SALT2: using distant supernovae to improve the use of Type Ia supernovae as distance indicators

0.63 MB·English

by  J. Guy

Astronomy&Astrophysicsmanuscriptno.salt2 (cid:13)c ESO2008
February4,2008
SALT2: using distant supernovae to improve the use of Type Ia
⋆
supernovae as distance indicators
J.Guy1,P.Astier1,S.Baumont1,D.Hardin1,R.Pain1,N.Regnault1,S.Basa2,R.G.Carlberg3,A.Conley3,S.Fabbro4,
D.Fouchez5,I.M.Hook6,D.A.Howell3,K.Perrett3,C.J.Pritchet7,J.Rich8,M.Sullivan3,P.Antilogus1,E.Aubourg8,
G.Bazin8,J.Bronder6,M.Filiol2,N.Palanque-Delabrouille8,P.Ripoche5,V.Ruhlmann-Kleider8
7
1 LPNHE,CNRS-IN2P3andUniversite´sParisVI&VII,4placeJussieu,75252ParisCedex05,France
0
2 LAM,CNRS,BP8,TraverseduSiphon,13376MarseilleCedex12,France
0
3 DepartmentofAstronomyandAstrophysics,UniversityofToronto,50St.GeorgeStreet,Toronto,ONM5S3H8,Canada
2
4 CENTRA-CentroM.deAstrofisicaandDepartmentofPhysics,IST,Lisbon,Portugal

n 5 CPPM,CNRS-IN2P3andUniversite´Aix-MarseilleII,Case907,13288MarseilleCedex9,France
a 6 UniversityofOxfordAstrophysics,DenysWilkinsonBuilding,KebleRoad,OxfordOX13RH,UK
J 7 DepartmentofPhysicsandAstronomy,UniversityofVictoria,POBox3055,Victoria,BCVSW3P6,Canada
9 8 DSM/DAPNIA,CEA/Saclay,91191Gif-sur-YvetteCedex,France
2
ReceivedMonthDD,YYYY;acceptedMonthDD,YYYY

1
ABSTRACT
v
8
Aims.WepresentanempiricalmodelofTypeIasupernovaespectro-photometricevolutionwithtime.
2
Methods.Themodelisbuiltusingalargedatasetincludinglight-curvesandspectraofbothnearbyanddistantsupernovae,thelatter
8
beingobservedbytheSNLScollaboration.WederivetheaveragespectralsequenceofTypeIasupernovaeandtheirmainvariability
1
componentsincludingacolorvariationlaw.Themodelallowsustomeasuredistancemoduliinthespectralrange2500−8000Å
0
withcalculableuncertainties,includingthosearisingfromvariabilityofspectralfeatures.
7
Results.Thanks to the use of high-redshift SNe to model the rest-frame UV spectral energy distribution, we are able to derive
0
improveddistanceestimatesforSNeIaintheredshiftrange0.8 < z < 1.1.Themodel canalsobeusedtoimprovespectroscopic
/
h identificationalgorithms,andderivephotometricredshiftsofdistantTypeIasupernovae.
p
Keywords.supernovae:general-cosmology:observations
-
o
r
t
s 1. Introduction shifts,andprovideapreciseestimator,withatypicaldispersion
a
: of7%ondistance,whennotlimitedbythemeasurementuncer-
v Theevolutionofluminosityorangulardistancewithredshiftis tainties(seee.g.Astieretal.2006,hereafterA06).
i anessentialobservabletoconstraintheequationofstateofdark
X As the number of SNe Ia in Hubble diagrams increases,
energy,responsibleforthe accelerationofthe expansionof the
systematic uncertainties are becoming the main limitation to
r Universe. Type Ia supernovae (SNe Ia) are, today, among the
a theaccuracyofmeasurementsofcosmologicalparameterswith
best distance indicators. They can be observed up to high red-
SNeIa.Amongthepotentiallyserioussystematicuncertainties,
thedominantonesareapossibleevolutionofthesupernovapop-
Sendoffprintrequeststo:[email protected] ulation,thephotometriccalibration,themodelingoftheinstru-
⋆ BasedonobservationsobtainedwithMegaPrime/MegaCam,ajoint ment response and the uncertainties arising from SN Ia large
project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii spectral features, including their possible supernova to super-
Telescope(CFHT)whichisoperatedbytheNationalResearchCouncil novavariations.In thispaper,we aim at addressingprincipally
(NRC) of Canada, the Institut National des Sciences de l’Univers of
the latter although the proposed method makes it also easy to
the Centre National de la Recherche Scientifique (CNRS) of France,
accountforthemodelingoftheinstrumentandtopropagatethe
and the University of Hawaii. This work is based in part on data
modeluncertainties.
products produced at the Canadian Astronomy Data Centre as part
of the Canada-France-Hawaii Telescope Legacy Survey, a collabora- Various approaches to distance estimation have been pro-
tiveprojectofNRCandCNRS.Basedonobservationsobtainedatthe posed, using light-curve shape parameters (∆m15 or a stretch
EuropeanSouthernObservatoryusingtheVeryLargeTelescopeonthe factor,seee.g.Phillips1993;Riessetal.1995;Perlmutteretal.
CerroParanal(ESOLargeProgramme171.A-0486).Basedonobserva- 1997) or color information (Wangetal. 2003b, Wangetal.
tions (programs GN-2004A-Q-19, GS-2004A-Q-11, GN-2003B-Q-9, 2005), or both(Riessetal. 1996, Tripp1998, Guyetal. 2005).
andGS-2003B-Q-8)obtainedattheGeminiObservatory,whichisoper-
Noneofthesemethodsreallyaddresstheproblemofuncertain-
atedbytheAssociationofUniversitiesforResearchinAstronomy,Inc.,
ties due to the variability of the large features of SNe Ia spec-
under acooperative agreement withtheNSFonbehalf oftheGemini
tra.In thisrespect,mostmethodsrelyonthe spectralsequence
partnership: the National Science Foundation (United States), the
providedbyNugentetal.(2002)(hereafterN02).Thisisaseri-
ParticlePhysicsandAstronomyResearchCouncil(UnitedKingdom),
the National Research Council (Canada), CONICYT (Chile), the ousconcernsincethiscouldpossiblyresultinsizable(common)
AustralianResearchCouncil(Australia),CNPq(Brazil)andCONICET systematic effects in the distance measurements. In Guyetal.
