Constraints on Omega_B, Omega_m and h from MAXIMA and BOOMERANG

We analyse the BOOMERANG and MAXIMA results in the context of models with $\Omega_{\rm Total} = 1$ and $n_s = 1$. We attempt to constrain three other parameters---$h$, $\Omega_B$, and $\Omega_m$---from these observations. We show that: (a) the value of $\O

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aConstraintson B, mandhfromMAXIMAandBOOMERANGT.PadmanabhanInter-UniversityCentreforAstronomy&Astrophysics,PostBag4,Pune411007,Indiapaddy@iucaa.ernet.inandShivK.SethiMehtaResearchInstitute,ChhatnagRoad,Jhusi,Allahabad211019,Indiasethi@mri.ernet.inABSTRACTWeanalysetheBOOMERANGandMAXIMAresultsinthecontextofmodelswith Total=1andns=1.Weattempttoconstrainthreeotherparameters—h, B,and m—fromtheseobservations.Weshowthat:(a)thevalueof Bh2istoohightobecompatiblewithprimordialnucleosynthesisobservationsat95%con dencelevel(b)universewithagegreaterthan12Gyrisruledoutat95%con denceleveland(c)thevalueof mhistoohightobecompatiblewiththeshapeofthepowerspectrumofgravitationalclustering.Ine ect,ouranalysisshowsthatmodelswith Total=1andns=1areruledoutbyBOOMERANGandMAXIMAobservations.PrecisedeterminationofCMBRanisotropieshaslongbeenexpectedtogiveaccuratevaluesofcosmologicalparameters(seee.g.Bond,Efstathiou&Tegmark1997andreferencestherein).

ThesecosmologicalparametersincludeparametersofbackgroundFRWmodel( Total, Λ,h, B,etc.),parametersthatdeterminetheformationofstructureintheuniverse(σ8,scalarspectralindexns,etc.),andtheparametersrelatedtothere-ionizationoftheuniverse(theopticaldepthtothelastscatteringsurface,τetc.).

WhilethefutureexperimentsMAPandPlanck1,largelyowingtotheirall-skycoverage,areexpectedtodeterminemostoftheseparameterswithafewpercentaccuracy(Jungemanetal.1996,Zaldarriagaetal.1997,Prunetetal.1999),recentobservationshavealreadybeguntogiveimportantcluesaboutsomeoftheseparameters(Milleretal.1999,Mauskopfetal.2000,Netter ledetal.1997,forasummaryofobservationupto1998andparameterestimationfrom

We analyse the BOOMERANG and MAXIMA results in the context of models with $\Omega_{\rm Total} = 1$ and $n_s = 1$. We attempt to constrain three other parameters---$h$, $\Omega_B$, and $\Omega_m$---from these observations. We show that: (a) the value of $\O

theseobservations,seeLineweaver&Barbosa1998).RecentballoonexperimentsBOOMERANGandMAXIMAreportedCMBRanisotropymeasurementsatangularscalesbetween 10 and 10′(deBernardisetal.2000,Hananyetal.2000).Theseexperimentsobservednearly1%oftheskywithangularresolution 10′.Forboththeseexperimentsthecosmicvariancewassmallenough(owingtotheskycoverage)todeterminepreciselythepositionofthe rstDopplerpeak( 200)oftheCMBRanisotropies.BothBOOMERANGandMAXIMAresultsgavestrongevidencethat Total=1(deBernardisetal.2000,Hananyetal.2000),whichwasalreadyindicatedbyotherobservations(Netter eldetal.1997).

Whilethepositionofthe rstDopplerpeakgiveunambiguousevidenceaboutthegeometryoftheuniverse,determinationofothercosmologicalparametersismoredi cult.Thisisbecausevariationinseveraldi erentparametersgivethesamechangeinmeasuredanisotropies,e.g.theheightof rstDopplerpeakisnearlydegeneratein B,h, Λandns.Someofthisdegeneracycanbeliftedwiththemeasurementofanisotropiesatevensmallerangularscales.BOOMERANGandMAXIMAprobewithangularscalescorrespondingtomultipoles max {600,700},respectively,whichisuptoorbeyondtheexpectedpositionofthesecondDopplerpeak.Thoughtheresultsoftheseexperimentshavenotbeenableto ndthepositionofthesecondDopplerpeak,accuratemeasurementofanisotropiesatsuchangularscalesisexpectedtobreaksomeofthedegeneracywhichmeasurementsnearthe rstDopplerpeakalonecannot.

