Table Of ContentMNRAS000,000–000(0000) Preprint23January2017 CompiledusingMNRASLATEXstylefilev3.0
Photometric classification and redshift estimation of LSST
Supernovae
Mi Dai,1(cid:63) Steve Kuhlmann,2 Yun Wang,1,3 and Eve Kovacs2
1Homer L. Dodge Department of Physics & Astronomy, University of Oklahoma, 440 W Brooks Street, Norman, OK 73019, USA
2Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA
3IPAC, Mail Code 314-6, California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, USA
7
1 23January2017
0
2
n ABSTRACT
a Supernova(SN)classificationandredshiftestimationusingphotometricdataonlyhave
J become very important for the Large Synoptic Survey Telescope (LSST), given the
0 largenumberofSNethatLSSTwillobserveandtheimpossibilityofspectroscopically
2 following up all the SNe. We investigate the performance of a SN classifier that uses
SN colors to classify LSST SNe with the Random Forest classification algorithm.
] Our classifier results in an AUC of 0.98 which represents excellent classification. We
O
are able to obtain a photometric SN sample containing 99% SNe Ia by choosing a
C probability threshold. We estimate the photometric redshifts (photo-z) of SNe in our
. sample by fitting the SN light curves using the SALT2 model with nested sampling.
h
p Weobtainameanbias((cid:104)z −z (cid:105))of0.012withσ zphot−zspec =0.0294without
- phot spec 1+zspec
o using a host-galaxy photo-z prior, and a mean bias ((cid:16)(cid:104)zphot−zsp(cid:17)ec(cid:105)) of 0.0017 with
r σ zphot−zspec = 0.0116 using a host-galaxy photo-z prior. Assuming a flat ΛCDM
st 1+zspec
a m(cid:16)odel with Ω(cid:17)m = 0.3, we obtain Ωm of 0.298±0.008 (statistical errors only), using
[ the simulated LSST sample of photometric SNe Ia (with intrinsic scatter σ =0.07)
int
derived using our methodology without using host-galaxy photo-z prior. Our method
1
will help boost the power of SNe from the LSST as cosmological probes.
v
9
Key words: cosmology: observations–supernovae: general
8
6
5
0
1. 1 INTRODUCTION reliable and accurate typing and redshift estimation using
0 the photometric data of the SNe only.
Since the accelerating expansion of the Universe was dis-
7
covered by observing distant Type Ia supernovae (SNe Ia)
1
(Riess et al. 1998; Perlmutter et al. 1999), SNe Ia have
:
v beenplayinganimportantroleinconstrainingtheunknown
i cause behind the observed cosmic acceleration, or what we
X
refertoasdarkenergy.TheSNeweobserveneedtobecor- In this paper, we use realistic LSST SN simulations to
r
a rectly typed and accurate redshift information needs to be studytheperformanceofSNclassificationwiththeRandom
obtained,beforetheSNeIacanbeusedtoconstraincosmo- Forest classification algorithm, using SN colors as features
logicalmodels.Withasamplesizeof<1000,itispossibleto for the first time, together with parameters from a general
obtain correct types and redshifts of SNe via spectroscopy. functionfitofthelightcurves.Weareabletoobtainapho-
OngoingandplannedsurveyssuchastheDarkEnergySur- tometrically classified SN Ia sample that is 99% pure. We
vey (DES) (Bernstein et al. 2012), and the Large Synoptic derive the photometric redshifts of the SNe by fitting their
Survey Telescope (LSST) (LSST Science Collaboration et lightcurvesusingtheSALT2modelwithanestedsampling
al. 2009), will observe a dramatically increased number of algorithm, and study the performance of this photometri-
SNe,makingitimpossibletospectroscopicallyfollowupall cally classified sample with photometric redshifts in con-
theSNe.SNcosmologywillrelyonphotometrictypingand strainingcosmology.WedescribeoursimulationsinSection
redshift estimation. It is important to derive methods for 2,ourclassificationmethodinSection3,ourphoto-zestima-
tor in Section 4. The construction of a photometric Hubble
Diagram is described in Section 5, followed by a summary
(cid:63) Email:[email protected] and discussion in Section 6.
(cid:13)c 0000TheAuthors
2 Dai et al.
2 SN SIMULATIONS classification algorithms when they are used together with
the two parameters from the general parametrization that
WeusetheSNANA1 (Kessleretal.2009)softwaretogener-
describes the rising and falling time of the SN light curves.
aterealisticSNlightcurvesimulations;itprovidesasimula-
tionlibrarythatcontainsthesurveyconditioninformation,
and the LSST filter transmissions. Our simulation is based
3.2 General parametrization of SN light curves
on 10 years of operation on 10 deep drilling fields of LSST.
