethedotsrepresentinteractiontermswhicharesuppressedbyoneortwofactorsof1=p N.Thesetermswillbeignoredhenceforth.Equation 2-116 maybewrittenas i@t0B@b~kby)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k1CA=0B@A(t)B(t))]TJ /F3 11.955 Tf 9.3 0 Td[(B(t))]TJ /F3 11.955 Tf 9.3 0 Td[(A(t)1CA0B@b~kby)]TJ /F4 7.97 Tf 6.16 2.11 Td[(~k1CA;(2-117)whereA(t)=(t))]TJ /F3 11.955 Tf 11.95 0 Td[((t),B(t)=)]TJ /F3 11.955 Tf 9.3 0 Td[((t)with (t)=k2 2mt t4 3;(t)=1 3t2t t2 32m k2:(2-118)Weperformatime-dependentBogoliubovtransformation 0B@b~kby)]TJ /F4 7.97 Tf 6.16 2.11 Td[(~k1CA=0B@c(t)s(t)s(t)c(t)1CA0B@~ky)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k1CA:(2-119)Thetransformationiscanonicalprovidedjc(t)j2)-251(js(t)j2=1,andprovidedc(t)ands(t)donotdependonthesignof~k.(The~kdependenceofA;B;;;c;sissuppressedtoavoidclutteringtheequationsunnecessarily.)Thenewoperatorssatisfy i@t0B@~ky)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k1CA=0B@A(t)B(t)B(t)A(t)1CA0B@~ky)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k1CA;(2-120) 45

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where A(t)=(jcj2+jsj2)A+(cs+cs)B)]TJ /F3 11.955 Tf 11.95 0 Td[(i(c_c)]TJ /F3 11.955 Tf 11.96 0 Td[(s_s)B(t)=2csA+(c2+s2)B)]TJ /F3 11.955 Tf 11.96 0 Td[(i(c_s)]TJ /F3 11.955 Tf 11.95 0 Td[(s_c): (2-121) TheJeanslength,Eq. 2-103 ,increasesast1 2,whereasthewavelengthassociatedwitheachwavevector~kincreasesasa(t)/t2 3.Hence,thereis,foreachwavevector~k,atimeoforder tk=k6t4 (2m)3(2-122)beforewhichtheperturbationswiththatwavevectorarestableandafterwhichtheyareunstable.Considermodesthataredeeplyintheunstableregimeatthetimetunderconsideration,i.e.tkt.ThesearethemodesthatobeyEq. 2-104 intheclassicaldescription.WemaysetA=0bychoosingc=coshu;s=sinhuwith tanh(2u)=)]TJ /F3 11.955 Tf 10.86 8.09 Td[(A B=)]TJ /F1 11.955 Tf 9.3 0 Td[(1+ :(2-123)Since,uislargeandnegative.Wehave,toleadingorder, u=1 4ln 2=1 4ln"3 2tk t2 3#;(2-124)andtherefore B=)]TJ 9.3 10.42 Td[(p B2)]TJ /F3 11.955 Tf 11.96 0 Td[(A2)]TJ /F3 11.955 Tf 11.96 0 Td[(i_u=)]TJ /F10 11.955 Tf 9.3 10.8 Td[(p 4Gmn0(t)+i 6t:(2-125)Theequationofmotionforthe~koperatorsisthus i@t~k= )]TJ /F10 11.955 Tf 9.3 18.79 Td[(r 2 3+i 6!1 ty)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k:(2-126)TheHamiltonianforthemodesofwavevector~kand)]TJ /F3 11.955 Tf 8.69 3.15 Td[(~k,withtkt,isthus H~k= )]TJ /F10 11.955 Tf 9.29 18.78 Td[(r 2 3+i 6!1 ty)]TJ /F4 7.97 Tf 6.16 2.11 Td[(~ky~k+ )]TJ /F10 11.955 Tf 9.3 18.78 Td[(r 2 3)]TJ /F3 11.955 Tf 14.08 8.08 Td[(i 6!1 t)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k~k:(2-127) 46

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Itmaybewrittenas H~k=)]TJ /F1 11.955 Tf 12.61 8.09 Td[(1 2t(~k~k+y~ky~k)+1 2t(0~k0~k+0y~k0y~k)(2-128)intermsofthecanonicalvariables ~k=1 p 2(~k+)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k)0~k=1 p 2(~k)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k) (2-129) andtheconstantq 2 3+i 6=jjeiwithjj=5 6andsin=1 5.Thephaseofcanbeabsorbedintoaredenitionofthe~kand0~koperators.WethusconsiderthedynamicsimpliedbyaHamiltonianoftheform H(t)= 2t)]TJ /F3 11.955 Tf 5.48 -9.68 Td[((t)(t)+(t)y(t)y(2-130)whereisarealpositiveconstant.Theequationofmotion i@t(t)= t(t)y(2-131)issolvedby (t)=1 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[((t))]TJ /F3 11.955 Tf 11.96 0 Td[(i(t)yt t+1 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[((t)+i(t)yt t:(2-132)Equation 2-132 impliesaninstability,albeitonlyapowerlawinstability.Toseeitsimplications,considertheevolutionofstatesintheSchrodingerpicture.TheSchrodingerpictureHamiltonianis Hs(t)= 2t)]TJ /F3 11.955 Tf 5.48 -9.69 Td[((t)(t)+(t)y(t)y;(2-133)andthetimeevolutionoperator U(t;t)=exp)]TJ /F3 11.955 Tf 9.3 0 Td[(i 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[((t)(t)+(t)y(t)ylnt t:(2-134) 47

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Wehave (t)=U(t;t)y(t)U(t;t):(2-135)Asanexample,considertheevolution js(t)i=U(t;t)js(t)i(2-136)ofthestatedenedby (t)js(t)i=0:(2-137)CombiningEqs. 2-135 2-137 and 2-132 ,wehave U(t;t)(t)U(t;t)yjs(t)i="1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[((t))]TJ /F3 11.955 Tf 11.95 0 Td[(i(t)yt t+1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[((t)+i(t)yt t#js(t)i=0: (2-138) Eq. 2-138 yieldsarecursionrelationbetweenthecoecientsintheexpansion js(t)i=1Xn=0cn(t)jni;(2-139)where jni=1 p n!)]TJ /F3 11.955 Tf 5.48 -9.68 Td[((t)ynj(t)i:(2-140)Therecursionrelationimplies cn(t)=0(fornodd)=()]TJ /F3 11.955 Tf 9.3 0 Td[(itanhs)ps (2p)]TJ /F1 11.955 Tf 11.96 0 Td[(1)!! 2pp!c0(t)(forn=2p); (2-141) wheresisdenedby e)]TJ /F4 7.97 Tf 6.59 0 Td[(s=t t:(2-142)Thenormalizationconditionhs(t)js(t)i=1yieldsthen jc0(t)j2=1 coshs:(2-143) 48

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Theaverageoccupationnumberandaverageoccupationnumbersquaredare hN(t)i=(t)j(t)y(t)j(t)=sinh2s;N(t)2=(t)j(t)y(t)(t)y(t)j(t)=sinh4s+2sinh2scosh2s: (2-144) Therootmeansquaredeviationoftheoccupationnumberfromitsaverageisthus N(t)=q hN(t)2i)-222(hN(t)i2=1 p 2sinh2s:(2-145)Boththeaverageoccupationnumberanditsrootmeansquaredeviationincreaseast t2=t t5 3for=jj=5 6. 2.3.3DurationofClassicalityTothelowestorderintheperturbations,thedensityoperatoris n(~r;t)= (~r;t)y (~r;t)=X~k;~k0u~k(~r;t)a~k(t)yu~k0(~r;t)a~k0(t)=Nu~0(~r;t)u~0(~r;t)+p NX~k6=0hu~0(~r;t)u~k(~r;t)b~k(t)+u~0(~r;t)u~k(~r;t)b~k(t)yi+O(1=N)=n0(t)+n0(t) p NX~k6=0b~k(t)+b)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k(t)yei~k~r a(t)+O(1=N): (2-146) Since b~k(t)+b)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k(t)y=(c(t)+s(t))(~k(t)+)]TJ /F4 7.97 Tf 6.16 2.11 Td[(~k(t)y)=eu p 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(~k(t)+0~k(t)+~k(t)y)]TJ /F3 11.955 Tf 11.96 0 Td[(0~k(t)y/t)]TJ /F7 5.978 Tf 7.78 3.25 Td[(1 6(t5 6andt)]TJ /F7 5.978 Tf 7.79 3.25 Td[(5 6)=t2 3andt)]TJ /F9 7.97 Tf 6.59 0 Td[(1; (2-147) weseethattheperturbationsgrowatthesamerateasintheclassicaldescription,Eq. 2-104 .Themaindierenceisthattheperturbationsareseededinthequantumdescription,whereasintheclassicaldescription,theyarenot. 49

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Theb~k(t)annihilationoperators(for~k6=0)aregivenintermsofthe~k(t)and0~k(t)by b~k(t)=c(t) p 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(~k(t)+0~k(t)+s(t) p 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(~k(t)y)]TJ /F3 11.955 Tf 11.95 0 Td[(0~k(t)y=1 2p 2t t5 6h(c(t))]TJ /F3 11.955 Tf 11.96 0 Td[(ieis(t)))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(~k(t)+0~k(t)+(s(t)+ie)]TJ /F4 7.97 Tf 6.59 0 Td[(ic(t)))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(~k(t)y)]TJ /F3 11.955 Tf 11.95 0 Td[(0~k(t)yi (2-148) whereweusedEqs. 2-119 and 2-129 ,andEq. 2-132 with=jj=5 6for(t)=ei=20~k(t)and(t)=iei=2~k(t).Wekeptgrowingtermsonly.Forttk,wehave c(t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(s(t)=1 2e)]TJ /F4 7.97 Tf 6.59 0 Td[(u=1 22 31 4t tk1 6(2-149)inviewofEq. 2-124 .Therefore b~k(t)=1 4p 22 31 4t t5 6t tk1 6h(1+iei))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(~k(t)+0~k(t))]TJ /F1 11.955 Tf 9.3 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(ie)]TJ /F4 7.97 Tf 6.59 0 Td[(i))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(~k(t)y)]TJ /F3 11.955 Tf 11.95 0 Td[(0~k(t)yi (2-150) fortk
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instatejiattimet.Weusedsin=1 5.Asanalternativetojiweconsideredthestatej0idenedby b~k(t)j0i=0(2-153)forall~k6=0,andveriedthattheaverageoccupationnumberforany~k6=0modeislargerinstatej0ithaninstatejiforlarget.Thestatejithushasthelargerdurationofclassicalityandisthestateweconsiderhenceforth.Instateji,thetotalnumberofquantathathaveleftthe~k=0condensateattimetis Nev(t)=XkkJ(t)d3k (2)32m k2t6#1:3VkJt3 202t t20:26NGm2p mtt t2; (2-154) whereweusedEqs. 2-103 2-113 and 2-122 .Theintegralover~kinEq. 2-154 shouldberestrictedtok>a(t)H(t)sincethemodesareunstableonlyforwavelengthsthatarewithinthehorizon.However,thisrestrictionisirrelevantsincetheintegralisdominatedbyvaluesofknearkJ(t).Afteratimeoforder tclt1 (Gm2p mt)1 2(2-155)the~k=0condensateislargelydepletedandtheclassicaldescriptionbecomesinvalid. 51

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CHAPTER3QUANTUMANDCLASSICALDESCRIPTIONSOFANINHOMOGENEOUSCONDENSATEWITHREPULSIVECONTACTSELF-INTERACTIONSInSection 2.2 ofthepreviouschapter,wehavecalculatedthedurationofclassicalityofahomogeneouscondensatewithattractivecontactself-interactions.Inthischapter,wedothesameforaninhomogeneouscondensatewithrepulsivecontactself-interactions.Thedynamicsofthesysteminthenon-relativisticregimeisdescribedbyEqs. 2-1 2-5 with>0. 3.1ClassicalDescription 3.1.1GenericFirstOrderSolutionsTheclassicalevolutionofthecondensateisgivenbytheSchrodinger-Gross-Pitaevskii(SGP)equation,Eq. 2-11 i@t=)]TJ /F1 11.955 Tf 15.62 8.09 Td[(1 2mr2+ 8m2jj2:(3-1)Thecomplexwavefunction(~x;t)describesauidofnumberdensityn(~x;t)andvelocityeld~v(~x;t)givenby n(~x;t)=j(~x;t)j2~v(~x;t)=~r)]TJ /F1 11.955 Tf 11.95 0 Td[(~r 2imjj2: (3-2) n(~x;t)and~v(~x;t)satisfythecontinuityequation,Eq. 2-15 andtheEuler-likeequation,Eq. 2-16 .Thehomogeneouscondensateisdescribedby 0(t)=p n0e)]TJ /F4 7.97 Tf 6.59 0 Td[(i!t;(3-3)wheren0isitsnumberdensityand !=n0 8m2:(3-4) basedonRef.[ 22 ]. 52

