Rates and delay times of Type Ia supernovae in the Dark Energy Survey

Rates and delay times of Type Ia superno v ae in the Dark Energy Survey

P. Wiseman ,

^1‹

M. Sulli v an ,

¹

M. Smith,

^1,2

C. Frohmaier ,

^1,3

M. Vincenzi ,

³

O. Graur ,

³

B. Popovic,

⁴

P. Armstrong,

⁵

D. Brout,

⁶

† T. M. Davis ,

⁷

L. Galbany,

⁸

S. R. Hinton,

⁷

L. Kelsey,

¹

R. Kessler,

^9,10

C. Lidman,

^5,11

A. M ¨oller,

¹²

R. C. Nichol,

³

B. Rose,

⁴

D. Scolnic ,

⁴

M. Toy,

¹

Z. Zontou,

¹

J. Asorey,

¹³

D. Carollo,

¹⁴

K. Glazebrook,

¹⁵

G. F. Lewis,

¹⁶

B. E. Tucker,

⁵

T. M. C. Abbott,

¹⁷

M. Aguena,

¹⁸

S. Allam,

¹⁹

F. Andrade-Oliveira,

^18,20

J. Annis,

¹⁹

D. Bacon,

³

E. Bertin,

^21,22

D. Brooks,

²³

E. Buckley-Geer,

^9,19

D. L. Burke,

^24,25

A. Carnero Rosell,

^18,26,27

M. Carrasco Kind,

^28,29

J. Carretero,

³⁰

M. Costanzi,

^31,32,33

L. N. da Costa,

^18,34

M. E. S. Pereira,

³⁵

S. Desai,

³⁶

H. T. Diehl,

¹⁹

P. Doel,

²³

S. Everett,

³⁷

I. Ferrero,

³⁸

B. Flaugher,

¹⁹

P. Fosalba,

^39,40

J. Frieman,

^10,19

J. Garc ´ıa-Bellido,

⁴¹

E. Gaztanaga,

^39,40

T. Giannantonio,

^42,43

D. Gruen,

^24,25,44

R. A. Gruendl,

^28,29

J. Gschwend,

^18,34

G. Gutierrez,

¹⁹

D. L. Hollowood,

³⁷

K. Honscheid,

^45,46

B. Hoyle,

^47,48

D. J. James,

⁶

E. Krause,

⁴⁹

K. Kuehn,

^50,51

N. Kuropatkin,

¹⁹

M. A. G. Maia,

^14,34

J. L. Marshall,

⁵²

P. Martini,

^45,53,54

F. Menanteau,

^28,29

R. Miquel,

^30,55

R. Morgan,

⁵⁶

R. L. C. Ogando,

^18,34

A. Palmese,

^10,19

F. Paz-Chinch ´on,

^28,42

D. Petravick,

²⁸

A. Pieres,

^18,34

A. A. Plazas Malag ´on,

⁵⁷

A. K. Romer,

⁵⁸

E. Sanchez,

¹³

V. Scarpine,

¹⁹

M. Schubnell,

³⁵

S. Serrano,

^39,40

I. Sevilla-Noarbe,

¹³

M. Soares-Santos,

³⁵

E. Suchyta,

⁵⁹

M. E. C. Swanson,

²⁸

G. Tarle,

³⁷

D. Thomas,

³

C. To,

^24,25,44

T. N. Varga,

^48,60

and A. R. Walker

¹⁷

(DES Collaboration)

Affiliationsarelistedattheendofthepaper

Accepted2021July2.Received2021June29;inoriginalform2021May25

ABSTRACT

Weuseasampleof809photometricallyclassifiedTypeIasupernovae(SNeIa)discoveredbytheDarkEnergySurvey(DES) alongwith40415fieldgalaxiestocalculatetherateofSNeIapergalaxyintheredshiftrange0.2<z<0.6.Werecoverthe knowncorrelationbetweenSNIarateandgalaxystellarmassacrossabroadrangeofscales8.5≤log(M_∗/M)≤11.25.We findthattheSNIarateincreaseswithstellarmassasapowerlawwithindex0.63±0.02,whichisconsistentwiththeprevious work.Weuseanempiricalmodelofstellarmassassemblytoestimatetheaveragestarformationhistories(SFHs)ofgalaxies acrossthestellarmassrangeofourmeasurement.CombiningthemodelledSFHswiththeSNIaratestoestimateconstraints onthe SN Iadelay timedistribution(DTD), we find that the dataare fit well byapower-law DTDwithslope indexβ =

−1.13±0.05andnormalizationA=2.11±0.05×10⁻¹³SNeM⁻¹yr⁻¹,whichcorrespondstoanoverallSNIaproduction efficiencyN_Ia/M_∗ =0.9⁺₋⁴₀^._.⁰₇×10⁻³SNeM⁻¹ .UponsplittingtheSNsamplebypropertiesofthelightcurves,wefindastrong dependenceonDTDslopewiththeSNdeclinerate,withslower-decliningSNeexhibitingasteeperDTDslope.Weinterpret thisasaresultofarelationshipbetweenintrinsicluminosityandprogenitorage,andexploretheimplicationsoftheresultinthe contextofSNIaprogenitors.

Keywords: supernovae:general– whitedwarfs– galaxies:evolution.

1 I N T R O D U C T I O N

TypeIa supernovae(SNeIa)are explosions ofwhitedwarf stars (WDs).AlthoughSNeIashowdiversityintheirobservedproperties, a largefraction of them (‘non-peculiar’ SNe Ia) display a small dispersionintheirpeakbrightnesseswhichcanbereducedfurther through empirical relations between brightness and light-curve

E-mail:p.s.wiseman@soton.ac.uk

†NASAEinsteinFellow.

properties,suchasdeclinerate(stretch)oropticalcolour(Rust1974; Pskovskii1977;Phillips1993;Tripp1998).Thesepropertieshave ledSNeIatobeusedextensivelybycosmologiststomeasurerelative distancesintheUniverse(Riessetal.1998;Perlmutteretal.1999).

DespiteusingtheserelationstocorrectSNIapeakbrightnessesto withinadispersionof∼0.1maginsampleswithoveronethousand SNe (Scolnicet al.2018), the exactnature ofthe progenitors of SNe Ia isyet to beconfirmed.Whileit islikely that SNeIa are causedbymasstransferontoaWDfrom,orviolentmergerwith, acompanionstar,multiplepossiblescenariosexistforthenatureof thatcompanionstar(seeMaoz,Mannucci&Nelemans2014;Ruiter

© 2021TheAuthor(s).

PublishedbyOxfordUniversityPressonbehalfofRoyalAstronomicalSociety.ThisisanOpenAccessarticledistributedunderthetermsoftheCreative

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2020forreviews).Theleadingmodelsinvolveamain-sequence(MS) starinthesingle-degenerate(SD)scenario(Whelan&Iben1973; Nomoto1982),asecondarywhitedwarfin thedouble-degenerate (DD)scenario(Tutukov&Iungelson1976;Iben&Tutukov1984; Webbink1984),ormoreexoticmodelssuchasthecore-degenerate (CD)scenariowhichinvokeWDsmergingwiththecoresofmassive stars(Ilkov&Soker2011;Kashi&Soker2011)

Theprogenitorsofseveralcore-collapseSNe(CCSNe)havebeen identified in high-resolution pre-explosion images (Smartt 2009; Eldridgeetal.2013),allowingdetailedanalysisofthelatterstages ofmassivestarevolution.However,thisapproachhasnotyetbeen successfulinidentifyingaprogenitorofaSNIa(e.g.Graur,Maoz

&Shara2014a;Kellyetal.2014;Graur&Woods2019,although seeMcCullyetal.(2014)forevidenceofaprogenitorforaTypeIax SN).Conversely,therehavebeensearchesforsurviving remnants of the binary system – main sequence stars or WDs leftbehind attheSNlocationorkickedoutathighvelocity(e.g.Schaefer&

Pagnotta2012;Ruiz-Lapuenteetal.2018;Kerzendorfetal.2018, 2019).PromisingrecentresultsincludethatfromShenetal.(2018) who discovered three high-velocityWDs consistentwith models ofdouble-degenerate,double-detonationSNeIa.However,without an unambiguous observation of an SN Ia progenitor system or remnant companion, there is as yet no direct evidence that any progenitorchannelfromprovidesasignificantcontribution tothe totalpopulationofSNeIa.

