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Axion BEC

Permanent Link: http://ufdc.ufl.edu/UFE0044589/00001

Material Information

Title: Axion BEC a Model beyond CDM
Physical Description: 1 online resource (113 p.)
Language: english
Creator: Yang, Qiaoli
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: axion -- bbn -- bec -- cdm -- cmbr -- cosmology -- qcd
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Qiaoli Yang.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Sikivie, Pierre.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044589:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044589/00001

Material Information

Title: Axion BEC a Model beyond CDM
Physical Description: 1 online resource (113 p.)
Language: english
Creator: Yang, Qiaoli
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: axion -- bbn -- bec -- cdm -- cmbr -- cosmology -- qcd
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Qiaoli Yang.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Sikivie, Pierre.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044589:00001


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