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Axion BEC
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Title:
Axion BEC a Model beyond CDM
Creator:
Yang, Qiaoli
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (113 p.)

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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Sikivie, Pierre
Committee Members:
Fry, James N
Thorn, Charles B
Sullivan, Neil S
Hamann, Fredrick
Graduation Date:
8/11/2012

Subjects

Subjects / Keywords:
Baryons ( jstor )
Cosmic rays ( jstor )
Dark matter ( jstor )
Flux density ( jstor )
Magnetic fields ( jstor )
Momentum ( jstor )
Neutrinos ( jstor )
Photons ( jstor )
Protons ( jstor )
Signals ( jstor )
Physics -- Dissertations, Academic -- UF
axion -- bbn -- bec -- cdm -- cmbr -- cosmology -- qcd
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:
I started work on the field of dark matter and cosmology with Dr. Sikivie three years ago with a goal to distinguish observationally axions or axion-like particles (ALPs) from other dark matter candidates such as weakly interacting massive particles (WIMPs) and sterile neutrinos. The subject is exciting because if one can determine the identity of the dark matter, it will be a mile-stone of physics beyond the standard model. On the high energy frontier, the  standard model with three generation fermions is firmly established. However, it is not complete because the theory does not contain a plausible dark matter candidate, with properties required from observation, and the theory has fine-tuning problems such as the strong CP problem. On the cosmology and astrophysics frontiers, new observations of the dynamics of galaxy clusters, the rotation curves of galaxies, the abundances of light elements, gravitational lensing, and the anisotropies of the CMBR reach unprecedented accuracy. They imply cold dark matter (CDM) is $23\%$ of the total energy density of the universe. Although many "beyond the standard model" theories may provide proper candidates to serve as CDM particles, the axion is especially compelling because it not only serves as the CDM particle, but also solves the strong CP problem. The axion was initially motivated by the strong CP problem, namely the absence of CP violation in the strong interactions. Peccei and Quinn solved the problem by introducing a new $U_{PQ}(1)$ symmetry, and later Weinberg and Wilczek pointed out that the spontaneous breaking of $U_{PQ}(1)$ symmetry leads to a new pseudoscalar particle, the axion. Axion models were proposed in which the symmetry breaking scale may be much larger than the electroweak scale, in which case the axion is very light and couples extremely weakly to ordinary matter. Furthermore, it was realized that the cold axions, produced by the misalignment mechanism during the QCD phase transition, have the right properties to be cold dark matter. It was realized that the existence of axions or axion like particles (ALPs) can be probed experimentally by exploiting their coupling to photons. The ADMX experiment is a realization of the concept of the axion haloscope, in which halo axions in an electromagnetic cavity permeated by magnetic field are induced to convert to microwave photons, which may then be picked up by an antenna. The CERN Axion Solar Telescope (CAST) and the Tokyo Helioscope are axion helioscopes which convert axions from the Sun into X-rays in a magnetic field. A third type of experiment is called photon regeneration. In these experiments photons in a laser beam are converted to axions in a magnetic field. The axions travel unimpeded through a wall, behind which is an identical setup of magnets, where some axions are converted back to photons which can be detected. The three major candidates for CDM, axions/ALPs, WIMPs, and sterile neutrinos, were thought until recently to be indistinguishable by purely astronomical and cosmological observations. However, axions/ALPs are very different from the other two in terms of statistical mechanics properties. Axions/ALPs are spin 0 particles and form a highly degenerate Bose fluid while the typical WIMPs such as neutralinos and sterile neutrinos are fermions and are not degenerate. It was recently found that cold axions are not only a highly degenerate bose fluid, but also form a Bose-Einstein condensate (BEC). Therefore, if the CDM particles are indeed axions/ALPs, there is an opportunity to distinguish them from the other candidates on observational grounds. In this thesis, chapter 1 is an introduction to cosmology and the evidence for CDM. Chapter 2 introduces axion/ALPs physics. Chapter 3 provides an introduction to the axion/ALPs detection experiments and the constraints on axion/ALP parameter space. Chapter 4 shows quantitatively that cold axions do form a BEC. Finally, chapter 5 discusses the observational consequences of axion BEC. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Sikivie, Pierre.
Statement of Responsibility:
by Qiaoli Yang.

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Copyright Yang, Qiaoli. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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AXIONBEC:AMODELBEYONDCDMByQIAOLIYANGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012QiaoliYang 2

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ACKNOWLEDGMENTS IthankProfessorSikivieforhismentoring,allmembersatthephysicsdepartmentofUFfortheirkindnessassistance.Ithankmyparentsfortheirloveandsupport. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 ABSTRACT ......................................... 7 1OVERVIEWOFCOSMOLOGY ........................... 10 1.1THEEXPANDINGUNIVERSE ........................ 10 1.1.1GeneralRelativityandFriedmann'sEquation ............ 10 1.1.2Redshift,DistancesandHorizon ................... 13 1.2BIGBANGNUCLEOSYNTHESIS ...................... 15 1.2.1ThermalizationandDecoupling .................... 16 1.2.2CosmologicalConstituents ....................... 17 1.2.3ThePrimordialAbundanceofTheLightElements .......... 18 1.3COSMICMICROWAVEBACKGROUNDRADIARION ........... 21 1.3.1Recombination ............................. 21 1.3.2PhotonDecoupling ........................... 22 1.4CMBRTEMPERATUREANISOTROPIES .................. 24 1.4.1AnisotropyObservables ........................ 24 1.4.2BoltzmannandEinsteinEquationsinThePerturbedUniverse ... 25 1.4.3CMBRAnisotropiesRevealCDM ................... 32 1.5EVIDENCEFORCDMFROMGALACTICROTATIONCURVES ...... 32 2INTRODUCTIONTOAXIONPHYSICS ...................... 33 2.1THESTRONGCPPROBLEM ........................ 33 2.1.1LagrangianofTheStandardModel .................. 33 2.1.2TheU(1)AProblem ........................... 35 2.1.3TheVacuumandInstantons ..................... 36 2.1.4SolutionstoTheStrongCPProblem ................. 39 2.1.5AxionModels .............................. 40 2.2AXIONPROPERTIES ............................. 43 2.2.1TheAxionMass ............................. 43 2.2.2TheAxionCouplings .......................... 44 2.3AXIONASTROPHYSICSANDCOSMOLOGY ............... 44 2.3.1ConstraintDuetoCosmology ..................... 44 2.3.2ConstraintDuetoAstrophysics .................... 45 3AXIONSEARCHES ................................. 49 3.1AXIONDARKMATTERSEARCH ....................... 49 3.2SOLARAXIONSEARCHES ......................... 51 3.2.1SolarAxionProduction ......................... 51 3.2.2DetectorUsingBraggScattering ................... 52 3.2.3AxionHelioscope ............................ 52 4

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3.3LASEREXPERIMENTS ............................ 53 3.3.1PhotonRegeneration .......................... 53 3.3.2PolarizationExperiments ........................ 54 4AXIONBOSE-EINSTEINCONDENSATION .................... 55 4.1REVIEWOFCOLDAXIONPROERTIES .................. 55 4.2AXIONTHERMALIZATIONINTHEPARTICLEKINETICANDCONDENSEDREGIMES .................................... 56 4.2.1EvolutionEquationsforNonrelativisticAxions ............ 56 4.2.2TheParticleKineticRegime ...................... 60 4.2.3TheCondensedRegime ........................ 62 4.2.4ColdAxionsFormaBEC ....................... 65 4.3THERMALCONTACTWITHOTHERSPECIES ............... 65 4.3.1EvolutionEquationsforOtherSpecies ................ 65 4.3.1.1Baryons ............................ 67 4.3.1.2HotAxions .......................... 68 4.3.1.3Photons ............................ 69 4.3.2PossibleOutcomes ........................... 69 5IMPLICATIONSFOROBSERVATION ....................... 72 5.1ANON-RETHERMALIZINGAZIONBECBEHAVESASORDINARYCDM 72 5.2TIDALTORQUING,INNERCAUSTICSANDAXIONBEC ......... 75 5.3AXIONBECANDCOSMOLOGICALPARAMETERS ............ 76 5.3.1PossibilityofPhotonCooling ...................... 78 5.3.2EffectonTheOtherLightElementPrimordialAbundances ..... 80 5.3.3EffectiveNumberofNeutrinoSpecies ................ 81 6CONCLUSIONS ................................... 83 APPENDIX ADETECTIONOFAXION-LIKEPARTICLESBYINTERFEROMETRY ...... 84 A.1INTRODUCTION ................................ 84 A.2DESIGNOFEXPERIMENT .......................... 87 A.3DISCUSSIONANDCONCLUSIONS ..................... 92 BCOSMICRAYPROTONSILLUMINATEDARKMATTERAXIONS ........ 95 B.1INTRODUCTION ................................ 95 B.2THEORETICALANALYSISOFPHOTONEMISSIONBYPROTONSINAPSEUDOSCALARFIELD .......................... 96 B.2.1MatrixElements ............................ 97 B.2.2DifferentialCrossSection ....................... 99 B.2.3EmissionRateofThePhotons .................... 99 B.2.4EnergySpectrumofPhotons ..................... 101 5

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B.3OBSERVATIONALCONSEQUENCES .................... 103 B.3.1InsufcientSensitivitytoDetectTheQCDAxion ........... 103 B.3.2ConstrainingTheParameterSpaceofALPs ............. 103 B.4CONCLUSIONS ................................ 105 REFERENCES ....................................... 107 BIOGRAPHICALSKETCH ................................ 113 6

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyAXIONBEC:AMODELBEYONDCDMByQiaoliYangAugust2012Chair:PierreSikivieMajor:Physics IstartedworkontheeldofdarkmatterandcosmologywithDr.Sikiviethreeyearsagowithagoaltodistinguishobservationallyaxionsoraxion-likeparticles(ALPs)fromotherdarkmattercandidatessuchasweaklyinteractingmassiveparticles(WIMPs)andsterileneutrinos.Thesubjectisexcitingbecauseifonecandeterminetheidentityofthedarkmatter,itwillbeamile-stoneofphysicsbeyondthestandardmodel. Onthehighenergyfrontier,thestandardmodelwiththreegenerationfermionsisrmlyestablished.However,itisnotcompletebecausethetheorydoesnotcontainaplausibledarkmattercandidate,withpropertiesrequiredfromobservation,andthetheoryhasne-tuningproblemssuchasthestrongCPproblem. Onthecosmologyandastrophysicsfrontiers,newobservationsofthedynamicsofgalaxyclusters,therotationcurvesofgalaxies,theabundancesoflightelements,gravitationallensing,andtheanisotropiesoftheCMBRreachunprecedentedaccuracy.Theyimplycolddarkmatter(CDM)is23%ofthetotalenergydensityoftheuniverse. AlthoughmanybeyondthestandardmodeltheoriesmayprovidepropercandidatestoserveasCDMparticles,theaxionisespeciallycompellingbecauseitnotonlyservesastheCDMparticle,butalsosolvesthestrongCPproblem.TheaxionwasinitiallymotivatedbythestrongCPproblem,namelythepuzzlewhythereisnoCPviolationinthestronginteractions.PecceiandQuinnsolvedtheproblembyintroducinganewUPQ(1)symmetry,andlaterWeinbergandWilczekpointedoutthat 7

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thespontaneousbreakingofUPQ(1)symmetryleadstoanewpseudoscalarparticle,theaxion[ 1 ][ 2 ][ 3 ].Axionmodelswereproposedinwhichthesymmetrybreakingscalemaybemuchlargerthantheelectroweakscale,inwhichcasetheaxionisverylightandcouplesextremelyweaklytoordinarymatter.Furthermore,itwasrealized[ 4 ]thatthecoldaxions,producedbythemisalignmentmechanismduringtheQCDphasetransition,havetherightpropertiestobecolddarkmatter. Itwasalsorealizedthattheexistenceofaxionsoraxionlikeparticles(ALPs)canbeprobedexperimentallybyexploitingtheircouplingtophotons[ 30 ].TheADMXexperimentisarealizationoftheconceptoftheaxionhaloscope,inwhichhaloaxionsinanelectromagneticcavitypermeatedbymagneticeldareinducedtoconverttomicrowavephotons,whichmaythenbepickedupbyanantenna.TheCERNAxionSolarTelescope(CAST)andtheTokyoHelioscopeareaxionhelioscopeswhichconvertaxionsfromtheSunintoX-raysinamagneticeld.Athirdtypeofexperimentiscalledphotonregeneration[ 99 ].Intheseexperimentsphotonsinalaserbeamareconvertedtoaxionsinamagneticeld.Theaxionstravelunimpededthroughawall,behindwhichisanidenticalsetupofmagnets,wheresomeaxionsareconvertedbacktophotonswhichcanbedetected. ThethreemajorcandidatesforCDM,axions/ALPs,WIMPs,andsterileneutrinos,werethoughtuntilrecentlytobeindistinguishablebypurelyastronomicalandcosmologicalobservations.However,axions/ALPsareverydifferentfromtheothertwointermsofstatisticalmechanicsproperties.Axions/ALPsarespin0particlesandformahighlydegenerateBoseuidwhilethetypicalWIMPssuchasneutralinosandsterileneutrinosarefermionsandarenotdegenerate.Itwasrecentlyfound[ 5 ]thatcoldaxionsarenotonlyahighlydegenerateboseuid,butalsoformaBose-Einsteincondensate(BEC).Therefore,iftheCDMparticlesareindeedaxions/ALPs,thereisanopportunitytodistinguishthemfromtheothercandidatesonobservationalgrounds. 8

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Inthisthesis,Chapter1isanintroductiontocosmologyandtheevidenceforCDM.Chapter2introducesaxion/ALPsphysics.Chapter3providesanintroductiontotheaxion/ALPsdetectionexperimentsandtheconstraintsonaxion/ALPparameterspace.Chapter4showsquantitativelythatcoldaxionsdoformaBEC.Chapter5discussestheobservationalconsequencesofaxionBEC.Finally,Chapter6drawsabriefconclusionoftheimplicationsofaxionBEC.Inaddition,appendixAinvestigatesanewmethodtodetecttheaxionlikeparticlesandappendixBstudiesthesignalsproducedbycosmicraypropagatinginanaxionbackground. 9

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CHAPTER1OVERVIEWOFCOSMOLOGY 1.1THEEXPANDINGUNIVERSE WiththediscoveryofthetheoryofGeneralRelativityandHubble'slaw,moderncosmologywasborn.Peoplerealizedthattheuniverseisexpandingandwasoncemuchhotteranddenser.ThispictureofanexpandingUniverseisnowcalledtheBigBangtheoryandiswidelyacceptedasthestandardmodelofcosmologythankstoabundantobservationalsupport. 1.1.1GeneralRelativityandFriedmann'sEquation GeneralRelativityisageometrictheoryofgravity.Ithastwopillars:1)themetricofspace-timedeterminedbythestress-energytensorofeverythingpresent;2)geodesicmotionofallparticlesmovinginthatspace-time.TheEinsteinequationistherstpillar: G=8GT(1) where G=R)]TJ /F4 11.955 Tf 13.15 8.08 Td[(1 2R)]TJ /F4 11.955 Tf 11.96 0 Td[(.(1)RistheRiccitensor,Risthescalarcurvature,andisthecosmologicalconstant. Forahomogeneousisotropicuniverse,onecanshowthatthemetrictensor(FLRWmetric)hasthegeneralform: ds2=)]TJ /F3 11.955 Tf 9.3 0 Td[(dt2+a2(t)(dr2 1)]TJ /F3 11.955 Tf 11.95 0 Td[(kr2+r2d2+r2sin2d2),(1) wherea(t)isafunctionoftime,andkisarealnumber.FromEq.(1-3),wecanseethatthespatialdistancebetweentwopointsisproportionaltoa(t),soa(t)iscalledthescalefactor.(Thescalefactordescribestheevolutionofuniverseinabsenceofdensityperturbation.)Itcanbeshownthatkisthespatialcurvature.Forexample,whenk=0,spaceisat. 10

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Providedthecontentoftheuniversecanbedescribedasaperfectuid,thestress-energytensoratRHSofEq.(1-1)canbewrittenas: T=(+p)+p,(1) whereandparetheenergydensityandpressure.CombiningEq.(1-1),Eq.(1-3),andEq.(1-4)onegetstwoindependentequations: 3 a2(_a2+k)=8G+(1) and a a=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(4G 3(+3p)+ 3.(1) Eq.(1-5)iscalledtheFriedmann'sequationandtakesthestandardform: (_a a)2=8 3G)]TJ /F3 11.955 Tf 15.21 8.09 Td[(k a2+ 3.(1) Eq.(1-5),Eq.(1-6)canbecombinedtoget: d dt(a3)+pda3 dt=0.(1) BycombiningEq.(1-8)withequationsrelatingthepressuretotheenergydensity,onegetsthedependenceofthedensityuponthescalefactor.Fordustlikematterp=0andtherefore m(t)=m,0=a3. (1) Forradiationp==3andhence r(t)=r,0=a4. (1) ThereforeonemayrewriteEq.(1-6)as: (_a a)2=8 3G(r,0 a4+m,0 a3))]TJ /F3 11.955 Tf 15.21 8.09 Td[(k a2+ 3.(1) 11

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Weseethat,asa(t)increasesrsttheradiationtermdominates,thenthemattertermdominates,thenthespace-curvaturetermk,andnallythecosmologicalconstantdominates.TheHubbleparameterisdenedas:H=_a(t)=a(t).RewritetheFriedmann'sequationas: k=(8G 3+ 3)]TJ /F3 11.955 Tf 11.95 0 Td[(H2)a2=8Ga2 3(+ 8G)]TJ /F6 11.955 Tf 11.96 0 Td[(c) (1) wherec=3H2=8Gisthecriticaldensity.Letusdenethetotalenergydensity:t=+ 8Gand=t=c.Onecanseethat: if>1,k>0,=1,k=0,<1,k<0. Thusthevalueofthecosmologicalparameterdeterminesthegeometryofuniverse.If>1,theuniverseisclosed;=1correspondstoaatuniverse;andfor<1,theuniverseisopenandhashyperbolicgeometry.FromEq.(1-12),wecanidentifythreetypesofenergydensitywhichcontributeto:radiationr(t);matterm(t);anddarkenergyd= 8G/constant.FromobservationsofCMBR,itisconcludedthatt=1.0023+0.0056)]TJ /F9 7.97 Tf 6.58 0 Td[(0.0054.Therefore,wecantakek=0.Thus: (_a a)2=8 3G(r,0 a4)(1) fortheradiationdominatederainwhichcasethescalefactora(t)r/t1=2;and (_a a)2=8 3G(m,0 a3)(1) forthematterdominatederainwhichcasethescalefactora(t)m/t2=3;and (_a a)2=8 3G()(1) forthedarkenergydominatederainwhichcasethescalefactora(t)/et. 12

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1.1.2Redshift,DistancesandHorizon Photonpropagationsatisestheon-shellcondition 0=)]TJ /F3 11.955 Tf 9.3 0 Td[(dt2+a2(t)(dr2 1)]TJ /F3 11.955 Tf 11.95 0 Td[(kr2+r2d2+r2sin2d2).(1) Thereforewehave: dt a(t)=)]TJ /F3 11.955 Tf 28.21 8.09 Td[(dr p 1)]TJ /F3 11.955 Tf 11.95 0 Td[(kr2.(1) Consideracomovingsourceemittingphotonsoffrequency=1=dtfromtimettot+dtandthephotonsreachacomovingobserverwithfrequency0=1=dt0betweentimet0andt0+dt0.Thenitfollows: Zt0tdw a(w)=Zt0+dt0t+dtdw a(w).(1) Sincedtanddt0aresmallnumbers,wehave: dt dt0=a(t) a(t0), (1) whichleadsto 0 =a(t) a(t0).(1) Incosmologyliterature,peopleusez=0 )]TJ /F4 11.955 Tf 11.96 0 Td[(1todenotetheredshift.Sowend: 1+z=a(t0) a(t),(1) whichshowsthatthephotonsareredshiftedwhiletheypropagatefromthesourcetoobserverduetotheexpansionofuniverse. Ontheotherhand,onecandenetheluminositydistancebetweenthesourceandobserver: dL=r L 4B(1) whereListheemittingpowerofsource,andBistheobservedapparentluminosity. 13

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ConsiderasourcewhichemitsNphotonswithfrequencyduringtimeperioddt, L=Nh dt.(1) Theobservedluminosityattimet0is: B=Nh 4a(t0)2r2dt0.(1) CombiningEq.(1-23)andEq.(1-24)wehave: dL=a(t0)r(1+z),(1) whichrelatesdLandz. Sincedt=da=_a,a(t0)=1anda(t1)=(1+z))]TJ /F9 7.97 Tf 6.59 0 Td[(1,therelationbetweenrandzis )]TJ /F11 11.955 Tf 11.95 16.27 Td[(Z0rdr p 1)]TJ /F3 11.955 Tf 11.96 0 Td[(kr2=Zt0tdt a(t)=1 a(t0)Z1(1+z))]TJ /F12 5.978 Tf 5.75 0 Td[(1da a_a. (1) CombingtheFriedmannequation,Eq.(1-25),Eq.(1-26)andthefactthatr0<<1wehave: H0dL=1+z jkj1=2sinnfjkj2Zz0dx (1+x2)(1+m0x))]TJ /F3 11.955 Tf 11.96 0 Td[(x(2+x)g(1) whereH0=_a(t0)=a(t0)istheHubbleconstant,k=1)]TJ /F4 11.955 Tf 11.95 0 Td[(m0)]TJ /F4 11.955 Tf 11.95 0 Td[(,and sinn=sin,ifk>0=1,ifk=0=sinh,ifk<0. (1) Ifzissmall,onemayexpandtheRHSofEq.(1-27): H0dL=z+1 2(1+)]TJ /F4 11.955 Tf 11.95 0 Td[(m0=2)z3+...(1) wheretheleadingtermisHubble'slaw.Whenzisnotsmall,dLandzwillnolongerbealinearfunctionofz.Thereforebyttingtheluminositydistance-redshiftrelation 14

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functionwithobservations,andm0canbedetermined.Inthelate90'sPerlmutter,Schmidt,andRiessetal.measuredtheluminositydistance-redshiftrelationoftypeIasupernovaeandgotveryaccurateresults.FromCMBR,weknowthattheuniverseisat,som0+=1andk=0.Whenthedistance-redshiftcurveistted,theresultshowsadarkenergydominateduniverse,with74%.Anotherimportantconceptistheeventhorizon.Theeventhorizonisthedistancelighttraveled,soforeventsbeyondthatdistance,causalconnectionisimpossible.Thesizeoftheeventhorizonisveryimportantfortheevolutionofinhomogeneitiesandstructureformationaswewillsee.Therelationbetweenhorizoncoordinaterhandphotonspropagationtimet0is: Zt00dt a(t)=)]TJ /F11 11.955 Tf 11.29 16.27 Td[(Z0rhdr p 1)]TJ /F3 11.955 Tf 11.96 0 Td[(kr2.(1) Sothesizeofeventhorizonisa(t0)dh: dh=a(t0)Zrh0dr p 1)]TJ /F3 11.955 Tf 11.96 0 Td[(kr2=a(t0)Zt00dt a(t).(1) Forexample,duringmatterdominatedera,wehavea(t)/t2=3;sotheeventhorizonisoforder3t. 1.2BIGBANGNUCLEOSYNTHESIS Fromtheprevioussection,weseethatthedarkenergy=74%andthematterm0=26%.Itisanaturalquestiontoaskwhataretheconstituentsofmatterm0intoday'suniverse.Partofmatterisbaryonic.Howevertheenergydensityofbaryonicmatterisorderof4%casinferredfrombigbangnucleosynthesis(BBN).Thepartofmattermthatisnotbaryoniciscalleddarkmatter.Partofdarkmattercanbemassiveneutrinoshowevertheyarenotthemajorcomponentbecausetheyaretoohotforstructureformationaswewillseebelow.Soanewkindofmatterisrequiredthatdoesnotexistinthestandardmodel. Beforegoingtothedetailsofthecosmicconstituents,letusgiveabriefviewofthehistoryoftheearlyuniverseaccordingtotoday'sunderstanding: 15

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1.t=10)]TJ /F9 7.97 Tf 6.59 0 Td[(43s(T=1019Gev)Planckepochduringwhichthetheoryofquantumgravityisnecessarytounderstandphysics. 2.t=10)]TJ /F9 7.97 Tf 6.59 0 Td[(43)]TJ /F4 11.955 Tf 12.56 0 Td[(10)]TJ /F9 7.97 Tf 6.59 0 Td[(7s(T=1019GeV-1GeV)GeneralRelativityisvalid.Howeverphysicsbeyondthestandardmodelisrequiredtodescribephenomenasuchasthebaryonasymmetryandination. 3.t=10)]TJ /F9 7.97 Tf 6.59 0 Td[(7s(T=1GeV)Quarksandgluonsbecomeconned,coldaxionsareproduced. 4.t=0.2s(T=2)]TJ /F4 11.955 Tf 12.07 0 Td[(1MeV)Neutrinosdecoupleandtheratioofneutronstoprotonsfreezesout. 5.t=1s(T=0.5MeV)Electron-positronpairsannihilate,increasingthephotontemperaturecomparedtothetemperatureofneutrinos. 6.t=200)]TJ /F4 11.955 Tf 11.96 0 Td[(300s(T=0.05MeV)Nuclearreactionsproducelightelements. 7.t=1011s(T=1eV)Matter-radiationequality. 8.t=1012)]TJ /F4 11.955 Tf 11.96 0 Td[(1013sTheuniversebecomestransparentduetorecombination. 9.t=1016)]TJ /F4 11.955 Tf 11.96 0 Td[(1017sStructureformation,darkenergydomination. 1.2.1ThermalizationandDecoupling Tounderstandthethermalevolutionofuniverse,oneneedstoknowwhenparticlespeciesbecomedecoupledfromtheotherconstituents.Theconditionofparticlesremaininginthermalequilibriumis: )]TJ /F6 11.955 Tf 10.1 0 Td[(>>1 tH.(1) Therefore,thermalizationmeanstheparticlecollisionrateismuchbiggerthantheexpansionrateofuniverse.Intheparticlekineticregime,thecollisionrateofparticlesis: )-277(=n(1) wherenistheparticlenumberdensity,visparticle'srelativevelocityand(v)isthecross-section.<>meansaverageovervelocitiesforgiventemperature.Thereforethe 16