Anindirectmethodofinferringprogenitorchannelsisthestudy oftheSNIadelaytimedistribution(DTD).TheDTDdescribesthe rateatwhichSNeIaoccurasafunctionofthedelaytimeτsincean episodeofstarformation,andthuscarriescharacteristicsignatures oftheprogenitorchannel(seeWang& Han2012forareviewof thetheory,andMaoz&Graur2017forareviewofobservations).

TheSDscenarioproducesabroadrangeoffunctionalformsforthe DTD,mostofwhichfailtoaccountforlongdelaytimes(Grauretal.

2014b).Ontheotherhand,mostvariantsoftheDDscenariopredict apowerlawoftheformτ^β,withβ∼ −1(e.g.Ruiter,Belczynski

&Fryer2009;Mennekensetal.2010),althoughseee.g.Yungelson

&Kuranov(2017)forDDmodelsthatdeviatefroma−1power-law slope.

Asastatisticaldistributionactingontime-scalesfromtensofMyr toseveralGyrafteranepochofstarformation,theDTDisnon-trivial tomeasureandvarioustechniqueshavebeendevelopedtoinferit indirectly.OnesuchapproachistomeasureglobalpropertiesofSNIa hostgalaxies.Simpleexamplesofsuchanalysesarecomparisonsof theratesofSNeIainhoststhathavebeensplitbysomeobservational property,suchasmorphologyorcolour,andhaveshownthatSNIa rateperunitstellarmass(SNuM)issignificantlylargerinlate-type spiral galaxiesas well as in bluer galaxies(e.g. Mannucci etal.

2005).TheseobservationswereinterpretedasshowingthatSNeIa arestronglyinfluencedbyrecentorongoingstarformation,andthus thatthemajorityofSNeIaexplodeaftershortdelaytimes.Onthe otherhand,theSNuMinE/S0andredgalaxiesisalsonon-negligible, suggestingthatthereisasecondary,mucholdercomponenttothe DTD(Sullivanetal.2006;Lietal.2011;Smithetal.2012).The DTDwasthusapproximatedtofirstorderbyatwo-component(A +B)model,whereAandBarethenormalizationsoftheDTDfor

‘prompt’(i.e.proportionalto instantaneousSFR) and‘tardy’(i.e.

proportionaltooverallstellarmass)SNe,respectively.

Acomplementarytechniqueinvolvesmeasuringtheevolutionof thevolumetricrateofSNeIaasafunctionofredshiftandcomparing thisevolutiontotheaveragecosmicstarformationhistory(CSFH) oftheUniverse(Dahlenetal.2004;Gal-Yam&Maoz2004;Strolger etal.2004;Dahlen,Strolger&Riess2008;Grauretal.2011;Graur

& Maoz 2013;Rodneyetal. 2014;Frohmaier etal.2019). This technique(knownasthevolumetricratemethod)isalsoapplicable togalaxyclusters,forwhichitisassumedthattheSFHsarestrongly peakedatsomepastepochandthusthattherateofSNeIainclusters asafunctionofredshiftisamoredirectmeasureoftheDTD(Maoz

& Gal-Yam2004;Maoz,Sharon& Gal-Yam2010;Friedmann&

Maoz2018;Freundlich&Maoz2021).Thesestudieshave,almost ubiquitously,foundβtobeconsistentwith−1.

Insteadofcomparingvolumetricrates tothe cosmic SFH,it is alsopossibletoestimatetheSFHofindividualgalaxiesthroughthe modellingoftheirstellarpopulationsviaspectralenergydistribution (SED)fitting.WorkssuchasTotanietal.(2008),Maozetal.(2011), Maoz,Mannucci&Brandt(2012),Graur&Maoz(2013),andGraur, Bianco&Modjaz(2015)estimatedtheDTDbymeasuringSFHsfor asampleoffieldgalaxiesandcomparingthemtothenumberofSNe detectedineachgalaxy(theSFHRmethod),andhaveledtoresults thatsuggestaDTDpowerlawwithβconsistentwith−1.

Childress,Wolf&Zahid(2014,hereafterC14)showedthattheA +Bapproximationarisesasadirectconsequenceofthecombination ofapower-lawDTDwiththeaverageSFHsofgalaxies– theprompt componentproportionaltotheamountofongoingstarformationat theepochofobservation,andthetardycomponentcausedbythefact thatmassivegalaxiesexperiencedhighSFRsseveralGyrago.Recent advancesinintegralfieldspectroscopy(IFS)allowforanextension of the SFHRmethod,by reconstructingthe SFHforhundredsor thousands oflocal regionsof eachSNIa host galaxy. Usingthis method,Castrilloetal.(2020)findapower-lawslopeof−1.1±0.3, whileChen,Hu&Wang(2021)find−1.4±0.3.

ThemajorityofobservationalevidencebasedonstudiesofSNIa populationsthuspointstowardsaDDscenarioformost,ifnotall, SNeIa.However,findingself-consistentprogenitorandexplosion models that recreatethe observed luminosityfunction aswell as correlations betweenluminosity, light-curveparameters, and host galaxypropertieshasprovendifficult.Inparticular,simulationsbased aroundexplosionsofMChWDs(linkedstronglywiththeSDscenario butalsowithmanyDDscenarios)finddifficultyinreproducingthe lightcurvesof‘normal’SNeIaaswellas‘peculiar’objects(Ropke etal.2007;Simetal.2013;seeMaozetal.2014;Blondinetal.

2017;Jha,Maguire&Sullivan2019foroverviews).Inrecentyears, attentiondirectedtowardsexplosionsofsub-M_Ch WDstriggeredby doubledetonations(primarilyrelatedtoaDDscenario)hasledto promisingresults(e.g.Shen,Toonen&Graur2017;Shenetal.2018; Townsleyetal.2019;Gronowetal.2020;Shenetal.2021)although theystillstruggletomatchobservationsatlatetimesinthelight-curve evolution(Gronowetal.2021;Shenetal.2021).Anadditionalfactor insupportofthesub-MChmodelisthattheSNluminosityisrelated tothemassoftheprimaryWD,whichitselfislikelytoberelated to itsage(althoughthisrelationisprobablycomplicatedbyother factorssuchasaccretionrate,metallicity,andthecompositionofthe companion),therebyprovidinganexplanationforobservedrelation betweenlight-curvestretchandstellarage(Rigaultetal.2013,2020; Rose,Garnavich&Berg2019;Nicolasetal.2021).Otherproposed scenariosincludehybridmodelsinwhichstandardCOWDsmerge withhybridhelium-COWDs(Zenati,Toonen&Perets2019).With many models showing promising similaritiesto observationsbut each subject to its own drawbacks,it isbecoming accepted that morethanone progenitorscenariomaycontribute significantlyto theoverallpopulationof‘normal’SNeIa;detailedobservationsare thusrequiredinordertoplaceconstraintsontherelativefractionsof eachpossibleprogenitorchannel.

