RevisitingSupernovaProjectdataMohamedElashriAugust8,2019AbstractInthisproject,wearegoingtoanalyzethesupernovadata-settoextractimportantphysicalquantitiesofgreatinterestwhichareHubble’sconstantandpercentageofmattervsdarkenergydensitiesintheuniverse.Wealsoinvestigatedifferentcosmologicalmodelstoseewhichmodelfitsbestthedata.Wearegoingtousedataanalysistoolsandfittechniquestostudythisdataandproducegraphsrepresentthephysicsbeyondourdataset.WehaveadatasetofsupernovasurveyprojectwhichcontainstheapparentmagnitudeandredshiftsofthelargesetofTypeIasupernovaswhichactedasstandardcandles.FromthesedatawerunstatisticalanalysistomeasuresomeoftheimportantcosmologicalparameterswhichareHubbleconstantanddensityofmatteranddarkenergyintheuniverse.Wetestdifferentmodeltofitthedataandthesignificanceoftheparametersmeasurementindifferentdataranges.1TheoreticalBackgroundSupernovatypeIaarestandardcandles(orcanbemadeso),socanbeusedtomeasurethecon-tentsoftheUniverse.Standardcandlesareaclassofastrophysicalobjects,suchassupernovaeorvariablestars,whichhaveknownluminosityduetosomecharacteristicqualitypossessedbytheentireclassofobjects.Incosmology,agivencosmologymodel,withagivensetofdensityparameters,σiwhicharethedensitiesofuniversecomponents,willgiveusafunctionDL(z).Thisfunctioniscalledlu-minositydistancewhichcanbewrittenbasedonsomephysicalreasoningintermsofred-shiftsasfollowsDL(z)=(1+z)x(1)TheideabehindusingSNdataisthatweknowfromlocalmeasurementsthattheluminosityofSNehaveawellknowntimedependence:Theyrise,reachamaximumluminosityLmax.thenfadebackagain.LmaxisfoundtobethesameforallSNe(uptosomecorrections,)whichmakeSNegoodcandidatestobeconsideredstandardcandles.Therefore,ifwemeasurethered-shiftzofaSN,thenwecanpredictitsfluxF=L4πD2L(z|σi)andcompareitwiththemeasuredflux.IftheymatchformanySNeinarangeofredshifts,thenweknowwehavechosentherightdensityparametersσi.Thequestionthenbecomestoquantifythephrase“iftheymatch.”Andthatisthe1
useofstatistics!Wewillusetheχ2statisticstoquantifyhowcloselyourdatamatchwithourmodel.Wearealsointerestedinestimatingthematteranddarkenergydensityparametersbasedonthisrealdata.alsowewanttomodeldataandseethebestfitmodel.2analysisanddiscussionInthisproject,usingsupernovaprojectdata[1]weIncorporatedthedataanalysistechniquesofminimizationofχ2totestmodelsandhowthisstatisticallyaffectsthephysicsofinterest.weobtainedthedatathenwetrytoreadininausefulform.wehavearangeofredshiftvaluesfrom0.01toaround1.5andamuchwiderrangefordistancemodulusm-M.Soweusedverticallogplotforconvenientrepresentationfordatawhichshowninplot1(witherrorbarsfromerrorestimationindistancemodulusincludedindata).Figure1:Plotofdistancemodulusvsred-shiftsforourdataNowwewanttotestourtheoreticalmodelsagainstthisdata,basedonthetheoreticalbackgrounddiscussionfortherelationbetweenmodulusdistanceandluminositydistancewhichhaveseveralmodelstocalculateintermsofredshiftz.wearetestthreedifferentmodelsinthiswork.2
D1=czH0D2=czH01+zD3=cz(1+z)H0pΩm(1+z)3ΩL(1+z)dzNowwehaveoneparametertoestimateinmodelone(Linearmodel)andmodeltwo(NonLinearmodel)whichistheHubble’sconstantH0andtwoparametersinModelthree(FRWmodel)whichisthematteranddarkenergydensitiesintheuniverseΩmandΩLrespectively.WewillusetheH0estimationfromlinearandnon-linearmodelswhenweareworkingonFRWmodel.Thereasonforthatisshowninfigure2,thisplotisafitplotforthethreemodelsagainstthedataandweseethatforsmallz,thethreemodelsfitthedataandworkingfinesowecangetestimationforH0fromthisrange.Tables1and2summarizethevaluesforourestimationbasedondifferentred-shiftsranges.Figure2:Plotofdistancemodulusvsred-shiftsforourdatawiththethreemodels3
