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High Resolution Search for Dark Matter Axions in Milky Way Halo Substructure


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ThisworkisbasedonresearchperformedbytheAxionDarkMattereX-periment(ADMX).IamgratefultomyADMXcollaboratorsfortheireorts,particularlyinrunningtheexperimentandprovidingthehighresolutiondata.Withouttheseeorts,thisworkwouldnothavebeenpossible.Ithankmyadvisor,PierreSikivie,forhissupportandguidancethoughoutgraduateschool.Ithasbeenapriviegetocollaboratewithhimonthisandotherprojects.IalsothankDaveTannerforhisassistanceandadviceonthiswork.Iwouldliketothanktheothermembersofmyadvisorycommittee,JimFry,GuenakhMitselmakher,PierreRamond,RichardWoodardandFredHamann,fortheirrolesinmyprogress.IamalsogratefultotheothermembersoftheUniversityofFloridaPhysicsDepartmentwhohavecontributedtomygraduateschoolexperience.Iamespeciallygratefultomyfamilyandfriends,bothnearandfar,whohavesupportedmethroughthislongendeavor.SpecialthanksgotoLisaEverettandEthanSiegel. ii

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page ACKNOWLEDGMENTS ............................. ii LISTOFTABLES ................................. v LISTOFFIGURES ................................ vi ABSTRACT .................................... viii CHAPTER 1INTRODUCTION .............................. 1 2AXIONS .................................... 7 2.1Introduction ............................... 7 2.2TheStrongCPProblem ........................ 7 2.3TheAxion ................................ 11 2.3.1Introduction ........................... 11 2.3.2ThePeccei-QuinnSolutiontotheStrongCPProblem .... 11 2.3.3TheAxionMass ......................... 15 2.3.4TheAxionElectromagneticCoupling ............. 19 2.4AxionsinCosmology .......................... 21 3DISCRETEFLOWSANDCAUSTICSINTHEGALACTICHALO .. 27 3.1Introduction ............................... 27 3.2Existence ................................ 27 3.3Densities ................................. 37 3.4Discussion ................................ 39 4HIGHRESOLUTIONSEARCHFORDARKMATTERAXIONS .... 41 4.1Introduction ............................... 41 4.2AxionDarkMattereXperiment .................... 42 4.3AxionSignalProperties ........................ 45 4.4NoiseProperties ............................. 49 4.5RemovalofSystematicEects ..................... 53 4.6AxionSignalSearch .......................... 61 4.7Results .................................. 64 4.8Discussion ................................ 69 iii

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...................... 72 REFERENCES ................................... 74 BIOGRAPHICALSKETCH ............................ 79 iv

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Table page 4{1Eectivepowerthresholdsforalln-binsearches,withthefrequencyres-olutions,bnandcorrespondingmaximumowvelocitydispersions,vn,foraowvelocityof600km/s. ....................... 67 4{2Numericallycalculatedvaluesoftheformfactor,C,andampliernoisetemperatures,Tel,fromNRAOspecications. ............... 67 v

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Figure page 3{1A2-Dsliceof6-Dphase-space.Thelineisacross-sectionofthesheetofwidthvonwhichthedarkmatterparticlesliepriortogalaxyfor-mation.Thewigglesarethepeculiarvelocitiesduetodensityperturba-tions.Whenoverdensitiesbecomenon-linear,thesheetbeginstowindupclockwiseinphase-space,asshown. ................... 30 3{2Thephase-spacedistributionofdarkmatterparticlesinagalactichaloataparticulartime,t.Thehorizontalaxisisthegalactocentricdistance,r,inunitsofthehaloradius,R,andvistheradialvelocity.Sphericalsymmetryhasbeenassumedforsimplicity.Particleslieonthesolidline. 31 3{3Thecross-sectionofthetricuspring.Eachlinerepresentsaparticletra-jectory.Thecausticsurfaceistheenvelopeofthetriangularfeature,in-sidewhichfourowsarecontained.Everywhereoutsidethecausticsur-face,thereareonlytwoows.IllustrationcourtesyofA.Natarajan. ... 33 3{4Thetricuspringcaustic.Axialsymmetryhasbeenusedforillustrativepurposes.IllustrationcourtesyofA.Natarajan. ............. 34 4{1Schematicdiagramofthereceiverchain. .................. 45 4{2SketchoftheADMXdetector. ........................ 46 4{3Powerdistributionforalargesampleof1-bindata. ............ 51 4{4Powerdistributionforalargesampleof2-bindata. ............ 52 4{5Powerdistributionforalargesampleof4-bindata. ............ 53 4{6Powerdistributionforalargesampleof8-bindata. ............ 54 4{7Powerdistributionforalargesampleof64-bindata. ........... 55 4{8Powerdistributionforalargesampleof512-bindata. ........... 56 4{9Powerdistributionforalargesampleof4096-bindata. .......... 57 4{10AnenvironmentalpeakasitappearsintheMRsearch(top)andthe64-binHRsearch.Theunitfortheverticalaxisisthermspoweructua-tionineachcase. ............................... 58 vi

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.............. 59 4{12Sample4096-binspectrumbeforecorrectionforthecavity-ampliercou-pling.Thelineisthetobtainedusingtheequivalentcircuitmodel. .. 60 4{13Thesame4096-binspectrumofFig. 4{12 aftercorrectionforthecavity-ampliercoupling. .............................. 61 4{14Illustrationoftheadditionschemeforthe2,4and8-binsearches.Thenumberscorrespondtothedatapointsofthe1-binsearch.Numberswithinthesameboxarebinsaddedtogethertoformasingledatuminthen-binsearcheswithn>1. ........................... 62 4{1597.7%condencelevellimitsfortheHR2-binsearchonthedensityofanylocalaxiondarkmatterowasafunctionofaxionmass,fortheDFSZandKSVZacouplingstrengths.AlsoshownisthepreviousADMXlimitusingtheMRchannel.TheHRlimitsassumethattheowveloc-itydispersionislessthanv2givenbyEq.( 4{13 ). ............. 68 4{1697.7%condencelevellimitsfortheHR4-binsearchonthedensityofanylocalaxiondarkmatterowasafunctionofaxionmass,forDFSZandKSVZacouplingstrengths.Densitiesabovethelinesareexcluded.Forcomparison,thepredicteddensityoftheBigFlowisalsoshown.TheHRlimitsassumethattheowvelocitydispersionislessthanv2givenbyEq.( 4{13 ). ................................. 70 vii

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viii

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ix

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1 2 ].Thedarkmattercomponentistheconcernofthisdissertation.DarkmatterwasrstpostulatedbyFritzZwickyin1933[ 3 ].WhileobservingtheComaclusterofgalaxies,henotedthattheamountofvisiblematterwastoosmallforthesystemtobegravitationallybound.Giventheobservedgalacticvelocities,thesystemshouldyapart.Zwickyproposedadditionalmatterthatwasnotvisibletoprovidethenecessarygravitationalpotentialenergytobindthesystem.Thestrongestevidencefordarkmattertodayisprovidedbytherotationcurvesofspiralgalaxies.Plotsofobservedcircularvelocityagainstradialdistance 1

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areattolargedistances.Thecontributionstothiscurvefromthediskarenotenoughtosupportthisrotationcurve.Itisthusbelievedthatspiralgalaxiesconsistofavisiblediskembeddedinamuchlargerellipticaldarkmatterhalo.Forareviewofevidencefordarkmatter,seeBertoneetal.[ 4 ].Darkmatterparticlesmusthavethefollowingtwoproperties:(1)Theyareeectivelycollisionlessasfarasstructureformationisconcerned;i.e.,theonlysignicantlong-rangeinteractionsaregravitational,and(2)Thedarkmattermustbecold;i.e.,itmustbenon-relativisticwellbeforetheonsetofgalaxyformation.Therstpropertymeansthatdarkmattercaninteractonlyweaklywithbaryonicmatter.Thesecondpropertyisnecessarytoformthestructureintheuniversethatweobservetoday.Ifdarkmatterwasmoreenergetic,itwouldbeabletofreelystreamoutoftheinitialdensityperturbationsthathaveformedintogalaxies.Whatthedarkmatterconsistsofisstillunknown,despiteknowledgeofthepropertiesitmustpossess.Thestandardmodelofparticlephysicsdoesnotcontainaparticlethatcanprovidethedarkmatteroftheuniverse.Extensionstothestandardmodeldo,however,provideviableparticlecandidates.Theleadingdarkmatterparticlecandidatesareaxionsandweakly-interactingmassiveparticles(WIMPs).Theaxionisthepseudo-Nambu-GoldstonebosonfromthePeccei-QuinnsolutiontothestrongCPproblem[ 5 6 7 8 ].Theaxionmass,ma,isconstrainedtolieintherange106to102eV[ 9 10 11 ].Therearetwobenchmarkaxionmodelsthatareminimalextensionsofthestandardmodel:theKim-Shifman-Vainshtein-Zakharov(KSVZ)[ 12 13 ]modelandtheDine-Fischler-Srednicki-Zhitnitsy(DFSZ)[ 14 15 ]model.Intheearlyuniverse,onepopulationofaxionsisproducedbythermalprocessesandhastemperatureoforder1Ktoday.Inaddition,coldaxionpopulationsarisefromvacuumrealignment[ 16 17 18 ]and

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stringandwalldecay[ 19 20 21 22 23 24 25 26 27 28 29 30 31 ].WhichmechanismscontributedependsonwhetherthePeccei-Quinnsymmetrybreaksbeforeorafterination.Thecoldaxionswereneverinthermalequilibriumwiththerestoftheuniverse.WIMPsareaclassofdarkmattercandidates:heavyparticlesthatinteractviaforcesofweak-scalestrength.WIMPsoccurinmanymodels,particularlyinextensionsofthestandardmodelwhichincludeanewparitysymmetrytopreventprotondecay.ThemostpopularWIMPcandidateisarguablythelightestsuper-symmetricparticle(LSP),i.e.,thelightest,as-yet-undetectedparticleprovidedbytheminimalsupersymmetricextensiontothestandardmodel(MSSM).Incontrasttoaxions,WIMPsarethermalrelics.Theybeganinthermalequibriumwiththeprimordialheatbath.WhentheinteractionrateofWIMPswiththerestoftheheatbathfallsbelowtheexpansionrateoftheuniverse,theseparticlesdecoupleor\freezeout."Theirevolutionisthengovernedbytheuniverse'sexpansionandgravitationalinteractions.Muchworkiscurrentlyunderwaytodetectdarkmatteranddeduceitsparticleproperties.ADMXisadirectdetectionexperimentsearchingfordarkmatteraxions[ 32 33 34 35 ].ThisexperimentusesatunableSikiviemicrowavecavity[ 36 ]tosearchforaxions.Whentheresonantfrequencyofthecavity,,correspondstotheenergy,Ea,ofaxionspassingthroughthecavity(i.e.,=Ea=h),resonantconversionofaxionstophotonswilloccur.Thesignalisapeakintheenergyspectrumoftheoutputfromthecavity.ManydirectdetectionexperimentsarealsosearchingforWIMPs.WIMPdirectdetectionisbasedonlookingfornuclearrecoilsfromtheelasticscatteringofpassingWIMPs.Additionally,attemptsarebeingmadetodetectdarkmatterindirectlyusingastrophysicalsignatures.TheseprimarilyfocusondetectingtheproductsofWIMPannihilations:neutrinos,positrons,anti-protonsandgamma-rays.

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Thesesignalsaredependentonthedarkmatterdensity.Thepoweroutputfromresonantaxionconversiontophotonsisproportionaltothelocalaxiondensity.ForWIMPs,therateofnuclearrecoileventsfromWIMPscatteringisproportionaltothelocalWIMPdensity.Also,theuxofannihilationproductssearchedforinindirectdetectionisproportionaltothesquareoftheWIMPdensityatthesiteofannihilation.Thusitisnecessarytomakeassumptionsaboutthedistributionofdarkmatterinourgalactichalo.Inparticular,galactichalosubstructureisofinterestfordarkmatterdetection.Thepresenceofsubstructureinagalactichalomeansthattherewillberegionsofenhanceddarkmatterdensity,improvingdetectionprospectsduetothesignaldependenceondensity.Whilethepowerinanaxionsignalobservedbyamicrowavecavitydetectorisproportionaltothelocalaxiondensity,thesignalwidthiscausedbythevelocitydispersionofdarkmatteraxions.Insearchingforaxions,itisthusalsonecessarytomakeassumptionsabouttheirvelocitydistributionintheMilkyWayhalo.AnumberofmodelsareusedtoguideADMX'ssearch.Thesearetheisothermalmodel,theresultsfromN-bodysimulations[ 37 38 ]andadescriptionofgalactichalosintermsofdiscreteowsfromlateinfallofdarkmatterontothegalaxy.Aspecicmodelwhichconsiderslateinfallisthecausticringmodel[ 39 40 ].Intheisothermalmodel,itisassumedthatthedarkmatterhalohasther-malizedviavirializationandthushasaMaxwell-Boltzmannvelocitydistribution.ADMX'smediumresolution(MR)channel[ 34 ]searchesforsuchaxions,assumingthatthevelocitydispersionisO(103c)orless,wherecisthevelocityoflight.(Theescapevelocityfromourgalaxyisapproximately2103c.)Ofparticularinteresttothisworkistheexistenceofcoldowsofdarkmatteraxionswithinthehalo.SuchowsareassociatedwiththetidaldisruptionofsubhalospredictedbyN-bodysimulationsandwithlateinfallofdarkmatterontothegalactichalo.

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Numericalsimulationsindicatethathundredsofsmallerclumps,orsubhalos,existwithinthelargerhalo[ 37 38 ].Tidaldisruptionofthesesubhalosleadstoowsintheformof\tidaltails"or\streams."TheEarthmaycurrentlybeinastreamofdarkmatterfromtheSagittariusAdwarfgalaxy[ 41 42 ].Non-thermalizedowsfromlateinfallofdarkmatterontothehaloarealsoexpected[ 43 44 ].Darkmatterthathasonlyrecentlyfallenintothegravitationalpotentialofthegalaxywillhavehadinsucienttimetothermalizewiththerestofthehalo.Thisdarkmatterwillbepresentinthehalointheformofdiscreteows.Therewillbeoneowofparticlesfallingontothegalaxyforthersttime,oneduetoparticlesfallingoutofthegalaxy'sgravitationalpotentialforthersttime,onefromparticlesfallingintothepotentialforthesecondtime,etc.Furthermore,wherethegradientoftheparticlevelocitydiverges,particles\pileup"andformcaustics.Inthelimitofzeroowvelocitydispersion,causticshaveinniteparticledensity.Thevelocitydispersionofcoldaxionsatatime,t,priortogalaxyformationisapproximatelyva31017(105eV=ma)(t0=t)2=3[ 40 ],wheret0isthepresentageoftheuniverse.Thus,aowofdarkmatteraxionswillhaveasmallvelocitydispersion,leadingtoalarge,butnitedensityatthelocationofacaustic.ThecausticringmodelpredictsthattheEarthislocatednearacausticfeature[ 45 ].Thismodel,ttedtobumpsintheMilkyWayrotationcurveandatriangularfeatureseenintheIRASmaps,predictsthattheowsfallinginandoutofthehaloforthefthtimecontainasignicantfractionofthelocalhalodensity.Thepredicteddensitiesare1:71024g/cm3and1:51025g/cm3[ 45 ],comparabletothelocaldarkmatterdensityof9:21025g/cm3predictedbyGatesetal.[ 46 ].Theowofthegreatestdensityisreferredtoasthe\BigFlow."ThepossibleexistenceofdiscreteowsprovidesanopportunitytoincreasethediscoverypotentialofADMX.Adiscreteaxionowproducesanarrowpeakinthe

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spectrumofmicrowavephotonsintheexperimentandsuchapeakcanbesearchedforwithhighersignal-to-noisethanthesignalfromaxionsinanisothermalhalo.Ahighresolution(HR)channelhasbeenbuilttotakeadvantageofthisopportunity.Ifasignalisfound,theHRchannelwillalsoprovidedetailedinformationonthestructureoftheMilkyWayhalo.TheHRchannelisthemostrecentadditiontoADMX,implementedasasimpleadditiontothereceiverchain,runninginparallelwiththeMRchannel.ThischannelandthepossibleexistenceofdiscreteowscanimproveADMX'ssensitivitybyafactorofthree[ 35 ],signicantlyenhancingitsdiscoverypotential.Thisworkisarrangedasfollows.BackgroundinformationontheaxionisgiveninChapter 2 .ThestrongCPproblemofthestandardmodelofparticlephysics,themotivationfortheaxion,isdescribedandthePeccei-Quinnsolution,resultingintheaxion,isdiscussed.Propertiesimportanttoaxiondetection,suchasitsmassandcoupling,arealsoreviewed.Someastrophysicalandcosmologicalconsequencesoftheaxionareoutlined,particularlytheproductionofcoldaxionpopulations.AreviewofdiscreteowsandcausticsinthegalactichaloispresentedinChapter 3 .Chapter 4 describesADMXandprovidesdetailsoftheHRanalysis.Thenewresult,improvingADMX'ssearchsensitivitybyafactorofthree,isalsoshown.AsummaryandconclusionsarepresentedinChapter 5

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5 6 7 8 ].Itisalsoagoodcandidateforthedarkmatteroftheuniverse.Thischapterprovidesbackgroundinformationonaxions.ThestrongCPproblemisdescribedinSection 2.2 .InSection 2.3 ,thePeccei-Quinnsolutiontothisproblemisoutlined,usingtheoriginalPeccei-Quinn-Weinberg-Wilczekaxionmodel.Axionsareshowntobeanaturalconsequenceofthissolution.Axionpropertiesimportanttodetectionarealsoreviewed.Section 2.4 discussescosmologicalaspectsofaxions,specicallyhowaxiondarkmatterarisesandthelimitsthatcosmologyandastrophysicsplaceontheaxionmass. 7

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TheLagrangianofQCDis 4GaGa+NXj=1[i qjDqj(mjqyLjqRj+h:c:)]+g2 2{1 )isthe\-term."Theangle,,isaparameterandeGaisthedualtensortothegluoneldstrength,denedby 2Ga;(2{4)with,theLevi-Civitatensor.TheparametersofQCDarethusg,mjand.Thecolorcoupling,gisenergydependentandindeningthetheory,itisnormallyexchangedfortheQCDconnementscale,QCD,oforder200MeV.Theparameter,,istheQCDvacuumangle.ThisparameterisnecessarytofullydescribeQCDbecausetheSUC(3)gaugesymmetryisnon-Abelian.Non-Abeliangaugepotentialshavedisjointsectors,labelledbyanintegertopologicalwindingnumber.Thesesectorsaredisjointastheycannotbetransformedcontinuouslyintoeachother.Thereexistsa

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vacuumcongurationcorrespondingtoeachn,betweenwhichquantumtunnellingcanoccur.ThegaugeinvariantQCDvacuumstateisthusasuperpositionofvacuaofdierentn,i.e., 47 48 ]meansthatthissymmetryisnotpresentinthequantumtheory.Inthefullquantumtheory,includingquarkmasses,thephysicsofQCDremainsunchangedunderthetransformations, Whilethephysicsremainsthesame,thisisnotasymmetrybecausetheparameterhaschanged.ThetransformationsofEq.( 2{6 )throughEq.( 2{8 )canbeusedtomovephasesbetweenthequarkmassesand.However,thequantity,

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However,thisisnotthecase;CPviolationhasbeenobservedintheelectroweaksector.Consequently,thereisnoapparentreasonwhythe-termwouldnotbepresentinthestandardmodel.WhileCPviolationispresentintheelectroweaksectorofthestandardmodel,ithasnotbeenobservedinQCD.AnelectricdipolemomentfortheneutronisthemosteasilyobservedconsequenceofstrongCPviolation.The-termresultsinaneutronelectricdipolemomentof[ 9 49 10 11 ], 50 ] 5 6 ]resultsinthepresenceofanaxion[ 7 8 ],whichhastheadditionalmotivationofbeingagoodcandidateforthedarkmatteroftheuniverse.Thissolutionisoutlinedindetailinthefollowingsection.OthersolutionsincludetheupquarkmassbeingzeroandthatCPisspontaneouslybroken.Ifthebareupquarkmassiszero,the

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engineeredspontaneousCPviolation[ 51 52 ].WefocusononlythePQsolutioninthefollowingsection. 2.3.1IntroductionThissectionprovidesimportantbackgroundinformationforaxiondetection.InSection 2.3.2 ,wediscussthePQsolutiontothestrongCPproblem.TheoriginalPeccei-Quinn-Weinberg-Wilczekaxionmodelisusedforillustration,butotheraxionmodelsarealsodiscussed.AderivationoftheaxionmassisgiveninSection 2.3.3 ,usingthemethodsoflowenergyeectivetheory.InSection 2.3.4 ,theaxion-electromagneticcouplingisreviewed.Thiscouplingisthebasisforaxiondetectionexperiments.Theresultingpowerdevelopedinamicrowavecavitydetector,usingthiscoupling,isalsogiven.