In thiswork, wecombineparts of thetraditional methodsand deriveanewmeasurementoftheSNIaDTD.Insteadofmeasuring

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thevolumetricrateofSNeIa,wemeasuretheratepergalaxyasa functionofstellarmass,asperSullivanetal.(2006),Smithetal.

(2012),andBrownetal.(2019).Weusethestellarmassassembly model of C14 to predict the stellar age distribution of galaxies fora given stellar mass atthe mean redshift of ourSN sample, andthenforwardmodeltheDTDbyconvolvingitwiththestellar agedistributionandcomparingthepredictedratestotheobserved rates.

InSection2,weintroduceourlargesampleofSNeIafromthe Dark EnergySurvey (DES) as well as ourdeep sample of field galaxiesthatprovideaneffectivelycompletecontrolsamplefrom whichtomeasuretherate.WedescribeourdetailedhandlingofSN andgalaxyincompletenessinSection3.WepresenttheSNIarate pergalaxyinSection4andshowthatitisconsistentwithprevious works.WeoutlineourmodellingofSFHsandournovelconstraints ontheDTDparametersinSection5.InSection6,weinvestigatehow theDTDdiffersamongsubpopulationsofSNewithdifferentlight- curvecharacteristics,in particularthestretch,and showstretchto bestronglydependentonprogenitorage.WeconcludeinSection7 by discussingthe implications of theresults. Where relevant,we assumeaspatiallyflatCDMcosmologywithm=0.3andH0

=70kms⁻¹ Mpc⁻¹ .Unlessotherwisestated,weassumeGaussian measurementuncertaintiesquotedatthe1σlevel,andwequotethe posteriormedianand1σ credibleintervalsonderivedparameters.

MagnitudesarequotedintheABsystem(Oke&Gunn1983).

2 DATA

InordertomeasuretherateofSNeIapergalaxyperyear,werequire asampleofSNeIa,aswellasasampleofallofthepossiblegalaxies (‘field’galaxies)thatthoseSNecouldhaveexplodedin.Inpractice, SN and galaxysurveys do not cover the sameskyarea, redshift ranges, times,and havedifferentselection biases.In thissection, weintroduceourSNandfieldgalaxysamplesandinSection3we describeourmethodofcorrectingtheselectioneffectsforboththe SNandfieldgalaxysamples.

2.1 DarkEnergySurveysupernovaprogramme

To derive our sample of SNe and field galaxies, we make use of the DES. The DES Supernova Programme (DES-SN) was a transient survey basedon five six-monthseasonsof observations of tenSouthern hemispherefields withthe Dark EnergyCamera (DECam; Flaugher et al. 2015). The SN survey was designed primarily to measure the light curves of SNe Ia foruse as cos- mological distance indicators. Transients were detected and pro- cessed using a difference imaging pipeline (Kessler et al. 2015) and image-subtraction artefacts were rejected using a machine learning(ML)algorithm (Goldsteinetal.2015).Spectralfollow- up of live SN candidates was performed on a suite of large optical telescopes (Smith et al. 2020a), leading to the initial publication of a measurement of cosmological parameters using 207spectroscopicallyconfirmedSNeIa(DESCollaboration2018, and references therein). Spectroscopic redshifts forhost galaxies come from the OzDES Global Redshift Catalog (GRC;¹ Yuan et al. 2015; Childress et al. 2017; Lidman et al. 2020), which comprises galaxies targeted as DES-SN hosts with the OzDES programmeaswellasredshiftsfromlegacycataloguesintheDES- SNfields.

1https://docs.datacentral.org.au/ozdes/overview/dr2/

2.2 Supernovae

2.2.1 Photometricclassificationandqualitycuts

Withover30000discoveredtransientsitwasnotpossibletoobtain spectralfollow-upofeveryobject.Insteadwemakeuseofphotomet- ricclassificationintheformoftherecurrentneuralnetworkclassifier superNNova(SNN;M¨oller&deBoissi`ere2019),followingthe approachofScolnicetal.(2020).Afulldescriptionofthetraining andfittingofSNNwillbepresentedinVincenzietal.(inpreparation) and M¨oller et al. (in preparation); we provide a brief overview here.

Wetrain SNN using a large suite of simulated multibandSN light curves and their associated host galaxy redshifts, using its default architecture. SNe Ia are simulated based on the SALT2 model(Guyetal.2007) trainedon theJointLightcurveAnalysis (JLA)data set(Betouleetal.2014) usinganidenticalmethod to thatdescribedindetailinVincenzietal.(2020),andbasedonthe workflowoutlinedinKessleretal.(2019).WesimulateSNeusing the SuperNova ANAlysis software (SNANA; Kessler etal. 2009) integrated into the pippin framework (Hinton & Brout2020).

SALT2parametersx1(stretch)andc(colour)aredrawnrandomly from theintrinsic asymmetricGaussian distributionsdescribedin Scolnic&Kessler(2016)and‘blurred’followingtheintrinsicscatter model of Guyet al. (2010), whileredshifts are drawn following thevolumetricrateevolutionofFrohmaieretal.(2019).Synthetic CCSNelightcurvesaregeneratedfromthetemplatesofVincenzi etal.(2019),witharatefollowingtheCSFHofMadau&Dickinson (2014)andnormalizedbythelocalUniverserateofFrohmaieretal.

(2020).

Giventhe trainingset ofsyntheticlightcurves,SNN returnsa classification(SNIa,CCSN,orpeculiar)forobservedlightcurves.

Beforeclassifying,weremovetransientswithvariabilityinmultiple seasons(likelyactivegalacticnucleiorsuperluminousSNe).SNN is then run on every transient with a host galaxy redshift (see Section2.2.2).WedefineSNewithathresholdprobabilityofP(Ia)

≥0.5asphotometricallyclassifiedSNeIa,althoughouranalysisis notsensitivetothischoice,sincethevastmajorityofSNereceive classifications closeto 1or0.ThephotometricallyclassifiedSNe Ia are then passedthrough aSALT2 light-curvefitting code and are subject to qualitycutson x₁ and cin anidentical manner to Vincenzietal.(2020):−3≤x₁ ≤3,−0.3≤c≤0.3.Thisselection helpstoreducethepotentialcontaminationfromcore-collapseSNe (CCSNe)andoutlyingthermonuclearSNe,aswellasremovingthose in extremely dusty environments.We imposefurthercuts on the quality of those measuredparameters, whichare: the uncertainty onx1:σx₁<1;theuncertaintyonthedateatwhichthelightcurve haspeakbrightness:σ_{t peak}<2d;theSALT2fitprobability² >0.01.

ThenumberofSNepassingeachstageofthesecutsisdisplayedin table2ofVincenzietal.(2020)andreducesthe∼30,000objectsto 1604SNe.Withtheadditionofthephotometricclassifierweareleft with1441SNewithwhichweproceedtoinspecttheirhostgalaxies.

Whiletheinclusionofstringentcutssignificantly reducesthesize of the sample, it greatly enhances the purity and is in line with previousratesanalysesofthisnature(Sullivanetal.2006)aswell asbeingconsistentwithotherDES-SNanalyses(e.g.Kelseyetal.

2021).ThecontaminationofthissamplefromCCSNeaccordingto the simulationsofVincenzietal.(2020) isexpectedto bebelow 3percent,whichisclosetothatofspectroscopicsamples(Rubin etal.2015).

2Basedonlight-curvefitχ²andnumberofdegreesoffreedom.

