Bayesian constraints on dark matter halo properties using gravitationally-lensed supernovae

1.3 MB·English

by  N. V. Karpenka

Mon.Not.R.Astron.Soc.000,1–13(2012) Printed17July2012 (MNLATEXstylefilev2.2)
Bayesian constraints on dark matter halo properties using
gravitationally-lensed supernovae
N.V. Karpenka1⋆, M.C. March2, F. Feroz3 and M.P. Hobson3
1TheOskarKleinCentreforCosmoparticlePhysics,DepartmentofPhysics,StockholmUniversity,AlbaNova,SE-10691Stockholm,Sweden
2AstronomyCentre,UniversityofSussex,BrightonBN19QH,UK
2 3AstrophysicsGroup,CavendishLaboratory,JJThomsonAvenue,CambridgeCB30HE,UK
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2
Accepted—.Received—;inoriginalform13July2012

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ABSTRACT
6
AhierarchicalBayesianmethodisappliedtotheanalysisofType-IAsupernovae(SNIa)ob-
1
servations to constrain the properties of the dark matter haloes of galaxies along the SNIa

lines-of-sightviatheirgravitationallensingeffect.Thefulljointposteriordistributionofthe
]
O dark matter halo parametersis explored using the nested sampling algorithm MULTINEST,
C which also efficiently calculates the Bayesian evidence, thereby facilitating robust model
comparison. We first demonstrate the capabilities of the method by applying it to realistic
.
h simulatedSNIadata,basedonthereal3-yeardatareleasefromtheSupernovaLegacySurvey
p (SNLS3).Assumingtypicalvaluesforthehaloparametersinoursimulations,wefindthata
- catalogueanalogoustotheexistingSNLS3datasetisincapableofdetectingthelensingsignal,
o
butacataloguecontainingapproximatelythreetimesasmanySNIadoesproducerobustand
r
t accurateparameterconstraintsand modelselection resultsfor two halo models:a truncated
s
a singularisothermalsphere(SIS)andaNavarro–Frenk–White(NFW)profile,therebyvalidat-
[ ing our analysis methodology.In the analysis of the real SNLS3 data, contrary to previous

studies,weobtainonlyaverymarginaldetectionofalensingsignalandweakconstraintson
1
the halo parametersfor the truncated SIS model, althoughthese constraintsare tighter than
v
8 those obtained from the equivalent simulated SNIa data set. This difference is driven by a
0 preferred value of η ≈ 1 in the assumed scaling-law σ ∝ Lη between velocity dispersion
7 andluminosity,whichissomewhathigherthanthecanonicalvaluesofη = 1 andη = 1 for
4 3
3 early and late-type galaxies, respectively,and leads to a strongerlensing effect by the halo.
. NodetectionofalensingsignalismadefortheNFWmodel.
7
0
Key words: gravitational lensing: weak – methods: data analysis – methods: statistical –
2
supernovae:general–galaxies:haloes
1
:
v
i
X
r 1 INTRODUCTION Nonetheless, the gravitational lensing of SNIa can itself be
a usedtoconstraincosmology(Metcalf1999,Dodelson&Vallinotto
In using Type Ia supernovae (SNIa) as ‘standardizable’ can-
2006, Zentner&Bhattacharya 2009) and/or the properties of the
dles toconstrain cosmological parameters, one typicallyassumes
lensing matter (Rauch 1991, Metcalf&Silk 1999). In the latter
that the universe is homogeneous and isotropic, and therefore
case,oneperformsacomplementaryanalysistocosmologicalpa-
one ignores gravitational lensing effects due to cosmic struc-
rameter estimation by instead assuming a particular background
ture along the line-of-sight to each SNIa. In conventional cos-
cosmological model (i.e. fixing the cosmological parameters to
mological SNIa analyses, this effect is usually regarded as an
someconcordancevalues)andusingtheobserveddistancemoduli
additional source of uncertainty, which adds extra scatter to the
toconstrainthenatureofthecosmicstructure,suchastheproper-
brightness of SNIa that increases with redshift (Kantowskietal.
tiesofdarkmatterhaloes,alongthelines-of-sighttotheSNIa.In
1995,Frieman1996,Wambsganssetal.1997,Holz&Wald1998,
principle,onemightevenhopetoperformajointanalysistocon-
Bergstro¨metal. 2000). Fortunately, owing to flux conservation,
strain the background cosmological parameters and the nature of
theeffectsof gravitational magnificationand demagnification av-
thecosmicstructuresimultaneously,butsuchanapproachislikely
erage out and are therefore expected to lead to negligible bias
tosufferfromstrongdegeneraciesbetweenparameters.
incosmological parameter estimates(see,e.g.,Sarkaretal.2008,
Jo¨nssonetal.2008). An early tentative detection of gravitational lensing of SNIa
was made by Jo¨nssonetal. (2007) using a sample from the
Great Observatories Origins Deep Survey (GOODS; Riessetal.
⋆ E-mail:[email protected] 2004, Strolgeretal. 2004, Riessetal. 2007). More recently,
(cid:13)c 2012RAS
2 N.V. Karpenka et al.
Kronborgetal.(2010)focussedonthedetectionofagravitational 2 BAYESIANINFERENCE
lensing signal by assuming properties for dark matter haloes and
OuranalysismethodologyisbuiltupontheprinciplesofBayesian
reportedapositiveresultatthe99percentconfidencelevel.More-
inference,whichprovideaconsistentapproachtotheestimationof
over, Jo¨nssonetal. (2010a) used 24 high-redshift (0.4 . z .
asetofparametersΘinamodel(orhypothesis)HforthedataD.
1.8) SNIafrom GOODS to constrain the properties of dark mat-
Bayes’theoremstatesthat
ter haloes of galaxies also contained within GOODS. This study
wasextendedinJo¨nssonetal.(2010b)byusing175high-redshift Pr(D Θ,H)Pr(ΘH)
Pr(ΘD,H)= | | , (1)
(0.1.z .1)SNIafromthe3-yeardatareleaseoftheSupernova | Pr(DH)
|
LegacySurvey(SNLS3;Astieretal.2006)toconstrainthehaloes
where,for brevity, wedenotePr(ΘD,H) P(Θ)asthepos-
of galaxies in the deep Canada-France-Hawaii Telescope Legacy terior probability distribution of the|paramet≡ers, Pr(DΘ,H)
Survey(CFHTLS)fields.AlthoughtheSNIainSNLSaretypically (Θ) as the likelihood, Pr(ΘH) π(Θ) as the|prior, an≡d
notasdistantasthosefromGOODS,theyarefarmorenumerous LPr(DH) astheBayesiane|viden≡ce.
and selected in a more homogeneous way. Jo¨nssonetal. (2010b) | ≡Z
In parameter estimation, the normalising evidence factor is
report thedetection ofagravitationallensingsignal atthe92per
usuallyignored, sinceitisindependent of theparametersΘ,and
centconfidencelevel.
inferences are often obtained by taking samples from the (un-
In this paper, we also use high-redshift SNIa from SNLS3
normalised) posterior using standard MCMC sampling methods,
to constrain the properties of dark matter haloes of galaxies in
whereatequilibriumthechaincontainsasetofsamplesfromthe
theCFHTLSfieldsthatintersecttheSNIalines-of-sight.Oursta-
parameterspacedistributedaccordingtotheposterior.Thisposte-
tistical methodology differs greatly, however, from that used by
riorconstitutesthecompleteBayesian inferenceof theparameter
Jo¨nssonetal.(2010a,b)andotherpreviousstudies.Asrecentlydis-
values,andcanbemarginalisedovereachparametertoobtainin-
cussed by Marchetal. (2011), the usual χ2-method used to con-
dividualparameterconstraints.
straincosmological parametersand/or thenatureofcosmicstruc-
In contrast to parameter estimation problems, for model se-
turefromlightcurvefitstoSNIaobservations(seee.g.Astieretal.