(Argentina). (2005)(hereafterSALT),wehaveappliedbroadbandcorrections
2 J.Guyetal,SNLSCollaboration:SALT2
tothespectralsequenceN02asafunctionofphase,wavelength datemaximumluminosityinB-band,c=(B−V) −hB−Vi.
MAX
and a stretch factor so that the spectra integrated in response Thisparametrizationmodelsthepartof thecolorvariationthat
functionsmatchtheobservedlight-curves.Thisprovideduswith is independentof phase, whereasthe remainingcolorvariation
atooltofittheobservedlight-curveswithoutcorrectingthedata withphaseisaccountedforbythe linearcomponents. x isthe
0
pointssincetheK-correctionswerenaturallybuiltintothemodel normalizationoftheSEDsequence,andx fork>0,arethein-
k
(see e.g. N02 for a definition of K-corrections).This approach trinsicparametersofthisSN(suchasastretchfactor).Tosum-
hasbeenquitesuccessfulwhenappliedtoestimatingSNeIadis- marize,whereas(M )andCLarepropertiesoftheglobalmodel,
k
tancesathighredshift.Howeveritdidnotaddressthe problem (x )andcareparametersofagivensupernovaandhencediffer
k
ofvariabilityofspectralfeatureseither. fromoneSNtoanother.
Inthispaper,weuseasimilarframework.Whereasweonly Except for the color exponential term, Eq. 1 is equivalent
usedmulti-bandlightcurvestotrainthemodelinSALT,herewe to a principal component decomposition. However, a princi-
include spectroscopic data to improve the model resolution in pal component analysis cannot be used since this would re-
wavelengthspace,befullyindependentofthespectralsequence quirehavinganhomogeneousanddensesetofobservationsfor
N02, and more generally extract the maximum amount of in- each SN, namely one spectro-photometric spectrum every 4–5
formation from the current data sets. Modeling the supernova days, which is not presently available (note that current ongo-
signalinspectroscopicspaceensuresthatthe K-correctionsare ing SN programs such as the SNfactory, Alderingetal. 2002,
treated in a consistent manner since there is a single model to the Carnegie Supernova Program, Hamuyetal. 2006, the CfA
addressbothlight-curvesandspectra.Italsopermitsacoherent Supernovaprogram1andtheLOTOSSproject2,shouldprovide
propagationoferrors,fromthefitoflight-curvestodistancees- suchdatainthecomingyears).Soweresortedtousingamethod
timate.Themodelisallowedtovaryasafunctionofphaseand abletodealwithmissingdata.Themethodusedisdescribedin
wavelength with a small number of a priori unknown intrinsic thenextsection.
parametersandacolorvariationlawwhichisalsoadjusteddur-
ing the training process. The main goal of this approach is to
2.1.Modelimplementation
provide the best “average” spectral sequence and the principal
componentsresponsible for the diversity of SNe Ia, so that the Thephasespacethatwewanttomodel(wavelengthrangetimes
model can account for possible variations in SNe Ia spectra at phase range) is not covered by the set of observations of any
anygivenphase. given supernova. We typically have for each supernova a lim-
Thefluxnormalizationofeachsupernovaisafreeparameter itedsetoflight-curvespointsobservedwithdifferentfiltersand,
ofthemodel.Hence,wedonotneedtoknowtheirdistancesto forsomesupernovae,oneorseveralspectraatdifferentphases.
trainthemodel.ThisallowsustousebothnearbySNewhichare However,whenusinganensembleofSNe,thisphasespacecan
notintheHubbleflowandhigh-redshiftoneswithoutanyprior becorrectlysampledandifthedatasetislargeenough,several
oncosmology.Usinghigh-redshiftsupernovaepermitstomodel componentscanbeextracted.
therest-frameUVemissionwhichisinvaluabletoimprovedis- Inordertolinkthemodeldefinedbyalimitedsetofparam-
tanceestimatesofsupernovaefoundatredshiftslargerthan0.8. eters and the SNe observations, we used a basis of functions,
In Sect. 2 we present the model implementation. The su- as function of phase and wavelength f(p,λ) . We used third
pernova data sets used for training the model are described in order B-splines (to ensure continuous(cid:2)seicond d(cid:3)erivatives). The
Sect. 3. Some technical aspects of the training procedure are actualchoiceofthebasisisirrelevantinthephasespaceregions
then given in Sect. 4, and in Sect. 5, we present some qualita- whicharedenselycoveredbydata,aslongasitprovidesasuffi-
tive aspectsof the resulting model.In an effortto improvedis- cientresolutiontofollowtheobservedvariabilityoftheSEDse-
tanceestimates fora cosmologyapplicationto SNe Iasurveys, quenceasafunctionofphaseandwavelength.Choosinganother
wequantifytheremainingvariabilitybeyondtheprincipalcom- basis will modify the model in regions where it is poorly con-
ponents extracted in Sect. 6. We show how distance estimates strained,suchasveryearlyspectra(p<−15days).Asdescribed
ofSNLSdistantSNeareimprovedwiththisapproachinSect.7 in section 6, those poorly constrained phase space regions are
anddiscussseveralotherpossibleuseofthemodelinSect.8. identifiedafterthetrainingusingajack-knifetechnique.Inthis
framework, a model is a linear combination of the basis func-
tionsandcanbedescribedbyavectorM.Eachmeasurementat
2. TheTypeIasupernovaspectralsequencemodel
agivenphaseandwavelengthm(p ,λ )isthencomparedtothe
m m
We aim at modeling the mean evolution of the spectral energy modelwithavectorHm (withvaluesHm,i = fi(pm,λm)),sothat
distribution (SED) sequence of SNe Ia and its variation with a theexpectedvalueforthemodelat(pm,λm)isthescalarproduct
fewdominantcomponents,includingatimeindependentvaria- HTM.
m
tionwithcolor,whetheritisintrinsicorduetoextinctionbydust
in the host galaxy(or both).The followingfunctionalform for
2.2.Theuseofspectralinformation
thefluxisused
Most spectra of SNe Ia available in the literature are not cali-
F(SN,p,λ)= x × M (p,λ)+x M (p,λ)+...