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collisionrateisafunctionoftemperature.Forexample,theneutrinosarecoupledtoleptonsandbaryonsvia: +$l+l+l$+lp+$n+ln+$p+l. (1) Thecross-sectionisG2FT2,whereGFistheFermicoupling.Thevelocityofneutrinosisv1andtheirparticledensityisnT3.Thereforethecollisionrateis:)]TJ /F2 11.955 Tf 10.1 0 Td[(G2FT5.TheHubblerateisH=_a=a1=tG1=2T2.Sowehave: )]TJ ET q .478 w 186.19 -295.87 m 195.59 -295.87 l S Q BT /F3 11.955 Tf 186.19 -307.05 Td[(HG2FG)]TJ /F9 7.97 Tf 6.58 0 Td[(1=2T3.(1) FromEq.(1-35)wecanseethattheratioofthecollisionratetotheHubblerateisdecreasingasthetemperaturedropsandwemayestimatethedecouplingtemperatureofneutrinosasTG)]TJ /F9 7.97 Tf 6.58 0 Td[(2=3FG1=61MeV. 1.2.2CosmologicalConstituents Formasslessparticles,theenergydensityforgiventemperatureandzerochemicalpotentialis: =gZd3p (2)3p ep=T+1(1) wheregisthenumberofspindegreesoffreedom,Tisthetemperature,)]TJ /F1 11.955 Tf 12.62 0 Td[(forbosonsand+forfermions.(Thechemicalpotentialiszeroorapproximatelyzerowhentheparticlenumberisnotconservedintheearlyuniverse.Furthermoreforphotons,the 17

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CMBRspectrumgivesalimit=T<910)]TJ /F9 7.97 Tf 6.58 0 Td[(5.)Afterintegrationonegets: b=2 30gbT4forbosons (1) f=2 307 8gfT4forfermions. (1) Sincewealreadyshowedthatforrelativisticparticles/1=a4,theirtemperatureT/1=a. Thephotonsarehotterthantheneutrinosbecauseannihilationofelectronsandpositronsinjectenergyintothephotonswhentheneutrinoshavealreadydecoupled.Onecancalculatethetemperatureratiobetweenphotonsandneutrinosbymeansofentropyconservation: T T=(g1 g0)1=3=(4 11)1=3 (1) whereg0andg1aretheeffectivenumberofdegreesoffreedombeforeandafterelectron-positronannihilation.Therearethreegenerationsofneutrinos.Thereforethetotalradiationenergydensityis: r=2 15(1+3.0467 8(4 11)4=3)T4.(1) ThephotontemperaturetodayisT=2.73K.Sowecaneasilycalculatetheenergydensityofradiationtodayr4.710)]TJ /F9 7.97 Tf 6.59 0 Td[(34g=cm3.Sincethecriticaldensitytodayisc=3H20 8G210)]TJ /F9 7.97 Tf 6.58 0 Td[(29g=cm3,theratioofradiationenergydensitytocriticaldensityisoforder10)]TJ /F9 7.97 Tf 6.59 0 Td[(5. 1.2.3ThePrimordialAbundanceofTheLightElements ThemostabundantbaryonicmatterintheuniverseisHydrogenfollowedbyHelium-4andotherlightelementssuchasDeuterium,Helium-3andLithium.Afterthetemperaturedropsto0.05MeV,theprimordialnucleosynthesisoflightelementsbegins. 18

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Typicalreactionsare: n+$p+e, (1) n+e$p+, (1) forproductionofneutrons,and p+n$D+,D+D!3He+n,3He+n!3T+p,3He+2D!4He+p,4He+3T!7Li+,... (1) Letusrstestimatetheratioofneutronstoprotonsatthebeginningofnucleosynthesis.Theneutronsareunstablewithameanlifetime=887s,sonn=np=nn(td)=np(td)e)]TJ /F9 7.97 Tf 6.58 0 Td[(t=,wheretdisthefreeze-outtimeofneutrons,andt100sisdurationbetweenfreeze-outtimeandthebeginningofnucleosynthesis. WhenthetemperatureisorderofMeV,bothprotonsandneutronsarenonrelativisticparticles.Thenumberdensityofsuchparticlesis: n'g(Tm 2)3=2exp()]TJ /F3 11.955 Tf 10.49 8.09 Td[(m)]TJ /F6 11.955 Tf 11.96 0 Td[( T)(1+15T 8m)(1) whereg=2forprotonsandneutrons.Whenthenumberdensityofneutronsfreezesout,neutronsandprotonshavethesametemperatureandchemicalpotential.Thereforewehave: nn(td) np(td)'(mn mp)3=2exp()]TJ /F3 11.955 Tf 10.49 8.09 Td[(mn)]TJ /F3 11.955 Tf 11.95 0 Td[(mp Tf)'exp()]TJ /F3 11.955 Tf 10.5 8.09 Td[(mn)]TJ /F3 11.955 Tf 11.96 0 Td[(mp Tf) (1) 19

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whereTf0.8MeVisthetemperatureofneutrinosattheirfreezeout,andmn)]TJ /F3 11.955 Tf 12.26 0 Td[(mp=1.29MeVisthemassdifferencebetweenneutronsandprotons.Weconcludethattheneutronstoprotonsratioisorderofe)]TJ /F9 7.97 Tf 6.59 0 Td[(1.29=0.8=0.199whenneutronsfreeze-out,andXnnn=(nn+np)=0.166.AmoredetailedcalculationshowsXn'0.158[ 11 ]. Thebaryonexcessremainsafterbaryonanti-baryonannihilationsintheearlyuniverse.Asthetemperaturedropsbelowtheelectronmass,photoncollisionsnolongerproduceelectronpositronpairs.Letusdenetheratioofbaryonnumbertothenumberofphotonsas=nB=n.Itisconvenienttointroduce10=1010sinceisaverysmallnumber. Thecollisionbetweenneutronsandprotonscanformdeuteronsbutthedeuteronsalsodissociateduetosurroundinghighenergyphotons.Afterthetemperaturehasdroppedsufcientlybelowthebindingenergyofthedeuteron,thenewlyformeddeuteronsarenolongerbrokenbythephotonsandnetdeuteronsareproduced.Oncedeuteronsareproduced,3He,4He,canalsobequicklyproducedbyprocessesdescribedinEq.(1-43)andnally,stable7Liisproduced.Thereisnoprimordialproductionofstableelementsheavierthan7Li. Insummary,primordialnucleosynthesisproducesstableelementssuchasD,3He,4Heand7Li.Thereareintotaleightparticlesinvolvingintheprocesses:n,p,2D,3T,3He,4He,7Li,and7Be.Theevolutionequationsoftheirnumberdensitiesare: dni=dt=)]TJ /F4 11.955 Tf 9.29 0 Td[(3Hni+Xa,jnanj,(1) wheretheniarethenumberdensitiesofrespectiveparticles.Theequationsarevalidonlyafterthedeuteronsarenolongerdissociatedbyphotons,whichtimedependsontheparameter10. Theequationscanonlybesolvednumerically.FromWMAP,Spergel.etal.derive10=6.30.3.Withthisinput,onendsthatYP=0.24850.0008,yLi=10)]TJ /F9 7.97 Tf 6.59 0 Td[(10=4.670.64,yD=10)]TJ /F9 7.97 Tf 6.59 0 Td[(5=2.450.20,y3H=10)]TJ /F9 7.97 Tf 6.59 0 Td[(5=1.030.04.Thepredictedabundances 20

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areroughlyconsistentwithobservations,butdonottexactly.Wewillseethattheexistenceofcoldaxionsmayprovidethekeytosolvetheconictbetweenobservationsandpredictions. 1.3COSMICMICROWAVEBACKGROUNDRADIARION Thediscoveryofcosmicmicrowavebackgroundradiation(CMBR)markedthebirthofmoderncosmology.TheCMBRisradiationleftfromthehotplasmaoftheearlyuniverse.Itprovidesevidencethattheuniversewasoncemuchhotteranddenser. 1.3.1Recombination Chargedelectronsandnucleonsbecomeboundbetween105and3105yearsafterthebigbang.Radiationdecouplesfrommatterthen.Themainrecombinationprocessis: e)]TJ /F4 11.955 Tf 9.74 -4.93 Td[(+p$H+.(1) Theprocessof( 1 )isreversiblewhichmeansthatforgiventemperature,bothneutralandionizedatomsexist,andtheratioofthetwodependsonthephotontemperature.Letusdenetheionizationratio: Xp=np np+nH,(1) whereforsimplicityweneglectotherlightelementssuchasHe.1)]TJ /F3 11.955 Tf 12.56 0 Td[(Xpistheneutralhydrogenpercentage.WhenXpissmallenough,onecanregardtherecombinationprocesscomplete.Noticethatthebaryontophotonratio=nN nisaconstantsincenNandnbothdecreaseasa)]TJ /F9 7.97 Tf 6.59 0 Td[(3(t),so10)]TJ /F9 7.97 Tf 6.58 0 Td[(10todayaswellasintheearlyuniverse.Now,letusndtherelationshipbetweenXpandT. IntheSahaapproximation,protons,freeelectronsandneutralhydrogenatomssatisfytheBoltzmanndistribution: np=2(mpT 2)3=2ep)]TJ /F14 5.978 Tf 5.76 0 Td[(mp T (1) ne=2(meT 2)3=2ee)]TJ /F14 5.978 Tf 5.76 0 Td[(me T (1) nH=4(mHT 2)3=2eH)]TJ /F14 5.978 Tf 5.76 0 Td[(mH T, (1) 21

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whereiaretheirchemicalpotentials,miaretheparticlemasses;p+e=H,np=ne.Theseequationsassumethermalequilibrium,whichatlatertimeofrecombinationisnolongeraccurate,Oneneedsamoredelicatemethodsuchasnumericalsimulationtocompletelydescribetherecombinationprocess.Usingthedenitionofandphotonnumberdensityn=2.4T3=2,wehave: np+nH=n=(2.4 2T3). (1) Combining( 1 ),( 1 )andp+e=Hwehave: n2p=nH=npne=nH=(meT 2)3=2e)]TJ /F14 5.978 Tf 7.78 4.62 Td[(mp+me)]TJ /F14 5.978 Tf 5.75 0 Td[(mH T.(1) From( 1 ),( 1 )onegets: 1)]TJ /F3 11.955 Tf 11.95 0 Td[(Xp X2p=1.110)]TJ /F9 7.97 Tf 6.59 0 Td[(8e13.6eV=T(T=eV)3=2.(1)( 1 )tellsusexplicitlytherelationbetweenphotontemperatureandionizationfractionofmatter.UsingtheWMAPvalueofwendXpis0.1whenT0.3eV,thereforeonecanregardtherecombinationcompleteatTrec=0.3eV.Since: 1+z=a(t0)=a(t)=T=T0, (1) andT0=2.73K=2.3510)]TJ /F9 7.97 Tf 6.58 0 Td[(4eV,wehavezrec=1251. 1.3.2PhotonDecoupling Inthelastsection,wediscussedtherecombinationofchargedparticles.Thephotonscanpropagatefreelyoncerecombinationiscomplete.Therearetwomajorprocessesinvolved: +e)]TJ /F2 11.955 Tf 10.4 -4.94 Td[(!e)]TJ /F4 11.955 Tf 9.74 -4.94 Td[(++p+!p++. (1) 22

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SincetheThomsonscatteringcross-sectionisinverselyproportionaltothemasssquared,onlyphoton-electronscatteringmattershere.Thecollisionrateofphotonsandelectronsis: )-278(=cne,(1) wherec=1isthespeedoflight,and=1.7103GeV)]TJ /F9 7.97 Tf 6.59 0 Td[(2.Duringrecombination,thenumberdensityoffreeelectronsdropsdown,sothecollisionratedecreasesaswell.Whenthecollisionrateissmallerthenthehubblerate: )]TJ /F6 11.955 Tf 6.77 0 Td[(=H<1,(1) thephotonsdecouplefrommatter. Sincetherecombinationhappensduringthematterdominatedera,wecanwritetheFriedmannequationas: H2=8G 3m=H20m0m m0, (1) wherem0isthematterdensitytoday.Usingm=m0=T3=T30,wecanrewrite( 1 )as: H2=m0H20(T=T0)3.(1) Puttingnumbersto( 1 )onegets: H=8.110)]TJ /F9 7.97 Tf 6.59 0 Td[(43(T T0)3=2GeV.(1)( 1 )canalsobewrittenintermsofT,T0: )-278(=Xen=5.410)]TJ /F9 7.97 Tf 6.59 0 Td[(36Xp(T T0)3GeV. (1) Therefore,byusing( 1 ),( 1 )and)]TJ /F6 11.955 Tf 6.78 0 Td[(=H<1wendthatphotonsdecouplewhenXp410)]TJ /F9 7.97 Tf 6.59 0 Td[(3whichisequivalenttoTdec=0.25eVorzdec=1061.Afterthatphotonspropagatefreely.ThephotonfrequencyspectrumisaPlanckdistributionbecausetheyareinthermalequilibriumwiththematterplasmaatthelastscattering 23

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surface.Thereforewhenz=zdec,oneexpectsthespectrumofphotonstobe:n()=23(e=Tdec)]TJ /F4 11.955 Tf 12.36 0 Td[(1))]TJ /F9 7.97 Tf 6.59 0 Td[(1.Thefrequencyofphotonsredshiftsaccordingto=(1+zdec)dec.Thereforetheeffectivetemperatureofphotonstodayis: T0=Tdec 1+zdec. (1) PenziasandWilsonrstobservedthisrelicradiationoftheearlyuniverse.Theygotthe1978NobelPrizeinphysics. 1.4CMBRTEMPERATUREANISOTROPIES 1.4.1AnisotropyObservables TheobservedCMBReffectivetemperatureTcanbewrittenasafunctionofdirectionT[^n(,)].Theuctuationofthetemperatureis:T=T0+T(^n),whereT0=2.728K.Letusdene: (^n)T(^n) T0.(1)(^n)canbeexpendintosphericalharmonics: alm=ZdYlm(^n)(^n).(1) Thealmhavetheproperty: =ll0mm0Cl,(1) whereweassumethattheuctuationshavestatisticalisotropy,sotheClareindependentofm.From( 1 )-( 1 )wehave: =T20Xl2l+1 4ClPl(cos),(1) whereistheangularseparationbetween^n1and^n2;thePl(x)aretheLegendrepolynomials. 24

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1.4.2BoltzmannandEinsteinEquationsinThePerturbedUniverse Inchapters1.1-1.3wediscussedtheevolutionofthehomogeneousuniverse.TodiscussanisotropiesoftheCMBR,oneneedstoaddperturbationstotheenergy-stresstensorandtothemetric.Thedecompositiontheoremstatesthattheperturbationscanbedividedintoscalar,vectorandtensor,andthateachtypeofperturbationevolvesindependently.Thevectorandtensorperturbationsplayasubdominantroleinstructureformation.Thereforeweonlyconsiderscalarperturbationshere.Itcanbeshownthatforscalarperturbationsthemetriccanalwaysbewrittenintheform: ds2=)]TJ /F3 11.955 Tf 9.3 0 Td[(a2()[(1+2 )d2+(1+2)d~x2],(1) whereisconformaltimewhichisdenedby:=Rdt=a(t),aisthescalefactor,istheNewtonianpotentialand istheperturbationtothespatialcurvature.ThischoiceofmetriciscalledconformalNewtongauge. LetusconsidertheBoltzmannequationforthephotonsrst.TheBoltzmannequationisdf=dt=C[f]whereC[f]isthecollisionterm,and: df dt=@f @t+@f @xidxi dt+@f @pdp dt+@f @^pid^pi dt,(1) where^pisunitvectorinthedirectionofmomentum~p.Sinceboth@f=@^piandd^pi=dtarerstorder,wecanneglectthelasttermof 1 ,asitisofsecondorder. Forphotonsp2=0andhencedxi=dt=pi=p0=^pip 1+2 =ap 1+2'^pi(1+ )]TJ /F6 11.955 Tf 11.99 0 Td[()=a.Thegeodesicequationleadstodp=pdt=)]TJ /F3 11.955 Tf 9.3 0 Td[(H+@=@t)]TJ /F4 11.955 Tf 12.13 0 Td[(^pi@ =a@xi.Sowehave: df dt=@f @t+@f @xi^pi a)]TJ /F3 11.955 Tf 11.96 0 Td[(p@f @p(H+)]TJ /F6 11.955 Tf 9.3 0 Td[(@ @t+^pi@ a@xi),(1) wherewehaveneglected@f=@xi^pi( )]TJ /F6 11.955 Tf 12.56 0 Td[()=asinceitisahigherorderterm.Letusassumetheperturbedphotonphasespacedistributionfunctionfmaybewritten: f(~x,~p,)=[expp T()[1+(~x,^p,)])]TJ /F4 11.955 Tf 11.95 0 Td[(1])]TJ /F9 7.97 Tf 6.59 0 Td[(1,(1) 25

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wheretheperturbationischaracterizedby,andisfunctionof~x,,^p. Byexpandingthedistributionfunctionnear=0wehave: f=f0)]TJ /F4 11.955 Tf 11.96 0 Td[((~x,^p,)p@f0 @p,(1) wheref0=(exp[p=T])]TJ /F4 11.955 Tf 12.2 0 Td[(1))]TJ /F9 7.97 Tf 6.59 0 Td[(1.Putting 1 to 1 ,andusingT@f0=@T=)]TJ /F3 11.955 Tf 9.3 0 Td[(p@f0=@p,T/1=a,onends: df dt=)]TJ /F3 11.955 Tf 9.3 0 Td[(p@f0 @p[@ @t+^pi a@ @xi+@ @t+^pi a@ @xi],(1) uptorstorder. FortheC[f]terms,wehave[ 17 ][ 18 ]: C[f]=X~l~m~qjAmplitudej2ff(~l)f(~p)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(fe(~q))(1+f(~m)))]TJ /F3 11.955 Tf 11.96 0 Td[(f(~q)f(~m)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(fe(~l))(1+f(~p))g=1 2pZd3qd3q1d3p1 (2)923E(q)E(q1)E(p1)jMj2(2)44(p+q)]TJ /F3 11.955 Tf 11.95 0 Td[(p1)]TJ /F3 11.955 Tf 11.95 0 Td[(q1)F, (1) wherejMj2isthematrixelementofThompsonscatteringand F=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(fe(~q))(1+f(~p))fe(~q1)f(~p1))]TJ /F4 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(fe(~q1))(1+f(~p1))fe(~q)f(~p)[fe(~q1)f(~p1))]TJ /F3 11.955 Tf 11.96 0 Td[(fe(~q)f(~p)], (1) sincefe(electrondistributionfunction)ismuchlessthan1.fe(~q1)fe(~q)fortheepochofinterestsincephotonmomentumismuchsmallerthanthatofelectrons.The3functioncanbeeasilyintegratedsinceEemeandjMj2isaconstant: C= 8m2epZD~qD~p11 p1[p+q2 2me)]TJ /F3 11.955 Tf 10.18 0 Td[(p1)]TJ /F4 11.955 Tf 11.38 8.09 Td[((~q+~p)]TJ /F6 11.955 Tf 11.39 .5 Td[(~p1)2 2me]jMj2(fe(~q+~p)]TJ /F6 11.955 Tf 9.62 .5 Td[(~p1)f(~p1))]TJ /F3 11.955 Tf 10.18 0 Td[(fe(~q)f(~p)),(1) whereD~q=d3q=(2)3.Nextsincefe(~q+~p)]TJ /F6 11.955 Tf 11.39 .49 Td[(~p1)fe(~q)wehave: C= 8m2epZD~qD~p11 p1fe(q)jMj2[(p)]TJ /F3 11.955 Tf 9.64 0 Td[(p1)+(~p)]TJ /F6 11.955 Tf 11.4 .5 Td[(~p1)~q med(p)]TJ /F3 11.955 Tf 11.96 0 Td[(p1) dp1](f(~p1))]TJ /F3 11.955 Tf 9.63 0 Td[(f(~p)),(1) 26

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whereweexpandthefunctionaroundp)]TJ /F3 11.955 Tf 12.09 0 Td[(p1andq2)]TJ /F4 11.955 Tf 12.09 0 Td[((~q+~p)]TJ /F6 11.955 Tf 11.52 .5 Td[(~p1)22(~p1)]TJ /F6 11.955 Tf 11.52 .5 Td[(~p)~qforthescattering.PuttingjMj2=8Tme2andnoticingthatfe(~q)ne(~q)]TJ /F3 11.955 Tf 11.95 0 Td[(me~vb),(~vbisthebulkvelocityoftheelectrons)weconclude: C=2neT pZD~p11 p1[(p)]TJ /F3 11.955 Tf 11.96 0 Td[(p1)+(~p)]TJ /F6 11.955 Tf 11.39 .49 Td[(~p1)~vbd(p)]TJ /F3 11.955 Tf 11.95 0 Td[(p1) dp1](f(~p1))]TJ /F3 11.955 Tf 11.96 0 Td[(f(~p)),(1) whereneiselectronnumberdensity.Letusdene0=1=(4)Rd(^p,~x,t).Combining( 1 ),( 1 ),wehave: C=neT pZp1dp1[(p)]TJ /F3 11.955 Tf 11.62 0 Td[(p1)()]TJ /F3 11.955 Tf 9.3 0 Td[(p1@f0 @p10+p@f0 @p(^p))+~p~vbd(p)]TJ /F3 11.955 Tf 11.95 0 Td[(p1) dp1(f0(p1))]TJ /F3 11.955 Tf 11.62 0 Td[(f0(p))].(1) Thenintegratingthesecondtermbypartswenallyobtain: C=)]TJ /F3 11.955 Tf 9.3 0 Td[(p@f0 @pneT[)]TJ /F4 11.955 Tf 9.3 0 Td[((^p,~x,)+0+^p~vb].(1) ThereforewecanwritetherstorderBoltzmannequationforphotons: @(+) @+~p~r(+ )=neTa[0)]TJ /F4 11.955 Tf 11.96 0 Td[(+^p~vb],(1) whereweuseconformaltimeforconvenience.ItiseasiertosolvethislinearpartialdifferentialequationinitsFouriermodes,sincedifferentmodesaredecoupled.Letusdene(~x,^p,)=Rd3k=(2)3ei~k~x(~k,^p,),andassume(~k,^p,)=(~k,,)where=^k^p.Inrstorderforscalarperturbations,~vb(~k)=~kvb(~k).Thenwehave: d(~k,,)=d+ik+d(~k,)=d+ik (~k,)=neTa[0)]TJ /F4 11.955 Tf 11.96 0 Td[(+vb(~k)].(1) TheBoltzmannequationforneutrinoscanbeobtainedbyfollowingsimilarstepsasforthephotonsbutwithC=0andf=[exp(p=T)+1])]TJ /F9 7.97 Tf 6.59 0 Td[(1,noticingthatfortheepochofinteresttheneutrinosarerelativistic.Thereforetheequationofneutrinosis: d=d+ik+d(~k)=d+ik (~k)=0,(1) whereisdenedsimilarlyasforphotons. 27

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Colddarkmatterandbaryonsarenon-relativisticduringtheeraweareinterestedin.Fornon-relativisticparticles,wedonotneedtoassumeaparticularformofthermaldistributionfunctionbecausethethermalmotionoftheparticles,whichisorderofTmv21m,canbeneglected.Weuseonlyn(~x,t)and~v(~x,t)andthecorrespondingequationsofmotiontodescribetheevolutionofsuchparticlesystems. Thedarkmatterparticlesaremassive.Sowehave:gpp=)]TJ /F3 11.955 Tf 9.3 0 Td[(m2,whichgivesaconstraintonthefourmomenta.LetuschooseEand^pasindependentvariablesso: df dt=@f @t+@f @xidxi dt+@f @EdE dt+@f @^pid^pi dt,(1) wherethelasttermcanbeneglectedsinceitisofsecondorder.Byusingtheon-shellconditionandgeodesicequationformassiveparticles,weget: df dt=@f @t+@f @xi^pi ap E)]TJ /F6 11.955 Tf 14.36 8.09 Td[(@f @E(Hp2 E+p2 E@ @t+p^pi@ a@xi).(1) Wecanmultiply 1 bythephasespacevolumeandintegrate: Zd3p (2)3[@f @t+@f @xi^pi ap E)]TJ /F6 11.955 Tf 14.36 8.08 Td[(@f @E(Hp2 E+p2 E@ @t))]TJ /F6 11.955 Tf 14.36 8.08 Td[(@f @Ep^pi@ a@xi]=0.(1) Forthersttermin 1 weusen=R[d3p=(2)3]f.Wealsohavenvi=R[d3p=(2)3]f[p^pi=E]forthesecondterm.SincedE=dp=p=E,thethirdtermcanbeintegratedbypart:R[d3p=(2)3][p2=E]@f=@E=R[d3p=(2)3]p@f=@p=)]TJ /F4 11.955 Tf 9.3 0 Td[(3n.Thefourthtermcanbeneglectedsincetheintegraloverthedirectionvectorisnonzeroonlyfortheperturbedpartoff.Thusitisahigherorderterm.Togetherwehave: @n @t+1 a@(nvi) @xi+3[H+@ @t]n=0.(1) 28

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Lettingndm=n0dm[1+(~x,t)],onegetsthezeroorderandrstorderequationsforcolddarkmatter: @n0dm @t+3Hn0dm=0 (1) @ @t+1 a@vi @xi+3@ @t=0. (1) Assuming~v(~k)=^kv(~k),wehavetherstorderequationinkspace d d+ikv+3d d=0. (1) Insimilarfashion,wecanmultiply 1 byd3p(p=E)^pj=(2)3andintegrate.Neglectingalltermsoforder(p=E)2orhigherwehavetherstmomentoftheBoltzmannequation: @(nvj) @t+4Hnvj+n a@ @xj=0.(1) Combinedwiththezeroorderequation, 1 leadsto: @vj @t+Hvi+1 a@ @xj=0,(1) orinFourierspace dv d+da adv+ik =0.(1) Thisequationtogetherwith 1 completestheBoltzmannequationsforcolddarkmatter. Theelectronsandprotonsaregenerallynamedbaryonsincosmologyliteraturealthoughelectronsareactuallyleptons.Letusdenetheirperturbations: e)]TJ /F6 11.955 Tf 11.95 0 Td[(0e 0e=p)]TJ /F6 11.955 Tf 11.96 0 Td[(0p 0pb.(1) HereweusedthefactthattheCoulombscatteringrateismuchlargerthantheexpansionrateatallepochsofinterest,soelectronsandprotonsaretightlycoupled.Thesamereasoningalsoimplies~ve=~vp~vb. 29