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Letusconsiderasmallperturbationabout0(t) (~x;t)=0(t)+1(~x;t)(3-5)andexpand1(~x;t)inFouriermodesas 1(~x;t)=0(t)X~kC~k(t)ei~k~x:(3-6)InsteadofworkingwithC~k(t)andC)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k(t),wedenes~k(t)andr~k(t)as s~k(t)=1 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(C~k(t)+C)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k(t)r~k(t)=1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(C~k(t))]TJ /F3 11.955 Tf 11.96 0 Td[(C)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k(t): (3-7) Thedenitionsimplys)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k(t)=s~k(t)andr)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k(t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(r~k(t).TheSGPequation,Eq. 3-1 ,impliestheequationsofmotionfors~k(t)andr~k(t) i@ts~k(t)=k2 2mr~k(t)+2!X~k0r~k)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k0(t)s~k0(t)i@tr~k(t)=k2 2m+2!s~k(t)+!X~k0)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(3s~k)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k0(t)s~k0(t))]TJ /F3 11.955 Tf 11.96 0 Td[(r~k)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k0(t)r~k0(t) (3-8) wherewehavekeptuptoquadratictermssincetheperturbationsaresmall.Inrstorder,thesolutionsofs~k(t)andr~k(t)aregivenby s~k(t)=d~ke)]TJ /F4 7.97 Tf 6.59 0 Td[(i!(~k)t+d)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~kei!(~k)t (3-9) andr~k(t)=2im k2@ts~k(t) (3-10) where !(~k)=s k2 2mk2 2m+2!:(3-11) 53

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Thecomplexconstants,d~kandd)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k,aredeterminedbytheinitialdensityandvelocityeld.Inrstorder,s~k(t)andr~k(t)arerelatedton(~x;t)and~v(~x;t)by 2s~k(t)=1 VZd3xn(~x;t))]TJ /F3 11.955 Tf 11.95 0 Td[(n0 n0e)]TJ /F4 7.97 Tf 6.58 0 Td[(i~k~x~kr~k(t)=1 VZd3xm~v(~x;t)e)]TJ /F4 7.97 Tf 6.58 0 Td[(i~k~x: (3-12) Oneshouldnotethat,incaseofrepulsiveinteractions,theperturbationsarestableandoscillatingforallmodesinrstorder,unlikethecaseofattractiveinteractions. 3.1.2OneModeInhomogeneityTostudythedepartureofthequantumevolutionfromtheclassicalone,weconsidertheinitialstatetobeahomogeneouscondensatewithplanewaveperturbationinonlyonemode~p.FollowingEq. 3-9 ,wechoosed~k=0forall~k6=~p.Intherstorderclassicaldescription,onlythemode(~k=~p)whichwasinitiallypresentpersistsandoscillateswithangularfrequency!(~p).Thedensityis n(~x;t)=n0)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(1+2d~pei(~p~x)]TJ /F4 7.97 Tf 6.59 0 Td[(!(~p)t)+2d~pe)]TJ /F4 7.97 Tf 6.58 0 Td[(i(~p~x)]TJ /F4 7.97 Tf 6.59 0 Td[(!(~p)t):(3-13)Now,d~pcanalwaysbeconsideredtoberealsinceitsphasecanbecanceledbyaredenitionoftheinitialtime.Whend~p=d~p,wehave n(~x;t)=n0(1+4d~pcos(~p~x)]TJ /F3 11.955 Tf 11.95 0 Td[(!(~p)t))and~v(~x;t)=!(~p) p2~p4d~pcos(~p~x)]TJ /F3 11.955 Tf 11.96 0 Td[(!(~p)t): (3-14) Therefore,4d~pistheamplitudeofthedensitycontrast(n(~x;t))]TJ /F3 11.955 Tf 11.95 0 Td[(n0)=n0.Inthesecondorderclassicaldescription,inadditionto~k=~pmode,themodewith~k=2~pappearsandoscillateswithangularfrequency2!(~p)andwithmagnitudeproportionalto(d~p)2: s(2)2~p(t)=(d~p)22m! P2[cos(2!(~p)t))]TJ /F1 11.955 Tf 11.95 0 Td[(1+isin(2!(~p)t)]:(3-15) 54

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Toallorder,onlymodeswith~k=j~p(j=1;2;3;:::)appearandoscillatewithangularfrequencyj!(~p)andwithamplitudeoforder(d~p)j.Inthenextsection,weshowthecontrastingquantumevolutionofthehomogeneouscondensatewithonemodeinhomogeneity. 3.2QuantumDescriptionInordertostudythedeviationofthequantumevolutionfromtheclassicalone,wewritetheeldoperator (~x;t)as (~x;t)=(~x;t)+(~x;t);(3-16)where(~x;t)istheclassicalwavefunctionfoundinthelastsectionand(~x;t)theoperatorcontainingallthequantumcorrections.ReplacingEq. 3-16 intoEq. 2-5 andneglectingnonlineartermsin,wend i@t=)]TJ /F1 11.955 Tf 15.61 8.09 Td[(1 2mr2+! n0)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(2jj2+2y:(3-17)Weexpandintheform (~x;t)=e)]TJ /F4 7.97 Tf 6.59 0 Td[(i!t p VX~k(6=~0)b~k(t)ei~k~x;(3-18)whereb~k(t)areoperatorssatisfyingequaltimecanonicalcommutationrelations [b~k(t);b~k0(t)]=0;[b~k(t);b~k0(t)y]=~k0~k(3-19)andVisthevolumeofspacewherethetheoryisdened.Theb~k(t)'ssatisfythefollowingequationsofmotion i@tb~k=k2 2m+!b~k+!by)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+2!X~k0hs~k0(t)2b~k)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k0(t)+by)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+~k0(t)+r~k0(t)by)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+~k0(t)i;(3-20)wheres~k(t)andr~k(t)aregivenbyEquations 3-6 and 3-7 55

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WeperformaBogoliubovtransformation 0B@b~kby)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k1CA=0B@u(~k)v(~k)v(~k)u(~k)1CA0B@~ky)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k1CA;(3-21)whereu(~k)2)]TJ /F3 11.955 Tf 12.01 0 Td[(v(~k)2=1,forthetransformationtobecanonical.Weassumeu(~k)andv(~k)toberealandwriteu(~k)=cosh~kandv(~k)=sinh~k.Choosing~ksuchthat tanh2~k=)]TJ /F3 11.955 Tf 24.76 8.09 Td[(! k2 2m+!;(3-22)Eq.( 3-20 )transformsto i@t~k(t)=!(~k)~k(t)+2!X~k0(6=~0)hP~k0~k(t)~k)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k0(t)+Q~k0~k(t)y)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+~k0(t)i;(3-23)where P~k0~k(t)=s~k0(t)sinh(~k+~k)]TJ /F4 7.97 Tf 6.16 2.11 Td[(~k0)+2cosh(~k+~k)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k0))]TJ /F3 11.955 Tf 11.96 0 Td[(r~k0(t)sinh(~k)]TJ /F3 11.955 Tf 11.96 0 Td[(~k)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k0) (3-24) Q~k0~k(t)=s~k0(t)cosh(~k+~k)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k0)+2sinh(~k+~k)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k0)+r~k0(t)cosh(~k)]TJ /F3 11.955 Tf 11.95 0 Td[(~k)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k0): (3-25) 3.2.1OneModeInhomogeneityNowweconsiderthesimplestcaseinwhichthereisaplanewaveperturbationonlyinmode~p,i.e. s~p(t)=s)]TJ /F4 7.97 Tf 6.69 0 Td[(~p(t)=d~pe)]TJ /F4 7.97 Tf 6.58 0 Td[(i!(~p)tandr~p(t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(r)]TJ /F4 7.97 Tf 6.69 0 Td[(~p(t)=d~p2m!(~p) p2e)]TJ /F4 7.97 Tf 6.58 0 Td[(i!(~p)t (3-26) andallothers~k(t)andr~k(t)arezero.Intermsofoperators~k(t)denedas ~k(t)=~k(t)ei!(~k)tforall~k;(3-27) 56

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Eq. 3-23 becomes i@t~k(t)=2!d~phP+(~k;)]TJ /F3 11.955 Tf 9.36 0 Td[(~p)ei(~k;)]TJ /F4 7.97 Tf 6.7 0 Td[(~p)t~k+~p(t)+P)]TJ /F1 11.955 Tf 7.08 1.8 Td[((~k;~p)ei(~k;~p)t~k)]TJ /F4 7.97 Tf 6.69 0 Td[(~p(t)+Q+(~k;)]TJ /F3 11.955 Tf 9.37 0 Td[(~p)ei(~k;)]TJ /F4 7.97 Tf 6.69 0 Td[(~p)t)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k)]TJ /F4 7.97 Tf 6.69 0 Td[(~p(t)y+Q)]TJ /F1 11.955 Tf 7.09 1.79 Td[((~k;~p)ei(~k;~p)t)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k+~p(t)yi (3-28) where P(~k;~k0)=sinh(~k)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k0+~k)+2cosh(~k)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k0+~k)sinh(~k)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k0)]TJ /F3 11.955 Tf 11.96 0 Td[(~k) cosh2~k0+sinh2~k0Q(~k;~k0)=cosh(~k)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k0+~k)+2sinh(~k)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k0+~k)cosh(~k)]TJ /F4 7.97 Tf 6.16 2.11 Td[(~k0)]TJ /F3 11.955 Tf 11.96 0 Td[(~k) cosh2~k0+sinh2~k0; (3-29) and (~k;~k0)=!(~k)+!(~k0))]TJ /F3 11.955 Tf 11.96 0 Td[(!(~k)]TJ /F3 11.955 Tf 11.36 3.15 Td[(~k0)(~k;~k0)=!(~k))]TJ /F3 11.955 Tf 11.95 0 Td[(!(~k0))]TJ /F3 11.955 Tf 11.95 0 Td[(!(~k)]TJ /F3 11.955 Tf 11.35 3.15 Td[(~k0)(~k;~k0)=!(~k)+!(~k0)+!(~k)]TJ /F3 11.955 Tf 11.35 3.16 Td[(~k0)(~k;~k0)=!(~k))]TJ /F3 11.955 Tf 11.95 0 Td[(!(~k0)+!(~k)]TJ /F3 11.955 Tf 11.36 3.15 Td[(~k0): (3-30) (~k;)]TJ /F3 11.955 Tf 9.36 0 Td[(~p),(~k;~p)and(~k;~p)approaching0implycertainscatteringprocessesofthequasi-particleslabeledbymomentum~kandenergy!(~k)asshowninSub-sections 3.2.1.1 and 3.2.1.2 .Theprocessesare Process1:(~k;)]TJ /F3 11.955 Tf 9.36 0 Td[(~p)=0=)~0+(~k+~p)$~k+~pProcess2:(~k;~p)=0=)~0+~k$~p+(~k)]TJ /F3 11.955 Tf 12.02 0 Td[(~p)Process3:(~k;~p)=0=)~0+~p$~k+(~p)]TJ /F3 11.955 Tf 11.35 3.16 Td[(~k):(~k;)]TJ /F3 11.955 Tf 9.36 0 Td[(~p)neverapproaches0unless~k=~p=~0whichisphysicallyuninterestingbecauseitimpliesscatteringprocess,~0+~0$~0+~0.Inourinitialstate,onlythequasi-particlestates 57