0 (cid:2) 0 1 1 (cid:3) brated photometrically. Their flux calibration have broad-band
× exp[cCL(λ)] (1) systematicuncertainties.
Onewaytocircumventthisdifficultyconsistsinphotometri-
where p is the rest-frame time since the date of maximum lu-
cally”re-calibrating”agivenspectrumusingtheavailablelight-
minosity in B-band (the phase), and λ the wavelength in the
curves for this particular SN. However since the full SED se-
rest-frameoftheSN. M (p,λ)istheaveragespectralsequence
0
whereasMk(p,λ),fork > 0,areadditionalcomponentsthatde- 1 CfASupernovaGroup:
scribe the main variability of SNe Ia. CL(λ) represents the av- cfa-www.harvard.edu/oir/Research/supernova/index.html
erage color correction law. As for SALT, the optical depth is 2 The Lick Observatory and Tenagra Observatory Supernova
expressedusingacoloroffsetwithrespecttotheaverageatthe Searches:astro.berkeley.edu/bait/lotoss.html
J.Guyetal,SNLSCollaboration:SALT2 3
quencemodelis neededto derivean accurate interpolationbe- maximumtoensureagoodestimateoftheluminosityatpeak.A
tweenlightcurvepointsatthedateofthespectroscopicobserva- largefractionofthoseSNelight-curvescomefromHamuyetal.
tion,andthespectraareneededaswelltoaccuratelymodelthe (1996b); Riessetal. (1996) and Jhaetal. (2006). We did not
spectralfeaturesofSNe,thephotometric”re-calibration”ofthe consider 1991bg-like SNe Ia (with very low stretches). They
spectrahastobeincludedintheglobalminimizationprocedure. havesuchdifferentlight-curvesandspectrathatthelinearmodel
We have chosen to parameterize the ”re-calibration” function weconsidercannotfitthosealongwithotherSNeIa.Thisisnot
withtheexponentialofapolynomial(toforcepositivecorrected aproblemsinceweaimatmodelingthebulkoftheSNeIapop-
fluxes),withthedegreeofthepolynomiallimitedbythenumber ulation(andwedonotexpecttodetectmanyofthoseobjectsat
oflightcurvesfortheSN,andthewavelengthrangeofthespec- highredshift).
trum. This re-calibration function is applied to the model, for We donotuseanyspectrawithoutphotometricdataforthe
which the SN parameters are mostly constrained by the light- sameSN(atleasttwolight-curvesindifferentfilters),sothatthe
curves. Thanks to the simultaneous use of a large amount of date of maximum,color and (x ) can be determined.However,
k
SNe data, we do not need to have photometric observationsat since spectra are calibrated on the model and not on the pho-
thesameepochasspectroscopicones. tometricdata,wedonotneedsimultaneousphotometricobser-
Statistical errors are rarely provided with the spectra. We vations; we just need enough photometric observations to de-
haveevaluatedthemusingthefactthatallSNespectralfeatures rive the SN parameters. From the sample of 52 SNe, we were
arebroadenedduetothekinematicsoftheejectedmatter.Sowe abletogather264spectrafor16SNe.Thereare10spectralse-
expectanintrinsiccorrelationlengthgreaterthan30Å(forave- quences (with more than 10 spectra), namely 1989B, 1990N,
locityrangelargerthanabout2000km.s−1)whichpermitsoneto 1991T, 1992A, 1994D, 1996X, 1998aq, 1998bu, 1999ee, and
evaluatethephotonnoiseinspectra(assumedtobewhitenoise). 2002bo (see Table 2 for the complete list of spectra and their
Nonetheless,wescalederrorssothattheweightofspectrawas references).
of order of that of lightcurvesfor which we expectlower sys- All available UV spectra from the InternationalUltraviolet
tematic errors.Thisweighting,alongwith the resolutionof the Explorer(INES2006)wereincluded.Thisisveryhelpfulsince
re-calibrationfunction,is a bitarbitrarybutcannotbeavoided mosthigh-redshiftSNe spectra which coverthe rest-frame UV
atthisstageduetothequalityofcurrentlyavailabledatasets. range have a low signal to noise ratio. For all spectra from
ground-based observations, we do not consider any measure-
mentbelow 3400Å becauseof thestrengthof the atmospheric
3. Thetrainingsupernovadatasets absorptioninthisspectralregion.
In this section we describe the data sets used for training the
model.
3.2.Thehigh-redshiftsupernovasample
In the proposed model (Eq. 1), the overall SED sequence
normalizationofagivenSNisafreeparameter(x ).Asaconse- We used a set of 121TypeIa supernovaelight-curvesobtained
0
quence,itispossibletouseboththeverynearbysupernovaedata bytheSupernovaLegacySurvey(SNLS)duringthefirst2years
thatarenotintheHubbleflow(z<0.001)andthehighredshift ofthesurvey(seeTable2).Thelightcurveswereobtainedwith
oneswithoutadoptingvaluesofcosmologicalparameters. the same reduction pipeline as described in A06, but with new
Nearby supernovae have much higher signal to noise than imagesinthephotometricfit,sothatlight-curveshavemoredata
their distant counterparts over a much wider range of phases. pointsandwithimprovedstatisticalaccuracysincethereference
One important difficulty however in using nearby SNe is that data used to anchor the estimate of the galaxy brightness be-
theysufferfrompotentiallylargesystematic errorsinthe ultra- come deeper with time (thanks to the rolling search observing
violet(UV),since the atmosphericextinctionis stronganddif- strategy).Allthe71SNeusedforcosmologyinA06wereused,
ficult to model in this wavelength range. Including SNe from with50additionalones.
