Constraining Cosmological Parameters by the Cosmic

arXiv:astro-ph/0504471v222Aug20051ConstrainingCosmologicalParametersbytheCosmicInversionMethodNoriyukiKogo1,2,∗),MisaoSasaki2,∗∗)andJun’ichiYokoyama3,∗∗∗)1DepartmentofEarthandSpaceScience,GraduateSchoolofScience,OsakaUniversity,Toyonaka560-0043,Japan2YukawaInstituteforTheoreticalPhysics,KyotoUniversity,Kyoto606-8502,Japan3ResearchCenterfortheEarlyUniverse,SchoolofScience,TheUnivresityofTokyo,Tokyo113-0033,JapanWeinvestigatethequestionofhowtightlywecanconstrainthecosmologicalparametersbyusingthe“cosmicinversion”methodinwhichwedirectlyreconstructthepowerspectrumofprimordialcurvatureperturbations,P(k),fromthetemperatureandpolarizationspectraofthecosmicmicrowavebackground(CMB).Inapreviouspaper,wesuggestedthatitmaybepossibletoconstrainthecosmologicalparametersusingthefactthatthereconstructedP(k)doesnotdependonhowmanypolarizationdataweincorporateinourinversionprocedureifandonlyifthecorrectvaluesofthecosmologicalparametersareused.TheadvantageofthisapproachisthatweneednoassumptionregardingthefunctionalformofP(k).Inthispaper,weestimatetypicalerrorsinthedeterminationofthecosmologicalparameterswhenourmethodisappliedtothePLANCKobservation.Weinvestigateconstraintsonh,Ωb,Ωm,andΩΛthroughMonteCarlosimulations.§1.IntroductionThecosmicmicrowavebackground(CMB)isoneofthemostpowerfultoolstodeterminethevaluesofthecosmologicalparametersandexaminethepropertiesoftheprimordialfluctuations.AnumberofobservationshavebeencarriedoutsincetheCosmicBackgroundExplorer(COBE)observation.1)Inparticular,arecentpreciseobservationmadeusingtheWilkinsonMicrowaveAnisotropyProbe(WMAP)hasconfirmedthatouruniverseisconsistentwithaspatiallyflatΛCDMuniversewithGaussian,adiabatic,andnearlyscale-invariantprimordialfluctuations,aspredictedbyasimpleslow-rollinflationmodel.2)–5)Nevertheless,ithasbeensuggestedthattheprimordialpowerspectrumofthecurvatureperturbation,P(k),mayhavesomenontrivialfeatures,suchasalackofpoweronlargescales,runningofthespectralindex,andoscillatorybehaviorofthespectrumonintermediatescales.Therefore,aconventionalparameter-fittingmethod,inwhichoneoftenassumesasimplefunctionalformofP(k),suchasapower-lawspectrum,isunsatisfactory.InsteadweneedtoexamineP(k)directlyfromobservationswithoutanytheoreticalprejudice.Forthispurpose,therehavebeenseveralattemptstoreconstructtheprimordialspectrumusingtheWMAPdata∗)E-mail:[e-mail]∗∗)E-mail:[e-mail]∗∗∗)E-mail:[e-mail],M.Sasaki,andJ.Yokoyamausingmodel-independentmethods.6)–12)CosmicinversionisamethodtoreconstructtheprimordialspectrumdirectlyfromCMBanisotropies.ItwasoriginallyproposedbyMatsumiyaetal.(2002,2003).13),14)WeappliedthismethodtotheWMAPfirst-yeardataandshowedthattherearepossiblenontrivialfeaturesofP(k).15)OurmethodcanreproducefinefeaturesofP(k)witharesolutionofΔk≃3.7×10−4Mpc−1,whichroughlycor-respondstoΔℓ≃5intheangularpowerspectrum,Cℓ.Inapreviouswork,weimprovedourmethodinsuchamannerthatwecanusetheauto-correlationsofbothCMBtemperaturefluctuations(TT)andE-modepolarization(EE).16)Withpolarization,wehaveshownthatlargenumericalerrorsinthereconstructedP(k)duetothesingularitiesintheinversionformula,whichcorrespondtozeropointsofthetransferfunctions,canbesuppressed.Asaresult,wewereabletoreconstructP(k)withanerrorofafewpercentintheidealsituationthatobservationalerrorsdonotexist.Wehavealsofoundthatitmaybepossibletoconstrainthecosmo-logicalparametersbyvaryingthecontributionofthepolarizationintheinversionformulaandrequiringthattheresultantP(k)isindependentofthecontributionofthepolarization.Inaconventionalparameter-fittingmethod,itappearsthatonecandeterminethecosmologicalparametersuptoafewpercentfromrecentWMAPdata.However,thisisbecausethefunctionalspaceofP(k)isrestrictedbytheas-sumptionofasimplefunctionalform.Asaresult,thesevaluesofthecosmologicalparametersdependonthisassumption.Ontheotherhand,ifweregardP(k)asafreefunctiontobereconstructed,thereremainsthedegeneracythatvaryingtheshapeofP(k)cancompensateforthevariationofthecosmologicalparameters.17),18)Therefore,itisimportanttoinvestigatehowwellwecandeterminethecosmologicalparameterswithoutanyassumptiononthefunctionalformofP(k).Inthispaper,weexaminethepropositionofdeterminingthecosmologicalpa-rametersbyusingthecosmicinversionmethod.Wefirstconfirmthatitispossibletodeterminethecosmologicalparametersquiteaccuratelywhenthereisnoobser-vationalerror.ThenweaddartificialobservationalerrorsassumingthePLANCKobservation∗)andestimateprobabilitydistributionsofthecosmologicalparametersbyperformingMonteCarlosimulationsforeachsetofthecosmologicalparameters.Thispaperisorganizedasfollows.In§2,wereviewourcosmicinversionmethodthatemploysboththeCMBtemperatureandpolarizationspectra.Wealsoextendittoanonflatuniverseandexaminetheeffectofobservationalerrorsonthere-constructedP(k).In§3,wedescribeanewmethodtoconstrainthecosmologicalparametersbyusingourcosmicinversionmethodandreporttheresultsofsimula-tionstoestimatetheerrorsonthecosmologicalparameters.Finally,wepresentourconclusionin§4.∗)=PLANCKConstraini