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ofmomentum~0and~pareoccupied.Inthatcase,onlyprocess3isimportantasweshowinthenextsubsection. 3.2.1.1Processes1and2:OscillationsConsidermodes~ksuchthat(~k;)]TJ /F3 11.955 Tf 9.36 0 Td[(~p)0.Forthose~k's,(~k;~p),(~k;)]TJ /F3 11.955 Tf 9.36 0 Td[(~p)and(~k;~p)cannotbezeroandarealwaysmuchlargerthan(~k;)]TJ /F3 11.955 Tf 9.37 0 Td[(~p).IgnoringthefastoscillatingtermsinEq. 3-28 ,weget i@t~k(t)2!d~pP+(~k;)]TJ /F3 11.955 Tf 9.37 0 Td[(~p)ei(~k;)]TJ /F4 7.97 Tf 6.7 0 Td[(~p)t~k+~p(t):(3-31)As~kiscoupledwith~k+~p,usingEqs. 3-23 and 3-27 ,wewritetheequationfor~k+~p(t) i@t~k+~p(t)2!d~pP)]TJ /F1 11.955 Tf 7.08 1.79 Td[((~k+~p;~p)ei(~k+~p;~p)t~k(t)(3-32)wherewehaveignoredfastoscillatingterms.Since(~k+~p;~p)=)]TJ /F3 11.955 Tf 9.3 0 Td[((~k;)]TJ /F3 11.955 Tf 9.36 0 Td[(~p)0andP)]TJ /F1 11.955 Tf 7.08 1.79 Td[((~k+~p;~p)=P+(~k;)]TJ /F3 11.955 Tf 9.36 0 Td[(~p),Eqs. 3-31 and 3-32 implythat @2t~k)]TJ /F3 11.955 Tf 11.95 0 Td[(i(~k;)]TJ /F3 11.955 Tf 9.36 0 Td[(~p)@t~k+2!d~pP+(~k;)]TJ /F3 11.955 Tf 9.36 0 Td[(~p)2~k0(3-33)whichshowsthat,for~kwith(~k;)]TJ /F3 11.955 Tf 9.36 0 Td[(~p)0,~koscillateswithangularfrquencyj2!d~pP+(~k;)]TJ /F3 11.955 Tf 9.37 0 Td[(~p)j.Similarly,onecanshowthat,for~kwith(~k;~p)0,~koscillateswithangularfrquencyj2!d~pP)]TJ /F1 11.955 Tf 7.08 1.79 Td[((~k;~p)j.Processes1and2occursinthebathofquasiparticlesonlyasaresultofquantumuctuations. 3.2.1.2Process3:ParametricresonanceConsidermodeslabeledby~ksuchthat(~k;~p)0.UsingEq. 3-28 andignoringthefastoscillatingterms,weget i@t~k(t)2!d~pQ)]TJ /F1 11.955 Tf 7.09 1.79 Td[((~k;~p)ei(~k;~p)t)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+~p(t)y:(3-34)As~kiscoupledwith)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+~p(t)y,usingEqs. 3-23 and 3-27 ,wewritetheequationfor)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k+~p(t)y: ()]TJ /F3 11.955 Tf 9.29 0 Td[(i)@t)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k+~p(t)y2!d~pQ)]TJ /F1 11.955 Tf 7.09 1.79 Td[(()]TJ /F3 11.955 Tf 8.7 3.16 Td[(~k+~p;~p)e)]TJ /F4 7.97 Tf 6.59 0 Td[(i()]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k+~p;~p)t~k(t)(3-35) 58

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wherewehaveignoredfastoscillatingtermsandused()]TJ /F3 11.955 Tf 8.69 3.16 Td[(~k+~p;~p)=(~k;~p)0,andQ)]TJ /F1 11.955 Tf 7.09 1.8 Td[(()]TJ /F3 11.955 Tf 8.7 3.15 Td[(~k+~p;~p)=Q)]TJ /F1 11.955 Tf 7.08 1.8 Td[((~k;~p).Tomakethecoecientstimeindependent,weperformthefollowingtransformation ~~k(t)=e)]TJ /F8 5.978 Tf 7.78 3.26 Td[(it 2~k(t)~)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k+~p(t)y=eit 2)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+~p(t)y (3-36) where=()]TJ /F3 11.955 Tf 8.7 3.15 Td[(~k+~p;~p)=(~k;~p).Equations 3-34 and 3-35 imply i@t0B@~~k~y)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k+~p1CA=1 20B@ 22!d~pQ)]TJ /F1 11.955 Tf 9.3 0 Td[(2!d~pQ)]TJ /F4 7.97 Tf 10.9 4.71 Td[( 21CA0B@~~k~y)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k+~p1CA(3-37)whereQ=Q)]TJ /F1 11.955 Tf 7.08 1.8 Td[(()]TJ /F3 11.955 Tf 8.69 3.15 Td[(~k+~p;~p)=Q)]TJ /F1 11.955 Tf 7.09 1.8 Td[((~k;~p).Tosolvetheequationabove,weperformanotherBogoliubovtransformation 0B@~~k~y)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k+~p1CA=0B@cosh~sinh~sinh~cosh~1CA0B@~~k~y)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k+~p1CA:(3-38)When~isgivenby tanh2~=)]TJ /F3 11.955 Tf 9.3 0 Td[( 4!d~pQ;(3-39)theoperators~'ssatisfy i@t~~k=~y)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+~pi@t~y)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k+~p=)]TJ /F3 11.955 Tf 9.3 0 Td[(~~k (3-40) where =r (2!d~pQ)2)]TJ /F10 11.955 Tf 11.95 13.27 Td[( 22:(3-41)Thesolutionoftheoperatorequation 3-40 is ~~k(t)=cosht~~k(0))]TJ /F3 11.955 Tf 11.96 0 Td[(isinht~y)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+~p(0)~y)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+~p(t)=cosht~y)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+~p(0)+isinht~~k(0): (3-42) 59

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Thesolutionsshowthatmodes~kand()]TJ /F3 11.955 Tf 8.7 3.16 Td[(~k+~p)areunstableifisreal.Theinstabilitywindowisgivenby )-222(j2!d~pQj< 2
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UsingthereversetransformationsofEqs. 3-38 3-36 and 3-27 ,onemaywrite ~k(t)=e)]TJ /F4 7.97 Tf 6.59 0 Td[(i!(~k)t+it 2h(cosht+isinh2~sinht)~k(0))]TJ /F3 11.955 Tf 19.26 0 Td[(icosh2~sinht)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k+~p(0)yi)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k+~p(t)y=ei!(~k)]TJ /F4 7.97 Tf 6.69 0 Td[(~p)t)]TJ /F8 5.978 Tf 7.78 3.26 Td[(it 2hicosh2~sinht~k(0)+(cosht)]TJ /F3 11.955 Tf 11.95 0 Td[(isinh2~sinht))]TJ /F4 7.97 Tf 6.16 2.11 Td[(~k+~p(0)yi: (3-44) Ontheotherhand,since()]TJ /F3 11.955 Tf 9.37 0 Td[(~p)modewasnotoccupiedinitially,~k)]TJ /F4 7.97 Tf 6.69 0 Td[(~p(t)and)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k(t)ydonotexhibitanyparametricresonance: ~k)]TJ /F4 7.97 Tf 6.7 0 Td[(~p(t)=e)]TJ /F4 7.97 Tf 6.58 0 Td[(i!(~k)]TJ /F4 7.97 Tf 6.7 0 Td[(~p)t~k)]TJ /F4 7.97 Tf 6.7 0 Td[(~p(0))]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k(t)y=ei!(~k)t)]TJ /F4 7.97 Tf 6.16 2.11 Td[(~k(0)y: (3-45) Equations 3-21 3-44 and 3-45 canbeusedtondthesolutionofb~k(t) b~k(t)=cosh~ke)]TJ /F4 7.97 Tf 6.58 0 Td[(i!(~k)t+it 2h(cosht+isinh2~sinht))]TJ /F1 11.955 Tf 5.48 -9.68 Td[(cosh~kb~k(0))]TJ /F1 11.955 Tf 11.96 0 Td[(sinh~kb)]TJ /F4 7.97 Tf 6.17 2.1 Td[(~k(0)y)]TJ /F3 11.955 Tf 9.3 0 Td[(icosh2~sinht)]TJ /F1 11.955 Tf 11.29 0 Td[(sinh~k)]TJ /F4 7.97 Tf 6.7 0 Td[(~pb~k)]TJ /F4 7.97 Tf 6.69 0 Td[(~p(0)+cosh~k)]TJ /F4 7.97 Tf 6.69 0 Td[(~pb)]TJ /F4 7.97 Tf 6.16 2.1 Td[(~k+~p(0)yi+sinh~kei!(~k)t)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F1 11.955 Tf 11.29 0 Td[(sinh~kb~k(0)+cosh~kb)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k(0)y: (3-46) Unliketheclassicalevolution,inthequantumevolution,modeswith~ksatisfying!(~p)=!(~k)+!(~p)]TJ /F3 11.955 Tf 11.36 3.16 Td[(~k)growexponentially. 3.2.2DurationofClassicalityIntheclassicalevolution,onlymodeswith~k=j~p(j=1;2;3;:::)appearandoscillatewithangularfrequencyj!(~p).However,inthequantumevolution,modeswith~ksatisfying!(~p)=!(~k)+!(~p)]TJ /F3 11.955 Tf 11.89 3.15 Td[(~k)areunstableandtheiroccupationnumbersgrowexponentiallyasthequasi-particlesscatterfrommodes~0and~pintomodes~kand(~p)]TJ /F3 11.955 Tf 11.53 3.15 Td[(~k).Thedurationofclassicality(tcl)isdenedasthetimeafterwhichalmostallofthequantahaveleftthe~k=~0and~k=~pmodes.Thistime(tcl)dependsontheinitialstateofthesystem. 61

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Thedurationofclassicalityislongestwhentheinitialstateisjisuchthat ~~k(0)ji=~)]TJ /F4 7.97 Tf 6.16 2.11 Td[(~k+~p(0)ji=0:(3-47)Theoccupationnumberofamode~kwithintheinstabilitywindow(seeEq. 3-43 )isgivenby N~k(t)=hjby~k(t)b~k(t)ji=cosh2~k)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(sinh2~cosh2t+cosh2~sinh2t+1 2sinh2~k (3-48) where~k,~andaredenedinEqs. 3-22 3-39 and 3-41 .Aftersucientlylongtimesuchthatt1, N~k(t)1 4cosh2~cosh2~ke2t:(3-49)Thetotalnumberofquantathathaveleftthe~k=~0and~k=~pmodesaftertimetisgivenby Nev(t)=X~k(6=~0;~p)N~k(t)=V 42Zk2dkdN~k(t)(3-50)where=coswithbeingtheanglebetween~kand~p.Wechangethevariablesfrom(k;)to(!1;!2)where!1=!(~k)and!2=!(~k)]TJ /F3 11.955 Tf 12.02 0 Td[(~p).Theintegralnowtakestheform Nev(t)=V 42Zd!1d!22mq !21+!2)]TJ /F3 11.955 Tf 11.96 0 Td[(!@(k;) @(!1;!2)N~k(t):(3-51)Theintegralgetscontributionsonlyfromtheunstablemodes.AsN~k(t)growsexponentially(seeEq. 3-48 ),weusethesaddle-pointapproximation.hasamaximumwhen!1=!2and=0intherstorder.Thisleadsto!1=!2=!(~p) 2.Aftertediousbutstraightforwardalgebra,weobtain Nev(t)=N n01 !t(m!)3 2F()eQ()!tn n0(3-52) 62

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where=!(~p) !andF()isadimensionlessfunctionofgiveninAppendix B andshowninFig. 3-2 .Q()isgivenby Q()=1 p 2+4+p 2+1)]TJ /F1 11.955 Tf 11.95 0 Td[(3:(3-53) Figure3-2. PlotofF()whichisadimensionlessfunctionof=!(~p)=!. ThedurationofclassicalitytclisthetimetsuchthatNev(t)N: tcl1 Q()!n n0ln 1 F()n0 (m!)3 2!:(3-54)Thedurationofclassicalityisinverselyproportionaltotheinhomogeneityofthecondensate.TheargumentofthelogarithmisproportionaltoNforaxedvolume.Forthesimplestcaseofinhomogeneouscondensatewithrepulsiveinteractions,thedurationofclassicalityscalesaslnN (n=n0).. 63

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CHAPTER4ASTROPHYSICALSIGNATURESOFCAUSTICRINGS:PREDICTIONSInthischapter,weintroducecausticringsofdarkmatterandbrieyexplainhowtheyarerelatedtothequantumeectsindarkmatteraxions.InSections 4.3 and 4.4 ,westudyhowthedistributionsofstarsandofinterstellargasareaectedwhenacausticringofdarkmatterpassesthroughthem. 4.1CausticRingsasManifestationsoftheQuantumBehaviorofAxions 4.1.1CausticRingsintheDarkMatterHaloSincedarkmatterparticlesarecollisionless,theyaredescribedinsix-dimensionalphasespace.Inthelimitofzerovelocitydispersion,darkmatterparticleslieonathree-dimensionalhypersurfaceembeddedinsix-dimensionalphasespace.Oneinevitableconsequenceofthisistheformationofcaustics[ 24 ][ 25 ][ 26 ][ 27 ][ 28 ][ 29 ].Astheparticlesinagalactichaloarehugeinnumber,theycanbelabeledbyasetofthreecontinuousparameters,~=(1;2;3).Let~x(~;t)bethepositionofparticle~attimet.Foranarbitraryphysicalpoint~r,lettheequation~x(~;t)=~rhavesolutions~j(~r;t)withj=1;2;:::;Nf(~r;t).Nf(~r;t)isthenumberofowsthrough~rattimet.Ifd3N d3(~)isthenumberdensityofparticlesinthechosenparameterspace,themassdensityinphysicalspaceisgivenby[ 27 ] d(~r;t)=mNf(~r;t)Xj=1d3N d3(~)1 jD(~;t)j~=~j(~r;t)(4-1)wheremisthemassofeachparticleandjD(~;t)jjdet@~x(~;t) @~jistheJacobianofthemapfrom~to~x.Causticsarelocationsinphysicalspacewherethedensityd(~r;t)divergesbecausethemapissingular,i.e.D(~;t)=0. partofthischapterisbasedonRef.[ 23 ]. 64