a largeredshiftrangehelps to sample homogeneouslythe rest- In addition to the light-curve points, we used 39 high-
framevisiblewavelengthrangewithbothphotometricandspec- redshift SNLS spectra obtained at VLT (Basaetal. 2007) and
troscopic data, especially in the rest-frame UV. Indeed, if only Gemini (Howelletal. 2005) during the regular SNLS observa-
nearby SNe are used, photometric data do not cover the gaps tionprograms,whichaimattypingandmeasuringtheredshifts
between the central wavelength of the filter set used (mostly of SN candidates. Obviously more spectra were recorded (at
Johnson-CousinsUBVRI),andonereliesonlyonspectraforin- leastoneforeachSN)butwechoosetouseonlythosewithneg-
terpolation between those bands, which may introduce (weak) ligibleresidualcontaminationfromthehostgalaxy.Thecontri-
systematiceffectsonK-corrections. butionfromthehostgalaxywasremovedinthereductionproce-
One may argue that possible evolution of the SNe Ia with dure of the VLT spectra (Baumont 2007). For all spectra, the
redshift might cause some problems with the modeling since remaining contamination was evaluated a posteriori using the
objects at all redshifts are used to obtain the model. Actually, modelitselfwiththefollowingprocedure:usingtheSNparame-
themodeldescribesanaverageSNIaatanaverageredshiftbut tersretrievedfromthefitoflight-curves,themodelwasfitonthe
evolutioncanstillbestudied.Forinstance,withoutanyapriori SN spectrum with re-calibration parameters, and an additional
ontheeffectofevolutiononSEDsequence,onecanlookatthe contribution of the host galaxy. We used for this purpose tem-
variationofthe(x )parameterswithredshift. platesofelliptical,S0,Sa,SbandScgalaxies,theactualgalaxy
k
typewasfittedatthesametimeasitsnormalization.Allspectra
with a non zero contribution of the galaxy (at 68% confidence
3.1.Thenearbysupernovasample
level)werenotusedinthetrainingsample.
We use a sample of 52 nearby supernovae (without restricting Figure1showsthe(p,λ)phase-spaceregioncoveredbythe
ourselvestoverynearbyonesasinSALT)listedtable2.Those photometric and spectroscopic data sets. Since we do not use
SNewereselectedfromthequalityoftheirlight-curvesampling infra-redphotometricdata,there-calibrationofspectramaynot
where we basically require measurements before the date of bereliableforrest-framewavelengthslargerthan8000Å,which
4 J.Guyetal,SNLSCollaboration:SALT2
of the SALT modelSED sequence with respect to all previous
components.
8000 102 We endup with morethan 3000parametersto fit, with ob-
viousnon-linearities,sothatweusedtheGauss-Newtonproce-
dure,whichconsistsin:
1. Approximatinglocallytheχ2 byaquadraticfunctionofthe
)6000
Å 10 parameters.
(
λ 2. Solvingalargelinearsystemtogetanincrementofthepa-
rameters(δP).
i
3. Incrementtheparametersanditerateuntiltheχ2 decrement
4000
withrespecttothepreviousiterationbecomesnegligible.
1
First, the average model is estimated along with the color-
law, calibration coefficients for spectra, and parameters of the
2000
-20 0 20 40 SNe((x),c).Whenthesystemhasconverged,weaddanother
i
p component,andalltheparametersarefittedagain(components,
color-law,SNparameters).Theconvergencealgorithmisinsen-
sitivetotheinputsetofcomponents.
8000
4.2.Regularization
Theremightbesomedegeneracyin partofthe phasespace for
the given data set. For instance, if a phase×wavelength region
)6000
Å
isonlycoveredbyphotometryandnotspectroscopy,wedonot
(
λ have enough data to constrain the combinations of parameters
thatmodelspectralfeatures,whereaswecanstillmodelapho-
4000 tometric measurement,since the signal is integrated on a large
spectralband.Addingaregularizationtermintheχ2 solvesthis
issue. If its contributionis low enough,it will not alter signifi-
cantlythedeterminationofparametersthatareaddressedbythe
2000 data, while putting some limitation on the parameters that are
-20 0 20 40
not.Wehavechosentominimizesecondderivativeswithrespect
p
tophaseandwavelength(onceagain,effectiveonlywhenthere
isnotenoughdata).Theregularizationtermisthefollowing:
Fig.1Phase-spacemappingbyphotometricdata(top)andspec-
tra (bottom).For photometricobservations,the rest-framecen-
tralwavelengthofthefilterisconsidered. χ2REGUL =n×XMTkDTDMk
whereM isthevectordescribingcomponentk,Disthederiva-
k
tivematrixandnanormalizationthatcontrolstheweightofthis
isthecentralwavelengthofthe I-bandfilter. Also,wehavelit- regularizationwithrespecttodata.Sincesuchatermintroduces
tle spectroscopic informationin the UV for phases earlier than abiasintheestimator(departurefromthemaximumlikelihood
−10daysorgreaterthan10dayssincethespectroscopicobser- estimator),wehavetoquantifyitinordertoadjustthenormal-
vations of the SNLS are designed to be as close as possible to ization n. For this purpose, we used a simulated dataset. This
the date of maximum luminosity. The few late UV spectra we simulationhelpsus to define the resolutionof the model.Each
haveinoursamplecomefromIUEdatabase(INES2006). SNofthetrainingsamplewasadjustedusingtheSALTmodel,
then fake light-curvesandspectra were computedby replacing
each true measurement of the SN by the best fit value of the
4. Trainingthemodel model.Thetrainingprocedureappliedtothisdatasetgivesare-
sult that is slightly biased due to the regularizationterm in the
4.1.Thetrainingprocedure χ2 in the UV wavelengthregion.The weight of the regulariza-
The convergenceprocess consists in minimizing a χ2 that per- tion term (normalization n) was chosen so that the bias in K-
correctionsissmallerthan0.005magforallwavelength,which
mitsthecomparisonofthefulldatasetwiththemodelofequa-
issignificantlylessthanthestatisticaluncertainties.