The actual number and location of the deep drilling fields, The SN light curves (both Type Ia and non-Ia) are found
as well as the observing cadence, are still under study. The to be well fitted using a general functional form; it has
observing cadence will affect the final number of SNe that nospecificphysicalmotivation,butdescribesthelightcurve
wecanobserveandthequalityofthelightcurves.Wegener- shapes in a model-independent manner. Following Bazin et
atemixed-typeSNebyusingSNratesforbothTypeIaand al. (2009), we use the following equation to fit the SN light
non-Ia (core-collapse (CC) SN). The SN rate is in the form curves (we refer to this as the “Bazin function”):
of α(1+z)β. For SNe Ia, we use the rate measured by Dil-
dayetal.(2008),withα =2.6×10−5SNeMpc−3h3 yr−1
and βIa =1.5; for the coIrae-collapse SNe, we assume7α0CC = f(t)=A exp−(t−t0)/tfall +B (2)
6.8 × 10−5SNeMpc−3h3 yr−1 and β = 3.6, following 1+exp−(t−t0)/trise
70 CC
Bernstein et al. (2012); LSST Science Collaboration et al. In this equation, t and t measure the declining
fall rise
(2009). The light curves are generated using the extended and rising time of the light curve, A is the normalization
SALT2model,whichextendsthestandardSALT2modelon constant, and B is aconstantterm. We shouldnote thatt
0
bothendsof thespectra.Theuse ofthis extendedmodel is isnotexactlythedateofmaximumflux(dateatthepeakof
essential in generating light curves in all six LSST bands, thelightcurve),andsimilarlyAisnotexactlythemaximum
sincewiththestandardSALT2modelsomebandswithcer- flux.Bycalculatingthederivativeofthefunctionweareable
tain redshifts will not be generated if they fall out of the to obtain the functional form of date at the maximum flux
wavelength range of the standard model. However, since and the value of the maximum flux:
thereislittlefluxoutofthewavelengthrangeofthestandard
SALT2model,thelightcurvesgeneratedinthosebandsare
t
basically constant noise. Section 4.1 gives a brief review of t =t +t ln( fall −1) (3)
max 0 rise t
the SALT2 model. We make a quality cut during the simu- rise
lation by requiring that the photometry in at least 3 bands
have a maximum signal-to-noise-ratio (SNR) greater than f(t )=Axx(1−x)1−x+B,x= trise (4)
5. A total of 144246 SNe are obtained, including 62147 Ia, max t
fall
67631 II, and 14468 Ibc. The redshift range of our simula-
The equation above also indicates that we should set
tionisfrom0.01to1.2.Wealsoapplyotherqualitycutson trise <1inordertohaveameaningfult .Thisconstraint
thissampleindifferentstages,whicharediscussedinsection tfall max
is useful in excluding some of the bad fits. More details on
3.3.
the quality cuts are described in Section 3.3.
The SN spectra are near zero for a rest-frame wave-
length that is less than 2000 Angstroms. So for higher red-
3 SN CLASSIFICATION shift SNe, there is little flux in the u, g band. At the other
end of the spectra, the Y band data are usually noisy. We
3.1 Using SN colors for classification
notice that the Bazin function does not fit well for these
InWangetal.(2015),SNcolorsareusedtobuildananalytic bandswithlowSNR.Sowefitthelightcurvewithdifferent
photo-z estimator and are shown to have very good perfor- forms depending on the SNR of the band being fitted. For
mance. Inspired by the ability of using SN colors only to SNR>5,thelightcurveisfittedusingEq.2,otherwisethe
estimate the SN photo-z, we expand the usage of SN colors light curve is fitted to a constant f(t)=B.
inSNclassification.TheSNlightcurvesarefirstfittedintoa Weutilizethecurvefitprocedureinpythonscipy2.Itis
generalfunctionalformthatisdescribedindetailinSection necessary to set initial values and limits for the parameters
3.2, so that a relatively accurate peak for each band is ob- being fitted. To achieve better results, we do the fit in two
tained. The peak magnitudes (converted from peak fluxes) steps with different initial conditions and parameter limits.
are then calculated using the fitted parameters (Eq. 3 and We list the initial values and parameter limits in Table 1.