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ThesimplestwaytointroduceadditionaldegreesoffreedomisviaanextraHiggsdoublet.WeassumethatoneoftheHiggsdoublets,u,couplestotheup-typequarksandtheother,d,couplestothedown-typequarks.Wedistinguishbetweentheup-anddown-typequarksbylabellingthemuianddi,respectively(ratherthanqi,asintheprevioussection).AsthereareNquarks,thereareN=2up-typequarksanddown-typequarks.TheleptonscanacquiremassthroughYukawacouplingstoeitheroftheHiggsdoubletsortoathirdHiggsdoublet.Weignorethiscomplicationhereandsimplyexaminethecouplingstoquarks.ThequarksacquiretheirmassesfromtheexpectationvaluesoftheneutralcomponentsoftheHiggs,0uand0d.Themassgeneratingcouplingsare

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NotethatitispossibletowritedownleptoncouplingswhichalsoobservethePQsymmetry.Itisnecessarythatthesecouplingsdoso,otherwiseapotentialtermforwillresult,destroyingthePQmechanism.Whentheelectroweaksymmetrybreaks,theneutralHiggscomponentsacquirevevs: OnelinearcombinationoftheNambu-Goldstoneelds,PuandPd,isthelongi-tudinalcomponentoftheZ-boson,asperelectroweaksymmetrybreakinginthestandardmodel.Thiscombinationis UsingEqs.( 2{20 ),( 2{21 ),( 2{24 )and( 2{25 )inEq.( 2{13 ),theaxioncouplingstoquarksarisefrom vuauRi+mdidyLieicosv vdadRi+h.c.;(2{26)wheremui=yuivuandmdi=ydivd.TheaxionelddependencecanberemovedfromthemasstermsusingthetransformationsofEqs.( 2{17 ),( 2{18 )and( 2{19 ).DirectcouplingsbetweentheaxionandquarkswillstillremainintheLagrangian,

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throughtheassociatedchangeinthequarkkineticterm.Theresultingchangein 2{27 ).Dening vu=vd+vd=vu;(2{28)the-termofEq.( 2{1 )isreplacedby

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exist:theDine-Fischler-Srednicki-Zhitnitsky(DFSZ)andKim-Shifman-Vainshtein-Zhakharov(KSVZ)models.InboththeKSVZandDFSZmodels,anaxionwithpermissablemassandcouplingsresults.IntheKSVZmodel,onlytheHiggsdoubletofthestandardmodeloccurs.Theaxionisintroducedasthephaseofanadditionalelectroweaksingletscalareld.Theknownquarkscannotbedirectlycoupledtosuchaeld,astheYukawacouplingswouldleadtounreasonablylargequarkmasses.Instead,thisscalariscoupledtoanadditionalheavyquark,alsoanelectroweaksinglet.Theaxioncouplingsaretheninducedbytheinteractionsoftheheavyquarkwiththeotherelds.TheDFSZmodelhastwoHiggsdoublets,asinthePQWWmodel,andanadditionalelectroweaksingletscalar.ItistheelectroweaksingletscalarwhichacquiresavevatthePQsymmetrybreakingscale.Thescalardoesnotcoupledi-rectlytoquarksandleptons,butthroughit'scouplingstothetwoHiggsdoublets.Thus,itispossiblefortheexistenceofanaxiontosolvethestrongCPproblem.Whilesignicantforthatalone,theaxionalsoprovidesaninterestingcandidateforthecolddarkmatteroftheuniverse.

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condensateacquiresexpectationvalue f;(2{35)

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wherefisthedecayconstant.TheeectiveLagrangianis 2@@+f2 f+h.c.)+1 8m2f2cos2 f; wherethenaltermisthepotentialforrotationsintheUA(1)direction.Whenthe-termisincluded,theexpectationvalueofthecondensateremainsthatgiveninEq.( 2{35 ),exceptthat fisreplacedby f+ f! f: Inanaxionmodel,isreplacedby+Na va,wherevaisthescaleatwhichthePQsymmetrybreaks.TheconstantNisdenedbytheanomaly, 2@a@a+1 2@@+f2 f+h.c.)+1 8m2f2cos2 f++Na va: Thevariabledenestheoriginoftheaxioneld,sowemaychoosethistobezero.Thequarkmassmatrixcanbewrittenasarealmatrixtimesaphaseand

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wemayrotatetomoveontothequarkmassterm.Thisillustratresthatthedependenceisalwaysadependenceon 2@a0@a0+f2 withcorrespondingmasses =f2m2 Usingthestandardvaluesform,f,muandmd,theaxionmasscanbeexpressedas[ 9 49 10 11 ]

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36 ].Wediscusshowthiscouplingarisesineectiveeldtheoryandreviewthepowerdevelopedinamicrowavecavityexperimentinthissection.Theaxionelectromagneticcouplingisduetomixingbetweentheaxion,neutralpionandeta.ThecouplingsoftheLagrangianforanyoftheseparticlestodecaytotwophotonsis 3 f+Ne vaFeF:(2{52)Thecoecientsintheaboveequationarisefromthetraceovertheanomalyloop.TheconstantNeisgivenby 2{44 )and( 2{47 ),theresultingaxioncouplingtotwophotonsis faFeF;(2{54)where 2Ne 3mdmu

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ThefullLagrangianfortheinteractionofaxionswithphotonsinfreespaceis[ 36 ] 4F+g faFeF+1 2@a@a1 2m2aa21+Oa2 2{56 )canbewrittenas 2(E2B2)+1 2@a@a1 2m2aa2g faEB(2{57)Inacavitypermeatedbyastrong,inhomogeneousmagneticeld,resonantconversionofaxionstophotonscanbeinducedifthecavityfrequencycorrespondstothatoftheaxionenergy.Theresultingpowerdevelopedinamicrowavecavitydetectoris mamin(Q;Qa):(2{58)whereVisthecavityvolume,B0isthemagneticeldstrength,aisthelocaldensityofaxionswithenergycorrespondingtothecavityfrequency,QisthequalityfactorofthecavityandQaistheratiooftheenergyofhaloaxionstotheirenergyspread,equivalenttoa\qualityfactor"forthehaloaxionsignal.Themode-dependentformfactor,C,isgivenby 105;

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whereaistheaxionenergyfrequency.Thus,whensuchacavityistunedtothecorrectfrequency,resonantcon-versionofaxionstophotonsresults.Thisconversionisobservedasapeakinthefrequencyspectrumofthedetectoroutput. 16 17 18 ],stringdecay[ 19 20 21 22 23 24 25 26 27 28 29 ]anddomainwalldecay[ 29 30 31 ].Wediscussthehistoryoftheaxioneldastheuniverseexpandsandcoolstoseehowandwhenthesemechanismsoccur.Wealsoreviewtheprocessofvacuumrealignmentindetail,astherewillalwaysbeacontributiontothecoldaxionpopulationsfromthatmechanismand,asdiscussedbelow,itispossiblethatthisprovidestheonlycontribution.Therearetwoimportantscalesintheproblemofaxionsasdarkmatter.TherstisthetemperatureatwhichthePQsymmetrybreaks,TPQ.Whichoftheabovemechanismscontributesignicantlytothecoldaxionpopulationdependsonwhetherthistemperatureisgreaterorlessthantheinationaryreheatingtemperature,TR.Thesecondisthetemperatureatwhichtheaxionmass,arisingfromnon-perturbativeQCDeects,becomessignicant.Athightemperatures,thelattereectsarenotsignicantandtheaxionmassisnegligible[ 53 ].Theaxion

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massbecomessignicantatacriticaltime,t1,whenmat11[ 16 17 18 ].ThecorrespondingtemperatureisT1'1GeV.Atinitialearlytimes,thePQsymmetryisunbroken.AtTPQ,itbreaksspon-taneouslyandtheaxioneld,whichisproportionaltothephaseofthecomplexscalareldaquiringavev,mayhaveanyvalue.Thephasevariescontinously,changingbyorderonefromonehorizontothenext.Axionstringsappearastopo-logicaldefects.IfTPQ>TR,theaxioneldishomogenizedoverhugedistancesandthestringdensityisdilutedbyination,tothepointwhereitisextremelyunlikelythatanyaxionstringsremaininourvisibleuniverse.InthecaseTPQ1,thedomainwallproblemoccurs[ 54 ]becausethevacuumismultiplydegenerateandthereisatleastonedomainwallperhorizon.ThedomainwallsendupdominatingtheenergydensityandcausetheuniversetoexpandasS/t2,whereSisthescalefactor.Althoughothersolutionstothedomainwallproblemhavebeenproposed[ 29 ],weassumeherethatN=1orTPQ>TR.Thus,ifTPQTR,thehomoge-nizationfrominationwillresultinasinglevalueoftheaxioneldoverourvisibleuniverse.Non-perturbativeQCDeectscauseapotentialfortheaxioneld.When

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theseeectsbecomesignicant,theaxioneldwillbegintooscillateinthepo-tential.Theseoscillationsdonotdecayandcontributetothelocalenergydensityasnon-relativisticmatter.Thus,acoldaxionpopulationresultsfromvacuumrealignment,regardlessoftheinationaryreheatingtemperature.Tounderstandthecontributionfromvacuumrealignment,consideratoyaxionmodelwithonecomplexscalareld,(x),inadditiontothestandardmodelelds.Letthepotentialfor(x)be 53 ] 2{63 ),theeectiveLagrangian

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is 2@a@am2a(T)v2a va:(2{67)InaFriedmann-Robertson-Walkeruniverse,theequationofmotionis +3H(t)_1 2m2av2a2;(2{69)andthus, sin(N)'N:(2{70)Wenowrestrictthediscussiontothezeromomentummode.ForTPQ>TR,thiswillbetheonlymodewithsignicantoccupation,sothenalenergydensitycalculatedwillbeforthiscase.Inthecase,TR>TPQ,highermodeswillalsobeoccupied.Forthezeromomentummode,theequationofmotionreducesto +3H(t)_+m2a(t)=0;(2{71)i.e.,theeldsatisestheequationforadampedharmonicoscillatorwithtime-dependentparameters.Asnoinitialvalueofispreferred,themostgeneralsolutionis 2;(2{72)where1and2areconstants.Thus,atT>>TQCD,isapproximatelyconstant.Theeldwill,however,begintooscillateinitspotentialwhentheuniversecoolstothecriticaltemperature,T1,denedby[ 55 ] 2t1:(2{73)

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Astheaxioneldcanrealignonlyasfastascausalitypermits,thecorrespond-ingmomentumofaquantumoftheaxioneldis v2a:(2{76)Asaxionsarenon-relativisticanddecoupled, (2{78) =1 2f2ama(t1)1

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Theenergydensityinaxionstodayis (2{80) =1 2f2a1 whereS(t)isthescalefactorattime,t.Eq.( 2{81 )impliestheaxionenergydensity, a'0:15fa 621:(2{82)Astheaxioncouplingsareverysmall,thesecoherentoscillationsdonotdecayandmakeagoodcandidateforthedarkmatteroftheuniverse.ThemassisrelatedtothePeccei-Quinndecayconstant,fa,byEq.( 2{51 )andthecouplingsoftheaxionmassareinverselyproportionaltofa.Thuslimitsonanyoftheaxionmass,axioncouplingsorPQdecayconstantisalsoarestrictionontheothertwo.Sincea106eV.Thisisthelowerboundontheaxionmassrange.Iftheaxionmasswereanygreater,toomuchdarkmatterwouldbeproducedviatherealignmentmechanism.Theupperlimitontheaxionmassis102eV,fromobservationsofSN1987a.ThenumberofneutrinosobservedonEarthduetothissupernovaeanditsdurationareingoodagreementwithmodelsofsupernovae.Lightparticles,suchasaxions,presentnovelcoolingmechanismsthatcanalterthedurationofsupernovae.Iftheaxionmassislessthan102eV,axionsarenotproducedinsignicantnumberstoaectsupernovae.However,forarangeofaxionmassesabovethis,axionproductionandescapefromsupernovaewillsignicantlyshortenthesupernovadurationbyecientlytransportingenergyaway.Aboveapproximately0:5eV,themeanfreepathofanaxionwillbetooshortforsignicantnumbersofaxionstoescapefromsupernovae.Atthispoint,otherastrophysicalprocesses,suchasthelifetimeofredgiantsforbidaxionmassesinhigherranges[ 11 ].

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3.2 ,wereviewliteraturedemonstratingthatdiscreteowsareanaturalconsequenceofacolddarkmattercosmology.Section 3.3 discussesthedensitiesofsuchows.Asignicantfractionofthelocalhalodensityshouldbecontainedindiscreteows,whichisimportantwhensearchingforthem,asthesignalobservedisproportionaltothedensity.Abriefdiscussionofevidenceforowsanddetectionofaxionsintheseowsconcludesthischapter,inSection 3.4 27

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Whileweareinterestedinthissubstructurefromthepointofviewofaxiondetection,itshouldbenotedthattheexistenceofdiscreteowsandcausticsisindependentofthetypeofcolddarkmatter.Theonlyrequirementforowsandcausticstoformistheassumptionofcolddarkmatteritself.Colddarkmatterparticlesareassumedtopossessthefollowingproperties:(1)Theparticlesmustbecollisionless,i.e.,theonlysignicantinteractionsoftheseparticlesaregravitational.Thispropertyexplainswhytheparticlesaredarkmatter.(2)Theparticleshavenegligibleinitialvelocitydispersion,wheretheinitialconditionsarethosewhenthedarkmatterrstfallsintoagalaxy'sgravitationalpotential.Thisisdiscussedfurtherinthefollowing.TheprimordialvelocitydispersionofbothaxionsandWIMPsisnegligible[ 56 ]asfaraslargescalestructureformationisconcerned.ForWIMPs,theprimordialvelocitydispersionisdeterminedbythetemperature,TD,atwhichtheydecouplefromtheprimordialheatbath.ConsideringHubbleexpansiontobetheonlysignicanteecttoaltertheWIMPvelocitydispersion,thevelocitydispersion,vW,ofaWIMPofmassmWfallingintoagalaxytodayis 2S(tD)

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axioneldisinhomogeneousonthescaleofthehorizonsize(t1)whenthemassbecomessignicantandhence, 3{2 ).Theprimordialvelocitydispersionofdarkmatterparticlesfallingontoagalaxyatanytime,t,canbeobtainedbysubstitutionofS(t0)forthescalefactor,S(t).ForbothaxionsandWIMPs,weseethattheinitalvelocitydispersionissosmallastobenegligible.Theformationofdiscreteowsandcausticscanbeunderstoodbyconsideringthephase-spacedistributionofdarkmatterparticlesfallingintoagravitationalpotential.Atearlytimes,priortotheonsetofgalaxyformation,theseparticleswilllieonathin3-dimensional(3D)sheetin6Dphase-space,asillustratedinFig. 3{1 .Thethicknessofthesheetisproportionaltothelocalvelocitydispersionofthedarkmatterparticles,v,andthusthesheetisthin.Thissheetwillalsobecontinuous,asthenumberdensityofparticlesisverylargeoverthescaleatwhichthesheetisbentinphasespace.Asdarkmatterparticlesarecollisionless,theevolutionofthesheetisde-terminedbytheinuenceofgravityonly.Wheredensityperturbationsbecomenon-linear,the3Dsheetwillbeginto\windup"clockwiseinphase-space.Whereaspreviously,inthelinearregime,thesheetcoveredphysicalspaceonlyonce,itwillnowbegintocoverphysicalspacemultipletimes.Aftermuchtime,thephase-spaceparticledistributionwilllookasshowninFig. 3{2 .Asparticlesfallintoagrav-itationalpotential,therewillbeanumberofdiscreteowspresentateachpointatanytime[ 43 ].Therewillbeoneowofparticlesfallinginforforthersttime,

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A2-Dsliceof6-Dphase-space.Thelineisacross-sectionofthesheetofwidthvonwhichthedarkmatterparticlesliepriortogalaxyformation.Thewigglesarethepeculiarvelocitiesduetodensityper-turbations.Whenoverdensitiesbecomenon-linear,thesheetbeginstowindupclockwiseinphase-space,asshown.

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Thephase-spacedistributionofdarkmatterparticlesinagalactichaloataparticulartime,t.Thehorizontalaxisisthegalactocentricdistance,r,inunitsofthehaloradius,R,andvistheradialvelocity.Sphericalsymmetryhasbeenassumedforsimplicity.Particleslieonthesolidline.

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oneowofparticlesfallingoutforthersttime,oneofparticlesfallingoutforthesecondtime,etc.Also,atthelocationswherethesheetfolds,causticsform.Therearetwotypesofcausticsthatoccurwithinagalactichalo:\outer"and\inner."Outercausticsformnearwhereaowofparticlesfallingoutofthehalo'sgravitationalpotentialturnaroundandfallbackin.Thesecausticsaretopologicallyspheres.Innercausticsformwhereparticlesfallingintothepotentialreachtheirdistanceofclosestapproachtothecenterofthegalaxy.Whentheinitialvelocityofinfallingparticlesisdominatedbyarotationalcomponent,innercausticsarea\tricuspring"[ 40 ],whosecross-sectionisaD4catastrophe.Thecross-sectionisillustratedinFig. 3{3 andtheringshowninFig. 3{4 .Axialsymmetryhasbeenusedinthesegures,butisnotaneccesaryconditionfortheformationofcaustics.Weproceedtoreviewthemathematicalargumentsfortheexistenceofouterandinnercaustics[ 44 ].Parametrizetheparticlesonthephase-spacesheetusing=(1;2;3).Thisparametrizationmaybechosenasconvenient.Letx(;t)bethephysicalpositionoftheparticlelabelledattimet.Atearlytimes,beforegalacticevolutionbecomesnon-linear,themapping!xwillbeone-to-one.Atlatetimes,whenthesheetcoversphysicalspacemultipletimes,foranygivenphysicallocationrtherewillbe,ingeneral,multiplesolutionsj(r;t)withj=1;2;;n(r;t),tor=x(;t).Thatis,therewillbeparticleswithdierentatthesamephysicallocation,r.Thenumberofowsatrattimetisn(r;t).Thenumberdensityofparticlesonthesheetisd3N d3.Itfollowsthatthemassdensityinphysicalspaceis[ 40 ] d3()1

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Thecross-sectionofthetricuspring.Eachlinerepresentsaparticletrajectory.Thecausticsurfaceistheenvelopeofthetriangularfeature,insidewhichfourowsarecontained.Everywhereoutsidethecausticsurface,thereareonlytwoows.IllustrationcourtesyofA.Natarajan.

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Thetricuspringcaustic.Axialsymmetryhasbeenusedforillustrativepurposes.IllustrationcourtesyofA.Natarajan.

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wheremistheparticlemassand 3{3 )isthesumoverthemassdensityineachdiscreteowatr.CausticsoccurwhereD=0andthemapissingular[ 39 ].Atthesepoints,themappingfromphase-spacetophysicalspacechangesfromn-to-oneto(n2)-to-one.Thephysicaldensityatthelocationofcausticsbecomesverylarge,asthephase-spacesheetistangenttovelocityspace.Inthelimitofzeroinitialvelocitydispersion,thedarkmatterparticledensitydivergesatthelocationofacaustic.Inreality,theseowswillhaveasmallvelocitydispersionandthusthecausticswillhavealarge,butnite,density.ThepresenceofoutercausticsiseasilyseenfromFig. 3{2 .NatarajanandSikivie[ 44 ]demonstratedthatinnercausticsmustalsobepresentinthegalactichalo.ConsideracontinuousowofcolddarkmatterparticlesfallinginandoutofagravitationalpotentialandasphericalsurfaceofradiusRsurroundingthepotentialwell.Usingtheparametrization,=(;;),whereandarethepolarcoordinateswhereaparticlefallingintothepotentialcrossesthesphereattime,.Thenx(;;;t)givestheparticle'spositionattime,t.NatarajanandSikiviedemonstratedthat

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crossingtimeofparticlescurrentlycrossingthesphereonthewayin(out).Thesphere'scenterischosentolieattheorigin,x=0.Thedistancefromthesphere'scentertoaparticle'spositionis @;;out(;)<0and@r @;;in(;)>0:(3{7)Thus,forall(;)thereexistsa0(;)suchthat @;;0(;)=x @0=x @0=x 3{9 ),( 3{10 )and( 3{11 )implythat@x

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@@ @2:(3{15)AsD=0at=0,theoriginisthelocationofacausticinthisspecialcase.Inthiscase,thecaustichascollapsedtoapoint.Thus,bothinnerandoutercausticsmustbepresentinagalactichalo.Discreteowsandcausticsareanaturalconsequenceofacolddarkmattercosmology.

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ThelocaldensityoftherstfewowswasrstestimatedbySikivieandIpser[ 43 ],forcasesbothwithoutandwithangularmomentum.Wereviewtheirestimatesbelow.Theinitialestimatewascalculatedfortherstow,i.e.,theowofparticlespassingEarthforonlythersttime.Theseparticleshadamaximumgalactocen-tricdistanceofrm1Mpc,whichwasreached5109yearsago.Thedensityatthislocationisestimatedtobetheaveragecosmologicaldarkmatterdensitytoday,CDM(t0).Inthecaseofnoangularmomentum,thelocaldensityoftherstowwillbethedensityatrmmultipliedbytheappropriategeometricalfocussingfactor,i.e.,(rm=r)2,thus, 57 ]conrmthisestimateandprovideestimatesofdensitiesofthesameorderofmagnitudefortheotherows.Theircalculationsshowthateachofthersteightinandoutowshavedensitiesoftheorderof2%ofthelocalhalodensity(assumingalocaldarkmatterdensityof9:21025g/cm3[ 46 ]).Thus,theseestimatesleadustoexpectthatowscontainasignicantfractionofthelocaldarkmatterdensity.Atthelocationofacaustic,thedarkmatterdensitywillbegreatlyenhanced.Thiswillbereectedbyrisingbumpsinthegalacticrotationcurveattheselocations.FittingthecausticringmodeltorisesintheMilkyWayrotationcurveandtoatriangularfeatureintheIRASmappredictsthattheowsfallingin

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andoutforthefthtimecontainasignicantfractionofthehalodensityatthelocationofoursolarsystem.Thepredicteddensitiesare1:71024g/cm3and1:51025g/cm3[ 45 ].Theowofthegreatestdensityiscalledthe\BigFlow."Thisowispredictedtohaveavelocitydispersionof53m/sandvelocityofapproximately300km/srelativetotheSun.Thusthisowisofparticularinterestforaxiondarkmatterdetection. 3.2 ,discreteowsandcausticsareanecessaryconsequenceofcolddarkmattercosmology.Itissignicantinthisregardthatcausticsofluminousmatterarealsobelievedtoexistandhavebeenobservedinbrightellipticalgalaxies.MalinandCarterrstobservedripplesinthedistributionoflightinthesegalaxies[ 58 ].Computersimulationsdemonstratethatwhenasmallgalaxyfallsintothexedgravitationalpotentialofalargeellipticalgalaxy,thesmallgalaxyistidallydisruptedanditsstarsenduponathinribboninphase-space.Thesephase-spaceribbonsarelikethephase-spacesheetsofdarkmatterdiscussedearlier,exceptforbeinglimitedinspatialextent.Thefoldingofthesephase-spaceribbonswillleadtotheobservedripplesinthelightdistributionofanellipticalgalaxywhichhasswallowedasmallergalaxy[ 59 60 61 ].Thereisnoexplanationotherthantheexistenceofcausticsforthepresenceoftheseripplesinellipticalgalaxies.Theexistenceofcausticsofvisiblematterfurthersupportstheexpectationthatdarkmattercausticsarepresentingalactichalos.Whilevirializationwillthermalizethehaloanddestroytheoldestows,owswillbepresenttodayfromparticleswhichhaveonlylatelyfallenontothehalo.Theseparticleswillnothavehadsucienttimetothermalizewiththerestofthehalo.