TheBOOMERANGandMAXIMAdatahavebeenusedtodeterminevariouscosmologicalparameters(Balbietal.2000,Ja eetal.2000,Langeetal.2000,Tegmark&Zaldarriaga2000).Combinedwithotherindependentmeasurementsofcosmologicalparameters(e.g.measurementof Bh2fromelementabundance,measurementofhfromnearbyobservationsorinferenceaboutthevaluesof Λand mfromtheSN1adata,etc.)thesedataareexpectedtoleadtoauniquepicture.However,owingtodegeneraciesinparameterestimation,thevalueofestimatedparametersandtheirerrorsdependsensitivelyonvariousassumptionrelatedtotheassumedallowedrangeofparameters,i.e.onthepriorsontheparameters.

Inthisletter,weperformalikelihoodanalysisontheband-powersreportedbytheMAXIMAandBOOMERANGexperiments.However,insteadofofdoingamultipleparameteranalysis,we xthevaluesofmostparametersfromotherconsiderations(andprejudices!)andthenattempttoestimatejustthreeparameters— B,h,and massumingweakpriorsontheirallowedvalues.Inthenextsection,weexplainourchoiceofparametersandthemethodweuseinbrief.Inthethirdsection,wepresentandsummariseourresults.

1.CosmologicalparametersandCMBRdata

Mostgenericmodelsofin ationaryscenariogivetwouniquepredictions: Total=1andns 1(seee.g.Steinhardt1995).The rstofthesepredicationsiscon rmedbyBOOMERANGandMAXIMA.TheanalysisofCOBE-DMRdataisconsistentwithns=1(Bennettetal1996,Bunn

We analyse the BOOMERANG and MAXIMA results in the context of models with $\Omega_{\rm Total} = 1$ and $n_s = 1$. We attempt to constrain three other parameters---$h$, $\Omega_B$, and $\Omega_m$---from these observations. We show that: (a) the value of $\O

&White1996).Thereforeitisreasonabletobelievethatthecurrentdataisingoodagreementwiththesepredictions.We xthevalueofthesetwoparametersbasedontheseconsiderationsanduse Total=1andns 1throughout.(Itshouldbepointedoutthat Total=1isastricterrequirementofin ationthanns=1;onegetsns=1onlyforexponentialin ation;seee.g.Steinhardt1995.)Alsonotethatwearenotconcernedwiththeoriginofthevalues Total=1andns=1.Forexample,thesecouldarisemerelyfromtherequirementofscaleinvarianceforthebackgrounduniverse(giving Total=1)andtheperturbations(givingns=1)withoutinvokingin ation—aswasoriginallydonebyHarrisonandZeldovich,yearsbeforein ationwasinvented(Harrison1970,Zeldovich1972).But,ofcourse,in ationarymodelsmadetheseparametervaluesfairlywellaccepted.Otherparameterslike Λ,hand Bcannotbe xedbytheoreticalconsiderationsalone.Inouranalysisweassumeanon-zero Λbecauserecenthigh-zSN1aobservationssuggestanon-zerocosmologicalconstant(Perlmutteretal.1999,Riessetal.1998).WedonotconsiderCMBRanisotropiesfromtensorperturbationsasmostmodelsofin ationgivenegligiblecontributionfromtensorperturbationsforns=1(Steinhardt1995).Re-ionizationofuniversecanalsoalterprimaryCMBRanisotropiessigni cantly.Presentobservationssuggestthattheuniverseisionizeduptoz 5.FortheCMBRanisotropiestobesigni cantlyalteredbyre-ionization,theminimumre-ionizationredshiftshouldliebetween10and20(seee.g.Bond1996).Therefore,keepingtheconstraintsfrompresentobservationsinmind,weneglectthee ectofre-ionizationonthemeasuredCMBRanisotropies.

We xthevalueof m+ Λ= Total=1,andcomputethecon dencelevelsonthebest- tvaluesof Bandh,and mfromMAXIMAandBOOMERANGobservations.Therangeofparametersinwhichtheminimumofχ2issearchedis:0.01≤ B≤0.15,0.4≤h≤1.1and0.1≤ m≤0.95.

Theχ2forthemodelcomparisonwithobservationsisgivenby:

χ2=N obs C Cth

i=1

We analyse the BOOMERANG and MAXIMA results in the context of models with $\Omega_{\rm Total} = 1$ and $n_s = 1$. We attempt to constrain three other parameters---$h$, $\Omega_B$, and $\Omega_m$---from these observations. We show that: (a) the value of $\O

insteadofband-powersfordoingthemaximumlikelihoodanalysis.Wedonotuseithere.However,whilequotingerrorswedonotusetheFishermatrixapproachwhichcangivemeaningfulresultsonlyfortheGaussiancase.Insteadwedirectlygivethecon dencelevelson χ2.

2.Results

OurresultsareshowninFigures1and2.Theχ2for22datapointsfromBOOMERANGandMAXIMAwiththree ttedparameters(i.e.19degreesoffreedom)is22.4,whichisanexcellent t

.003(Goodness-of- tprobability,Q=0.26).Thebest tvaluesand1σerrorsare: B=0.075+0

0.015,

.08+0.04h=0.62+0

0.02and m=0.86 0.21.Thebest tmodelalongwithBOOMERANGandMAXIMAdatapointsisshowninFigure3.