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2.2.2 Hostgalaxyselection

Hostgalaxies forall transientsinDES-SNare retrievedfrom the DES-SNDeepcatalogue,agalaxycataloguefromtheg,r,i,z-band coaddedimagesofthetenDES-SNfieldswhichwaspresentedin Wiseman et al.(2020).SNe are associated to galaxies using the directionallight radius(DLR) method (e.g. Sullivan et al.2006; Gupta et al. 2016).As is standard acrossDES-SN analyses, we requireaSN-hostseparationtoDLRratio(d_DLR )lessthan4inorder toclassifyagalaxyasahost.Inthecasewheremultiplegalaxieslie withindDLR<4,theobjectwiththelowerorlowestvalueistaken tobethehost.

2.3 Fieldgalaxies

To calculate the SN Ia rate per galaxy as a function of stellar mass we require a representative sample of the global galaxy population, which we call our field galaxy sample. In order to maintainconsistencybetweentheSNhostandfieldgalaxysample selections,weobtainafieldgalaxysamplefromthesamecatalogue asthatfromwhichtheSNearematchedtoobtainhostgalaxies(SN Deep;Wisemanetal.2020).Thestepsrelevanttothisanalysisare outlinedinthefollowingsections.

2.3.1 Photometricredshifts

SincethevastmajorityofobjectsdetectedintheSNDeepcatalogue do not have a spectroscopic redshift measurement, we rely on photometricredshifts(photo-zs).Photo-zsaretaken,fromtheDES Y3A2DEEPcatalogueofHartleyetal.(2020).Y3A2DEEPmakes useofthesamedeepDESopticalphotometrythatwasusedinSN Deep fora subsetofthe fields (SN-X3,SN-C3 and SN-E2), but addsDECamu-banddataandnear-infrared(NIR)J,H,andK-band imagingfromtheultraVISTAandVIDEOsurveys.Thedeepoptical photometryinY3A2DEEPwasstackedusingadifferenttechnique to that of SN Deep, but the resulting photometry is consistent within 1σ uncertainties(Wiseman etal.2020, Meledorf etal. in preparation).Hartleyetal.(2020)estimatedphoto-zsusingtheeaZy code(Brammer,vanDokkum&Coppi2008).Photo-zaccuracyfrom Hartleyetal.(2020) at17 < i<24 isestimatedaround0.03as quantifiedbytheNormalizedMedianAbsoluteDeviation(NMAD).

Thisaccuracyisdegradedto0.07at17<i<26,arangethatincludes allgalaxiesinoursample,butismorethanadequateforthepurposes ofthiswork,whereallfieldgalaxieswillbegroupedintostellarmass binsoflog( M_∗/M)=0.25.Adetaileddescriptionofthephoto-z accuracyanditsdependenceonredshiftandapparentmagnitudeis presentedinHartleyetal.(2020).

2.4 Qualitycuts

Inordertorefinethesampleofhostandfieldgalaxiesfortherate analysisweperformaseriesofqualitycuts:

(i)objectsmustbedetectedandhaveaKronmagnitudemeasure- mentinallfourDESopticalbands;

(ii) fieldgalaxiesarelimitedto unmaskedregionoftheSN-X3 field(Hartleyetal.2020)whichhasanareaof1.52deg² ;

(iii)objectsmustnotbewithinagivennumber(20inx,50iny) ofpixelsoftheCCDedge,astheco-additionofslightlymisaligned imagesintroducedaregionofsignificantnoise inthispartof the detector;

Table1. NumbersoffieldgalaxiesandSNhostspassingvarious quality cuts.SNearederived frommust alreadyhave passedthe maskedchipsandgoodredshiftcutsbeforereachingthisstage.

Cut N(fieldgalaxies) N(SNhosts)

Chosenfields 1364311 1441

Kronmaginallbands 1069004 1439

Maskedchips 816950 -

HasugrizJHKphotometry 545748 -

Edgeofchip 481731 1401

Star/galaxyseparation 400051 1259

Hasgoodredshift 395034 -

SNR≥3 338256 1259

mr≤24.5 196109 1254

0.2≤z≤0.6 48177 809

(iv)tominimizestellarcontamination,objectsmusthaveavalue of <0.95in atleastonebandforthestar/galaxy(S/G)separation metric CLASSSTARprovidedbySource Extractor(Bertin

&Arnouts1996);

(v)fieldgalaxiesmustbecoveredbyugrizJHKphotometry,with aphotometricfluxmeasurementorupperlimitpresentinallbands;

(vi)objectsmusthaveaspectroscopicredshift measurementor aphotometricredshiftestimatewithawell-definedpeakinredshift probabilitydensity;

(vii)objects must be detected at signal-to-noise ratio (SNR) greaterthan3intherband;

(viii)galaxies must bebrighterthan m_r ≤ 24.5, as this is the magnitudeofthefaintestSNhostwithaspectroscopicredshift;

(ix)galaxiesmustbewithintheredshiftrange0.2≤z≤0.6(see Section3.1foranexplanationofthiscut).

Thenumbersof SNhostsandfieldgalaxies passingthesecuts arelistedinTable1.Thefinalsamplescomprise809SNeandtheir hostgalaxiesand40415fieldgalaxies.Thevolume-weightedmean redshiftsare0.50forbothSNeandfieldgalaxies.

2.5 Galaxyproperties

WeestimateglobalgalaxypropertiesforboththeSNhostandfield galaxies that passthe quality cutsby fittingthe photometry with stellarpopulationtemplatesinamethodoutlinedbySullivanetal.

(2006) and consistent to that usedin previous DES-SN analyses (Wisemanetal.2020;Smithetal.2020b;Kelseyetal.2021).ForSN hoststheredshiftisfixedatthespectroscopicallydeterminedvalue, whileforfieldgalaxieswefixitateitherthespectroscopicvalueifone existsintheOzDESGRC,ormorecommonlythephotometrically derivedvalue.WeusethestellarpopulationtemplatesofBruzual&

Charlot (2003)and adoptaChabrier (2003) initial massfunction (IMF). The fitting procedure returns a best-fitting template and correspondingstellarmass(M_∗).Upperandlowerboundsonstellar massestimatesaretakenastheextremevaluesthatcorrespondto templatesthatare consistentwiththe data(giventhephotometric uncertainties) accordingtotheirχ² statistic,as perSullivanetal.

(2006).Tocheckforbiascausedbytemplatechoices,wealsofitthe galaxiesusingthePEGASE.2´ templates(Fioc&Rocca-Volmerange 1997;LeBorgne&Rocca-Volmerange2002)andaKroupa(2001) IMF,andfindresultsconsistentwithinmeasurementuncertainties.

ThisisconsistentwiththefindingsofSmithetal.(2020b).

Theresults ofourSED fittingare shownin Fig. 1. Thefigure showcases the vastly different distributions of the two samples.

SN hostsare preferentially high-mass galaxies, whereas the field

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Figure1. Upper:DistributionofthestellarmassoftheSNhostgalaxies.The raw(greendashed)andspectroscopic-efficiency-weighted(bluedot–dashed;

Section3.2)histogramsarebothnormalized.ComparisonsamplesarePTF (purple;Panetal.2014)andPantheon(orange;Scolnicetal.2018).Lower:

Asupperbutforfieldgalaxies,andwithgalaxyfrequencyshownonalog scale.ThedistributioncloselymatchesdatafromtheZFOURGEsurveyat 0.5<z<0.75(magentadots).Thedifferencebetweentherawcounts(grey dotted)andtheVmaxcorrectedcounts(orangesolid;seeSection3)isminimal.

galaxydistributionincreasesdowntolowermasses,peakingaround 10^8.5 M. TheSN host stellarmassdistribution isplotted twice:

onceasrawcounts;onceweightedbythehostgalaxyspectroscopic efficiency(Section3.2).Asshownin Fig.1,thehoststellarmass distributionoftheDES-SNsampleusedinthisanalysisisqualita- tivelysimilartothatfromthelowredshiftPalomarTransientFactory (PTF)sample(Pan etal. 2014)aswell asthelargecosmological Pantheonsample(Scolnicetal.2018).Thisconsistencyreflectsthat seeninthesmallerspectroscopicallyconfirmedsamplespresented inWisemanetal.(2020)andSmithetal.(2020b).Thefieldgalaxy stellarmassdistributionisshownonalogscale,alongwiththestellar massfunction(SMF)fromtheZFOURGEsurvey(Tomczaketal.

