The Coincidence Problem and Interacting Holographi

arXiv:0801.1755v3[astro-ph]14May2008ThecoincidenceproblemandinteractingholographicdarkenergyKhampheeKarwanTheoreticalHigh-EnergyPhysicsandCosmologyGroup,DepartmentofPhysics,ChulalongkornUniversity,Bangkok10330,ThailandAbstract.WestudythedynamicalbehaviouroftheinteractingholographicdarkenergymodelwhoseinteractiontermisQ=3H(λdρd+λcρc),whereρdandρcaretheenergydensityofdarkenergyandCDMrespectively.TosatisfytheobservationalconstraintsfromSNIa,CMBshiftparameterandBAOmeasurement,ifλc=λdorλd,λc0,thecosmicevolutionwillonlyreachtheattractorinthefutureandtheratioρc/ρdcannotbeslowlyvaryingatpresent.Sincethecosmicattractorcanbereachedinthefutureevenwhenthepresentvaluesofthecosmologicalparametersdonotsatisfytheobservationalconstraints,thecoincidenceproblemisnotreallyalleviatedinthiscase.However,ifλc6=λdandtheyareallowedtobenegative,theratioρc/ρdcanbeslowlyvaryingatpresentandthecosmicattractorcanbereachednearthepresentepoch.Hence,thealleviationofthecoincidenceproblemisattainableinthiscase.ThealleviationofcoincidenceprobleminthiscaseisstillattainablewhenconfrontingthismodeltoSDSSdata.Keywords:DarkEnergy.Thecoincidenceproblemandinteractingholographicdarkenergy21.IntroductionObservationssuggestthattheexpansionoftheuniverseisaccelerating[1,2].Theaccelerationoftheuniversemaybeexplainedbysupposingthatthepresentuniverseisdominatedbyamysteriousformofenergywhosepressureisnegative,knownasdarkenergy.Oneproblemofthedarkenergymodelisthecoincidenceproblem,whichistheproblemwhythedarkenergydensityandmatterdensityareofthesameorderofmagnitudeinthepresentepochalthoughtheydifferentlyevolveduringtheexpansionoftheuniverse.Apossiblewaytoalleviatethecoincidenceproblemistosupposethatthereisaninteractionbetweenmatteranddarkenergy.Thecosmiccoincidencecanthenbealleviatedbyappropriatechoiceoftheformoftheinteractionbetweenmatteranddarkenergyleadingtoanearlyconstantratior=ρc/ρdduringthepresentepoch[3,4,5]orgivingrisetoattractorofthecosmicevolutionatlatetime[6,7].Sincetheexistenceofthecosmicattractorimpliesconstantrbuttheattractordoesnotalwaysoccuratthepresentepoch,wefirstfindarangeofdarkenergyparametersforwhichtheattractoroccursandthenchecktheevolutionofrduringthepresentepoch.Basedonholographicideas[8,9],onecandeterminethedarkenergydensityintermsofthehorizonradiusoftheuniverse.Thistypeofdarkenergyisholographicdarkenergy[10]-[14].BychoosingHubbleradiusasthecosmologicalhorizon,thepresentamountofdarkenergydensityagreeswithobservations.Nevertheless,darkenergyevolveslikematteratpresent,soitcannotleadtoanacceleratedexpansion.However,iftheparticlehorizonischosentobethecosmologicalhorizon,theequationofstateparameterofdarkenergycanbecomenegativebutnotnegativeenoughtodriveanacceleratinguniverse.Thesituationisbetterwhenoneusestheeventhorizonasthecosmologicalhorizon.Inthiscase,darkenergycandrivethepresentacceleratedexpansion,andthecoincidenceproblemcanberesolvedbyassuminganappropriatenumberofe-foldingsofinflation.Roughlyspeaking,thecoincidenceproblemcanberesolvedbecausethesizeofthecosmologicalhorizonduringthepresentepochdependsontheamountofe-foldsofinflation,andtheamountofholographicdarkenergydependsonthehorizonsize.Nevertheless,thesecondlawofthermodynamicswillbeviolatedifwd−1[9,15].Hence,wdshouldnotcrosstheboundarywd=−1.Theboundarywd=−1canbecrossedifdarkenergyinteractswithmatter.Sincenowthehorizonsizehasadependenceontheinteractionterms,thealleviationofcosmiccoincidenceshouldalsodependontheinteractionterm.Inthisworkwesupposethattheholographicdarkenergyinteractonlywithcolddarkmatter(CDM)andtreatbaryonsasnon-interactingmattercomponent.OurobjectiveistocomparetheregionofdarkenergyparametersforwhichthecosmicevolutionhasanattractorwithintheparameterregionthatsatifiestheobservationalconstraintsfromcombinedanalysisofSNIadata[16],CMBshiftparameter[17]andBAOmeasurement[18].Theresultsofthecomparisoncantellusabouttherangeofparametersthatalleviatethecosmiccoincidence.Thecoincidenceproblemandinteractingholographicdarkenergy32.TheautonomousequationsInthissection,wederivethefirstorderdifferentialequationsthatdescribetheevolutionofradiation,baryon,CDManddarkenergydensitiesintheuniverse.Byanalyzingtheseequations,onecanestimatetheasymptoticevolutionoftheuniverse.Toproceed,westartfromtheFriedmannequationH2+Ka2=13¯mp(ρr+ρb+ρc+ρd),(1)whereHistheHubbleparameterandthesubscriptsr,b,canddcorrespondtotheradiation,baryons,CDManddarkenergyrespectively.TheparameterKdenotesthecurvatureoftheuniverse,whereK=−1,0,+1fortheclose,flatandopenuniverserespectively.TheaboveequationcanbewrittenintermsofthedensityparametersΩK=K/(a2H2)andΩα=ρα/(3¯m2pH2)as1+ΩK=Xα=r,b,c,dΩα=Ωr+Ωb+Ωc+Ωd.(2)Theindexαrunsoverthe4species,namelyradiation,baryons,CDManddarkenergy.WenowderivetheautonomousequationsforthedynamicalvariablesΩKandΩα.DifferentiatingΩK=K/(a2H2)withrespecttolna,wegetΩ′K=−2KH˙aa3H2+˙Ha2H3!=−2ΩK1+˙HH2!,(3)whereprimeanddotdenotederivativewithrespecttolnaandtimerespectively.Fromthedefinitionofthedensityparameter,onecanshowthatΩ′α=Ωα˙ραHρα−2˙HH2!.(4)Tostudytheevolutionoftheuniverseatlatetime,wewillsearchforthefixedpointso