4), and the colors are calculated as the difference between The initial values of the 2nd fit are calculated using the
the peak mag of the two adjacent bands. resultsofthe1stfit.Wedefinea“successfulfit”assatisfying
the following conditions: t > t > 1. The successfully
fall rise
fitted bands in the 1st fit are kept and the median of the
cij =mp,i−mp,j (1) parametersofsuchbandsareusedastheinitialvalueinthe
2ndfit.A2ndfitisalsoperformedwhentheconstantfitting
where i and j represents two of the adjacent bands of the
returnsaB valuelargerthan5;thisusuallyhappensforthe
LSST filters (ugrizY).
Y band where the SNR is lower than 5 but there is indeed
WefindthattheSNcolorscanbeusedinSNclassifica-
signal and can be fitted with Eq. 2.
tionwithverygoodperformanceusingthemachinelearning
1 http://snana.uchicago.edu/ 2 https://www.scipy.org/
MNRAS000,000–000(0000)
SN photometric classification and photo-z estimation 3
The peak magnitude in each band is calculated as: algorithm to demonstrate the performance of classification
usingSNcolors.Weadoptacodesimilartotheoneusedin
m =−2.5log (f )+zero-point (5)
p 10 max hostgalaxyidentificationbyGuptaetal.(2016),andmodi-
where fmax is calculated using Eq. 4. For any band that is fiedittosuitourneeds.Fordetailsaboutthealgorithm,see
fittedtoaconstant,tfall,trise,andmparesetto0.Thecolors Breiman(2001).Therearealsomanyothermachinelearning
arecalculatedusingEq.1,regardlessofthevalueofmp.We algorithmsthatcanbeused,mostofwhichareveryeasyto
notice that the colors can be 0, or the opposite of the peak implement.Thecomparisonofperformanceforseveralcom-
magnitude in one band, instead of actual colors, when the monly used machine learning algorithms can be found in
zero peak magnitude is used in calculating the colors. We Lochner et al. (2016).
treat this as a property of our sample and pass it to the As described in Section 3.1, a total number of 17 fea-
classifier. tures are passed to our classifier. The features are: 12
Bazin-fitparameterst (u),t (u),t (g),t (g),t (r),
fall rise fall rise fall
t (r),t (i),t (i),t (z),t (z),t (Y),t (Y),and
3.3 Quality cuts rise fall rise fall rise fall rise
5 colors c , c , c , c , c .
ug gr ri iz zY
In order to obtain a high quality sample, we apply several We now summarize the concepts that are commonly
quality cuts, both before and after the general function fit. used in presenting the classification results.
Before the fitting, we require that the max SNR is greater
than5foratleast3bands,andthatatleast3bandshave1
pointbeforethepeakand2pointsafterthepeak,including 3.4.1 Confusion matrix
at least one of the SNR>5 bands. These cuts ensure that
For a binary classification problem, a confusion matrix is
the light curve has a well-defined peak and not too noisy
definedinTable2.Inourcase,thetwoclassesare‘Ia’(Yes)
in at least one band so that at least one successful fit is
and ‘non-Ia’ (No).
achieved in the first step described above.
After all the pre-fitting quality cuts are applied, the
fitting program is able to return a set of parameter values
3.4.2 Receiver operating characteristic (ROC) curves
for most of the SNe, although some of the values are not in
areasonablerange.Sowemakeaseriesofcutsbasedonthe We define the true positive rate (TPR) and the false posi-
parameter distributions. The cuts we used are listed below: tiverate(FPR)asthefollowing(accordingtotheconfusion
matrix):
• t >1, and t not close to 1 with tolerance = 0.01
rise rise
• −20<B <20
• χ2/d.o.f <10 TP
TPR= (6)
• t <150 TP+FN
fall
• t <t
rise fall
• A<5000
FP
• A <100 FPR= (7)
err FP+TN
• t <50
0,err
• t <100 By varying the probability threshold within a classifier
fall,err
• t <50 in determining the class, different values of TPR and FPR
rise,err
• A(Y),A(u)<1000 are returned. The ROC curve is defined as TPR vs FPR,
since we would expect an excellent classifier to have high
These cuts also serve to exclude some of the non-Ia’s,
TPR with low FPR. Another value that is often used in
especiallytypeII’swhichhavearatherlargert value.The
fall comparing classification results is the area-under-the-curve
remainingfractionsaftercutsforeachofthethreeSNtypes
(AUC) of a ROC curve. For perfect classification, AUC =
inthesimulationsareIa,IbcandIIare55%,38%and15%,
1, the ROC curve behaves as a step function, while for a
respectively.
random classification, AUC = 0.5, the ROC curve behaves
This after-cut sample is used to test our classification
as a diagonal line. An AUC that is larger than 0.9 usually
algorithm,whichcontains68%Ia’s,11%Ibc’sand20%II’s.
represents excellent classification.