2014)scaledtomatchatlog(M/M)=10,whosestellarmassesare derivedwiththesamephoto-zcode,IMFandtemplate libraryas usedhere.Above10^8.5M,theDESfieldgalaxydistributionclosely followsthe ZFOURGESMF for0.5< z< 0.75,indicating that theDESsampleisrepresentativeoffieldgalaxiesinthismassand redshiftrange.

3 I N C O M P L E T E N E S S C O R R E C T I O N S

The simple ratio of the number of SN hosts and field galaxies presentedattheendoftheprevioussectionprovidesafirstapproxi- mationoftheSNratepergalaxy,andtheinclusionofafactorequal tothesurveydurationnormalizestheratetoperyear.However,both theSNandfieldgalaxysamplesintroducedinSections2.2and2.3 areaffectedbyincompleteness,whichislikelytobethedominant systematiceffect inthe analysis.In this section, wedescribeour methodofcorrectingforvarioussources ofincompletenessinthe data.

3.1 Supernovae

Incompletenessin aSNsurveyarisesfrom anumber ofsources.

Theprimarysourceofincompletenessiscausedbythemagnitude

Figure2. SNdetectionefficienciessplitbyDES-SNfield,x1andc.They

−axisrepresentsthefractionofsimulatedSNeinagivenredshiftbinthat wouldhavebeendetectedbyDES-SNandpassedlight-curvequalitycuts.

Linesarepolynomialfitsthatapproximatetheefficiencycurves.

limitofthesurvey:SNewithapparentmagnitudesbelowthesurvey limit willnotbedetected.SinceSNe Iaare relativelyuniform in absoluteluminosity,thisformofincompletenessisprimarilyredshift dependent. The largeredshift range of DES-SN also means that SNe are probed atdifferent regions of theirrest-frame SED that varysignificantlyin luminosity.Additionally, DES-SN comprises 10separatepointings,eachwithdifferentvisibilityandthusairmass throughouttheobservingseason.Thesedifferencesleadtodifferent detectionefficienciesacrossthefields.

Tocorrectfortheseincompleteness,wefollowasimilarmethod to thatusedin the PTFrates analysesofFrohmaieretal. (2019) and Frohmaier etal. (2020),outlined in Frohmaieret al.(2017).

Wesimulate 1.1×10⁶ SNe in the redshift range0.05≤ z_SN ≤ 1.3.TheSNearegeneratedinthesamewayasforthetrainingof SNN(Section2.2.1).TheSNearesimulatedwithexplosionepochs t₀ uniformlydistributedbetweentwomonthsbeforeDES-SNbegan andtwomonthsafteritfinishedinordertoaccountforallSNethat couldhavebeen observedby DES-SN. Werunmockversionsof theDES-SNsurvey,usingtheexactcadence,conditions,andzero- pointsfromthesurveyitself.AllofthedetectedsimulatedSNeare passedthroughthelight-curvefitofSALT2(Betouleetal.2014), aspertheimplementationinSNANA,andthosethatfailthelight- curvecutsoutlinedinSection2.2.1arediscarded.Weareleftwitha fractionoftheoriginalsimulatedSNe,andthatfractionisdependent onacombinationofskylocation,explosionepoch,redshift,stretch, and colour.ThefractionofrecoveredSNe(theefficiency)isthus describedbyafive-dimensionalsurface.FortheithSNtheefficiency ηSN,iinfieldF,explodingattimet0atredshiftz,withstretchx1and colourc,is

η_{SN ,i} (F_i ,z_i ,t_{0 ,i} ,x_{1 ,i} ,c_i )= N_obs

F_i ,z_i ,t_{0 ,i} ,x_{1 ,i} ,c_i N_sim

F_i ,z_i ,t_{0 ,i} ,x_{1 ,i} ,c_i

. (1)

In practice,the gradient of the efficiency function is strongest betweendifferentDESfieldsandasafunctionofredshift,whileSN stretchandcolourhavesmallereffects.Weintegratetheefficiencies acrossthefullsimulatedtimerange,andassuchtheefficiencyis limited to∼0.5due tothe 6-monthnatureofthe DESobserving seasons. Thedistributionofefficiency asafunctionofredshiftis shown in Fig. 2. It is evident that the deep fields (X3, C3) are

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Figure3. SNhostgalaxyspectroscopicefficienciesasafunctionofredshift z.Efficiencycurvesaregroupedbyshallow({F}12)anddeep({F}3)fields, andaveragedoverthe5yrofthesurvey.Thelowerandupperredshiftlimits forthisanalysisareindicatedbyzminandzmax,respectively.

sensitivetoSNeathigherredshifts,whilethereisnodrasticshifts betweenefficienciesin theeightshallowfields.There isevidence thattheSNefficiencydependsweaklyonx₁ andmorestronglyonc, whichisexpectedsincethecolour-correctiontermislargerthanthe stretchcorrectiontermintheSNIastandardizationformula(Tripp 1998).BlueSNearerecoveredmorereadilythanredSNeasthey aregenerallybrighter.

Thedeep fieldsshownon-zeroefficienciesapproaching z= 1, whereastheshallowfieldstypicallyreachz=0.8.Sincefractional uncertaintyislargeatsuchlowefficiencies,wechoosetomakea redshiftcutofz=0.6wheretheefficiencyiswellabove0.1inall fields.

3.2 Supernovahosts

AfurtherlimitingfactorintheSNhostsampleistherequirement ofaspectroscopicredshift.ThemajorityofSNhostspectroscopic redshifts in DES areprovided bythe dedicatedfollow-up survey OzDES, for which the limiting magnitude is around 24 to 24.5 maginthe rband(Lidmanetal.2020).Therateatwhichahost redshiftissuccessfullymeasuredgivenanapparentmagnitudehas beenextensivelymodelledby Vincenzietal.(2020)whoprovide thespectroscopicredshiftefficiency_{z spec}(m^host _r,i )asafunctionofhost r-bandmagnitude,hostgalaxycolour,andtheyearinwhichtheSN wasdiscoveredinordertoallowforalongerpossiblespectroscopic exposuretimeforhostsofSNediscoveredearlierinthesurvey.To assesshow _{z spec}(m^host _r,i ) translatesinto anefficiency as afunction ofredshift_{z spec}(z)withintheredshiftlimitsofoursurvey,weusea simulationofhostgalaxiesasdescribedinVincenzietal.(2020).The simulationusesgalaxyphotometryfromDESScienceVerification images, and then selectsthem as potentialSN hostsaccording a weighting driven by previously measured SN rate - host galaxy relationships.Wenotethatthisisfordisplaypurposesonly,sothe relationshipsbuiltintothesimulationarenotpropagatedtothefinal ratemeasurements.Forbinsinredshift,wefindthemeanefficiency formeasuringaspec-zofallgalaxiesinthebinvia_{z spec}(m^host_r,i ).The resulting_{z spec}(z) curvesare displayedin Fig.3 andshow ahigh spectroscopicefficiencywithintheredshiftlimitsofthisanalysis.

