Gravitational Lensing Constraints on Dynamical and

arXiv:0803.1640v2[astro-ph]9Apr2008GravitationalLensingConstraintsonDynamicalandCoupledDarkEnergyGLaVacca1,3andLPLColombo2,3,41DipartimentodiFisicaTeoricaeNucleare,Universit`adiPavia,viaA.Bassi,6I-27100Pavia,Italy2DipartimentodiFisica“GOcchialini”,Universit`adiMilano-Bicocca,PiazzadellaScienza,3I-20126Milano,Italy3INFNSezionediMilano-Bicocca4DepartmentofPhysics&Astronomy,UniversityofSouthernCalifornia,LosAngeles,CA90089-0484E-mail:[e-mail](WL)surveyscanbeusedtoconstrainDarkEnergy(DE)properties,namelyiftomographictechniquesareusedtoimprovetheirsensitivity.Inthiswork,weuseaFishermatrixtechniquetocomparethepowerofCMBanisotropyandpolarizationdatawithtomographicWLdata,inconstrainingDEparameters.AddingWLdatatoavailableCMBdataimprovesthedetectionofallcosmologicalparameters,buttheimpactisreallystrongwhenDE–DMcouplingisconsidered,asWLtomographycanthensucceedtoreducetheerrorsonsomeparametersbyfactors10.1.IntroductionThefirstdatasystemrequiringDarkEnergy(DE)concernedcosmicacceleration,detectedthroughhigh-redshiftsupernovae[1].CMB[2]anddeepsample[3]datasupportedalsotheDEcase,showingthatthedensityparameterfornon–relativisticmatterΩ0,m∼0.3,whilethetotaldensityparameterΩ0∼1.Inthemostpopularscenario,DEisascribedtoacosmologicalconstantΛ.Alternativeoptionsincludeaself–interactingscalarfield,φ(quintessenceordynamicalDE[4,5])andmodificationsofGeneralRelativity[6].ItisknownthatmodelswithΛ(ΛCDM)apparentlyaccommodateallavailabledatasystems.TheproblemisthephysicaloriginofΛ,whichcanbeafalsevacuum;thishowevercauseswellknownfinetuningandcoincidenceproblems.TheformerproblemispartiallyeasedindynamicalDE(dDE)scenarios,whenselfinteractionisduetoatrackingpotentialV(φ)[7].IfV(φ)isSUGRA[5],thefitwithdataisatleastasgoodasforΛCDM[8].Intheattempttoeasethecoincidenceproblem,DM–DEinteraction(e.g.,[9,10])wasalsoconsidered,yieldinganenergytransferbetweenthedarkcomponents,soGravitationalLensingConstraintsonDynamicalandCoupledDarkEnergy2allowinga(quasi)–parallelscalingofDMandDEfromafairlyhighredshiftuntilthepresent.WhilelaboratorydatasetnosignificantconstraintonDM–DEinteractionstrength,parametrizedbyβ(seebelow),recentworksplacedconstraintsonpossiblecouplings,byusingSNIadata[11]ortheredshiftevolutionoftheHubbleparameter,H[12].Accordingly,β0.12–0.15[13,14]ishardlyconsistentwithobservations.Unfortunately,suchalowcouplinglevelnolongereasesthecoincidenceproblem[15],but,oncethegeniehascomeoutfromthelamp,itishardtoputitbackinside.Thepointiswhetherlowvaluesofβ,asallowedbycurrentdata,caninterferewithfuturedataanalysis.Inparticular,whenweallowfornon–zeroβ,howdoerrorsonotherparametersbehave?InthisworkwetriedtoanswerthisquestionbyusingaFishermatrixtechnique.Weconsideredtwodifferentmodels,setbysimilarvaluesofcosmologicalparameters,withoutandwithcoupling.Inthelattercase,wetookβ=0.1.Startingfromthesemodels,weevaluatedtheexpectederrorsoncosmologicalparameters,asobtainedwhendataconcernjustCMBanisotropyandpolarizationorincludetomographicweaklensing(WL).Asamatteroffact,incoupledmodels,thetimeevolutionofthedarkcomponentsisnon–standard.IfsuchmodelsareconsideredinaNewtonianapproximation,itisasthoughDMparticleshadaφ–dependentmass.Alsoforquitelowβ’s,thisanomalousscalingleavesanimprintonboththeexpansionhistoryoftheUniverse,andthegrowthof(matter)fluctuations,atthelinearandnon–linearlevels(e.g.[16]).However,anydetectedevolutionofHcanbereproducedthroughasuitableredshiftdependenceofDEdensityρdeandstateparameterwφ,whenφapproachesmp(thePlanckmass).Ariskisthat,ifmatteranddarkenergyarecoupled,fittingobservationsleadstoanestimateofaphantomequationofstate(wφ−1),evenifwφ−1atallredshifts[17].Inprinciple,thisriskcanbeexcludediftheredshiftdependenceofthegrowthfactorG(z)isalsotested,throughtheincreaseinnumberandconcentrationofboundsystems.DataprovidinginformationbothonH(z)andG(z)arethereforeabletodiscriminatebetweencoupledanduncoupledmodels.Experiments,orcombinationsofexperiments,probingH(z)andG(z)arethenneeded.CMBdata,usedtoconstraincoupling[13,18],placeonlyupperlimitsonβ.TheanalysisofLy–αandthematterpowerspectrumofthe2dFandSDSSsurveys[14]doesnotleadtogreatimprovements.Attheavailablesensitivitylevel,suchdatasystemsprovidejustweightedintegralsofH(z)andG(z),whichremainconsistentwitharatherwidesetofoptions.Onthecontrary,gravitationallensing,aloneorincombinationwithCMBdata,wasalreadyshowntobeapowerfultoolfortheanalysisofDE.WLtomographyprobesthepowerspectrumP(k)atdifferentredshiftsandisthuswellsuitedtoconstrainG(z).Inthisworkweaimtoputtheseconceptualpointsonamorequantitativebasisandtodeepenthecaseofcoupling,byperformingaFisheranalysisoffutureWLsurveysandCMBexperiments.GravitationalLensingConstraintsonDynamicalandCoupledDarkEnergy3Theoutlineofthisworkisasfollow.InSec.2wereviewthebasicpropertiesanddefinitionsofdDEmodelsandWL,inSec.3weshowtheresultsoftheFisheranalysis,inSec.4wediscussthemandinSec.5wesummarizeourfindingsanddrawourconclusions.2.Modelsanddefinitions2.1.InteractingDarkEnergyWeconsideracosmologicalmodelwheretheDEfieldφinteractswiththecoldDMcompon