tion 1. For each SN, the parameters are the normalization and
coordinates along the principal components (x ), a color and
k
re-calibration parameters for spectra if any. The actual com- 4.3.Modelresolution
ponents (M ) and the parameters of the color-law CL(λ) also
k
have to be estimated. This procedure requires a first guess for Thechoiceofthemodelresolutionisimposedbythedatasetwe
the model components (Mk), for a first estimate of normaliza- have. We used 10×120 parametersfor M0 (10 along the time
tion,spectrare-calibrationandcolor.We usedtheSALTmodel axis and 120 for wavelength), in a phase range of [−20,+50]
SED sequence for a SN with stretch = 1 for M and the dif- daysandaspectralrangeof[2000,9200]Å.Thisgivesaspectral
0
ference of SED sequence of a SN with stretch = 1.1 and the resolutionoforderof60Åwhichissufficientforthemodeling
previousoneforM (i.e.a linearizedversionof SALTmodel). ofSNewithbroadlinesduetothevelocityoftheejecta.ForM ,
1 1
Additionalcomponentswhereinitiatedwiththeorthogonalpart wechoosetousealowerresolution(10×60parameters).The
J.Guyetal,SNLSCollaboration:SALT2 5
time axis is remapped so that the time resolution at maximum ultravioletmaynotbereliable,whereasUVspectraobtainedby
and+20daysaftermaximumisafactortwobetterthanat−20 the InternationalUltravioletExplorerhavea verylowsignalto
and +50 days (approximately4.5 and 9 days respectively). As noiseratioforwavelengthslargerthan3000Å.Thisfeaturecan
inSALT, weusedonlytwofreecoefficientstomodelthe color hardlybeapproximatedbyabroad-bandcolorevolution.Hence
lawCL(λ)(thirdorderpolynomial,withtwocoefficientsfixedso we expect a net improvementof the accuracy of distance esti-
thatCL(λB)=0andCL(λV)=0.4log(10),seeEq.1).Usingthis matesintheUVrangewithrespecttoSALTorotherequivalent
numberofparameters,whenthemodelistrainedwiththesim- methodswhichrelyonasinglespectralsequence.Asanexam-
ulateddatasetdescribedabove,wefoundthatthelimitedreso- pleofthemodelingofthevariabilityofspectralfeatures,figure5
lution introduces a scatter in colors of only 0.01 mag standard presentsthevariationoftheR(SiII),asdefinedinNugentetal.
deviation(it is a scatter rather than a systematic effect because (1995),asafunctionof∆m retrievedfromthemodel.Itiscom-
15
of the varying epochs of photometric observations),which has paredto a compilationof observationsbyBenettietal. (2004).
anegligibleondistanceswhencomparedtotheintrinsicdisper- Thereisagoodmatchinthe∆m rangeofthetrainingsample
15
sionofSNeIaluminosities. (∆m <1.6).
15
One must however evaluate carefully the statistical signifi-
canceofthetrendsinthemodel.Statisticalerrorsofthismodel
5. Resultofthetraining
rely on the weighting applied to spectra, and are sensitive to
We decided to consider only two components for the current theerrorsassumedforphotometricmeasurementswhicharenot
analysis since additionalcomponentsare poorlyconstrained in verysecureformostnearbysupernovae.Hencethisaccuracyof
most of the phase space and marginally significant in the re- themodelingmustbeevaluatedwithdistributionsofresidualsas
gionofgooddatacoverage.Asthedatasetsimprovesowillthe describedinthenextparagraph.
powerto extractadditionalcomponents.As a consequence,for
each SN, we endedup with fourparameters,a date of B−band
6. Theremainingvariability
maximum, a normalization, the parameter x and a color. The
1
averagevalueofx anditsscalearearbitrarysincewecanmod- In order to assert the predictability of the model we need an
1
ifythecomponentsinconsequence.Weadopted< x >= 0and independentdata set that is not used in the training procedure.
1
< x2 >=1. However,sincewedonothavealargenumberofmeasurements
1
Figure 2 shows the variation of the UBVRI light-curvesas available, we resorted to a jackknife procedure: for each SN,
a functionof parameter x . We find thatmostofthe variability we trained the model using all SNe from Table 2 but this one,
1
can be described by a simple stretching of light-curvesdespite andlookedatresidualsoftheSNmeasurementstotheretrieved
thefactthatwedidnotforcesuchbehaviorinthemodel.More model,fittingonlytheparametersconcerningthisparticularSN,
quantitatively,the parameter x can be convertedinto a stretch i.e.thedateofmaximum,(x ),andcolor.
1 0,1
factor3,whoseactualvaluedependsonthereferencelight-curve Theresidualsobtainedbythismethodareapriorihighlycor-
template used, here the one of SALT and of Goldhaberetal. related,andthiscorrelationisdifficulttoestimatefromfirstprin-
(2001)(B-bandlightcurvetemplate“Parab-18”,G01);orinto ciples.Howeverwewouldliketousethisinformationtoextract
∆m (Phillips1993)usingthefollowingtransformations: someintrinsicvariabilityoftheSNe,beyondtheprincipalcom-
15
ponentsof the model,in orderto weightdata accordingto this
variabilityinadditiontomeasurementerrors.Weresortedtothe
s(SALT) =0.98+0.091x +0.003x2−0.00075x3
1 1 1 followingsimplification,usingtwokindsofresiduals(ormodel
s(G01) =1.07+0.069x −0.015x2+0.00067x3 errors):i) Diagonalerrorsofthemodelareestimatedfromfits
1 1 1
of light-curves with an independent normalization for each of
∆m =1.09−0.161x +0.013x2−0.00130x3
15 1 1 1 them. The residuals obtained are of course still correlated, but
weallowourselvestotreatthemasindependent,assumingthat
Since there is not a perfect match of the non-linear stretch
and∆m modelswiththisone,thosetransformations(obtained the correlationlength(alongtime axis)is smallerthan thedata
15 samplingformostlight-curves4.ii) K-correctionuncertainties,
withsimulations)varywiththeweightattributedtoeachphase
which can be estimated using the difference between the peak
(thescatterforstretchisabout0.02).