Wenoticethattheχ2 cuteliminatesalmostallofthelow-z
(z < 0.3) SNe, since the low-z SNe usually have very high
SNR and have a second peak in the redder bands, which
3.4.3 Efficiency and Purity
cannot be well-fitted using Eq. 2, and thus result in very
large χ2 per d.o.f. We can also define the efficiency and purity using the con-
DetailedlistsoftheremainingnumberofSNeaftereach fusion matrix:
cut are shown in Appendix A. TP
efficiency= (8)
TP+FN
3.4 SN classification with random forest
algorithm TP
purity= (9)
TP+FP
Machine learning algorithms are used in SN classification
recently (Lochner et al. 2016; Mo¨ller et al. 2016), and have To achieve higher purity usually means sacrificing the
excellent performance when the features used in the classi- efficiency, and vice versa, given that the probability thresh-
fierarecarefullyselected.HerewechoosetheRandomForest old is varied.
MNRAS000,000–000(0000)
4 Dai et al.
Table 1.2-stepgeneralparametrizationfit:initialconditionsandparameterlimits
1ststep 2ndstep
initialvalue limits initialvalue limits
A fluxatpeak [0,inf] fluxatpeak [0,inf]
t0 timeatpeak [-inf,inf] median(t0) fixed
t 15 [0,inf] median(t ) [1,inf]
fall fall
tfall 5 [0,inf] median(trise) [1,inf]
B 0 [-inf,inf] 0 [-inf,inf]
Table 2.Confusionmatrixforabinaryclassification
PredictedClass
Yes No
Yes TurePositive(TP) FalseNegative(FN)
ActualClass
No FalsePositive(FP) TrueNegative(TN)
3.5 Training sample size determination
1.0
Theclassificationalgorithmsrelyonatrainingsamplewith
knowntypestopredicttypesforthetestsample.Thereare 0.8
generally two ways that a training sample can be obtained:
e
one is to use a spectroscopic sample from the same survey, Rat
0.6
butthesamplesizecanberelativelysmall,comparedtothe ve
large number of SNe that LSST can observe; the other is siti
o
simplyusingrealisticsimulations,sothesamplesizecanbe eP0.4
u
as large as we need, representing good statistics of the test Tr
sample. 0.2
We compare the effects on the classification results by
ROCcurve(area=0.98)
varying the training sample size as a fraction of 0.05, 0.1,
0.0
0.3, 0.5 and 0.9 of the total sample. We find that a train- 0.0 0.2 0.4 0.6 0.8 1.0
ingsetof0.05or0.1fractionofthewholesampleresultsin FalsePositiveRate
an AUC of 0.96, while a fraction greater or equal to 0.3 re-
sultsinanAUCof0.98,whichindicatesthatalargeenough Figure 1.ROCcurveforourclassificationresult.
trainingsamplesizeisrequiredtobestrepresentthesample
and thus leads to better classification performance. How-
ever,asoursamplesizeafterallthequalitycutsis∼39000 Section3.1.ThisAUCvalueiscomparabletorecentstudies
including all types, a 0.1 fraction with ∼ 3900 SNe is al- (Lochner et al. 2016; Mo¨ller et al. 2016) with different data
ready larger than current spectroscopically confirmed data sample or simulations.
sets.Thismeansthataspectroscopictrainingsetforclassifi- Fig. 2 shows the purity and efficiency curves as the
cationwillbechallengingtoobtain.Wealsoconcludethata threshold probability varies. High purities can be obtained
trainingsamplewithcomparablesizetothetestsamplewill by sacrificing some efficiency. We notice that a 90% - 95%
result in best performance. In this paper, we show the clas- puritycanbeeasilyachievedwithefficiencylargerthan90%.
sification result with the 0.3-fraction training sample. We Whileweaimatapurityof99%forthefollowingcosmolog-
use the same sample for the following analysis, in order to ical analysis, the efficiency dropped to 74%. Note that this
obtainaslargeasamplesizeaswecanforthecosmological efficiency is the classification only efficiency, not including
analysis. When dealing with real data in the future, a sim- the quality cuts through all the procedures. Our analysis
ulated sample with the same size as the real sample can be results in a photometric sample with 13744 SNe with 99%
used for best performance. purity.