lectiontheevidencetakesthecentralroleandissimplythefactor
2006;Kowalskietal.2008;Conleyetal.2011)suffersfromsome requiredtonormalizetheposterioroverΘ,
severalshortcomingsintermsofitsstatisticalfoundationsandro-
bustness,includingnotallowingforrigorousmodelcheckingand = (Θ)π(Θ)dDΘ, (2)
not providing a reliable framework for the evaluation of system- Z L
Z
atic uncertainties. Consequently, we instead analyse the SALT- whereD isthedimensionalityoftheparameterspace.Astheav-
II lightcurve fits of the SNIa observations using the statistically- erageof thelikelihoodover theprior, theevidence islargerfor a
principled and rigorous Bayesian hierarchical method (BHM) of model if more of its parameter space is likely and smaller for a
Marchetal.(2011)toobtainarobusteffectivelikelihoodfunction model with large areas in its parameter space having low likeli-
givingtheprobabilityofobtainingtheobservedSNIadata(i.e.the hoodvalues,evenifthelikelihoodfunctionisveryhighlypeaked.
parametervaluesobtainedintheSALT-IIlightcurvefits)asafunc- Thus,theevidence automaticallyimplementsOccam’srazor.The
tionoftheparametersofthedarkmatterhalomodelsassumedfor questionofmodel selectionbetweentwomodelsH andH can
0 1
thegalaxiesalongthelines-of-sighttotheSNIa.Moreover,rather thenbedecidedbycomparingtheirrespectiveposteriorprobabili-
thanexploringtheparameterspaceofthedarkmatterhalomodels tiesgiventheobserveddatasetD,asfollows
usingsimplegriddingmethods(seee.g.Jo¨nssonetal.2010a,b),we
Pr(H D) Pr(DH )Pr(H ) Pr(H )
insteadsamplefromthefulljointposteriordistributionofthedark R= 1| = | 1 1 = Z1 1 , (3)
Pr(H D) Pr(DH )Pr(H ) Pr(H )
matterhaloparametersusinganestedsamplingalgorithm(Skilling 0 0 0 0 0
| | Z
2004;Feroz&Hobson2008;Ferozetal.2009).Thisenablesusto wherePr(H )/Pr(H )istheaprioriprobabilityratioforthetwo
1 0
explore all the halo model parameters simultaneously and allows models, whichcanoftenbesettounitybut occasionally requires
for straightforward marginalisation over subsets of them. The al- furtherconsideration.
gorithmalsoefficientlycalculatestheBayesianevidence, thereby Evaluation of the multidimensional integral in Eq. (2) is a
facilitatingrobustmodelcomparison. challenging numerical task. Standard techniques like thermody-
Theoutlineof thispaper isasfollows.InSection2wegive namic integration are extremely computationally intensive which
abriefsummaryofBayesianinferencemethods,followedinSec- makes evidence evaluation at least an order of magnitude more
tion3byadescriptionofourBayesianmethodologyforusinggrav- costly than parameter estimation. Some fast approximate meth-
itationallensingofSNIatoconstrainthepropertiesofdarkmatter ods have been used for evidence evaluation, such as treating the
haloesofthegalaxiesintersectingthelines-of-sight.InSection4, posterior asa multivariateGaussian centered at itspeak (seee.g.
wedescribetheSNLS3supernovae andgalaxiesdatausedinour Hobson&McLachlan 2003), but this approximation is clearly a
analysis.WetesttheperformanceofourBayesianmethodologyin pooroneformultimodalposteriors(exceptperhapsifoneperforms
Section5byapplyingtorealisticsimulateddatabasedonthereal a separate Gaussian approximation at each mode). The Savage-
SNLS3data,beforeanalysingtherealdatasetsandpresentingour Dickeydensityratiohasalsobeenproposed(seee.g.Trotta2007)
resultsinSection6.WegiveourconclusionsinSection7. asanexact,andpotentiallyfaster,meansofevaluatingevidences,
Throughoutthepaper,weassumeaspatially-flatconcordance but is restricted to the special case of nested hypotheses and a
ΛCDM background cosmology, characterised by the parameters separableprioronthemodelparameters.Variousalternativeinfor-
Ω ,Ω ,H = 0.27,0.73,0.7 .Finally,wenotethat mationcriteriaforastrophysical modelselectionarediscussedby
m,0 Λ,0 0
C ≡ { } { }
thispapermaybeconsideredascomplementarytoourcompanion Liddle(2007),buttheevidenceremainsthepreferredmethod.
paper(Marchetal.2012),inwhichweusetheBHMtoanalysethe Thenestedsamplingapproach,introducedbySkilling(2004),
SNLS3 catalogue, together with additional low-redshift SNIa, to isaMonteCarlomethodtargetedattheefficientcalculationofthe
constrainthebackgroundcosmologicalmodel,assumingnogravi- evidence, but also produces posterior inferences as a by-product.
tationallensingalongthelines-of-sighttotheSNIa. Feroz&Hobson(2008)andFerozetal.(2009)builtonthisnested
(cid:13)c 2012RAS,MNRAS000,1–13
Constrainingdarkmatterhaloesusingsupernovae 3
sampling framework and have introduced the MULTINEST algo- covariance matrix Ci. The ‘observed’ distance modulus µoibs for
rithmwhichisveryefficientinsamplingfromposteriorsthatmay eachSNisthenmodelledas
contain multiple modes and/or large (curving) degeneracies and b
µobs =mˆ∗ M +αxˆ βcˆ, (6)
also calculates the evidence. This technique greatly reduces the i B,i− i 1,i− i
computational cost of Bayesian parameter estimation and model
whereM isthe(unknown)B-bandabsolutemagnitudeoftheSN,
i
selection and has already been applied to a number of problems
and α, β are (unknown) nuisance parameters (assumed the same
inastrophysics(seee.g.Feroz&Hobson2011;Ferozetal.2011;
forallSN)controllingthestretchandcolourcorrections.
Bridgesetal. 2009; Ferozetal. 2010). Weemploy this technique
It should benoted that a sophisticated Bayesian hierarchical
inthispaper.
methodhasrecentlybeenproposedbyMandeletal.(2009,2010)
tofitopticalandinfraredlightcurvedata.Thismayprovideamore
robusttechniquefordefiningthebasicSNIadatathatweuseinour
3 ANALYSISMETHODOLOGY subsequentanalysis,butweleavetheinvestigationofthisissueto
afuturework.
3.1 DefinitionoftheSNIadata
In practice, there are no perfect astronomical standard candles.
In particular, SNIa have absolute magnitudes that vary by about
3.2 Computingthepredicteddistancemoduli
0.8magintheB-bandduetophysicaldifferencesinhow each
±
supernovaistriggeredandalsoduetoabsorptionbyitshostgalaxy. In using SNIa to constrain cosmic structure, the predicted dis-
Nonetheless, SNIa do constitute a set of ‘standardizable’ can- tancemodulusmust includetheeffect ofgravitational magnifica-
dles, since by applying small corrections to their absolute mag- tionduetocosmicstructurealongtheline-of-sighttoeachsuper-
nitudes,derivedbyfittingmulti-wavelengthsobservationsoftheir nova. In particular, we assume this magnification is due to dark
lightcurves, one can reduce the scatter considerably, to around matterhaloesassociatedwithknowngalaxiesintersectingtheline-
0.15 mag in the B-band. In essence, SNIa with broader light of-sight.Thus,foreachSNIa,thepredictedmagnificationdepends
±
curvesandslowerdeclineratesareintrinsicallybrighterthanthose onthesetsofparameters g,h ,where
{ }
withnarrowerlightcurvesandfastdeclinerates(Phillips1993).