WealsonoticetheU −Bcolorvariationasafunctionof B- magnitude obtained from a single light-curve fit, and the one
band light-curve broadening.The value of U − B for phase=0 predictedbythemodelinthesamefilter,fittingalllight-curves.
does not vary with x (see Fig. 2), but when the flux is inte-
1
gratedin the phase range-10, +10days, we find thatU − B ∝
6.1.Diagonaluncertainties
−0.2×s(SALT).ThisisabouthalfthevalueobtainedwithSALT.
However,wefindacolorlawveryclosetotheoneobtainedwith Weusethecorrelationsbetweentheestimatesofthecomponents
SALT (see Fig. 3), despite the factthatthe supernovamodelis givenbythecorrelationmatrixretrievedattheendofthetrain-
significantlydifferentandthetrainingsetmuchlarger. ing procedure.However,we allow ourself to scale these statis-
TheseresultsconfirmthemainfindingsofSALT.Moreinter- tical uncertainties on the model in p,λ bins to account for the
estingisthevariabilityofspectrawiththefirstcomponentasdis- remainingvariability.
playedin Figure4 forthreephasesaboutmaximum.Itappears Themodelvariancecanthenbedefinedas:
thatthevariabilityinU-bandatmaximumidentifiedwithlight-
V (x ,p,λ) = S(p,λ)×V (x ,p,λ)
MODEL 1 MEAN 1
curvescanbeattributedtoasharpvariationofthespectrumfor
V (x ,p,λ) = HTV H +x2HTV H +2x HTC H
wavelengthslowerthan3400Å.Itispossibletoidentifysucha MEAN 1 0 0 0 1 1 1 1 1 0 0,1 1
featurethankstothehighredshiftSNLSspectra.Indeed,thecal- whereH (p,λ)andH (p,λ)arethe vectorsdefinedinSec. 2.1
0 1
ibrationofground-basedspectroscopicobservationsinthenear forcomponents0and1,V ,V andC arethefullvarianceand
0 1 0,1
3 Ofcourse,sincethemodelisalinearcombinationoftwocompo- 4 It is actually the purpose of the principal component analysis to
nents,wedonotretrieveexactlythestretchmodel. extractallthecorrelationsbetweenobservablesforagivenSN
6 J.Guyetal,SNLSCollaboration:SALT2
covariancematricesofthecomponents,andS(p,λ)thescaling (consistentwiththeoneobtainedinA06,Fig.11.),butanuncer-
function. taintyofonly0.022magnitudehastobeaddedtothestatistical
Ineach p,λbin,S(p,λ)isevaluatedsothat errorstomatchtheobserveddispersion.Diagonaluncertainties
and K-correctionuncertaintiesaretakenintoaccountinthefits
1Xhfi−x0(cid:16)HT0,iM0+x1HT0,iM1(cid:17)i2 =1 performedinthefollowingsection.
n σ2+x2S(p,λ)V (x ,p,λ)
i 0 MEAN 1
7. ImprovingthedistanceestimatesofdistantSNe
where (f) and (σ) are the measurements and their associated
i i
uncertainties. We evaluated separately these errors for light- Despite the fact that our modeling is less accurate in the UV
curvesand spectra, in order to take into accountthe correlated range,itisstillveryusefulfordistanceestimatesofhigh-redshift
errorsalongthewavelengthaxiswhendealingwithphotometric SNe(z > 0.8)forwhich,asinthecaseofSNLS,therest-frame
data. BandV-bandobservationsoftenhaveaverypoorsignaltonoise
Photometric residuals of the jackknife-like procedure are ratio.
shown figure 6. The data set are split in four rest-frame wave-
length ranges, [3200,3900], [3900,5000], [5000,5700] and
7.1.Light-curvefitofdistantsupernovae
[5700,7300] Å, that roughly correspond to U, B, V, R and I
bandsrespectively.Themodeluncertaintiesbasicallyfollowthe Figure 8 shows the SN Ia SNLS-04D3gx at z = 0.91 fitted by
statisticalerrorsofdata,withverygoodaccuracyatpeakbright- the model. All four light-curves (g,r,i,z) are well described by
ness and a poor quality at early and late phases, especially in thebestfitmodelforwhichonlyfourparameterswereadjusted
the U−band. In the rise time region, the large errorsare partly (dateofmaximum,normalization,colorandx ).Theχ2 perde-
1
duetothelimitingresolutionofthemodel.Thoseerrorsarealso gree of freedom (d.o.f.) of the fit is 0.76 (for 50 d.o.f.) when
displayedinfigure2for|x1|=2. diagonal and K-correction uncertainties of the model are con-
When fitting spectra, we imposed the values of the date of sidered.Table1illustratesthegainintheaccuracyofthecolor
maximum, x1 and color obtained with the light-curvesfit. The estimateforSNLS-04D3gx.
only remaining free parameters were those used to photomet-
rically ”re-calibrate” spectra. We found that model accuracy is
pooratearlyphasesandintheUVregion(Fig.4). bands λmin color
i,z 3980 −0.220±0.180(s)±0.033(d)±0.005(k)
These estimates of the model errors can be accounted for
r,i,z 3250 −0.147±0.051(s)±0.026(d)±0.049(k)
when fitting the light-curves or spectra. It gives more reliable
g,r,i,z 2520 −0.172±0.047(s)±0.023(d)±0.047(k)
statistical errors for parameters (peak brightness, color, x and
1
dateofmaximum)thanwhenonlystatisticalerrorsofmeasure- Table1ErroronthecolorestimateofSNLS-04D3gxasafunc-
mentsareconsidered. tionofthenumberoflight-curvesincludedinthefit.Thecontri-
butionstotheerroraremeasurementstatisticalerrors(s),diago-
nalmodelerrors(d)andK-correctionerrors(k).
6.2.K-correctionuncertainties
A direct approach to access the quality of K-corrections is to
comparetheobservedpeakmagnitudeofalight-curveinagiven Thetotaluncertaintyonthecolorparameterisreducedbya
filterwiththeonepredictedbythemodelusingafitoftheother factor 2.5 when g and r-band (rest-frame UV) light-curves are
light-curves. Figure 7 presents the differences between the ob- includedinthefit.Weseethatthemodeluncertaintiesarelarge
served and predicted magnitudes as a function of the effective inthiswavelengthrangeandthereforemustbepropagated.
rest-framewavelengthoftheinstrumentresponseused.