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Discreteowsareexpectedtocontainasignicantfractionofthelocalhalodensity,asdiscussedinSection 3.3 .Discreteowsproduceadistinctsignalinanaxiondetector.Aseriesofnarrowpeaks,oneperow,willappearinthespectraoutput.Thewidthofeachpeakisproportionaltothevelocitydispersionofthecorrespondingow.Thepowerineachpeakisdirectlyproportionaltothedensityofaxionsintheow.Suchnarrowpeakshavehighersignal-to-noiseratioinahighresolutionaxionsearch.Thus,ifasignicantfractionofthelocalhalodensityconsistsofaxionsinsuchows,ahighresolutionaxionsearchincreasestheexperimentsensitivitytoaxions.Furthermore,ifasignalisfound,itwillprovidedetailedinformationonthestructureofaxiondarkmatterwithinourgalaxy.

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33 62 63 64 65 66 ].Initspresentsearchmode,theADMXdetectorspendsapproximately50secondsateachcavitysetting.Asaresultitcanlookforfeaturesintheaxionfrequencyspectrumwitharesolutionoforder20mHz.ThispotentialhasrecentlybeenrealizedbybuildingtheHRchannel,whichbecamefullyoperationalinAugust2002.Itoerstheopportunitytoimprovethesensitivityoftheexperimentbysearchingforthespectralfeaturesexpectedfromthepresenceofdiscreteowsofdarkmatteraxions.IthasbeendemonstratedthattheHRchannelincreasesADMX'ssensitivitytoanaxionsignalbyafactorofthree[ 35 ].ADMXcanoperateitstwochannelssimultaneously.TheMRchannelsearchesforbroadsignals,withwidthoforder1kHzandaMaxwell-Boltzmannenergydis-tribution.TheHRchannelsearchesfornarrowsignalsarisingfromdiscreteaxionows.Eachdiscreteowproducesapeakintheaxionsignal.Thefrequencyatwhichapeakoccursisindicativeofthesquareofthevelocityofthecorrespondingowinthelaboratoryframe.Insearchingforcoldowsofaxions,itisassumedthattheowsaresteady,i.e.,theratesofchangeofvelocity,velocitydispersionandowdensityareslowcomparedtothetimescaleoftheexperiment.Theas-sumptionofasteadyowimpliesthatthesignalwearesearchingforisalwayspresent.Evenso,thesignalfrequencywillchangeovertimeduetotheEarth'srotationandorbitalmotion[ 67 68 ].Inadditiontoasignalfrequencyshiftindatatakenatdierenttimes,apparentbroadeningofthesignaloccursbecauseits 41

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frequencyshiftswhilethedataarebeingtaken.TheHRchannelhasafrequencyresolutionof0.019Hz.Toconductasearchwithoutmakingassumptionsaboutowvelocitydispersions,searchesareconductedforpeakpowerspreadacrossseveralbins.Werefertotheassociatedsumofpoweracrossnsinglebinsasn-binsearches.Thesesearchesareperformedforn=1,2,4,8,64,512and4096.ThischapterisonADMX'sHRchannelsearch[ 69 ].TheexperimentisdescribedinSection 4.2 .InSection 4.3 ,thesignalexpectedfromamicrowavecavitydetectorobservingacoldowofaxionsisdiscussed.ThedetectornoisecharacteristicsareanalyzedinSection 4.4 .Section 4.5 containsdetailsofthesystematiccorrectionsperformedonthedata.ThecompleteanalysisandaxionsignalsearchprocedureareinSection 4.6 .TheHRsearchhascoveredtheaxionmassrange1.98{2.17eV.Noaxionsignalwasfoundinthisrange.Exclusionlimitsonthedensityofaxionsinlocaldiscreteows,basedonthisresult,arepresentedinSection 4.7 .AdiscussionoftheresultsisinSection 4.8

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IntheKSVZmodel,g=0:97,whereasintheDFSZmodel,g=0:36.Theaxiondecayconstantisrelatedtoitsmassby 36 ].Asaxionsinthegalactichaloarenon-relativistic,theenergyofanysingleaxionwithvelocity,v,is 2mav2:(4{4)Theaxion-to-photonconversionprocessconservesenergy,i.e.,anaxionofenergy,Ea,convertstoaphotonoffrequency,=Ea=h.Whenfallswithinthebandwidthofacavitymode,theconversionprocessisresonantlyenhanced.Thesignalisapeakinthefrequencyspectrumofthevoltageoutputofthedetector.Thepowerdevelopedinthecavityduetoresonantaxion-photonconversionis[ 36 ] mamin(Q;Qa);(4{5)whereVisthecavityvolume,B0isthemagneticeldstrength,aisthedensityofgalactichaloaxionsatthelocationofthedetector,Qaistheratiooftheenergyofthehaloaxionstotheirenergyspread,equivalenttoa\qualityfactor"forthehaloaxionsignal,andCisamodedependentformfactorwhichislargestinthefundamentaltransversemagneticmode,TM010.Cisgivenby

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numerically.Eq.( 4{5 )canberecastintheconvenientform, 105: AschematicofADMX,showingboththeMRandHRchannels,isgiveninFig. 4{1 .Amoredetailedillustrationofthemagnet,cavityandcryogeniccomponentsisshowninFig. 4{2 .Themicrowavecavityhasaninnervolume,V,of189L.ThefrequencyoftheTM010modecanbetunedbymovingapairofrodsinside.Therodsmaybemetalordielectricandcanbereplacedasnecessarytoreachthedesiredfrequencyrange.Thecavityislocatedintheboreofasuperconductingsolenoid,whichgeneratesamagneticeld,B0,of7.8T.Thevoltagedevelopedacrossaprobecoupledtotheelectromagneticeldinsidethecavityispassedtothereceiverchain.Astheexperimentoperateswiththecavityatcriticalcoupling,halfthepowerdevelopedinthecavityislosttoitswallsandonlyhalfispassedtothereceiverchain.Duringoperation,thequalityfactorofthecavity,Q,isapproximately7104andthetotalnoisetemperaturefortheexperiment,Tn,isconservativelyestimatedtobe3.7K,includingcontributionsfromboththecavityandthereceiverchain.TherstsegmentofthereceiverchainiscommontoboththeMRandHRchannels.ItconsistsofacryogenicGaAsHFETamplierbuiltbyNRAO,acrystalbandpasslterandmixers.Attheendofthissegment,thesignaliscenteredat35kHz,witha50kHzspan.TheMRsignalissampleddirectlyafterthispartofthereceiverchain.TheHRchannelcontainsanadditionalbandpasslterandmixer,resultinginaspectrumcenteredat5kHzwitha6kHzspan.Timetracesofthevoltageoutputfromthereceiver,consistingof220datapoints,aretakenwithsamplingfrequency20kHzintheHRchannel.Thisresultsinadatastreamof52.4sinlength,correspondingto0.019Hzresolutioninthe

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Schematicdiagramofthereceiverchain. frequencyspectrum.ThedatawereprimarilytakeninparallelwiththeoperationsoftheMRchanneloveraperiodbeginninginNovember,2002andendingMay,2004.ContinuousHRcoveragehasbeenobtainedandcandidatepeakeliminationperformedforthefrequencyrange478{525MHz,correspondingtotheaxionmassrange1.98{2.17eV.DatawithQlessthan40000and/orcavitytemperatureabove5Kwerediscarded.Whenthiswasthecase,additionaldataweretakentoensurecoverageofthefullrange. 4{4 )andthecorrespondingfrequencychangeovertimeduetotheEarth'srotationalandorbitalmotions.Inadditiontoasignalfrequencyshift

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SketchoftheADMXdetector.

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indatatakenatdierenttimes,apparentbroadeningofthesignaloccursbecauseitsfrequencyshiftswhilethedataarebeingtaken.UsingEq. 4{4 ,oneseesthatratiooftheshiftinfrequency,f,tothebasefrequency,f,duetoachangeinvelocity,v,is f f=mavv mac2+1 2mav2(4{8) =)f=fvv c2:(4{9)ThevelocityofadarkmatterowrelativetotheEarthwillbeintherange1001000km/s.Wechosev=600km/sasarepresentativevalueforthepurposeofestimation.Afrequencyoff=500MHzischosenastypicalforthedataunderconsideration.ThemagnitudeofthevelocityonthesurfaceofEarthattheequatorduetotheEarth'srotationisvR=0:4km/s.Itislessthanthisatthelocationoftheaxiondetector,butthisvalueisusedforthepurposeofillustration.AssumingtheextremecaseofalignmentoftheEarth'srotationalvelocitywiththeowvelocity,v=2vR.Theresultingdailysignalmodulationis3Hz.ApproximatingtheEarth'sorbitascircular,themagnitudeofit'svelocitywithrespecttotheSunisvT=30km/s.Again,consideringtheextremecaseofvelocityalignment,thefrequencymodulationduetotheorbitofEartharoundtheSunisatmost200Hz.ThebandwidthoftheHRchannelis6kHz.Afteridentifyingcandidatefrequencies,theyarereexaminedtoseeiftheysatisfythecriterionofaconstantlypresentsignal.Thus,ifthespectrumiscenteredonthecandidatefrequencywhenitisreexamined,thesignalwillstillbewithinthedetectorbandwidthasitwillmoveatmost200Hzfromitsoriginalfrequency.Inaddition,boththerotationoftheEarthanditsmotionaroundtheSunwillresultinasmallchangeintheowvelocityrelativetothedetectorwhile

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eachspectrumistakenandasubsequentincreaseinthesignalline-widthrelativetowhatwouldbeexpectedinthestaticcase.SimilarlytoEq.( 4{9 ),thesignalbroadening,f,duetoachangeintheowvelocity,v,is c2:(4{10)Takingthetimeofintegrationtobet'50s,thechangeinrelativevelocityisatmost T(4{11)whereTistheperiodofthemotion(diurnalorannual)andvmistherespectivevelocity(vRorvT).Theline-widthisincreasedby4103HzduetotheEarth'srotation.TheEarth'sorbitalmotionincreasestheline-widthby103Hz.ThespectralresolutionoftheHRchannelis0.019Hz,largeenoughtomaketheseeectsnegligible.Forowsofnegligiblevelocitydispersion,thesensitivityoftheexperimentisproportionaltothefrequency,f,andthetimeofintegration,t,providedtheresolution,B=1=t,islessthantheshiftofthesignalfrequencyduringmeasurement.Thisrequirementallowsameasurementintegrationtimeaslongas 2:(4{12)Thissuggeststhatforthedatathisnoteisbasedon,amoresensitivelimitcouldhavebeenachievedwithalongerintegrationtimethantheactual52s.Thevelocitydipersionoftheowmay,however,bealimitingfactor.WhilenovalueforvelocitydispersionisassumedinperformingtheHRanalysis,forillustrativepurposes,letusconsideraparticularcase:the\BigFlow,"discussedbySikivie[ 45 ].Theupperboundonthevelocitydispersionofthisowisv.50m/s.ThisleadstoamaximumlinebroadeningoffBF.8102Hz,i.e.,asignalfromaxionsintheBigFlowisspreadoverfourfrequencybinsinthe

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detectorspectrumifthelimitv.50m/sissaturated.Letusemphasize,however,thatthereisnoreasontobelievethisboundissaturated.Ingeneral,wedonotknowthevelocitydispersionofthecoldaxionowsforwhichwesearch.Subsequently,wedonotknowthesignalwidth.Tocompensate,searchesareperformedatmultipleresolutionsbycombining0.019Hzwidebins.Thesesearchesarereferredtoasn-binsearches,wheren=1,2,4,8,64,512and4096.Forf=500MHzandv=600km/s,thecorrespondingowvelocitydispersionsare 4.6 4{15 )hasbeenveriedexperimentallybyallowingthecavitytowarmandobservingthatisproportionaltoTC.ThenoiseintheHRchannelhasanexponentialdistribution.Thenoiseina1-binisthesumofindependentsineandcosinecomponents,asnoaveragingisperformedinHRsampling.Theenergydistributionshouldbeproportionaltoa

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Boltzmannfactor,exp(E=kT),andnon-relativisticandclassicalenergies,suchasE=(1=2)mv2orE=(1=2)kx2,areproportionaltosquaresoftheamplitude.Thus,thenoiseamplitude,a,forasinglecomponent(i.e.,sineorcosine)hasaGaussianprobabilitydistribution, da=1 dpn=2nYi=1Z1dai!exp(1 22aP2nj=1a2j) (p dpn=pnn1 dp1=1 4{19 ),thenoisepowerdistributionfunctionbecomes dp1=1 4{20 ), ep ;(4{21)

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Powerdistributionforalargesampleof1-bindata. whereNisthetotalnumberoffrequencies.As lnNp=p +lnNp ;(4{22)istheinverseoftheslopeofthelnNpversuspplot.Figure 4{3 demonstratesthatthedataisingoodagreementwiththisrelationforplessthan20.ThedeviationofthedatafromEq.( 4{22 )forpgreaterthan20isduetothefactthatthebackgroundisnotpurenoise,butalsocontainsenvironmentalsignalsofanon-statisticalnature.Aswecombineanincreasingnumberofbins,thenoisepowerprobabilitydistributionapproachesaGaussian,inaccordancewiththecentrallimittheorem.Theright-handsideofEq.( 4{18 )approachesaGaussianinthelimitoflargen.Wehaveexaminedalargesampleofnoiseineachn-binsearchandveriedthatitisdistributedaccordingtoEq.( 4{18 ).Figures 4{4 through 4{9 showthe

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Powerdistributionforalargesampleof2-bindata. progressionfromtheexponentialdistributionofFig. 4{3 toanearGaussiancurveforthe4096-binsearch.Inadditiontoexaminingthebehaviorofthenoisestatistics,wehaveper-formedacross-calibrationbetweentheHRandMRchannels.Thesignalpowerofanenvironmentalpeak,observedat480MHzandshowninFig. 4{10 ,wasexaminedinboththeHRandMRchannels.TheobservedHRsignalpowerwas(1:80:1)1022W,wheretheerrorquotedisthestatisticaluncertainty.TheMRchannelobservedsignalpower1:71022W,inagreementwiththeHRchannel.NotethattheMRsignalwasacquiredwithamuchlongerintegrationtimethanthatoftheHRsignal(2000forMRversus52sforHR).Thecombinationofthecalibrationofthenoisepowerwithcavitytemperature,theconsistencybetweenexpectedandobservednoisestatisticsandtheagreement

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Powerdistributionforalargesampleof4-bindata. ofsignalpowerobservedinboththeHRandMRchannels,makesuscondentthatthesignalpowerisaccuratelydeterminedintheHRchannel. 4{1 .TwopassbandltersarepresentontheHRreceiverchain:onewithbandwidth35kHzonthesharedMR-HRsectionandapassiveLClterofbandwidth6kHz,seenbytheHRchannelonly.Thecombinedresponseofboththeseltershasbeenanalyzedandremovedfromthedata.Thesecondsystematiceectisduetothefrequency-dependentresponseofthecouplingbetweenthecavityandtherstcryogenicamplier.Thiseectisremovedusingtheequivalentcircuitmodeldescribedlater.Thecombinedpassbandlterresponsewasdeterminedbytakingdatawithawhitenoisesourceattherfinputofthereceiverchain.Atotalof872timetraces

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Powerdistributionforalargesampleof8-bindata. wererecordedoveratwodayperiod.Inordertoachieveareasonablysmoothcalibrationcurve,512binsinthefrequencyspectrumforeachtimetracewereaveragedgiving9.77Hzresolution.ThecombinedaverageofalldataisshowninFig. 4{11 .ThismeasuredresponsewasremovedfromalldatausedintheHRsearch,asfollows.Therawpowerspectrahavefrequency0{10kHz,wherethecenterfrequencyof5kHzhasbeenmixeddownfromthecavityfrequency.Eachrawpowerspectrumiscroppedtotheregion2{8kHztoremovethefrequenciesnotwithintheLClterbandwidth.Eachremainingfrequencybinisthenweightedbyafactorequaltothereceiverchainresponseatthegivenfrequencydividedbythemaximumreceiverchainresponse.Interpolationforfrequencypointsnotspecicallyincludedinthecalibrationcurveisperformedbyassumingthateachpointonthecalibrationcurvewasrepresentativeof512binscenteredonthatfrequency,soallpowercorrespondingtofrequencieswithinthatrangeisnormalized

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Powerdistributionforalargesampleof64-bindata. bythesamefactor.Asthecalibrationcurvevariesslowlywithfrequencywithinthewindowtowhicheachspectraiscropped,thisisanadequatetreatmentofthenormalization.IntheMRchannel,theeectofthecavity-ampliercouplingisdescribedusinganequivalent-circuitmodel[ 70 ].ThismodelhasbeenadaptedforuseintheHRchannel.Thefrequencydependentresponseofthecavityampliercouplingismostevidentinthe4096-binsearch,thusthisisthedatausedtoapplytheequivalentcircuitmodel.AsamplespectrumbeforecorrectionisshowninFig. 4{12 .Intheequivalent-circuitmodel,eachfrequencyisgivenby,thenumberofbinsitisosetfromthebinofthecenterfrequency,measuredinunitsofthe4096-binresolution,i.e.b4096=78:1Hz.Theequivalent-circuitmodelpredictsthatthepower(inunitsofthermsnoise)attheNRAOamplieroutput(thepoint

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Powerdistributionforalargesampleof512-bindata. labelled\RF"inFig. 4{1 )inthe4096{binsearchatthefrequencyosetis (4{27) Intheaboveexpressions,TCisthephysicaltemperatureofthemicrowavecavity,TIandTVarethecurrentandvoltagenoise,respectively,contributedbythe

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Powerdistributionforalargesampleof4096-bindata. amplier,Tnisthenoisetemperaturecontributedfromallcomponents,bisthefrequencyresolutionoftheHRchannel,i.e.0.019Hz,Listheelectrical(cable)lengthfromthecavitytotheHFETamplifer,f0isthecavityresonantfrequency,fcenisthecenterfrequencyofthespectrumandkisthewavenumbercorrespondingtofrequencyfcen+b.Thefactorb4096=bappearsintheparametersa1,a3anda4asitisanoverallfactorwhichresultsfromnormalizingthepowertothesinglebinnoisebaseline.Inpractice,theparametersa1througha5areestablishedbytting.ThelineinFig. 4{12 showsthetobtainedusingtheequivalentcircuitmodel.Largepeaksinthedata,e.g.anaxionsignalorenvironmentalpeak,arere-movedbeforettingtopreventbias.The4096-binspectrumisusedtoperformthetandthentheoriginal1-binspectrumiscorrectedtoremovethesystematiceect.TheweightingfactorsarecalculatedusingEq.( 4{23 )andthettedpa-rameters,a1througha5,atthecenterofeachbinofwidthb4096.Thesefactorsare

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AnenvironmentalpeakasitappearsintheMRsearch(top)andthe64-binHRsearch.Theunitfortheverticalaxisisthermspoweructuationineachcase.

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HRlterresponsecalibrationdata(512binaverage).Thepowerhasbeennormalizedtothemaximumpoweroutput.

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Sample4096-binspectrumbeforecorrectionforthecavity-ampliercoupling.Thelineisthetobtainedusingtheequivalentcircuitmodel.

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Thesame4096-binspectrumofFig. 4{12 aftercorrectionforthecavity-ampliercoupling. theratioofthetatagivenpointtothemaximumvalueofthet.Each1-binismultipliedbythefactorcalculatedforthebinofwidthb4096withinwhichitfalls.Figure 4{13 showsthespectrumofFig. 4{12 afterremovalofsystematiceects.Theremovalofthecavity-ampliercouplingandthepassbandlterresponseusingthetechniquesdescribedaboveresultsinatHRspectra. 4{10 )),thelatterbeingthemostuncertainvariable.n-binsearches,wherenisthenumberofadjacent1-binsaddedtogether(n=1,2,4,8,64,512and4096),areconductedtoallowforvarious

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1{binsearch: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2{binsearch: 12 34 56 78 910 1112 1314 1516 23 45 67 89 1011 1213 1415 4{binsearch: 1234 5678 9101112 13141516 3456 78910 11121314 8{binsearch: 12345678 910111213141516 56789101112 Figure4{14. Illustrationoftheadditionschemeforthe2,4and8-binsearches.Thenumberscorrespondtothedatapointsofthe1-binsearch.Num-berswithinthesameboxarebinsaddedtogethertoformasingledatuminthen-binsearcheswithn>1. velocitydispersions.Forsearcheswithn>1,thereisanoverlapbetweensuccessiven-binssuchthateachn-binoverlapswiththelasthalfofthepreviousandrsthalfofthefollowingn-bin.Thisschemeisillustratedforthe2,4and8-binsearchesinFig. 4{14 .Thesearchforanaxionsignalisperformedbyscanningeachspectrumforpeaksaboveacertainthreshold.Allsuchpeaksareconsideredcandidateaxionsig-nals.Thethresholdsaresetatalevelwherethereisonlyasmallprobabilitythatapurenoisepeakwilloccurandsuchthatthenumberoffrequenciesconsideredascandidateaxionpeaksismanageable.Thecandidatethresholdsusedwere20,25,30,40,120,650and4500,inincreasingorderofn.Alltimetracesareanalyzedinthesamemanner.AfastFouriertransformisperformedandaninitialestimateofisobtainedbyttingthe1-binnoisedistributiontoEq.( 4{22 ).Systematiceectsarethenremoved,i.e.thecorrectionsdescribedinSection 4.5 forthelterpassbandresponseandcavity{ampliercouplingareperformed.\Large"peaksnotincludedintheequivalentcircuitmodeltforthecavity-amplierresponsearedenedtobethosegreaterthan120%ofthesearchthresholdforeachn{binsearch.Aftertheremovalofsystematiceects,the1-binnoisedistributionisagainttedtoEq.( 4{22 )toobtainthetruevalueofandthesearchforpeaksabovethethresholdstakesplace.