Several methods are available for fitting SNIa lightcurves g={zg1al,θg1al,MB1,τ1,...,zgNaglal,θgNaglal,MBNgal,τNgal} (7)
(and constraining cosmological parameters), including, amongst
containstheredshift,skyposition,absoluteB-bandmagnitudeand
others, the Multi-Colour Lightcurve Shape (MCLS) strategy
spectraltypeoftheN galaxiesthatintersecttheline-of-sightto
(Jhaetal.2007),CMAGIC(Wangetal.2003;Conleyetal.2006), gal
theSNIa,andhcontainstheparametersoftheassumeddarkmat-
andtheSpectrallyAdaptiveLightcurveTemplate(SALT)method
terhalomodelforthesegalaxies(seeSection3.3).Ingeneral,the
(Guyetal. 2007), the current version of which is SALT-II. The
parametersgare(naturally)differentforeachSNandareassumed
relativemeritsofthesemethodsisatopicofmuchdebate,butthe
known; one wishesto place constraints on theunknown halo pa-
SALT-IImethod isparticularly attractivefor our purposes, since
rametersh,which(perhapsunrealistically)areassumedcommon
(unlikeMCLS)itfirstfitseachSNIalightcurvetoobtainthreepa-
toalltheforegroundgalaxies.
rameterscontrollingtheSNmagnitude,stretchandcolourcorrec-
TocomputethepredictedmagnificationofaSNIa,weusethe
tionstoatemplate‘learned’fromnearbyanddistantSN;onlyina
weak-lensingapproximation (see,e.g..Schneideretal.1992),the
second,separatesteparethesefitsusedtoconstraincosmological
validityof whichwaschecked byJo¨nssonetal.(2010b)by com-
parameters. Following Marchetal. (2011), we may therefore use
paring it with a ray-tracing algorithm. For the SNIa sample we
theproductsofthefirststepastheinputstoastatisticallyrigorous
considerhere(seeSection4.1),whichhaverelativelylowredshifts,
Bayesianhierarchicalmodel.
theweak-lensingapproximationwasfoundtobeaccuratetowithin
Ouranalysistakesplaceaftertheselectioncuts,lightcurvefit-
5percent.Inthisapproximation,thepredicteddistancemodu-
tingandMalmquistcorrectionhavebeenperformed.Foreachse- ∼
lusof aSNIa,expressed intermsofmagnitudes, isrelatedtothe
lectedSNIa,inadditiontoanestimatezˆofitsredshiftandanasso-
convergenceκalongitsline-of-sightby
ciateduncertaintyσ ,derivedfromobservationsofitshostgalaxy,
z
wetakeasourbasicdatatheoutputfromtheSALT-IIlightcurve µ(z, ,g,h) µ (z, ) 2.17[κ (g,h) κ (h)], (8)
fittingalgorithm,whichproducesthebest-fitvalues:mˆ∗,therest C ≈ 0 C − los − b
B
frameB-bandapparentmagnitudeofthesupernovaeatmaximumn where µ0(z, ) is the predicted distance modulus for our as-
C
luminosity;xˆ ,thestretchparameterrelatedtothewidthofthefit- sumedcosmologicalparameters ,neglectinggravitationallensing,
1 C
tedlightcurve;andcˆ,thecolourexcessintheB-bandatmaximum κlos(g,h)isasumoverthecontributionstotheconvergencefrom
luminosity.Thesearesupplementedbythecovariancematrixofthe eachgalaxyalongtheline-of-sight,sothatκ = Ngalκj ,and
los j=1 gal
uncertaintiesintheestimatedlightcurveparameters,namely κ (h)representsthecompensatingeffectofthebackgroundden-
b
P
σm2∗B σm∗B,x1 σm∗B,c stoityaahnodmaocgtesnaesoausnournmivaelrisseattioonbealcloowmipnugtethde(mJo¨angsnsoifinceattiaoln.2re0l1a0tibv)e.
C = σm∗B,x1 σx21 σx1,c . (4) Fluxconservationimpliesthathκi=0andtoensurethiscondition
b  σm∗B,c σx1,c σc2  liastseadtiussfiiendgwaelasregteκnbu(mhb)e=rohfκralonsd(ogm,hly)is,ewlehceterdetlhineelsa-totfe-rsiigshcta.lcu-
 
Therefore,ourbasicinputdataforeachSN(i=1,...,N )are
SN
∗
D zˆ,mˆ ,xˆ ,cˆ , (5)
i ≡{ i B,i 1,i i}
3.3 Halomodels
and we assume (as is implicitly the case throughout the SNe lit-
erature) that the vector of values (mˆ∗ ,xˆ ,cˆ) for each SN is Ourprimarygoalistoconstraintheparametershthatdescribethe
B,i 1,i i
distributed as a multivariate Gaussian about the true values, with properties of the dark matter haloes associated with the galaxies
(cid:13)c 2012RAS,MNRAS000,1–13
4 N.V. Karpenka et al.
along thelines-of-sight totheSNIa.Weassume that each galaxy providedR 6 r .Thus,forexample,onehasaconstantmass-to-
t
contributestotheconvergencebyanamount lightratioM/L L0ifη=0.5.
∝
For the truncation radius, we again follow Jo¨nssonetal.
Σ(ξ)
κgal(ξ)= Σ , (9) (2010b)andassumeitobeysascalinglawoftheform
c
wherethesurfacedensity,Σ(ξ),isobtainedbyprojectingthemat- σ γ L ηγ
terdistributionontoalensplane, rt =r∗ σ∗ =r∗ L∗ , (17)
(cid:18) (cid:19) (cid:18) (cid:19)
∞
Σ(ξ)= ρ(ξ,y)dy, (10) where r∗ is a fiducial truncation radius. Since we only include
whereξisavectorintheplanZe−a∞ndyisacoordinatealongtheline tgraulnacxaietisonlorcaadteiudsahadsisatannecfefeθcctfornolmyitfhretp/oDsditio<nθocf.tFhoerSθNc =Ia,6t0h′e′
ofsight;itisatthispointthatthedensityprofileρ(r)ofthedark andzgal <1thiscorrespondstort .300h−1kpc.
Thus,forthetruncatedSIShalomodel,wewishtoconstrain
matter halo enters the calculation. The denominator in (9) is the
criticalsurfacedensity, thefourparametersh= γ,η,σ∗,r∗ .
{ }
c2 D
Σ = s , (11)
c 4πGDD 3.3.2 Navarro–Frenk–Whiteprofile
l ls
which,inturn,dependsontheangulardiameterdistancesbetween
TheradialdensitydistributionoftheNavarro–Frenk–White(NFW)
theobserverandthesource,D ,theobserverandthelens,D,and
s l profileisgivenby
the lens and the source, D . These distances are computed from
ls
theredshiftsoftheSNandgalaxyrespectively,assumingourcon- ρ(r)= δcρc , (18)
cordancebackgroundcosmology. (r/rs)(1+r/rs)2
Inthispaper,weinvestigatetwodifferentmodelsfortheden-
whereρ = 3H2(z)/8πGisthecriticaldensityforclosureofthe
sity profile ρ(r) of the dark matter halo: the truncated singular c
universeattheredshift,z,ofthehaloandH(z)istheHubblepa-
isothermalsphere(SIS)andtheNFWprofile(Navarroetal.1997),
rameter at that same redshift. The scale radius r = r /c is a
bothofwhichwenowdescribe. s 200
characteristicradiusofthecluster,wherecisadimensionlessnum-
berknownastheconcentrationparameter,and
3.3.1 Truncatedsingularisothermalsphere
200 c3
δ = (19)
Theradialdensitydistributionofasingularisothermalsphere(SIS) c 3 ln(1+c) c(1+c)−1
−
isgivenby
isacharacteristicoverdensityforthehalo.Thevirialradius,r ,
200
σ2 1 isdefinedastheradiusinsidewhichthemassdensityofthehalois
ρ(r)= , (12)
2πGr2 equalto200ρc.Theprofilethereforedependsontwofreeparame-
ters:thevirialradiusr andtheconcentrationparameterc.The
whichdependsonthesinglefreeparameterσ,theone-dimensional 200
massofanNFWhalocontainedwithintheradiusr is
velocitydispersionofitsconstituentparticles.SincetheSISprofile 200
hasadivergenttotalmass,wetruncateitataradiusr =rt,which M M(r6r )= 800πρ r3 = 800π ρmr3 , (20)
isthusasecondfreeparameter. 200 ≡ 200 3 c 200 3 Ωm 200
ThesurfacedensityofatruncatedSISiseasilycalculatedus-
whereρ isthemeanmatterdensityoftheuniverseandΩ isthe
m m
ing(10)andfoundtobe
matterdensityparameterattheredshiftzofthehalo.