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Thelefthandsidesof 1 and 1 arethesameforbaryonsasforcolddarkmattersincebothofthemarenon-relativistic.Therighthandsidesaretherelevantcollisionterms.LetCe[f]denoteelectron-photonscattering.Itisgivenby 1 .Proton-photonscatteringisnegligiblebecausetheirComptoncross-sectonismuchsmaller.LetCep[f]termdenoteelectron-protonscattering.Itcanbewrittenas 1 withCoulombscatteringamplitudejMjep.Becauseprotonsaremuchheavierthanelectrons,theFtermcanbesimpliedto 1 byusingasimilarargumentasforelectronphotonscattering.TheBoltzmannequationsforelectronsandprotonsarerespectively: dfe=dt=Ce[f]+Cep[f] (1) dfp=dt=Cpe[f]. (1) Let<>denoteintegrationoverthemomentumappearingintheC[f]term.Since=0,thecontinuityequationforbaryonsis db d+ikvb+3d d=0. (1) FortherstmomentoftheBoltzmannequation,werstadd 1 to 1 inordertoannihilatetheCoulombscatteringtermafterintegration.Theremainingtermcanbecalculatedbyusing 1 ,whichwerewritehereforconvenience. C=)]TJ /F3 11.955 Tf 9.3 0 Td[(p@f0 @pneT[)]TJ /F4 11.955 Tf 9.3 0 Td[((^p,~x,)+0+^p~vb].(1) InFourierspaceitis: C=)]TJ /F3 11.955 Tf 9.3 0 Td[(p@f0 @pneT[)]TJ /F4 11.955 Tf 9.3 0 Td[((^p,~k,)+0+vb(~k)].(1) Wehave: ==)]TJ /F3 11.955 Tf 9.29 0 Td[(neTZdp 22p4@f0 @pZ1)]TJ /F9 7.97 Tf 6.59 0 Td[(1d 2[0)]TJ /F4 11.955 Tf 11.95 0 Td[(()+vb]=neT4(Z1)]TJ /F9 7.97 Tf 6.59 0 Td[(1du 2()+vb=3). (1) 30

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Dene1=i=2Rd().FinallywehavetherstmomentumoftheBoltzmannequationforbaryons: dvb d+da advb+ik =)]TJ /F3 11.955 Tf 9.3 0 Td[(neTa4 3b[3i1+vb],(1) whereweuseconformaltimeforconvenience. TheEinsteinEquationisG=8GT.WecancalculatetheperturbedEinsteintensorintermsof .Forthetime-timecomponentinkspace G00=)]TJ /F4 11.955 Tf 9.3 0 Td[(6H@ =@t+6 H2)]TJ /F4 11.955 Tf 11.95 0 Td[(2k2=a2.(1) Thetime-timecomponentoftheenergy-momentumtensoris T00=)]TJ /F11 11.955 Tf 11.29 11.35 Td[(XigiZd3p (2)3Ef(~p,~x,)=)]TJ /F6 11.955 Tf 9.3 0 Td[(0[1+40])]TJ /F6 11.955 Tf 11.96 0 Td[(0[1+40])]TJ /F6 11.955 Tf 11.96 0 Td[(0dm[1+])]TJ /F6 11.955 Tf 11.96 0 Td[(0b[1+b], (1) sowecanwritetheperturbedtime-timecomponentoftheEinsteinEquation: k2 +3_a a(_)]TJ /F6 11.955 Tf 11.96 0 Td[( _a a)=4Ga2[400+400+0dm+0bb],(1) where0isdenedsimilarlyto0,_a=da=d. WenowconsiderthespatialpartoftheEinsteinequations.ThelongitudinaltracelesspartofGijinkspaceis: (^ki^kj)]TJ /F6 11.955 Tf 11.95 0 Td[(ij=3)Gij=2 3a2k2(+ ).(1) ThensamepartofTijis: (^ki^kj)]TJ /F4 11.955 Tf 11.95 0 Td[((1=3)ji)Tij=giZd3p (2)32p2)]TJ /F4 11.955 Tf 11.96 0 Td[(1=3p2 E(p)(f(~p,~x,)+f(~p,~x,). (1) WecanwritetheperturbedspatialpartoftheEinsteinEquation: k2( +)=)]TJ /F4 11.955 Tf 9.3 0 Td[(32Ga2[02+02],(1) 31

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where2=)]TJ /F11 11.955 Tf 11.29 9.64 Td[(RdP2()(k,,),2=)]TJ /F11 11.955 Tf 11.29 9.64 Td[(RdP2()(k,,),P2=3=2(2)]TJ /F4 11.955 Tf 12.05 0 Td[(1=3).Weseethatexceptforthemulti-polecontributedbyphotonsandneutrinosthetwoscalarperturbationsareequalandopposite.Soforthematterdominatedera =)]TJ /F6 11.955 Tf 9.3 0 Td[(. 1.4.3CMBRAnisotropiesRevealCDM TocalculatetheCMBRanisotropies,onechoosesappropriateinitialconditionsandasetofcosmologicalparameterssuchasmatterdensitym,baryondensityb,cosmologicalconstant...andsolvesnumericallythedynamicalequations.DifferentsetsofcosmologicalparametersgivedifferentanisotropiesofCMBR.ByttingthesemodelswithdataobtainedbyWMAP,oneobtainstheabundanceofbaryonsandofnon-relativisticmatterintheUniverse: bh2=0.0240.001Mh2=0.140.02,(1) whichisclearevidencefortheexistenceofnon-baryoniccolddarkmatter. 1.5EVIDENCEFORCDMFROMGALACTICROTATIONCURVES Therotationcurvesofgalaxiesshowevidencefordarkmatter.Theobservedrotationcurvesareapproximatelyatatlargedistancesbeyondtheedgeofthegalacticdisk.Usingclassicaldynamicstherotationvelocityis: v(r)=r GM(r) r(1) forasphericallysymmetricmassdistribution,whereM(r)isthemassinsideradiusr.Thereforethevelocitiesofstarsshouldbehavelike1=p riftheluminousmatterweretheonlymatterpresent.Theobservedv(r)isapproximatelyconstantsoitindicatesaninvisiblehalowithmassproleM(r)r. 32

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CHAPTER2INTRODUCTIONTOAXIONPHYSICS 2.1THESTRONGCPPROBLEM 2.1.1LagrangianofTheStandardModel Thestandardmodelofelementaryparticlesisaverysuccessfultheorywhichdescribesthefundamentalinteractionsandtheconstituentsofmatter.ItsuccessfullypredictsorexplainsallphenomenaobservedinthelaboratoryuptotheTeVenergyscaletoday.Thestandardmodelisbasedonquantizedrelativisticeldtheory.Thefundamentallagrangianismadeoffourparts:theYang-MillspartLY,theWeyl-DiracpartLW,theHiggspartLH,andtheYukawacouplingpartLYu.TheYang-Millspartis: LY=)]TJ /F4 11.955 Tf 16.34 8.09 Td[(1 4g23XGAGA)]TJ /F4 11.955 Tf 19 8.09 Td[(1 4g22XFaFa)]TJ /F4 11.955 Tf 18.99 8.09 Td[(1 4g21BB,(2) wheregiaredimensionlessconstants,A=1...8,a=1...3and GA=@AA)]TJ /F6 11.955 Tf 11.96 0 Td[(@AA)]TJ /F3 11.955 Tf 11.96 0 Td[(fABCABAC (2) Fa=@Fa)]TJ /F6 11.955 Tf 11.95 0 Td[(@Fa)]TJ /F6 11.955 Tf 11.95 0 Td[(abcWbWc (2) BA=@B)]TJ /F6 11.955 Tf 11.95 0 Td[(@B, (2) whereAA,FaandBaregaugeeldsforSU(3)c,SU(2)LandU(1).fABCandabcarethestructurefunctionsofSU(3)andSU(2)respectively. TheWeyl-Diracpartdescribesthefermioneldsandtheircouplingtogaugeelds.Letusrstdene: W=1 2Waa (2) A=1 2AAA, (2) whereaarethePaulimatricesandAaretheGell-Mannmatrices.Theleft-handedfermionsformSU(2)doublets,theright-handedfermionsareSU(2)singlets,thequarksareSU(3)tripletsandtheleptonsareSU(3)singlets.Onecanwritethecovariant 33

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derivatives: DLi=(@+iW)]TJ /F3 11.955 Tf 14.34 8.09 Td[(i 2B)Li (2) DQi=(@+iA+iW+i 6B)Qi (2) Dei=(@+iB)ei (2) Dui=(@)]TJ /F3 11.955 Tf 11.96 0 Td[(iA)]TJ /F4 11.955 Tf 13.15 8.09 Td[(2i 3B)ui (2) Ddi=(@)]TJ /F3 11.955 Tf 11.95 0 Td[(iA+i 3B)di, (2) whereLi=)]TJ /F8 7.97 Tf 5.48 -4.38 Td[(ieiL,Qi=)]TJ /F5 7.97 Tf 5.48 -4.38 Td[(uidiLand = cLfor =u,d,e.i=1,2,3runsoverthethreefamilies.ThentheWeyl-DiracLagrangianis: LW=LyiDLi+eyiDei+QyiDQi+uyiDui+dyiDdi.(2) NowletusconsidertheHiggspart.ThesimplestmodelcontainsaHiggsdoublet:)]TJ /F5 7.97 Tf 5.48 -4.38 Td[(H1H2.TheLagrangianis: LH=(DH)y(DH)+2HyH)]TJ /F6 11.955 Tf 11.96 0 Td[((HyH)2,(2) whereDH=(@+iW+i 2B)H.FinallythegeneralformoftheYukawapartis: LYu=iLTi2ejHYeij+iQTi2djHYdij+iQTi2uj2HYuij+c.c.(2) wheretheYukawacouplingsYijare33matrices.SinceanymatrixcanbewrittenasM=U1DU2,whereU1,2areunitarymatricesandDisarealdiagonalmatrix,onecanalwaysmakeYeijarealdiagonalmatrixbyredeningtheleptonelds.ForthequarkparttherearetwotermsYdYuin 2 ,soonecanonlydiagonalizeonematrix.Onecanwrite: LYuQuark=iQTi2ydiidiH+iQTi2Ujiyujjuj2H+c.c.(2) wheretheyd,uiiarerealandUjiistheCabibbo-Kobayashi-Maskawa(CKM)matrix.TheCKMmatrixcontainsthreemixinganglesandonephase.Therefore,thestandardmodel 34

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Lagrangiancontains3gaugecouplingsg1,g2,g3,fourparametersfortheCKMmatrix,9massesandandtermsfortheHiggspart.Itisatotalof18parameters,butwewillseethatthereisonemoretermwhichisproducedbyquantumeffectsintheQCDvacuum.These19parametersformthecompleteparameterspaceofthestandardmodelofparticlephysics. 2.1.2TheU(1)AProblem Themassesofthetwolightquarksu,dareorderofMeVwhichissmallerthantheQCDscaleQCD=217MeV.Onemaytreatthetwoquarksasmassless.Thisiscalledthechirallimit.ThelightquarkQCDLagrangianisthen: LQCD')]TJ /F4 11.955 Tf 23.11 8.08 Td[(1 4GAGA+2Xi=1qiDqi,(2) wheretheqiareDiracfermions.TheLagrangianhassymmetry:SUL(2)SUR(2)U(1)UA(1),underwhich: SUL,R(3):q!UL,RqL,R;U(1):qL,R!eiqL,R; (2) UA(1):qL!eiqL;qR!e)]TJ /F5 7.97 Tf 6.59 0 Td[(iqR (2) wheretheUL,RareSU(2)matrices.InlowenergyQCDthechiralsymmetrySUL(2)SUR(2)isspontaneouslybrokenduetothecondensationofquarkpairsinthevacuum=ij3.ThishoweverleavesthevectorsymmetrySUL+R(2)unbroken.Sowehave: SUL(2)SUR(2)!SUL+R(2).(2) SincethreegeneratorsarebrokenwehavethreeNambu-Goldstonebosonsinthechirallimit.Intherealworld,thelightquarksarenotmassless.Thereforetherearethreelightmesons+,)]TJ /F4 11.955 Tf 7.08 -4.34 Td[(,0.Thispredictiontstheexperimentverywell.However,thequarkcondensatealsobreaksUA(1).ThusafourthGoldstonebosonisproduced.Weinberg[ 19 ]obtainedamassboundofthefourthgoldstoneboson:m


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Asexplainedbelow,itissolvedbyintroducingtheeffectsofinstantons.TheU(1)AcurrenthastheABJanomaly.TheABJanomalytermisatotaldivergence.However,fornon-abelianstrongcouplinggaugeelds,itcontributestothephysics.Therefore,U(1)AisexplicitlybrokenduetoQCD.SotheU(1)Aproblemissolved.Ontheotherhand,theanomalycreatesthestrongCPproblem. 2.1.3TheVacuumandInstantons Letusconsidernon-abeliangaugeeldtheory.ThegaugetransformationsareA!UAUy)]TJ /F3 11.955 Tf 12.74 0 Td[(i=gU@Uy,wheregisthegaugecouplingandUisaunitarymatrix.OneoftheclassicaleldcongurationscorrespondingtothegroundstateisA=0.SoA=i=gU@Uyareclassicaleldcongurationsdescribingthevacuum.LetususethegaugeA0=0.SowerestricttoU=U(~x)whichisindependentoftime.AlsoweimposeaboundaryconditionU(~x)!Constantwhenj~xj!1.ForU(~x)satisfyingaboveconditions,itturnsoutthatnoteveryUcansmoothlydeformtotheotherswithoutpassingthrougheldcongurationswithnon-zeroenergy.Any22specialunitarymatrixmaybewrittenU=a4+i~a~wherea2=1.Wecanseta0=(~x2)]TJ /F6 11.955 Tf 12.02 0 Td[(2)=(~x2+2)and~a=2i~x=(~x2+)whereisaparameter.Thewindingnumberoftheabovemapisone.Agaugeeldcannotbesmoothlydeformedintootherswithdifferentwindingnumberwithoutpassingenergybarriers.Thewindingnumbercanbecalculatedas n=)]TJ /F4 11.955 Tf 19.63 8.09 Td[(1 242Zd3xijkTr[(U@iUy)(U@jUy)(U@kUy)].(2) ForatimedependenteldA(~x,t),thePontryaginindex q=g2 162Zd4xTr(F~F),(2) isthedifferenceinwindingnumbersbetweenthecongurationatt=andt=+1.Sonon-abeliangaugetheoryhasaninnitenumberofeldcongurationswithzeroenergy.Theyaredistinguishedbywindingnumbersnandseparatedbyenergybarriers.Althoughtheyareseparatedbyenergybarriers,congurationofdifferentwinding 36

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numbercantunneltoeachotherduetoinstantons.Soforthephysicalvacuumwehavetoincludeeldcongurationswithallpossiblewindingnumbern. Nowletusdiscussbrieytheinstantons.Fortwoquantumstatesseparatedbyanenergybarrier,thetunnelingamplitudebetweenthemcanbecalculatedbythepathintegral:e)]TJ /F5 7.97 Tf 6.59 0 Td[(S,whereSistheEuclideanactionwithboundaryjn1>att=andjn2>att=+1.Inmostcases,Sisinnitesinceforeldtheoryoneintegratesovertheentirespacetime.Sothetunnelingamplitudevanishesandthetwostatesremainexactlydegenerate.Forstrongcouplingnon-abeliangaugeeldtheory,therearesolutionsthatmediatebetweenstateswithdifferentwindingnumber(n16=n2)andtheactionSofthesesolutionsisniteevenintheinnitevolumelimit.Solutionsthatrelatejn1>att=andjn2>att=+1withn2)]TJ /F3 11.955 Tf 12.26 0 Td[(n1=1aretheinstantons.ThesimplestinstantonsolutionwasdiscoveredbyBelavin,Polyakov,Schwartz,andTyupkin(BPST): A=if(r)g)]TJ /F9 7.97 Tf 6.59 0 Td[(1(x)@g(x) (2) f(r)=r2 r2+2 (2) g(x)=)]TJ /F3 11.955 Tf 11.3 8.09 Td[(i rx, (2) where=(~,i)andistheinstantonsize.TheactionofthisinstantonisS=jn2)]TJ /F3 11.955 Tf 12.37 0 Td[(n1j82=g2=82=g2.Theinstantondescribestunnelingbetweentwostateswithwindingnumberdifferingbyone. Nowletusdiscussthephysicalvacuumj!>.LetTbeagaugetransformationthatchangeswindingnumberbyone:Tjn>=jn+1>.Forthephysicalvacuum,wehaveTj!>=eij!>.Sincewehaveseenthatj!>hastoincludestateswithallpossiblewindingnumbersn,oneconcludesthatthephysicalvacuumhastheform:j!>=Pneinjn>.Thusthephysicalvacuumisdenedbyaparameter2[0,2].ThevacuumproducesannewterminthestandardmodelLagrangian.Letusconsiderthe 37

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vacuumtovacuumtransitionamplitude: <1je)]TJ /F5 7.97 Tf 6.59 0 Td[(Htj>=Xn1Xne)]TJ /F5 7.97 Tf 6.58 0 Td[(i(n11)]TJ /F5 7.97 Tf 6.59 0 Td[(n)=Xn1e)]TJ /F5 7.97 Tf 6.58 0 Td[(in1(1)]TJ /F8 7.97 Tf 6.59 0 Td[()Xn1)]TJ /F5 7.97 Tf 6.58 0 Td[(nZ[DA]n1)]TJ /F5 7.97 Tf 6.59 0 Td[(nexp[)]TJ /F11 11.955 Tf 11.29 16.28 Td[(Zd4xL)]TJ /F3 11.955 Tf 11.95 0 Td[(i(n1)]TJ /F3 11.955 Tf 11.95 0 Td[(n)]=(1)]TJ /F6 11.955 Tf 11.95 0 Td[()Z[DA]qexp[)]TJ /F11 11.955 Tf 11.29 16.27 Td[(Zd4x(L+ 322Fa~Fa)], (2) whereR[DA]n1)]TJ /F5 7.97 Tf 6.59 0 Td[(ndenotesfunctionalintegrationwithrespecttogaugecongurationsofPontryaginindexn1)]TJ /F3 11.955 Tf 12.42 0 Td[(n=q.Weseeanadditionaleffectiveinteractionisobtained:L==322Fa~Fa.BecauseF~FisCPodd,QCDisnotCPinvariantif6=0. Noether'stheoremandtheABJanomalyimplythatthephysicsofQCDisunchangedunderthetransformation: F~F!()]TJ /F11 11.955 Tf 11.95 11.36 Td[(Xi)F~F (2) qimiqi!qie(ii5=2)mie(ii5=2)qi (2) WecanchoosePi=togetridofthetermattheexpenseofadditionalCPviolatingmassterms: X(micosiqiqi+misiniqi5qi).(2) TheiareconstrainedbytherequirementthattheCPviolatingLagrangianshouldnotcausearealignmentofvacuumandtheQCDvacuumisaavorsinglet,i.e.=

=.Baluni[ 24 ]nds: Ximi=mumdms mums+mdms+mumd.(2) Sotheeffectiveinteractionis: L=mumdms (mu+md)ms+mumd(ui5u+di5d+si5s),(2) forsmall.Herethemassesareassumedtobereal.Ifthemassesarecomplex,theparametercontrollingCPviolationinthestronginteractionis=+argdetM.The 38

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CPviolationgivesacontributiontotheneutronelectricdipolemoment.BaluniusedtheMITbagmodelandfounddn=2.710)]TJ /F9 7.97 Tf 6.58 0 Td[(16ecm;Crewtheretal.usingcurrentalgebrandd=5.210)]TJ /F9 7.97 Tf 6.59 0 Td[(16ecm.Theupperboundoftheneutronelectricdipolemomentis0.610)]TJ /F9 7.97 Tf 6.58 0 Td[(24ecm.Sohasanaupperboundjj<1.210)]TJ /F9 7.97 Tf 6.59 0 Td[(9.Suchasmallneedsanexplanation,whichisthestrongCPproblem. 2.1.4SolutionstoTheStrongCPProblem TosolvethestrongCPproblem,therearethreesuggestions: 1.Theultravioletupquarkmassiszero,sotheRHSof 2 vanishes.However,latticeQCDsimulationssuggestanon-zeroupquarkmass. 2.CPisspontaneouslybroken,Inthatcaseisniteandcanbearrangedtobesufcientlysmall. 3.ThePeccei-Quinnsolution.Ifisadynamicalvariable,itwillnaturallygotothevaluethatminimizestheenergy,whichiszeroaswewillsee.Thisextravariablewillproduceaquasi-Nambu-Goldstonebosonwhichiscalledtheaxion. Asimpleprooftoshowthattheenergyisminimizedwhen=0wasgivenbyC.VafaandE.Witten[ 20 ].LetusconsiderthepathintegralinEuclideanspace.TheQCDlagrangianwiththetermisL=)]TJ /F4 11.955 Tf 9.3 0 Td[(1=4g2Tr(GG)+Pq(D+mi)qi+i=322Tr(G~G).Integratingoutthefermionsonehas: e)]TJ /F5 7.97 Tf 6.59 0 Td[(VE=ZDAdet(D+M)eRd4x[1=4g2TrGG)]TJ /F5 7.97 Tf 6.59 0 Td[(i=322G~G]. (2) InQCDdet(D+M)ispositive.Thisisbecause(iD)ishermitianinEuclideanspaceand5anti-commuteswith.Thisimpliesthatforevery(imaginary)eigenvalueof(D)thereisaneigenvalueofoppositesign.NotealsothatiG~Gispureimaginary,soitonlyreducesthetotalvalueofthepathintegral.Thuswhenisnotzero,theenergyisincreased. 39

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Figure2-1. Theenergyduetothevacuumangle IfisdynamicalvariableinaLagrangionoftheform: L=)]TJ /F4 11.955 Tf 9.3 0 Td[(1=4g2Tr(GG)+Xq(D+mi)qi+=322TrG~G+1=2@a@a+a=(fa322)TrG~G, (2) theshiftingofthedynamicaleldawillautomaticallydrivethevacuumenergytoitsminimumsothat+a=fa!0andtheCPviolationofQCDnolongerexists.Onecanshowthatthelowenergyeffectivetheoryhastheform 2 ifthereisaUPQ(1)symmetryinthestandardmodel.UPQ(1)symmetrymusthavethefollowingproperties:1)itisasymmetryoftheclassicalactiondensity.2)itisspontaneouslybroken.3)itisexplicitlybrokenbytheQCDchiralanomaly. 2.1.5AxionModels Theaxiontsintovariousextensionsofthestandardmodel.ThePeccei-Quinn-Weinberg-Wilczek(PQWW)axionistheearliestmodel.ThistypeofmodelassumesanadditionalHiggsdoubletandaspontaneouslybrokenglobalsymmetry,thePeccei-Quinnsymmetry.AppropriateU(1)PQchargesareassignedtothequarkssothatUPQ(1)isexplicitlybrokenbytheQCDchiralanomaly.Thistypeofaxionhasfa250GeV,whichistheelectrowakscale.Astrophysicalandexperimentalsearcheshaveruledoutthistypeofaxion.Thereforewewillnotdiscusssuchmodelsindetail. 40

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ThePQWWaxiontiedPeccei-Quinnsymmetrybreakingscalewiththeelectroweakscaleandwasruledout.Later,J.Kimetal.separatedthesetwoscalestogiverisetoanewtypeofaxion,whichisoftencalledtheinvisibleaxion.Therearetwomajorbenchmarkmodelsofthistype,oneistheKim-Shifman-Vainshtein-Zakharov(KSVZ)axion,andtheotheristheDine-Fishler-Srednicki-Zhitnitskii(DFSZ)axion. LetusrstconsidertheKSVZaxion.TheKSVZmodelintroducesannewcomplexscalareldandanewheavyquarkQ.TheU(1)PQtransformationis U(1)PQ:a!a+fa (2) !exp(iq) (2) QL!exp(iQ=2)QL (2) QR!exp()]TJ /F3 11.955 Tf 9.29 0 Td[(iQ=2)QR (2) whereqisthePQchargeforthescalareldandwewillassumeq=1.ThepotentialforisU(1)PQinvariant.UPQ(1)isspontaneouslybrokenbythevacuumexpectationvalue<>=v.Thescalareldmaybewrittenas: =(v+)exp(ia v).(2) Weseethatfa=v.TopreservethePQsymmetrytheheavyquarkcannothaveabaremass.TherelevantYukawacouplingandHiggspotentialare LYu=)]TJ /F3 11.955 Tf 9.3 0 Td[(fQyLQR)]TJ /F3 11.955 Tf 11.96 0 Td[(fQyRQR (2) V=)]TJ /F6 11.955 Tf 9.3 0 Td[(2+()2 (2) wheref,areparameters.WeseethatthemassoftheheavyquarkQdependsonv. AdrawbackoftheKSVZmodelisitintroducesanewheavyquark.TheDFSZmodelavoidsthisattheexpenseofanadditionalHiggsdoublet.IntheDFSZmodel,stillisacomplexscalar.HowevercouplestotheHiggsdoublets,whichthencoupleto 41