3.6 Classification results 4 SN PHOTOMETRIC REDSHIFT
Wenowpresenttheclassificationresultsusingtheconcepts Accurateredshiftinformationisessentialinconstructingthe
defined in Section 3.4. Fig. 1 shows the ROC curve for our Hubble Diagram and constraining cosmology. Analyses of
classification,withanAUCof0.98,indicatingthatwereach past and ongoing surveys rely on spectroscopic redshifts –
excellentclassificationbyusingthefeatureswedescribedin eitherfromtheSNspectraorthehost-galaxyspectra.With
MNRAS000,000–000(0000)
SN photometric classification and photo-z estimation 5
4.2 2-stage fit using nested sampling
1.0
P]t Wechoosetousethenestedsamplingmethodforlightcurve
≥0.9
a fittingusingSALT2.Wefindthatnestedsamplingresultsin
P[I betterphoto-zestimates,comparedtothenormalmaximum
urity 0.8 likelihood method using MINUIT.
P
ncy, 0.7 The model is not well characterized in the UV region
ficie withadramaticallyincreaseduncertainty,whichcanleadto
Ef a wrong fit if the parameter limits are not set correctly.
a 0.6
NI efficiency
S purity We take advantage of a 2-stage fit again in the SALT2
0.5 fit. Now we describe the two stages in detail:
0.0 0.2 0.4 0.6 0.8 1.0
ThresholdProbabilityforClassification,Pt
Fortheinitialfit,weaimatlocatingtherightrangesof
theparameters.Themodelcovarianceisnotusedinthisfit,
Figure 2. Purity and efficiency curves for our classification re-
onlythestatisticalerrorsfromthephotometryareincluded.
sult, red solid line shows the purity curve with respect to the
We set the parameter limits as follows: 0.01 < z < 1.2,
threshold probability chosen, blue dashed curve shows the effi-
|x |<5,|c|<0.5,t −15<t <t +15;andthebound
ciencycurve.Theverticallineindicatesthethresholdprobability 1 min 0 max
oftheamplitudeparameter,x ,isdeterminedinternallyby
for99%purity. 0
SNCosmo.
Thesecondfitislimitedtoasmallerrangedetermined
LSST, it is impossible for us to obtain spectroscopic red- by the 1st fit, with x , c, t limited to a 3-sigma range
1 0
shifts for all the SNe; it is also unclear whether we will be fromthemeanvalueofthe1stfit,whilex boundsarestill
0
able to obtain spectroscopic redshifts for the host galaxies “guessed” by SNCosmo. We limit the redshift to a rather
of all the SNe; thus it is useful to develop methods for SN larger range: z ±10σ , in order to better estimate the fi-
ini z
redshift estimations using the photometric data. Currently naluncertaintyofz.Wefindthatthe3σ limitfortheother
two kinds of approaches are proposed for SN photo-z esti- threeparametersisnecessaryforobtainingagoodfit,since
mation: one is analytic utilizing the multi-band SN colors the fit can easily be trapped in an unreasonable parameter
(Wang2007;Wangetal.2007,2015),theotheristemplate- regionwheretheχ2 isverysmallduetoalargevalueofun-
based by fitting the light curves into SN Ia models (Kessler certainty in the UV bands. This fitting deficiency has been
et al. 2010; Palanque-Delabrouille et al. 2010). We adopt observedinanotheranalysis(Dai&Wang2016),whichuses
thetemplateapproachinthispaper,byfittingtheSNlight Markov Chain Monte Carlo (MCMC) method to fit light
curves using the commonly used SALT2 model, using the curves to the SALT2 model. In Dai & Wang (2016), the
nested-sampling method. Our fits are performed using the model covariance is kept fixed to mimic and reproduce the
SNCosmo3 package. Wealso investigatetheperformanceof original SALT2 result. When we are simultaneously fitting
our photo-z estimator using a host-galaxy photo-z prior. theredshift,fixingthecovarianceisnotapplicable.Wealso
useanotherconditioninthefittingthathelpstoreducethis
fittingbias:forSNewithredshifts(fromtheinitialfit)that
arelessthan0.65,all6bandsareusedinthefitting;forthose
4.1 The SALT2 model thathaveredshiftsgreaterthan0.65,theuandgbandsare
The SALT2 model provides an average spectral sequence excludedfromthefitting.The0.65lineisdeterminedbyob-
and its higher order variations, as well as a color variation serving the fitted results using all 6 bands and determining
law,whichcanbeusedtofitSNIalightcurvesusingseveral wherethebiasstartstooccur.Abettercharacterizedmodel
parameters. The model flux is given as: in the UV band can be very useful in all aspects.