σ2 arctan r2/ξ2 1 ifξ6r The surface density of the NFW profile is given by
Σ(ξ)= πGξ t − t (13) (Bartelmann1996)
( 0 p ifξ>r ,
t
whichcan,inturn,besubstitutedinto(9)toobtaintheconvergence 2rsδcρc 1 2 arctanh 1−x (x<1)
(x2−1) − √1−x2 1+x
κgal(ξ)duetothegalaxydarkmatterhalo.  (cid:20) q (cid:21)
luminToosiatyllowanfdorvaenldociintyvesdtiigspateerstihoen,relwateionfoshlliopwbetJwo¨nesesnongaeltaxaly. Σ(ξ)= 2rs3δcρc (x=1)
(2010b)andadopttheFaber-Jackson(Faber&Jackson1976)and 2rsδcρc 1 2 arctan x−1 (x>1)
Tully-Fisher-like(Tully&σ=Fisσh∗er19L77)ηsc,alinglaw (14) wherex=ξ/(rxs2−is1a)d(cid:20)im−ens√ioxn2l−es1sprojecteqdxra+d1ial(cid:21)distancein(2th1e)
L∗ lensplane.Onceagain,thisexpressionmaybesubstitutedinto(9)
(cid:18) (cid:19)
where L∗ is a fiducial luminosity, which we take to be to obtain the convergence κgal(ξ) due to the galaxy dark matter
L∗ =1010h−2L in the B-band, and σ∗ is the corresponding halo.
fiducial velocity⊙dispersion. In terms of absolute B-band magni- By analogy with the Tully–Fisher and Faber–Jackson rela-
tudes,withwhichwewillbeworking,thescalingrelationbecomes tions, to allow for and investigate the relationship between the
σ=σ∗10−η(MB−MB∗)/2.5, (15) gthaelaxtryunlcuamteidnoSsiItSy parnodfilteh,ewveirifaolllroawdiuJso¨nisnsoanseitmali.la(r20m1a0nbn)erantdo
whereM∗ = 19.52+5log handhistheHubbleconstantin Kleinheinrichetal.(2004)andadoptthescalinglaw
unitsof1B00km−s−1 Mpc−1.T10he(aperture)mass-to-lightratioof
thegalaxiesisdeterminedbyη,since r =r∗ L λ, (22)
200 200 L∗
M(rL6R) = 2σ∗2(LG/LL∗)2ηR ∝L2η−1, (16) where, once again, L∗ isafiducia(cid:18)l lum(cid:19)inosity that wetake to be
(cid:13)c 2012RAS,MNRAS000,1–13
Constrainingdarkmatterhaloesusingsupernovae 5
L∗ =1010h−2L in the B-band, and r2∗00 is the corresponding Nuisanceparameter Symbol Value
fiducialvirialrad⊙ius.Thevirialmass-to-lightratioisthusgivenby SNIapopulation
Dispersionofabsolutemagnitude σint U(−3,0)onlogσint
M200 = 800πρ r2300 L3η−1. (23) Lightcurve
L 3 c L ∝ Stretchmultiplier α U(0,4)
For example, a constant mass-to-light ratioM/L L0 requires Colourmultiplier β U(0,4)
λ= 1. ∝ Dispersionofstretchcorrection Rx U(−5,2)onlogRx
T3hus,fortheNFWhalomodel,wewishtoconstrainthethree Dispersionofcolourcorrection Rc U(−5,2)onlogRc
parametersh={λ,c,r2∗00}. Table1.Priorsonnuisanceparameters,whereU(a,b)denotesauniform
distributionbetweenthelimitsaandb.
3.4 Likelihoodfunction
Haloparameter Symbol Prior
To construct the likelihood function for the SNIa data, we adopt
TruncatedSIShalo
theBayesianhierarchicalmodelproposed byMarchetal.(2011). Fiducialvelocitydispersion(kms−1) σ∗ U(0,300)
Thishasbeenshowntodelivertighterconstraintsoncosmological
Fiducialtruncationradius(Mpc) r∗ U(0,0.4)
parametersthantheusualχ2-method,andsimultaneouslyprovides Exponentofσ–Lpowerlaw η U(−5,5)
arobuststatisticalframeworkforthefullpropagationofsystematic Exponentofrt–σpowerlaw γ U(−5,5)
uncertaintiestothefinalinferences. NFWprofilehalo
FortheithSN,werequirethelikelihoodoftheinputdata(5) Concentration c U(0,100)
giventheparametersofourmodel,namely Fiducialvirialradius(Mpc) r∗ U(0,0.4)
200
Pr(mˆ∗ ,xˆ ,cˆ,zˆ h,α,β,σ ), (24) Exponentofr200–Lpowerlaw λ U(−2,2)
B,i 1,i i i| int
Table2.PriorsonhaloparametersforthetruncatedSISandNFWmodels
which also depends on the cosmological parameters , the
C respectively.
covariance matrix C of the uncertainties on the input data
i
(mˆ∗ ,xˆ ,cˆ),andtheuncertaintyσ intheestimatedredshift
B,i 1,i i z,i
zˆi,allofwhichareabssumedknown.Inparticular,weseektocon- 4 SUPERNOVAEANDGALAXIESDATASETS
strain the unknown halo parameters h, global colour and stretch
To apply the Bayesian analysis methodology described above to
correctionmultipliersαandβ,andtheintrinsicdispersionσ of
int
constrainthedarkmatterhaloesofgalaxies,weneedobservations
SNIaabsolutemagnitudes(alloftheseareassumedtobe‘global’
both of supernovae and foreground galaxies. As inJo¨nsson et al.
parameters,i.e.commontoeverySN).Finally,thefulllikelihood
(2010b),weusehigh-redshift(0.1.z .1)SNIafromthe3-year
functionisgivenbytheproductofthelikelihoods(24)foreachSN.
SNLSdataset(Guyetal.2010)toconstrainthepropertiesofdark
FollowingMarchetal.(2011),wecomputethelikelihood(24)
matter haloes of galaxies in the CFHTLSfields that intersect the
for each SN by first introducing the hidden variables M , x , c
i i i
SNIalines-of-sight.
and z , which are, respectively, the true (unknown) values of its
i
TheSNLSconsistsbothofphotometricandspectroscopicob-
absolute B-band magnitude, stretch and colour corrections, and
servations. The photometry is obtained as part of the deep com-
redshift. These are then assigned priors, which themselves con-
ponentofCFHTLSwiththeonesquare-degreeimagerMegaCam
tain further nuisance parameters, and all the introduced parame-
(Bouladeetal. 2003). The deep part of CFHTLS comprises four
ters are marginalised over to obtain the likelihood (24). The de-
fields(D1,D2,D3andD4),each 1deg2 insize,imagedinu∗,
tails of this procedure are given in Appendix A of our compan-
g′, r′, i′ and z′ filtersapproximat≈ely every 4–5 days during dark
ion paper (Marchetal. 2012). By assuming separable Gaussian
andgreytime,suitablefordetectingsupernovaeandbuildinglight
priors on the hidden variables and nuisance parameters, one can
curves(Astieretal.2006;Guyetal.2010).Thespectroscopicob-
perform all the marginalisations analytically, except for two nui-
servationsareusedtodeterminethenatureofthesupernovaecandi-
sance parameters R and R that must be marginalised over nu-
x c
datesandmeasuretheirredshifts(Howelletal.2005;Bronderetal.
merically.Thefulllikelihoodfunctionthusdependsontheparam-
eters Θ = h,α,β,σ ,R ,R , and is 9-dimensional for the 2008;Ballandetal.2009).
int x c
{ }
truncatedSIShalomodeland8-dimensional fortheNFWprofile
model.