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Inagalactichalo,bothouterandinnercausticsareformed.Asoutercausticsappearintheouterregionsofthegalactichalo(e.g.atadistanceoforder100kpcfromtheMilkyWaycenter),theyhardlyaectthestellardynamicsinthedisk.Hereweonlydiscusstheinnercausticswhichformwhentheparticlesareattheirclosestapproachtothegalacticcenter(seeFig. 4-1 ).Thenthinnercausticformsintheowofparticlesexperiencingtheirnthinfallinthegalacticpotentialwell.Ifthetotalangularmomentumofdarkmatterparticlesisdominatedbyanetoverallrotation,eachinnercausticisaclosedtubewhosecrosssectionhastheshapeofatricusp[ 30 ][ 27 ](seeFig. 4-2 ).Thisstructureiscalledacausticringofdarkmatter.Thecausticringmodel[ 27 ][ 29 ][ 31 ]isaproposalforthefullphasespacedistributionofcolddarkmatterhalos.Itisaxiallysymmetric,reectionsymmetricandhasself-similartimeevolution.Itpredictsthatcausticringslieinthegalacticplaneandthattheirradiian(t)increaseoncosmologicaltimescaleasan(t)/t4=3.Thereisobservationalevidenceinsupportofthecausticringmodel[ 32 ][ 33 ].Furthermore,itwasshownthatnetoverallrotation,self-similarityandaxialsymmetryaretheexpectedoutcomesoftherethermalizationofBose-Einsteincondensedaxiondarkmatter[ 18 ][ 4 ]beforeitfallsontoagalactichalo[ 34 ][ 2 ].IntheMilkyWay,thepresentradiusofthenthcausticringisapproximately40kpc n.SincetheSolarSystemisabout8:5kpcawayfromthegalacticcenter[ 35 ],then=1,2,3and4causticringshavepassedthesolarorbitwhilethen=5causticringisapproaching. 4.1.2StructureoftheInnerCausticsandVelocityFieldSincedarkmatterparticlesfallinandoutofthegravitationalpotentialwellofthegalaxymanytimes,thereisasetofinnerandoutercaustics.Innercausticsformwheretheparticlesareclosesttothegalacticcenter,whileoutercausticsoccurwheretheyareattheirouterturnaroundbeforefallingbackin.Theshapeandstructureoftheinnercausticsaredeterminedbythevelocityeldoftheparticlesattheirlastturnaround,i.e.atthetimewhentheirradiallyoutwardHubbleowwaslaststoppedbythegravitational 65

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attractionofthegalaxy.Ashellcontainingsuchparticlesiscalledtheturnaroundsphere.Iftheinstantaneousvelocityeldontheturnaroundsphereisdominatedbynetoverallrotation,with~r~v6=~0,theinnercaustictakestheshapeofaclosedtubewhosecross-sectionhasthreecusps(seeFig. 4-2 )[ 29 ].Suchinnercausticiscommonlyreferredasacausticring.Ontheotherhand,ifthevelocityeldisirrotational(~r~v=~0),theinnercaustichasatent-likestructurewhichisquitedistinctfromacausticring[ 29 ].AswediscussinChapter 5 ,thereisevidenceforthecausticrings,notthetent-likestructures,intheMilkyWayandinotherisolatedspiralgalaxies. 4.1.3ClassicalTheoryisInadequateConventionaldarkmatterparticleslikeWIMPsandsterileneutrinosbehaveascollisionlessclassicaluidsonthelengthscalesofobservationalinterests.Theirvelocityeldsatises @~v @t(~r;t)+(~v(~r;t)~r)~v(~r;t)=)]TJ /F3 11.955 Tf 10.43 3.02 Td[(~r(~r;t)(4-2)where(~r;t)isthegravitationalpotential.Thevelocityeldofdarkmatterparticlesattheirrstturnaroundisexpectedtobeirrotationalbecausetherotationalmodesdecaywiththeexpansionoftheuniverse.Equation 4-2 impliesthat,if~r~v=~0initially,itiszeroatalllatertimes.Theinitialvelocityeldisirrotationalevenwhentheparticlesinagalaxygainangularmomentumviatidaltorquingfromnearbyprotogalaxiesduringearlystagesofstructureformation[ 29 ].Therefore,conventionalcolddarkmatterparticleswouldyieldatent-likestructurefortheinnercaustics,whereastheobservationsimplytheexistenceofcausticrings.Thepuzzleisresolvedifcolddarkmatteriscomposed,atleastinpart,byarethermalizingaxionBose-Einsteincondensate(BEC)[ 18 ]. 4.1.4QuantumBehaviorofDarkMatterAxionsUnlikeothercolddarkmatterparticles,axionsoraxion-likeparticleshavehugequantumdegeneracy(1061forQCDaxionofmasseV)andverysmallvelocitydispersion(10)]TJ /F9 7.97 Tf 6.59 0 Td[(17c).IthasbeenshowninRef.[ 18 ]thattheaxionsformaBECbecausetheyareidenticalbosonswithconservedtotalnumber,theyarehighlycondensedinphase 66

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spaceandtheythermalize.Whentheythermalize,almostallofthemgotothelowestenergystateavailablethroughthethermalizinginteractions(seeChapter 1 foradetaileddiscussion).AxionsarenotcollisionlessinthecontextofEq. 4-2 becausetheyinteractwitheachothergravitationally[ 4 ].Thermalizationofaxionsiscausedbythegravitationalscatteringbetweenmodesofverylowrelativemomenta.Bose-Einsteincondensationisaquantumphenomenonandcannotbedescribedclassically.AxionsformaBECwhenthephotontemperaturereachesoforder500eV(f=1012GeV)1 2,wherefistheaxiondecayconstant[ 18 ][ 34 ].ItwasshowninRef.[ 18 ]that,atthistime,thethermalizationrateviagravitationalself-interactionsbecomeslargerthantheHubbleexpansionrate.Becausetheaxionscontinuallyrethermalize,theytrackthelowestenergystateavailablewhiletheuniverseexpands.Thelowestenergyforaxedtotalangularmomentumacquiredfromtidaltorquingisachievedwhenthevelocityeldoftheturnaroundsphereisthatofrigidrotation[ 2 ][ 18 ].Therefore,thenetoverallrotationrequiredforcausticringformationisaccommodatedwhenthedarkmatterisarethermalizingaxionBEC.Nootherphenomenonisknowntoproducesuchalargenetoverallrotation.Correctionsfromgeneralrelativitymaygiveanon-zerocurltothevelocityeld.However,thosecorrectionsaretoosmalltoproducethecausticringsforwhichobservationalevidencehavebeenfound[ 34 ]. 4.2CausticRings 4.2.1FlowsNearaCausticRingEachparticleinanaxiallysymmetricowofcolddarkmatterislabeledbytwoparameters(;).Thethirdparameterlabelingaparticleisitsazimuthwhichisirrelevantinthecaseofaxialsymmetry.istheanglefromthez=0planeatthetimeoftheparticle'smostrecentturnaround,i.e.= 2)]TJ /F3 11.955 Tf 12.15 0 Td[(whereisthepolarangleinsphericalcoordinates.Foreachvalueof,=0isdenedasthetimewhentheparticlecrossesthez=0plane,i.e.z(;=0)=0.Thecoordinates(;z)ofaparticlenearthe 67

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causticringaregivenby[ 27 ] =a+1 2u()]TJ /F3 11.955 Tf 11.96 0 Td[(0)2)]TJ /F1 11.955 Tf 13.15 8.09 Td[(1 2s2 (4-3) z=b (4-4) wherea;u;0;sandbareconstantsforagivencaustic.ThecausticoccurswheretheJacobianjD2(;)jjdet@(;z) @(;)jis0.Itslocationinthe-zplaneasafunctionoftheparameterisgivenby =a+1 2u()]TJ /F3 11.955 Tf 11.95 0 Td[(0)(2)]TJ /F3 11.955 Tf 11.96 0 Td[(0) (4-5) z=br u s3(0)]TJ /F3 11.955 Tf 11.96 0 Td[(): (4-6) Equations( 4-5 )and( 4-6 )denethetricusp(seeFigs. 4-1 and 4-2 ).Itssizeispinthe-directionandqinthez-direction:p=1 2u20;q=p 27 4b p usp.Thedensityinphysicalspaceis[ 27 ] d(;z)=1 Nf(~r;t)Xj=1dM dd(;)cos jD2(;)jj(;z);j(;z);(4-7)wheredM ddisthemassofdarkmatterparticlesfallinginperunitsolidangleperunittime. 4.2.2Self-similarityTheevolutionofdarkmatterparticlesinagalactichaloisself-similarifthereisnospecialtimeinitshistory[ 36 ][ 37 ][ 38 ].Self-similarityimpliesthatthephasespacedistributionremainsidenticaltoitselfexceptforanoverallrescalingofitsdensity,sizeandvelocity.AspointedoutinRef.[ 31 ],self-similaritydoesnotrequireanysymmetry.Intheself-similarmodel,theradiusofeachcausticringincreasesoncosmologicaltimesaleasa(t)/twhere=2 3+2 9.Fromtheslopeofthepowerspectrumofdensityperturbationongalacticscales,isdeterminedtobeintherange0:25<<0:35[ 39 ].Thevalue=1 3(hence=4 3)isconsistentwiththeobservationalevidenceforcausticrings,theslopeofthepowerspectrumongalacticscalesandwiththevaluefromtidaltorquing[ 34 ]. 68

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Figure4-1. Flowsofcolddarkmatterparticlesina-zcrosssectionofacausticring.XandZarerescaledcoordinatesX=)]TJ /F4 7.97 Tf 6.59 0 Td[(a pandZ=z p. Figure4-2. TheenvelopeofthetrajectoriesshowninFig. 4-1 .Theshapeinthisgureisusuallycalledatricusp.Theradiusaandthetransversesizes(p;q)areindicatedinthegure.ThegureisreprintedwithpermissionfromRef.[ 31 ]. 69

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4.2.3GravitationalFieldofaCausticRingThegravitationaleldinsideandnearacausticringhasbeencalculatedinRefs.[ 27 ]and[ 40 ].Assumingthesize(p;q)ofthecrosssectionoftheringtobemuchsmallerthantheradiusaofthering,thegravitationaleldintermsoftherescaledvariables)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(X=)]TJ /F4 7.97 Tf 6.59 0 Td[(a p;Z=z pisgivenby ~g=)]TJ /F1 11.955 Tf 9.3 0 Td[(8GdM dd1 b[I(X;Z)^+Iz(X;Z)^z];(4-8)where I=1 2Z1dAZ1dTX)]TJ /F1 11.955 Tf 11.95 0 Td[((T)]TJ /F1 11.955 Tf 11.95 0 Td[(1)2+A2 (X)]TJ /F1 11.955 Tf 11.95 0 Td[((T)]TJ /F1 11.955 Tf 11.95 0 Td[(1)2+A2)2+(Z)]TJ /F1 11.955 Tf 11.95 0 Td[(2AT)2;Iz=1 2Z1dAZ1dT(Z)]TJ /F1 11.955 Tf 11.96 0 Td[(2AT) (X)]TJ /F1 11.955 Tf 11.95 0 Td[((T)]TJ /F1 11.955 Tf 11.95 0 Td[(1)2+A2)2+(Z)]TJ /F1 11.955 Tf 11.95 0 Td[(2AT)2and=su b2.Thevariables(;)havebeenreplacedby)]TJ /F3 11.955 Tf 5.47 -9.69 Td[(A=b u0;T= 0.InRef.[ 40 ],wastakentobeunityandtheintegralswerecalculatedbothanalytically,usingresiduetheory,andnumericallywithconsistentresults.Equation( 4-8 )givesthegravitationaleldofthewholeowformingthecaustic.Thegravitationaleldduetotheowswithoutcausticsisthatofasmoothhalowhichresultsinaatrotationcurve.Weareinterestedonlyinthemodicationofthegravitationaleldduetotheformationofthecaustic.Inthecausticringmodel,onecannotremoveaowwithoutremovingthecausticintheow.Toseparatethegravitationaleldofacausticringfromthatoftheowofwhichitispart,weintroducealongdistancedampingfactor, ~gc(;z)=exp)]TJ /F3 11.955 Tf 14.9 8.09 Td[(s2 R2~g(;z)(4-9)wheres2=()]TJ /F3 11.955 Tf 10.99 0 Td[(a)]TJ /F4 7.97 Tf 12.18 5.25 Td[(p 4)2+z2isthedistanceofthepoint(;z)fromthecenterofthecausticandR1:5pisthedistancescaleoverwhichtheeectsofthecausticringaresignicant.TheparametersforvariouscausticringshavebeenlistedinRef.[ 31 ].ThevaluesfortheinfallratedM ddgiveninRef.[ 31 ]arebasedontheassumptionofisotropicinfallofdarkmatterparticles.However,Ref.[ 2 ]arguesthat,becausethevorticesintheaxion 70