A more elaborated approach consists in modeling K-
7.2.Improvingcosmologicalresults
correction errors with a parametric function of wavelength
whichvaluevanishesforwavelengthcorrespondingtotherest- Theparametersretrievedfromthelight-curvefitcanbeusedto
frame B andV−bands(errorson B and V magnitudesat maxi- estimatedistancesusingthesameprocedureasdescribedinA06.
mumenterinthenormalizationandcolorevaluation).Foreach Thedistanceestimatorisalinearcombinationofm∗,x andc:
B 1
SNwithenoughlight-curves,those K-correctionadditionalpa-
rameters can be estimated and their standard deviation used to µ =m∗ −M+α ×x −β×c
B B x 1
derive a model of K-correction errors. Such a model is repre-
sented by the solid line figure 7 and is given by the following with m∗, x and c derived from the fit to the light curves, and
B 1
formula: α 5,βandtheabsolutemagnitudeM areparameterswhichare
x
fittedbyminimizingtheresidualsintheHubblediagram.Asin
σK(λ)= 0.022(cid:16)λλU−−λλBB(cid:17)3 forλ<λB A06,we introducean additional”intrinsic” dispersion(σint) of
= 0.018 λ−λV 2 forλ>λ SNabsolutemagnitudestoobtainareducedχ2 ofunityforthe
(cid:16)λR−λV(cid:17) V bestfitsetofparameters.
We minimize the following functional form, which gives
Since the estimate of σ is based on the fit of the normal-
K negligiblebiasestotheestimatesofα andβ:
ization of light-curves,it measures a dispersion of colorsaver- x
aged over the phase range defined by the data set. Clearly, the
VTX −M−5log d (θ,z)/10pc
K-correction errors are large in the UV range. Also, those er- χ2 = s 10(cid:0) L (cid:1)
X
rors must be added to the statistical errors on normalizations, s VT C(Xs)V
but they do notaccountfor the whole observedscatter. For in-
stance,forλU ≃3600Å,wefindadispersionof0.04magnitude 5 NotethatthedefinitionofαxdiffersfromthatofαinSALT
J.Guyetal,SNLSCollaboration:SALT2 7
with Light-curvescanbefittedwithallthemodelsfromthejack-
m∗ 1 knifeprocedure,sothatwecanderiveasmanyestimatesofΩ
 B   M
Xs =xc1, V=−αβx aosftthheisredaisrteriSbNuteioinnitshe0.t0ra0i3n,insogsthaamtpwlee.eWxepefcotuanddethvaiatttihoenRfrMomS
θ stands for the cosmological parameters that define the fitted a modeltrained with an infinitenumberofSNe of the orderof
modelanddListheluminositydistance.C(Xs)isthecovariance 0.0015onΩM(foraflatΛCDMcosmology),avaluethatisneg-
matrixof the parametersX for whichwe haveincludedin the ligiblecomparedtotheothersourcesofsystematicerrorsinsuch
s
variance of m∗ the intrinsic dispersion and an error of the dis- ananalysis.
B
tance modulus due to peculiar velocities, which we take to be
300km.s−1.
8. Otherapplications
To estimate the systematic effect due to modeling of the
SN Ia SED sequence, we fit the data set of A06, which con- Theproposedmodelprovidesatoolforspectroscopicandpho-
sistsin44nearbySNeIaand71SNLSSNe.Whereasweused tometric identification,and photometricredshiftdetermination.
improvedphotometryforthetrainingofthemodel,weconsider Adetailedanalysisofthepurityandefficiencyofaphotometric
here exactly the same data set as in A06, in order to ease the identificationtoolwithrespecttoSNeIb,IcandIIisbeyondthe
comparisonwiththeresultsobtainedwithSALT. scopeofthispaper.
Thankstotheevaluationofthemodelerrors,wecansafely
use all available light-curves in the fit. Especially the r−band
8.1.Spectroscopicidentification
dataatredshiftsgreaterthan0.8(effectiverest-framewavelength
lowerthan3440Å)areveryusefultoconstrainthecolorofthe Themodelproposedallowssimultaneousfitsoflight-curvesand
supernova. With this additional information, the uncertainty in spectra (with additional galaxy templates to evaluate the host
distancemoduliissignificantlyreducedathighredshift,yielding contamination as mentioned in Sec. 3.2). This can turn out to
abetterresolutiononcosmologicalparameters.ForflatΛCDM beveryusefulforidentificationofTypeIasupernovae.Indeed,
cosmology,weobtain: someTypeIcSNecanpresentlight-curvesandspectrathatlook
qualitatively like SNe Ia, and in the most extreme cases, pho-
Ω = 0.240±0.033
M tometricandspectroscopicidentificationstakenseparatelymay
α = 0.13±0.013
x failtotagthisobjectcorrectly.Adetailedanalysisisbeyondthe
β = 1.77±0.16 scopeofthispaper.Wewillshowonlytwoexamples.
First, figure 10 shows the observed light-curves and spec-
with σ = 0.12 (which is smaller than the value of 0.13 in
int trumofSNLS-03D4agata redshiftof0.285alongwiththere-
A06 because we have considered uncertainties in the model).
sultofthesimultaneousfit.Four“re-calibration”parametersfor
TheRMSoftheresidualsaroundthebestfitHubblerelationis
thespectrumwereconsidered(sincetherearefourlight-curves).
0.161mag(comparedto0.20inA06,seeFig.9).