4.1 Supernovae
Our initial sampleconsisted of 230 high-redshift (0.1 . z . 1)
3.5 Priorsonthesampledparameters
SNIa from the 3-year SNLS data set (Guyetal. 2010). We note
Todetermine the Bayesian inference problem completely, it only thatthisis13fewerthanusedbyJo¨nssonetal.(2010b),sincethese
remains to specify the prior π(Θ) on the parameters to be sam- SNIawereexcludedfromtheSALT-IIfitsforvariousreasonsde-
pled. The choice of prior isparticularlyimportant for weak lens- scribedinConleyetal.(2011),anddonot appear inthetablesin
inganalyses,sincetheproblemisinherentlyunderconstrainedand that paper. The sample of 230 SNIa is further reduced, however,
thereforeanypriorinformationavailableisextremelyuseful.One byedgeeffectsandbysomepartsofthedeepCFHTLSfieldsbe-
should, however, becareful inthe choice of priorsnot to impose ingcoveredbybrightstarsthathavetobemasked(Sullivanetal.
toostrongassumptions,whichmayleadtoerroneousinferences. 2006).SNIalocatedtooclosetotheboundary ofthefieldortoa
FollowingMarchetal.(2011),weadopttheseparablepriors maskedregionareremovedfromthesamplebecauseofthelackof
listedinTable 1on each of the ‘non-halo’ (nuisance) parameters observationsofforegroundgalaxies.Detailsofthisprocedureare
σ ,α,β,R andR .Onthehaloparametersh,thepriorsforeach given inJo¨nssonetal. (2010b). Only162 of theinitial sample of
int x c
modelarelistedinTable2,andcorrespondtobroad,conservative 230SNIafulfilledthisselectioncriterion.
assumptions. Tocomputethepredictedmagnificationduetotheforeground
(cid:13)c 2012RAS,MNRAS000,1–13
6 N.V. Karpenka et al.
galaxies along the line-of-sight toeach SNIa, werequire an esti- Haloparameter Symbol Value
mateofitsskylocationθ andredshiftzˆ .Theskylocationis TruncatedSIShalo
obtainedfromtheCFHTLSSNi′-bandphotomSeNtry.Whenavailable,zˆ Fiducialvelocitydispersion σ∗ 120kms−1
istaken tobe thespectroscopic redshift of the SNIahost galaxy; Fiducialtruncationradius r∗ 0.07Mpc
Exponentofσ–Lpowerlaw η 1
otherwisethespectroscopicredshiftoftheSNIaitselfisused. 3
Selectioncuts,SALT-IIlightcurvefittingandMalmquistcor- Exponentofrt–σpowerlaw γ 0
NFWprofilehalo
rectionsaremadebytheSNLS3teamandarealreadyimplemented
Concentration c 20
inthesupplieddatafiles.AsdescribedinSection3.1,theSALT-II
Fiducialvirialradius r∗ 0.15Mpc
fittingalgorithmisappliedtoeachSNIalightcurvetoobtainbest- 200
fitestimatesof:mˆ∗,therestframeB-bandapparentmagnitudeof Exponentofr200–Lpowerlaw λ 31
B
thesupernovaeatmaximumnluminosity;xˆ ,thestretchparameter Table 3. Halo parameter values used in the generation of simulated
1
ofthefittedlightcurve;andcˆ,thecolourexcessintheB-bandat gravitationally-lensedSNIadata.
maximum luminosity. These are supplemented by the covariance
matricesCoftheuncertaintiesintheestimatedlightcurveparame-
byassumingeachgalaxytohaveahalodescribedbyeitheraSIS
ters,whicharetakenfromConleyetal.(2011).
orNFWprofilewiththeparameterslistedinTable3.
b
FortheSISmodel,thechoiceofthevelocitydispersionσ∗and
4.2 Galaxies truncation radius r∗ correspond to thebest-fit values obtained by
Jo¨nssonetal.(2010b).Thevalueη = 1 correspondstotheTully–
Our foreground galaxies are taken from the SNLS galaxy cata- 3
Fisher relation for late-type galaxies, which should make up the
logues in the deep CFHTLS fields. As discussed in Section 3.2,
majorityoftheSNLSsample,sincenearly95percentofthemare
to perform our analysis we require for each galaxy: the redshift
star-forminggalaxies.Wealsoassumeγ = 0,whichcorresponds
z ,skypositionθ ,absoluteB-bandmagnitudeM ,andspec-
gal gal B to a constant truncation radius rt = r∗, i.e. independent of the
traltypeτ.Thetechniquesusedtoobtainthesegalaxyproperties
velocitydispersionandluminosityofthegalaxy.
are described in detail in Sullivanetal. (2006) and Jo¨nssonetal. FortheNFWmodel,ourchoicesofvaluesforcandr∗ are
(2010b). 200
motivatedinpartbySpringeletal.(2008),whoperformedasim-
ulation of Milky Way sized haloes and calculated the c and r
200
valuesforthem.Ther valuesforthesehaloesaregivenintheir
200
5 APPLICATIONTOSIMULATEDSUPERNOVAEDATA table1,andcvalueforthemainsub-haloisgivenintheirtable2.
Typically,theyfindr &0.2Mpcandc 10 16.Weassume
InordertotestourBayesiananalysismethodology,wefirstapply 200 ≈ −
that most of galaxies in the SNLS sample are somewhat smaller
ittosimulatedSNIadata. First,SNIaphotometric dataintheab-
thantheMilkyWay,andsinceitiswellknownthatcisadecreas-
sence ofgravitational lensing weresimulatedandfittedusingthe
ingfunction ofmass, weassume c = 20, whichyields alensing
publiclyavailableSNANApackage(Kessleretal.2009),inaiden-
signalofsimilarstrengthtothatofourassumedtruncatedSIShalo.
ticalmannertothatdescribedinourcompanionpaper(Marchetal.
Itshouldbenoted,however,thatthevalueofcinthesimulations
2012). Insummary, thedata weresimulated tomatchclosely the
ofSpringeletal.(2008)varieswithzandgalaxymassM,whereas
SNLS3 data set (Guyetal. 2010) by using the SNLS3 co-added
weareassumingafixedvalueforc,whichisanaiveassumption.
simulationlibraryfiles(whicharepubliclyavailableaspartofthe
Wealsoassumethatλ= 1,whichcorrespondstoaconstantvirial
SNANA package), a coherent magnitude smearing of 0.12, and 3
mass-to-lightratioM/L L0.
coloursmearing.Thecoloursmearingeffect,orbroad-bandcolour ∝
Foranassumedsetofhaloproperties,thelensingconstribu-
dispersionmodel,implementedinthedatasimulationistheEXP-
tionfromthehaloesalongtheline-of-sighttoeachsimulatedSNis
POLmodeldescribedbyfig.8ofGuyetal.(2010),andthesimu-
thenaddedtotheSNapparentmagnitude,anditsuncertaintyisleft
latedMalmquistbiasisbasedonfig.14ofPerrettetal.(2010).
unchanged.