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Bose-Einsteincondensateattracteachother,numeroussmallervorticesjointoformahugevortexalongtherotationaxisofthegalaxy.Asaresult,axionsfallinpreferentiallyalongthegalacticplaneandcausticringsareenhanced.ThisexplainswhythebumpsintheMilkyWayrotationcurveatthelocationsofthecausticringsaretypicallyafactorof4largerthanthatattributedonlytothecausticringswithisotropicinfall[ 32 ].WethereforemultiplytheinfallratesinRef.[ 31 ]byafactor4toaccountfortheformationofthe`bigvortex',givingthemthevalues dM ddn:n=1;2;3;4;5;:::(210;95;60;40;32;:::)M sterad-yr:(4-10)Unlessmentionedotherwise,weusethefollowingparameterstodescribeacausticringpassingthroughthesolarneighborhood:p=0:5kpc,b=523km/s,V=da dt=1kpc/GyranddM dd=32M sterad-yr.InFig. 4-3 ,weshowtheradialandverticalcomponents(gc,gcz)ofthegravitationaleldatZ=z p=0:25asafunctionofX=)]TJ /F4 7.97 Tf 6.59 0 Td[(a p. 4.2.4GravitationalPotentialofaCausticAssumingthesizeofthecrosssectionofacausticringtobemuchsmallerthanitsradius(i.e.p;qa;seeFig. 4-2 ),thegravitationalpotential(X;Z)nearthecausticringisgivenby (X;Z))]TJ /F1 11.955 Tf 11.96 0 Td[((X0;Z0)=2GdM ddp baJ(X;Z;X0;Z0)(4-11)where J(X;Z;X0;Z0)=Z1dAZ1dTln(X)]TJ /F1 11.955 Tf 11.96 0 Td[((T)]TJ /F1 11.955 Tf 11.96 0 Td[(1)2+A2)2+(Z)]TJ /F1 11.955 Tf 11.96 0 Td[(2AT)2 (X0)]TJ /F1 11.955 Tf 11.96 0 Td[((T)]TJ /F1 11.955 Tf 11.96 0 Td[(1)2+A2)2+(Z0)]TJ /F1 11.955 Tf 11.95 0 Td[(2AT)2:(4-12)Wechoosethecenterofthetricuspasthereferencepoint(X0;Z0).Figure 4-4 showsthetwo-dimensionalplotofthecausticpotential c(X;Z)=(X;Z))]TJ /F1 11.955 Tf 11.96 0 Td[((X0=0:25;Z0=0):(4-13)Itlookssmoothandcontinuous,althoughitssecondderivativer2cdivergesatthecaustic. 71

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A BFigure4-3. Theradialandverticalcomponentsofgravitationaleld~gc(X;Z=0:25)[seeEq.( 4-9 )]duetoacausticringasafunctionofX=)]TJ /F4 7.97 Tf 6.59 0 Td[(a pforZ=z p=0:25.Theparametersofthecausticringhavebeenchosenasa=8:0kpc,p=0:5kpc,b=523km/s,dM dd=32M sterad.yr.Thechosenparametersaresimilartothatofthen=5causticring,theoneclosesttous. 72

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Figure4-4. Two-dimensionalplotofthegravitationalpotentialc(X;Z)ofacausticringasafunctionoftherescaledcoordinatesX=)]TJ /F4 7.97 Tf 6.58 0 Td[(a pandZ=z p.TheparametersofthecausticringarethesameasthoseinFig. 4-3 .Thecenterofthetricusp(X0=0:25,Z0=0)ischosenasthereferencepoint.Thepotentialhasatriangularshapeinheritedfromthetricusp. 4.3EectsonStars 4.3.1ASingleStarForthestarsnearthegalacticdisk,smallradialandverticaloscillationscanbetreatedindependently[ 35 ]intheabsenceofacaustic.Theeectivepotentialforradialmotionconsistsofalogarithmicgravitationalpotentialandtheangularmomentumbarrier: e()=v2rotln+l2 22(4-14)wherel=2_=constant.Forsmallradialoscillationsabouttheminimum0=l vrotoftheeectivepotential,theangularfrequencyis!=p 2vrot 0.Forverticalmotion,wechoosethepotentialz(z)ofanisothermalstellardiskwithvelocitydispersionzandscaleheightz0[ 35 ] z(z)=2zlnhcosh2z 2z0i:(4-15) 73

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Inthepresenceofapassingcausticring,theradialandverticalaccelerationsofastararegivenby a(;z)=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(v2rot +l2 3+gc(;z;a(t)) (4-16) az(;z)=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(2z z0tanhz 2z0+gcz(;z;a(t)) (4-17) wheregc(;z;a(t))andgcz(;z;a(t))arethecontributions[seeEqs.( 4-8 )and( 4-9 )]fromthecausticringwithradiusa(t).Tovisualizetheeectsofthecausticringonthestar,wedeneradialandverticalenergiesperunitmass E=1 2v2+e())]TJ /F1 11.955 Tf 11.95 0 Td[(e(0); (4-18) Ez=1 2v2z+z(z); (4-19) wheree()andz(z)aregivenbyEqs.( 4-14 )and( 4-15 )and0istheminimumofe().EandEzremainconstantintheabsenceofacausticbecauseradialandverticalmotionsareindependent.Wenumericallysolvetheequationsofmotionofastarinthe-zplanewitharbitraryinitialconditionsasthecausticringpassesthroughitsorbit.Thevelocitydispersionsofthestarsinthesolarneighborhoodare40km/sintheradialdirectionand20km/sintheverticaldirection,andthescaleheightz0=0:5kpc[ 35 ].Asanexample,wechooseastarthatorbitsthegalaxyat0=8kpcwithsmallradialandverticaloscillationssuchthatvmax=10km/sandvmaxz=5km/s.InFig. 4-5 ,weshowtheenergies,EandEz,ofthestarasthecausticringpassesthroughitsorbit.Theuctuationsintheenergiesarelargeandoccuronatimescaleofapproximately2Gyr.TheuctuationsaresmallerforlargerinitialvaluesofEandEz.Thestarswhicharemostaectedbyapassingcausticringarethosewithnearlycircularorbits.InFig. 4-6 ,weplottheradialcoordinateofastar,andthelocationsoftherear(a)andfront(a+p)ofthetricuspofthecausticringasafunctionoftime.Initially,thestarinFig. 4-6 (a)hasanexactlycircularorbitwithradius0=8kpc, 74

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A BFigure4-5. Variationoftheradialandverticalenergies,EandEz(seeEqs. 4-18 and 4-19 ),ofastarwhenacausticringpassesthroughitsorbit.Thestarwasinitiallyorbitingthegalaxyat0=8kpcoscillatingintheverticaldirectionwithvmaxz=5km/sandintheradialdirectionwithvmax=10km/s.ThedottedlinesindicatethevaluesofEandEzintheabsenceofthecausticring.Fluctuationsintheenergiesareshownastheradiusa(t)ofthecausticringchangesfrom6kpcto10kpc.Asthecausticringmovesveryslowly(1kpc/Gyr),thetimescaleoverwhichtheuctuationsoccurisverylarge(approximately2Gyr). 75

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whereasthestarinFig. 4-6 (b)hasanalmostcircularorbitwith0=8kpcandvmax=vmaxz=5km/s.Thetricuspmovesradiallyoutwardwithaspeedda dt=1kpc/Gyrwitha(t=0)=6kpc.Inbothcases,astheguresshow,thestarisrstattractedtowardsthetricuspandthenmoveswithandoscillatesaboutthetricuspforapproximately1Gyrbeforereturningtoitsinitialorbitduetoconservationoftheangularmomentum.Forthechosencausticringparameters,wendthatallthestarswithvmax;vmaxz10km/sexhibitsuchbehavior.Theintermediatephaseoffollowingthetricuspcausesstellaroverdensitiesaroundthecausticring. 4.3.2DistributionofStarsWhenacausticringpassesthrougharelaxeddistributionofstars,itgeneratesbulkvelocitiesofthestarsandperturbsthedensityproleofthestellarpopulation. 4.3.2.1BulkvelocitiesFromthecontinuityequation,@d @t+~r:(d~v)=0,thebulkvelocitiesofthestarsareoforder, vd dx t;(4-20)whered distherelativeoverdensitycausedbythecausticand,xandtarethelengthandtimescalesoverwhichthestellardistributionchanges.Foracausticringwithradiusaandtricuspofsizep,xpandtp VwhereVda dtisthespeedofthetricuspintheradialdirection.AllthecausticringsmoveslowlywithspeedV1km/s.Hence,thebulkvelocitiesarev1km/s)]TJ /F9 7.97 Tf 6.68 -4.97 Td[(d d.Eveniftheoverdensitiesareaslargeas100%,thebulkvelocitiesinducedbythecausticringscannotbemorethanafewkm/s.Thisistoosmalltoexplaintherecentlyobserved[ 41 ][ 42 ][ 43 ]bulkvelocitiesoforder10km/sforthestarsinthesolarneighborhood. 76

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A BFigure4-6. Variationoftheradialcoordinateofastarasacausticringwithp=0:5kpcpassesthroughitsorbit.Intheabsenceofthecausticring,thestarinFig.(a)hasaninitiallycircularorbitofradius0=8kpcwhereasthestarinFig.(b)hasaninitiallyalmostcircularorbitwith0=8kpcandvmax=vmaxz=5km/s.Thethickdottedlines,aanda+p,indicatetherearandfrontofthetricusp(seeFig. 4-2 )movingwithspeed1kpc/Gyr.Inbothcases,thestarisinitiallyattractedtowardsthetricusp,thenmoveswithandoscillatesaboutthetricuspforapproximately1Gyrbeforecomingbacktoitsinitialorbit. 77

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4.3.2.2OverdensitiesAstellarpopulationwithsmallervelocitydispersionismoresusceptibletoapassingcausticring.Thepresentlocationsofthecausticringsaregivenby (an:n=1;2;3;4;5;:::)(40;20;13:3;10;8;:::)kpc:(4-21)Theradialvelocitydispersionofthestarsatadistancefromthegalacticcenterdecaysexponentiallywith[ 44 ][ 45 ]: ()(40km/s)exph)]TJ /F1 11.955 Tf 13.15 8.09 Td[(()]TJ /F1 11.955 Tf 11.96 0 Td[(8:5kpc) 8kpci:(4-22)Therefore,theradialvelocitydispersionsofthestarsneartherstvecausticringsare: (n:n=1;2;3;4;5;:::)(1;10;20;30;40;:::)km/s:(4-23)Theverticalvelocitydispersionszaretypicallyhalfoftheradialones.TheinfallratesforthecausticringsaregivenbyEq.( 4-10 ).Causticringswithsmallernhavestrongergravitationaleld[seeEq.( 4-8 )]andaresurroundedbystarswithsmallervelocitydispersions.Inthepresenceofacausticring,thestellardynamicsintheradialandverticaldirectionsarenotindependent.Simulatingthedynamicsofalargenumberofstarsinthe-zplanenearthecausticringiscomputationallyexpensive.Foreachcausticring,wesimulateone-dimensionalmotionofthestarsintheradialandverticaldirectionsindependently.Todeterminehowthestaroverdensitieschangewithradial(vertical)coordinates,wesimulatetheradial(vertical)motionsandsuppressthevertical(radial)dynamicsofthestars.Theinitialmotionintheradialdirectionisassumedtobesimpleharmonic,i.e.astaroscillatingabout=0movesinaharmonicpotentialwith!=p 2vrot 0wherevrot=220km/s[seeEq.( 4-14 )].Fortheradialmotionofthestarsneareachcausticring,wegeneratearelaxeddistributionof500;000starswithphasespacedensityf(v;)exp)]TJ /F4 7.97 Tf 14.34 6.86 Td[(v2+!2()]TJ /F4 7.97 Tf 6.58 0 Td[(0)2 22.Theinitialverticalmotionis 78

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determinedbythepotentialz(z)inEq.( 4-15 )andthephasespacedensityisgivenbyf(vz;z)sech2z 2z0exp)]TJ /F4 7.97 Tf 15.83 5.7 Td[(v2z 22z.Intheabsenceofthecaustics,eachstellardistributionremainsinequilibrium;i.e.thenumberdensityproledeq()ordeq(z)doesnotchangewithtime.Forthenthcausticring,theparametersfortheradialandverticaldynamicsofthestellardistributionsarechosenasfollows:=2z=n[seeEq.( 4-23 )],hi=0=an[seeEq.( 4-21 )]andz0=0:5kpc.Tominimizetheerrorduetothenitesizeofthestellarpopulation,wechoosethesizepofthetricuspofeachcausticringtobemuchsmallerthanthesizeofthecorrespondingstellarpopulation.Whilesimulatingthedynamicsofthestarsasthecausticringspassthroughthem,wetakeseveralsnapshotsofeachstellardistributionanddeterminetherelativeoverdensityd d=d)]TJ /F4 7.97 Tf 6.59 0 Td[(deq deq.Wendthat,aslongasthesizepofthetricuspismuchsmallerthanthespreadofthestellarpopulation,theoverdensityisindependentofthesizep.InFig. 4-7 ,weplotthestellaroverdensitiesd dduetotheradialmotionatthelocationsofdierentcausticringsasfunctionsoftherescaledradialcoordinateX=)]TJ /F4 7.97 Tf 6.59 0 Td[(a p.InFig. 4-8 ,weplotthesamefortheverticalmotionasfunctionsoftherescaledverticalcoordinateZ=z p.Wedidnotconsiderthen=1causticringat40kpc.Inthelinearapproximation,thetotaloverdensitynearthecausticisthesumoftheoverdensitiesobtainedforbothdirections.Wendthemaximumtotaloverdensitytobeoforder120%;45%;30%and15%forthen=2;3;4and5causticrings.Ifthestaroverdensitynearacausticringislarge,itenhancestheeectsofthecausticbyattractingmorestarsandinterstellargas.Wedidnotincludesuchbackreactionsinoursimulationhere.Thelargestaroverdensitiesnearthen=2and3causticringsmayexplaintheexistenceoftheMonocerosRing[ 46 ][ 47 ][ 48 ][ 49 ]at20kpcandtheBinneyandDehnenRing[ 50 ]at13:6kpc.TheMonocerosRinghasaverticalscaleheightoforder10kpc[ 46 ].AccordingtoFig. 4-8 ,theverticalscaleheightoftheoverdensityduetothen=2causticringisoforder4p.So,toformanoverdensityofverticalsize10kpc,then=2causticringisrequiredtohaveasizep2:5kpc.Thesizesofthen=1and2 79