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Theaxionmassisnotknown,requiringthatarangeoffrequenciesmustbeexamined.FullHRcoveragehasbeenobtainedfortheregion478{525MHz,correspondingtoaxionmassesbetween1.98and2.17eV.Theselectedfrequencyrangeisexaminedinthreestagesforaxionpeaks,asfollows:Stage1:Datafortheentireselectedfrequencyrangeistaken.Thefrequencystepbetweensuccessivespectraisapproximately1kHz,i.e.thecenterfrequencyofeachspectrumdiersfromthepreviousspectrumby1kHz.Frequenciesatwhichcandidateaxionpeaksoccurarerecordedforfurtherexaminationduringstage2.Stage2:Multipletimetracesaretakenateachcandidatefrequencyfromstage1.Thesteadyowassumptionmeansthatapeakwillappearinspectratakenwithcenterfrequencyequaltothecandidatefrequencyfromstage1ifsuchapeakisanaxionsignal.Thefrequenciesofpersistentpeaks,i.e.peaksthatappearduringbothstage1and2areexaminedfurtherinstage3.Stage3:Frequenciesofpersistentpeaksundergoathree-partexamination.Therststepistorepeatstage2,toensurethepeaksstillpersist.Secondly,thewarmportattenuatorisremovedfromthecavityandmultipletimetracestaken.Ifthepeakisduetoexternalradiosignalsenteringthecavity(anenvironmentalpeak),thesignalpowerwillincreasedramatically.Ifthesignaloriginatesinthecavityduetoaxion-photonconversion,thepowerdevelopedinthecavitywillremainthesameasthatforthenormalconguration.Thethirdstepistouseanexternalantennaprobeasafurtherconrmationthatthesignalisenvironmental.Somedicultieswereencounteredwiththeantennaprobe,duetopolarizationofenvironmentalsignals.However,thesecondstepisadequatetoconrmthatpeaksareenvironmental.Ifapersistentpeakisdeterminedtonotbeenvironmental,analtestwillconrmthatitisanaxionsignal.Thepowerinsuchasignalmustgrowproportionallywiththesquareofthemagneticeld(B0inEq.( 4{7 ))anddisappearwhenthemagneticeldisswitchedo.

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Noaxionpeakswerefoundintherange478{525MHzusingthisapproach.Theexclusionlimitcalculatedfromthisdataisdiscussedinthefollowingsection. 4{13 ).SeveralfactorsreducethepowerdevelopedinanaxionpeakfromthatgiveninEq.( 4{7 ).Theexperimentisoperatednearcriticalcouplingofthecavitytothepreamplier,sothathalfthispowerisobservedwhenthecavityresonancefrequency,f0,ispreciselytunedtotheaxionenergy.Iff0isnotatthecenterofa1-bin,thepowerisspreadintoadjacentbins,asdiscussedbelow.Whentheaxionenergyiso-resonance,butstillwithinthecavitybandwidthatafrequencyf,theLorentziancavityresponsereducesthepowerdevelopedbyanadditionalfactorof 1+4Q2f f012:(4{29)Tobeconservative,wecalculatethelimitsatpointswheresuccessivespectraoverlap,i.e.atthefrequencyosetfromf0thatminimizesh(f).Ifanarrowaxionpeakfallsatthecenterofa1-bin,allpowerisdepositedinthat1-bin.However,ifsuchapeakdoesnotfallatthecenterofa1-bin,thepowerwillbespreadoverseveral1-bins.Wenowcalculatetheminimumpowerinasinglen-bincausedbyarandomlysituated,innitelynarrowaxionline.Thedatarecordedisthevoltageoutputfromthecavityasafunctionoftime.ThevoltageasafunctionoffrequencyisobtainedbyFouriertransformationandthensquaredtoobtainaraw\power"spectrum.Theactualpowerisobtainedbycomparisontothermsnoisepower.Thedataaresampledforaniteamountoftimeandthus,

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theFouriertransformationoftheoutput,F(f),willbeofthevoltagemultipliedbyawindowingfunction,i.e., 4{30 )isequivalentto 4{32 )andinsertingEq.( 4{33 ),wehave 2)b)sin((f b(m+1 2))) b(m+1 2));(4{34)wherebisthefrequencyresolutionoftheHRchannel,2Npointsaretakenintheoriginaltimetrace,andthecenterfrequencyofthejth1-binis(j+1=2)b.Thus,foranaxionsignaloffrequencyffallingin1-binj,afractionofthepower,

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Themaximumpowerlossoccurswhenasignalinthe1-binsearchfallsexactlybetweenthecenterfrequencyoftwoadjacent1-bins.Inthiscase,when==2,Eq.( 4{35 )showsthat40.5%ofthepowerwillbedepositedineachoftwo1-bins.Inn-binseacheswithn2,notasmuchpowerislosttoothern-bins,duetotheoverlapbetweensuccessiven-bins.Theminimumpowerdepositedinann-binis81%forn=2,87%forn=4and93%forn=8.Forn=64,512and4096,theamountofpowernotdepositedinasinglen-binisnegligible.Forthen-binsearcheswithn=64,512and4096,abackgroundnoisesub-tractionwasperformedwhichwillleadtoexclusionlimitsatthe97.7%condencelevel.Theselimitsarederivedusingthepoweratwhichthesumofthesignalpowerandbackgroundnoisepowerhavea97.7%probabilitytoexceedthecandi-datethresholds.Wecallthispowerthe\eective"thresholdforeachsearch.Theeectivethresholdsareobtainedbyintegratingthenoiseprobabilitydistribution,Eq.( 4{18 ),numericallysolvingforthebackgroundnoisepowercorrespondingtothe97.7%condencelevelforeachnandsubtractingthesevaluesfromtheoriginalcandidatethresholds.Forn=64,512and4096,theeectivethresholdsare71,182and531,respectively.Forsmallervaluesofn,backgroundnoisesubtractiondoesnotsignicantlyimprovethelimitsandtheeectivethresholdwastakentobethecandidatethreshold.Table 4{1 summarizesthisinformationandshowsthefrequencyresolutionofeachsearchwiththecorrespondingmaximumowvelocitydispersionfromEq.( 4{13 )forv=600km/s.Ourexclusionlimitswerecalculatedforanaxionsignalwithpowerabovetheeectivethresholdreducedbytheappropriatefactors.Thesefactorsarisefromthecriticalcoupling,theLorentziancavityresponseandthemaximumpowerlossduetothepeaknotfallinginthecenterofann-bin,asoutlinedabove.Equations( 4{7 )and( 4{15 )wereused,forbothKSVZandDFSZaxioncouplings.Thecavityvolume,V,is189L.Measuredvaluesofthequalityfactor,Q,themagneticeld,

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Table4{1. Eectivepowerthresholdsforalln-binsearches,withthefrequencyresolutions,bnandcorrespondingmaximumowvelocitydispersions,vn,foraowvelocityof600km/s. 1200.01962250.038104300.076208400.155064711.24005121829.8300040965317820000 Table4{2. Numericallycalculatedvaluesoftheformfactor,C,andampliernoisetemperatures,Tel,fromNRAOspecications. Frequency(MHz)CTel(K) 4500.431.94750.421.95000.411.95200.381.95500.362.0 4{2 .Theelectronicnoisetemperature,Tel,wasconservativelytakenfromthespecicationsoftheNRAOamplier,thedominantsourceofnoiseinthereceiverchain,althoughourmeasurementsindicatethatTelislessthanspecied.ThesevaluesarealsogiveninTable 4{2 .LinearinterpolationbetweenvaluesatthefrequenciesspeciedwasusedtoobtainvaluesofCandTelatallfrequencies.The2-binsearchdensityexclusionlimitobtainedusingthesevaluesisshowninFig. 4{15 .Forvaluesofnotherthann=2,theexclusionlimitsdierbyonlyconstantfactors.Theconstantfactorsare1.60,1.00,1.12,1.39,2.53,5.90and17.2forn=1,2,4,8,64,512and4096,respectively.

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97.7%condencelevellimitsfortheHR2-binsearchonthedensityofanylocalaxiondarkmatterowasafunctionofaxionmass,fortheDFSZandKSVZacouplingstrengths.AlsoshownisthepreviousADMXlimitusingtheMRchannel.TheHRlimitsassumethattheowvelocitydispersionislessthanv2givenbyEq.( 4{13 ).

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4{15 ,isfromthe2-binsearch.Foraowvelocityof600km/srelativetothedetector,the2-binsearchcorrespondstoamaximumowvelocitydispersionof10m/s.The1-binsearchlimitislessgeneral,inthatthecorrespondingowvelocitydispersionishalfthatofthe2-binlimit.Itisalsolessstringent;muchmorepowermaybelostduetoasignaloccurringawayfromthecenterofabinthaninthen=2case.Forn>2,thelimitsaremoregeneral,butthelargerpowerthresholdofthesearchesmakethemlessstringent.Thelargestowpredictedbythecausticringmodelhasdensity1:71024g/cm3(0:95GeV/cm3),velocityofapproximately300km/srelativetothedetector,andvelocitydispersionlessthan53m/s[ 45 ].UsingEq.( 4{13 )withTable 4{1 andtheinformationdisplayedinFig. 4{15 multipliedbytheappropriatefactorsof1.12toobtainthe4-binlimit,itcanbeseenthatthe4-binsearch,correspondingtomaximumvelocity50m/sforv=300km/s,woulddetectthisowifitconsistedofKSVZaxions.ForDFSZaxions,thisowwouldbedetectedforapproximatelyhalfthesearchrange.TheselimitsandtheBigFlowdensityareillustratedinFig. 4{16 .Figure 4{15 demonstratesthatthehighresolutionanalysisimprovesthedetectioncapabilitiesofADMXwhenasignicantfractionofthelocaldarkmatterdensityisduetoowsfromtheincompletethermalizationofmatterthathasonlyrecentlyfallenontothehalo.TheadditionofthischanneltoADMXprovidesanimprovementofafactorofthreeoverourpreviousmediumresolutionanalysis.Itispossiblethatanevenmoresensitivelimitcouldhavebeenachievedwithalongerintegrationtime,asdiscussedinSection 4.3 .Thisissueshouldbeconsidered

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97.7%condencelevellimitsfortheHR4-binsearchonthedensityofanylocalaxiondarkmatterowasafunctionofaxionmass,forDFSZandKSVZacouplingstrengths.Densitiesabovethelinesareexcluded.Forcomparison,thepredicteddensityoftheBigFlowisalsoshown.TheHRlimitsassumethattheowvelocitydispersionislessthanv2givenbyEq.( 4{13 ).

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atthebeginningoffuturedatarunsinordertomaximizethediscoverypotentialoftheHRchannel.

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72

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resolutionchannelsearchesforaxionsinathermalizedcomponentoftheMilkyWayhalo.TheseaxionshaveaMaxwellianvelocitydistribution.Axionsindiscreteowshaveasmallvelocitydispersion,resultinginanarrowpeakinthespectrumoutputbythecavitydetector.Thehighresolutionchannelcansearchforthesepeakswithahighsignal-to-noiseratio,improvingdetectorsensitivity.Discreteowsareexpectedtobepresentinthehalofromtidaldisruptionofdwarfgalaxiesandfromlateinfallofdarkmatterintothegravitationalpotential.Darkmatterwhichhasonlyrecentlyfallenintothepotentialwillnothavehadsucienttimetothermalizewiththerestofthehalo.Examiningthephase-spacestructureofsuchparticlesshowsthatdiscreteowswilloccurduetothislateinfall.Therstanalysisforthischannelhasbeensuccessfullycompleted.Afteranalysisofthenoisebackgroundandremovalofsystematiceects,noaxionsignalwasfoundinthemassrange1.97{2.17eV.Abroadrangeofowvelocitydispersionswasconsideredbysearchingforsignalsacrossmultiplebinsbyaddingadjacentbinstogether.ThenewexclusionlimitsobtainedfromthehighresolutionchannelincreasethesensitivityoftheADMXdetectorbyuptoafactorofthreeoverthepreviousmediumresolutionresult.ThehighresolutionchannelthusenhancesADMX'sdetectionability.Shouldanaxionsignalbefound,thehighresolutionchannelwillalsoyieldvaluableinformationaboutthephase-spacestructureoftheMilkyWaygalactichalo.

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LeanneDelmaDuywasbornonAugust3,1975,inSydney,Australia.In1980,herfamilymovedtotheGoldCoast,Australia,wheresheattendedMiamiStatePrimarySchoolandMerrimacStateHighSchool.Aftergraduatingfromhighschool,shemovedtoBrisbane,Australia,wheresheobtainedaBachelorofSciencewithhonoursinphysicsfromTheUniversityofQueenslandin1997.Shespenttwoyearsworkingforanenvironmentalconsultingrm,PacicAir&Environment,PtyLtd,andthenmovedtoGainesville,Florida,tostudyforaDoctorofPhilosophyinphysics. 79


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Title: High Resolution Search for Dark Matter Axions in Milky Way Halo Substructure
Physical Description: Mixed Material
Copyright Date: 2008

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HIGH RESOLUTION SEARCH FOR DARK MATTER AXIONS IN MILKY
WAY HALO SUBSTRUCTURE
















By

LEANNE DELMA DUFFY


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

UNIVERSITY OF FLORIDA


2006















ACKNOWLEDGMENTS

This work is based on research performed by the Axion Dark Matter eX-

periment (ADMX). I am grateful to my ADMX collaborators for their efforts,

particularly in running the experiment and providing the high resolution data.

Without these efforts, this work would not have been possible.

I thank my advisor, Pierre Sikivie, for his support and guidance throughout

graduate school. It has been a priviege to collaborate with him on this and other

projects. I also thank Dave Tanner for his assistance and advice on this work.

I would like to thank the other members of my advisory committee, Jim Fry,

Guenakh Mitselmakher, Pierre Ramond, Richard Woodard and Fred Hamann,

for their roles in my progress. I am also grateful to the other members of the

University of Florida Physics Department who have contributed to my graduate

school experience.

I am especially grateful to my family and friends, both near and far, who have

supported me through this long endeavor. Special thanks go to Lisa Everett and

Ethan Siegel.















TABLE OF CONTENTS
page

ACKNOW LEDGMENTS ............................. ii

LIST OF TABLES ................................. v

LIST OF FIGURES ..................... ......... vi

ABSTRACT ................... .............. viii

CHAPTER

1 INTRODUCTION .................... ....... 1

2 AXIONS. .......... ................... ..... 7

2.1 Introduction . . . . . . . 7
2.2 The Strong CP Problem .......... ............. 7
2.3 The Axion ...................... ........ 11
2.3.1 Introduction ........................... 11
2.3.2 The Peccei-Quinn Solution to the Strong CP Problem .. 11
2.3.3 The Axion Mass ............... .... .. 15
2.3.4 The Axion Electromagnetic Coupling . . ..... 19
2.4 Axions in Cosmology .................. ....... .. 21

3 DISCRETE FLOWS AND CAUSTICS IN THE GALACTIC HALO 27

3.1 Introduction .................. ............ .. 27
3.2 Existence .................. ............. .. 27
3.3 Densities .................. .............. .. 37
3.4 Discussion .. ... .. .. .. .. .. .. .. .. .. .... .. .. 39

4 HIGH RESOLUTION SEARCH FOR DARK MATTER AXIONS .... 41

4.1 Introduction .................. ............ .. 41
4.2 Axion Dark Matter eXperiment ............. ... .. .. 42
4.3 Axion Signal Properties .................. .... .. 45
4.4 Noise Properties .................. .......... .. 49
4.5 Removal of Systematic Effects ...... ......... .. 53
4.6 Axion Signal Search ............... . 61
4.7 R results . . ... . . . ..... . 64
4.8 Discussion ............... ............. .. 69









5 SUMMARY AND CONCLUSION ............ ....... .... 72

REFERENCES ................................... 74

BIOGRAPHICAL SKETCH ........ .................... 79















LIST OF TABLES


Table page

4-1 Effective power thresholds for all n-bin searches, with the frequency res-
olutions, b, and corresponding maximum flow velocity dispersions, 6v',
for a flow velocity of 600 km/s. ............. .... 67

4-2 Numerically calculated values of the form factor, C, and amplifier noise
temperatures, T1, from NRAO specifications. .. . ..... 67















LIST OF FIGURES
Figure page

3-1 A 2-D slice of 6-D phase-space. The line is a cross-section of the sheet
of width 6v on which the dark matter particles lie prior to galaxy for-
mation. The wiggles are the peculiar velocities due to density perturba-
tions. When overdensities become non-linear, the sheet begins to wind
up clockwise in phase-space, as shown. .................... .. 30

3-2 The phase-space distribution of dark matter particles in a galactic halo
at a particular time, t. The horizontal axis is the galactocentric distance,
r, in units of the halo radius, R, and v is the radial velocity. Spherical
symmetry has been assumed for simplicity. Particles lie on the solid line. 31

3-3 The cross-section of the tricusp ring. Each line represents a particle tra-
jectory. The caustic surface is the envelope of the triangular feature, in-
side which four flows are contained. Everywhere outside the caustic sur-
face, there are only two flows. Illustration courtesy of A. Natarajan. 33

3-4 The tricusp ring caustic. Axial symmetry has been used for illustrative
purposes. Illustration courtesy of A. Natarajan. ........... .. 34

4-1 Schematic diagram of the receiver chain. ................ 45

4-2 Sketch of the ADMX detector. .................. ..... 46

4-3 Power distribution for a large sample of 1-bin data. .......... ..51

4-4 Power distribution for a large sample of 2-bin data. .......... ..52

4-5 Power distribution for a large sample of 4-bin data. .......... ..53

4-6 Power distribution for a large sample of 8-bin data. .......... ..54

4-7 Power distribution for a large sample of 64-bin data. ......... ..55

4-8 Power distribution for a large sample of 512-bin data ........... ..56

4-9 Power distribution for a large sample of 4096-bin data. . .... 57

4-10 An environmental peak as it appears in the MR search (top) and the 64-
bin HR search. The unit for the vertical axis is the rms power fluctua-
tion in each case. . . . . .. . .. .. .58









4-11 HR filter response calibration data (512 bin average). The power has
been normalized to the maximum power output. ............. ..59

4-12 Sample 4096-bin spectrum before correction for the cavity v- ii1.1 ifr v cou-
pling. The line is the fit obtained using the equivalent circuit model. 60

4-13 The same 4096-bin spectrum of Fig. 4-12 after correction for the cavity-
amplifier coupling. .................. .. ...... 61

4-14 Illustration of the addition scheme for the 2, 4 and 8-bin searches. The
numbers correspond to the data points of the 1-bin search. Numbers within
the same box are bins added together to form a single datum in the n-
bin searches with n > 1. .................. .... 62

4-15 97.7'. confidence level limits for the HR 2-bin search on the density of
any local axion dark matter flow as a function of axion mass, for the DFSZ
and KSVZ a77 coupling strengths. Also shown is the previous ADMX
limit using the MR channel. The HR limits assume that the flow veloc-
ity dispersion is less than 6v2 given by Eq. (4-13). ............ ..68

4-16 97.7'. confidence level limits for the HR 4-bin search on the density of
any local axion dark matter flow as a function of axion mass, for DFSZ
and KSVZ a77 coupling strengths. Densities above the lines are excluded.
For comparison, the predicted density of the Big Flow is also shown. The
HR limits assume that the flow velocity dispersion is less than 6v2 given
by Eq. (4-13). . . . . . .. . . 70















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

HIGH RESOLUTION SEARCH FOR DARK MATTER AXIONS IN MILKY
WAY HALO SUBSTRUCTURE

By

Leanne Delma Duffy

August 2006

C('! r: Pierre Sikivie
M, 1 ri Department: Physics

The axion is one of the leading particle candidates for the universe's dark

matter component. Despite possessing very small couplings, the axion's interaction

with photons can be utilized to search for it using a microwave cavity detector. The

Axion Dark Matter eXperiment (ADMX) uses such a detector to search for axions

in our galactic halo.

ADMX has recently added a new, high resolution channel to search for axions

in discrete flows. ADMX's medium resolution channel searches for axions in the

thermalized component of the halo.

We review the motivation for the axion and its properties which make it

a good dark matter candidate. We also review the arguments for the existence

of discrete flows in galactic halos. A flow of discrete axions with small velocity

dispersion will appear as a very narrow peak in the output of a microwave cavity

detector. A high resolution search can detect such a peak with large signal to noise.

We have performed such a search.

The details of the high resolution axion search and analysis procedure are

presented. In this search, no axion signal was found in the mass range 1.98-2.17









peV. We place upper limits on the density of axions in local discrete flows based on

this result.















CHAPTER 1
INTRODUCTION

This work is on a new search for axion dark matter. The Axion Dark Matter

eXperiment (ADMX) has achieved improved sensitivity by implementing a high

resolution channel to search for axions in halo substructure. In this chapter, we

give background information on dark matter and outline the contents of this work.

The i i i i ily of the total energy density of the universe is contributed by com-

ponents that are not understood. Only approximately !.' of the energy budget is

contributed by baryonic matter, that is, particles which interact electromagnetically

and can thus be observed by radiation. The remaining contributions to the energy

budget come from components that are called dark matter and dark energy. The

dark matter component acts as matter, but interacts only gravitationally with the

observable baryonic matter. This component contributes approximately 2"'-. to

the total energy density. Dark energy acts as a fluid with negative pressure and

contributes the remaining 7!' The dark energy component is causing the recent

epoch of accelerated expansion of the universe [1, 2]. The dark matter component

is the concern of this dissertation.

Dark matter was first postulated by Fritz Zwicky in 1933 [3]. While observing

the Coma cluster of galaxies, he noted that the amount of visible matter was too

small for the system to be gravitationally bound. Given the observed galactic

velocities, the system should fly apart. Zwicky proposed additional matter that

was not visible to provide the necessary gravitational potential energy to bind the

system.

The strongest evidence for dark matter tod i- is provided by the rotation

curves of spiral galaxies. Plots of observed circular velocity against radial distance









are flat to large distances. The contributions to this curve from the disk are not

enough to support this rotation curve. It is thus believed that spiral galaxies

consist of a visible disk embedded in a much larger elliptical dark matter halo. For

a review of evidence for dark matter, see Bertone et al. [4].

Dark matter particles must have the following two properties: (1) They are

effectively collisionless as far as structure formation is concerned; i.e., the only

significant long-range interactions are gravitational, and (2) The dark matter must

be cold; i.e., it must be non-relativistic well before the onset of galaxy formation.

The first property means that dark matter can interact only weakly with

baryonic matter. The second property is necessary to form the structure in the

universe that we observe tod i-. If dark matter was more energetic, it would be

able to freely stream out of the initial density perturbations that have formed into

galaxies.

What the dark matter consists of is still unknown, despite knowledge of the

properties it must possess. The standard model of particle physics does not contain

a particle that can provide the dark matter of the universe. Extensions to the

standard model do, however, provide viable particle candidates. The leading dark

matter particle candidates are axions and weakly-interacting massive particles

(WIMPs).