TheSNANASNIadatasimulationisatwo-stageprocessthat
mimicstherealdatacollectionandanalysisprocess.Thefirststage
isthesimulationofphotometricdatainaccordancewiththecharac-
5.1 Resultsfor162simulatedsupernovae
teristicinstrumentandsurveypropertiesoftheSNSL3surveyusing
the SNLS3simulation library filesmentioned above. The second To match the real SNLS3 data to be analysed in Section 6, our
stageisthelightcurvefittingprocessinwhichthephotometricdata firstsetofsimulationscontain162SNIa.Indeed,inthiscase,the
arefittedto SALT-IItemplatestogiveestimatesoftheSNIaab- positionsontheskyofthesimulatedSNIaarefixedtothoseofthe
soluteB-bandmagnitudemˆ ,lightcurvestretchxˆ andcolour cˆ. realSNLS3SNIa.
B 1
Atthislightcurvefittingstage,basiccutsaremadetodiscardSNIa Table4liststheBayesianlog-evidenceforeachcombination
withalowsignal-to-noiseratioand/ortoofewobservedepochsin ofsimulationandanalysismodel,relativeineachcasetothenull
sufficientbands. Afterthelightcurvefittingstagewemakeared- (no-lensing) model. Foreach simulation model, one sees that the
shift dependent magnitude correction to correct for the Malmqist analysismodelpreferredbytheBayesianevidenceisthatassuming
bias;thecorrectionistakenfromasplineinterpolationoftable4in nolensing. Thissuggeststhat thequantityandqualityof thereal
Perrettetal.(2010). SNLS3datatobeanalysedinSection6areinsufficenttoobtaina
TheresultingsimulatedSNIaarerandomlydistributedonthe detection of the lensing signal, and certainly do not allow one to
skyacrossthefour SNLS3fields,outsideof our masked regions, distinguishbetweendifferenthalomodels.
andhavearedshiftdistributionappropriatefortheSNLS3survey. ForthefourcasesdenotedbyasterisksinTable4,weplotthe
Thefinalstageofoursimulationprocessaddsthelensingcontribu- corresponding haloparameterconstraintsinFigs1and2,respec-
tionduetothe(assumed) galaxyhaloesalongtheline-of-sightto tively. From Fig. 1, one sees that, for the truncated SIS analysis
eachSN.ThisisperformedusingtherealSNLSgalaxycatalogue, model,theparameterconstraintsderivedfromsimulateddatawith
(cid:13)c 2012RAS,MNRAS000,1–13
Constrainingdarkmatterhaloesusingsupernovae 7
2 2
η 0 η 0
−2 −2
300 300
m/s] 200 m/s] 200
σ [k*100 σ [k*100
0 0
0.4 0.4
Mpc] 0.2 Mpc] 0.2
r [* r [*
0 0
−5 0 5 −2 0 2 0 100 200 300 0 0.2 0.4 −5 0 5 −2 0 2 0 100 200 300 0 0.2 0.4
γ η σ* [km/s] r* [Mpc] γ η σ* [km/s] r* [Mpc]
Figure1.1Dand2Dmarginalisedposteriorsdistributionsfortheparametersh={γ,η,σ∗,r∗}ofthetruncatedSIShalomodel,derivedfromtheanalysis
of162simulatedSNIadatageneratedassumingnolensing(left)andatruncatedSISmodel(right).Intheright-handpanel,trueparametersareindicatedby
verticallinesandcrossesin1Dand2Dplotsrespectively.
100 100
80 80
60 60
c c
40 40
20 20
0 0
0.2
0.1
Mpc] Mpc] 0.15
*r [2000.05 *r [200 0.1
0.05
0 0
−0.5 0 λ0.5 1 0 5c0 100 0 r02*.0005 [Mpc]0.1 0 λ0.5 1 0 5c0 100 0 r2*000. 1[Mpc] 0.2
Figure2.1Dand2Dmarginalisedposteriorsdistributionsfortheparametersh = {λ,c,r∗ }oftheNFWhalomodel,derivedfromtheanalysisof162
200
simulatedSNIadatageneratedassumingnolensing(left)andatruncatedSISmodel(right).Intheright-handpanel,trueparametersareindicatedbyvertical
linesandcrossesin1Dand2Dplotsrespectively.
Simulationmodel and without a truncated SIS lensing signal are very similar, indi-
Analysismodel Nolensing tSIS NFW catingthat aSNLS-qualitycatalogue containing just 162 SNIais
Nolensing 0.0±0.2 0.0±0.2 0.0±0.2 insufficient to constrain the halo properties. In particular, we see
NtSFISW −−31..18±±00..22∗∗ −−21..61±±00..22∗ −−21..33±±00..22∗ thatthe1Dmarginalsforσ∗ andr∗ bothpeakatzero(whichcor-
respondstonolensingsignal).Anothernoteworthyfeatureisthat,
Table4.∆logZ(log-evidencevaluerelativetothenullevidence)forthe whereasthevalueofηisunconstrained,thereisquiteastrongcon-
analysis ofsimulated datawith162SNIa.Asterisks denotethecases for straint restricting γ to be negative. From (17), one sees that this
whichthecorrespondinghaloparametersconstraintsareplottedinFigs1 corresponds to a rather curious truncation radius scaling-law, for
and2,respectively. whichgalaxieswithlargervelocitydispersionshavesmallertrun-
cationradii.Moreover,providedηispositive,thisalsocorresponds
(cid:13)c 2012RAS,MNRAS000,1–13
8 N.V. Karpenka et al.
Simulationmodel
Analysismodel Nolensing tSIS NFW
Nolensing 0.0±0.2 0.0±0.2 0.0±0.2
tSIS −1.7±0.2 4.5±0.2∗ 6.9±0.2
NFW −3.1±0.2 3.6±0.2 7.2±0.2∗
Table5.∆logZ(log-evidencevaluerelativetothenullevidence)forthe 2
analysis ofsimulated datawith500SNIa.Asterisks denotethecases for
whichthecorrespondinghaloparametersconstraintsareplottedinFigs3 η 0
and4,respectively.
−2
300
to more luminous (and presumably more massive) galaxies hav-
ingsmaller truncation radii.Thisphenomenon wasalsonoted by m/s] 200
Jo¨nssonetal.(2010b),butwaserroneouslyinterpretedasbeinga σ [k*100
constraintonthehalopropertiesderivedfromtheSNIadata,which 0
isclearlynotthecasesinceitoccursevenintheanalysisofsimula-
0.4
tionscontainingnolensingsignal.Rather,from(13)and(17),one
seesthatanegativevalueofγallowsforasmallervalueofthetrun- Mpc] 0.2
cationradiusrtto‘offset’anincreaseinthevelocitydispersionσ, r [*
therebyreducingthelensingsignalproducedbytheputativehalo,
0
−5 0 5 −2 0 2 0 100 200 300 0 0.2 0.4
asrequiredtobeconsistentwithsimulateddatacontainingnolens- γ η σ* [km/s] r* [Mpc]
ingsignal.
FromFig.1,onealsoseesthattheconstraintsonNFWhalo Figure3.1Dand2Dmarginalisedposteriorsdistributionsfortheparame-
parametersarealmostidenticalwhenanalysingdatawithandwith- tersh={γ,η,σ∗,r∗}ofthetruncatedSIShalomodel,derivedfromthe
analysis of500simulated SNIadatagenerated assumingatruncated SIS
outthelensingsignal.Oncemore,itisnotpossibletoplacecon-
model.Trueparameters areindicated byvertical lines andcrosses in1D
traintsonthehaloparameters.Oneshouldnote,however,thatthe
and2Dplotsrespectively.
exponentλofthescalinglawbetweenvirialradiusandluminosity
doesappearrelativelywellconstrainedaroundλ 1,whichcorre-
≈ 3
spondstoaconstantmass-to-lightratioM/L L0.Nonetheless, thecorresponding halo parameter constraints in Figs3and 4, re-
∝
one sees a similar constraint when analysing simulated data con-
spectively.FromFig.3,whichcorrespondstoatruncatedSIShalo
tainingnolensingsignal;essentially,largervaluesofλ(throughits model inboth thesimulationandtheanalysis, oneseesthateach
dependenceonr200)resultinastrongerlensingsignalthatisruled ofthe1Dand2Dmarginalisedposteriorshasawell-definedpeak
outbythedata.