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Figure4-7. Relativeoverdensitiesofthestarsd d=d)]TJ /F4 7.97 Tf 6.59 0 Td[(deq deqduetoradialmotionnearthen=2;3;4and5causticringsasfunctionsoftherescaledradialcoordinateX=)]TJ /F4 7.97 Tf 6.59 0 Td[(a p.Stellaroverdensitiesarehighernearthecausticringswithsmallernbecausetheyhavestrongergravitationaleldandaresurroundedbystarswithlowervelocitydispersions. causticringsarenotknownfromthebumpsintherotationcurvewhilethesizeofthethirdcaustichasbeendeterminedtobe1kpc[ 31 ].Sincetheoverdensitiesnearthen=4and5causticringsoccurwithinadistanceoffewkpcfromtheSun,theymaybeobservedinupcomingastronomicaldatasuchasfromGAIA. 4.4EectsonInterstellarGasThebulkpropertiesofadistributionofstarsinthesolarneighborhoodareaectedbythecausticringsonlyatthe15%levelbecausethesquareofthevelocitydispersion(2)ofthestarsislargerthanthedepthofthepotential(c)ofacausticring.Sincegasanddustintheinterstellarmediumhavesmallervelocitydispersions[ 35 ],theirbulkproperties,e.g.density,areexpectedtobeaectedmore.Assumingthegastobeinthermalequilibriuminthegravitationalpotentialofboththecausticandthegasitself,westudyitsdensityprolenearthen=5causticring.Thecausticistakentobestaticas 80

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Figure4-8. Relativeoverdensitiesofthestars(sameasFig. 4-7 )duetoverticalmotionasfunctionsoftherescaledverticalcoordinateZ=z p. thegasdynamicsisfastcomparedtothetimescaleoverwhichtheradiusofthecausticringchanges.Intheabsenceofthecausticring,theinterstellargasisassumedtobeinthermalequilibriumthroughself-gravitationalinteractions.Itspotentialanddensityaregivenby[ 35 ]: g(;z)=2glnhcosh2z 2zgi (4-24) dg(;z)=d0gsech2z 2zg (4-25) wherezg=g p 8Gd0gisthescaleheightandgisthevelocitydispersion.Theparametersinthesolarneighborhoodarethefollowing:zg65pc,g5km/sandd0g0:05M=pc3[ 35 ].Sincethescaleheightofthegasismuchsmallerthanthatofdiskstars(300pcforthinand500pcforthickdisks),weignorethegravityofthestars.Thepotentialanddensityaretakentobeindependentoftheradialcoordinatesincetheydonotchangesignicantlywithoverthesizeofthetricusp. 81

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Wechangeourcoordinatesystemfrom(;z)to(X;Z)=()]TJ /F4 7.97 Tf 6.58 0 Td[(a p;z p)asthelatterismoreconvenient.Inthepresenceofacausticring,thepotential(X;Z)duetogasisthesolutionofPoisson'sequation, r2(X;Z)=4Gdg(X;Z)(4-26)where dg(X;Z)=dg(X0;Z0)exp)]TJ /F1 11.955 Tf 13.15 8.08 Td[((X;Z)+c(X;Z) 2g(4-27)assuming(X0;Z0)=c(X0;Z0)=0.Wecalculatethenewpotential(X;Z)usingatwo-dimensionalGreen'sfunction.IntheX-Zplane,wechooseasucientlylargeregionaroundthetricuspsuchthat,attheboundaryoftheregion,(X;Z)tendstobeg(X;Z)[seeEq.( 4-24 )].ChoosingaGreen'sfunctionG(X;Z;X0;Z0)thatvanishesattheboundary,wehave (X;Z)=ZdX0ZdZ0G(X;Z;X0;Z0)4Gdg(X0;Z0)+Idl0g(X0;Z0)@G @n0(X;Z;X0;Z0):(4-28)Wesolvetheaboveequationnumerically.Thesolutionfor(X;Z)convergesafterafewiterations.InFig. 4-9 ,weplotthedensityofgasdg(X;Z)nearthetricuspofthen=5causticringwithsizep=150pc.Asevidentfromthegure,thedensityprolehasatriangularshapeasdoesthecausticpotentialshowninFig. 4-4 .IntherecentGAIAskymapoftheMilkyWay[ 51 ][ 52 ],triangularfeaturesareobserved[ 53 ]inbothtangentdirectionstothefthcausticring.WediscussaboutthemindetailinSection 5.3 .However,theobservedfeatureshavesharperedgesthanthetriangularshapeofFig. 4-9 82

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Figure4-9. Two-dimensionalplotofthedensityofgasdg(X;Z)(Eqs. 4-27 4-28 )whenthegasisinthermalequilibriuminthegravitationalpotentialofthecaustic(showninFig. 4-4 )andofthegasitself.Thevelocitydispersiongofthegasischosentobe5km/s.Thedensityprolehasatriangularshapereectingthatofthetricusp. 83

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CHAPTER5ASTROPHYSICALSIGNATURESOFCAUSTICRINGS:EVIDENCEInthischapter,westudysomerecentastrophysicalobservationsthatsupporttheexistenceofcausticrings. 5.1SummaryofPreviousEvidenceInthissection,wemakealistofastrophysicalobservationsthatareconsistentwiththepredictionsofthecausticringmodel.ThissectionisbasedonRefs.[ 32 ]and[ 31 ]. 5.1.1FlatRotationCurveatLargeRadiiThecausticringmodelpredicts[ 31 ]thatthedensityofdarkmatterd(r)isapproximatelyproportionalto1 r2atlargerradii,allthewayuptotheturnaroundradiusRwhered(R)istheaveragevalueofcosmologicaldarkmatterdensity.Incontrast,thecomputersimulations[ 54 ][ 55 ]ofconventionaldarkmattercandidateslikeWIMPspredictsthatd(r)/1 r3atlargeradii,whichimpliesthattherotationvelocitydecreasesas1 p r.TherotationcurveofMilkyWayisatuptoradiusoforder20kpc,thelargestradiuswhereithasbeenmeasured.Ingeneral,thespiralgalaxieshaveatrotationcurves[ 56 ][ 57 ]uptothelargestradiiwheretheyhavebeenmeasured.Studiesofthedynamicsoftheobservedsatellitegalaxies[ 58 ]alsoimplythatthed(r)/1 r2behaviorprevailsuptor200kpc. 5.1.2BumpsintheRotationCurvesSincethecausticringslieonornearthegalacticplane,theycausesharprisesingalacticrotationcurves.Acausticringwithradiusaandwidthpcauses[ 27 ][ 23 ]anupwardkinkatr=aandadownwardkinkatr=a+p.InRef.[ 32 ],rotationcurvesof32dierentgalaxieswereanalyzedandtwobumpswerefoundwithsignicance3:0and2:6nearthepredictedlocationsoftherstandsecondcausticrings,respectively.Nootherphenomenonisknowntocausethebumpsatsuchlargeradiiexceptforcausticrings.AnumberofrisesintheMilkyWayrotationcurveconsistentwiththecausticringmodelare basedonRef.[ 53 ]. 84

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alsoobserved[ 31 ].ThewidthsofvariouscausticringswereestimatedinRef.[ 31 ]bytheseparationbetweentheupwardanddownwardkinks. 5.1.3TriangularFeatureintheIRASSkyMapAcausticringmayleaveitsimprintontheinterstellarmediumviaitsgravitationaleld[ 32 ][ 23 ].Suchimprintsarebestvisiblealongthetwotangentdirectionstoacausticring.AtriangularfeaturesimilartothetricuspcrosssectionofacausticringwasobservedintheIRASmapoftheMilkyWay[ 32 ]alonggalacticcordinates(l;b)(80;0).Theorientationofthetrianglewithrespecttothegalacticcenterandgalacticplaneisconsistentwiththecausticringmodel.Also,thepositionofthetrianglematchesthepositionofasharpriseintherotationcurvebetween8:28and8:43kpccausedbythe5thcausticring. 5.1.4StellarOverdensitiesBysimulatingthedynamicsofstarsundertheinuenceofcausticrings,Ref.[ 23 ]predictsstellaroverdensitiesof120%and45%nearthe2ndand3rdcausticringsatradiiof20and13:3kpc,respectively.Relativeoverdensityoforder100%nearthe2ndcausticringwasalsopredictedinRef.[ 59 ].TheMonocerosringofstars[ 46 ][ 47 ][ 48 ][ 49 ]atr20kpcmaybeinterpretedasevidenceforthe2ndcausticring,whereastheBinneyandDehnenRing[ 50 ]at13:6kpcasevidenceforthe3rdcausticring. 5.2M31RotationCurveOurnearestmajorgalaxy,M31isviewededge-onfromourlocationanditsrotationcurveisoneofthebestmeasuredones.AsFig. 5-1 shows,thereareseveralbumpsintherotationcurvesforboththerecedingandapproachingsides[ 60 ].Therstoroutermostbumpat30kpciswelloutsidethelastobservedspiralarm(r25kpc)andthebumpsfrombothsidesmatchperfectly.Thisstronglyindicatesthepresenceofaring-likestructureinthedarkhalo.Thelocationsoftherst,secondandthirdbumpsat30,15and10kpc,respectively,areconsistentwiththepredictionofthecausticringmodelthattheradiusofthenthcausticringanisproportionalto1 n.Then=2bumpsonbothsides 85

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Figure5-1. HIrotationcurveofM31withlocationsofthebumpsmarkedwitharrows.Bluedownwardtrianglesarettedfortheapproachingsidewhiletheredupwardtrianglesarefortherecedingside.Filleddiamondsareforbothsidesttedsimultaneously.ThegureisadaptedfromRef.[ 60 ]. dierslightlyfromeachotherwhichmayoccurifthecenterofthecausticringisdisplacedfromthecenterofthegalaxy.Then=3bumpisabsentfromtherecedingside.ItisworthmentioningthattherearevariousmeasurementsoftherotationcurveofM31atsmallerradiiandtheydonotallagreewitheachother. 5.3TriangularFeaturesintheGAIASkyMapandTheirImplications 5.3.1LeftandRightTrianglesAsmentionedinSubsection 5.1.3 ,atriangularfeaturewasobservedintheIRASskymapalonggalacticcoordinates(l;b)(80;0)[ 32 ]whichisconsistentwiththepredictedlocationofthe5thcausticringandwithoneoftheobservedbumpsintheMilkyWayrotationcurve.NosuchfeaturewasfoundintheIRASskymapalongtheothertangentaround(l;b)()]TJ /F1 11.955 Tf 9.3 0 Td[(80;0).IntheGAIAskymap[ 51 ][ 52 ],atriangularfeaturewithshapeandorientationconsistentwiththepredictionofcausticringmodelisobservedalong 86

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Figure5-2. ThepanoramicviewofthegalacticplanefromtheGAIAskymap.Here,darkerregionsoccurasthelightraysfromstarsareobscuredbytheinterstellardust.Locationsofthetwotriangularfeaturesaremarkedinredcircles.ThegureisadaptedfromRef.[ 52 ]. A BFigure5-3. TheleftandrighttriangularfeaturesfoundintheGAIAskymap[ 52 ]. (l;b)()]TJ /F1 11.955 Tf 9.3 0 Td[(91;0).TheGAIAskymapalsocontainsthetrianglealong(l;b)(80;0)previouslyobservedintheIRASskymap.Figure 5-2 showsthepanoramicviewofthegalacticplanefromtheGAIAskymapwithcirclesaroundthetriangles,andFig. 5-3 showsthetwoareassurroundingthetriangles.Forthelefttriangle,theverticesoftheinneredgearelocatedat(l1;b1)(77:8;)]TJ /F1 11.955 Tf 9.29 0 Td[(2:4)and(l2;b2)(77:8;3:4),andtheoutervertexisat(l3;b3)(83:0;0:4).Fortherighttriangle,theverticesoftheinneredgeareat(l1;b1)()]TJ /F1 11.955 Tf 9.3 0 Td[(89:6;)]TJ /F1 11.955 Tf 9.3 0 Td[(2:4)and(l2;b2)()]TJ /F1 11.955 Tf 9.3 0 Td[(89:6;1:4),andtheoutervertexat(l3;b3)()]TJ /F1 11.955 Tf 9.3 0 Td[(92:7;)]TJ /F1 11.955 Tf 9.29 0 Td[(0:7).Thenewlyobservedrighttriangleislocatedfurtherfromthegalacticcenter(0;0)ascomparedtothelefttriangle,whichimpliesthecenterofthe5thcausticringisdisplacedfromthegalacticcenter.Assumingthecausticringtobeanexactcircle,wendthat 87