Theeffects ofthe host galaxyspectroscopicefficiency aredis- playedintheupperpanelofFig.1,wherethenormalizedhistogram weighted bythe efficiencyisskewedtowardslowermasseswhen comparedtotheunweighteddistribution.Sothatwereduceanybias towardsSNeinbrighthoststhatareeasiertoobtainspectroscopic redshiftsfor,thehostgalaxyspectroscopicredshiftefficiencyismul- tipliedbythatderivedfromtheSNdetectionefficiency(Section3.1) toarriveatthefinalefficiencyfortheithSN:

η_{SN ,i} =η_{SN ,i} (F_i ,z_i ,t_{0 ,i} ,x_{1 ,i} ,c_i )×_{z spec} m^host _r,i

. (2)

3.3 Fieldgalaxies

3.3.1 Apparentmagnitudelimits

Asweusephotometric,ratherthanspectroscopic,redshiftsforthe fieldgalaxies,theydonotsufferfromspectroscopicincompleteness as theSN hostsdo.Instead,theinclusion ofany givengalaxyin the surveyareainthe sampleisdetermined simplyby whetherit is detected above a prescribedthreshold in SNR,i.e. the sample ismagnitude-limitedmodulothecutsdescribedinSection2.4.To determinetheapparentmagnitudelimitforgalaxiesineachofthe opticalbandsweemploythemethodofJohnston,Teodoro&Hendry (2007),Teodoro,Johnston&Hendry(2010),andJohnston,Teodoro

& Hendry(2012) (hereafter Completeness I, II, III respectively).

Theseworksprovidetwocomplimentarystatisticsinordertofind the limiting magnitude based on ranking of galaxies’ absolute magnitudes ina givenbandM anddistance moduliZ,namedTC

andT_V ,respectively.GalaxiesarefirstplacedintheM−Zplane, whichislimitedtodistancemodulithatcorrespondtotheredshift cuts0.2<z< 0.6introducedin Section3.1.Eachgalaxyinthe surveyisranked byitsZ (M)comparedto all othergalaxies that fall within a sliceof widthδM (δZ) – wefind that δM = δZ = 0.02magprovidesthebestbalancebetweenhighsamplingresolution and an adequate number of galaxies within each slice. Wethen iterativelytestdifferenttriallimitingapparentmagnitudesm_{lim, trial} . For agalaxy of apparent magnitude m< mlim,trial observed in a surveycomplete tomagnitudem_{lim, true} wherem_{lim, trial} ≤ m_{lim, true} , theexpectationvalueoftherankforarandomis0.5.However,ifa triallimitingmagnitudem_{lim, trial} ≥m_{lim, true} ,therewillbealackof observed faintobjects,suchthattheexpectationvalueofthe rank drops.

CompletenessI,II,andIIIshowthatforsurveyswithsharp,well- definedmagnitudelimitstheteststatisticsTCandTVarestableand flatasafunctionofapparentmagnitudeupuntilthemagnitudelimit, where they drop sharply. Fig. 4 shows the values of TC and TV

intheDESX3field.Thevaluesoftheteststatisticsincreasewith mlim,trialuntilapeak,beforedecreasingtostablevaluesatmagnitudes far beyond the limit of the survey, contraryto the sharp dropin anidealsurvey.Theshapeatbrighterm_{lim, trial} islikely causedby incompleteness at the bright end: due to the small sky area, we simply donot probeenoughvolume to samplethe brightend of the galaxyluminosityfunctionwellenoughforthestatisticstobe robust.

Toapproximateanefficiencyfunction,wefitthepeakofthestatis- ticswithapolynomialfunction,andthennormalizebythemaximum andminimumvalues.Wetheninterpolatebetweenthepeakandthe faint-magnitudefloortofindthe50percentcompletenesslimitat thepointwherethenormalizedvalueoftheteststatisticis0.5.These valuesareconsistentwiththevalueatwhich50percentofthetrue sourcesaredetectedbuthavetheadvantageofbeingderivedfrom thedatawithoutasimulationthatisbasedonassumptionsandthus

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Figure4. Measuringthecompletenessofthefieldgalaxysample.Upper:the TCgalaxycompletenessstatistic(whichiscalculatedbyrankingobjectsby absolutemagnitudeinslicesofdistancemodulus)asafunctionofapparent magnitude;middle:asperupper,butfortheTVstatisticwhichismeasuredby rankinggalaxiesbydistancemodulusinslicesofabsolutemagnitude;lower:

thecombined,normalizedcompletenessineachband.

susceptibletobias.Ourfinallimitingmagnitudeforeachbandisthe meanofthatfoundfromeachofTCandTV.

3.3.2 V_max correction

To correct for incompleteness caused by the magnitude-limited natureofthe surveywefollowthe prescription ofSullivanetal.

(2006) andSmith etal. (2012) byusingaV_max method basedon Schmidt(1968).For eachgalaxyin the sample wecalculatethe maximumvolumewithinwhichitwouldhavebeenobservedgivenits absolutemagnitudeandk−correction,andtheapparentmagnitude limitscalculated using the T_C and T_V statistics. A correction of Vsurvey/Vmax is applied to all galaxies for which Vmax < Vsurvey, whereV_{surv e y} is the maximum volume reachedby the survey. In ourcasethiscorrespondsto thevolumeatz=0.6. Fig.5shows the distribution of the correction among galaxies in the sample.

Thevast majorityofgalaxiesrequirenocorrection,meaningthey wouldhavebeenobservedbeyondthemaximumvolumeconsidered.

Roughly1percentofobjectshaveacorrectiongreaterthan1,with thedistributionwelldescribedbyapower-lawfunction.Theeffect of the V_max correctionon the total galaxy counts in each stellar mass bin can beseen in Fig. 1; indeed, the two histograms are visiblyindistinguishableabove10⁹ M whichindicatesasample thatiseffectivelycompleteinthismassrangeintheredshiftrange ofinterest.

4 T H E P E R - G A L A X Y R AT E O F T Y P E I A S U P E R N OVA E

TherateofSNepergalaxy(R_G )peryearinatransientsurveycanbe approximatedbytheequation:

R_G = NSN

NG

VG

VSN

1

T, (3)

whereNSNandNGaretherespectivenumbersofSNeandgalaxies detected,V_SN andV_G thevolumesfromwhichtheSNeandgalaxy samplesweretaken,andTthedurationoftheSNsurveyinyears.

Figure5. DistributionofVmaxcorrectionsappliedtofieldgalaxiesinthe DESsampleinordertocorrectforincompleteness.Themajorityofgalaxies havenocorrection,andthedistributionofthosethatdofollowsapowerlaw.

Asdescribed in Section 3, the values of NSN and NG that we observeareunderestimatesofthetruenumbersduetoobservational incompleteness– wedonotdetectandcountallSNeinthevolume V_SN ,nordowedetectandcountallgalaxiesinthevolumeV_G .Wethus estimatetheintrinsicnumbersofSNeandgalaxiesbymultiplying theobservednumbersbytheirrespectiveincompletenesscorrections calculatedinSections3.1and3.2forSNeandSection3.3forfield galaxies:

N_{SN , intrinsic} =

n SN

i

η

F,z,t₀ ,x₁ ,c,m^host _r

i , (4)

and NG,intrinsic=

n G

j

η(Vmax)_j (5)

foreachSNiandgalaxyj.ThevolumesV_SN andV_G arecalculated fromtheskyareasfromwhichtherespectivesamplesweretaken:

1.52 deg² in the case of the field galaxies (Hartleyetal. 2020);

23deg² forSNe(Smithetal.2020a).Theseareasarecombinedwith theredshiftinterval[0.2,0.6]todeterminethetotalvolumes.