(awayfromzero),indicatingarobustdetectionofthelensingsig-
nal.ThissupportstheconclusiondrawnabovefromtheBayesian
evidence resultsin Table 5. One also sees that all theparameters
5.2 Resultsfor500simulatedsupernovae
have been estimated with reasonable accuracy, although some of
Totestthatouranalysisprocedureiscapableofdetectingthegrav- the2Dmarginalsexhibitpronounceddegeneraciesbetweenthepa-
itationallensingsignalandplacingthecorrectconstraintsonhalo rameters,particularthoseinvolvingtheparameterr∗,whichisthe
parametersinpresenceofmoredata,wealsoanalyseasimulated leastwellconstrained.
SNLS3-stylecataloguecontaining500SNIa.Inthiscase,theSNIa Similar conclusions can be drawn from the parameter con-
are randomly distributed on the sky over the four SNLS3 fields, straintscorrespondingtoaNFWhalomodelinboththesimulation
outsideofourmasked regions.Thissimulatedcataloguecontains andanalysis,whichareshowninFig4.Oneseesthatthe1Dand
approximatelythreetimesthenumberofSNIaasconsideredpre- 2Dmarginaliseddistributionshavereasonablywell-definedpeaks
viouslyandisthusrepresentativeofwhatcouldbeachievedbythe thatcontainthetrueparametervalues,althoughagaindegeneracies
SLNSprogrammeinatotalofabout9yearsofobservation. This do exist between parameters. One also sees that the 1D posteri-
isclearlyratherunrealisticintermsoftherequiredobservingtime orsforbothcandr2∗00 peakwellawayfromzero,supportingthe
andresources,butstillprovidesausefulinsightintothequantityof conclusiondrawnfromtheBayesianevidencesinTable5thatone
SNIadatarequiredtomakearobustdetectionofthelensingsignal obtainsastrongdetectionofthelensingsignal.
andconstrainhaloproperties.
Table5liststheBayesianlog-evidenceforeachcombination
ofsimulationandanalysismodel,relativeineachcasetothenull
6 APPLICATIONTOREALSUPERNOVAEDATA
(no-lensing)model.Fortheno-lensingsimulationmodel,onesees
thattheno-lensinganalysismodelisagaincorrectlypreferredover WenowapplyourBayesiananalysismethodologytotherealSNIa
bothlensinghalomodels.Moreimportantly,onenowfindsthatfor and galaxies data sets described in Section 4. For the real data
both simulations containing a lensing signal, the analysis models we again consider both the truncated SIS and NFW halo models
alsocontainingalensingsignalarepreferredathighsignificance, in our analysis. For each case, we also investigate dividing the
thereby indicating a robust detection of the gravitational lensing foreground galaxiesin theSNLScatalogue into passive and star-
effect.Moreover,wenotethatineachcasethecorrecthalomodelis forming classes, each having halo model parameters that are al-
preferred,althoughlog-evidencedifferencesbetweenhalomodels lowedtobeindependent.
arequitesmall,sothisdoesnotconstitutearobustselectionofthe Table 4 lists the Bayesian log-evidence for each analysis
correcthalomodel. model,relativeineachcasetothenull(no-lensing)model.Inthe
For the two cases denoted by asterisks in Table 5, we plot caseofonegalaxytype,oneseesthatthereisaverymarginalpref-
(cid:13)c 2012RAS,MNRAS000,1–13
Constrainingdarkmatterhaloesusingsupernovae 9
2
100 η 0
80
60 −2
c
40 300
20 m/s] 200
0 σ [k*100
0
0.2
0.4
*r [Mpc]200000..01.155 r [Mpc]*0.2
0 0
0.2 0λ.4 0.6 0 5c0 100 0 r2*000. 1[Mpc] 0.2 −5 0γ 5 −2 η0 2 0 1σ0*0 [km2/0s0] 300 0 r* [0M.2pc] 0.4
Figure4.1Dand2Dmarginalisedposteriorsdistributions fortheparam- Figure5.1Dand2Dmarginalisedposteriorsdistributionsfortheparame-
eters h = {λ,c,r∗ }ofthe NFWhalomodel, derived fromthe anal- tersh={γ,η,σ∗,r∗}ofthetruncatedSIShalomodel,derivedfromthe
200
ysisof500simulatedSNIadatageneratedassumingaNFWmodel.True analysisofrealSNLS3data.
parametersareindicatedbyverticallinesandcrossesin1Dand2Dplots
respectively.
1galaxytype 2galaxytypes
parametersconstraintsliemid-waybetweenthoseobtainedforthe
Model (hp=hsf) (hp6=hsf)
simulatedSNIadatawith162and500supernovaerespectively,as
TruncatedSIS 0.2±0.2∗ 0.5±0.2∗
discussedinSection5.
NFW −2.5±0.2∗ −2.2±0.2
Focussing on the 1D marginal posterior distribution of each
Table6.∆logZ (log-evidencevaluerelativetothenullevidenceforno haloparameter,wefirstnotethestrongpreferencefornegativeval-
lensing signal) for the analysis of the real SNIa data. hp = hsf indi- uesofγ,asweobservedintheanalysisofsimulationscontaining
cates that the halo parameters for passive and star-forming galaxies are nolensingsignal.AsexplainedinSection5,thisismerelyaconse-
constrainedtobethesame,whereasforhp 6= hsf theyareallowedtobe
quenceofthelensingsignal(ifany)inthedatabeingveryweakand
independent.Asterisksdenotethecasesforwhichthecorrespondinghalo
doesnotconstituteameaningfulconstraintonthehaloproperties.
parametersconsraintsareplottedinFigs5,6and7,respectively.
Conversely,thereisarelativelystrongpreferenceforpositive
t values of η, corresponding to a positive correlation between the
halovelocitydispersionandluminosity.Indeed,themarginaldis-
tribution peaks at η 1, which is considerably larger than the
erenceofjust0.2log-evidenceunitsforalensingsignalproduced canonical values of η≈p = 1 and ηsf = 1, corresponding to the
byatruncatedSIShalomodel.Thisis,however,onlyatthelevel 4 3
Faber–JacksonandTully–Fisherrelations,validforearlyandlate-
of the uncertainty in the evidence calculation, and so this model
type galaxies, respectively. Thisconstraint also differsnoticeably
andtheno-lensingmodelarebroadlyequallyfavoured.Alensing
from that shown in the left-hand panel of Fig. 1, obtained from
signal produced by a NFWhalo model is, however, strongly dis-
simulateddatacontainingnolensingsignal,whichissuggestiveof
favoured with an evidence of 2.5 log-evidence units relative to
− amarginallensingsignalbeingpresentintherealdata.
thenolensingmodel.
The possibility of a faint lensing signal being present in the
Whenonesplitstheforegroundgalaxiesintotheirpassiveand
dataisalsosuggestedbythe1Dmarginaldistributionforthefidu-
star-formingspectraltypes,oneseesthattheevidenceforalensing
cialvelocitydispersionσ∗,whichhasaclearpeakawayfromzero.
signal produced by atruncated SISincreases very slightlyand is
Indeed,thepeakiscentredonσ∗ 120kms1,whichwasalsothe
marginalpreferredoverthenolensingmodelby0.5log-evidence ≈
value assumed inthe simulations analysed inSection 5. The fact
units.Onceagain,alensingsignalproducedbyaNFWhalomodel
thattheparameterconstraintobtainedfromtherealdataagainap-
isquitestronglydisfavouredby 2.2log-evidenceunits.