dF Byapplyingthe2-stepfitdescribedabove,weobtaina
(p,λ)=x ×[M (p,λ)+x M (p,λ)+...]×exp[cCL(λ)] (cid:16) (cid:17)
dλ 0 0 1 1 set of photo-z’s with an accuracy σ zphot−zspec =0.0294,
(10) 1+zspec
and a mean bias ((cid:104)z −z (cid:105)) of 0.0120, after applying
where p is the phase (rest-frame time from the maximum phot spec
a cut on the reduced χ2 of the SALT2 fit (χ2 <1.5). The
slipgehctt)r,aλlsiesquthenecreeastn-dfraitmsfierwstaovredleenrgvtahr,iaMtio0na,nCdLMis1thaerecotlhoer resultsareshowninFig.3.Theoutlier((cid:12)(cid:12)(cid:12)zp1h+ortez−dspzescpec(cid:12)(cid:12)(cid:12)>0.1)
law. x , x and c are light curve parameters that describe fraction is 1.12%. We also show that our method results in
0 1
theamplitude,stretchandcolorofthelightcurve;theyare accurate photo-z errors, as illustrated in Fig. 4. In Fig. 4,
fitted in a fitting process in which the date of maximum t thehistogramsof(z −z )/z areplottedindifferent
0 phot spec err
and the redshift z can also be fitted simultaneously. redshift ranges, where the z is the error in the photo-z
err
In this paper, we use an extended model that covers whichisoutputfromtheSALT2fitting.Thehistogramsare
wider wavelength ranges than the standard SALT2 model. fittedwithaGaussianfunction.Withthefittedσcloseto1,
we conclude that photo-z error estimation from the SALT2
fittingisaccurate.Suchafitcanonlybeachievedwhenthe
SALT2 model covariance is included in the fitting and the
3 https://sncosmo.readthedocs.io/ parameter limits are carefully chosen.
MNRAS000,000–000(0000)
6 Dai et al.
outcome.Notethatoursimulationisgeneratedusingamore
30 complicated intrinsic scatter model, so using this constant
25 rhmzpshoztp1h−+otz−zspzsepscpeecci==00..00219240 σint mayintroduceabias.Onlythestatisticaluncertainties
20 (cid:12)zp1h+otz−spzescpec(cid:12)>0.1:1.12% aCroesmcoonMsiCde(rLedewiins &thiBsrfiidtl.eT20h0e2)fi4tsiosftpwearrfeo.rmed using the
(cid:12) (cid:12)
(cid:12) (cid:12) Before fitting to cosmological models, a bias correction
15
termiscalculatedandappliedtothedistancemodulusµin
10 Eq.11.Thebiascorrectiontermisdeterminedin20redshift
binsbytakingthemeanofthedifferencesbetweenthefitted
5
distancemoduli,µ ,andthetruedistancemoduli,µ of
fit true
0 the SNe in each bin:
0.20 0.15 0.10 0.050.00 0.05 0.10 0.15 0.20 0.25
− − − −
(zphot−zspec)/(1+zspec)
∆µ(z )=(cid:104)µ −µ (cid:105) (14)
Figure 3. Distribution of (zphot −zspec)/(1+zspec), with no i fit true zi
host-galaxyphoto-zprior whereµ iscalculatedfromtheinputcosmologicalmodel,
true
and µ is calculated from the fitted SALT2 parameters,
fit
withαandβ takenfromafitwithoutsubtractinganybias.