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thelightquarks.TherelevantYukawainteractionsandscalarpotentialare: LYu=)]TJ /F3 11.955 Tf 9.3 0 Td[(f(u)ijqyLj2uRi)]TJ /F3 11.955 Tf 11.95 0 Td[(f(d)ijqyLj1dRi+c.c. (2) V=(ay11+by22)+c(T122+h.c.)+djT12j2+ejT2j2+1(y11)]TJ /F3 11.955 Tf 11.95 0 Td[(v21=2)2+2(y22)]TJ /F3 11.955 Tf 11.95 0 Td[(v22=2)2+()]TJ /F3 11.955 Tf 11.95 0 Td[(v2=2)2, (2) whereTdenotestranspose,abcdareparameters,v1,2arethevacuumexpectationvaluesofthetwoHiggsdoubletsrespectivelyandi,jarethefamilyindices.UnderthePeccei-Quinnsymmetry: U(1)PQ:1!exp()]TJ /F3 11.955 Tf 9.3 0 Td[(iQ)1 (2) 2!exp()]TJ /F3 11.955 Tf 9.3 0 Td[(iQ)2 (2) uR!exp(iQ)uR (2) dR!exp(iQ)dR (2) !exp(iq). (2) Wehave 2=+. (2) where =2x x+x)]TJ /F9 7.97 Tf 6.59 0 Td[(1 (2) =2x)]TJ /F9 7.97 Tf 6.58 0 Td[(1 x+x)]TJ /F9 7.97 Tf 6.58 0 Td[(1, (2) withx=v2=v1. 42

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Figure2-2. Axiongluongluoncouplingduetoanomaly Figure2-3. Axionmassduetomixingwiththe0meson. 2.2AXIONPROPERTIES 2.2.1TheAxionMass Axionsarequasi-Goldstonebosonsandtheyobtainamassduetothepotentialgeneratedbytheinstantons.OnecanremovetheaG~Gtermbyintroducingtermsthatmixpseudoscalarmesonsandaxions.Thereforethemassofaxionsisrelatedtothemassofthepions.Goingtothemasseigenstates,onends: mam0f fa(2) wherefisthepiondecayconstantandwehaveneglectedthefactorsdependingonthequarkmassratios. 43

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Figure2-4. Axion-Photoncoupling 2.2.2TheAxionCouplings Theaxion-photoncouplingisthesumoftwopieces:oneisduetotheABJanomalyandtheotherisduetoaxionmixingwithmesonsandthecouplingofmesonstophotons.Bothpiecesaresuppressedbythesymmetrybreakingscale.Wehave: La faaF~F.(2) Theaxioncouplestothefermionstoo Laqiimi faaqi5qi,(2) whereidenotedifferentfermions.Thesecouplingsarealsosuppressedbythesymmetrybreakingscalefa. 2.3AXIONASTROPHYSICSANDCOSMOLOGY 2.3.1ConstraintDuetoCosmology Oneofthemostnotablefeaturesofaxionsisthattheyareanexcellentcandidateforcolddarkmatter.Darkmeansnegligiblyinteractingexceptforgravitationalinteractions.ThedarkmatterparticlesshouldalsobestablecomparedtotheageofUniverse.Theaxionscoupletootherparticlesveryweaklysotheynaturallysatisfythesetwo 44

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conditions.Forexample,thelifetimeTofaxonsisapproximately T'1050sfa 1012GeV5.(2) Axionscanbehotdarkmatterorcolddarkmatterdependingontheirorigin.Hotdarkmatteraxionsarecreatedfromthermalprocessesinthehotplasmaintheearlyuniverse,andtheirpresentnumberdensityis na(t0)=(3) 2T3D(aD a0)3,(2) whereTDistheaxiondecouplingtemperature.aDanda0arethescalefactorsatdecouplingandtoday,respectively. Thecoldaxionsareproducedbythemisalignmentmechanism[ 21 ]andtheirnumberdensitytodayis naf2a 2t1(a1 a0)3,(2) wheret1isdenedbyma(t1)t1=1andisorderof10)]TJ /F9 7.97 Tf 6.59 0 Td[(7sec(fa=1012GeV)1=3.Ifallthedarkmatteriscomposedofcoldaxions,themassofaxionsisoforderma10)]TJ /F9 7.97 Tf 6.59 0 Td[(6eV. 2.3.2ConstraintDuetoAstrophysics Theaxionschangetheevolutionofstarsbecausestarsemitaxionsfromtheirbulkwhiletheyemitphotonsonlyfromtheirsurfaces[ 22 ].Adetailedstudyofourstar,theSun,showsthataxionemissionduetothePrimakoffprocessmodiesthetemperaturedistributionproleoftheSunandthereforechangestheneutrinoux.ThemeasuredneutrinouxfromtheSungivestheconstraintga.510)]TJ /F9 7.97 Tf 6.59 0 Td[(10GeV)]TJ /F9 7.97 Tf 6.59 0 Td[(1[ 23 ]. Studiesofredgiantsandhorizontalbranchstarsinglobularclustersalsogiveveryimportantconstraintsonaxioncouplings.Agiantstarisastarwithlargerradiusandluminositythanamainsequencestarofthesamesurfacetemperature,andaredgiantisagiantstarwith0.5-10solarmassesinitslatephaseofevolution.ThehorizontalbranchstarsarestarsintheevolutionaryphaseimmediatelyaftertheredgiantphaseforstarswithamasssimilartothemassofSun.Axionsareproducedmoreefcientlyin 45

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Figure2-5. ThecosmologyandastrophysicalconstraintonALPs/axions Figure2-6. ThePrimakoffProcessconvertphotonstoaxions 46

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thehorizontalbranchstarsthanintheredgiants.Theadditionalaxionuxinhorizontalbranchstarsexhauststheirnuclearfuelandthereforedecreasestheirlifetime.Thiswouldimplythattheredgiantsaremoreabundantinglobularclusterscomparedtothepredictionofstandardstarevolution.Theobservationsagreewithin10%ofthestandardstarevolutionprediction,andthereforeconstraintheadditionalenergy-losschannel.Consequently,thecouplingislimitedto[ 26 ] ga<10)]TJ /F9 7.97 Tf 6.58 0 Td[(10GeV)]TJ /F9 7.97 Tf 6.59 0 Td[(1.(2) Awhitedwarfisthenalevolutionphaseofastarwithamassnothighenoughtobecomeaneutronstar.Theenergylossesofawhitedwarfinthestandardstarevolutionmodelareduetoneutrinoemissionfromthecoreandphotonemissionfromthesurface.Ifaxionsexist,therewillbeanadditionalchannelofenergylossduetoaxionemissionduetoCompton-likescatteringinvolvingthecouplingsofaxionstoelectrons.Thestudiesofthecoolingrateofwhitedwarfsconclude[ 27 ]: gaee<10)]TJ /F9 7.97 Tf 6.58 0 Td[(13(2) Asupernovaistheexplosionofastarthatformsaneutronstarorablackhole.Theexplosionexpelsthematterofthestarintoanexpandingshellwithaspeedof10%thatofthespeedoflight.Thecoredensityishighenoughtodiffusethepropagationofneutrinos.Ifaxionsexist,additionalenergylosschannels,dominatedbyaxionnucleonbremsstrahlung,willbepresentbesidesthechannelduetoneutrinodiffusivetransportation,sothatthedurationoftheneutrinoburstfromsupernovascanbeshortened.Ifthecouplingistooweakthedurationisnotchangedsincetoofewaxionsareproduced.Toostrongacouplingalsodoesnotchangethedurationbecauseaxionscannotfreesteamfromthecore.Thestudyofsupernova1987Aconstrainstheaxion 47

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nucleoncoupling[ 28 ]: 10)]TJ /F9 7.97 Tf 6.58 0 Td[(10gann. (2) 48

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CHAPTER3AXIONSEARCHES 3.1AXIONDARKMATTERSEARCH Inchapter1wesawthat23%oftheuniverse'stotalenergydensitytodayiscontributedbydarkmatter.Inchapter2wesawthatQCDaxionsareoneoftheleadingcandidatesforcolddarkmatter[ 21 ]becausetheQCDaxionsareeffectivelycollisionlessandthemisalignmentmechanismproducesaverycoldpopulationofaxionswhichhavetherequiredenergydensity. TheAxionDarkMattereXperiment(ADMX)[ 29 ][ 30 ]isarealizationofaxionhaloscopeinwhichaxionsinthehaloofourgalaxyareinducedtoconvertinacavitytomicrowavephotonsthatarethenpickedupbyanantenna.Thepoweroftheaxionsignalisproportionaltothelocalaxiondensity,andthesignalwidthisproportionaltotheenergydispersionofthedark-matteraxions.Therefore,thesignalpropertiesobservedbyADMXdependonthestructureofthegalactichalo. Theaxionscoupletophotonsvia: La=g a(x) faF~F,(3) whereisthenestructureconstant,faistheaxiondecayconstant,andgisamodel-dependentcouplingoforderone(IntheKSVZmodelg)]TJ /F4 11.955 Tf 23.79 0 Td[(0.97,andintheDFSZmode(g0.36). Coldaxionsinthegalactichaloarenon-relativisticsothattheaxion-photonconversioninamagneticeldcreatesphotonswhoseenergyapproximatelyequalsthemassoftheaxions.Ifthemassiswithinthebandwidthofthecavity,theconversionisresonantandthepowerPofthephotonsis P=g fa2VB20aC mamin(Q,Qa),(3) 49

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whereVisthecavityvolume,B0isthemagneticeldinsidethecavity,aisthelocaldensityofhaloaxions,Qistheloadedqualityfactorofthecavity(denedascenterfrequencydividedbythefrequencybandwidth),Qaistheratiooftheenergyofaxionstotheirenergyspread,andCisaformfactorgivenby: C=RVd3xE!B02 B20VRVd3xjE!j2,(3) where~E!(~x)istheelectriceldofthecavitymodetheaxionsconvertinto.in 3 isthedielectricconstantofthemediuminsidethecavity.AtypicalCisorderofunity. 3 canberewrittenas: P=0.510)]TJ /F9 7.97 Tf 6.58 0 Td[(21W)]TJ /F5 7.97 Tf 13.39 -4.98 Td[(V 500L)]TJ /F5 7.97 Tf 15.78 -4.8 Td[(B0 7T2C)]TJ /F5 7.97 Tf 9.98 -3.82 Td[(g 0.362a 0.510)]TJ /F12 5.978 Tf 5.76 0 Td[(24g.cm)]TJ /F12 5.978 Tf 5.76 0 Td[(3)]TJ /F8 7.97 Tf 14.41 -4.97 Td[( 1GHzmin[Q,Qa] 105. (3) TheQCDaxionmassistherange10)]TJ /F9 7.97 Tf 6.59 0 Td[(6
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3.2SOLARAXIONSEARCHES StarsproduceabundantnumbersofaxionsintheircorebythePrimakoffeffect,andtheSunisthedominantsourceofthiskindofaxionsintheskyduetoitsrelativelyshortdistancetotheEarth.Afterproduction,theaxionspropagatefreelywithaspeedveryclosetothespeedoflightandreachtheEarthafter500s.Onecandetectthesolaraxionsbyusinganaxionhelioscope[ 30 ].Naturally,thesensitivityoftheaxionhelioscopedependsontheaxionuxfromtheSunandtheprobabilityofaxionphotonconversioninthedetector. 3.2.1SolarAxionProduction InthecoreofstarsthermalphotonswithenergiesofaboutakeVareconvertedintoaxionsintheelectriceldsofchargedparticles:+Ze$Ze+a.ThisprocessisknownasthePrimakoffeffect.ThetemperatureofelectronsandnucleiisaboutakeV,whichismuchsmallerthantheirmass,sothatthedifferentialcross-sectionforthisprocessisgivenby: d d=g2aZ2 8~k~ka2 j~qj4,(3) where~q=~k)]TJ /F6 11.955 Tf 11.81 3.16 Td[(~kaisthemomentumtransfer,Zisthechargeofanucleus,andisthenestructureconstant.Inaplasma[ 34 ],thedifferentialcrosssectionismodiedto: d d=g2aZ2 8~k~ka2 j~qj4~q2 +~q2,(3) whereistheDebye-Huckelscale: 2=4 TnB Ye+XjZ2jYj!,(3) wherenBisthebaryonnumberdensity,YeandYjarethepercentagesofelectronsandnucleonsintheplasmaandTistheplasmatemperature[ 35 ].Thetotalcrosssection 51

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canbeobtainedbyintegrating 3 andsummingoveralltargetspecies: =nB(Ye+XZ2jYj)Zdg2aZ2 8~k~ka2 j~qj4~q2 +~q2.(3) Forsmallaxionmass,mak,onends[ 36 ]: =g2aTk2s 321+k2s 4E2log1+4E2 k2s)]TJ /F4 11.955 Tf 11.96 0 Td[(1.(3) TheenergyspectrumofaxionsatEarthisthen: d dE=n 4d2=1 4d2ZR0d3rE eE=T(r))]TJ /F4 11.955 Tf 11.96 0 Td[(1, (3) wheredisthedistancebetweentheSunandtheEarth,RisthediameteroftheSunandT(r)isthetemperatureinsidetheSun.WeseethatthesolaraxionuxdensitydependsonT(r)whichcanbeobtainedfromsolarmodels.ForthewellestablishedSolarmodelonendstheaxionuxspectrum[ 22 ]. 3.2.2DetectorUsingBraggScattering Todetectsolaraxions,onemayusetheCoulombeldofnucleiinacrystaltoconvertaxionstophotonsbytheinversePrimakoffprocess.Theenergyofsolaraxionsis4keV,sothewavelengthofconvertedphotonsiscomparabletothelatticespacingofthecrystal.ThusthephotonsconvertedinacrystalwillproduceaBraggpattern.TheconstructiveinterferenceintheBraggpatterncanenhancethesignalbyorder104.ThereareseveralexperimentsusingthistechniqueoncrystalssuchasGermanium(SOLAX)andSodium-iodide(DAMA).Theresultingboundsontheaxion-photoncouplingareoforder10)]TJ /F9 7.97 Tf 6.58 0 Td[(9GeV)]TJ /F9 7.97 Tf 6.58 0 Td[(1[ 38 ]. 3.2.3AxionHelioscope Theaxionhelioscope[ 30 ]employsalaboratorymagneticeldtoconvertsolaraxionstolow-energyX-rays.InaregionoflengthL,magneticeldB,andbuffergas 52

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whoseabsorbtionrateis)]TJ /F1 11.955 Tf 6.78 0 Td[(,theconversionprobabilityofaxionsis[ 33 ] P=(gaBL=2)2 L2(q2+)]TJ /F9 7.97 Tf 18.73 3.45 Td[(2=4)1+e)]TJ /F9 7.97 Tf 6.59 0 Td[()]TJ /F5 7.97 Tf 4.82 0 Td[(L)]TJ /F4 11.955 Tf 11.96 0 Td[(2e)]TJ /F9 7.97 Tf 6.59 0 Td[()]TJ /F5 7.97 Tf 4.82 0 Td[(L=2cos(qL),(3) whereqisthemomentumtransfer: q=m2a)]TJ /F3 11.955 Tf 11.96 0 Td[(m2 2Ea,(3) andmistheeffectivephotonmassinthebuffergas: m=!p=r 4ne me.(3)neistheelectrondensityofthebuffergas.Thepurposeofthebuffergasistorestorethecoherenceofaxion-photonconversion(q=0)incasetheaxionsareheavy. TheTokyoAxionHelioscopeprovidesthelimit:ga6.010)]TJ /F9 7.97 Tf 6.59 0 Td[(10GeV)]TJ /F9 7.97 Tf 6.59 0 Td[(1forma.0.03eV[ 39 ].TheCERNAxionSolarTelescope(CAST)isthemostsensitiveaxion-helioscopeexperimenttoday. 3.3LASEREXPERIMENTS Bothphotonregenerationexperimentsandpolarizationexperimentsemploylasertechnology.TheydonotrelyonphysicalprocessesinsidetheSunorthehypothesisthataxionsarethedarkmatter.Soeverythingisunderlaboratorycontrol,buttheyaregenerallylesssensitivethanthehelioscopeandhaloscopeexperiments. 3.3.1PhotonRegeneration Inphotonregenerationexperimentsasmallfractionofphotonsinalaserbeamtraversesaregionpermeatedbyamagneticeld,whereitisconvertedtoaxions.Becauseoftheirweakcouplingtoordinarymatter,theaxionsthentravelessentiallyunimpededthroughawall,ontheothersideofwhichisanidenticalarrangementofmagnets,wheresomeoftheaxionsareinducedtoconvertbacktophotons,whichcanbedetected.Themajordrawbackofthiskindofexperimentisthatitssignalisveryweaksincetwostagesofconversionarerequired.Resonantlyenhancedphotonregeneration 53

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experimentsaddressthisproblembyaddingresonantcavitiesonbothsidesofthewallsothatthesignalisenhancedbyafactorofF2whereFisthenesseoftheFabry-Perotcavitieswhichcouldbeorderof105.TheimprovementinsensitivitytogaisafactorF1=2whichis300.Theconstraintsfromnon-resonantphotonregenerationexperiments,suchasthatbytheBrookhaven-Fermilab-Rutherford-Trieste(BFRT)collaboration,areoforder[ 42 ]: ga<10)]TJ /F9 7.97 Tf 6.59 0 Td[(7GeVwithma<10)]TJ /F9 7.97 Tf 6.58 0 Td[(3eV.(3) 3.3.2PolarizationExperiments Thepolarizationexperimentslookforbirefringenceanddichroismcreatedbyaxion-photonmixing.Startingwithabeamlinearlypolarizedat45degreeswithrespecttothedirectionofatransverseexternalmagneticeld,thebeamcanbeviewedasasuperpositionoftwobeams,oneparallelandonenormaltothemagneticeld.Thephotonspolarizedparalleltothemagneticeldmixwiththeaxions,whiletheotheronesareunperturbed.Thepolarizationvectorofthebeamemergingfromthemagneticregionrotatesduetotheamplitudereductionoftheparallelcomponent.Initiallylinearlypolarizedlightalsobecomesellipticallypolarizedduetothephaselagoftheparallelcomponent[ 44 ]: g2aB2!2 m4a[m2aL 2!)]TJ /F4 11.955 Tf 11.95 0 Td[(sin(m2aL 2!)].(3) However,QEDeffectscanalsoproduceaphaselagduetotheeffectiveinteraction2 90m4e[(FF)2+7 4(F~F)2].Themaindifcultyofthepolarimetryexperimentsistheunavoidableintrinsicbirefringenceoftheopticalelements.Inpracticethesensitivityofthepolarizationmeasurementsisseverelylimitedbythisfact.TheboundobtainedbytheBFRTcollaborationisga<3.610)]TJ /F9 7.97 Tf 6.59 0 Td[(7GeV)]TJ /F9 7.97 Tf 6.59 0 Td[(1withma<510)]TJ /F9 7.97 Tf 6.58 0 Td[(4eV[ 42 ]. 54

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CHAPTER4AXIONBOSE-EINSTEINCONDENSATION InthischapterwediscusstheaxionsBose-Einsteincondensationandthepossiblethermalizationofaxionswithotherparticlespecies.Thisdiscussionisbasedon[ 46 ]. 4.1REVIEWOFCOLDAXIONPROERTIES Coldaxions(withmass10)]TJ /F9 7.97 Tf 6.59 0 Td[(5eV)areoneoftheleadingcandidatesforcolddarkmatter(CDM)alongwithWIMPs(WeaklyInteractingMassiveParticleswithmass100GeV)andsterileneutrinos(withmassKeV).ColdaxionsarecreatedwhentheaxionmassturnsonatQCDtime:t1210)]TJ /F9 7.97 Tf 6.59 0 Td[(7sec.Thecoldaxionshavesmallvelocitydispersion,oforder: v(t)1 mt1a(t1) a(t),(4) wherea(t)isthescalefactor.Thevelocitydispersiondeterminestheeffectivetemperatureofcoldaxions.IncaseinationoccursafterthePeccei-Quinnphasetransition,visevensmallerbecausetheaxioneldgetshomogenizedduringination.Thenumberdensityofcoldaxionsis: n(t)41047 cm3f 1012GeV5 3a(t1) a(t)3.(4) Clearly,theaveragephasespacedensityNisveryhighduetothecombinationofhighnumberdensityandsmallvelocitydispersion.Wehave: Nn(2)3 4 3(mv)31061f 1012GeV8 3.(4) Inaddition,axionsinteractveryweakly.Forexample,theinteractions 4!'4andga'~E~Bhave10)]TJ /F9 7.97 Tf 6.58 0 Td[(48andga10)]TJ /F9 7.97 Tf 6.59 0 Td[(22eV)]TJ /F9 7.97 Tf 6.59 0 Td[(1. 55

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4.2AXIONTHERMALIZATIONINTHEPARTICLEKINETICANDCONDENSEDREGIMES 4.2.1EvolutionEquationsforNonrelativisticAxions Theactiondensityofaxionsis La=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2@@)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2m22+ 4!4+....(4) Thedotsrepresentinteractionsoftheaxionwithotherparticlesandaxionself-interactionswhicharehigherorderinanexpansioninpowersof.FortheaxionthatsolvesthestrongCPproblem[ 1 3 ],themassmandself-couplingaregivenby m=mf fap mumd mu+md'610)]TJ /F9 7.97 Tf 6.59 0 Td[(6eV1012GeV fa=m2 f2am3u+m3d (mu+md)3'0.35m2 f2a (4) wheremisthepionmass,f'93MeVthepiondecayconstant,andmuandmdaretheupanddownquarkmasses.Theformulafortheaxionmass[ 2 ]isobtainedbyexpandingtheeffectivepotentialforpionsandaxionstosecondorderinthephysicalaxioneld.Toobtain,simplyexpandtofourthorder. WeintroduceacubicboxofvolumeV=L3withperiodicboundaryconditionsatitssurface.Insidethebox,theaxioneld(~x,t)anditscanonicalconjugateeld(~x,t)areexpandedintoFouriercomponents (~x,t)=X~na~n(t)~n(~x)+ay~n(t)~n(~x)(~x,t)=X~n()]TJ /F3 11.955 Tf 9.3 0 Td[(i!~n)a~n(t)~n(~x))]TJ /F3 11.955 Tf 15.28 0 Td[(ay~n(t)~n(~x) (4) where ~n(~x)=1 p 2!~nVei~p~n~x,(4) 56

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and~n=(n1,n2,n3)withnk(k=1,2,3)integers,~p~n=2 L~n,and!=p ~p~p+m2.Thea~nanday~nsatisfycanonicalequal-timecommutationrelations: [a~n(t),ay~n0(t)]=~n,~n0,[a~n(t),a~n0(t)]=0.(4) NotethatwearequantizingintheHeisenbergpicture,nottheinteractingpicture. Providedtheaxionsarenon-relativistic,theHamiltonianis H=X~n!~nay~na~n+X~n1,~n2,~n3,~n41 4~n3,~n4s~n1,~n2ay~n1ay~n2a~n3a~n4(4) where ~n3,~n4s~n1,~n2=)]TJ /F6 11.955 Tf 22.67 8.08 Td[( 4m2V~n1+~n2,~n3+~n4.(4) ThepresenceoftheKroneckersymbol~n1+~n2,~n3+~n4expressesmomentumconservationforeachindividualinteraction.InEq.( 4 )wedroppedalltermsoftheformayayayay,aayayay,aaaa,andaaaay.Wearejustiedindoingsobecauseenergyconservationallowsonlyaxionnumberconservingprocessesattreelevel.Axionnumberviolatingprocessesoccurinloopdiagramsbutcanbesafelyignoredbecausetheyarehigherorderinanexpansioninpowersof1 fa.Infact,allaxionnumberviolatingprocesses,includingtheaxiondecaytotwophotons,occurontimescalesmuchlongerthantheageoftheuniverseintheaxionmassrange(10)]TJ /F9 7.97 Tf 6.59 0 Td[(5eV)ofinterest. IntheNewtonianlimit,thegravitationalself-interactionsoftheaxionuidaredescribedby Hg=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(G 2Zd3xd3x0(~x,t)(~x0,t) j~x)]TJ /F6 11.955 Tf 11.44 .5 Td[(~x0j(4) where=1 2(2+m22)istheaxionenergydensity.Becauseweneglectgeneralrelativisticcorrections,ourconclusionsareapplicableonlyforprocesseswellwithinthehorizon.Substitutingandbytheirexpansionsintermsofcreationandannihilationoperators,Eqs.( 4 ),anddroppingagainallaxionnumberviolatingterms,Eq.( 4 ) 57

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becomes Hg=X~n1,~n2,~n3,~n41 4~n3,~n4g~n1,~n2ay~n1ay~n2a~n3a~n4(4) where ~n3,~n4g~n1,~n2=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(4Gm2 V~n1+~n2,~n3+~n41 j~p~n1)]TJ /F6 11.955 Tf 11.39 .5 Td[(~p~n3j2+1 j~p~n1)]TJ /F6 11.955 Tf 11.39 .5 Td[(~p~n4j2.(4) Insummarysofar,theaxioneldisequivalenttoalargenumberMofcoupledoscillatorswithHamiltonianoftheform H=MXj=1!jayjaj+Xi,j,k,l1 4ijklaykaylaiaj.(4) Inparticular,thetotalnumberofquanta N=MXj=1ayjaj(4) isconserved.InEq.( 4 ),ijkl=ijlk=jikl=klij.Thequestionofinterestnowisthefollowing:startingwithanarbitraryinitialstate,howquicklywilltheaverageshNkioftheoscillatoroccupationnumbersNk=aykakapproachathermaldistribution?TheusualapproachtothisquestionusestheBoltzmannequation.However,wewillseethattheassumptionsunderlyingtheBoltzmannequationarenotvalidforthecoldaxionuid.Soweneedamoregeneralapproach. Itisinstructivetostartwithasystemofjustfouroscillators(M=4)andoneinteractionbetweenthem: H=4Xj=1!jayjaj+(ay1ay2a3a4+ay3ay4a1a2).(4) Wehaveinthiscase _a1=i[H,a1]=i()]TJ /F6 11.955 Tf 9.3 0 Td[(!1a1)]TJ /F4 11.955 Tf 11.95 0 Td[(ay2a3a4)(4) andtherefore _N1=i(a1a2ay3ay4)]TJ /F3 11.955 Tf 11.96 0 Td[(ay1ay2a3a4)(4) 58