− pearstighterthanthatobtainedfromthesimulationsof162SNIa
Wediscuss thehalo parametersconstraints for thesevarious
withnolensingsignal,shownintheleft-handpanelofFig.1,may
modelsinthefollowingthreesubsections.
bearesultoftherealdatapreferringη 1,asopposedthevalue
≈
η = 1 usedinoursimulations.From(13)and(17),oneseesthat
3
thelensingsignalproducedbyahalowithagivenσ∗ islargerin
6.1 TruncatedSISmodel
theformercase.
The1Dand2Dmarginalisedparameterconstraintsobtainedforthe Finally,wenotethatthe1Dmarginalforthefiducialtrunca-
truncatedSIShalomodelwithonegalaxytypeareshowninFig.5. tionradiuspeaksveryclosetozeroandyieldsnorealconstrainton
Asonemightexpectfromtherelativeevidenceforthismodel,the r∗,aswefoundinouranalysisofthe162simulatedSNIa.
(cid:13)c 2012RAS,MNRAS000,1–13
10 N.V. Karpenka et al.
those for star-forming galaxies. Indeed, for the passive galaxies,
themarginaldistributionscloselyresemblethoseshowninFig.1,
obtained from simulations containing 162 SNIa with no lensing
signal. Conversely, for star-forming galaxies, the constraints are
slightlytighterthanthoseshowninFig.5,obtainedfromthereal
dataassumingjustonegalaxytype,butresemblethemintheirmain
features.Inparticular,weagainseeastrongpreferencefornegative
100 valuesof γsf,but now the1D marginal distributionhasamodest
80 peakatγsf 3.Wealsorecoverastrongconstraintthatηsf is
60 positive,wit≈hth−e1Dmarginalpossessingasmallpeakatηsf 1.
c ≈
40 The1Dmarginalforσ∗sfpeaksstronglyawayfromzero,preferring
20 avalueofσ∗sf 150kms−1.Thereis,however,noconstrainton
≈
0 r∗sf,whichisinkeepingwithourpreviousfindings.
Finally, it is worth noting that the 2D marginal in the
(σ∗p,σ∗sf)-spaceshowninFig.7bearsapassingresemblancetothe
0.15
correspondingplot(figure6)inJo¨nssonetal.(2010b),butonlyif
Mpc] 0.1 σ∗pandσ∗sf areinterchanged.Giventherelativepercentagesofpas-
*r [2000.05 sive and star-forming galaxies in the catalogue, it seems sensible
that one should obtain a tighter constraint on σ∗sf than σ∗p, as we
0 findinFig.7.Bycontrast,figure6inJo¨nssonetal.(2010b)shows
0 λ0.5 1 0 5c0 100 0 0.0r52*00 [M0.1pc] 0.15 theopposite, which suggests that they have erroneously swapped
thegalaxytypesintheiranalysis.
Figure6.1Dand2Dmarginalisedposteriorsdistributionsfortheparame-
tersh={λ,c,r∗ }oftheNFWhalomodel,derivedfromtheanalysisof
200
realSNLS3data.
6.4 Inclusionofadditionallow-redshiftsupernovaedata
Wealsoinvestigatetheinclusionofadditional, low-redshift SNIa
6.2 NFWmodel data toour analysis as apotential means of enhancing the detec-
tion of a halo lensing signal. The rationale here is first to anal-
The1Dand2Dmarginalisedparameterconstraintsobtainedforthe
ysethelow-z SNIadataalone, using preciselythesamemethod-
NFWhalomodelforonegalaxytypeareshowninFig.6.Thecon-
ologyasfortheSNLS3data,butassumingnolensingfromfore-
straintsontheparametersareverysimilartothoseshowninFig2,
groundgalaxyhaloes.Theresultingposteriordistributionsderived
obtainedfrom162simulatedSNIawithandwithoutalensingsig-
for (most of) the nuisance parameters listed in Table 1 may then
nalproduced byaNFWhalomodelwiththeparameterslistedin
usedaspriorsontheseparametersinthesubsequentanalysisofthe
Table3.Oneseesthatitisnot possibletoplacemeaningful con-
(high-z)SNLS3data,therebyreplacingtheveryconservativepri-
straintson any of the parameters, indicating theabsence of ade-
orsassumedinTable1,whichmaybe‘diluting’thelensingsignal.
tectablelensingsignal.Thisisconsistentwiththelowrelativeevi-
Asourlow-z SNIadataset,weuseanupdatedversionofthe
denceforthismodellistedinTable6.
catalogue compiled by Sullivan et al. (2010). This consists of a
samplefromthecompilationofConleyetal.(2010),whichitself
includes SNIa from a variety of sources (primarily Hamuy et al.
6.3 Separatingpassiveandstar-forminggalaxies
1996a;Riessetal.1999;Jhaetal.2006;Hickenetal.2009a;Con-
So far we have assumed that the halo parameters for the trun- trerasetal.2009).Sullivanetal.applybulk-flowpeculiarvelocity
catedSISandNFWmodels,respectively,arethesamefordiffer- corrections to the SN magnitudes and redshifts, placing the red-
ent galaxytypes. Thismaybeanoversimplication and soitisof shiftsintheCMB-frame(z )followingNeilletal.(2007), but
cmb
interesttosplittheforegroundgalaxiesintotheirpassiveandstar- withupdatedmodels(Conleyetal.2010).OnlySNIainthesmooth
formingspectraltypes,bothofwhichareallowedtohaveindepen- Hubbleflow,definedasz > 0.01areused,andthesamelight
cmb
denthaloparameters. curvequality cutsasfor the SNLS3sampleareemployed. There
WeconcentrateonthetruncatedSISmodel,sincethisyielded are 123 low-redshift SN Ia in total, with redshifts in the range
asignificantlyhigherevidencethantheNFWprofileinouranalysis [0.01,0.08].
assumingjustasinglegalaxytype(seeTable6).Thegalaxiesinour Analysis of these data yields approximately Gaussian pos-
sampleareclassifiedaseitherpassiveorstar-formingdependingon teriors (truncated to positive values) on the nuisance parameters,
theirsSFR(seeSection4.2).Wethusallowallthehaloparameters given by: α (1.30,0.112), β (2.81,0.182), R
x
∼ N ∼ N ∼
tobedifferentforeachtypeofgalaxy,sothatthefullhaloparam- (0.119,0.0082) and R (0.074,0.052). These distribu-
c
eter space becomes h = γp,ηp,σ∗p,r∗p,γsf,ηsf,σ∗sf,r∗sf . The Ntionsareclearlymuchtighter∼thanNtheoriginalassumedpriorsgiven
{ }
fractionofpassivegalaxiesrangesfrom5percentintheCFHTLS in Table 1; note that we use the original prior on the nuisance
fieldsD1andD3to8percentinfieldD4.Thus,thevastmajority parameter σ , as this quantity is likely to differ between low-z
int
offoregroundgalaxiesarestar-forming. andhigh-z SNIa,sinceobservationaluncertaintiessuchasthees-
Theresulting1D and 2D marginalisedposteriorsfor thepa- timationof photometric errorscan lead tovariations of σ on a
int
rametersareshowninFig.7.Oneseesthattheconstraintsonthe sample-by-samplebasis(Marchetal.2012).Wefindthatusingthe
parameters for passive and star-forming galaxies are very differ- posteriorsderivedfromthelow-zSNIaaspriorsinthesubsequent
ent.Asonemightexpect,giventherelativepercentagesofpassive analysisofthehigh-zSNLS3datayieldsparameterconstraintsthat
and star-forming galaxies in the catalogue, the constraints on the arealmostidenticaltothosepresentedinFigs5–7.Moreover,the
haloparametersofthepassivegalaxiesareverymuchweakerthan log-evidences relativetothenull(no-lensing) model arefoundto
(cid:13)c 2012RAS,MNRAS000,1–13