4.3 Effect of host galaxy priors
The correction for each SN is then obtained using lin-
Weinvestigatetheeffectofusingahost-galaxyphoto-zprior ear interpolation. So the bias-corrected distance of each SN
intheSALT2fitting.WeapplyaGaussianpriorwithmean becomes:
and sigma values set as the host-galaxy photo-z and error
from a simulated host-galaxy library for LSST. The SNe
µ =µ −∆µ (15)
with host-galaxy photo-z smaller than 0.01 or greater than SN fit,SN SN
1.2 are dropped. Using this host-galaxy photo-z prior, we
(cid:16) (cid:17)
obtainasetofphoto-z’swithanaccuracyσ zp1h+otz−spzescpec = 5.1 Ellipse cut and other quality cuts
0.0116, and a mean bias ((cid:104)z −z (cid:105)) of 0.0017, after
phot spec We utilize an ellipse cut (Fig. 6) to exclude the SNe with
applying a cut on the reduced χ2 of the SALT2 fit (χ2 <
1.5).Theoutlier((cid:12)(cid:12)(cid:12)zp1h+otz−spzescpec(cid:12)(cid:12)(cid:12)>0.1)fractionis0.16%.reTdhe eaxstirnemBeazvianlueetsailn. (t2h0e11x)1-acnpdlaCnaem. Wpbeelaldeotpatl.a(2si0m13il)a.rTchuet
resultsareshowninFig.5.Usingahostgalaxyphoto-zprior ellipse we draw has semi-axes a = 3, and a = 0.25, cen-
x1 c
leads to significant improvement in the photo-z estimation, tered at (x ,c) = (−0.2,0). Note that this cut excludes a
1
although the currently available LSST host galaxy library higherfractionofnon-Ia’sthanIa’sinoursample,andthus
that we have used may have optimistic host galaxy photo-z imporves the purity of the final sample (99.7%).
errors. We will re-evaluate the performance of our photo- Wealsorequirethephoto-ztobegreaterthan0.2,since
z estimator with host-galaxy priors when a more realistic the low-z SNe are already excluded in the general-function-
LSST host galaxy library becomes available. fitstepbeforeclassification;werequirethephoto-zerrorto
be less than 0.1.
Our final sample for cosmological analysis has a total
5 FITTING COSMOLOGY numberof12618SNeincluding12586(99.7%)SNeIaand32
(0.3%)Core-collapseSNe.Weshowthemarginalizedmeans
Our final step is to examine the performance of using our
oftheparametersinTable3.TheHubbleDiagramisshown
photometric-onlySNIasample(withCCcontamination)in
in Fig. 7.
constrainingcosmology.WeusetheSALT2parameters(x ,
0
x , c) obtained from Section 4.2 to calculate the distance
1
modulus µ:
6 SUMMARY AND DISCUSSION
WehavedevelopedamethodforSNclassificationusingtheir
µ=−2.5log (x )+αx −βc+M, (11)
10 0 1 colors and parameters from a general function fit of their
where α, β and M are nuisance parameters. lightcurves,utilizingtheRandomForestclassificationalgo-
WehavesimulatedthedataassumingtheΛCDMmodel rithm. Our method is independent of the SN models, and
with Ω =0.3 and a flat Universe. The χ2 is calculated as: make no use of redshift information of the SN or its host
m
galaxy. We have achieved performance comparable to other
photometric classification methods. By varying the proba-
χ2 =(cid:88)(µi−µmodel,i)2 (12) bilitythresholdweareabletoobtainsampleswithdifferent
σ2
i purity as needed. A sample with 99% purity is chosen for
where our cosmological constraints study in this paper.
Wehaveobtainedphoto-z’sforourphotometricsample
σ2 =σm2B +σz2+σi2nt+α2σx21 +β2σc2 by fitting the SN light curves using the extended SALT2
+2αCov −2βCov −2αβCov (13) model, with the nested sampling method. We show that
mB,x1 mB,c x1,c
initial conditions and parameter limits need to be set very
andσ iscalculatedfromthecosmologicalmodelusingerror
z
propagation. We set σ to be a constant value and vary
int
it with different values to see whether it affects the fitting 4 http://cosmologist.info/cosmomc/
MNRAS000,000–000(0000)
SN photometric classification and photo-z estimation 7
z 0.65 z<0.65 z=all
1800 ≥ 500 2500
µ=0.36 µ=0.24 µ=0.33
1600
σ=0.97 σ=1.05 σ=1.00
1400 400 2000
1200
300 1500
1000
800
200 1000
600
400 100 500
200
0 0 0
10 5 0 5 10 10 5 0 5 10 10 5 0 5 10
− − − − − −
(zphot−zspec)/zerr (zphot−zspec)/zerr (zphot−zspec)/zerr
Figure 4.Distributionof(zphot−zspec)/zerr indifferentredshiftranges.Left:z≥0.65,middle:z<0.65,right:zinthewholesample
range.DashedlinesarefromGaussianfitswithbest-fitvalueshownintherightcorner.