Bybinningboththe SNhostsandfieldgalaxiesbytheirstellar masswemeasurethemeanSNIaratepergalaxyasafunctionof stellarmass,R_G (M_∗).WeemployabootstrapMonteCarloapproach in order to estimate the uncertainty in the value of the rate in each stellarmass bin.The probability densityfunction (PDF)of eachgalaxy’s stellarmassis representedby the sum of twohalf Gaussian distributions to represent the asymmetric positive and negativeuncertaintiesderivedintheSEDfittingstage(Section2).For eachSNhostandfieldgalaxy,wetake100samplesofitsstellarmass bydrawingatrandomfromitsPDF.Foreachofthe100sampleswe calculatetheSNratepergalaxyineachstellarmassbinviaequation (3),usingthe completeness-correctedvaluesofN_SN and N_G from equations(4)and(5)respectively.

TherateofSNeIapergalaxyasafunctionofstellarmassisshown inFig.6.TherelationshipbetweenSNIarateandgalaxystellarmass iswelldescribedbyalinearfunctioninlogspace,whichcorresponds to apowerlaw.Tofindtheslope andintercept thatbestdescribe the data weuse Bayesian inference. To sample the posterior we

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Figure6. Therate pergalaxy ofSNeIa asa functionof stellarmass.

Horizontalerrorbarsrepresentthewidthofthestellarmassbins.Vertical errorbarsareestimatedviaaMonteCarloresamplingoftherategiventhe uncertaintiesinstellarmass.Thelinearfitisbasedonallbutthelowestand highestmassbins,andtakesintoaccountuncertaintiesintherate.

usetheenhancedno-U-turnSampler(NUTS)algorithm(Betancourt 2017), which is a variant of Hamiltonian Monte Carlo (HMC), implementedin the Stanprogramminglanguage(Carpenter etal.

2017).WedescribethefittingprocedureinfulldetailinAppendixA.

Wemeasureaslope of0.63±0.02,whichisconsistentwiththe valuefoundinweaklystar-forminggalaxiesintheSupernovaLegacy SurveybySullivanetal.(2006),butalsoinpassivegalaxiesinSDSS bySmithetal.(2012),andinallgalaxiesintheLickObservatory Supernova Search(LOSS)by Li etal. (2011).Our slope isalso consistentwiththatmeasuredinthelow-redshiftAll-SkyAutomated Survey for Supernova (ASAS-SN) Bright Supernova Catalogues (Brownetal. 2019),which extendsto galaxieswith stellarmass aslowas10⁷ M.

Abest-fittingslopeoflessthanunitycorrespondstoapowerlaw and indicatesthat SNIa rate isnot uniquely determined by host galaxystellarmass.Indeed,therateisinsteadlikelytobedrivenby theDTDwhichisanonlinearfunctionofstellarage.Investigating theshapeofthisDTDformsthebasisforthelatterpartofthispaper inSections5and6.ThestraightlinefitinFig.6deviatessomewhat fromthecentralvaluesofthedataathighstellarmass,thereasonfor whichweinvestigateinSection5.4.

5 M O D E L L I N G T H E P E R - G A L A X Y R AT E O F T Y P E I A S U P E R N OVA E

InSection4andFig.6,weshowedthattheSNIaratepergalaxy asafunctionofstellarmassisapproximatedby,butnotperfectly describedby,apowerlaw.Inthissection,weintroduceaphysical modelin order to betterfit the observed rate versus stellarmass relationship.

Webeginby consideringthe delaytimedistributionofSNeIa.

Werepresent theSNIaDTDbyapowerlaw,whichweconsider effectiveaftersome‘prompttime’t_p ,beforewhichitsvalueissetto 0.tpisgenerallyinterpretedasthetimetakenforWDstoformafter theburstofstarformation.TheDTDisthus:

(τ)=

⎧⎨

⎩

0, τ <tp

A τ

Gyr

β

, τ≥t_p

(6)

where τ is the time sincea burstof star formation, and A is a normalizationatτ = 1Gyr.Ifallstarsin agalaxywere formed at asingle epochtf,the DTD would describethe rate ofSNe at anobservationtimeτ =t₀ .Howeverinrealitygalaxiesareformed bythegradualbuild-upofstellarmassoverseveralepochsofstar formation– thedistributionofthismassbuild-upisknownasthe SFH.TherateofSNeIaisthusthesumoftheDTDevaluatedfor eachepochofstarformation,multipliedbythestellarmassformed inthatepoch.Mathematicallythisisrepresentedbytheconvolution oftheDTDandSFH:

RG= t_f

t₀

ψ(t0−τ)(τ)dτ, (7)

wheret₀ istheepochatwhichthegalaxyisobserved,t_f isthetime atwhichthefirststarsinthegalaxyformed,andψ istheSFH.

Inpreviouswork(e.g.Strolgeretal.2004;Maozetal.2012)ithas beencommontodeterminetheSFHforeverygalaxyinthesurvey viaSEDfitting.Equation(7)isthenusedtocalculateanexpected numberofSNeineachgalaxygiventheeffectivesurveytime,which iscomparedtotheobservednumberinthatgalaxy(usually0and occasionally1)usingPoissonstatistics.Thismethodreliesoneither photometry coveringseveral wavelengthbands,or opticalspectra withahighSNR,inordertodistinguishaccuratelybetweenSFHs.

SuchaccuracyisnotpossiblefortheDESsample;wedonotpossess spectraofallgalaxiesinthefield,andinsomecasesonlyhavethe four opticalbandsavailable– andhaveamaximumofeightfrom NUVtoNIR– fromwhichtoinferanSFH.Thislackofdetailis compoundedbyourrelianceonphotometricredshifts,whichaddan extralayerofuncertaintytothecalculation.

Insteadof determiningindividualSFHs foreachgalaxyin our fieldsample,weuseanempiricalapproachtoestimatemeanSFHs forgalaxiesasafunctionoftheirstellarmassatanygivenredshift (orequivalently,timet₀ ),thatcanberepresentedasψˆ (t₀ −τ;M_∗). ThismethodpavesthewayformodellingtheSNIarateasafunction ofstellarmassbycombiningthemeanSFHsandtheDTDthrough equation(7),andthusplacingconstraintsontheDTD.

5.1 Modellingthestarformationhistoriesofgalaxies

To modeltheSFHof galaxiesas afunctionof theirstellarmass weadopttheprescriptionofstellarmassassemblyofC14,which drawsontheworkofZahidetal.(2012).Inthemodel,galaxiesare expectedtoevolvesmoothlyalongtheso-called‘mainsequenceof starformation’wherebytheSFRofagalaxyofstellarmassM_∗at redshiftzisdeterminedbyasimplerelationship(theSMzrelation, Zahid et al. 2012). Specifically, we implementtheir ‘fine-tuned’

model(equationA7inC14),wherebytheSMzflattensabovez∼2 inlinewithobservations(Starketal.2013):

(M_∗,z)= M_∗

10¹⁰ 0 . 7

exp(1.9z)

exp(1.7(z−2))+exp(0.2(z−2))

×[Myr⁻¹], (8) where M_∗ isgiven in units of M. As galaxies grow,their SFR begins to slowdownand eventually shutoff almostentirely in a process knownasquenching.Weadoptthequenchingpenaltyp_Q directlyfromC14:

p_Q (M_∗,z)= 1 2

1−erf

log(M_∗)−log(M_Q (z)) σ_Q

, (9)

where M_Q (z) describes how the quenching mass evolves with redshift, and σQis thetransitionscalewhichcontrols howfast a

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galaxyquenches. Weadopttheobservationallymotivatedformof thequenchingmassevolution(equationA8fromC14):

log(M_q (z)/M)=10.077+0.636z, (10) andatransitionscaleofσ_Q =1.1whichprovidesagoodfittodata fromtheGalaxyAndMassAssembly(GAMA)survey(Baldryetal.