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andsimilarequationsfortheother_ajand_Nj.WesolvetheequationsperturbativelyuptoO(2).Letusdene aj(t)=(Aj+Bj(t))e)]TJ /F5 7.97 Tf 6.59 0 Td[(i!jt+O(2)(4) whereAjaj(0)andBj(t)arerespectivelyzerothandrstorder,andBj(0)=0.Eq.( 4 )implies _B1=)]TJ /F3 11.955 Tf 9.3 0 Td[(iAy2A3A4e+it+O(2),(4) with!1+!2)]TJ /F6 11.955 Tf 11.95 0 Td[(!3)]TJ /F6 11.955 Tf 11.95 0 Td[(!4,andtherefore B1(t)=)]TJ /F3 11.955 Tf 9.29 0 Td[(iAy2A3A4e+it=22 sin(t 2)+O(2).(4) SubstitutingthisintoEq.( 4 ),wehave _N1=i(A1A2Ay3Ay4e)]TJ /F5 7.97 Tf 6.58 0 Td[(it)]TJ /F3 11.955 Tf 11.95 0 Td[(h.c.)+2[(Ay2A2A3Ay3A4Ay4+A1Ay1A3Ay3A4Ay4)]TJ /F3 11.955 Tf 19.26 0 Td[(A1Ay1A2Ay2A4Ay4)]TJ /F3 11.955 Tf 11.95 0 Td[(A1Ay1A2Ay2Ay3A3)e)]TJ /F5 7.97 Tf 6.58 0 Td[(it=22 sin(t 2)+h.c.]+O(3). (4) Eq.( 4 )mayberecastintheform _N1=i(A1A2Ay3Ay4e)]TJ /F5 7.97 Tf 6.59 0 Td[(it)]TJ /F3 11.955 Tf 11.95 0 Td[(h.c.)+2[N3N4(N1+1)(N2+1))-221(N1N2(N3+1)(N4+1)]2 sin(t)+O(3) (4) byrewritingthesecondorderterms. WenowgeneralizetoasystemwithanarbitrarilylargenumberMofcoupledoscillators,Eqs.( 4 ).ThecalculationisessentiallythesameasfortheM=4toymodel,exceptthatthereisamultiplicityofinteractiontermstokeeptrackof.Onends 59

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(l=1...M) _Nl=iMXi,j,k=11 2(klijAyiAyjAkAle)]TJ /F5 7.97 Tf 6.58 0 Td[(iklijt)]TJ /F3 11.955 Tf 11.95 0 Td[(h.c.)+MXk,i,j=11 2jklijj2[NiNj(Nl+1)(Nk+1))-222(NlNk(Ni+1)(Nj+1)]2 klijsin(klijt)+MXk,i,j=1MXp,m,n=1(p;m,n)6=(k;i,j)[1 2ijkllpmnAymAynAykApAiAjei(klij+mnlp=2)t1 mnlpsin(mnlp 2t)+h.c.]+MXk,i,j=1MXp,m,n=1(p;m,n)6=(l;i,j)[1 2ijklkpmnAylAymAynApAiAjei(klij+mnkp=2)t1 mnkpsin(mnkp 2t)+h.c.])]TJ /F5 7.97 Tf 28.41 14.95 Td[(MXk,i,j=1MXp,m,n=1(p;m,n)6=(j;l,k)[1 2ijlkmnipAylAykAypAmAnAjei(klij+ipmn=2)t1 ipmnsin(ipmn 2t)+h.c.])]TJ /F5 7.97 Tf 28.41 14.94 Td[(MXk,i,j=1MXp,m,n=1(p;m,n)6=(i;l,k)[1 2ijlkmnjpAylAykAypAiAmAnei(klij+jpmn=2)t1 jpmnsin(jpmn 2t)+h.c.]+O(3), (4) whereklij!k+!l)]TJ /F6 11.955 Tf 12.32 0 Td[(!i)]TJ /F6 11.955 Tf 12.32 0 Td[(!j.Thedoublesumsareabsentinthetoymodelbecausethereisonlyoneinteractioninthatcase.Atanyrate,thedoublesumswillnotplayanimportantroleinthediscussionthatfollows. 4.2.2TheParticleKineticRegime Inmostphysicalsystems,therateofchangeoftheoccupationnumberforaparticularstateissmallcomparedtotheenergyexchangedinthetransition,so klijt>>1.Letuscallthisthe`particlekineticregime'.Intheparticlekineticregime,therstordertermsandoffdiagonalsecondtermsaveragetozerointime.Energyisconservedineachtransitionbecause 2 klijsin(klijt)!2(klij)(4) 60

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forklijt!1.Wehavethen <_Nl>=+MXi,j,k=11 2jklijj2[NiNj(Nl+1)(Nk+1))-222(NlNk(Ni+1)(Nj+1)]2(klij),(4) wheretheaverageontheLHSofthisequationisatimeaverage. 4 isanoperatorequation. Aftersubstitutingtheinteractionsandtakingtheinnitevolumelimit,werecovertheBoltzmannequation[ 17 ].Forexample,the4self-interactionleadsto: <_N1>=1 2!1Zd3p2 (2)32!2Zd3p3 (2)32!3Zd3p4 (2)32!42(2)44(p1+p2)]TJ /F3 11.955 Tf 11.95 0 Td[(p3)]TJ /F3 11.955 Tf 11.95 0 Td[(p4)1 2[(N1+1)(N2+1)N3N4)-222(N1N2(N3+1)(N4+1)] (4) Inthenon-relativisticlimit,thea+a!a+across-sectiondueto4interactionis =1 j~v1)]TJ /F6 11.955 Tf 11.54 .5 Td[(~v2j1 2!11 2!2Zd3p3 (2)32!3Zd3p4 (2)32!421 2(2)44(p1+p2)]TJ /F3 11.955 Tf 11.96 0 Td[(p3)]TJ /F3 11.955 Tf 11.95 0 Td[(p4)'2 641 m2 (4) Theparticledensityinphysicalspaceis n=Zd3p (2)3N~p.(4) Ifmoststatesarenotoccupied, 4 4 and 4 implytheusualexpressionfortherelaxationrate )]TJ /F2 11.955 Tf 10.1 0 Td[(_N Nnv(4) wherevisameasureofthevelocitydispersionofthesystem. Ifthemomentumstatesarehighlyoccupied,suchasinthecaseofcoldaxions(N~p1061),therelaxationrateismultipliedbyonefactorofN: )]TJ /F2 11.955 Tf 10.09 0 Td[(_N NnvN.(4) 61

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4.2.3TheCondensedRegime Thecondensedregimereferstothecasewheretheenergyspreadofthehighlyoccupiedstatesissmallcomparedtotheevolutionrateofthesystem:!<<)]TJ /F1 11.955 Tf 6.78 0 Td[(.Fortransitionsbetweensuchcloselyspacedstates, e)]TJ /F5 7.97 Tf 6.59 0 Td[(iklijt=1.(4) Therstordertermsinthatequationnolongeraveragetozero.Letusdenecl(t)al(t)ei!lt.TheHamiltonianimplies _cl(t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(iMXk,i,j=11 2ijklcykcicjeiklijt.(4) Letusdenefurther cl(t)Cl(t)+dl(t)(4) wherethedl(t),likethecl(t),areannihilationoperatorssatisfyingcanonicalequaltimecommutationrelationsandtheCl(t)arecomplexc-numberfunctionswhichsatisfytheclassicalequationsofmotion _Cl(t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(iMXk,i,j=11 2ijklCkCiCjeiklijt.(4) Forthehighlyoccupiedcoldaxionstates,theClhavemagnitudeoforderp N.TherelaxationrateofthehighlycondensedcoldaxionsistheinverseofthetimescaleoverwhichthoseCl(t)changebyorderp N. SincethesuminEq.( 4 )isdominatedbytermsforwhichk,iandjlabelhighlyoccupiedaxionstates, _Cl(t))]TJ /F3 11.955 Tf 21.91 0 Td[(iKXk,i,j=11 2ijklCkCiCj.(4) For4self-interactions,wesubstituteEq.( 4 ).Thisyields _C~p1(t)+i 4m2VX~p2,~p31 2C~p2C~p3C~p4(4) 62

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where~p4=~p1+~p2)]TJ /F6 11.955 Tf 12.25 .5 Td[(~p3,andthesumisrestrictedtotheKhighlyoccupiedstatesforwhichp.pmax.WemaythinkofthetermsontheRHSofEq.( 4 )asstepsinarandomwalkincomplexspace.ThemagnitudeofeachstepisoforderN3 2andthenumberofstepsisoforderK2.Hence _C~p 4m2VKN3 2 4m2VNN1 2.(4) Henceourestimatefortherelaxationratedueto4self-interactionsinthecondensedregime: )]TJ /F8 7.97 Tf 6.78 -1.79 Td[(1 4nm)]TJ /F9 7.97 Tf 6.59 0 Td[(2(4) wheren=N=Visthedensityofparticlesinthehighlyoccupiedcloselyspacedstates.Likewise,usingEq.( 4 ),wendthatthroughgravitationalself-interactions _C~p1(t)+i4Gm2 VX~p2,~p31 2C~p2C~p3C~p41 j~p1)]TJ /F6 11.955 Tf 11.4 .5 Td[(~p3j2+1 j~p1)]TJ /F6 11.955 Tf 11.39 .5 Td[(~p4j2.(4) Thecorrespondingrelaxationrateis )]TJ /F5 7.97 Tf 6.77 -1.8 Td[(g4Gnm2`2(4) where`1 pmaxisthecorrelationlengthoftheparticles. Wenotethatattheboundarybetweentheparticlekineticandcondensedregimes,where!)]TJ /F1 11.955 Tf 6.77 0 Td[(,thetwoestimatesoftherelaxationrateagreewithoneanother.Indeedatthatboundary,uptofactorsofordertwoorso, vNvn (p)3n m2!n m2)]TJ 21.26 8.2 Td[(.(4) SubstitutingthisintoEq.( 4 )fortherelaxationratedueto4self-interactionsintheparticlekineticregimeyieldsEq.( 4 )whichisthecorrespondingestimateinthecondensedregime.Thesameholdstrueforthegravitationalself-interactions. LetusalsonotethatEqs.( 4 )and( 4 )arenotvalidwhenalmostallaxionsareinasinglestate(K=1),aswhentheBose-Einsteincondensationhasbeen 63

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completed.Indeed,ifK=1,thereisonlyoneterminthesumontheRHSofEqs.( 4 )and( 4 )thatisenhancedbylargeoccupationnumbers,i.e.thetermforwhichboth~p2and~p3equalthemomentumofthesinglehighlyoccupiedstate,anditdescribesaninteractionwithzeromomentumtransfer.Thus,oncetheBose-Einsteincondensationiscompleteandallaxionsareinthelowestenergystate,anyfurtherthermalizationissuppressed. Finallyconsidertransitionsa(~p1)+a(~p2)$a(~p3)+a(~p4)where~p2and~p4aremomentaofhighlyoccupiedstatesbut~p1and~p3arenot.Suchtransitionsareinthecondensedregimebecausethemomentumtransfer,andhencetheenergytransfer,issmall.Eqs.( 4 )and( 4 )applytosuchtransitionsandimplythattherateatwhichstateswithp>pmaxmodifytheiroccupationnumbersisalsogivenbyEqs.( 4 )and( 4 )withtheprovisothatthequantacanonlymovebetweenstatesdifferinginmomentumbylessthanpmax. 4 hasasimpleinterpretation.Theaxions,havingenergydensity=mnandcorrelationlength`,producegravitationaleldsg4G`.Thegravitationalforceonaparticleisoforderg!,where!istheenergyoftheparticle.Sincetheforceistherateofchangeoftheparticle'smomentum,therelaxationrateisoforder )]TJ /F5 7.97 Tf 6.77 -1.8 Td[(gg! p4Gnm`! p(4) wherepisthemomentumdispersionoftheparticles.Fortheaxionsthemselves,weobtaintherelaxationrateEq.( 4 )bysubstituting!=mandp`)]TJ /F9 7.97 Tf 6.59 0 Td[(1.Eq.( 5 )showsthatthemomentumdistributionofanyparticlespeciesismodiedbythegravitationaleldsofthecoldaxionuidandthereforethatgravitationalinteractionsmayproducethermalcontactbetweenthecoldaxionsandotherparticlespecies. 64

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4.2.4ColdAxionsFormaBEC Using 4 onendsthatthethermalizationcondition:)]TJ /F5 7.97 Tf 6.77 -1.79 Td[(g=H>1issatisedatatimetBECwhenthephotontemperatureisoforder TBEC500eVXfa 1012GeV1 2.(4) TheaxionsthermalizethenandformaBECasaresultoftheirgravitationalself-interactions.Thewholeideamayseemfar-fetchedbecauseweareusedtothinkthatgravitationalinteractionsamongparticlesarenegligible.Theaxioncaseisspecial,however,becausealmostallparticlesareinasmallnumberofstateswithverylongdeBrogliewavelength,andgravityislongrange.Bygravitationalself-interactionstheaxionsmodifytheirmomentumdistributiontilltheirentropyismaximizedfortheavailableenergy,whichinthiscasemeansthattheyformaBEC. AxionBECcausesthecorrelationlengthtoincrease.Indeedinaninnitevolume,whenallparticlesareinthelowestenergystate,themomentumdispersionistheoreticallyzeroandthecorrelationlengthinnite.ThisidealstateneveroccursbecausethermalizationandhenceBECformationareconstrainedbycausality.Theaxionsinonehorizonareunawareofthedoingsofaxionsinthenexthorizon.Henceweexpectthecorrelationlength`,whichmaynowbethoughtofasthesizeofcondensatepatches,tobecomeoforderbutlessthanthehorizon.Thegrowthinthecorrelationlengthcausesthethermalizationtoaccelerate.Oncelissomefractionoft,)]TJ /F5 7.97 Tf 6.77 -1.79 Td[(g(t)=H(t)/a(t))]TJ /F9 7.97 Tf 6.59 0 Td[(3t3,implyingthatthermalizationoccursonevershortertimescalescomparedtotheHubbletime. 4.3THERMALCONTACTWITHOTHERSPECIES 4.3.1EvolutionEquationsforOtherSpecies Ourpurposeinthissubsectionistoestimatethegravitationalinteractionratesofotherspecies-baryons,relativisticaxionsandphotons-withthecoldaxionuid. 65

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TheHamiltoniandescribinggravitationalinteractionsbetweenthecoldaxionsandanyotherspecieshasthegeneralform: H=MXj=1!jayjaj+SXr=1!rbyrbr+Xi,j,k,l1 4ijklaykaylaiaj+Xj,k,r,sjrbksaykbysajbr,(4) wherejrbks=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(ksbjr.Thebraretheannihilationoperatorsforquantaofthenewspecies.Theysatisfycanonicalcommutationoranti-commutationrelations.The!raretheenergiesofthosequanta.Asbefore,wequantizeinaboxofvolumeV=L3withperiodicboundaryconditions.Thelabelsofthenewparticlestatesarethenr=(~n,),givingtheirmomenta~p=2 L~nandtheirspin.Theirenergyis!=p ~p~p+m2bwherembisthemassofthenewspecies. Wedenecj(t)aj(t)ei!jtasbefore,andc0r(t)br(t)ei!rt.TheHeisenbergequationsofmotionforthec0r(t)arethen: _c0s=)]TJ /F3 11.955 Tf 9.3 0 Td[(iXj,k,rjrbkscykcjc0reiksjrt(4) whereksjr!k+!s)]TJ /F6 11.955 Tf 12.09 0 Td[(!j)]TJ /F6 11.955 Tf 12.09 0 Td[(!r.Because3-momentumisconservedineachinteraction,thejrbkshavetheform: jrbks=)]TJ /F6 11.955 Tf 9.29 0 Td[(jrbks~pk+~ps,~pj+~pr.(4) TheimportantcontributionsinthesumontheRHSofEq.( 4 )arefromtermsinwhichbothjandklabelhighlyoccupiedcoldaxionstates.Therefore _c0s+iKXk,j=1jrbksCkCjc0rei(!s)]TJ /F8 7.97 Tf 6.58 0 Td[(!r)t(4) with~pr=~ps+~pk)]TJ /F6 11.955 Tf 12.52 .5 Td[(~pj.Asbefore,KisthenumberofhighlyoccupiedcoldaxionstatesandtheCkaredenedbyEq.( 4 ).Again,thesumontheRHSofEq.( 4 )representsarandomwalkincomplexspace.ThenumberofstepsisK2andthetypicalstepsizeisbNc0,wherebisthetypicalvalueofjrbks.Therateatwhichallquantaofthenewspeciesmaymovetoneighboringstatesseparatedinmomentumspacebyless 66

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thanp1 `istherefore )]TJ /F5 7.97 Tf 6.77 -1.8 Td[(b,pKNb=bN(4) whereN=KNisthenumberofcoldaxionsinvolumeV.Therelaxationrateofthenewspeciesisthen )]TJ /F5 7.97 Tf 6.77 -1.8 Td[(bbNp pbN1 `p(4) wherepisthemomentumdispersionofthenewspeciespopulation.(Ifthemomentumdispersionisverydifferentintheinitialstatethaninthenalstate,pisthelargerofthetwo.)Eq.( 4 )assumesthatthebparticlesarebosonsornon-degeneratefermions.Iftheyaredegeneratefermions,theirrelaxationrateissuppressed,relativetoEq.( 4 ),byPauliblocking. Also,letusreiteratethatwhenmostcoldaxionsareinthelowestenergystate,implyingK=1,thermalizationissuppressedcomparedtotheestimateinEq.( 4 ),becausethereisonlyoneterminthesumofEq.( 4 )inthatcaseandthemomentumtransfervanishesforthatterm. 4.3.1.1Baryons Fornon-relativisticspecies,suchasbaryonsandWIMPs,thetermintheHamiltonianthatdescribesgravitationalinteractionswiththecoldaxionsis HB=)]TJ /F3 11.955 Tf 9.3 0 Td[(GZd3xd3x0(~x,t)B(~x0,t) j~x)]TJ /F6 11.955 Tf 11.43 .5 Td[(~x0j,(4) where B(~x,t)=mB VX~n,~n0,by~n,b~n0,ei(~p0)]TJ /F8 7.97 Tf 6.18 .33 Td[(~p)~x,(4) andmBisthemassofthenon-relativisticparticle.Thisyields ~n1,(~n2,)B~n3,(~n4,0)=+4GmmB Vq20,(4) 67

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where~q=~p1)]TJ /F6 11.955 Tf 12.28 .5 Td[(~p3isthemomentumtransfer.Sinceq`)]TJ /F9 7.97 Tf 6.58 0 Td[(1,theBparticleshaverelaxationrate )]TJ /F5 7.97 Tf 6.77 -1.79 Td[(B4GnmmB` pB,(4) wherepBistheirmomentumdispersion.Eq.( 4 )assumesthattheBparticlesarebosonsornon-degeneratefermions,asisthecaseforbaryonsandWIMPs. 4.3.1.2HotAxions Forrelativisticspecies,thetermthatdescribesgravitationalinteractionswiththecoldaxionuidis Hr=)]TJ /F11 11.955 Tf 11.29 16.27 Td[(Zd3x1 2hT(4) whereT(~x,t)isthestress-energy-momentumtensorofthisspeciesandhistheperturbationofthespace-timemetriccausedbythecoldaxions: h00(~x,t)=2GZd3x0(~x0,t) j~x)]TJ /F6 11.955 Tf 11.44 .49 Td[(~x0jh0k(~x,t)=0hkl(~x,t)=2GZd3x0(~x,t) j~x)]TJ /F6 11.955 Tf 11.44 .5 Td[(~x0j3(xk)]TJ /F3 11.955 Tf 11.96 0 Td[(x0k)(xl)]TJ /F3 11.955 Tf 11.95 0 Td[(x0l). (4) Notethath=0.Forascalareld(x) Hr=)]TJ /F11 11.955 Tf 11.29 16.27 Td[(Zd3x1 2h@@.(4) Aftersomealgebra,Eq.( 4 )yields ~n1,~n2r~n3,~n4=+4Gm Vq2p !2!4[!2!4+~p2~p4)]TJ /F4 11.955 Tf 11.95 0 Td[(2(~q~p2)(~q~p4) q2],(4) where~q=~p1)]TJ /F6 11.955 Tf 12.36 .5 Td[(~p3.Therelaxationrateforrelativisticscalarsthroughgravitationalinteractionswiththehighlyoccupiedlowmomentumaxionmodesisthusoforder )]TJ /F5 7.97 Tf 6.78 -1.8 Td[(r4Gnm`,(4) sinceq`)]TJ /F9 7.97 Tf 6.59 0 Td[(1andp!. 68

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4.3.1.3Photons Thetermthatdescribesthegravitationalinteractionsofphotonswiththecoldaxionuidis H=)]TJ /F11 11.955 Tf 11.29 16.27 Td[(Zd3x1 2hFF(4) whereFistheelectromagneticeldstrengthtensor,andtheharegivenbyEqs.( 4 )asbefore.Thisyields ~n1,(~n2,~2)~n3,(~n4,~4)=+8Gm Vq4p !2!4[!2!4(~2~q)(~4~q)+(~p2~2)~q(~p4~4)~q],(4) where~2and~4arethepolarizationvectorsoftheinitialandnalstatephotons.Wendtherefore )]TJ /F8 7.97 Tf 6.78 -1.79 Td[(4Gnm`(4) fortherelaxationrateofphotons.Itisthesameasforrelativisticaxions,Eq.( 4 ),inorderofmagnitude. 4.3.2PossibleOutcomes Therateatwhichnon-relativisticspeciessuchasbaryonschangetheirmomentumdistributionthroughgravitationalinteractionswiththecoldaxionfuidisgivenbyEq.( 4 ).Themomentumdispersionofbaryonsisoforderpp 3mBTwhereTisthephotontemperature.Wewillassumeherethatcoldaxionsarethebulkofthedarkmatter.TheFriedmannequationimpliesthen 4Gnm3 8t2t teq1 2(4) fort
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Photonsareinthermalcontactwiththebaryons,butthenatureanddegreeofthisthermalcontactarechangingatthetimeofaxionBECformation[ 51 ].BaryonsinteractwithelectronsbyCoulombscattering.ElectronsinteractwithphotonsbyComptonscattering,doubleComptonscatteringandbremsstrahlung.Aboveapproximately1keVphotontemperature,doubleComptonscatteringandbremsstrahlungassurechemicalequilibriumbetweenbaryonsandphotons(thenumberofphotonsisnotconservedintheseprocesses).Belowapproximately1keVphotontemperature,Comptonscatteringistheonlyimportantinteractionremaining.Itmaintainskinetic,butnotchemical,equilibriumbetweenbaryonsandphotonstillapproximately100eVphotontemperature.Below100eV,thedegreeofkineticequilibriumprogressivelydiminishestillapproximately2eV,whenitdisappearsaltogether. Inanycase,aslongasthereisonlythermalcontactbetweenbaryonsandafewlowmomentummodesoftheaxioneld,onlyaverysmallamountofenergycanbeexchangedbetweentheaxioneldandtheotherspecies.However,astimegoeson,higherandhighermomentummodesoftheaxioneldreachthermalcontactwithitshighlyoccupiedlowmomentummodes.Eq.( 4 )givestherelaxationrate)]TJ /F5 7.97 Tf 6.77 -1.8 Td[(roftheaxioneldasawhole,includingrelativisticstates.Therelaxationrateofphotons)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(,Eq.( 4 ),isofthesameorderofmagnitude.CombiningEqs.( 4 )and( 4 )withtheFriedmannequation,wehave )]TJ /F8 7.97 Tf 6.77 -1.79 Td[(=H)]TJ /F5 7.97 Tf 6.77 -1.79 Td[(r=H3 2H`a tot,(4) whereaisthecoldaxiondensityandtotthetotaldensity.Since`/t,Eq.( 4 )impliesthat)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(=Hand)]TJ /F5 7.97 Tf 6.77 -1.79 Td[(r=Hgrowproportionallytoa(t)tillequalityandremainconstantafterthat.Thecriticalparameteristheirvalueatequality )]TJ /F8 7.97 Tf 6.78 -1.79 Td[(=Hjteq)]TJ /F5 7.97 Tf 6.77 -1.79 Td[(r=Hjteq`(teq) teq.(4) 70

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ThermalcontactbetweenaxionsandphotonsisestablishediftheRHSofEq.( 4 )isoforderone,i.e.iftheaxioncorrelationlengthishorizonsizeatequality.ToshowthattheaxionBECcorrelationlengthbecomestrulyaslargeasthehorizonisaproblem,involvingbothout-of-equilibriumstatisticalmechanicsandgeneralrelativity,whichwedonotknowhowtosolveatpresent. Henceweconsideratthisstagetwopossibilities,whichwecallcasesAandB.IncaseA,)]TJ /F5 7.97 Tf 6.78 -1.8 Td[(r,=Hdonotreachonebeforeequality(because`,althoughproportionaltot,maybemuchlessthant,e.g.`=t/100),andhencethermalcontactgetsestablishedonlybetweenbaryonsandlowmomentummodesoftheaxioneld.IncaseB,)]TJ /F5 7.97 Tf 6.78 -1.79 Td[(r,=Hdoreachonebeforeequalityandthermalequilibriumisreachedbetweenbaryons,axionsandphotons.Thisequilibriumiskineticonlysincegravitationalinteractionsconserveparticlenumberforallthespeciesinvolved. Weshouldaskwhetherneutrinosmayalsoreachthermalcontactwiththehighlyoccupiedlowmomentumaxionmodes,inwhichcaseneutrinos,axions,baryonsandphotonswouldallreachthesametemperature.Webelievethispossibility,whichwecallcaseC,unlikelyforthefollowingreason.Eq.( 5 )doesnotapplytodegeneratefermionsbecauseofPauliblocking.Cosmicneutrinosaresemi-degeneratesincetheyhaveathermaldistributionwithzerochemicalpotential.BecauseofpartialPauliblocking,theirthermalizationisslowerthanthatofrelativisticaxions,Eq.( 4 ).Sincerelativisticaxionsonlybarelyreachthermalcontactwiththehighlyoccupiedlowmomentummodesoftheaxioneldiftheydosoatall,andthermalcontactbetweenthoselowmomentummodesoftheaxioneldandneutrinosisdelayedrelativetorelativisticaxions,itappearsmostlikelythatneutrinosremaindecoupledfromtheaxionsatalltimes. 71