The axion is the pseudo-Nambu-Goldstone boson from the Peccei-Quinn

solution to the strong CP problem [5, 6, 7, 8]. The axion mass, ma, is constrained

to lie in the range 10-6 to 10-2 eV [9, 10, 11]. There are two benchmark axion

models that are minimal extensions of the standard model: the Kim-Shifman-

Vainshtein-Zakharov (KSVZ) [12, 13] model and the Dine-Fischler-Srednicki-

Zhitnitsy (DFSZ) [14, 15] model. In the early universe, one population of axions

is produced by thermal processes and has temperature of order 1 K tod iv. In

addition, cold axion populations arise from vacuum realignment [16, 17, 18] and









string and wall decay [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Which

mechanisms contribute depends on whether the Peccei-Quinn symmetry breaks

before or after inflation. The cold axions were never in thermal equilibrium with

the rest of the universe.

WIMPs are a class of dark matter candidates: heavy particles that interact

via forces of weak-scale strength. WIMPs occur in many models, particularly in

extensions of the standard model which include a new parity symmetry to prevent

proton decay. The most popular WIMP candidate is arguably the lightest super-

symmetric particle (LSP), i.e., the lightest, ..--iet-undetected particle provided by

the minimal supersymmetric extension to the standard model ( LSSM). In contrast

to axions, WIMPs are thermal relics. They began in thermal equibrium with the

primordial heat bath. When the interaction rate of WIMPs with the rest of the

heat bath falls below the expansion rate of the universe, these particles decouple

or I. out." Their evolution is then governed by the universe's expansion and

gravitational interactions.

Much work is currently underway to detect dark matter and deduce its particle

properties. ADMX is a direct detection experiment searching for dark matter

axions [32, 33, 34, 35]. This experiment uses a tunable Sikivie microwave cavity

[36] to search for axions. When the resonant frequency of the cavity, v, corresponds

to the energy, Ea, of axions passing through the cavity (i.e., v = Ea/h), resonant

conversion of axions to photons will occur. The signal is a peak in the energy

spectrum of the output from the cavity. Many direct detection experiments are

also searching for WIMPs. WIMP direct detection is based on looking for nuclear

recoils from the elastic scattering of passing WIMPs. Additionally, attempts are

being made to detect dark matter indirectly using -1 i .i i cal signatures. These

primarily focus on detecting the products of WIMP annihilations: neutrinos,

positrons, anti-protons and gamma-rays.









These signals are dependent on the dark matter density. The power output

from resonant axion conversion to photons is proportional to the local axion

density. For WIMPs, the rate of nuclear recoil events from WIMP scattering is

proportional to the local WIMP density. Also, the flux of annihilation products

searched for in indirect detection is proportional to the square of the WIMP

density at the site of annihilation. Thus it is necessary to make assumptions about

the distribution of dark matter in our galactic halo. In particular, galactic halo

substructure is of interest for dark matter detection. The presence of substructure

in a galactic halo means that there will be regions of enhanced dark matter density,

improving detection prospects due to the signal dependence on density.

While the power in an axion signal observed by a microwave cavity detector is

proportional to the local axion density, the signal width is caused by the velocity

dispersion of dark matter axions. In searching for axions, it is thus also necessary

to make assumptions about their velocity distribution in the Milky Way halo. A

number of models are used to guide ADMX's search. These are the isothermal

model, the results from N-body simulations [37, 38] and a description of galactic

halos in terms of discrete flows from late infall of dark matter onto the galaxy. A

specific model which considers late infall is the caustic ring model [39, 40].

In the isothermal model, it is assumed that the dark matter halo has ther-

malized via virialization and thus has a Maxwell-Boltzmann velocity distribution.

ADMX's medium resolution (i lR) channel [34] searches for such axions, assuming

that the velocity dispersion is 0(10-3c) or less, where c is the velocity of light.

(The escape velocity from our galaxy is approximately 2 x 10-3c.)

Of particular interest to this work is the existence of cold flows of dark matter

axions within the halo. Such flows are associated with the tidal disruption of

subhalos predicted by N-body simulations and with late infall of dark matter onto

the galactic halo.






5


Numerical simulations indicate that hundreds of smaller clumps, or subhalos,

exist within the larger halo [37, 38]. Tidal disruption of these subhalos leads to

flows in the form of "tidal tails" or -I. ,I i- The Earth may currently be in a

stream of dark matter from the Sagittarius A dwarf galaxy [41, 42].

Non-thermalized flows from late infall of dark matter onto the halo are also

expected [43, 44]. Dark matter that has only recently fallen into the gravitational

potential of the galaxy will have had insufficient time to thermalize with the rest

of the halo. This dark matter will be present in the halo in the form of discrete

flows. There will be one flow of particles falling onto the galaxy for the first time,

one due to particles falling out of the galaxy's gravitational potential for the

first time, one from particles falling into the potential for the second time, etc.

Furthermore, where the gradient of the particle velocity diverges, particles "pile

up" and form caustics. In the limit of zero flow velocity dispersion, caustics have

infinite particle density. The velocity dispersion of cold axions at a time, t, prior to

galaxy formation is approximately 6va ~ 3 x 10-17(10-5eV/mT)(to/t)2/3 [40], where

to is the present age of the universe. Thus, a flow of dark matter axions will have a

small velocity dispersion, leading to a large, but finite density at the location of a

caustic.

The caustic ring model predicts that the Earth is located near a caustic feature

[45]. This model, fitted to bumps in the Milky Way rotation curve and a triangular

feature seen in the IRAS maps, predicts that the flows falling in and out of the

halo for the fifth time contain a significant fraction of the local halo density. The

predicted densities are 1.7 x 10-24 g/cm3 and 1.5 x 10-25 g/cm3 [45], comparable

to the local dark matter density of 9.2 x 10-25 g/cm3 predicted by Gates et al. [46].

The flow of the greatest density is referred to as the "Big Flow."

The possible existence of discrete flows provides an opportunity to increase the

discovery potential of ADMX. A discrete axion flow produces a narrow peak in the









spectrum of microwave photons in the experiment and such a peak can be searched

for with higher signal-to-noise than the signal from axions in an isothermal halo. A

high resolution (HR) channel has been built to take advantage of this opportunity.

If a signal is found, the HR channel will also provide detailed information on the

structure of the Milky Way halo.

The HR channel is the most recent addition to ADMX, implemented as a

simple addition to the receiver chain, running in parallel with the MR channel.

This channel and the possible existence of discrete flows can improve ADMX's

sensitivity by a factor of thi [;,], significantly enhancing its discovery potential.

This work is arranged as follows. Background information on the axion is given

in C'! lpter 2. The strong CP problem of the standard model of particle physics,

the motivation for the axion, is described and the Peccei-Quinn solution, resulting

in the axion, is discussed. Properties important to axion detection, such as its mass

and coupling, are also reviewed. Some .,-I i. .1li, -ii J1 and cosmological consequences

of the axion are outlined, particularly the production of cold axion populations. A

review of discrete flows and caustics in the galactic halo is presented in Chapter 3.

C'! lpter 4 describes ADMX and provides details of the HR analysis. The new

result, improving ADMX's search sensitivity by a factor of three, is also shown. A

summary and conclusions are presented in C'! Ipter 5.















CHAPTER 2
AXIONS

2.1 Introduction

The axion is the pseudo-Nambu-Goldstone boson implied by the Peccei-Quinn

solution to the strong CP problem [5, 6, 7, 8]. It is also a good candidate for the

dark matter of the universe. This chapter provides background information on

axions. The strong CP problem is described in Section 2.2. In Section 2.3, the

Peccei-Quinn solution to this problem is outlined, using the original Peccei-Quinn-

Weinberg-Wilczek axion model. Axions are shown to be a natural consequence

of this solution. Axion properties important to detection are also reviewed.

Section 2.4 discusses cosmological aspects of axions, specifically how axion dark

matter arises and the limits that cosmology and .i-1 |i1, ,!-ii place on the axion

mass.

2.2 The Strong CP Problem

Quantum chromodynamics (QCD) is the theory of the strong nuclear forces.

Its gauge symmetry is SUc(3), the color symmetry group. In nature, QCD is not a

stand-alone theory. It is embedded within the standard model of particle physics.

The full gauge symmetry of the standard model is SUc(3) xSUL(2)x Uy(1), i.e., the

direct product of the color, left-handed and hypercharge symmetries, respectively.

The unified SUL(2) x Uy(l) forms the electroweak symmetry. Breaking of the

electroweak symmetry down to UEM(1) via the Higgs mechanism results in the W'

and Z boson masses and quark and lepton masses. In this section, we outline the

strong CP problem and explain that both QCD and the electroweak effects that

give the quarks mass combine to create this problem.









The Lagrangian of QCD is

S(a a + N g ca -a ( )
LQCD = GG + [iij7Y D/,q (mjqIJtQJ + h.c.)] + 1G""" (2-1)
j=1

The qi are the quark fields, the subscript i indicating each of the N = 6 quark

flavors and the additional subscript L or R denoting a left- or right-handed field.

The mi are the quark masses, g is the color coupling and the 7, are the gamma

matrices. The notation "h.c." stands for hermitian conjugate. The gluon field

strength tensor is

CG, PA -P A ga fabcAb A (22)

where Aa is the gluon vector potential, the superscripts referring to the eight

possible gluon color assignments, and fabc are the structure constants of SUc(3).

Acting on a spinor field, i, the covariant derivative, D., is

Da
D,, = (, +gA- a ) (2-3)

where the A4 are the Gell-Mann matrices. The final term of Eq. (2-1) is the "0-

term." The angle, 0, is a parameter and G0a" is the dual tensor to the gluon field

strength, defined by

Gap" = -e""G a (2-4)
2 (24)

with d"P", the Levi-Civita tensor.

The parameters of QCD are thus g, mj and 0. The color coupling, g is energy

dependent and in defining the theory, it is normally exchanged for the QCD

confinement scale, AQCD, of order 200 MeV. The parameter, 0, is the QCD vacuum

angle. This parameter is necessary to fully describe QCD because the SUc(3)

gauge symmetry is non-Abelian. Non-Abelian gauge potentials have disjoint

sectors, labelled by an integer topological winding number. These sectors are

disjoint as they cannot be transformed continuously into each other. There exists a









vacuum configuration corresponding to each n, between which quantum tunnelling

can occur. The gauge invariant QCD vacuum state is thus a superposition of vacua

of different n, i.e.,

10) e nOI)n. (2-5)

This is the origin of 0.

In the limit of massless quarks, QCD has a classical chiral symmetry, UA(1).

However, this symmetry is anomalous. The existence of the Adler-Bell-Jackiw

anomaly [47, 48] means that this symmetry is not present in the quantum theory.

In the full quantum theory, including quark masses, the physics of QCD remains

unchanged under the transformations,


qi eiiy/2qi (2-6)

mi C e-iim (2-7)
N
0 0 aj. (28)
i= 1

While the physics remains the same, this is not a symmetry because the

parameter 0 has changed. The transformations of Eq. (2-6) through Eq. (2-8) can

be used to move phases between the quark masses and 0. However, the quantity,


S0 arg(mlm2...mvN) (2-9)


is invariant and therefore observable, unlike 0. This is commonly written as


0 0 arg det M (2-10)


where M is quark mass matrix.

The 0-term violates the discrete parity symmetry (P) and the combined

operation of a parity transformation followed by charge conjugation (CP). If CP

was a good symmetry of the standard model, the 0-term would not be permitted.









However, this is not the case; CP violation has been observed in the electroweak

sector. Consequently, there is no apparent reason why the 0-term would not be

present in the standard model.

While CP violation is present in the electroweak sector of the standard model,

it has not been observed in QCD. An electric dipole moment for the neutron is the

most easily observed consequence of strong CP violation. The 0-term results in a

neutron electric dipole moment of [9, 49, 10, 11],


|dn ~ 10-16 ecm, (2-11)


where e is the electric charge. The current experimental limit is [50]


Sdn, < 6.3 x 10-26 cm (212)


thus |81 < 10-9. However, there is no reason to expect that 0 should be so close

to zero. Since CP violation is introduced in the standard model by allowing the

quark mass matrices to have arbitrary complex entries, 0 is naturally expected to

be of order one. This is the strong CP problem, i.e. the question of why the angle 0

should be nearly zero, when CP violation is present in the standard model.

A number of solutions to the strong CP problem have been proposed. The

Peccei-Quinn (PQ) solution [5, 6] results in the presence of an axion [7, 8], which

has the additional motivation of being a good candidate for the dark matter of the

universe. This solution is outlined in detail in the following section. Other solutions

include the up quark mass being zero and that CP is spontaneously broken. If the

bare up quark mass is zero, the 0 dependence of the QCD Lagrangian disappears

and the strong CP problem is solved. This solution is, however, disfavored by

lattice calculations and by the success of first order chiral perturbation theory in

reproducing the pattern of pseudo-scalar meson masses. The Nelson-Barr model

is an example of a theory where the strong CP problem is solved by properly









engineered spontaneous CP violation [51, 52]. We focus on only the PQ solution in

the following section.

2.3 The Axion

2.3.1 Introduction

This section provides important background information for axion detection.

In Section 2.3.2, we discuss the PQ solution to the strong CP problem. The

original Peccei-Quinn-Weinberg-Wilczek axion model is used for illustration, but

other axion models are also discussed. A derivation of the axion mass is given in

Section 2.3.3, using the methods of low energy effective theory. In Section 2.3.4,

the axion-electromagnetic coupling is reviewed. This coupling is the basis for

axion detection experiments. The resulting power developed in a microwave cavity

detector, using this coupling, is also given.

2.3.2 The Peccei-Quinn Solution to the Strong CP Problem

The Peccei-Quinn solution to the strong CP problem promotes 0 from a

parameter to a dynamical variable. To implement this mechanism, a global

symmetry, U(1)pQ, is introduced. This symmetry has a color anomaly and is

spontaneously broken. The resulting pseudo-Nambu-Goldstone boson is the

axion. The axion field, a, can be redefined to absorb the parameter 0. The non-

perturbative effects which make QCD 0 dependent result in a potential for the

axion field, causing it to relax to the CP conserving minimum and solving the

strong CP problem.

To realize the PQ solution, it is necessary to add new fields to the standard

model, otherwise there are no degrees of freedom available to accommodate the

axion. In the original, Peccei-Quinn-Weinberg-Wilczek (PQWW) axion model an

additional Higgs doublet was introduced. We review this model to demonstrate the

Peccei-Quinn mechanism in this section.









The simplest way to introduce additional degrees of freedom is via an extra

Higgs doublet. We assume that one of the Higgs doublets, 0,, couples to the up-

type quarks and the other, Qd, couples to the down-type quarks. We distinguish

between the up- and down-type quarks by labelling them ui and di, respectively

(rather than qj, as in the previous section). As there are N quarks, there are N/2

up-type quarks and down-type quarks. The leptons can acquire mass through

Yukawa couplings to either of the Higgs doublets or to a third Higgs doublet. We

ignore this complication here and simply examine the couplings to quarks. The

quarks acquire their masses from the expectation values of the neutral components

of the Higgs, Q% and j0. The mass generating couplings are

S y"u. ",UR + yi d + h.c. (2-13)


Peccei and Quinn chose the Higgs potential to be

V (,1 ,-d) 2 t 2, 7i, t, > (2-14)
ij i,3

where the matrices (aij) and (bi) are real and symmetric and the sum is over

the two types of Higgs fields. With this choice of potential, the full Lagrangian,

including the kinetic term and 0-term, has the following global invariance, UpQ(1):


SC i2au (2-15)

Se d (216)

ui e-i"Y5Ui (2-17)


di e-id sd (2-18)

0 0-N(a+ ad) (2-19)










Note that it is possible to write down lepton couplings which also observe the PQ

symmetry. It is necessary that these couplings do so, otherwise a potential term for

a will result, d, -1i. i-; the PQ mechanism.

When the electroweak symmetry breaks, the neutral Higgs components acquire

vevs:


( 0)


SvueiPu/v

diPdv .
VdCe


(2-20)

(2-21)


One linear combination of the Nambu-Goldstone fields, P, and Pd, is the longi-

tudinal component of the Z-boson, as per electroweak symmetry breaking in the

standard model. This combination is


h = cos 3P, sin 3Pd .


The orthogonal combination is the axion field,


a = sin 3, P, + cos OPd .


sin pOa + cos po h

cos Oa sin 3h .


Using Eqs. (2-20), (2-21), (2-24) and (2-25) in Eq. (2-13), the axion couplings to

quarks arise from


i t sin 4a
-m =mu e u aURi + m die
i Li i Li


'dRi + h.c. ,


where m" = y"'iv and m = 11' The axion field dependence can be removed

from the mass terms using the transformations of Eqs. (2-17), (2-18) and (2-19).

Direct couplings between the axion and quarks will still remain in the Lagrangian,


(2-22)


Thus,


(2-23)


(2-24)

(2-25)


(2-26)









through the associated change in the quark kinetic term. The resulting change in 0

is

S- N(v,/vd + vd/v,)a/v (2-27)

where v = v + v. The axion field can be redefined to absorb 0 on the right-

hand side of Eq. (2-27). Defining

2v
VPQ = (2-28)
Vu/Vd + dVu,

the 0-term of Eq. (2-1) is replaced by

2
La G G""" (2-29)
167 2VpQ P

Non-perturbative QCD effects explicitly break the Peccei-Quinn symmetry, but

do not become important until the universe cools to the quark-hadron transition.

These effects give the axion field a potential and when they become important, the

field relaxes to the minimum, which conserves CP. Hence the PQ mechanism, which

replaces 0 with the dynamical axion field, solves the strong CP problem.

However, the PQWW axion has been ruled out by observation. Under the

PQWW scheme, the axion mass is inherently tied to the electroweak symmetry

breaking scale, v. As VpQ ~ v and v = 247 GeV, the axion mass is of the order

of 100 keV. Such a heavy axion would have been observed at particle colliders and

has thus been ruled out. The calculation of the axion mass is reviewed in the next

subsection.

While the PQWW axion model is not viable, this does not, however, eliminate

the possibility of an axion solving the strong CP problem. "Invi- il!. axion

models, named such for their extremely weak couplings, are still possible. In an

invisible axion model, the PQ symmetry is decoupled from the electroweak scale

and instead is spontaneously broken at a much higher temperature, decreasing

the axion mass and coupling strength. Two benchmark, invisible axion models









exist: the Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) and Kim-Shifman-Vainshtein-

Zhakharov (KSVZ) models. In both the KSVZ and DFSZ models, an axion with

permissable mass and couplings results.

In the KSVZ model, only the Higgs doublet of the standard model occurs.

The axion is introduced as the phase of an additional electroweak singlet scalar

field. The known quarks cannot be directly coupled to such a field, as the Yukawa

couplings would lead to unreasonably large quark masses. Instead, this scalar

is coupled to an additional heavy quark, also an electroweak singlet. The axion

couplings are then induced by the interactions of the heavy quark with the other

fields.

The DFSZ model has two Higgs doublets, as in the PQWW model, and an

additional electroweak singlet scalar. It is the electroweak singlet scalar which

acquires a vev at the PQ symmetry breaking scale. The scalar does not couple di-

rectly to quarks and leptons, but through it's couplings to the two Higgs doublets.

Thus, it is possible for the existence of an axion to solve the strong CP

problem. While significant for that alone, the axion also provides an interesting

candidate for the cold dark matter of the universe.

2.3.3 The Axion Mass

We review how the axion mass can be obtained from the low-energy effective

field theory, using the chiral Lagrangian. For this purpose, we consider only

the two lightest quarks, up and down. The chiral Lagrangian is invariant under

SUL(2) x SUR(2) x Uv(1). We will introduce an extra UA(1) symmetry, but break

it explicitly by giving a large mass to the eta particle. Indeed, the group UA(1) is

not actually a symmetry of QCD, as it is broken at the quantum level by instanton

effects. The symmetries are spontaneously broken down to SUL+R(2) x Uv(1) by

the quark condensation at the quark-hadron transition. At the scale, A, the quark









condensate acquires expectation value


(q<9 j)o -A3U() (2-30)

where 7r(x) is the pion field. The scale, A, is of the order of AQCD, but not equal to

it. The matrix U is given by

U() = exp ,i (2 31)

where r is the Pauli matrices and f, is the pion decay constant, equal to 93 MeV.

For the SUL+R(2) triplet, the pions, the effective Lagrangian is

f2
L, = Tr(0,UtaPU) + A3Tr(mU + h.c.) (2-32)

where mq is the diagonal quark mass matrix,


m, = (7 0 (2-33)
0 rrmd

Expansion of the Lagrangian shows the pion mass to be

2 3 Md+ (3d
m = 2A3 (2-34)

To find the axion mass, we also need to introduce the would-be Nambu-

Goldstone boson associated with the spontaneous breaking of the UA(1) symmetry.

We denote this particle as Tl (eta) in the following (this state is actually some linear

combination of the rl and rl' pseudo-scalar mesons). The expectation value of the

quark condensate becomes


(qqLu)o= --A3U(r) exp f) (2-35)
L \J 1 A





17


where f, is the T] decay constant. The effective Lagrangian is

1 f,2
L7,n ,T989" + Tr(, Ut8WU)

+A3Tr(mqUexp ,) + h.c.) + -m, cs (2-36)

where the final term is the potential for rotations in the UA(1) direction.
When the 0-term is included, the expectation value of the condensate remains
that given in Eq. (2-35), except that is replaced by + 2. Indeed, under a
UA(1) transformation,

qj eg q, (2-37)
to
rriq e 2Tmq (2 38)

f + (2-39)
fA 2 f,
In an axion model, 0 is replaced by 0 + N, where Va is the scale at which the PQ
symmetry breaks. The constant N is defined by the anomaly,

N = 2 ptf, (2-40)
f
where pf is the appropriate charge and tf is the second casimir operator of the
algebra. The axion decay constant, f, is defined by

a = (2-41)
N

The effective Lagrangian, including the 0-term and the axion field, is

1 1 f2
7na 2 0,aaa + 2989a + f-Tr(aUtaPU)

+A3Tr(,qU exp (f) + h.c.) + mi os 2 cos 0 + (2-42)

The variable 0 defines the origin of the axion field, so we may choose this to be
zero. The quark mass matrix can be written as a real matrix times a phase and









we may rotate to move 0 onto the quark mass term. This illustratres that the 0

dependence is alv--i a dependence on 0.