Theχ2 perdegreeoffreedomforthelight-curvesandthespec-
TheuncertaintyonΩ isimprovedby10%withrespectto
M trum are respectively 0.53 (for 28 d.o.f.) and 0.63 (for 1770
theA06(Ω =0.263±0.037,table3ofAstieretal.2006)anal-
M d.o.f.),taking into account the model errors 7), so that this SN
ysis. We find a difference of 0.023 on Ω which is consistent
M canbesafelyconsideredasatypicalnormalSNIa.
with the assumed systematic error due to modeling of 0.02 in
NowifweconsideranSNIc,forinstanceSNLS-03D4aaat
thatpaper.HoweverthesupernovaSNLS-03D1cmataredshift
aredshiftof0.166,themodelgivesaverybadfitofthedata(re-
of0.87nowappearsasasignificantoutlierasthedistancemod-
ducedχ2 of4.6(for10d.o.f.)and1.6(for1807d.o.f)forlight-
ulusresolutionimproved6.Thisobjecthasaspectrumwithlow
curvesandspectrarespectively),allowingforaclearrejectionof
signaltonoiseratioandwasclassifiedasprobableIa.Discarding
thisevent.
this object from the analysis gives Ω = 0.246±0.032 and a
M
standarddeviationofresidualsof0.154magnitude.
Since the currentmodelsignificantly improvesdistance es- 8.2.Photometricredshifts
timates at high redshifts, we obtain a greater improvement on
For photometric redshift determination, one can compare the
theestimationoftheequationofstateofdarkenergy.Asanex-
redshiftestimate based ona simultaneousfit of alllight-curves
ample, we may use the figure of merit proposed by the Dark
of a givensupernovaand the much moreprecise spectroscopic
EnergyTaskForce(Albrechtetal.2006),whichisinverselypro-
redshiftderivedfromthespectroscopicobservationofthehostof
portionaltotheareaofthe95%confidencelevelcontourinthe
theobject.Usingallavailablelight-curves(uptofourforSNLS)
plane (w ,w ), where the following parametrization is consid-
a p
allowsustorelaxassumptionsonthelightcurveshapeparame-
eredfortheequationofstateofdarkenergy:
ter(x )andthecolor,withoutanyuseoftheabsoluteluminosity,
1
w=wp+(ap−a)wa sothatwedonotrequireanyprioronthecosmologicalparame-
ters.
a being the scale factor, anda a referencescale factor chosen
p
WehaveappliedthismethodtoallSNeIafromTable2with
so thatthe estimates of w and w are uncorrelatedfor a given
p a
at least 3 light-curves, using for each SN the model obtained
experiment. Using the baryon acoustic oscillations constraints
from Eisensteinetal. (2005) (Eq. 4), a = 0.851, and we im- without this SN (always to avoid over-training). A Gaussian
p
priorforx wasassumedbasedonthestatisticsfromthetraining
prove this figure of merit by 35% with respect to the analysis 1
usingSALT(forwhicha isslightlydifferent). procedure,x1 =0±1.Nopriorwasappliedtocolor.
p
Theresultingphotometricredshiftis comparedto thespec-
6 SNLS-03D1cmis0.6±0.2magdimmerthanexpectedforthebest troscopic one on Figure 11. The RMS of the distribution of
fitcosmology,whichcorrespondstoa3σdeviationwhenincludingthe ∆z/(1 + z) is 0.02 with no significant bias. A Gaussian fit of
intrinsicdispersion.Thishasa27%probabilitytooccuratleastonceby
chanceforoursampleof115SNe,ifSNedistancesarescatteredabout 7 Withthemodelerrors,theaverageχ2perdegreeoffreedomforall
theHubblelawfollowingaGaussiandistribution. theSNeofthetrainingsampleisonebydefinition
8 J.Guyetal,SNLSCollaboration:SALT2
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8 available together with the fitter source code at
http://supernovae.in2p3.fr/∼guy/salt
9 LSST:www.lsst.org
J.Guyetal,SNLSCollaboration:SALT2 9
-19 0.2 B V
* -18
U
-17
0.1
-16 B
-10 0 10 20 30 40 A
phase - 0
λ
A
-20
-19 -0.1
U -18
-0.2
-17
4000 6000 8000
-16 λ (Å)
-10 0 10 20 30 40
phase
Fig.3 Thecolorlaw c×CL(λ) asa functionofwavelengthfor
-19 avalueofcof0.1(solidline).Thedashedcurverepresentsthe
extinctionwithrespecttoBband,(A −A ),fromCardellietal.
-18 λ B
B (1989) with RV = 3.1and E(B−V) = 0.1,and the dottedline
-17 is the color law obtained with SALT (very close to the result
-16 obtainedhere).
-10 0 10 20 30 40
phase
-19
-18
V
-17
-16
-10 0 10 20 30 40
phase
-19
R -18
-17
-16
-10 0 10 20 30 40
phase
-19
-18
I
-17
-16
-10 0 10 20 30 40
phase
Fig.2 The U∗UBVRI template light curves obtained after the
training phase for values of x of -2, 0, 2 (corresponding to
1
stretches of 0.8, 1.0,and 1.2;darkto light curves)and null B
V
colorexcess. U∗ is a synthetictop hatfilter in the range2500–
3500Å.Theshadedareascorrespondtotheonestandarddevia-
tionestimateasdescribedinsection6.1.
10 J.Guyetal,SNLSCollaboration:SALT2
0.6
)
Si II 0.4
(
R
2000 3000 4000 5000 6000 7000 8000
Wavelength (Å); phase=-10
0.2
1 1.5 2
∆ m
15
Fig.5 The solid curve presents the relation of
R(Si II) (Nugentetal. 1995) as a function of ∆m along
2000 3000 4000 5000 6000 7000 8000 15
Wavelength (Å); phase=0 withmeasurementsfromBenettietal.(2004).
2000 3000 4000 5000 6000 7000 8000
Wavelength (Å); phase=15
2000 3000 4000 5000 6000 7000 8000
Wavelength (Å); phase=30
Fig.4 Spectra at -10, 0, +15 and +35 days aboutB-band max-
imum for values of x of -2, 0, 2 (corresponding to stretches
1
of 0.8,1.0, 1.2;lightgray,blackcurve,darkgray) andnull B
V
color excess. The shaded areas correspond to the one standard
deviationestimateasdescribedinsection6.1.Thedashedcurves
representsthespectraofN02(version1.2).Allspectraarearbi-
trarilynormalized.