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Figure5-4. Thetriangleinscribedinthetricuspcrosssectionofthecausticring.Theobservedtriangularfeaturesareassumedtobetheinscribedtrangles,ratherthanthetricusps.Thehorizontalsizesarerelatedtoeachotherbyp=4 3p0andtheverticalsizesbyq=3 2q0. itscenterislocatedalong(l;b)()]TJ /F1 11.955 Tf 9.3 0 Td[(5:9;0)whichistheaverageofthemidpointsoftheinneredgesofthetwotriangles.Theangularseparationoftheinneredgesfromthecausticringcenterisli83:7.TakingourdistancefromthecausticringcenterRCas8:5kpc,wendtheradiusofthe5thcausticring: a=RCsinli8:45kpc:(5-1)Toestimatethetransversesizesofthetricusp,weassumetheobservedtrianglestobethetrianglesinscribedinthetricuspasshowninFig. 5-4 .Theangularseparationoftheouteredgeofthelefttrianglefromthecausticringcenterisl0(l)88:9.Thewidthofthetricuspnearthelefttangentisthen p(l)=4 3)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(RCsinl0(l))]TJ /F3 11.955 Tf 11.96 0 Td[(a66:3pc:(5-2) 88

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Thefactor4 3takescareofthedierencebetweentheverticesoftheinscribedtriangleandthecusps(seeFig. 5-4 ).Theouteredgeoftherighttriangleisseparatedbyl0(r)86:8fromthecausticringcenterandthewidthneartherighttangentis p(r)=4 3)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(RCsinl0(r))]TJ /F3 11.955 Tf 11.96 0 Td[(a50:7pc:(5-3)Therighttriangleappearstobesmallerthantheleftinsizewhichindicatesthattheaxialsymmetryisnotpreservedexactly.InRef.[ 29 ],itwasshownhowsmallperturbationstoaxialsymmetrycausevariationsoftransversesizesalongtheringwhiletheringstructureremainsstable.Estimatingtheverticalsizeqaccuratelyisdicultasweexplainbelow.Theapparentangularheight,i.e.thedierencebetweentheb'softhetwoverticesoftheinneredgeofthelefttriangle,isb(l)5:8.Usinggeometricalmethodsnaively,weget:q(l)=3 22RCcoslitanb(l) 2141:8pc.Apossiblewaytoimprovethisestimateistocomparetheobservedtriangleswiththesimulatedcolumndensitiesfordierentverticalsizes. 5.3.2PositionoftheSunwithrespecttotheTricuspIfthecausticringisassumedtobeacircle,itsradiusisasinEq. 5-1 .WeassumethatthetransversedimensionsofthetricuspneartheSunistheaverageofthoseneartheleftandrighttangents: p=1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(p(l)+p(r)58:5pc:(5-4)Theplaneofthecausticringisdenedbythetwomidpointsoftheinneredgesofthetriangleswhichare(77:8;0:5)and()]TJ /F1 11.955 Tf 9.3 0 Td[(89:6;)]TJ /F1 11.955 Tf 9.3 0 Td[(0:5).Thecausticplaneisfoundtobetiltedby0:5withrespecttothegalacticplane.Theouterverticesoftheleftandrighttrianglesareb(l)0:1andb(r)0:2belowthecausticplane,respectively.TheverticaldistanceoftheSunfromthecausticplaneisestimatedas zC1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(RCcosl0(l)sinb(l)+RCcosl0(r)sinb(r)1:0pc:(5-5) 89

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Theresultabovegetsanegligiblecorrectionwhentheseparationbetweenthevertexoftheinscribedtriangleandthecuspistakenintoaccount.Since RC)]TJ /F1 11.955 Tf 11.95 0 Td[((a+p))]TJ /F1 11.955 Tf 21.92 0 Td[(7:2pc;(5-6)alongtheradialdirection,theSunis7:2pcaway(towardthecausticringcenter)fromtheoutercusp.WhethertheSunisinsideoroutsidethetricuspdependsontheverticalsizeqofthetricuspneartheSun.WediscussthisindetailinSubsection 5.3.4 .TodeterminethepropertiesofdarkmatterowsthroughtheSun,thelinejoiningthecausticringcenterandtheoutercuspneartheSunistakenasthez=0plane.Forfuturereference,wedeterminetherescaledcoordinatesofthesun X=C)]TJ /F3 11.955 Tf 11.95 0 Td[(a p0:88Z=zC p0:017 (5-7) wheretheradialdistanceCistakenas8:5kpc. 5.3.3Parametersofthe5thCausticRingForthesakeofcompleteness,webrieydescribesomepropertiesofacausticring.Underaxialsymmetry,adarkmatterparticleisconvenientlylabelledbytwoparameters(;)whereisthetimewhentheparticlecrossedz=0planeandistheanglefromz=0planeatthetimeoftheparticle'smostrecentouterturnaround.Thecoordinatesoftheparticlesnearthecausticisgivenby[ 27 ]: =a+1 2u()]TJ /F3 11.955 Tf 11.96 0 Td[(0)2)]TJ /F1 11.955 Tf 13.15 8.09 Td[(1 2s2z=b (5-8) wherea;u;0;sandbareconstantforagivencaustic.Theyarereferredastheparametersofthecausticring.ThecausticoccurswheretheJacobianjD2(;)jjdet@(;z) @(;)jiszero.Inthe-zplane,ittakestheshapeofatricuspasshowninFig. 4-2 .Its 90

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sizeispinthe-directionandqinthez-direction: p=1 2u20q=p 27 4p; (5-9) where =b p us=4 p 27q p:(5-10)Asmentionedbefore,theverticalsizeqisnotknownwithcertainty.Takingtherotationspeedvrot=220km/sandthespeedofdarkmatterparticlesv=522km/s[ 31 ],wend u=v2)]TJ /F3 11.955 Tf 11.95 0 Td[(v2rot a27:6pc Myr2:(5-11)AnalyzingtheGAIAtriangles,wendthata8:45kpc,p58:5pc.Thenthevalueof0is 0=)]TJ /F10 11.955 Tf 9.3 18.78 Td[(r 2p u)]TJ /F1 11.955 Tf 21.92 0 Td[(2:06Myr(5-12)where0hasbeenassumedtobenegative.Numericalsimulationsareconsistentwiththis.Theparameterbisassumedtohavethesamemagnitudeasthespeed[ 27 ] bv=522km/s:(5-13)Theparametersisgivenby s=b2 u210:27kpc1:0 2:(5-14) 5.3.4DensitiesandVelocitiesoftheNearbyFlowsItisconvenienttowriteEqs. 5-8 intermsoftherescaledcoordinatesX=)]TJ /F4 7.97 Tf 6.59 0 Td[(a p,Z=z p: X=(T)]TJ /F1 11.955 Tf 11.96 0 Td[(1)2)]TJ /F3 11.955 Tf 11.96 0 Td[(A2Z=2AT (5-15) 91

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whereA=q s u20andT= 0.Foragivenpoint(X;Z)nearthecaustic,Tisfoundbysolving X=(T)]TJ /F1 11.955 Tf 11.96 0 Td[(1)2)]TJ /F3 11.955 Tf 21.26 8.09 Td[(Z2 42T2:(5-16)EachrealsolutionofTcorrespondstoaowofdarkmatterparticlesthroughthatpoint.Therearetwoows,`in'and`out',throughapointoutsidethetricusp.Forapointinside,therearetwoadditionalows:`up',`down',`in'and`out'.ForasolutionTjofEq. 5-16 ,thecorrespondingowhasthefollowingdensity: dj(X;Z)=4:110)]TJ /F9 7.97 Tf 6.59 0 Td[(24g cm3Dj522km/s b58:5pc p8:5kpc dM dd 32:0M steradyr!(5-17)where Dj=cosZ Tjr p 2s42T2j Z2+42T3j(Tj)]TJ /F1 11.955 Tf 11.96 0 Td[(1):(5-18)Wehaveconsideredtheenhancedinfallrateofdarkmatterduetotheformationofthe`bigvortex'(seeEq. 4-10 andtheprecedingdiscussion).Wewritethevelocityoftheowtakingthecausticringcenterastheorigin ~vj=vj^+vj^+vjz^z(5-19)where^pointsradiallyoutward,awayfromthecausticringcenter,^zistowardthegalacticnorthpoleand^isoppositetothegalacticrotation.Thecomponentsaregivenby vj=sgn(0)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tj)56:83km sj0j 2:06Myr u 27:6pc Myr2!vjz=sgn(0))]TJ /F3 11.955 Tf 9.3 0 Td[(Z 2Tj56:83km sj0j 2:06Myr u 27:6pc Myr2!vj=)]TJ /F10 11.955 Tf 9.3 13.73 Td[(q v2)]TJ /F3 11.955 Tf 11.96 0 Td[(v2j)]TJ /F3 11.955 Tf 11.96 0 Td[(v2jz (5-20) 92

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wheresgn(0)isthesignoftheparameter0andv=522km/sisthespeedoftheow[ 31 ].Intermsofthegalaxycenteredcoordinates,thevelocityoftheowiswrittenas ~vj=vjx^x+vjy^y+vjz^z(5-21)where^xpointstowardthegalacticcenter,^yisinthedirectionofgalacticrotationand^zpointstowardsthegalacticnorthpole.ThecomponentsinEqs. 5-21 arerelatedtothoseinEqs. 5-20 by vjx=)]TJ /F1 11.955 Tf 11.29 0 Td[(cosvj)]TJ /F1 11.955 Tf 11.95 0 Td[(sinvjvjy=sinvj)]TJ /F1 11.955 Tf 11.95 0 Td[(cosvj (5-22) where5:9istheanglebetweenthedirectionstowardsthecentersofthecausticringandthegalaxy.TheverticalcomponentsinEqs. 5-21 and 5-20 arethesame.ThenumberofowsthroughtheSunandtheirdensititesandvelocitiesaresensitivetotheparameter(seeEq. 5-10 ).Alargervalueofimpliesalargerverticalsizeqofthetricusp.ForaxedverticaldistanceoftheSunfromtheplaneofthecausticring,thevalueofdetermineswhethertheSunisinsideoroutsidethetricusp.UsingEq. 5-7 andtheparametersfromSubsection 5.3.3 ,wecalculatethedensityandthevelocitycomponentsofeachowatthelocationoftheSunforvariousvaluesoftheparameter(seeTable 5-1 ).Thedensityandvelocityoftheoutgoingow,i.e.theowcorrespondingtoT=1:938,ispracticallyindependentofthevalueof. 93

