Neutrinos and Dark Matter

arXiv:astro-ph/9904001v11Apr1999NEUTRINOSANDDARKMATTERCHUNG–PEIMADepartmentofPhysicsandAstronomy,UniversityofPennsylvaniaPhiladelphia,PA19104E-mail:[e-mail]ightsomekeyfeaturesofmassiveneutrinosinthecontextofcosmology.Ifirstreviewthethermalhistoryandthefree-streamingkinematicsoftheuniformcosmicbackgroundneutrinos.Ithendescribehowfluctuationsinthephasespacedistributionsofneutrinosandotherparticlesariseandevolveafterneutrinodecouplingaccordingtothelinearperturbationtheoryofgravitationalinstability.Thedifferentclusteringpropertiesofmassiveneutrinos(akahotdarkmatter)andcolddarkmatterarecontrasted.Thelastpartdiscussesthenonlinearstageofgravitationalclusteringandhighlightstheeffectsofmassiveneutrinosontheformationofcosmologicalstructure.1NeutrinoMassesTheselecturesdiscusshowtheuniverseservesasalearninggroundformassiveneutrinos.Beforedoingso,letusbrieflyreviewsomeexperimentalmeasure-mentsofneutrinomasses.Upperboundsonneutrinomassesfromkinematicmeasurementsinlabo-ratoriescontinuetoimprove.1Fortheτ-neutrino,mντ18.2MeVfromthedecaychannelτ→5π+ντ.Fortheμ-neutrino,mνμ170keVfromtwo-bodypiondecay.Fortheelectronneutrino,thequantitym2νeismeasuredintritiumbetadecaybyfittingtheshapeoftheenergyspectrumneartheend-point.Experimentsthusfarhaveyieldednonphysicalnegativevaluesform2νe,indicatingunexplainedsystematiceffectsinthemeasurements.Aconservativeupperboundisputatmνe≈15eV.Thespreadinarrivaltimesofneutrinosfromsupernovaexplosionsprovidesanindependentwaytoconstrainthemassoftheelectronneutrino.VariouslimitshavebeenreportedforSN1987A;aconservativeestimateismνe23eV.1,22PropertiesofCosmicBackgroundNeutrinos2.1TemperatureandDensityForabrief1secondafterthebigbang,neutrinosenjoybeingpartofthethermalbathcomposedofphotons,electrons,protons,neutrons,andtheas-sociatedanti-particles(afterthequark-hadronera).Theweakinteractionsatthisearlytimearerapidenoughtokeeptheseparticlesinthermalequilib-riumatasingletemperatureT.After1second,whenTdropsbelowabout11MeV,however,theneutrinointeractionratebecomesslowerthantheHubbleexpansion,andneutrinosbecomeeffectivelycollisionlessandfreely-streamingparticleswhosetrajectoriesaredeterminedbythegeodesicequations.Thiseventiscommonlyreferredtoas“neutrinodecoupling.”Astheuniverseex-pands,themomentaandtemperatureofneutrinosaresimplyredshifted,andtheneutrinotemperatureisgivenbythefamiliarformulasTν(a)=a−1Tν,0,Tν,0=4111/3Tγ,0=1.947K,(1)whereaisthecosmicscalefactor,thesubscripts0denotethepresent-dayvalues,andthecosmicbackgroundphotontemperatureistakentobeTγ,0=2.728K.3Animportantfeatureoftheneutrinodistributionafterdecouplingisthat,althoughweakinteractionsarenolongerrapidenoughtokeepneutrinosinthermalequilibriumwithotherparticlespecies,neutrinosretaintheirequi-libriumdistributionaslongasnootherphysicalprocesses(e.g.,gravitationalclustering;seeSec.3)arepresenttoalterit.Therefore,tozerothorderinden-sityandmetricperturbations,thephasespacedistributionf0ofthecosmicbackgroundneutrinosisofthesimpleFermi-Diracformf0(ǫ)=gsh3p1eǫ/kBTν,0+1,(2)whereǫ=a(p2+m2ν)1/2isthecomovingenergy,Tν,0istheneutrinotempera-turegivenbyEq.(1),gsisthenumberofspindegreesoffreedom,andhpandkBarethePlanckandtheBoltzmannconstants.Thesituationisfurthersimplifiedifneutrinomassesare≪1MeV.Suchneutrinosarehighlyrelativisticatdecoupling;theirenergyǫ,andhencethedistributionfunctionf0,areindependentofmνtoagoodapproximation.Onecaneasilyshowthat,aslongasmν≪1MeV,thenumberdensityofthecosmicbackgroundneutrinosisrelatedtotheneutrinotemperaturebynν(Tν)=7gs8π2ζ(3)kBTν¯hc3,(3)whereζ(3)≈1.202istheRiemannzetafunctionoforder3.Thisgivesapresent-daydensityof≈113cm−3foreveryneutrinospeciesindependentoftheirmasses.(Forcomparison,thepresent-dayphotondensityis≈412cm−3.)Italsofollowsthatthecontributionoftheseneutrinostothepresent-daymassdensityparameter,Ων,isrelatedtotheirmassesbythesimplerelationΩνh2=Σimi93eV,(4)2wheretheindexirunsoveralllight,stableneutrinospecies(e.g.,νe,νμ,andντ),andtheHubbleconstantisH0=100hkms−1Mpc−1.Onethenarrivesattheimportantconclusionthatinorderforneutrinosnottocloseuniverse(i.e.Ων≤1),thesumofneutrinomassesmustnotexceed93h2eV.Thisvalueisfarbelowthecurrentlaboratorylimits(seeSec.1).Cowsik&McClelland4werethefirsttousesuchcosmologicalargumentstoplaceanupperboundonneutrinomasses.(Unfortunately,these“hotdarkmatter”modelsinwhichthemassdensityisdominatedbymassiveneutrinoshavebeenfoundtoproduceexcessivelargevoidssurroundedbylargecoherentsheetsandfilamentsthatarenotseenintheobservableuniverse.5Modificationstothismodelwillbediscussedbelow.)Inthehighmassregime,mν≫1MeV,thereexistsanotherwindowwheretheneutrinocontributiontothemassdensityparameterΩoftheuniverseissubcritical.Theargumentisthatneutrinoswithmν≫1MeVbecomenon-relativisticlongbeforedecoupling.Neutrinoandanti-neutrinopairsceasetobecreatedinabundanceoncethethermaltemperaturedropsbelowmν,andtheneutrinodensityissuppressedbytheBoltzmannfactore−mν/kBT.Thislargereductionfactorintherelicabundanceallowsneutrinostohavelargemasseswithoutoverclosingtheuniverse.Amorecarefulcalculation