The eta, neutral pion and axion fields mix, as they all have the same quantum

numbers. Firstly, consider rf-a mixing. The physical rl field is

Naf,
lphys = r + f (2-43)
2va

and we use the redefinition
Nf
a' = a (244)
2va
As the minimum of the potential occurs when the cosine term is zero, we may set

rlphys to zero. The new Lagrangian is

1 f2 _iTa' + (
aa,a Oa'a' + Tr( ,UtaPU) + A3Tr(mUexp + h.c.) (245)
2 4 k 2Va

We find that the minimum of the potential occurs at r = 0 and a' = 0. The

physical neutral pion and axion fields are

md mf a'+ O(f (2-46)
md + m,2f f( )
Smd m f, o, (f
aphys =a + 0 (2-47)
md+m, 2fa f j

with corresponding masses

mo A md+ o (2-48)

m mumd o+ Of (2 49)
a f2i(m, +md)

S( (2o50)



expressed as [9, 49, 10, 11]


(10 12GeV (251)
ma 6 x 10-6 eV (012Ge (2-5 1)
\ Ja









2.3.4 The Axion Electromagnetic Coupling

Axion detection is based on its electromagnetic coupling [36]. We discuss how

this coupling arises in effective field theory and review the power developed in a

microwave cavity experiment in this section.

The axion electromagnetic coupling is due to mixing between the axion,

neutral pion and eta. The couplings of the Lagrangian for any of these particles to

decay to two photons is

a (r 5 rTI N, a) F\ P
Cw/o/o--,,_ 4 j + y + F JF^Mt' (2-52)
47 f7 3 f, 2 Va

The coefficients in the above equation arise from the trace over the anomaly loop.

The constant N, is given by


N, 2 pf(ef)2 (2-53)
f

where pf is the PQ charge of a right-handed quark field. Using the definition of the

physical axion field given in Eqs. (2-44) and (2-47), the resulting axion coupling to

two photons is

r,7 = g7 Faa FL"V p (2-54)
47 fa
where
12 NI 3 md + n(

and we have relabelled the physical axion field as a. In grand unified theories, N,

and N are related, with N/N = 8/3. In this case, g, 0.36. Both the PQWW

and DFSZ axion models are grand-unifiable. In the KSVZ axion model, this is not

the case. The introduction of an additional heavy neutral quark means that the

KSVZ axion model cannot fit within a grand unified theory. In this case, N, = 0,

as the up and down quarks carry no PQ charge, and g, = -0.97.









The full Lagrangian for the interaction of axions with photons in free space is

[36]

1 ac 1 1 a2
L = F, + g- FgF + -,a8l"a -m+a2 2 + O (2-56)
4 4fa 2 2 v

In terms of the electric and magnetic fields, E and B, and introducing a medium

with dielectric constant, c, Eq. (2-56) can be written as

1 1 1 22 a
= (cE2 B- 2) + -aaaa a- _m -g -E B (2-57)
2 2 2 47Tfa

In a cavity permeated by a -1I i.- inhomogeneous magnetic field, resonant

conversion of axions to photons can be induced if the cavity frequency corresponds

to that of the axion energy. The resulting power developed in a microwave cavity

detector is
= ( VB min(Q,Qa) (2-58)
\7jfa/ ma
where V is the cavity volume, Bo is the magnetic field strength, pa is the local

density of axions with energy corresponding to the cavity frequency, Q is the

quality factor of the cavity and Qa is the ratio of the energy of halo axions to

their energy spread, equivalent to a "quality factor" for the halo axion signal. The

mode-dependent form factor, C, is given by

|I d3xE, Bo 2
C B2V/f d3 E2 1 (2 59)
BV fv xe E,| '

in which E,(x) is the time-dependent electric field of the mode under consideration

and c is the dielectric constant of the medium inside the cavity. This is more

conveniently expressed as

P = 0.5 x 10-21W (V BO) x 1024g.cm
500L 7T 0.36 0.5 x -24g.-3
( va min [Q, Qa] (260)









where Va is the axion energy frequency.

Thus, when such a cavity is tuned to the correct frequency, resonant con-

version of axions to photons results. This conversion is observed as a peak in the

frequency spectrum of the detector output.

2.4 Axions in Cosmology

Axions may p1 iv an important role in cosmology. We focus on two aspects of

this here. Firstly, for a mass in the range 10-6 10-4 eV, the axion is an interesting

dark matter candidate. Secondly, we outline the restrictions that cosmology and

.1- i' r,-lics place on the axion mass and coupling.

Axions satisfy the two criteria necessary for cold dark matter: (1) a non-

relativistic population of axions could be present in our universe in sufficient

quantities to provide the required dark matter energy density and (2) they are

effectively collisionless, i.e., the only significant long-range interactions are gravita-

tional. There are three mechanisms via which cold axions are produced: vacuum

realignment [16, 17, 18], string decay [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]

and domain wall decay [29, 30, 31]. We discuss the history of the axion field as

the universe expands and cools to see how and when these mechanisms occur. We

also review the process of vacuum realignment in detail, as there will alv--b i be a

contribution to the cold axion populations from that mechanism and, as discussed

below, it is possible that this provides the only contribution.

There are two important scales in the problem of axions as dark matter. The

first is the temperature at which the PQ symmetry breaks, TpQ. Which of the

above mechanisms contribute significantly to the cold axion population depends

on whether this temperature is greater or less than the inflationary reheating

temperature, TR. The second is the temperature at which the axion mass, arising

from non-perturbative QCD effects, becomes significant. At high temperatures, the

latter effects are not significant and the axion mass is negligible [53]. The axion









mass becomes significant at a critical time, ti, when majt ~ 1 [16, 17, 18]. The

corresponding temperature is T1 1 GeV.

At initial early times, the PQ symmetry is unbroken. At TpQ, it breaks spon-

taneously and the axion field, which is proportional to the phase of the complex

scalar field acquiring a vev, may have any value. The phase varies continuously,

changing by order one from one horizon to the next. Axion strings appear as topo-

logical defects. If TpQ > TR, the axion field is homogenized over huge distances and

the string density is diluted by inflation, to the point where it is extremely unlikely

that any axion strings remain in our visible universe. In the case TpQ < TR, the

axion field is not homogenized and strings radiates cold, massless axions until

non-perturbative QCD effects become significant. At this time, the axion strings

become the boundaries of N domain walls. If N 1= the walls bounded by string

rapidly radiate cold axions and decay (domain wall decay). For N > 1, the domain

wall problem occurs [54] because the vacuum is multiply degenerate and there is at

least one domain wall per horizon. The domain walls end up dominating the energy

density and cause the universe to expand as S o t2, where S is the scale factor.

Although other solutions to the domain wall problem have been proposed [29], we

assume here that N = 1 or TpQ > TR. Thus, if TpQ < TR, string and wall decay

contribute to the axion energy density. If TR < TpQ, and the axion string density is

diluted by inflation, these mechanisms do not contribute significantly to the density

of cold axions. Then, only vacuum realignment will contribute a significant amount.

Vacuum realignment will result in a population of cold axions, independent

of TR. This is discussed in more detail below, however, an overview is as follows.

At TpQ, the axion field amplitude may have any value. If TpQ > TR, the homoge-

nization from inflation will result in a single value of the axion field over our visible

universe. Non-perturbative QCD effects cause a potential for the axion field. When









these effects become significant, the axion field will begin to oscillate in the po-

tential. These oscillations do not decay and contribute to the local energy density

as non-relativistic matter. Thus, a cold axion population results from vacuum

realignment, regardless of the inflationary reheating temperature.

To understand the contribution from vacuum realignment, consider a toy axion

model with one complex scalar field, O(x), in addition to the standard model fields.

Let the potential for O(x) be


V(0) = (2 4 v)2, (2-61)
4

When the universe cools to a temperature TpQ ~ Va, Q acquires a vacuum

expectation value,

() = a exp(ia(x)) (2-62)

The axion field is related to a(x), the phase of the scalar field, O(x), by


a(x) vaa(x) (2-63)


At T ~ A, non-perturbative QCD effects give the axion a mass. They produce

an effective potential
2
V() = m (T) (1 cos) (2-64)

where [53]

ma(T) O .lma A .1D (2-65)

The minimum of the potential occurs at


(x) = Na(x)= 0. (2-66)


The axion acquires mass, ma, due to the curvature of the potential at this mini-

mum. Given the definition of the axion field in Eq. (2-63), the effective Lagrangian









is

2 '2 \ a ) (267)

In a Friedmann-Robertson-Walker universe, the equation of motion is


a + 3H(t)a V2 + (T(t)) sin(Na) 0 (2-68)
R2(t)

Near the potential minima,
1 222
V(a) -mava2 (2-69)
2

and thus,

sin(Na) Na (2-70)

We now restrict the discussion to the zero momentum mode. For TPQ > TR,

this will be the only mode with significant occupation, so the final energy density

calculated will be for this case. In the case, TR > TpQ, higher modes will also be

occupied. For the zero momentum mode, the equation of motion reduces to


S+ 3H(t) + mi(t)a 0 (2-71)


i.e., the field satisfies the equation for a damped harmonic oscillator with time-

dependent parameters. As no initial value of a is preferred, the most general

solution is

a = al + a2t (2-72)

where ac and a2 are constants. Thus, at T >> TQCD, a is approximately constant.

The field will, however, begin to oscillate in its potential when the universe cools to

the critical temperature, T1, defined by [55]

3
ma(Ti(ti)) 3H(TI(ti)) -= (2-73)
2t1









As the axion field can realign only as fast as causality permits, the correspond-

ing momentum of a quantum of the axion field is

1
pa(ti) ~ 10-9eV (2-74)

for t1 ~ 2 x 10-' s, i.e. the age of the universe at which the quark-hadron transition

occurs. As discussed below, the axion mass is restricted to the range 10-6 10-2 eV

and thus this population is non-relativistic or cold.

This mechanism can produce a sufficient quantity of cold axions to provide

the dark matter of the universe. We show this by reviewing the energy density

of axions produced by the realignment mechanism. The energy density for a

homogeneous scalar field around its potential minimum is


2
p T +mt (2-75)


By the Virial theorem,

(2) =m2(a2) = (2-76)

As axions are non-relativistic and decoupled,

mra(t)
p oc M (2-77)
R3(t)

Thus, the number of axions per comoving volume is conserved, provided the axion

mass varies adiabatically.

The initial energy density in the coherent oscillations is


P1 = m(ti)a /2 (2-78)
12 / ) ( )2 (2-79)
-f2 jr na(tI) ( -JN (279)









The energy density in axions t.--l-v is

m,(to) S31(tl)
po = (2S80)
m (ti) S3(to)
-1 f2 1 fS31(t> 1\2
f (2-81)
2 tJ S3 (to)

where S(t) is the scale factor at time, t. Eq. (2-81) implies the axion energy

density,

~0.15 ( v) (2-82)

As the axion couplings are very small, these coherent oscillations do not decay and

make a good candidate for the dark matter of the universe.

The mass is related to the Peccei-Quinn decay constant, fa, by Eq. (2-51) and

the couplings of the axion mass are inversely proportional to fa. Thus limits on any

of the axion mass, axion couplings or PQ decay constant is also a restriction on the

other two. Since Q, < QCDM = 0.22, fa < 1012 GeV and thus, m, > 10-6 eV. This

is the lower bound on the axion mass range. If the axion mass were any greater,

too much dark matter would be produced via the realignment mechanism.

The upper limit on the axion mass is 10-2 eV, from observations of SN1987a.

The number of neutrinos observed on Earth due to this supernovae and its duration

are in good agreement with models of supernovae. Light particles, such as axions,

present novel cooling mechanisms that can alter the duration of supernovae. If the

axion mass is less than 10-2 eV, axions are not produced in significant numbers

to affect supernovae. However, for a range of axion masses above this, axion

production and escape from supernovae will significantly shorten the supernova

duration by efficiently transporting energy away. Above approximately 0.5 eV, the

mean free path of an axion will be too short for significant numbers of axions to

escape from supernovae. At this point, other .,-I i .!r,~ i-i I1 processes, such as the

lifetime of red giants forbid axion masses in higher ranges [11].















CHAPTER 3
DISCRETE FLOWS AND CAUSTICS IN THE GALACTIC HALO

3.1 Introduction

ADMX's high resolution channel searches for discrete flows of cold axions

passing the detector. As discussed in the introduction, such flows occur due to tidal

stripping of dwarf galaxies and late infall of dark matter onto the galactic halo.

In this chapter, we review the arguments why such flows are expected to occur in

cold dark matter cosmology, thus providing an interesting possibility to search for

axions.

In Section 3.2, we review literature demonstrating that discrete flows are a

natural consequence of a cold dark matter cosmology. Section 3.3 discusses the

densities of such flows. A significant fraction of the local halo density should be

contained in discrete flows, which is important when searching for them, as the

signal observed is proportional to the density. A brief discussion of evidence for

flows and detection of axions in these flows concludes this chapter, in Section 3.4.

3.2 Existence

This work demonstrates that searching for discrete flows of cold axions in the

galactic halo improves the sensitivity of a microwave cavity detector. However, it

is necessary that such halo substructure exists for us to benefit from this improved

detector sensitivity. Natarajan and Sikivie have shown that discrete flows and

caustics are a necessary consequence of cold dark matter cosmology. In this section,

we review the arguments for the presence of discrete flows and caustics in the

galactic halo. First, we describe why it is expected that such halo substructure

forms and then we review the mathematical proof for the existence of both inner

and outer caustics in galactic halos.









While we are interested in this substructure from the point of view of axion

detection, it should be noted that the existence of discrete flows and caustics is

independent of the type of cold dark matter. The only requirement for flows and

caustics to form is the assumption of cold dark matter itself. Cold dark matter

particles are assumed to possess the following properties:

(1) The particles must be collisionless, i.e., the only significant interactions of

these particles are gravitational. This property explains why the particles are dark

matter.

(2) The particles have negligible initial velocity dispersion, where the initial

conditions are those when the dark matter first falls into a galaxy's gravitational

potential. This is discussed further in the following.

The primordial velocity dispersion of both axions and WIMPs is negligible [56]

as far as large scale structure formation is concerned. For WIMPs, the primordial

velocity dispersion is determined by the temperature, TD, at which they decouple

from the primordial heat bath. Considering Hubble expansion to be the only

significant effect to alter the WIMP velocity dispersion, the velocity dispersion,

6vw, of a WIMP of mass mw falling into a galaxy tod i- is

(6- 2TD ( S(tD)\ (3-t
\mw ) S(to) I

where S is the scale factor, given at the time of decoupling, tD, and todi-, to. For a

WIMP with mass of 1 GeV that decoupled when the temperature was 10 MeV, the

velocity dispersion todci- is 6vw ~ 10-12

For axions, the primordial velocity dispersion is due to the inhomogeneities in

the axion field when the axion mass, m,, becomes significant, i.e., when m, ~ H

at temperature Ta, 1 GeV and time ta ~ 2 x 10-' s. The magnitude of the field

inhomogeneity depends on whether the Peccei-Quinn (PQ) symmetry breaks before

or after inflationary reheating. If the PQ symmetry is broken after 1,. 1 iir the









axion field is inhomogeneous on the scale of the horizon size (~ ti) when the mass

becomes significant and hence,

Va 1 (S(ta) 017 -5eV (32)
6v, -- ^ 10-17 x (3-2)
mat, S(to)) ma

If the PQ symmetry is broken before reheating, inflation homogenizes the axion

field over enormous distances and the velocity dispersion, 6va, due to quantum

mechanical fluctuations in the axion field, is even smaller than in Eq. (3-2).

The primordial velocity dispersion of dark matter particles falling onto a

galaxy at any time, t, can be obtained by substitution of S(to) for the scale factor,

S(t). For both axions and WIMPs, we see that the initial velocity dispersion is so

small as to be negligible.

The formation of discrete flows and caustics can be understood by considering

the phase-space distribution of dark matter particles falling into a gravitational

potential. At early times, prior to the onset of galaxy formation, these particles

will lie on a thin 3-dimensional (3D) sheet in 6D phase-space, as illustrated in

Fig. 3-1. The thickness of the sheet is proportional to the local velocity dispersion

of the dark matter particles, 6v, and thus the sheet is thin. This sheet will also be

continuous, as the number density of particles is very large over the scale at which

the sheet is bent in phase space.

As dark matter particles are collisionless, the evolution of the sheet is de-

termined by the influence of gravity only. Where density perturbations become

non-linear, the 3D sheet will begin to 'iiLd up" clockwise in phase-space. Whereas

previously, in the linear regime, the sheet covered physical space only once, it will

now begin to cover physical space multiple times. After much time, the phase-space

particle distribution will look as shown in Fig. 3-2. As particles fall into a grav-

itational potential, there will be a number of discrete flows present at each point

at any time [43]. There will be one flow of particles falling in for for the first time,






30














A *

























Figure 3-1. A 2-D slice of 6-D phase-space. The line is a cross-section of the sheet
of width 6v on which the dark matter particles lie prior to galaxy
formation. The v-i.- .--~ are the peculiar velocities due to density per-
turbations. When overdensities become non-linear, the sheet begins to
wind up clockwise in phase-space, as shown.






31
















4















S-2 10
-3

-4


10 10-2 10- 100
r/R

Figure 3-2. The phase-space distribution of dark matter particles in a galactic
halo at a particular time, t. The horizontal axis is the galactocentric
distance, r, in units of the halo radius, R, and v is the radial velocity.
Spherical symmetry has been assumed for simplicity. Particles lie on
the solid line.









one flow of particles falling out for the first time, one of particles falling out for

the second time, etc. Also, at the locations where the sheet folds, caustics form.

There are two types of caustics that occur within a galactic halo: "outer" and

"inner." Outer caustics form near where a flow of particles falling out of the halo's

gravitational potential turnaround and fall back in. These caustics are topologically

spheres. Inner caustics form where particles falling into the potential reach their

distance of closest approach to the center of the galaxy. When the initial velocity

of infalling particles is dominated by a rotational component, inner caustics are a

"tricusp imi; [40], whose cross-section is a D_4 catastrophe. The cross-section is

illustrated in Fig. 3-3 and the ring shown in Fig. 3-4. Axial symmetry has been

used in these figures, but is not a neccesary condition for the formation of caustics.



We proceed to review the mathematical arguments for the existence of outer

and inner caustics [44]. Parametrize the particles on the phase-space sheet using

a = (ca, a2, a3). This parametrization may be chosen as convenient. Let x(a; t)

be the physical position of the particle labelled a at time t. At early times, before

galactic evolution becomes non-linear, the mapping a -- x will be one-to-one.

At late times, when the sheet covers physical space multiple times, for any given

physical location r there will be, in general, multiple solutions aj(r, t) with

j = 1, 2, n(r, t), to r = x(a; t). That is, there will be particles with different a

at the same physical location, r. The number of flows at r at time t is n(r, t). The

number density of particles on the sheet is d- It follows that the mass density in

physical space is [40]

n(r,t) 3
p(r, t) da3(a) dD(at) (3-3)
1 (r,t)
j =1 = (r,t)





33
























S*.1 I-.








Figure 3-3. The cross-section of the tricusp ring. Each line represents a particle
trajectory. The caustic surface is the envelope of the triangular feature,
inside which four flows are contained. Everywhere outside the caustic
surface, there are only two flows. Illustration courtesy of A. Natarajan.





































0.002 -

z 0-

-0.002 -











tr .04
094




Figure 3-4. The tricusp ring caustic. Axial symmetry has been used for illustrative
purposes. Illustration courtesy of A. Natarajan.


1, i't l -7.-11 f 1 4 -,









where m is the particle mass and


D(a, t) detO (a ) (3-4)

The magnitude of D is the Jacobian of the map a x. Eq. (3-3) is the sum over

the mass density in each discrete flow at r.

Caustics occur where D = 0 and the map is singular [39]. At these points,

the mapping from phase-space to physical space changes from n-to-one to (n 2)-

to-one. The physical density at the location of caustics becomes very large, as the

phase-space sheet is tangent to velocity space. In the limit of zero initial velocity

dispersion, the dark matter particle density diverges at the location of a caustic. In

reality, these flows will have a small velocity dispersion and thus the caustics will

have a large, but finite, density.

The presence of outer caustics is easily seen from Fig. 3-2. Natarajan and

Sikivie [44] demonstrated that inner caustics must also be present in the galactic

halo. Consider a continuous flow of cold dark matter particles falling in and out

of a gravitational potential and a spherical surface of radius R surrounding the

potential well. Using the parametrization, a = (0, ), where 0 and Q are the

polar coordinates where a particle falling into the potential crosses the sphere at

time, 7. Then x(0, Q, T; t) gives the particle's position at time, t. Natarajan and

Sikivie demonstrated that

9(x,y, z) Ox Ox x\
D det 9(0, ) x x) (3-5)


vanishes at at least one point inside the sphere at any t. Thus, a caustic is present

within the sphere. Such a caustic is an inner caustic. We review their proof in the

following. The variable t will be suppressed.

For each (0, 0), the time at which a particle within the sphere crossed its

surface lies in the range -Tout(O, 4) < T < T-r(O, ) where nr,(Torut) is the initial









crossing time of particles currently crossing the sphere on the way in(out). The

sphere's center is chosen to lie at the origin, x = 0. The distance from the sphere's

center to a particle's position is


r(0, Q, r) = x(O, ) x(O, 0, r)


and
ar r
r < 0 and
Thus, fr all (0, ) there exists a ( ) such that
Thus, for all (0, 0) there exists a Too(0, 0) such that


(3-6)


>0.


(3-7)


r(0, To(0, )) -minr(0,,r) -- rm(0, ) (3-8)

where the minimum is over r for fixed (0, 0). The distance rmi,(0, Q) is the smallest

distance to the origin among all particles labelled (0, 0).

There are two cases to be considered: rm,i(O, Q) / 0 for some (0, 0) and

rmin(0, 4) = 0 for all (0, 0). In the first case,

Or x xo
or -x. -0 (3-9)
OT ,o(eO) r OT O,6,ro(0,6)

for all (0, 0) such that rmi,(0, Q) / 0. The distance r,in(O, Q) has a maximum value

over the sphere S2(0, ). C'! (0, 00) be such that rmin(0o, 0o) = max rmin(O, ).