Table3.Simulationinputparametervaluesandthemarginalizedmeansofthecosmologicalparametersobtainedusingthephotometric
SNIasamplederivedwithoutusinghost-galaxyphoto-zprior.
Ωm α β M χ2 d.o.f
Siminput 0.30 0.135 3.19 13.43 - -
σint=0.1 0.304±0.009 0.119±0.001 3.25±0.02 13.42±0.01 9850.6 12614
σint=0.08 0.302±0.008 0.122±0.001 3.29±0.02 13.42±0.01 11552.0 12614
σint=0.07 0.298±0.008 0.123±0.001 3.30±0.02 13.42±0.01 12539.5 12614
fitted value varies a little with different choice of the in-
70 trinsic scatter terms. With the small statistical uncertainty
60 rhmzpshoztp1h−+otz−zspzsepscpeecci==00..00101167 duetothelargesamplesize,thestudyofthesystematicef-
50 (cid:12)zp1h+otz−spzescpec(cid:12)>0.1:0.16% fceacptasbbileictyomoefscomnosrteraiimnipnogrtcaonstm.oHloegreywuesinagimthaetpshhoowtoimngettrhiec
(cid:12) (cid:12)
40 (cid:12) (cid:12) sample.Wewillleavethesystematicstudiesforfuturework.
30
20
10 ACKNOWLEDGEMENTS
0 Some of the computing for this project was performed at
0.20 0.15 0.10 0.050.00 0.05 0.10 0.15 0.20 0.25
− − − − the OU Supercomputing Center for Education & Research
(zphot−zspec)/(1+zspec)
(OSCER) at the University of Oklahoma (OU).
Figure 5.Distributionof(zphot−zspec)/(1+zspec),withhost-
galaxyphoto-zprior
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Figure 7. Hubble Diagram of our photometric SN Ia sample derived without using host-galaxy photo-z prior. Blue dots are true SNe
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APPENDIX A: QUALITY CUTS DETAILS
WepresentthenumbersofSNeremainingaftereachquality
cut in detail in Table A1 and A2.
MNRAS000,000–000(0000)
10 Dai et al.
Table A1.SummaryofnumberofSNeremainingforeachtypeaftereachqualitycut
Ia II Ibc
TotalnumberofSNebeforeanycuts 199400(1) 1941000(1)
MaxSNR>5for3bands 62147(0.31) 67631(0.035) 14468(0.007)
1pointbeforeand2afterthepeakafor
48298(0.24) 54900(0.028) 11468(0.006)
3bands,1ofwhichhasmaxSNR>5
Bazinfitsuccessb(all6bands) 48159(0.24) 52342(0.027) 11311(0.006)
Bazinparametercutsc 26616(0.13) 7960(0.004) 4354(0.002)
Finalfractiond 0.684 0.204 0.112
aHere “peak” refers to the highest flux point in the raw light curve whose SNR is greater
thanthemedianSNRofthatband.
bHere“success”referstoanyfitthatreturnsasetofvalues(doesnotreturna“failure”by
thecurvefitprogram),whethertheyareinareasonablerangeornot.
cDetailedinTableA2.
dFractionoftypesinthefinalsample(addedupto1).
eNumbersinparenthesesindicatethefractions.
Table A2.NumberofSNeremainingforeachtypeaftereachBazinparametercut
Ia II Ibc
Afterpre-fitcuts 48159(1) 52342(1) 11311(1)
trise>1,andtrise notcloseto1 40337(0.84) 34162(0.65) 8320(0.74)
withtolerance=0.01
−20<B<20 38317(0.80) 17439(0.33) 6228(0.55)
χ2/d.o.f <10 36041(0.75) 16308(0.31) 5995(0.53)
t <150 33580(0.70) 13038(0.25) 5742(0.51)
fall
trise<tfall 31248(0.65) 11201(0.21) 5219(0.46)
A<5000 31191(0.65) 11172(0.21) 5207(0.46)
Aerr<100 27806(0.58) 9612(0.18) 4627(0.41)
t0,err<50 27545(0.57) 9121(0.17) 4567(0.40)
t <100 26826(0.56) 8078(0.15) 4390(0.39)
fall,err
trise,err<50 26616(0.55) 7961(0.15) 4354(0.38)
A(Y),A(u)<1000 26616(0.55) 7960(0.15) 4354(0.38)
†Numbersinparenthesesindicatethefractions.
MNRAS000,000–000(0000)