2012).

Finally,attimet,sometimeτ sinceanepochofstarformation, afractionofstellarmassislost.Weadopttheparametrizationused byC14, namelythatofLeitner&Kravtsov(2011)foraChabrier (2003)IMF:

f_ml =0.046ln τ

0.276Myr+1

. (11)

The mass loss at any given time is thus the convolution of the equations(8)and(11):

M_∗= _{t f}

0 (M_∗(t−τ),z(t−τ))·(f_ml (τ+ t)−f_ml (τ))dτ.

(12) Thestellarmassformedineachtimet(andcorrespondingredshift z) is thus the combination of the mass- and redshift-dependent SFR (equation 8), reduced by the mass- and redshift-dependent quenchingpenalty(equation9),minusthemasslostineachtime-step (equation12):

M_∗(t+ t)−M_∗(t)

t =p_Q

M_∗(t_, z(t)

·(M_∗(t),z(t))− M_∗ t .

(13) AsperC14weplantseedgalaxieswithinitialmassesof10⁶ M atintervalsof50Myrforlook-backtimes1Gyr≤t_f ≤10Gyr,and 25Myrforlook-backtimes10Gyr≤t_f ≤13Gyrtoachieveamore detailedmodelatearlytimeswithhighSFR.Foreachseedgalaxy, weevaluateequation(13)attime-stepsof0.5Myrandrecordthe totalmassformedandlostineachofthetime-steps– thisdefines ourmodelSFHtobeusedasinputtoequation(7).

TheresultofourtoymodelisshowninFig.7.Galaxieswithhigh massesatz=0formedthevastmajorityoftheirstarsinthefirstfew Gyrand arenowlikely tobepassivewitholdstellarpopulations, whilelow-massgalaxiesarecurrentlystronglystar-formingwitha vastmajorityofyoungstars.Galaxieswithmassesaround10¹⁰ M arecomposedofamixtureofyoungandoldstellarpopulations.

5.1.1 Validatingthemassassemblymodel

Asnoted in C14, this toy modelby construction followsseveral empiricalrelationsderivedfromgalaxyobservations:modelgalaxies followtheSMzrelationandobeyquenchingprescriptionsmotivated byobservedgalaxies.Herewemaketwofurthercheckstovalidate thatthemodelcanbeusedtoapproximatethestellaragedistribution ofSNIahostgalaxiesinDES.

First, wefollowC14 by summingourSFHs andweightingby the galaxystellarmass functionat z= 0 to estimatethe CSFH.

Weapproximatethepresent-daystellarmassfunctionaccordingto adoubleSchechterfunctionwithparametersfromtheGalaxyand MassAssemblysurveyBaldryetal.(2018):log(M^∗/M)=10.66, φ^∗₁=3.96×10⁻³ , α₁ = −0.35,φ₂^∗=0.79×10⁻³ , α₂ = −1.47.

ThecomparisonofourapproximateCSFHwiththatmeasuredfrom observations(Madau&Dickinson2014)isshowninFig.8.Aswas foundinC14,thetwofunctionsshowagoodqualitativeagreement – infact,wefindaclosermatchbetweenthemodelandobserved

CSFH, accurately reproducingthe peak of cosmicstar formation aroundz∼2andtracingthedecliningSFRtothepresentday.

Secondly,weassesstheshapeofourSFHsandhowtheydepend onstellarmassbycomparingwithSFHmeasurementsfromtheCalar Alto Legacy Integral Field spectroscopy Area survey (CALIFA) asrevealedbyGonz´alezDelgadoetal.(2017).Ourmodelsagree qualitatively on numerous features of the SFHs (their Fig. 5):

high-mass galaxies form almost all their stars at high redshift;

intermediate-mass galaxies display a plateau in their SFR from

∼4Gyrtothepresentday;low-massgalaxiesshowanincreasing rateofstarformationatz=0.Themaindiscrepancywiththelowest- massbinoftheCALIFAsample8.6<log(M_∗/M)<9.8,which aswellasanincreasingSFRatlowredshiftshowsasecondpeakof starformationatlargelook-backtimes,whichwedonotseeinour model.However,thismassbinspansoveroneorderofmagnitudein stellarmass.Uponcloserinspectionofourmodel,theloweredgeof thismassbincorrespondstogalaxiesthatformatalook-backtime of5GyranddisplayarisingSFH.Attheupperedge,galaxiesdo indeedformattimes>10Gyranddisplayanearlypeakfollowed by exponentialdecline. Thecombination ofthesedifferentmodel SFHsinthelow-massbinthusexplainthedual-peakednatureofthe observedSFHsinCALIFA.

5.2 ConstraintsontheSNIadelaytimedistribution

Atthebeginningofthissection,weshowedhowtherateofSNeIais drivenbytheconvolutionoftheSFHofgalaxiesandtheSNIaDTD (equation7).Wethenprescribedamodeltoinferthe meanSFHs ofgalaxiesofanygivenstellarmass.Here,wemeasurethe DTD byforwardmodellingitthroughequation(7)atthestellarmasses correspondingtothecentresofthebinsinFig.6.

WeassumeaDTD(τ)withapower-lawformdescribedbyindex β(equation6),normalizationAandeffectiveafterprompttimet_p .As perSection4,weconstraintheparametersviaBayesianinference, using HMCto explorethe posteriordistribution. At eachstepof thesamplingprocedurethemodelisre-calculatedviaequation(7), we evaluate the likelihood assuming that the rate measurements R_{G ,M ∗} are described by a Gaussian PDF with mean Rˆ_{G ,M ∗} and standarddeviationσR_G,M∗. Forβ weadopt aGaussian priorwith hyper-parameters p(β)∼N(−1,0.5). Wefit for A in log space, and adopt aGaussian prior (inlog space) withhyper-parameters p(log(A))∼N(−12.7,0.5).

SNeIadonotoccurinstantaneouslyafteranepisodeofstarfor- mation.Insteadstarswithzero-agemain-sequence(ZAMS)masses less than8M taketimeto evolvealongthe mainsequenceand formWDsandthistimeisnotwellconstrained.Inmanyworksthis

‘prompttime’whichwedenotet_p isfixedtosomevalueexpected tobetheminimumtimeforastartoevolveoffthemainsequence and becomeaWD, suchas40 Myr(Maoz etal.2012;Graur &

Maoz 2013;Graur etal. 2014b). Otherstudies haveleft t_p as a freeparameter(Heringer,Pritchet&vanKerkwijk2019;Castrillo etal. 2020).Weperform three fits:one with t_p fixedat 40Myr, one with it as afree parameteralong withβ and A,and a third with βand A fixedand t_p free. Wefit t_p in logspace and adopt aGaussianpriorwithhyper-parametersp(log(tp))∼N(−1.3,0.5), whichcorrespondstobeingcentredaround50Myr.Choosingthis regime to fit allows more prior weight to be placed on shorter prompt timesas perthe majorityof the literature.Weuse theRˆ diagnostic of Vehtari et al. (2019) to assess the convergence of MC chains.When including tp asa free parameter,we findthat thestrongdegeneracybetweent_p andAcausethesamplertofailto convergewithRˆ ∼1.3forthechainsofthosetwoparameters.In

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