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CHAPTER5IMPLICATIONSFOROBSERVATION 5.1ANON-RETHERMALIZINGAZIONBECBEHAVESASORDINARYCDM Inthissection,weshowthataxionBECbehavesasordinaryCDMonallscaleofobservationalinterestaslongastheaxionsremaininthesamestate,ie.aslongastheydonotrethermalize[ 52 ]. TheaxioneldsatisestheHeisenbergequationofmotion: DD'(x)=g[@@)]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 6.77 4.94 Td[(@]'(x)=m2'(x).(5) where'(x)istheaxioneldoperator.Theselfinteractionterm)]TJ /F9 7.97 Tf 10.49 4.71 Td[(1 6'3isonlyimportantforaveryshortperiodafterQCDtime.Itisneglectedhere. Theaxioneldcanbeexpandedinparticlemodesas '(x)=X~[a~~(x)+ay~?~].(5) Exceptforatinyfraction,mostaxionsgotoasinglestatethatwelabelas~=0.0(x)isthecorrespondingwavefunction.ThestateoftheaxioneldisjN>=(1=p N!)(ay0)Nj0>wherej0>isthevacuumandNistheparticlenumber.InthespatiallyathomogeneousandisotropicRobertson-Walkerspace-time, 0=A a(t)3 2e)]TJ /F5 7.97 Tf 6.58 0 Td[(imt(5) whereAisaconstant.Thestress-energy-momentumtensorhasexpectationvalue: =N[@0@0+@0@0+g()]TJ /F6 11.955 Tf 9.3 0 Td[(@0@0)]TJ /F3 11.955 Tf 11.96 0 Td[(m200)]. (5) InaatMinkowskispace-timeandinthenon-relativisticlimit,neglectingtermsoforder(1=m)@tcomparedtotermsoforderone,( 5 )impliestheSchrodingerequation: i@t=)]TJ 11.17 8.09 Td[(r2 2m.(5) 72

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Thewavefunctioncanbewrittenas: (~x,t)=1 p 2mNB(~x,t)ei(~x,t).(5) IntermsofB(~x,t)and(~x,t)thedensityandvelocityeldsoftheaxionuidsare:=mB(~x,t),and~v(~x,t)=1 m~r(~x,t).Then( 5 )leadstothecontinuityequation,andtheequationofmotion: @tvk+vj@jvk=)]TJ /F6 11.955 Tf 10.44 3.02 Td[(~rq(5) where q(~x,t)=)]TJ 13.63 8.69 Td[(r2p 2m2p .(5) Followingthemotion,thestresstensoris Tjk=vjvk+1 4m2(1 @j@k)]TJ /F6 11.955 Tf 11.96 0 Td[(jkr2).(5) ForordinaryCDM,thelasttermsontheRHSof( 5 )and( 5 )areabsent.Inthelinearregimeofevolutionwithinthehorizon,neglectingsecondorderterms,( 5 )becomes Tjk=)]TJ /F6 11.955 Tf 9.29 0 Td[(jk0(t) 4m2r2(~x,t)(5) where0(t)isunperturbeddensityand(~x,t)(~x,t) 0(t).BecausetheRHSof( 5 )isagradientandRHSof( 5 )isproportionaltotheKroneckersymbol,thevectorandtensorperturbationsarenotaffectedbytheadditionalforcesassociatedwiththeaxionBECinthelinearregime.Onlythescalarperturbationsareaffected.ThescalarperturbationsareconvenientlydescribedinconformalNewtoniangaugewherethemetricis ds2=)]TJ /F4 11.955 Tf 9.3 0 Td[((1+2 (~x,t))dt2+a(t)2(1+2(~x,t))d~xd~x.(5) 73

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Conservationofenergyandmomentuminthisbackgroundimpliestherstorderequations @t+1 a~r~v=)]TJ /F4 11.955 Tf 9.29 0 Td[(3@t+3H 4m2a2r2@t~v+H~v=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 a~r +1 4m2a3~rr2 (5) whereH=1 ada dt.TheequationsforCDMarerecoveredbylettingm!1.TheRHSofEinstein'sequationsaremodiedbytheadditionofTjktothestresstensor,butthismodicationdoesnotplayaroleinourdiscussionbecauseitissuppressed,relativetotheleadingterms,bythefactorkph m2,wherekphisthephysicalwavevectoroftheperturbation.Forscaleswithinthehorizon,oneobtainsfrom( 5 ): @2t+2H@t)]TJ /F11 11.955 Tf 11.95 16.85 Td[(4G0)]TJ /F3 11.955 Tf 23.33 8.08 Td[(k4 4m2a4=0.(5)( 5 )impliesthattheaxionBEChasaJeanslength: k)]TJ /F9 7.97 Tf 6.59 0 Td[(1J=(16Gm2))]TJ /F12 5.978 Tf 7.78 3.25 Td[(1 4=1.021014cm10)]TJ /F9 7.97 Tf 6.58 0 Td[(5eV m1 210)]TJ /F9 7.97 Tf 6.59 0 Td[(29g=cm3 1 4. (5) TheJeanslengthissmallcomparedtothesmallestobservablescales(100kpc),thustheaxionBECandCDMareindistinguishableinthelinearregime.Inthenonlinearregimeofstructureformation,therelevantequationsofmotionare: @t+~r(~v)=0,~r~v=0@t~v+(~v~r)~v=)]TJ /F6 11.955 Tf 10.44 3.03 Td[(~r )]TJ /F6 11.955 Tf 13.09 3.03 Td[(~rq. (5) ( 5 )isequivalenttotheSchrodingerequationforparticlesinaNewtoniangravitationaleld.AxionBECandCDMdifferbythe~rqterm,whichindicatesalocalquantumeffectofaxionBEC.However,aswasshownbynumericalsimulationandisexpectedfromtheWKBapproximation,thedifferencesoccuronlyonlengthscalessmallerthanthede-Brogliewavelength[4],whichisoforder1=(mv)1=(m10)]TJ /F9 7.97 Tf 6.59 0 Td[(3c)10m. 74

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5.2TIDALTORQUING,INNERCAUSTICSANDAXIONBEC Inthissection,weshowthataxionBECbehavesdifferentlyfromordinaryCDMwhenfallingintogalactichalosbecauseaxionBECrethermalizesasitfallsin[ 46 ]. LetusconsideraregionofsizeLinsideofwhichtheaxionstatestopsbeingthelowestenergyavailablestatebecausethebackgroundistimedependent.WeexpectthattheaxionBECrethermalizesprovidedthegravitationalforcesproducedbytheBECarelargerthanthetypicalrate_pofchangeofaxionmomentarequiredfortheaxionstoremaininthelowestenergystate.Thegravitationalforcesareoforder4Gnm2`.Inthiscase,thecorrelationlength`mustbetakentobeoforderthesizeLoftheregionofinterestsincethegravitationaleldsduetoaxionBECoutsidetheregiondonothelpthethermalizationoftheaxionswithintheregion.Hencetheconditionis 4Gnm2L&_p.(5) Wenowapplythiscriteriontothequestionwhetheraxionsrethermalizesufcientlyquicklythattheyshareangularmomentumwhentheyareabouttofallintoagalacticgravitationalpotentialwell. Weusetheself-similarinfallmodelofgalactichaloformationtoestimateLand_p.LisoforderafewtimestheturnaroundradiusR(t),sayL(t)3R(t),whereasp(t)mvrot(t)jmaxwherevrotistherotationvelocityandjmaxisthedimensionlessnumbercharacterizingtheamountofangularmomentumofthehalo.Intheself-similarmodel,vrot(t)R(t)=tandR(t)/t2 3+2 9whereisintherange0.25to0.35[ 53 ].Assumingthatmostofthedarkmatterisaxions,theFriedmannequationimplies 4Gnm=3 2H(t)2=2 3t2(5) fort>teq.TheLHSofEq.( 5 )isthereforeoforder 2mR(t) t22mvrot(t)1 t(5) 75

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whereasitsRHSisoforder d dt"mjmaxvrot(t0)t t0)]TJ /F12 5.978 Tf 7.78 3.26 Td[(1 3+2 9#=mjmaxvrot(t)1 t(2 9)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 3).(5) Thetypicalvalueofjmaxis0.18.HenceEq.( 5 )issatisedatalltimesfromequalitytilltodaybyamarginoforder2 jmax(2 9)]TJ /F12 5.978 Tf 7.78 3.26 Td[(1 3)30. WeconcludethattheaxionBECdoesrethermalizebeforefallingintothegravitationalpotentialwellofagalaxy.Mostaxionsgotothelowestenergystateconsistentwiththetotalangularmomentumacquiredfromneighboringinhomogeneitiesthroughtidaltorquing[ 54 ].Thatstateisastateofrigidrotationontheturnaroundsphere,implying~r~v6=0where~visthevelocityeldoftheinfallingaxions.Incontrast,thevelocityeldofWIMPdarkmatterisirrotational.Theinnercausticsofgalactichalosaredifferentinthetwocases.AxionsproducecausticringswhereasWIMPsproducethe`tent-like'causticsdescribedinref.[ 71 ].Thereisevidencefortheexistenceofcausticringsinvariousgalaxiesattheradiipredictedbytheself-similarinfallmodel.Forareviewofthisevidenceseeref.[ 55 ].Itisshowninref.[ 70 ]thatthephasespacestructureofgalactichalosimpliedbytheevidenceforcausticringsispreciselyandinallrespectsthatpredictedbytheassumptionthatthedarkmatterisarethermalizingBEC. 5.3AXIONBECANDCOSMOLOGICALPARAMETERS InthissectionweshowthataxionBECmayprovideanexplanationtotheLithiumProblemandmaychangecosmologyparameterssuchastheeffectivenumberofneutrinospeciescomparingtothestandardcosmologyvalues[ 56 ]. TheagreementbetweenobservationsandtheBBNpredictionsfortheprimordialabundancesoflightelementsisoftentoutedasatriumphofthestandardCDMcosmologicalmodel.Undertheassumptionthattherearethreeneutrinospecies,BBNasatheoryrequiresessentiallyasingleinput:thebaryon-to-photonratio,usuallygivenbytheparameter10=1010nB=n[ 57 ].Ifonetakes10tobe6.1900.145,inaccordancewiththelatestWilkinsonMicrowaveAnisotropyProbe(WMAP)results, 76

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theinferredprimordialabundancesofthemajorityofthelightelements(D,4He,3He)areremarkablyconsistentwithBBNpredictions,saveoneexception:thatof7Liisapproximatelytwotothreetimeslessthanwhatthetheorypredicts.Thediscrepancyisdeemedstatisticallysignicant,andthereissofarnowidelyacceptedexplanationfortheanomaly.Intheliterature,thisisreferredtoastheLithiumProblem. OneofthemostdifcultissuesinvolvedintestingBBNishowreliablytoinfertheprimordialabundancesoflightelementsfrommeasurementsthatareavailabletous.SubsequenttoBBN,theoriginalrelicabundancesareallsubjecttofurthermodicationbycomplicatedstellarprocesses.7Li,forexample,canbebothdepletedandsynthesizedinstars,aswellasproducedbycosmic-raynucleosynthesis.Assuch,theabundanceof7Liisinferredprimarilyfromabsorptionlinesintheatmosphereofgalactichalostarswithlowmetallicity,sincethesestarsareveryoldandhaveexperiencedverylittlenuclearprocessing. Althoughthesepost-BBNeffectsleadtoconsiderablecomplication,theyalsoopenupmanydifferentavenuestoexplainthe7Lianomaly.Formanyyears,ithasbeenhopedthatbetterdeterminationofnuclearparameterswillgraduallynarrowthediscrepancy,thoughitwaseventuallyrealizedthatdoesnotseemachievable[ 60 ].Quitethecontrary,itwasfoundin[ 57 ]thatimproveddataontheneutronlife-timeandthecrosssectionsp(n,)dand3He(,)7Beincreasesthepredictedabundanceof7Li,worseningthedisagreement.Revisionstostellarevolution,asaconsequenceofsystematicerrorsintheeffectivetemperatureofthemetal-poorstars[ 61 62 ],andsurface7Lidepletionintheinteriorofstarsduetosomemixingordiffusiveprocesses[ 63 ],havealsobeeninvestigatedaspossiblesolutions,butarestillconsideredcontroversial[ 57 ]. Thefactthatthenuclearreactionsrelevanttotheproductionofbothprimordialandpost-BBN7Liarenowquitewellunderstoodhasledtospeculationsthattheanomalymightinsteadbecausedbynewphysics.Manyexplanationshavebeenproposed,such 77

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asthevariationintimeofthedeuteronbindingenergyandoffundamentalcouplings[ 64 65 ],andthedecayofarelativelylong-livedparticleinthecontextofsupersymmetry[ 66 ].Atthispoint,noneoftheseexplanationshavewongeneralacceptanceinthecosmologycommunity. OnewaytoremovetheconictbetweendataandtheorycanbethecoolingofphotonsbetweentheendofBBNanddecoupling.Processesthatdothisaredifculttocomeby.Indeed,typicalprocessesarisingfromnewphysicstendtoheatupthephotons,modifying10inthewrongdirection[ 57 ].However,therecentrealizationthatdarkmatteraxionsformaBECatapproximately500eVphotontemperature[ 67 ]providesapossiblemechanism[ 68 ].Essentially,thehighoccupationofaxionmodeswithverylowmomentagreatlyenhancesthestrengthoftheirgravitationalinteractions,suchthatanexchangeofenergybetweenthephotonsandthemuchcolderaxionsbecomespossible.Photoncoolingimpliesthat10,BBN<10,WMAP,whichhastheeffectofreducingtheproductionof7Li[ 59 ].Ifthermalequilibriumbetweenthephotonsandaxionsisachieved,the7Liabundanceisreducedbyapproximatelyafactor2(seebelow),alleviatingthediscrepancyandperhapsremovingitaltogether.However,ourproposalpredictsahigherabundanceforDthanpresentobservationsindicateandpredictsthattheeffectivenumberofthermallyexcitedneutrinodegreesoffreedomishigh:Ne=6.77. Photoncoolingbykineticmixingwithhiddenphotonswasproposedinref.[ 69 ]. 5.3.1PossibilityofPhotonCooling Thegravitationaleldsofthecoldaxionuidcausetransitionsbetweenmomentumstatesofotherparticlespeciespresent.Forparticleswhicharebosonsornondegeneratefermions,therelaxationratethroughgravitationalinteractionswiththecoldaxionsisoforder[ 68 ] )]TJ /F2 11.955 Tf 10.1 0 Td[(4Gmn`! p(5) 78

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where!isthetypicalenergyoftheparticlesandptheirmomentumdispersion.Forphotonstocoolsubstantiallyitisnecessarythatenergyistransferredfromthephotonstothelowmomentumhighlyoccupiedaxionstatesandfromthosetotherelativisticaxionstates.Forbothrelativisticaxionstatesandforphotons,p!andhencetheirrelaxationrate)]TJ /F5 7.97 Tf 6.78 -1.79 Td[(rthroughgravitationalinteractionswithcoldaxionsisoforder4Gnm`.UsingtheFriedmannequation,onendsthat)]TJ /F5 7.97 Tf 6.77 -1.8 Td[(r=H/a(t)beforeequalitybetweenmatterandradiationandremainsconstantafterthat.Atequality,)]TJ /F5 7.97 Tf 6.78 -1.8 Td[(r=Hjteq`(teq)=teq.If`=tisorderoneatequality,thephotonsreachthermalequilibriumwiththeaxionsandhencecool. Gravitationalinteractionsconserveparticlenumberandthereforeproduceonlykinetic(asopposedtochemical)equilibriumbetweenthespeciesinvolved.Also,after500eVphotontemperature,thecouplingbetweenphotonsandbaryonsisinthekinetic,ratherthanchemical,equilibriumregime[ 74 ].Uponcooling,thephotonsthatcannotbeaccommodatedinthermallyexcitedstatesenterthegroundstate,aplasmaoscillationwithzerowavevector.Sincethephotonchemicalpotentialremainszero,thenalphotonspectrumisPlanckian,consistentwithobservation. Eq.( 5 )doesnotapplytodegeneratefermionsbecauseofPauliblocking.Thecosmicneutrinosaresemi-degeneratesincetheyhaveathermaldistributionwithzerochemicalpotential.Theirthermalizationrateislessthanthat)]TJ /F5 7.97 Tf 6.77 -1.79 Td[(rofrelativisticbosons.Since)]TJ /F5 7.97 Tf 6.78 -1.8 Td[(r=H/tn`/t2a)]TJ /F9 7.97 Tf 6.59 0 Td[(3(t),thatratiodoesnotgrowafterequality.Sincetherelativisticaxionsmayonlyreachthermalcontactwiththecoldaxionsatequalityandtheneutrinosaredelayedrelativetotherelativisticaxions,webelieveitmostlikelythatneutrinosremaindecoupledfromtheaxions,photonsandbaryonsatalltimes. Itisstraightforwardtodeterminehowmuchthephotonscooliftheyreachthermalequilibriumwiththeaxions.Energyconservationimpliesi,=f,+f,abecausethecontributionstotheenergydensityoftheinitialaxionsandofthebaryonsarenegligible.Theratiobetweenthenalandinitialphotontemperatureisthus(2=3)1=4.Sincetheir 79

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numberdensityisproportionaltoT3,wend: 10,BBN=2 33=410,WMAP=4.570.11(5) using10,WMAP=6.1900.145.Becausethe7Liabundanceisproportionalto210,BBNintherangeofinterest,itisreducedbyapproximatelythefactor(2 3)3 2'0.55. 5.3.2EffectonTheOtherLightElementPrimordialAbundances WhetherphotoncoolingbyaxionBECsolvestheLithiumProblemremainstobeseen.Thedatahavebeentimedependentinadditiontotheusualuncertainties.InFig 5-1 ,weplotthevalueof10,BBNinthestandardcosmologicalmodel,labeled`WIMP',andinthescenariodescribedhere,labeled`axion',alongwiththevaluesinferredfromtheobservedlightelementabundancesaccordingtothereviewbyG.Steigmanin2005[ 58 ],thereviewbyF.Ioccoetal.in2008[ 76 ]andaprivatecommunicationfromG.Steigmanupdatinghis2005estimatesinthelightofrecentobservations[ 77 ].Theerrorbarsindicatetherangeof10,BBNconsistentwiththeestimated1-uncertaintiesintheobservations.Theaxionpredictionagreesverywellwiththe7LiabundanceatthetimeofSteigman's2005review(10,7Li=4.500.30).Howevermorerecentobservationsindicatealowerprimordial7Liabundance,worseningtheLithiumProblem. Perhapsmoreproblematicisthatasmaller10,BBNpredictsanoverproductionofdeuterium(D).Traditionally,Dhasbeentheprimechoiceasabaryometeramongthelightelements,duetoitssensitivityto10,BBNandsimplepost-BBNevolution(abundancemonotonicallydecreasing).ThemajordrawbackwithDisthatitsabundanceisinferredfromaverysmallsetof(seven)spectraofQSOabsorptionlinesystems[ 78 ].Worseyet,thesefewmeasurementshavealargedispersion,anddonotseemtocorrelatewithmetallicity,obscuringtheexpecteddeuteriumplateau.DuetothevariousinadequaciesintheDmeasurementsmentioned,wehavereservationsaboutthecommonpracticeofattachingmostsignicancetoDinthecomparisonbetweendataandBBNpredictions.Incomparison,7Liisinferredfromalargenumber 80

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Figure5-1. Valuesof10,BBNinferredfromtheabundancesof7Li,D,3Heand4He,andthepredictedvaluesinthestandardcosmologicalmodel(WIMP)andinourproposal(axion).Thedatainferredvaluesaretakenfromrefs.[ 58 ],[ 76 ]and[ 77 ].Theerrorbarsindicatethe10,BBNvaluesconsistentwiththeestimated1-uncertaintiesintheobservations. ofmeasurements,whicharemore-or-lessconsistent.Also,sinceDismoreeasilydestructiblethan7Li,itisconceivablethatunknownstellarprocessesfurtherdepleteD. Finallythe3Heand4Heinferred10,BBNvalueshavelargeerrorbarsandhencecarrylessstatisticalweight.The4Heinferredvaluehasincreasedrecently(5.5<10,4He<11accordingtoref.[ 76 ]and7.5<10,4He<20accordingtoref.[ 77 ])comparedtoitsacceptedvalueafewyearsago,creatingadditionaluncertainty. 5.3.3EffectiveNumberofNeutrinoSpecies Aftertheaxionsareheatedupandreachthesametemperatureasthephotons,mostofthemarestillinthegroundstate.Theaxionsinthegroundstatebehaveascold 81

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darkmatter.Theaxionsintheexcitedstatescontributeonebosonicdegreeoffreedomtoradiation.TheradiationcontentoftheuniverseiscommonlygivenintermsoftheeffectivenumberNeofthermallyexcitedneutrinodegreesoffreedom,denedby rad=[1+Ne7 84 114 3](5) whereradisthetotalenergydensityinradiationandistheenergydensityinphotonsonly.ThestandardcosmologicalmodelwithordinarycolddarkmatterpredictsNe=3.046,slightlylargerthan3becausethethreeneutrinosheatupalittleduringe+e)]TJ /F1 11.955 Tf -432.83 -28.25 Td[(annihilation.Takingaccountofthefactthatnotonlyisthereanextraspeciesofradiation(thermallyexcitedaxions)butalsothecontributionofthethreeordinaryneutrinosisboostedbecausethephotonshavebeencooledrelativetothem,theproposedscenariopredicts rad=+a+="1+1 2+3.0467 84 114 33 2#, (5) whichyieldsNe=6.77. Atpresent,themeasuredvaluesaresmallerthanthisprediction.TheWMAPcollaborationfoundNe=4.34+0.86)]TJ /F9 7.97 Tf 6.59 0 Td[(0.88(68%CL)basedontheir7yeardatacombinedwithindependentdataonlargescalestructureandtheHubbleconstant.Ananalysis[ 79 ]usingtheSloanDigitalSkySurvey(SDSS)datarelease7halopowerspectrumfoundNe=4.82.0(95%CL).TheAtacamaCosmologyTelescope(ACT)collaborationnds[ 80 ]Ne=5.31.3(68%CL)usingonlytheirCMBanisotropydataandNe=4.560.75(68%CL)whencombiningthatdatawithlargescalestructuredata.Thetendencyforthemeasuredvaluestobelargerthan3.046hasbeentakensufcientlyseriouslytopromptproposalsfornewphysicsinvolvingextraneutrinospeciesoraneutrinoasymmetry[ 81 ]. 82

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CHAPTER6CONCLUSIONS AlthoughthethreemajorcandidatesforCDM,axions,WIMPs,andsterileneutrinos,werethoughtuntilrecentlytobeindistinguishablebypurelyastronomicalobservations,axionsareverydifferentfromtheothertwointermsofstatisticalmechanicsproperties.TheaxionsareahighlydegenerateBoseuidwhiletheothertwoarenotdegenerate.Throughgravitationalself-interactions,axionsthermalizewhenthephotontemperaturedropsbelow500eV.TheythenformaBose-Einsteincondensate,i.e.almostallaxionsgotothelowestenergystate.Wendthatifthatstateistimeindependent,axionsbehaveastheothercolddarkmattercandidatesonallscalesofobservationalinterest.Howeverobservationaldifferencesoccurwhentheaxionsrethermalizeandtheaxionstatetracksthelowestenergystate.Wendthatcoldaxionsrethermalizewhentheyfallintoagalactichalo.Asaresultthecoldaxionsacquireastateofnetoverallrotation.Incontrast,ordinarycolddarkmatterfallsinwithanirrotationalvelocityeld.Theinnercausticsofthegalactichaloaredifferentinthetwocases.Theoccurrenceofcausticringsofdarkmatteringalactichalosisinconsistentwithordinarycolddarkmatter,butconsistentwithaxionBEC.Inaddition,coldaxionsmayreachthermalcontactwithphotonsandbaryonsatthetimeofequalitybetweenmatterandradiation.Thermalcontactbetweencoldaxions,photonsandbaryonschangescosmologicalparameters,specicallythebaryontophotonratioatthetimeofprimordialnucleosynthesisandtheeffectivenumberofneutrinospeciesatdecoupling.Thechangeinthebaryontophotonratioalleviatesthefamouslithiumproblem.Futurecosmologicalobservations,suchasbythePlanckmission,mayprovideprecisedataontheeffectivenumberofneutrinos. 83

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APPENDIXADETECTIONOFAXION-LIKEPARTICLESBYINTERFEROMETRY A.1INTRODUCTION Althoughastrophysicalobservationsandcosmologicalconsiderationsprovideusefulconstraintsontheaxionparameterspace,whetheraxionsreallyexistcanonlybesettlediftheyareactuallydetectedinthelaboratory,andasoftodaythehypotheticalparticleremainselusive.Initially,theprospectofdetectingsuchweaklyinteractingparticleswasdeemedunlikely,sinceaverylargefaimpliesthataxionscoupleveryweaklytoordinarymatter.However,itwaspointedoutthatwemaycatchglimpsesoftheelusiveparticlebyexploitingitscouplingtotwophotons,whichisgivenintheLagrangianby[ 30 ] La=ga 4aF~F.(A) Throughthiscoupling,theaxionandphotoncanmixwitheachotherinabackgroundmagneticeld.Itisthisprinciplethatunderliesallexistingaxiondetectionexperiments.TheADMXexperiment,forexample,isarealizationoftheaxionhaloscope,inwhichaxionsinthehaloareinducedtoconvertinacavitytomicrowavephotonsthatarethenpickedupbyanantenna.TheCERNAxionSolarTelescopeandtheTokyoHelioscope,ontheotherhand,arearealizationofthehelioscopeandaimtodetectaxionsoriginatingfromtheSun,byconvertingthemintoX-raysinastrongmagneticeld.Thephoton-axionmixingcanalsomanifestitselfinthebirefringenceanddichroisminthevacuum,resultinginrotationandelliptizationofthepolarizationoflightinthepresenceofamagneticeld.Suchsignalisactivelybeingsought,asinthePVLASexperiment. Anothertypeofexperimentthatmakesuseofthismixingisphoton-regeneration(orlightshiningthroughawall)[ 99 ],inwhichasmallfractionofthephotonsinalaserbeamtraversingaregionpermeatedbyamagneticeldisconvertedtoaxions.Becauseoftheirweakcouplingtoordinarymatter,theaxionstravelunimpededthroughawall, 84