Then


Or x OX
0 o r o00
and
Or x Ox


where ao (0o, o, ro(0o, Qo)) and x(ao) / 0. Eqs.

that ao' Ox and o are all perpendicular

are linearly dependent and D(ao) = 0. Thus, xo is


(3-10)


(3-11)


(3-9), (3-10) and (3-11) imply

to xo, i.e. these three vectors

the location of a caustic. As









xo depends on the choice of origin, such a caustic is spatially extended, which is as

expected; caustics are generically surfaces.

In the special case, rni(0, ) = 0 for all (0, ), x(O, To((0, 0)) =0 for all (0, )

and thus, for T near To(O, 0):


x(0, Q, T) (, )(T To(0, 4)) + 0((T To(O, 0))2) (3-12)

where
89x
v(O,) aX (3-13)
O ,T e,(to(Oe6)
Using the reparametrization, 0' = 0, = and T' = T To(0, Q) and relabelling,

(," (0) 0,T),

x(0, Q, 7) v(8, )r + 0(r2) (314)

Hence,

D(0,0,T) -v(0,). x2 (3-15)

As D = 0 at T7 0, the origin is the location of a caustic in this special case. In

this case, the caustic has collapsed to a point.

Thus, both inner and outer caustics must be present in a galactic halo.

Discrete flows and caustics are a natural consequence of a cold dark matter

cosmology.

3.3 Densities

In order to be detectable by a microwave cavity experiment, flows in the

galactic halo must have sufficient density. In this section, we review arguments that

demonstrate that discrete flows are expected to contain a significant fraction of

the local dark matter density. In particular, the flow density is enhanced near the

location of a caustic. Evidence sl.-.-, -1 that the Earth is located near a caustic

feature.









The local density of the first few flows was first estimated by Sikivie and

Ipser [43], for cases both without and with angular momentum. We review their

estimates below.

The initial estimate was calculated for the first flow, i.e., the flow of particles

passing Earth for only the first time. These particles had a maximum galactocen-

tric distance of rm ~ 1 Mpc, which was reached 5 x 109 years ago. The density at

this location is estimated to be the average cosmological dark matter density .I-iv,

pCDM(t0). In the case of no angular momentum, the local density of the first flow
will be the density at rm multiplied by the appropriate geometrical focussing factor,

i.e., (rF/re)2, thus,


pi(r., to) ) CDM (t ~ l0-25g/cm3 (3-16)

When angular momentum is included, not all particles falling into the galaxies will

pass through the center. Defining d as the average distance of closest approach for

particles falling in for the first time, the estimated density is


pi (r, to) ~ PCDM (t) ( )2 ( 2 -g/cm3 (3 17)

The more detailed calculations of Sikivie, Tkachev and Wang [57] confirm this

estimate and provide estimates of densities of the same order of magnitude for the

other flows. Their calculations show that each of the first eight in and out flows

have densities of the order of ''- of the local halo density (assuming a local dark

matter density of 9.2 x 10-25 g/cm3 [46]). Thus, these estimates lead us to expect

that flows contain a significant fraction of the local dark matter density.

At the location of a caustic, the dark matter density will be greatly enhanced.

This will be reflected by rising bumps in the galactic rotation curve at these

locations. Fitting the caustic ring model to rises in the Milky Way rotation curve

and to a triangular feature in the IRAS map predicts that the flows falling in









and out for the fifth time contain a significant fraction of the halo density at the

location of our solar system. The predicted densities are 1.7 x 10-24 g/cm3 and

1.5 x 10-25 g/cm3 [45]. The flow of the greatest density is called the "Big Flow."

This flow is predicted to have a velocity dispersion of 53 m/s and velocity of

approximately 300 km/s relative to the Sun. Thus this flow is of particular interest

for axion dark matter detection.

3.4 Discussion

In this section, we discuss evidence for discrete flows and caustics and the

consequences for microwave cavity detection of axion dark matter.

As demonstrated in Section 3.2, discrete flows and caustics are a necessary

consequence of cold dark matter cosmology. It is significant in this regard that

caustics of luminous matter are also believed to exist and have been observed in

bright elliptical galaxies. Malin and Carter first observed ripples in the distribution

of light in these galaxies [58]. Computer simulations demonstrate that when a small

galaxy falls into the fixed gravitational potential of a large elliptical galaxy, the

small galaxy is tidally disrupted and its stars end up on a thin ribbon in phase-

space. These phase-space ribbons are like the phase-space sheets of dark matter

discussed earlier, except for being limited in spatial extent. The folding of these

phase-space ribbons will lead to the observed ripples in the light distribution of

an elliptical galaxy which has swallowed a smaller galaxy [59, 60, 61]. There is no

explanation other than the existence of caustics for the presence of these ripples in

elliptical galaxies. The existence of caustics of visible matter further supports the

expectation that dark matter caustics are present in galactic halos.

While virialization will thermalize the halo and destroy the oldest flows, flows

will be present today from particles which have only lately fallen onto the halo.

These particles will not have had sufficient time to thermalize with the rest of the

halo.









Discrete flows are expected to contain a significant fraction of the local halo

density, as discussed in Section 3.3. Discrete flows produce a distinct signal in

an axion detector. A series of narrow peaks, one per flow, will appear in the

spectra output. The width of each peak is proportional to the velocity dispersion

of the corresponding flow. The power in each peak is directly proportional to the

density of axions in the flow. Such narrow peaks have higher signal-to-noise ratio

in a high resolution axion search. Thus, if a significant fraction of the local halo

density consists of axions in such flows, a high resolution axion search increases the

experiment sensitivity to axions. Furthermore, if a signal is found, it will provide

detailed information on the structure of axion dark matter within our galaxy.















CHAPTER 4
HIGH RESOLUTION SEARCH FOR DARK MATTER AXIONS

4.1 Introduction

ADMX uses a microwave cavity detector to search for axions in our galactic

halo [33, 62, 63, 64, 65, 66]. In its present search mode, the ADMX detector

spends approximately 50 seconds at each cavity setting. As a result it can look

for features in the axion frequency spectrum with a resolution of order 20 mHz.

This potential has recently been realized by building the HR channel, which

became fully operational in August 2002. It offers the opportunity to improve the

sensitivity of the experiment by searching for the spectral features expected from

the presence of discrete flows of dark matter axions. It has been demonstrated

that the HR channel increases ADMX's sensitivity to an axion signal by a factor of

three [35].

ADMX can operate its two channels simultaneously. The MR channel searches

for broad signals, with width of order 1 kHz and a Maxwell-Boltzmann energy dis-

tribution. The HR channel searches for narrow signals arising from discrete axion

flows. Each discrete flow produces a peak in the axion signal. The frequency at

which a peak occurs is indicative of the square of the velocity of the corresponding

flow in the laboratory frame. In searching for cold flows of axions, it is assumed

that the flows are steady, i.e., the rates of change of velocity, velocity dispersion

and flow density are slow compared to the time scale of the experiment. The as-

sumption of a steady flow implies that the signal we are searching for is alv-- i

present. Even so, the signal frequency will change over time due to the Earth's

rotation and orbital motion [67, 68]. In addition to a signal frequency shift in

data taken at different times, apparent broadening of the signal occurs because its









frequency shifts while the data are being taken. The HR channel has a frequency

resolution of 0.019 Hz. To conduct a search without making assumptions about

flow velocity dispersions, searches are conducted for peak power spread across

several bins. We refer to the associated sum of power across n single bins as n-bin

searches. These searches are performed for n =1, 2, 4, 8, 64, 512 and 4096.

This chapter is on ADMX's HR channel search [69]. The experiment is

described in Section 4.2. In Section 4.3, the signal expected from a microwave

cavity detector observing a cold flow of axions is discussed. The detector noise

characteristics are analyzed in Section 4.4. Section 4.5 contains details of the

systematic corrections performed on the data. The complete analysis and axion

signal search procedure are in Section 4.6. The HR search has covered the axion

mass range 1.98-2.17 peV. No axion signal was found in this range. Exclusion

limits on the density of axions in local discrete flows, based on this result, are

presented in Section 4.7. A discussion of the results is in Section 4.8.

4.2 Axion Dark Matter eXperiment

The microwave cavity detector utilizes the axion-electromagnetic coupling to

induce resonant conversion of axions to photons. The relevant interaction is


,,ay 9 ,y,a E B (4-1)


where a is the axion field, E and B are the electric and magnetic fields, respec-

tively, and ga,, the axion-electromagnetic field coupling. The coupling depends on

the fine structure constant, c, the axion decay constant, f,, and a model dependent

factor, g,:

iafa (4 2)
9 /fo









In the KSVZ model, g, -0.97, whereas in the DFSZ model, g, 0.36. The axion

decay constant is related to its mass by


m e 6x 10-6( 12Gef eV. (4-3)


This coupling allows resonant conversion of axions to photons to be induced in a

microwave cavity permeated by a strong magnetic field [36].

As axions in the galactic halo are non-relativistic, the energy of any single

axion with velocity, v, is

E = mac2 + mav2 .(4 4)
2
The axion-to-photon conversion process conserves energy, i.e., an axion of energy,

Ea, converts to a photon of frequency, v = Ea/h. When v falls within the

bandwidth of a cavity mode, the conversion process is resonantly enhanced. The

signal is a peak in the frequency spectrum of the voltage output of the detector.

The power developed in the cavity due to resonant axion-photon conversion is

[36]
2 VBopaC
P ga V amin(Q, a), (4-5)
ma
where V is the cavity volume, Bo is the magnetic field strength, pa is the density

of galactic halo axions at the location of the detector, Qa is the ratio of the energy

of the halo axions to their energy spread, equivalent to a "quality factor" for the

halo axion signal, and C is a mode dependent form factor which is largest in the

fundamental transverse magnetic mode, T.I,,,, C is given by

S Jdx E, Bo 2
C B0Vf d3xE, 2 (4-6)
B VfV Ud3x |E,|2

in which E,(x'. is the time dependent electric field of the mode under considera-

tion, Bo(x) is the static magnetic field in the cavity and c is the dielectric constant

of the medium inside the cavity. The frequ'-ii- -dependent form factor is evaluated









numerically. Eq. (4-5) can be recast in the convenient form,


P -0.5 x 10-21W (V) (B 2 C 2 )
500 L 7 T 0.36 0.5x10-24 g.cm3

x (- ) mini(Q') (4-7)

A schematic of ADMX, showing both the MR and HR channels, is given

in Fig. 4-1. A more detailed illustration of the magnet, cavity and cryogenic

components is shown in Fig. 4-2. The microwave cavity has an inner volume, V,

of 189 L. The frequency of the T.I,,,, mode can be tuned by moving a pair of

rods inside. The rods may be metal or dielectric and can be replaced as necessary

to reach the desired frequency range. The cavity is located in the bore of a

superconducting solenoid, which generates a magnetic field, Bo, of 7.8 T. The

voltage developed across a probe coupled to the electromagnetic field inside the

cavity is passed to the receiver chain. As the experiment operates with the cavity

at critical coupling, half the power developed in the cavity is lost to its walls and

only half is passed to the receiver chain. During operation, the quality factor of

the cavity, Q, is approximately 7 x 104 and the total noise temperature for the

experiment, T,, is conservatively estimated to be 3.7 K, including contributions

from both the cavity and the receiver chain.

The first segment of the receiver chain is common to both the MR and HR

channels. It consists of a cryogenic GaAs HFET amplifier built by NRAO, a crystal

bandpass filter and mixers. At the end of this segment, the signal is centered at 35

kHz, with a 50 kHz span. The MR signal is sampled directly after this part of the

receiver chain. The HR channel contains an additional bandpass filter and mixer,

resulting in a spectrum centered at 5 kHz with a 6 kHz span.

Time traces of the voltage output from the receiver, consisting of 220 data

points, are taken with sampling frequency 20 kHz in the HR channel. This results

in a data stream of 52.4 s in length, corresponding to 0.019 Hz resolution in the










IMAGE
REJECT 10.7MHz MIXER
MIXER #1 IF #2
RFI t 35kHz AF 125Hz BIN
FFT
r------ --- ------
1.3K HFET MIXER
AMPLIFIER #3
5kHz AF 0.02FHF BIN

r u DISK

Bo

Sf MAGNET



CAVITY &
TUNING RODS

Figure 4-1. Schematic diagram of the receiver chain.


frequency spectrum. The data were primarily taken in parallel with the operations

of the MR channel over a period beginning in November, 2002 and ending VT ic,

2004. Continuous HR coverage has been obtained and candidate peak elimination

performed for the frequency range 478-525 MHz, corresponding to the axion mass

range 1.98-2.17 peV. Data with Q less than 40 000 and/or cavity temperature

above 5 K were discarded. When this was the case, additional data were taken to

ensure coverage of the full range.

4.3 Axion Signal Properties

The HR channel is used to search for narrow peaks caused by flows of cold

axions through the detector. It is assumed that the flows are steady, i.e., the

rates of change of velocity, velocity dispersion and density of these flows are slow

compared to the time scale of the experiment. The assumption of a steady flow

implies that the signal we are searching for is ahv-w present. Even so, the kinetic

energy term in Eq. (4-4) and the corresponding frequency change over time due to

the Earth's rotational and orbital motions. In addition to a signal frequency shift

















Stepping motors
TTT Stepping motors


Cryostat vessel


" Cavity LHe reservoir


Magnet LHe reservoir


1.3K J-T refrigerator

Cavity vacuum chamber
Amplifiers

Tuning mechanism

Microwave Cavity


Diplpotrir tnnino rnd


Figure 4-2. Sketch of the ADMX detector.









in data taken at different times, apparent broadening of the signal occurs because

its frequency shifts while the data are being taken.

Using Eq. 4-4, one sees that ratio of the shift in frequency, Af, to the base

frequency, f, due to a change in velocity, Av, is

f mavc2 + m(4 8)
f mi2 +!n#,,V2


Af v V (4-9)

The velocity of a dark matter flow relative to the Earth will be in the range

100 1000 km/s. We chose v = 600 km/s as a representative value for the purpose

of estimation. A frequency of f = 500 MHz is chosen as typical for the data under

consideration.

The magnitude of the velocity on the surface of Earth at the equator due to

the Earth's rotation is vR = 0.4 km/s. It is less than this at the location of the

axion detector, but this value is used for the purpose of illustration. Assuming the

extreme case of alignment of the Earth's rotational velocity with the flow velocity,

Av = 2vR. The resulting daily signal modulation is 3 Hz. Approximating the

Earth's orbit as circular, the magnitude of it's velocity with respect to the Sun

is VT = 30 km/s. Again, considering the extreme case of velocity alignment, the

frequency modulation due to the orbit of Earth around the Sun is at most 200 Hz.

The bandwidth of the HR channel is 6 kHz. After identifying candidate

frequencies, they are reexamined to see if they satisfy the criterion of a constantly

present signal. Thus, if the spectrum is centered on the candidate frequency when

it is reexamined, the signal will still be within the detector bandwidth as it will

move at most 200 Hz from its original frequency.

In addition, both the rotation of the Earth and its motion around the Sun

will result in a small change in the flow velocity relative to the detector while









each spectrum is taken and a subsequent increase in the signal line-width relative

to what would be expected in the static case. Similarly to Eq. (4-9), the signal

bN .I.. 1,i,,- 6f, due to a change in the flow velocity, 6v, is


6f f (4-10)
c2

Taking the time of integration to be At 50 s, the change in relative velocity is at

most

6v 27,, ( ) (4 11)

where T is the period of the motion (diurnal or annual) and v, is the respective

velocity (vO or vT). The line-width is increased by 4 x 10-3 Hz due to the Earth's

rotation. The Earth's orbital motion increases the line-width by 10-3 Hz. The

spectral resolution of the HR channel is 0.019 Hz, large enough to make these

effects negligible.

For flows of negligible velocity dispersion, the sensitivity of the experiment

is proportional to the frequency, f, and the time of integration, At, provided

the resolution, B = 1/At, is less than the shift of the signal frequency during

measurement. This requirement allows a measurement integration time as long as


t < 160s 50 (4-12)

This -i,-.- -r that for the data this note is based on, a more sensitive limit could

have been achieved with a longer integration time than the actual 52 s.

The velocity dipersion of the flow may, however, be a limiting factor. While

no value for velocity dispersion is assumed in performing the HR analysis, for

illustrative purposes, let us consider a particular case: the "Big Flow," discussed

by Sikivie [45]. The upper bound on the velocity dispersion of this flow is 6v < 50

m/s. This leads to a maximum line broadening of 6fBF < 8 x 10-2 Hz, i.e.,

a signal from axions in the Big Flow is spread over four frequency bins in the









detector spectrum if the limit 6v < 50 m/s is saturated. Let us emphasize, however,

that there is no reason to believe this bound is saturated.

In general, we do not know the velocity dispersion of the cold axion flows for

which we search. Subsequently, we do not know the signal width. To compensate,

searches are performed at multiple resolutions by combining 0.019 Hz wide bins.

These searches are referred to as n-bin searches, where n = 1, 2, 4, 8, 64, 512

and 4096. For f = 500 MHz and v =600 km/s, the corresponding flow velocity

dispersions are

6nm/s (600 km/s (4 13)

Further details of the n-bin searches are given in Section 4.6.

4.4 Noise Properties

The power output from the HR channel is expressed in units of a, the rms

noise power. This noise power is related to the noise temperature, T,, via


S- kBT, A (414)

where kB is Boltzmann's constant and b is the frequency resolution. The total noise

temperature, T, = Tc + Tel, where Tc is the physical cavity temperature and

Tei is the electronic noise contribution from the receiver chain. As no averaging is

performed in HR sampling, b = I/At. Thus, the rms noise power is


a = kBbT (4-15)


Output power is normalized to a and T, is used to determine this power. Eq. (4

15) has been verified experimentally by allowing the cavity to warm and observing

that a is proportional to Tc.

The noise in the HR channel has an exponential distribution. The noise in

a 1-bin is the sum of independent sine and cosine components, as no averaging is

performed in HR sampling. The energy distribution should be proportional to a





50


Boltzmann factor, exp(-E/kT), and non-relativistic and classical energies, such
as E = (1/2)mv2 or E = (1/2)kx2, are proportional to squares of the amplitude.
Thus, the noise amplitude, a, for a single component (i.e., sine or cosine) has a
Gaussian probability distribution,

dP 1 / a2 (41)
da = a exp (416

where Ja is the standard deviation.
As there are two components per bin, the addition of n bins is that of 2n
independent contributions. The probability distribution, dP/dpn, of observing noise
power p, in an n-bin is

dP f1 o exp( a) a (1
(- 2n 2 2ndai) W6n k (4-17)
dPn i 1 a (-co a 2n7 k21 2

Evaluating the above expression,
dP n-P l n2) (418)
dn p- p,
dp ( -. exp 418

For n = 1,
dP 1
dp exp (4-19)

which is indeed a simple exponential, as expected.
Using this noise distribution, we can easily see that the average (rms) noise
power in the one bin search is a = ac2. Substituting this in (4-19), the noise power
distribution function becomes

dP 1 pl
d exp (4-20)
dpi a

For each HR spectrum, a is determined by plotting the number of frequency
bins, Np, with power between p and p + Ap against p. According to Eq. (4-20),

N Ap (4-21)
Np a C (421)






51




108 .-- .........08




C%
10 6 104 ........... ... ......... --------........................................................






102
1 0 2 .. ...................................... ... ..................................................



102J-----'-------'------------ -----i-----i--- ----I------

100 ,
0 10 20 30 40
Power (c)

Figure 4-3. Power distribution for a large sample of 1-bin data.

where N is the total number of frequencies. As

In N+ln = NA (4-22)

a is the inverse of the slope of the In Np versus p plot. Figure 4-3 demonstrates

that the data is in good agreement with this relation for p less than 20J. The

deviation of the data from Eq. (4-22) for p greater than 20a is due to the fact

that the background is not pure noise, but also contains environmental signals of a

non-statistical nature.

As we combine an increasing number of bins, the noise power probability

distribution approaches a Gaussian, in accordance with the central limit theorem.

The right-hand side of Eq. (4-18) approaches a Gaussian in the limit of large

n. We have examined a large sample of noise in each n-bin search and verified

that it is distributed according to Eq. (4-18). Figures 4-4 through 4-9 show the






52



x 108
*

6


5












0 2 4 6 8 10.

Power (T)

Figure 4-4. Power distribution for a large sample of 2-bin data.


progression from the exponential distribution of Fig. 4-3 to a near Gaussian curve

for the 4096-bin search.

In addition to examining the behavior of the noise statistics, we have per-

formed a cross-calibration between the HR and MR channels. The signal power

of an environmental peak, observed at 480 MHz and shown in Fig. 4-10, was

examined in both the HR and MR channels. The observed HR signal power was

(1.8 0.1) x 10-22 W, where the error quoted is the statistical uncertainty. The MR

channel observed signal power 1.7 x 10-22 W, in agreement with the HR channel.

Note that the MR signal was acquired with a much longer integration time than

that of the HR signal (2000 for MR versus 52 s for HR).

The combination of the calibration of the noise power with cavity temperature,

the consistency between expected and observed noise statistics and the agreement
6 . . . . . . . . . .

5 , , , ,

2

I
QS


































the consistency between expected and observed noise statistics and the agreement






53



x 108
2.5



*


S1.5

C1
0.5



S;*




0 5 10 15
Power (o)

Figure 4-5. Power distribution for a large sample of 4-bin data.


of signal power observed in both the HR and MR channels, makes us confident that

the signal power is accurately determined in the HR channel.

4.5 Removal of Systematic Effects

There are two systematic effects introduced in the receiver chain shown in

Fig. 4 1. Two passband filters are present on the HR receiver chain: one with

bandwidth 35 kHz on the shared MR-HR section and a passive LC filter of

bandwidth 6 kHz, seen by the HR channel only. The combined response of both

these filters has been analyzed and removed from the data. The second systematic

effect is due to the frequency-dependent response of the coupling between the

cavity and the first cryogenic amplifier. This effect is removed using the equivalent

circuit model described later.

The combined passband filter response was determined by taking data with a

white noise source at the rf input of the receiver chain. A total of 872 time traces






54



x 107




6

5 ..,.. ....
C)
-6-
*o4 S
:114
3-

----- ..................................................