Table1:Table1ForLinearmodelredshiftrangeH0χ2z<0.0568.3944.043z<0.0968.2884.005z<0.1567.5573.756z<0.266.9613.569z<0.2566.1883.353Table2:Table2ForNon-linearmodelredshiftrangeH0χ2z<0.0569.3221.051z<0.0969.4631.045z<0.1569.1191.031z<0.268.9731.018z<0.2568.9730.995Basedonthesevalues,weseethatthevalueofHubble’sconstantdecreasesforbothmodelswhenweincreasetherangeofredshiftthatweincludeinthecalculation.Wealsoseefromthethesecondfigurethatthebestplaceistherange0to0.05redshiftsowewillusethevalueforthisrangeinourestimation.WealsonoticewithincreasingztheHubble’sconstantinthelinearmodelbegintodecreasemorequicklythannon-linearone.Thisisconsistentbecauseweseefromtheplotthatthenon-linearmodelhasalargerrangeoffitfordatathanthelinearmodel.Sowhenwegoalittlebeyondthefirstfewpointswearegettingaworseestimationforbothmodelsbutthenon-linearmodelisbetterinthiscase.ButwecansticktothefirstfewpointsastheyrevealsthesamephysicsofHubble’sconstantastheyarefromnearbysupernovas.WeseethatLinearmodelhasaχ2decreasingmorethanthenon-linearmodel(whichhassmallervalueoverall)whichrevealsthefactthatthenon-linearisabetterfitfortheseranges.Thefigures3and4givesshowsthefitandHubble’sconstantvaluesforthelinearandnon-linearmodels.Wealsotrytousinginterpolationtocalculatetheχ2forthistwomodels.Althoughthatwedon’tseethelinearmodelofgreatinterestweexpecttogethighχ2whichistrueandwegetavalueof4.043attherange0to0.05z.4
Figure3:LinearModelFigure4:Non-linearModel5
NowweareinterestinginFRWmodelwhicharethegeneralequationofluminositydistanceinstandardcosmologytheory.Wearegoingtominimizetheχ2toestimatethedensityratioassumingthatweonlyhavematterwithdarkenergy(flatuniverse).SoourbasicassumptionisΩmL=1.SothisreducestheproblemtoestimatingonlyoneparameterwhichwillbeΩmandthenwecangetthevalueofΩLusingourconstrain.byinterpolatingourmodelanddefineχ2wearegoingtoplottheχ2perdegreeoffreedomagainstΩmtoseewhichvalueofthematterdensitycorrespondingtominimumχ2insteadofdoingitbyiterationandputitintable(wehavetoomanyvaluesactually).Doingthiswillgiveusfigure5.Figure5:Plotofχ2vsΩmTheminimumχ2valueis0.9768whichcanbereaddirectlyfromthisgraphortheplotsoftwarecanhelpreaditoff.ThisvaluecorrespondingtoΩmof0.3093andΩLof0.6907.Thesevaluesareconsistentwiththeothermeasurements[2,3]ThelastthinginthisanalysisisthatwewanttoplotacontourplotforΩm-ΩLplanewhichcontaintheχ2confidencelevel.Thisisimportantforphysicalinterpretation.thisputsthelimitsofthemodelsandundertheassumptionofbigbangtheorywithaflatuniversewecangetinformationaboutthestructureoftheuniverse.Thisshowsupinfigure6.6
Figure6:contourplotinΩm-ΩLplanefigure6showsthattheconfidenceintervalsintheΩm-ΩLplaneareconsistentwiththeotherworkforthesupernovaproject[4]andHigh-zsupernova[5]teams.WehadsomehowatighterconstrainsthantheHigh-zsupernovaproject.Butfromthisfigurewecanruleoutthepossibilityforthecaseofopenuniversewhere(Ωm,ΩL)=(1,0).becausewehadtheconfidencelevelsruleoutthis.Howeverweshouldmentionthatinthisanalysistherearemainsourcesofsystematicerrorthatweshouldconsiderwhendiscussinghowsignificancethisresult.WehadestimatedtheHubble’sconstantusingthelowzvalues.ButinrealitythereareintrinsicdifferencesbetweentypeIasupernovaathighandlowred-shifts.withtakingintoaccountthisweestimatedthatthiseffectdoesnotmakeasignificanceeffectaswesawintable2fornon-linearmodel.wehaveusedtheestimationforHubble’sconstantfromthismodelasithastheleastχ2changeacrosstheredshiftdifferentranges.Therearealsosomeproblemsthatneedmoretimetobedone,isthestatisticalstudyofcorrela-tionbetweenthepropertiesofhighandlowred-shiftssupernovaandourparameterestimation.Althoughwefindtheeffectissmallbutforalargersetofdataweneedtounderstandthiseffectmore.AlsowecanincorporatemodelsthatcontaincurvatureofuniversesothereisanotherΩkbutthisleadtoalessconstrainedconditionsforparameterestimationthatwecouldnothandleinthisshortperiodoftime.7
References[1]http://supernova.lbl.gov/union/[2]arXiv:1303.5062[astro-ph.CO][3]arXiv:1807.06209[astro-ph.CO][4]S.Perlmutteretal.[SupernovaCosmologyProjectCollaboration],Nature391,51(1998)[astro-ph/9712212][5]A.G.Riessetal.[Hi-ZSupernovaTeamCollaboration],Astron.Journ.116,1009(1998)[astro-ph/9805201].8