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ontheothersideofwhichisanidenticalarrangementofmagnets.Theresomeoftheaxionsareinducedtoconvertbacktophotons,whichcanbedetected.Theprimaryadvantageofphoton-regenerationexperimentsistheirgreatercontroloverexperimentalconditions.Sincethelaserbeamispreparedinthelaboratory,onedoesnothavetorelyonextraterrestrialaxionsources.Themajordrawbackisthatthesignalisveryweak(/g4a),sincetwostagesofconversionarerequired.Atthemoment,photonregenerationexperimentsdonothavesufcientsensitivitytodetecttheQCDaxion,althoughtheyareinprinciplecapableofdetectingotherparticlesthatcouplemorestronglytothephotoninananalogousmanner.Hence,theirprimaryobjectiveistodetectaxion-likeparticles(ALPs),ratherthanaxions. ALPsarepredictedtoexistgenericallyinstringtheory[ 88 ].WhilepseudoscalarALPscoupletophotonsasaxionsdo,scalarALPscoupletophotonsviaaaFFtermintheLagrangian,sotheycanbeproducedbyphotonswhosepolarizationisperpendiculartothebackgroundmagneticeld.Ingeneral,thereisnoapriorirelationshipbetweenthemassandcouplingsofALPs;hencetheirparameterspaceisalotlessconstrainedcomparedtoaxions. WeproposeanewexperimentalmethodbasedoninterferometrytodetectALPs.Alaserbeamissplitintotwobeamsofequalintensity.Oneofthemactsasareferencebeam,whiletheothertraversesaregionpermeatedbyamagneticeldwhichinducesconversionintoALPs,justasinthersthalfofphoton-regenerationexperiments.However,insteadofhavingasecondstagebehindawallwhereALPsareconvertedbacktophotons,thebeamisrecombinedwiththereferencebeam.Ifphoton-ALPsconversionhasoccurred,thebeamemergingfromtheconversionregionwouldhaveaslightlyreducedamplitudeandaphaseshiftrelativetothereferencebeam.Thisleadstoachangeinthecombinedintensity,whichcanthenbemeasuredbyadetector.Becauseonlyonestageofconversionisneeded,thesignalintensityisproportionaltoonlyg2a,insteadofg4aforthephoton-regenerationexperiment.This,however,does 85

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notstraightforwardlyimprovesensitivitytogaduetothepresenceofshotnoiseinanylightsource;wewillexpoundonthislater. Inordertoavoidhavingthesignalbeingoverwhelmedbythebackground,thetwobeamsarearrangedtotraversepathsofdifferentlengths,suchthattheywouldbeoutofphasebyatthedetectorwhenthemagneticeldisswitchedoff.Thus,withoutanyconversionthetwobeamswouldinterferedestructivelyatthedetector.ThedetectionoflightwouldsignaltheoccurrenceofALPsproduction.Unfortunately,atthedarkfringethesignalisreducedtoasecond-ordereffect(O(g4a)),soitisnecessarytomodulatetheamplitude(orfrequency)ofthelaserbyusingaPockelscell.ThepresenceofthetwosidebandsinadditiontothecarriergivesrisetoacomponentinthepoweroutputthatisofO(g2a),whichcanthenbeisolatedanddetectedbytheuseofamixer. Anycoherentlightsourceisaffectedbyshotnoise.ForanincominglaserbeamofNphotons,theHeisenberguncertaintyprincipleimpliesanuctuationofp Ninthephotonnumber.ThisreducesourabilitytoplacealimitontheALPs-photoncoupling:gaB)]TJ /F9 7.97 Tf 6.59 0 Td[(1L)]TJ /F9 7.97 Tf 6.58 0 Td[(1N)]TJ /F9 7.97 Tf 6.58 0 Td[(1=4,whichisthesameasthatinordinaryphotonregeneration(whereB,L,andNarethemagneticeld,lengthofconversionregion,andnumberofphotonsrespectively).Fortunately,ourdesignadmitsastraightforwardimplementationoflightsqueezing,whichcanreduceshotnoisebyanorderofmagnitudewithcurrenttechniques. Furthermore,byemployingopticaldelaylines,wecanenhancethesignalbyafactorofn,wherenisthenumberoftimesalaserbeamisfolded.Sowecanimproveourconstraintongabyn1=2102.5.Bycomparison,theuseofopticaldelaylineinphoton-regenerationresultsinamuchweakerimprovementofordern1=4. Wealsopointoutthatinrecentyearstherehasbeenaproliferationofhypothesizedparticles,manyofwhichcoupletotwophotonsasALPsdo,sotheycouldalsopotentiallybediscoveredinourproposedexperiment.Someexamplesincludechameleons,massivehiddenphotons,andlightminichargedparticles[ 91 95 ].In 86

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FigureA-1. Schematicdiagramofourproposedexperiment.Alaserbeam,whoseamplitudeismodulatedbyaPockelscell,issplitintotwobeamsofequalintensity(B1andB2).ThebeamB2(vertical)traversesaregionpermeatedbyamagneticeld,wherephotonsconverttoaxions(andotherparticleswithatwo-photonvertex).ItisthenrecombinedatthedetectorwiththebeamB1(horizontal),whichactsasareference.Thetwoarmsaredifferentinlength,sothatthetwobeamsareoutofphasebyintheabsenceofamagneticeld.Achangeinintensityregisteredbythedetectorwouldsignaltheoccurrenceofaconversion.Toextractthecomponentoftheoverallsignalthatisproportionaltog2a,wemixtheoutputwiththeoscillatorvoltagethatdrivesthePockelscell. particular,usingresultsin[ 91 ],itisstraightforwardtogeneralizeouranalysistothedetectionofminichargedparticles. A.2DESIGNOFEXPERIMENT Photon-axionmixinginamagneticeldisbasedontheaF~Fcoupling,whereoneofthephotonlegsisavirtualphotoninthemagneticeld.Ifthepolarizationofthephotonisparalleltothemagneticeld,theprobabilityofconversioncanbeobtainedfromthe 87

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crosssectionofthisprocess: !a=1 4va(gaBL)22 qLsinqL 22,(A) wherevaisthevelocityoftheaxion,Bthemagneticeld,Lthelengthoftheconversionregion,andqthemomentumtransfer.Sincema!,thefrequencyofthelaserbeamphotons,va1.ForL10m,thisalsoimpliesthatqL10)]TJ /F9 7.97 Tf 6.59 0 Td[(51,so( A )canbeapproximatedby !a1 4(gaBL)2.(A) IfweuseB10T,L10m,andf)]TJ /F9 7.97 Tf 6.59 0 Td[(1a10)]TJ /F9 7.97 Tf 6.59 0 Td[(12GeV)]TJ /F9 7.97 Tf 6.59 0 Td[(1,theprobabilityofphoton-axionconversionisofO(10)]TJ /F9 7.97 Tf 6.59 0 Td[(26)toO(10)]TJ /F9 7.97 Tf 6.58 0 Td[(25).Aftertheconversion,theamplitudeAofthephotonisreducedtoA)]TJ /F6 11.955 Tf 11.96 0 Td[(A,where A!a=A!a 2g2aB2L2A 8.(A) WenotethatthediscussionhereisapplicabletopseudoscalarALPs,sincetheycoupletothephotoninexactlythesameway.Ifthephotonpolarizationisinsteadperpendiculartothemagneticeld,theanalysisisalsovalidforscalarALPs,astheycoupletophotonsviaaFF~B~Binstead. Whenaphotonentersaregionpermeatedbyamagneticeld,thedispersionrelationforthecomponentorthogonalwithrespecttothemagneticeldremains!2=k2.However,ifaxionproductionoccurs,thatoftheparallelcomponentismodied: !2=k2+1 2m2a+g2aB2q (m2a+g2aB2)2+4g2ak2B2. (A) ForB10Tandga10)]TJ /F9 7.97 Tf 6.59 0 Td[(12GeV)]TJ /F9 7.97 Tf 6.58 0 Td[(1,thevalueofg2aB2ismuchlessthanm2a.Undertheseassumptions,theadditionalphaseacquiredisthenapproximately !ag2aB2m2aL3 48k.(A) 88

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TheeffectofthephaseshiftisnegligibleincomparisonwithA=A. Inourproposedexperiment,alaserbeamrstentersaPockelscell(withapolarizerbehind)tomodulateitsamplitude(thepurposeofthemodulationwillbeexplainedbelow).Subsequently,itisdividedbyabeamsplitterintotwobeams(whichwelabelB1andB2inFigure A-1 )withequalintensity.B2isessentiallythelaserbeamusedinthersthalfoftheshining-light-through-the-wallexperiment:itpassesthrougharegionpermeatedbyaconstantmagneticeld,whereasmallfractionofthephotonsareconvertedintoaxionswhichcarryenergyawayfromthebeam.Forsimplicity,wewillconsiderherethatthecarrierofthemodulatedbeam(bothB1andB2)islinearlypolarizedinthedirectionofthemagneticeld,soouranalysisintheprevioussectionapplies(ForthedetectionofscalarALPs,thepolarizationshouldbeperpendiculartothemagneticeldinstead).Thetwobeamsarethenrecombinedatthedetector,andinthepresenceofaconversion,theslightamplitudereductionandphaseshiftwouldleadtointerference,whichcanbedetected. ThelengthofthepathtraversedbybeamB1isbydesignslightlydifferentfromthatbyB2,sothatatthedetectorthetwobeamswouldbeoutofphasebyifthemagneticeldhasbeenabsent.Operationally,thiscanbeachievedbyadjustingoneofthepathlengthsuntildestructiveinterferenceisobservedatthedetectorwhenthemagneticeldisturnedoff.Hence,intheabsenceofthesidebands,thetwobeamswouldinterferedestructivelyatthedetector.Thepurposeforthisarrangementistoreducethebackground,therebyenhancingthesignal-to-noiseratioandminimizingshotnoise. LetthepathlengthsofthetwoarmsbeLxandLy(correspondingtobeamsB1andB2),andthestateofthelaserafterpassingthroughthePockelscellbedescribedby ~Ein=~E0(1+sin!mt)ei!t,(A) 89

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whereisaconstant,~E0theinitialelectriceldatt=0,and!isthefrequencyofthelaser.Theamplitudeismodulatedatafrequency!m.Thiscanberecastas ~Ein=~E0ei!t+ 2iei(!+!m)t)]TJ /F6 11.955 Tf 14.6 8.09 Td[( 2iei(!)]TJ /F8 7.97 Tf 6.58 0 Td[(!m)t,(A) wherethersttermisreferredtoasthecarrier,andthelattertwoassidebands. Forsimplicity,weignorethecontributionoftheadditionalphaseinthepresentanalysis,sinceitisnegligibleincomparisontothatofA.Inthiscase,thestateofthecarrierafterrecombinationatthedetectorisgivenby ~Ecarrier=)]TJ /F6 11.955 Tf 11.09 11.25 Td[(~E0 2ei(!t+2kL)2isinkL)]TJ /F6 11.955 Tf 13.15 8.09 Td[(A Ae)]TJ /F5 7.97 Tf 6.58 0 Td[(ikL, (A) wherek=!=cisthewavenumberofthelaserphotons,A=j~E0j,L=Lx)]TJ /F3 11.955 Tf 12.72 0 Td[(Lyisthelengthdifferencebetweenthetwoarms,andL=(Lx+Ly)=2istheaverage.Asmentioned,wewillchoosekL=,sothatthedetectoroperatesatadarkfringe,inordertoeliminatethebackgroundsignal.Thisleadsto ~Ecarrier=ei(!t+2kL)A 2A~E0.(A) Notethatwithouttheaidofthesidebands,thiswouldbetheentiresignal.Whilethebackgroundiseliminated,theintensity(~E2)isofO(g4a)(foraxions).Thislossinsensitivity,aswewillsee,canberecoveredbyusingthesidebands. Meanwhile,thesidebands(secondandthirdtermsof( A ))aredescribedby ~E=~E0ei(!t+2kL)ei(!mt+2!mL=c)sin!mL ciA 2Aei!mL=c, (A) wherethesubscripts+and)]TJ /F1 11.955 Tf 12.62 0 Td[(denoterespectivelythesidebandcomponentsoffrequency!+!mand!)]TJ /F6 11.955 Tf 11.95 0 Td[(!m. 90

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Ifweset!mc=2L,thetotalelectriceldatthedetectorisobtainedbyaddingthatofthecarrierandsidebands: ~E=~E0ei(!t+2kL)A 2A+2)]TJ /F6 11.955 Tf 13.15 8.08 Td[(A Acos!mt+2!mL c. (A) Notethatthisparticularvalueof!mischosentomaximizethesignal.Since!m!n!mandkL!n(fornanoddinteger)areequallyvalidchoices,theexperimenterhasmuchfreedominchoosingasuitablevaluefor!mthatisexperimentallyfeasible. Hence,thepowerPthatfallsonthedetectoris P=Pin(A=A)2 4+2(4)]TJ /F4 11.955 Tf 11.96 0 Td[(4(A=A)+(A=A)2) 2+A A(2)]TJ /F6 11.955 Tf 13.15 8.09 Td[(A A)cos!mt+2L c+2(4)]TJ /F4 11.955 Tf 11.95 0 Td[(4A A+A2 A2) 2cos2!mt+2L c. (A) Thusthepowerhasadccomponent(rstline),andtwoaccomponentswithfrequencies!mand2!m.IfwemultiplythiswiththeoscillatorvoltagethatdrivesthePockelscell(plusanappropriatephaseshift)viaamixer,wecanextractthecomponentoffrequency!m.Neglectingthesecond-ordercontributions,thetime-averagedoutputpowerofthemixerisgivenby Pout=1 TZT2PinGA Acos2(!mt) (A) =PinGA A (A) whereGisthegainofthedetectorandTistakentobesufcientlylongtoensurethatthetime-averagingisaccurate.Hence,theoutputsignalisproportionaltog2aforaxionsandGforgravitons. Inthisanalysiswechoosetomodulatetheamplitude,ratherthanthephase,ofthephotonsbecausethereductioninamplitudehascomparativelyamuchlargereffect.In 91

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principle,wecouldinsteadmodulatethephase,inwhichcasethechangeinintensityregisteredbythedetectorwouldbeprimarilyaconsequenceofthephaseshiftinsteadoftheamplitudereduction.Thecorrespondinganalysisishighlyanalogousandwillnotberepeatedhere.Themajordifferenceisthatthecoefcientsforthesidebandsin( A ),=2i,arereplacedapproximatelybyJ1(),therst-orderBesselfunctionoftherstkind(higherharmonicsnowarealsopresent,butarenegligible).SinceJ1()isreal,ourearlieranalysiswouldworkifA=Aisreplacedbyi,whichispurelyimaginary.ThiscanbeimplementedbymanipulatingpolarizersadjacenttothePockelscell.Thusbyswitchingbetweenphaseandamplitudemodulation,wecaninferinformationonboththeamplitudereductionandphaseshift.Thisisoneconceivablewayofidentifyingtheparticlesthatthephotonshaveconvertedinto. A.3DISCUSSIONANDCONCLUSIONS Inthisappendix,weproposedanewmethodofALPdetectionbasedoninterferometry.Alasersourceissplitintotwobeams,whereoneisexposedtoamagneticeldpermeatingaconnedregion,withinwhichphoton-axionconversionoccurs.Thisresultsinaphaseshiftandreductioninamplitude,whichcanbemademanifestifthebeamisthenrecombinedandmadetointerferewiththeother,whichactsasareference.Becauseonlyonestageofconversionisneeded,thesignalgoesasg2a,whichisanimprovementoverthatofexistingphoton-regenerationexperiments.Thekeytotheimprovementistherealizationthatitisnotnecessarytoconverttheaxionsbacktophotonsfordetection;interferencewithareferencebeamcanrevealjustasmuch. However,whatmattersinpracticeisthesignaltobackgroundratio.Inordertoavoidthesignalbeingoverwhelmedbythebackground,itisnecessarytohavethedetectoroperateatadarkfringe.Unfortunatelythisalsoreducesthesignaltoasecond-ordereffect(O(g4a)).Thisreductioncanbenulliedbymodulatingthephotonamplitude,andmixingtheoutputsignalwiththeoscillatorvoltagethatdrivesthePockelscell. 92

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Despitetheimprovementinsignalsize,theuseofinterferometersisinevitablyaccompaniedbythepresenceofshotnoise,whichisamanifestationofthegranularnatureofthecoherentstateofphotonsinthelaserbeam.Thislimitstheresolutionoftheinterferometerthereforereducingthesensitivitytogainoursetup. ForalaserbeamconsistingofNincomingphotons,weexpecttheshotnoiseinoursetuptohaveamagnitudeofp NduetoPoissonstatistics.Thesignal-to-noiseratioisthusreducedto(gaBL)2N=p N.Inthecaseofanon-detection,thisallowsustoconstraintheaxion-photoncouplingtoga,max<(BL))]TJ /F9 7.97 Tf 6.58 0 Td[(1N)]TJ /F9 7.97 Tf 6.58 0 Td[(1=4,whichiswhatcanbeachievedbyconventionalphoton-regenerationexperiments.(Intheircase,thesignalismuchsmaller,ofO(g4aN),sodarkcountratecanbeaproblem.) Oursetupadmitsastraightforwardimplementationofsqueezedlightusingstandardopticaltechniques,whichcanhelpreduceshotnoise.Theideaisthatwecouldreducetheuncertaintyinanoperatorbyenhancingthatofitsconjugateoperator,sothattheHeisenberguncertaintyprincipleisstillsatised.A10dBsuppressionofshotnoisecanresultina101=2improvementoftheconstrainttoga.Tofurtherboostthesensitivity,wecanincorporateinoursetupopticaldelaylinestoenhancethesignalbyafactorofn,wherenisthenumberoftimesthelaserbeamisfolded.Theresultantimprovementinourabilitytoconstraingaisofordern1=2102.5v.s.n1=4101.25inphotonregenerationexperiment.Combined,theuseofsqueezedlightandopticaldelaylinesresultsinagaininthesensitivitytogaof102oversimplephotonregenerationexperiment. Ifweusen103,B10T,L10mwitha10W(=1m)laser,after240hoursrunning,theexperimentcanexcludeALPswithga>2.810)]TJ /F9 7.97 Tf 6.59 0 Td[(11GeV)]TJ /F9 7.97 Tf 6.58 0 Td[(1to5signicance.Ifonealsoemployssqueezed-lightlaserwhichimprovessignal-to-noiseratioby10dBwithsimilarsetup,theexclusionlimitcanreachga10)]TJ /F9 7.97 Tf 6.59 0 Td[(12GeV)]TJ /F9 7.97 Tf 6.58 0 Td[(1. FinallywepointoutthatwhilewehaveasourprincipalaimthedetectionofALPs,ourdesignistheoreticallyapplicabletoanyparticlewithatwophotonvertex,sothat 93

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mixinginthepresenceofanexternalmagneticeldispermitted.Giventhepossibilitythatmorethanonesuchparticleexists,itisimportanttoidentifywhatthephotonshaveconvertedinto.Wesuggesttwomethodsthatcanhelpshedlightonthisissue.First,wecouldrepeattheexperimentbymodulatingthephaseinsteadoftheamplitudeofthelaser,asthiswouldrevealinformationaboutthephaseshiftaswell.Secondly,scalarandpseudoscalarALPscanbedistinguishedbymodifyingthepolarizationofthelaser.Conversioncanonlyoccurifthepolarizationisparallel(perpendicular)totheexternalmagneticeldforpseudoscalar(scalar)ALPs. Evenwiththeincorporationofsqueezedlightandanopticaldelayline,thesensitivityofourexperimentstillfallsshortforthedetectionoftheQCDaxionandthegraviton.Thisisexpectedinlightofthefeeblenessoftheircouplingstophotons.Itishopedthatfutureimprovementsinthetechnologyoflightsqueezingandtheadventofmorepowerfullasersmightsomedayhelpbridgethegapinsensitivityrequired.Ifachievable,ourexperimentmightserveasanexcellentcomplementtoexistingexperiments,suchasADMXandLIGO. 94

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APPENDIXBCOSMICRAYPROTONSILLUMINATEDARKMATTERAXIONS B.1INTRODUCTION WeproposeanewobservationalprobeofaxionsorALPs,basedonthefactthatchargedparticlespropagatinginatime-dependentaxioneldemitphotons.Theaxioneldcanbeviewedasasourceofenergy(duetoitstimedependence),butnotthree-momentum(duetoitshomogeneity).Consequently,someprocessesthatarekinematicallyforbidden(energyandmomentumconservationcannotbesatisedsimultaneously)beforenowbecomepossible.Admittedly,becausetheaxion'scouplingtoordinarymatterisverysmall,therateofphotonemissionisextremelytiny.Infact,withcurrenttechnologies,performingalaboratoryexperimenttoobservetheemissionisdenitelynotfeasible.Aroughestimateshowsthatanelectronacceleratorwithareasonablelengthandelectronuxwillhavetobeinoperationformorethantheageoftheuniversebeforetheemissionofasinglephoton.Fortunately,thisphenomenonoccursnaturallyincosmology:bycosmicrays(primarilyprotons)propagatinginatime-dependentaxioneld.Aswewilldemonstrate,theabundanceofaxions(orALPs)andcosmicraysinourgalaxymightcompensateforthesmallnessofthecoupling.Withtheaidofadetectorwithacollectingareaof1010cm2(forexample,theSquareKilometreArraycurrentlyunderconstruction),thiscangiverisetoaweakbutdetectablesignalforALPS(butnotfortheQCDaxion). Becausegalacticcosmicraysalsogeneratediffusegalacticradiation,itmightappeardifculttodisentangleoursignalfromthebackground,whichisdominant.Fortunately,itturnsoutthattheenergyspectrumofthephotonshasawell-denedpeak,whichislocatedapproximatelyatthemassoftheaxionorALP.ThisisexpectedtobeofordereV,whilediffusegalacticradiationtendstobemuchmoreenergetic(GeV).Theexistenceofsuchapeakcanbeunderstoodsincekinematicsdictatesthatcosmicrayprotonsinthelowenergyendoftheirspectrumcanonlyproducephotons 95

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whosefrequencyisapproximatelythemassoftheaxionorALP.Sinceboththecrosssectionandcosmic-rayenergyspectrumdecreasewithincreasingenergy,photonswhosefrequencyliesinthevicinityoftheaxion(orALP)massaremostabundantlyproduced. Thepropagationofchargedparticlesinaspatiallyhomogeneous,buttime-dependent,pseudoscalarbackgroundwasinvestigatedin[ 105 ],inwhichtheauthorsmaketheassumptionthatthetime-varyingbackgroundbetreatedasaconstantintheLagrangian(thisisessentiallytheLorentz-violatingChern-SimonstermconsideredbyCarrolletal.in[ 106 ]).Thisadditionaltermhastheeffectofmodifyingthedispersionrelationofthephotons.Asaconsequence,theprocessofphotonemissionisdescribedatleadingorderbyasingleFeynmandiagramwithonevertex.Accordingto[ 105 ],photonemissionbychargedparticlesisthenpossibleonlyifthephotonismassive,andtheemissionangleissmall.Incontrast,ourcalculationincorporatesthetimedependenceoftheaxioneld,whichleadstothreeFeynmandiagramswithtwoverticeseach.Usingthismethod,wendinsteadthatemissionispossibleatallanglesandformasslessphotons. B.2THEORETICALANALYSISOFPHOTONEMISSIONBYPROTONSINAPSEUDOSCALARFIELD TheLagrangianthatdescribesthedynamicsofprotons( )propagatinginanaxion(orALP)eld()isgivenby L=)]TJ /F3 11.955 Tf 9.29 0 Td[(igap 5 + 4gaF~F+LQED,(B) whereLQEDistheusualQEDLagrangianthatdescribesprotonsandphotons.Thecouplingsgapp=cappmp=faandga=ca=(2fa)arerespectivelytheaxion-protonandaxion-photoncoupling,wherecappandcaaredimensional-lessmodel-dependentparameters,typicallyoforderunity[ 107 ].Inthispaper,weassumethatcapp=ca=1.TheparameterfaessentiallymeasuresthestrengthoftheALP'scouplingtothephoton 96

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FigureB-1. M1 FigureB-2. M2 FigureB-3. M3 andproton.FortheQCDaxion,faisknownastheaxiondecayconstant,anditisconstrainedto109
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ThematrixelementforprocessIIinwhichaproton,subsequenttoemittingarealphoton,interactswiththeaxioneld,is iM2=iegak 2(lp)(l)u(q)5(2p)]TJ /F6 11.955 Tf 11.96 0 Td[(l)u(p).(B) ThematrixelementforprocessIIIinwhichaprotoninteractsrstwiththeaxionbackground,thenemitsarealphoton,isgivenby iM3=)]TJ /F3 11.955 Tf 16.17 8.09 Td[(iegak 2(lq)(l)u(q)(2q+l)5u(p).(B) Tocalculatethedifferentialcrosssection,weneedtorstcomputejM1j2,jM2j2,jM3j2,M1M2,M1M3,M2M3.SquaringM1,averagingoverinitialprotonspins,andsummingovernalphotonpolarizations,wehave 1 2XspinsjM1j2=)]TJ /F4 11.955 Tf 12.72 9.17 Td[(4g2ae2 (p)]TJ /F3 11.955 Tf 11.95 0 Td[(q)44m2p(ql)(pl))]TJ /F4 11.955 Tf 11.96 0 Td[(4(pq)(pl)(ql)+4m2p(la)2)]TJ /F4 11.955 Tf 11.96 0 Td[(2(la)2(pq). (B) Meanwhile,squaringM2andM3yields 1 2XspinsjM2j2=4e2g2ak (lp)2(pl)(ql)+m2p(al))]TJ /F3 11.955 Tf 11.96 0 Td[(m2p(pq)+m4p,(B) and 1 2XspinsjM3j2=4e2g2ak (lq)2(pl)(ql)+m2p(al))]TJ /F3 11.955 Tf 11.96 0 Td[(m2p(pq)+m4p.(B) Thecrosstermscanlikewisebestraightforwardlycomputed: 1 2XspinsM1M2=2impe2gagak(al)2 (lp)(p)]TJ /F3 11.955 Tf 11.96 0 Td[(q)2 (B) 1 2XspinsM1M3=2impe2gagak(al)2 (lq)(p)]TJ /F3 11.955 Tf 11.96 0 Td[(q)2 (B) 1 2XspinsM2M3=)]TJ /F4 11.955 Tf 22.78 8.09 Td[(2e2g2ak (lp)(lq)(lp)(lq)+(pq)(la))]TJ /F4 11.955 Tf 11.95 0 Td[((pq)2+m2p(pq). (B) 98