2i


0 5 10 15 20 25
Power (o)

Figure 4-6. Power distribution for a large sample of 8-bin data.


were recorded over a two dv period. In order to achieve a reasonably smooth

calibration curve, 512 bins in the frequency spectrum for each time trace were

averaged giving 9.77 Hz resolution. The combined average of all data is shown

in Fig. 4-11. This measured response was removed from all data used in the HR

search, as follows. The raw power spectra have frequency 0-10 kHz, where the

center frequency of 5 kHz has been mixed down from the cavity frequency. Each

raw power spectrum is cropped to the region 2-8 kHz to remove the frequencies

not within the LC filter bandwidth. Each remaining frequency bin is then weighted

by a factor equal to the receiver chain response at the given frequency divided

by the maximum receiver chain response. Interpolation for frequency points not

specifically included in the calibration curve is performed by assuming that each

point on the calibration curve was representative of 512 bins centered on that

frequency, so all power corresponding to frequencies within that range is normalized










x 106
3.5




2.5 0




5 .. . . . . . . .. . . .. . .



.5 ............

a***

30 40 50 60 70 80 90 100
Power (a)

Figure 4-7. Power distribution for a large sample of 64-bin data.


by the same factor. As the calibration curve varies slowly with frequency within

the window to which each spectra is cropped, this is an adequate treatment of the

normalization.

In the MR channel, the effect of the cavi' i i-,plnif. v coupling is described

using an equivalent-circuit model [70]. This model has been adapted for use in the

HR channel. The frequency dependent response of the cavity amplifier coupling

is most evident in the 4096-bin search, thus this is the data used to apply the

equivalent circuit model. A sample spectrum before correction is shown in Fig. 4-

12.

In the equivalent-circuit model, each frequency is given by A, the number

of bins it is offset from the bin of the center frequency, measured in units of the

4096-bin resolution, i.e. b4096 = 78.1 Hz. The equivalent-circuit model predicts

that the power (in units of the rms noise) at the NRAO amplifier output (the point






56



x 104
14




10 .
*

.
8
O *
o_




2 6 .. ....... ....... .................................
... i .. . . . . ... . . . .... .... .
2-^^^^^^^^


00 450 500 550 600 650
Power (o)

Figure 4-8. Power distribution for a large sample of 512-bin data.


labelled "RF" in Fig. 4-1) in the 4096-bin search at the frequency offset A is

S 8a3+ S A st 2 4
P(A) + 8a3 + 4a4 ( Aa2 (4-23)
1+ 4 ( )a5
( a2)

where the parameters al through a5 are


a, = (b4096/b)(T + T, + Tv)/T, (4-24)

a2 f /(b4096Q) (4-25)

a3 (b4096/b)(T + Tv + (T T) cos(2kL))/T,, (4-26)

a4 (b4096/b)((T Tv) sin(2kL))/T, and (4-27)

a5 (fo fcen)/b4096 (4-28)


In the above expressions, Tc is the physical temperature of the microwave cavity,

T1 and Tv are the current and voltage noise, respectively, contributed by the









7000


6 0 0 0 .............................. ........ .....






3000
000 -----------

S... . . . . . . . . . . . . . . . . .





1000 ....... .


3%00 3900 4000 4100 4200 4300 4400
Power (G)

Figure 4-9. Power distribution for a large sample of 4096-bin data.

amplifier, T, is the noise temperature contributed from all components, b is the

frequency resolution of the HR channel, i.e. 0.019 Hz, L is the electrical (cable)
length from the cavity to the HFET amplifier, fo is the cavity resonant frequency,

fcen is the center frequency of the spectrum and k is the wavenumber corresponding
to frequency fen + bA. The factor b4096/b appears in the parameters a1, a3 and a4

as it is an overall factor which results from normalizing the power to the single bin

noise baseline. In practice, the parameters al through a5 are established by fitting.
The line in Fig. 4-12 shows the fit obtained using the equivalent circuit model.

Large peaks in the data, e.g. an axion signal or environmental peak, are re-

moved before fitting to prevent bias. The 4096-bin spectrum is used to perform
the fit and then the original 1-bin spectrum is corrected to remove the systematic

effect. The weighting factors are calculated using Eq. (4-23) and the fitted pa-

rameters, al through as, at the center of each bin of width b4096. These factors are





















'- 30-
0

t20-


3 10-
o


20
a-
Q -


479.994


479.997 480.000


BW= 1.21 Hz
t = 52 sec


479.9966 479.9970 479.9974
Frequency (MHz)


Figure 4-10.


An environmental peak as it appears in the MR search (top) and the
64-bin HR search. The unit for the vertical axis is the rms power
fluctuation in each case.


BW= 125 Hz
t =2000 sec







1 --- -





















1.0


0.8


0.6
0

0.4


0.2


0.0
0 2000 4000 6000 8000 10000
FFT frequency (Hz)

Figure 4-11. HR filter response calibration data (512 bin average). The power has
been normalized to the maximum power output.






60
















4500 --
0





S. . . . . . . . . . . . . . . .
.
4400 i


S2,
*0 6
4 2 0 0 .............................. ..... ................... .................. .O.o.. ..
4300 -
4 *
\ *
4 2 0 0 .................... ... ... ..... ...... ..... ...... ..... .... ..

4100 *


4 0 0 0 .. ...... 0- .. ...
*. '

499.805 499.807 499.809 499.811
Frequency (MHz)

Figure 4-12. Sample 4096-bin spectrum before correction for the cavity-amplifier
coupling. The line is the fit obtained using the equivalent circuit
model.






61



4350


4300 0*


4250 .. .
o O




4 1 5 0 ............................ *...................... .. ..... ....
4 ** *. 0*
4 0* _* o
S 6 *
4100 ** *
O0*
4050 .


4 O .805 499.807 499.809 499.811
Frequency (MHz)

Figure 4-13. The same 4096-bin spectrum of Fig. 4-12 after correction for the
cavi i,- ,lin l.ift. r coupling.

the ratio of the fit at a given point to the maximum value of the fit. Each 1-bin is

multiplied by the factor calculated for the bin of width b4096 within which it falls.

Figure 4-13 shows the spectrum of Fig. 4-12 after removal of systematic

effects. The removal of the cavi i i:plifi. r coupling and the passband filter

response using the techniques described above results in flat HR spectra.

4.6 Axion Signal Search

We now describe the search for an axion signal and summarize the analysis

performed on each time trace.

The width of an axion signal is determined by the signal frequency, axion

velocity and flow velocity dispersion (Eq. (4-10)), the latter being the most

uncertain variable. n-bin searches, where n is the number of .,lIi ient 1-bins added

together (n = 1, 2, 4, 8, 64, 512 and 4096), are conducted to allow for various









1-bin search: 2 4 5 6 7 8 9 10 11 12 13 14 1 16
2-bin search: 12 3 4 5 6 7 8 910 11 12 13 14 15 16
23 345 67 S89 10 11 1213 1415
4bin search: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
34 56 78910 11121314
8 bin search: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
S5 6 78 9 10 1112

Figure 4-14. Illustration of the addition scheme for the 2, 4 and 8-bin searches.
The numbers correspond to the data points of the 1-bin search. Num-
bers within the same box are bins added together to form a single
datum in the n-bin searches with n > 1.

velocity dispersions. For searches with n > 1, there is an overlap between successive

n-bins such that each n-bin overlaps with the last half of the previous and first half

of the following n-bin. This scheme is illustrated for the 2, 4 and 8-bin searches in

Fig. 4-14.

The search for an axion signal is performed by scanning each spectrum for

peaks above a certain threshold. All such peaks are considered candidate axion sig-

nals. The thresholds are set at a level where there is only a small probability that

a pure noise peak will occur and such that the number of frequencies considered as

candidate axion peaks is manageable. The candidate thresholds used were 20, 25,

30, 40, 120, 650 and 4500 a, in increasing order of n.

All time traces are analyzed in the same manner. A fast Fourier transform

is performed and an initial estimate of a is obtained by fitting the 1-bin noise

distribution to Eq. (4-22). Systematic effects are then removed, i.e. the corrections

described in Section 4.5 for the filter passband response and cavity-amplifier

coupling are performed. "L ,;, peaks not included in the equivalent circuit model

fit for the cavity-amplifier response are defined to be those greater than 12 1' of

the search threshold for each n-bin search. After the removal of systematic effects,

the 1-bin noise distribution is again fitted to Eq. (4-22) to obtain the true value of

a and the search for peaks above the thresholds takes place.









The axion mass is not known, requiring that a range of frequencies must

be examined. Full HR coverage has been obtained for the region 478-525 MHz,

corresponding to axion masses between 1.98 and 2.17 peV. The selected frequency

range is examined in three stages for axion peaks, as follows:

Stage 1: Data for the entire selected frequency range is taken. The frequency

step between successive spectra is approximately 1 kHz, i.e. the center frequency of

each spectrum differs from the previous spectrum by 1 kHz. Frequencies at which

candidate axion peaks occur are recorded for further examination during stage 2.

Stage 2: Multiple time traces are taken at each candidate frequency from stage

1. The steady flow assumption means that a peak will appear in spectra taken with

center frequency equal to the candidate frequency from stage 1 if such a peak is

an axion signal. The frequencies of persistent peaks, i.e. peaks that appear during

both stage 1 and 2 are examined further in stage 3.

Stage 3: Frequencies of persistent peaks undergo a three-part examination.

The first step is to repeat stage 2, to ensure the peaks still persist. Secondly, the

warm port attenuator is removed from the cavity and multiple time traces taken.

If the peak is due to external radio signals entering the cavity (an environmental

peak), the signal power will increase dramatically. If the signal originates in the

cavity due to axion-photon conversion, the power developed in the cavity will

remain the same as that for the normal configuration. The third step is to use an

external antenna probe as a further confirmation that the signal is environmental.

Some difficulties were encountered with the antenna probe, due to polarization of

environmental signals. However, the second step is adequate to confirm that peaks

are environmental. If a persistent peak is determined to not be environmental, a

final test will confirm that it is an axion signal. The power in such a signal must

grow proportionally with the square of the magnetic field (Bo in Eq. (4-7)) and

disappear when the magnetic field is switched off.









No axion peaks were found in the range 478-525 MHz using this approach.

The exclusion limit calculated from this data is discussed in the following section.

4.7 Results

Over the frequency range 478-525 MHz, we derive an upper limit on the

density of individual flows of axion dark matter as a function of the velocity

dispersion of the flow. The corresponding axion mass range is 1.97-2.17 /eV. Each

n-bin search places an upper limit on the density of a flow with maximum velocity

dispersion, 6v,, as given by Eq. (4-13).

Several factors reduce the power developed in an axion peak from that given

in Eq. (4-7). The experiment is operated near critical coupling of the cavity to

the preamplifier, so that half this power is observed when the cavity resonance

frequency, fo, is precisely tuned to the axion energy. If fo is not at the center of a

1-bin, the power is spread into .,1i ,i:ent bins, as discussed below. When the axion

energy is off-resonance, but still within the cavity bandwidth at a frequency f, the

Lorentzian cavity response reduces the power developed by an additional factor of

1
h(f) (4-29)


To be conservative, we calculate the limits at points where successive spectra

overlap, i.e. at the frequency offset from fo that minimizes h(f).

If a narrow axion peak falls at the center of a 1-bin, all power is deposited

in that 1-bin. However, if such a peak does not fall at the center of a 1-bin, the

power will be spread over several 1-bins. We now calculate the minimum power in a

single n-bin caused by a randomly situated, infinitely narrow axion line. The data

recorded is the voltage output from the cavity as a function of time. The voltage

as a function of frequency is obtained by Fourier transformation and then squared

to obtain a raw "pc.v, i spectrum. The actual power is obtained by comparison to

the rms noise power. The data are sampled for a finite amount of time and thus,









the Fourier transformation of the output, F(f), will be of the voltage multiplied by
a windowing function, i.e.,


F(f) v(t)w(t) exp(i27rft)dt (4-30)
--o0

where v(t) is the measured output voltage and w(t) is the windowing function for a

sampling period T,
w(t) 1 if T/2 < t < T/2 ,
w t) = (4-31)
0 otherwise.

Eq. (4-30) is equivalent to


F(f) V(k)W(f k)dk, (4-32)

where V(f) and W(f) are the Fourier transforms of the output voltage, v(t), and

the windowing function, w(t), i.e., F(f) is the convolution of V(f) and W(f),

given by
W(f) sin(fT) (433)
W(f)= (4 33)
7f
Discretizing Eq. (4-32) and inserting Eq. (4-33), we have

1 sin((f (m + 1+
F(f) V((m + -)b) b s( ( ( 434)
mO 2 'r(- (m +))
where b is the frequency resolution of the HR channel, 2N points are taken in the
original time trace, and the center frequency of the jth 1-bin is (j + 1/2)b. Thus,

for an axion signal of frequency f falling in 1-bin j, a fraction of the power,

g(m) sin(mr + 5 ) (4-35)
Sm7 +6 )

is lost to the mth 1-bin from 1-bin j, where 6 = 7(m + 1/2 f/b). If 6 = 0, i.e, the
axion signal frequency is exactly equal to a 1-bin center frequency, all the power is
deposited in a single 1-bin. However, if this is not the case, power is lost to other
1-bins. In setting limits, we assume that the power loss is maximal.









The maximum power loss occurs when a signal in the 1-bin search falls exactly

between the center frequency of two .,-li i:ent 1-bins. In this case, when 6 = 7/2,

Eq. (4-35) shows that 40.5'. of the power will be deposited in each of two 1-bins.

In n-bin seaches with n > 2, not as much power is lost to other n-bins, due to the

overlap between successive n-bins. The minimum power deposited in an n-bin is

81 for n = 2, ,7', for n = 4 and 9 ;', for n = 8. For n = 64, 512 and 4096, the

amount of power not deposited in a single n-bin is negligible.

For the n-bin searches with n = 64, 512 and 4096, a background noise sub-

traction was performed which will lead to exclusion limits at the 97.7'. confidence

level. These limits are derived using the power at which the sum of the signal

power and background noise power have a 97.7'. probability to exceed the candi-

date thresholds. We call this power the "effectli, threshold for each search. The

effective thresholds are obtained by integrating the noise probability distribution,

Eq. (4-18), numerically solving for the background noise power corresponding to

the 97.7'. confidence level for each n and subtracting these values from the original

candidate thresholds. For n = 64, 512 and 4096, the effective thresholds are 71,

182 and 531 a, respectively. For smaller values of n, background noise subtraction

does not significantly improve the limits and the effective threshold was taken to

be the candidate threshold. Table 4-1 summarizes this information and shows the

frequency resolution of each search with the corresponding maximum flow velocity

dispersion from Eq. (4-13) for v = 600 km/s.

Our exclusion limits were calculated for an axion signal with power above the

effective threshold reduced by the appropriate factors. These factors arise from the

critical coupling, the Lorentzian cavity response and the maximum power loss due

to the peak not falling in the center of an n-bin, as outlined above. Equations (4-7)

and (4-15) were used, for both KSVZ and DFSZ axion couplings. The cavity

volume, V, is 189 L. Measured values of the quality factor, Q, the magnetic field,









Table 4-1. Effective power thresholds for all n-bin searches, with the frequency
resolutions, bn and corresponding maximum flow velocity dispersions,
6v,, for a flow velocity of 600 km/s.


n Effective b, 6v,
threshold (a) (Hz) (m/s)
1 20 0.019 6
2 25 0.038 10
4 30 0.076 20
8 40 0.15 50
64 71 1.2 400
512 182 9.8 3000
4096 531 78 20000

Table 4-2. Numerically calculated values of the form factor,
temperatures, Te1, from NRAO specifications.

Frequency (\!I.:) C Tei (K)
450 0.43 1.9
475 0.42 1.9
500 0.41 1.9
520 0.38 1.9
550 0.36 2.0


C, and amplifier noise


Bo, and the cavity temperature, Tc, are recorded in each data file. Numerically

determined values of the form factor, C are given in Table 4-2. The electronic

noise temperature, Te1, was conservatively taken from the specifications of the

NRAO amplifier, the dominant source of noise in the receiver chain, although our

measurements indicate that Te1 is less than specified. These values are also given in

Table 4-2. Linear interpolation between values at the frequencies specified was used

to obtain values of C and Te1 at all frequencies.

The 2-bin search density exclusion limit obtained using these values is shown

in Fig. 4-15. For values of n other than n = 2, the exclusion limits differ by only

constant factors. The constant factors are 1.60, 1.00, 1.12, 1.39, 2.53, 5.90 and 17.2

for n = 1, 2, 4, 8, 64, 512 and 4096, respectively.




































2.00 2.05 2.10 2.15
axion mass (geV)


Figure 4-15.


97.7'. confidence level limits for the HR 2-bin search on the density of
any local axion dark matter flow as a function of axion mass, for the
DFSZ and KSVZ a77 coupling strengths. Also shown is the previous
ADMX limit using the MR channel. The HR limits assume that the
flow velocity dispersion is less than 6v2 given by Eq. (4-13).


en
E
U




-0
C
-a

N
in
LL
r)


1 M
E
U1





U
-.


*0"









4.8 Discussion

We have obtained exclusion limits on the density in local flows of cold axions

over a wide range of velocity dispersions. The most stringent limit, shown in

Fig. 4-15, is from the 2-bin search. For a flow velocity of 600 km/s relative to the

detector, the 2-bin search corresponds to a maximum flow velocity dispersion of 10

m/s. The 1-bin search limit is less general, in that the corresponding flow velocity

dispersion is half that of the 2-bin limit. It is also less stringent; much more power

may be lost due to a signal occurring .1 -li,- from the center of a bin than in the

n = 2 case. For n > 2, the limits are more general, but the larger power threshold

of the searches make them less stringent.

The largest flow predicted by the caustic ring model has density 1.7 x

10-24 g/cm3 (0.95 GeV/cm3), velocity of approximately 300 km/s relative to the

detector, and velocity dispersion less than 53 m/s [45]. Using Eq. (4-13) with

Table 4-1 and the information di-1 i'-, '1 in Fig. 4-15 multiplied by the appropriate

factors of 1.12 to obtain the 4-bin limit, it can be seen that the 4-bin search,

corresponding to maximum velocity 50 m/s for v = 300 km/s, would detect this

flow if it consisted of KSVZ axions. For DFSZ axions, this flow would be detected

for approximately half the search range. These limits and the Big Flow density are

illustrated in Fig. 4-16.

Figure 4-15 demonstrates that the high resolution analysis improves the

detection capabilities of ADMX when a significant fraction of the local dark matter

density is due to flows from the incomplete thermalization of matter that has only

recently fallen onto the halo. The addition of this channel to ADMX provides an

improvement of a factor of three over our previous medium resolution analysis.

It is possible that an even more sensitive limit could have been achieved with a

longer integration time, as discussed in Section 4.3. This issue should be considered






70














DFSZ
KSVZ
--- Big Flow





-r --------- --


101





c 0


0





-1
10
C
|10-1





10-2


Figure 4-16.


97.7'. confidence level limits for the HR 4-bin search on the density
of any local axion dark matter flow as a function of axion mass, for
DFSZ and KSVZ a77 coupling strengths. Densities above the lines
are excluded. For comparison, the predicted density of the Big Flow is
also shown. The HR limits assume that the flow velocity dispersion is
less than 6v2 given by Eq. (4-13).


2.15


2.05 2.1
Axion Mass (peV)






71


at the beginning of future data runs in order to maximize the discovery potential of

the HR channel.















CHAPTER 5
SUMMARY AND CONCLUSION

This work demonstrates that the new, high resolution channel of the Axion

Dark Matter eXperiment improves its sensitivity for axion detection by a factor of

three, provided a large fraction of the local density is in a single cold flow.

Axions present an interesting candidate for the cold dark matter component

of the universe's energy density. The original motivation for the axion was to solve

the strong CP problem of the standard model of particle physics. The axion is

the pseudo-Nambu-Goldstone boson associated with breaking the Peccei-Quinn

symmetry, implemented to solve the strong CP problem. It was later realized that

the axion was also a good particle candidate for dark matter. The Peccei-Quinn

symmetry breaking scale is the parameter which governs the properties of the

axion and is inversely proportional to the axion mass and couplings. The axion

mass is constrained to lie between 10-6 10-2 eV, by cosmological and astrophysical

processes. Thus, the axion parameter space is bounded and we know in which

range to search for the axion.

While the axion has very small couplings, it is possible to search for them by

utilizing the axion-electromagnetic coupling. The Axion Dark Matter eXperiment

(ADMX) uses a tunable microwave cavity detector to search for axions. When the

magnetic field inside the cavity is tuned to the axion energy, resonant conversion

of axions to photons will occur, which can be observed as a voltage peak in the

output of the detector.

A new, high resolution channel has recently been added to the ADMX

detector. This channel was designed to improve detector sensitivity by teaching for

axions in a specific form of halo substructure: discrete flows. The original, medium









resolution channel searches for axions in a thermalized component of the Milky

Way halo. These axions have a Maxwellian velocity distribution. Axions in discrete

flows have a small velocity dispersion, resulting in a narrow peak in the spectrum

output by the cavity detector. The high resolution channel can search for these

peaks with a high signal-to-noise ratio, improving detector sensitivity.

Discrete flows are expected to be present in the halo from tidal disruption of

dwarf galaxies and from late infall of dark matter into the gravitational potential.

Dark matter which has only recently fallen into the potential will not have had

sufficient time to thermalize with the rest of the halo. Examining the phase-space

structure of such particles shows that discrete flows will occur due to this late

infall.

The first analysis for this channel has been successfully completed. After

analysis of the noise background and removal of systematic effects, no axion

signal was found in the mass range 1.97-2.17 peV. A broad range of flow velocity

dispersions was considered by searching for signals across multiple bins by adding

.,ili .ient bins together. The new exclusion limits obtained from the high resolution

channel increase the sensitivity of the ADMX detector by up to a factor of three

over the previous medium resolution result. The high resolution channel thus

enhances ADMX's detection ability. Should an axion signal be found, the high

resolution channel will also yield valuable information about the phase-space

structure of the Milky Way galactic halo.















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BIOGRAPHICAL SKETCH

Leanne Delma Duffy was born on August 3, 1975, in Sydney, Australia. In

1980, her family moved to the Gold Coast, Australia, where she attended Miami

State Primary School and Merrimac State High School. After graduating from

high school, she moved to Brisbane, Australia, where she obtained a Bachelor

of Science with honours in physics from The University of Queensland in 1997.

She spent two years working for an environmental consulting firm, Pacific Air &

Environment, Pty Ltd, and then moved to Gainesville, Florida, to study for a

Doctor of Philosophy in physics.