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Table5-1. DensitiesandvelocitiesoftheowsatthelocationoftheSunfordierentvaluesof=4 p 27q p.qisthesizeofthetricuspintheverticaldirectionusingEqs. 5-4 and 5-9 .TisarealrootofEq. 5-16 withX=0:88andZ=0:017.Therearetwo(four)owscorrespondingtothetwo(four)realsolutionsforT,whentheSunisoutside(inside)thetricusp.distheowdensityusingEq. 5-17 ;(v;v;vz)arethevelocitycomponentsincausticringcenteredcoordinatesand(vx;vy;vz)arethecomponentsingalaxycenteredcoordinates.SeeSubsection 5.3.4 fordetails. q(pc)Td(10)]TJ /F9 7.97 Tf 6.58 0 Td[(24g/cm3)v(km/s)v(km/s)vz(km/s)vx(km/s)vy(km/s) 1:076:0)]TJ /F1 11.955 Tf 9.3 0 Td[(0:021122:1)]TJ /F1 11.955 Tf 9.3 0 Td[(58:0)]TJ /F1 11.955 Tf 9.3 0 Td[(518:3)]TJ /F1 11.955 Tf 9.3 0 Td[(22:9+111:0+509:5+1:9382:5+53:3)]TJ /F1 11.955 Tf 9.3 0 Td[(519:3+0:2+0:4+522:01:291:2)]TJ /F1 11.955 Tf 9.3 0 Td[(0:017923:4)]TJ /F1 11.955 Tf 9.3 0 Td[(57:8)]TJ /F1 11.955 Tf 9.3 0 Td[(518:1)]TJ /F1 11.955 Tf 9.3 0 Td[(27:0+110:8+509:4+0:0277106:6)]TJ /F1 11.955 Tf 9.3 0 Td[(55:3)]TJ /F1 11.955 Tf 9.3 0 Td[(518:8+17:4+108:3+510:3+0:0521132:8)]TJ /F1 11.955 Tf 9.3 0 Td[(53:9)]TJ /F1 11.955 Tf 9.3 0 Td[(519:1+9:3+106:9+510:8+1:9382:5+53:3)]TJ /F1 11.955 Tf 9.3 0 Td[(519:3+0:2+0:4+522:01:4106:4)]TJ /F1 11.955 Tf 9.3 0 Td[(0:015624:6)]TJ /F1 11.955 Tf 9.3 0 Td[(57:7)]TJ /F1 11.955 Tf 9.3 0 Td[(517:9)]TJ /F1 11.955 Tf 9.3 0 Td[(31:0+110:6+509:2+0:021973:8)]TJ /F1 11.955 Tf 9.3 0 Td[(55:6)]TJ /F1 11.955 Tf 9.3 0 Td[(518:6+22:1+108:6+510:1+0:0556100:9)]TJ /F1 11.955 Tf 9.3 0 Td[(53:7)]TJ /F1 11.955 Tf 9.3 0 Td[(519:2+8:7+106:8+510:9+1:9382:5+53:3)]TJ /F1 11.955 Tf 9.3 0 Td[(519:3+0:2+0:4+522:01:6121:6)]TJ /F1 11.955 Tf 9.3 0 Td[(0:013825:4)]TJ /F1 11.955 Tf 9.3 0 Td[(57:6)]TJ /F1 11.955 Tf 9.3 0 Td[(517:6)]TJ /F1 11.955 Tf 9.3 0 Td[(35:0+110:5+509:0+0:018462:7)]TJ /F1 11.955 Tf 9.3 0 Td[(55:8)]TJ /F1 11.955 Tf 9.3 0 Td[(518:3+26:2+108:8+509:9+0:057490:2)]TJ /F1 11.955 Tf 9.3 0 Td[(53:6)]TJ /F1 11.955 Tf 9.3 0 Td[(519:2+8:4+106:6+510:9+1:9382:5+53:3)]TJ /F1 11.955 Tf 9.3 0 Td[(519:3+0:2+0:4+522:01:8136:8)]TJ /F1 11.955 Tf 9.3 0 Td[(0:012425:8)]TJ /F1 11.955 Tf 9.3 0 Td[(57:5)]TJ /F1 11.955 Tf 9.3 0 Td[(517:4)]TJ /F1 11.955 Tf 9.3 0 Td[(39:0+110:4+508:7+0:015955:8)]TJ /F1 11.955 Tf 9.3 0 Td[(55:9)]TJ /F1 11.955 Tf 9.3 0 Td[(518:1+30:4+108:9+509:6+0:058484:5)]TJ /F1 11.955 Tf 9.3 0 Td[(53:5)]TJ /F1 11.955 Tf 9.3 0 Td[(519:2+8:3+106:6+510:9+1:9382:5+53:3)]TJ /F1 11.955 Tf 9.3 0 Td[(519:3+0:2+0:4+522:0 94

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CHAPTER6SUMMARYAxionsoraxion-likeparticlesareusuallytreatedasclassicaleldsbecausetheyhavelargequantumdegeneracy.But,inthecontextofthermalization,theclassicalelddescriptionleadstoanultravioletcatastropheandaquantumdescriptionisnecessary.Weweremotivatedbythefollowingquestion:howlongcanahighlydegeneratequantumscalareldbedescribedaccuratelybyclassicaleldequations?.Intuitively,thistimescalecannotbelongerthanthethermalizationtimescale.AgenericformalismtocalculatethedurationofclassicalityispresentedinSection 2.1 .Theformalismisappliedtoahomogeneouscondensatewithattractivecontactinteractionsandonewithgravitationalself-interactionsinSections 2.2 and 2.3 ,respectively.Classically,thehomogeneouscondensatepersistsforever.Inquantumevolution,smallquantumuctuationsgrowexponentiallyduetoattractivenatureoftheinteractionsandthecondensatebecomesdepleted[ 1 ].Wecalculatedthetimescaleafterwhichclassicaldescriptionbecomesinvalidinbothcases.InChapter 3 ,weapplytheformalismtoacondensatewithrepulsivecontactinteractions.Weconsiderahomogeneouscondensatewithaplanewaveperturbationofmomentum~p.Intheclassicaldescription,thecondensateisstableuptosecondordercorrections.Inthequantumdescription,thequasi-particlesscatterthroughaprocess~0+~p!~k+(~p)]TJ /F3 11.955 Tf 11.57 3.15 Td[(~k)and,theprocessisenhancedbyparametricresonanceif~kliesonthesurfacedenedby!(~0)+!(~p)=!(~k)+!(~p)]TJ /F3 11.955 Tf 11.83 3.16 Td[(~k).Wedeterminearegionofinstabilityaroundthissurface.Weestimatethetimescaleafterwhichtheclassicaldescriptionisnolongervalid.Infuture,themethodsusedinChapters 2 and 3 maybeappliedtothestudyofthequantumbehavioroftwooverlappingcondensateswithoppositemomenta.Ourresultsimplythat,afteracertaintime,thequantumeectsaredominantandtheinterferencepatternswillbeblurred.Theframeworkmayalsobeusefultopredictthequantum 95

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behaviorofamoregenericsystem,e.g.acondensatewithgravitationalself-interactionsandarbitraryinhomogeneities.InChapter 4 ,westartwithadiscussionabouthowthestructuresoftheinnercausticsofdarkmatterhaloarerelatedtothequantumbehavioroftheaxionBose-Einsteincondensate(BEC).Thereisobservationalevidencethattheinnercausticsarecausticrings.Conventionaldarkmatterparticlescannotacquirethevelocityeldrequiredtoformthecausticrings.Ifthecolddarkmatteriscomposed,atleastpartially,ofarethermalizingBECofaxions,therequiredvelocityeldisanaturaloutcomesinceitisthatofthestateoflowestenergyforgiventotalangularmomentum.Thethermalizationofaxionsviagravitationalself-interactionshasadierentoutcomeinquantumeldtheorythaninclassicaleldtheory.Theradiianofcausticringsincreaseslowlyattherateofapproximately1km/s)]TJ /F4 7.97 Tf 11.76 -4.97 Td[(an 8kpc.InChapter 4 ,westudythedynamicsofstarsandinterstellargasasthecausticringspassthroughthem.Wendthatstellarorbitsarehighlyperturbedbyapassingcausticring.Astarmovinginanearlycircularorbitisrstattractedtowardsthecausticring,thenmoveswithandoscillatesaboutthecausticforapproximately1Gyr,andnallyreturnstoitsoriginalorbitasaresultofangularmomentumconservation.Weshowthattheinducedbulkvelocitiesofthestarscannotbemorethanafewkm/s.Thestaroverdensityaroundacausticringdependsuponthevelocitydispersionofthestars.Sinceacausticringwithsmallernhasstrongergravitationaleldandissurroundedbystarswithlowervelocitydispersion,itcausesalargerstellaroverdensity.Wealsostudythedistributionofinterstellargasnearthen=5causticringwhichistheoneclosesttous.Thegasdynamicsismuchfasterthanthetimescaleoverwhichtheradiusofacausticringchanges.Weconsiderthecausticringtobestaticandassumethegastobeinthermalequilibriuminthegravitationalpotentialofthecausticandofthegasitself.Thedensityproleofgasin-zplaneshowninFig. 4-9 hasatriangularshapeasaresultofthetricuspcrosssectionofthecausticring. 96

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InChapter 5 ,wediscussrecentobservationswhichmaybeinterpretedasevidenceforcausticrings.BumpsintherotationcurvesfrombothrecedingandapproachingsidesoftheM31galaxysupporttheexistenceofthen=1;2and3causticrings.IntheGAIAskymap,triangularfeaturesinbothtangentdirectionstothefthcausticringareobserved.Takingthosetrianglesasevidenceofthe5thcausticring,wedeterminealltheparametersofthe5thcausticringexceptitsverticalsizeq.Forvariousvaluesofq,wecalculatethedensitiesandvelocitiesofthedarkmatterowsatthelocationoftheSun.Interestingly,theobservedtriangularfeatureshaveedgesmuchsharperthanthoseobtainedundertheassumptionofthermalequilibrium.Itisworthinvestigatingwhetherthesharpfeaturesariseasaresultofdustgrainsbeingentrainedbytheaxionows.Ifthedustgetsentrainedandthecollisionsbetweendustandgasdonotdestroytheows,thefowsofdustwouldformacausticwhichisobservableandhassharpedges. 97

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APPENDIXADERIVATIONOFM~K0~KThisappendixprovidesaderivationofEq. 2-26 .TheexpressionforM~k0~k(t)inEq. 2-10 mayberewrittenas M~k0~k(t)=ZVd3x1 2u~k(~x;t))]TJ /F3 11.955 Tf 9.3 0 Td[(i@t)]TJ /F1 11.955 Tf 18.28 8.09 Td[(1 2mr2u~k0(~x;t)+)]TJ /F3 11.955 Tf 9.3 0 Td[(i@t)]TJ /F1 11.955 Tf 18.27 8.09 Td[(1 2mr2u~k(~x;t)u~k0(~x;t)(A-1)byintegratingbypartsandnotingthat ZVd3xhu~k(~x;t)@tu~k0(~x;t)+@tu~k(~x;t)u~k0(~x;t)i=0(A-2)inviewofEq. 2-23 .SubstitutingEq. 2-19 andusingthefactthat(~x;t)satisesEq. 2-11 ,oneobtains M~k0~k(t)=1 2NZVd3xei(~k0)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k)~(~x;t)h)]TJ /F3 11.955 Tf 20.15 8.09 Td[( 4m2j(~x;t)j4+j(~x;t)j2(~k0+~k)@t~)]TJ /F3 11.955 Tf 16.28 8.09 Td[(i m~rln~r(~k0~)+i m~rln~r(~k~)+i 2mr2((~k)]TJ /F3 11.955 Tf 12.75 3.16 Td[(~k0)~)+1 2m(~r(~k~))2+1 2m(~r(~k0~))2i: (A-3) Since ~rln=1 2~rlnn+im~v(A-4)wehave i mh~rln~r(~k~))]TJ /F3 11.955 Tf 13.09 3.02 Td[(~rln~r(~k0~)i=i 2m~rlnn~r((~k)]TJ /F3 11.955 Tf 12.75 3.16 Td[(~k0)~)+~v~r((~k+~k0)~): (A-5) As(@t+~v~r)~=0,Eq. A-3 simpliesto M~k0~k(t)=)]TJ /F3 11.955 Tf 17.49 8.09 Td[( 8m2~n(~k)]TJ /F3 11.955 Tf 12.75 3.15 Td[(~k0;t)+1 2NZVd3xn(~x;t)ei(~k0)]TJ /F4 7.97 Tf 6.17 2.11 Td[(~k)~(~x;t)h1 2m(~r(~k~))2+1 2m(~r(~k0~))2+i 2m~rlnn~r((~k)]TJ /F3 11.955 Tf 12.76 3.15 Td[(~k0)~)+i 2mr2((~k)]TJ /F3 11.955 Tf 12.76 3.15 Td[(~k0)~)i: (A-6) Uponintegratingthethirdterminbracketsbyparts,weobtainEq. 2-26 98

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APPENDIXBTHEFUNCTIONF()HereweprovidethedimensionlessfunctionF()inEq. 3-52 where=!(~p) !: F()=1 4p 2f()s Q() G()(B-1)where f()=)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(+p 2+4 8(2+4)p p 2+1)]TJ /F1 11.955 Tf 11.96 0 Td[(1;Q()=1 p 2+4+p 2+1)]TJ /F1 11.955 Tf 11.96 0 Td[(3;G()=16 3(2+4)3 2p 2+422+3)]TJ 11.96 10.46 Td[(p 2+1)]TJ /F1 11.955 Tf 11.95 0 Td[(2)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(2+2: (B-2) ThefunctionF()isplottedinFig. 3-2 99

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BIOGRAPHICALSKETCHSankhaS.ChakrabartycompletedhisschoolinginasmalltowncalledJoynagarMajilpurnearKolkata,India.HepassedB.Sc.withHonoursinphysicsfromtheUniversityofCalcuttainJuly,2011.HedidtheM.Sc.inphysicsfromIndianInstituteofTechnologyKanpurinJuly,2013.InAugustof2013,hecametotheUnitedStatestostarthisgraduatestudiesinphysicsattheUniversityofFlorida(UF)inGainesville.HecompletedtheM.S.inphysics(EnRoutetoPhD)fromUFinMay2015.SankhastartedworkingonaxiondarkmatterwithProf.PierreSikivieinsummer2014.InDecember2018,hewasawardedCharlesF.HooperJr.MemorialAwardfordistinctioninresearchandteaching.Inspring2019,hereceivedtheMcGintyEndowmentDissertationFellowshipbytheCollegeofLiberalArtsandSciences,UF.HereceivedhisPh.D.fromUFinAugust2019andjoinedtheUniversityofTurin,Italytocontinueresearchinastrophysics. 104