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B.2.2DifferentialCrossSection Forsimplicity,weevaluatethedifferentialcrosssectionintherestframeoftheaxioneld.Inthiscasethefour-momentaaregivenbyp=(Ep,0,0,p),a=(ma,~0),q=(Eq,qsin,0,qcos),l=(!,~!),whereEp=p p2+m2p,Eq=p q2+m2p.Withoutlossofgeneralitywealignthez-axiswiththedirectionofpropagationoftheinitialproton,andrestrictthescatteringtothex-zplane.thusdenotestheanglebetweenthedirectionoftheemittedphotonandthez-axis.Thephoton'sfrequency!is !=m2a+2maEp 2Ep+2ma)]TJ /F4 11.955 Tf 11.96 0 Td[(2jpjcos.(B) Inthisframe,thephasespaceforthenalstateparticlesisgivenby Zd3~q (2)32Eqd3~! (2)32!(2)4(4)(p+a)]TJ /F3 11.955 Tf 11.96 0 Td[(q)]TJ /F3 11.955 Tf 11.95 0 Td[(l),(B) whichyieldsthedifferentialcrosssection d dcos()=ma+2Ep 16(2Ep+2ma)]TJ /F4 11.955 Tf 11.96 0 Td[(2jpjcos)2p E2p)]TJ /F3 11.955 Tf 11.96 0 Td[(m2p1 2XjM1+M2+M3j2. (B) B.2.3EmissionRateofThePhotons Ourgalaxyisteemingwithcosmicrays,whoseprimaryconstituentisprotons,towhichwewillrestrictourattentioninthispaper(henceE=Epbelow).Ourcalculationofthephotonemissionrateisthusaconservativeestimate,asotherchargedconstituents(e.g.electrons)wouldalsocontributetotheprocess.Ascosmicrayprotonspropagateinthisbackgroundtime-dependentaxioneld,theyundergophotonemissionviaprocessesdescribedintheprevioussection. ToestimatethephotonuxonEarth,wemaketheassumptionthatcosmicrayprotonsarehomogeneousandisotropicwithinourgalaxy.ThisispredicatedontheobservationthatcosmicrayprotonsscatteroffinterstellarmediumandtraverserandomtrajectorieswithintheGalaxyforanaverageof107yearsinthegalaxy[ 105 ].Forour 99

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calculation,weadoptthefollowingenergyspectrumforcosmicrayprotons[ 110 111 ]: dF dEd=3.06 cm2ssrGeVE GeV)]TJ /F9 7.97 Tf 6.59 0 Td[(2.70,(B) whichweassumetoholdforE>50GeV.Sincetheuxisknownwithlesscertaintyforlow-energyprotons,weimposeacutoffinourcalculationanddisregardallprotonswithanenergybelow50GeVasaconservativemeasure.Wewillalsoneglectthecontributionofextragalacticcosmicrays,whichshouldbesubdominantascomparedtothegalacticones. ConsidernowaphotondetectoronEarth,withaeldofviewd.Withourassumptionsonthedistributionofcosmicrays,thedetectorcanpickupphotonsoriginatingfromcosmicraysllingaregionfromtheEarthtotheedgeoftheGalaxy,whoseangularboundaryisdeterminedbyd.LetVdenotethisregion,anddVrbeadifferentialvolumeelementinVatadistancerfromthedetectoronEarth.AteachpointinV,therearecosmicrayspropagatinginallpossibledirections.Locally,wedeneasphericalcoordinatesystem(withtheusualcoordinatesand),where,aswedenedearlier,denotestheanglebetweenthevelocityvectorofthecosmicrayandthelineconnectingdVrtothedetectoronEarth.Theothervariableistheusualazimuthalangleconnedtotheplaneperpendiculartothedirectionofpropagationofthecosmicray. ThenumberofphotonsemittedbythecosmicraysthatllupdVroveranintervaldtisgivenby dN=nanpdVrdvdt.(B) wherenaandnparethenumberdensityofaxionsandprotons,vthevelocityofthecosmicrays,anddthedifferentialcrosssection.Fromthis,wecancomputetheuxofphotonsperunittimeatthedetector: dN dtdA=nadnp dEdEd dAvdVr, (B) 100

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wheredA=r2d=r2dcosdisthedifferentialareaatthedetector.Usingthefactthat dnp dEd=1 vdF dEd,(B) thephotonuxsimpliesto dN dtdA=na r2dFp dEdd dcosdVrdEdcos.(B) Integratingoverthevolumethatthedetectorcansee,theprotonenergy,andcos,andusing( B ),weobtainthephotonuxatthedetector, dN dtdA=naZV1 r2dVrZdE"3.06 cm2ssrGeVE GeV)]TJ /F9 7.97 Tf 6.59 0 Td[(2.70Z1)]TJ /F9 7.97 Tf 6.58 0 Td[(1d(E,cos) dcosdcos#,(B) whichcanbecomputednumerically. B.2.4EnergySpectrumofPhotons Duetotheuncertaintiesinherentintheenergyspectrumoflow-energycosmicrayswithintheGalaxy,itisnotpossibletodeterminetheprecisespectralshapeoftheemittedphotons.Nonetheless,thespectrumpossessesarobustfeature:ithasapeaklocatedatapproximatelythemassoftheaxionorALP(moreaccurately,at!c(m2a+2mamp)=(2mp+2ma),obtainedbysettingEp=mpin( B )). Theexistenceofsuchapeakcanbeunderstoodasfollows.From( B ),weobservethatforagivenE,theemittedphoton'sfrequencyisconnedtotherange!)]TJ /F2 11.955 Tf 10.4 1.79 Td[(!!+,where !=m2a+2maE 2E+2ma2p E2)]TJ /F3 11.955 Tf 11.95 0 Td[(m2p.(B) Thefrequencies!+and!)]TJ /F1 11.955 Tf 10.41 1.79 Td[(correspondtoscatteringat=0andrespectively(seegure B-4 ).Hence,photonsoffrequency!canonlybeproducedbycosmicrayprotonswithanenergylargerthan E=1 2m2a)]TJ /F4 11.955 Tf 11.96 0 Td[(4ma!()]TJ /F3 11.955 Tf 9.3 0 Td[(m3a+3m2a!)]TJ /F4 11.955 Tf 11.95 0 Td[(2ma!2+q )]TJ /F4 11.955 Tf 9.3 0 Td[(4m2pm2a!2+m4a!2+8m2pma!3)]TJ /F4 11.955 Tf 11.96 0 Td[(4m3a!3+4m2a!4). (B) 101

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FigureB-4. Photonfrequency!versustheprotonenergyE,foranaxionmassof10)]TJ /F9 7.97 Tf 6.59 0 Td[(6eV,accordingto( B ).Totherightofthecurveistheregioninwhichphotongenerationbycosmicrayprotonsisallowed.Theturnaroundpointcorrespondsto!=!c,whichisveryneartheaxionorALPmass.Thetop(bottom)halfofthecurvewhichisincreasing(decreasing)withrespecttoEcorrespondstoforward(backward)scattering. ThefrequencyatwhichdE=d!vanishesis!=!c(equivalently,E=mp).Thisimpliesthatphotonswhosefrequencyisinthevicinityof!careproducedbycosmicraysoverthewidestrangeofenergy.Morespecically,photonswithafrequencynear!ccanbeproducedaslongastheprotonenergyisaboveitsrestmass.Ontheotherhand,photonswithaverylowfrequencynearma=2canonlybeproducedbyenormouslyenergeticprotonswhichback-scatterat=,whilethosewithafrequency!macanonlybeproducedbyprotonswhoseenergyexceedsapproximatelyp !m2p=(2ma).SincethatthecosmicrayuxandcrosssectiondecreasemonotonicallywithE,wethusconcludethatapeakispresentintheenergyspectrumofthephotons.Thishasbeenveriednumerically(bylinearlyextrapolatingtheprotonuxfromE=mpto50GeV);seegure B-6 foraplotofthespectrum,whichfeaturesasalientpeak,asexpected. 102

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B.3OBSERVATIONALCONSEQUENCES B.3.1InsufcientSensitivitytoDetectTheQCDAxion Wenumericallyintegrate( B )andndthat,forma=10)]TJ /F9 7.97 Tf 6.59 0 Td[(6eV(correspondingtoroughlytheexpectedmassoftheQCDaxion), dN dtdA10)]TJ /F9 7.97 Tf 6.58 0 Td[(21cm)]TJ /F9 7.97 Tf 6.59 0 Td[(2s)]TJ /F9 7.97 Tf 6.59 0 Td[(1.(B) Thisestimateisquiteconservative,asweonlyincludedenergyE>50GeVintheintegration,duetouncertaintiesinthespectrumoftheprotons.Ifweextendittoaslowas,say,10GeV,wegainaboostintheuxbyapproximatelyafactoroften(From[ 111 ],weknowthattheuxactuallyincreasesasEisdecreasedtoapproximately1.4GeV).Forna,weusea=ma,wherea=10)]TJ /F9 7.97 Tf 6.59 0 Td[(24g/cm3,theexpectedlocalhalodensityoftheGalaxy[ 109 ].FortheradiusroftheregionV,weadoptthevalue1022cm,whichisapproximatelyonetenthofthesizeofthestellardisk.ThecollectingareaandeldofviewofthephotondetectoraretakentobethatoftheSquareKilometreArray:1010cm2and200deg2.Evenwithsuchahugesurfacearea,weexpectonlyonephotonevery1011s.Clearly,itisnotyetpossibletodetecttheQCDaxion. B.3.2ConstrainingTheParameterSpaceofALPs AlthoughthedetectionoftheQCDaxionseemsoutofreachgivenexistingtechnologies,aninterestingconstraintcanstillbeplacedonthe(ma,f)]TJ /F9 7.97 Tf 6.59 0 Td[(1a)parameterspaceofALPs,undertheassumptionthattheyconstitutedarkmatter.Theirnumberdensityisthustakentobea=ma,wherea10)]TJ /F9 7.97 Tf 6.59 0 Td[(24gcm)]TJ /F9 7.97 Tf 6.59 0 Td[(3.ThesensitivityofthedetectorisassumedtobetennanoJanskys.Overafrequencyrangeof1011Hz,whichisroughlywhatweneedtoseethepeakinthespectrum,thistranslatestoaminimumrateof10)]TJ /F9 7.97 Tf 6.58 0 Td[(3photons/cm2s(fortheSquareKilometreArray,about107photonspersecond).NotethattheconstraintisonlyvalidforanALPwhosemassliesbetween2.910)]TJ /F9 7.97 Tf 6.59 0 Td[(7eVand4.110)]TJ /F9 7.97 Tf 6.59 0 Td[(5eV,sincetheSquareKilometreArraycanonlydetectphotonsinthisrangewithaeldofviewdof200deg2.Forhigherphotonenergies(intherange 103

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FigureB-5. PhotonenergyspectrumdF=dEd=d!versusphotonenergy!foranaxionmassof10)]TJ /F9 7.97 Tf 6.59 0 Td[(6eV,uptoanormalizationfactor.ForE>50GeV,theprotonspectrumisgivenby3.06(E/GeV))]TJ /F9 7.97 Tf 6.59 0 Td[(2.70.Formp
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FigureB-6. Hypotheticalexclusionlimitonthemassmaandthedecayconstantfaforaxion-likeparticlesoverthemassrange2.910)]TJ /F9 7.97 Tf 6.59 0 Td[(7eV
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KilometreArray),weexpectonlyapproximatelyonephotonevery1011s.ThusdetectionoftheQCDaxionbythismeanisoutofthequestionatthemoment. Nonetheless,thesamemechanismcanbeexploitedtoimposeexclusionlimitsontheparameterspaceofALPs.Sincetheirdecayconstantfacanbesmaller,ALPscancouplemorestronglytoordinarymatter,therebyincreasingtherateofphotonemission.UndertheassumptionsthatdarkmatterisprimarilyALPs,andthatadetectionrateof107photonsperdayissufcient,wenumericallyndthatanon-detectionofphotonsbytheSquareKilometreArraytranslatestoexclusionlimitsonfa(forthemassrange2.910)]TJ /F9 7.97 Tf 6.59 0 Td[(7eV
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REFERENCES [1] R.D.PecceiandH.Quinn,Phys.Rev.Lett.38(1977)1440andPhys.Rev.D16(1977)1791. [2] S.Weinberg,Phys.Rev.Lett.40(1978)223. [3] F.Wilczek,Phys.Rev.Lett.40(1978)279. [4] J.Preskill,M.WiseandF.Wilczek,Phys.Lett.B120(1983)127;L.AbbottandP.Sikivie,Phys.Lett.B120(1983)133;M.DineandW.Fischler,Phys.Lett.B120(1983)137. [5] P.SikivieandQ.Yang,Phys.Rev.Lett.103(2009)111301. [6] Hickenetal.,2009TheAstrophysicalJournal,700:1097-1140. [7] Perlmutteretal.1999ApJ517565. [8] K.A.Olive,G.Steigman,T.P.Walker,2000Phys.Rep.333-334. [9] Y.I.Izotov,T.X.Thuan,andG.Stasinska,Astrophys.J.662,15(2007). [10] S.Dodelson,ModernCosmology,AcademicPress(2003). [11] V.Mukhanov,PhysicalFoundationsofCosmology,Cambridge(2005). [12] J.C.Mather,etal.ApJL420,439(1994). [13] Fixsenetal.1996,AstrophysicalJournal,473,576. [14] http://map.gsfc.nasa.gov/ [15] E.Komatsu,etal.arXiv:1001.4538(2010). [16] K.Begeman,A.BroeilsandRSanders,1991,MNRAS,249,523. [17] D.V.SemikozandI.I.Tkachev,Phys.Rev.D55,489(1997). [18] O.Erken,P.Sikivie,H.Tam,andQ.Yang,Phys.Rev.D85,063520(2012). [19] S.Weinberg,Phys.Rev.D.11(1979)3583. [20] C.VafaandE.Witten,Phys.Rev.Lett.53,535(1984). [21] L.F.AbbottandP.Sikivie,Phys.Lett.B120,133(1983).J.Preskill,M.B.WiseandF.Wilczek,Phys.Lett.B120,127(1983).M.DineandW.Fischler,Phys.Lett.B120,137(1983). [22] G.G.Raffelt,arXiv:hep-ph/0611350. [23] J.N.Bahcall,A.M.Serenelli,andS.Basu,ApJ621L85(2005). 107

PAGE 108

[24] V.Baluni,Phys.Rev.D,2227(1979). [25] G.Raffelt,Lect.Not.Phys.741:51-71,(2008). [26] G.Raffelt,Ann.Rev.Nucl.Part.Sci.49,163(1999)[hep-ph/9903472] [27] J.Isern,E.Garca-Berro,Nucl.Phys.Proc.Suppl.114,107(2003) [28] A.Burrows,M.S.Turner,R.P.Brinkmann,Phys.Rev.D39,1020(1989);A.Burrows,M.T.Ressell,M.S.Turner,Phys.Rev.D42,3297(1990) [29] S.Asztalosetal.,Phys.Rev.D64092003(2001). [30] P.Sikivie,Phys.Rev.Lett.51,1415(1983)[Erratum-ibid.52,695(1984)]. [31] G.Carosi,K.v.Bibber,Lect.Notes.Phys.741:135-156(2008). [32] S.J.Asztalosetal.Phys.Rev.Lett.104,041301(2010). [33] vanBibber,K.andMcIntyre,P.M.andMorris,D.E.andRaffelt,G.G.Phys.Rev.D39,2089C2099(1989). [34] A.PeresandA.Ron,Phys.Rev.A13,417C425(1976). [35] G.G.Raffelt,Phys.Rev.D37,1356(1988). [36] G.G.Raffelt,Phys.Rev.D33,897(1986). [37] B.Beltran,etal.arXiv:hep-ex/0507007(2005). [38] Z.Ahmed,etal.Phys.Rev.Lett.103(14)141802(2009). [39] S.Moriyama,etal.Phys.Lett.B,434(1998)147. [40] M.Ariketal.Phys.Rev.Lett.107,261302(2011) [41] E.Ariketal.AIPConf.Proc.899,37(2006) [42] R.Cameron,etal.Phys.Rev.D47,3707(1993). [43] P.Sikivie,D.B.Tanner,andKarlvanBibber,Phys.Rev.Lett.98,172002(2007). [44] G.Raffelt,L.Stodolsky,Phys.Rev.D37,1237(1988). [45] M.Bregant,etal.Phys.Rev.D78,032006(2008). [46] O.Erken,P.Sikivie,H.TamandQ.Yang,Phys.Rev.D85(2012)063520. [47] G.BaymandL.P.Kadanoff,QuantumStatisticalMechanics,Benjamin1962. [48] K.Huang,StatisticalMechanics,Wiley1987. 108

PAGE 109

[49] Forareview,seeP.Sikivie,Lect.NotesPhys.741(2008)19.Recentpapersonaxionradiationbystringsinclude:O.Wantz,E.P.S.Shellard,Phys.Rev.D82,123508(2010);T.Hiramatsuetal.,arXiv:1012.5502. [50] E.Fermi,J.PastaandS.Ulam,StudiesofNonlinearProblems,DocumentLA-1940(May1955);G.P.BermanandF.M.Israilev,Chaos15(2001). [51] W.HuandJ.Silk,Phys.Rev.D48(1993)485;W.Hu,Ph.D.Thesis,1995. [52] P.SikivieandQ.Yang,Phys.Rev.Lett.103(2009)111301. [53] P.Sikivie,I.TkachevandY.Wang,Phys.Rev.Lett.75(1995)2911;Phys.Rev.D56(1997)1863. [54] P.J.E.Peebles,Ap.J.155(1969)2,andAstron.Ap.11(1971)377. [55] L.DuffyandP.Sikivie,Phys.Rev.D78(2008)063508. [56] O.Erken,P.Sikivie,H.TamandQ.Yang,Phys.Rev.Lett.108(2012)061304. [57] R.H.Cyburt,B.D.Fields,K.A.Olive,JCAP0811,012(2008). [58] G.Steigman,Int.J.Mod.Phys.E15,1(2006). [59] G.Steigman,Ann.Rev.Nucl.Part.Sci.57,463(2007). [60] A.Cocetal.,Ap.J.600,544(2004);R.H.Cyburt,B.D.Fields,K.A.Olive,Phys.Rev.D69,123519(2004);C.Anguloetal.,Ap.J.630,L105(2005). [61] J.Melendez,I.Ramirez,Ap.J.615,L33(2004). [62] B.D.Fields,K.A.Olive,E.Vangioni-Flam,Ap.J.623,1083-1091(2005). [63] S.Vauclair,C.Charbonnel,Ap.J.502372(1998);M.H.Pinsonneaultetal.,Ap.J.527,180-198(2002);M.H.Pinsonneaultetal.,Ap.J.574,398-411(2002);O.Richard,G.Michaud,J.Richer,Ap.J.619,538-548(2005);A.J.Kornetal.,Ap.J.442(2006)657. [64] V.F.Dmitriev,V.V.Flambaum,J.K.Webb,Phys.Rev.D69,063506(2004). [65] A.Cocetal.,Phys.Rev.D76,023511(2007). [66] K.Jedamzik,Phys.Rev.D70,063524(2004);J.L.Feng,S.Su,F.Takayama,Phys.Rev.D70,075019(2004);J.R.Ellis,K.A.Olive,E.Vangioni,Phys.Lett.B619(2005)30;K.Jedamziketal.,JCAP0607,007(2006);R.H.Cyburtetal.,JCAP0611(2006)014;T.Jittohetal.,Phys.Rev.D76,125023(2007);K.Jedamzik,M.Pospelov,NJP11,105028(2009). [67] P.Sikivie,Q.Yang,Phys.Rev.Lett.103,111301(2009). [68] O.Erken,P.Sikivie,H.Tam,Q.Yang,arXiv:1111.1157. 109

PAGE 110

[69] J.Jaeckel,J.Redondo,A.Ringwald,Phys.Rev.Lett.101131801(2008). [70] P.Sikivie,Phys.Lett.B695,22(2011). [71] A.Natarajan,P.Sikivie,Phys.Rev.D73,023510(2006). [72] J.A.Fillmore,P.Goldreich,Ap.J.281,1(1984);E.Bertschinger,Ap.J.Suppl.58,39(1985);P.Sikivie,I.Tkachev,Y.Wang,Phys.Rev.Lett.75,2911(1995)andPhys.Rev.D56(1997)1863. [73] L.D.Duffy,P.Sikivie,Phys.Rev.D78,063508(2008). [74] W.T.Hu,Ph.D.thesis,astro-ph/9508126andreferencestherein. [75] S.-J.Sin,Phys.Rev.D50,3650-3654(1994);W.Hu,R.Barkana,A.Gruzinov,Phys.Rev.Lett.85,1158-1161(2000);J.-W.Lee,S.Lim,JCAP1001,007(2010);E.W.Mielke,J.A.V.Perez,Phys.Lett.B671,174-178(2009);F.Ferrer,J.A.Grifols,JCAP0412,012(2004);.C.G.Boehmer,T.Harko,JCAP0706,025(2007). [76] F.Ioccoetal.,Phys.Rep.472(2009)1. [77] G.Steigman,privatecorrespondence. [78] M.Pettinietal.,MNRAS,391(2008)1499. [79] J.Hamannetal.,JCAP07(2010)022. [80] J.Dunkleyetal.,arXiv:1009.0866. [81] L.M.Krauss,C.Lunardini,C.Smith,arXiv:1009.4666;J.Hamannetal.,Phys.Rev.Lett.105(2010)181301. [82] K.Ichikawa,T.Sekiguchi,T.Takahashi,Phys.Rev.D78(2008)083526. [83] F.Moulin,D.Bernard,1,F.Amiranoff,Z.Phys.C72,607-611(1996). [84] A.Mirizzi,G.G.Raffelt,P.D.Serpico,Phys.Rev.D72,023501(2005).[astro-ph/0506078]. [85] D.EspriuandA.Renau,arXiv:1010.3580[hep-ph]. [86] P.W.GrahamandS.Rajendran,arXiv:1101.2691[hep-ph]. [87] O.Mena,S.Razzaque,F.Villaescusa-Navarro,JCAP1102,030(2011).[arXiv:1101.1903[astro-ph.HE]] [88] P.Svrcek,E.Witten,JHEP0606,051(2006).[hep-th/0605206]. [89] M.E.Gertsenshtein,Sov.Phys.JETP,64,84(1962). 110

PAGE 111

[90] M.Ahlers,J.Jaeckel,A.Ringwald,Phys.Rev.D79,075017(2009).[arXiv:0812.3150[hep-ph]]. [91] H.Gies,J.JaeckelandA.Ringwald,Phys.Rev.Lett.97,140402(2006)[arXiv:hep-ph/0607118]. [92] M.Ahlers,H.Gies,J.JaeckelandA.Ringwald,Phys.Rev.D75,035011(2007)[arXiv:hep-ph/0612098]. [93] B.Holdom,Phys.Lett.B166,196(1986). [94] S.A.AbelandB.W.Schoeld,Nucl.Phys.B685,150(2004)[arXiv:hep-th/0311051]. [95] B.BatellandT.Gherghetta,Phys.Rev.D73,045016(2006)[arXiv:hep-ph/0512356]. [96] J.JaeckelandA.Ringwald,Ann.Rev.Nucl.Part.Sci.60,405(2010)[arXiv:1002.0329[hep-ph]]. [97] P.Sikivie,arXiv:1012.1553[astro-ph.CO]. [98] F.HoogeveenandT.Ziegenhagen,Nucl.Phys.B358,3(1991). [99] K.vanBibber,N.R.Dagdeviren,S.E.Koonin,A.K.Kerman,andH.N.Nelson,Phys.Rev.Lett.59,759(1987);A.A.Anselm,Yad.Fiz.42,1480(1985)[Sov.J.Nucl.Phys.42,1480(1985)]. [100] L.Maiani,R.Petronzio,E.Zavattini,Phys.Lett.B175,359(1986). [101] S.J.Asztalos,R.F.Bradley,L.Duffy,C.Hagmann,D.Kinion,D.M.Moltz,L.JRosenberg,P.Sikivieetal.,Phys.Rev.D69,011101(2004).[astro-ph/0310042]. [102] I.G.Irastorzaetal.[CASTCollaboration],[astro-ph/0211606]. [103] M.Minowa,S.Moriyama,Y.Inoue,T.Namba,Y.Takasu,A.Yamamoto,Nucl.Phys.Proc.Suppl.72,171-175(1999).[hep-ex/9806015]. [104] J.Redondo,A.Ringwald,[arXiv:1011.3741[hep-ph]]. [105] D.Espriu,A.Renau,[arXiv:1010.3580[hep-ph]]. [106] S.M.Carroll,G.B.Field,R.Jackiw,Phys.Rev.D41,1231(1990). [107] M.Giannotti,L.D.Duffy,R.Nita,JCAP1101,015(2011).[arXiv:1009.5714[astro-ph.HE]]. [108] P.Sikivie,Lect.NotesPhys.741,19-50(2008).[astro-ph/0610440]. 111

PAGE 112

[109] E.I.Gates,G.Gyuk,M.S.Turner,Astrophys.J.449,L123-L126(1995).[astro-ph/9505039]. [110] T.Stanev,HighEnergyCosmicRays,(Springer2003). [111] O.Adrianietal.[PAMELACollaboration],Science332,69-72(2011).[arXiv:1103.4055[astro-ph.HE]]. 112

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BIOGRAPHICALSKETCH QiaoliYangwasborninKunming,China,whereheattendedprimaryschool,middleschoolandhighschool.Allthreeschoolswerewithinonemilefromhisfamilyhome.Aftergraduatingfromhighschool,hewenttothecityofNanjing,China,whereheobtainedhisbachelordegreefromNanjinguniversity.HecametotheU.S.in2005andenrolledintheUniversityofKentuckywhereheobtainedMasterofSciencein2007,andthentransferredtotheUniversityofFloridatostudyforaDoctorofPhilosophyinPhysics.HeiscurrentlyworkinginthehighenergygroupwherehisPh.D.advisorisProfessorPierreSikivie.Hiscurrentresearchinterestisindarkmatter,axionphysicsandcosmology.HeobtainedhisPh.D.in2012. 113