Therangeofallowedhisinfairagreementwiththemeasurementofhfromlocalobservations(forarecentsummaryoftheseresults,seePrimack2000).RecentSN1asuggestthatthatthe

.09+0.052universeis atwith Λ+ m=1with m=0.28+0

0.08 0.04(1σ).Thisiswithin χ=2ofthe

valueinferredbyouranalysis.

InFigure1weshowthecon dencelevelsinthe B–hplane.Theregionboundedbythecontourscorrespondto χ2≤2.3and χ2≤6.17,which,forGaussianerrors,correspondsto68%(1σ)and95.4%(2σ)fortwo-parameter ts.Wealsoshowthe95%regionallowedbyprimordialnucleosynthesisobservations(forarecentreviewseeTytleretal2000).Asseeninthe gure,theregionallowedbyCMBRobservationsisatvariancewiththepredictionsofprimordialnucleosynthesisatatleast95%level.Thisresultisinagreementwithotheranalysesonthesamedataset(Tegmark&Zaldarriaga2000,Ja eetal2000).

Theregioncorrespondingtooneandtwoσinthe m–hplaneisshowninFigure2.OurresultsshowthatthecurrentCMBRobservationsfavourauniversewithage≤11Gyrandareincompatiblewithauniverseofage≥12Gyrat 95%level.Thoughthisisonthelowersideoftheexpectedageoftheuniversefromestimatedagesofglobularclusters,etc.,thisisnotindisagreementwiththoseobservations(forarecentstatusreportseePrimack2000).Anotherimportantconstraintonthevaluesof mandhcomesfromtheshapeofthepowerspectrumofgalaxyclustering(seee.g.Bond1996).Theseobservations,withinthecontextofa atmodelwithcosmologicalconstant,give mh 0.25±0.05.Weplotthisregioninthe m–hplaneinFigure2.Ifwerequirethath≤0.75,asmostrecentmeasurementsofhsuggest(Primack2000),thenCMBRobservationsexcludetheregionrequiredtosatisfythegalaxyclusteringobservationsbymorethan95%con dencelevel.

Inconclusion,recentBOOMERANGandMAXIMAobservations,withinthecontextofsim-plestin ationarymodels( Total=1andns 1),imply:(1)toolargeaofvalueof Bh2tobecompatiblewithprimordialnucleosynthesisobservations,(2)anageofuniverset0≤12Gyr,and

(3)toolargeavalueof mhtobeinagreementwiththeshapeofthepowerspectrumofgalaxyclustering.Inviewofthis,itissafetoconcludethatthemodelswith Total=1andns=1areruledoutbytheseobservations.

We analyse the BOOMERANG and MAXIMA results in the context of models with $\Omega_{\rm Total} = 1$ and $n_s = 1$. We attempt to constrain three other parameters---$h$, $\Omega_B$, and $\Omega_m$---from these observations. We show that: (a) the value of $\O

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We analyse the BOOMERANG and MAXIMA results in the context of models with $\Omega_{\rm Total} = 1$ and $n_s = 1$. We attempt to constrain three other parameters---$h$, $\Omega_B$, and $\Omega_m$---from these observations. We show that: (a) the value of $\O

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We analyse the BOOMERANG and MAXIMA results in the context of models with $\Omega_{\rm Total} = 1$ and $n_s = 1$. We attempt to constrain three other parameters---$h$, $\Omega_B$, and $\Omega_m$---from these observations. We show that: (a) the value of $\O

Fig.1.—Thecontourscorrespondtoallowed1and2σregionsbyCMBRobservations(seetextfordetail).Thecross-hatchedregioncorrespondtothe95%region( 2σ)fromprimordialnucleosynthesis(Tytleretal.2000).

We analyse the BOOMERANG and MAXIMA results in the context of models with $\Omega_{\rm Total} = 1$ and $n_s = 1$. We attempt to constrain three other parameters---$h$, $\Omega_B$, and $\Omega_m$---from these observations. We show that: (a) the value of $\O

Fig.2.—Thecontourscorrespondtoallowed1and2σregionsbyCMBRobservations(seetextfordetail).Thehatchedregioncorrespondsto mh=0.25±0.05.Othercurvesshowtheageoftheuniverset0(valueindicatedinthe gurelegend)

We analyse the BOOMERANG and MAXIMA results in the context of models with $\Omega_{\rm Total} = 1$ and $n_s = 1$. We attempt to constrain three other parameters---$h$, $\Omega_B$, and $\Omega_m$---from these observations. We show that: (a) the value of $\O

Fig.3.—Thebest tmodelisplottedalongwithBOOMERANGandMAXIMAdatapoints,whichcorrespondtoemptyand lledpolygons,respectively.

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