Improved Treatment of Host-Galaxy Correlations in Cosmological Analyses With Type Ia Supernovae
Brodie Popovic, Dillon Brout, Richard Kessler, Dan Scolnic, Lisa Lu
PPreprint typeset using L A TEX style emulateapj v. 12/16/11
IMPROVED TREATMENT OF HOST-GALAXY CORRELATIONS IN COSMOLOGICAL ANALYSES WITHTYPE IA SUPERNOVAE
Brodie Popovic , Dillon Brout , Richard Kessler , Dan Scolnic , Lisa Lu Department of Physics, Duke University, Durham, NC, 27708, USA. Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA NASA Einstein Fellow Department of Astronomy and Astrophysics, The University of Chicago, Chicago, IL 60637, USA. Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA. and Independent Scholar (Dated: February 4, 2021)
ABSTRACTImproving the use of Type Ia supernovae (SNIa) as standard candles requires a better approachto incorporate the relationship between SNIa and the properties of their host galaxies. Using aspectroscopically-confirmed sample of ∼ γ ) to within 0 .
004 mags, which is a factor of 5 improvement over the previous methodthat results in a γ -bias of ∼ .
02. We adapt BBC for a novel dust-based model of intrinsic brightnessvariations, which results in a greatly reduced mass step for data ( γ = 0 . ± . γ = 0 . ± . w ,vary from ∆ w = 0 . . . w -bias using previous BBC methods that ignore SNIa-host correlations. INTRODUCTION
The standardisation of measurements of Type Ia Su-pernovae (SNIa) led to the discovery that the universe isexpanding at an increasing rate (Riess et al. 1998; Perl-mutter et al. 1999). A possible cause of this expansion,called ‘dark energy’, remains an unsolved mystery in cos-mology to this day. In the intervening decades since thediscovery of dark energy, SNIa cosmology has grown fromsample sizes of tens to thousands. When combined withconstraints from the Cosmic Microwave Background, thestatistical uncertainties in measurements of the dark en-ergy equation-of-state parameter, w , are on the order of ∼ .
04, (Scolnic et al. 2018; Jones et al. 2018; Brout et al.2019b). Therefore, understanding systematic uncertain-ties on the level of ∼ .
01 is needed to better measurethe nature of dark energy. To further reduce systematicuncertainties, here we improve the treatment of the cor-relation between supernova properties and host-galaxyproperties.Most SNIa cosmology analyses use the SALT2 (Guyet al. 2010) framework that relies on two parameters tostandardise the SNIa brightness: a colour ( c ) describ-ing the wavelength-dependent luminosity and a light-curve stretch ( x ) describing the luminosity dependenceon light-curve duration. In addition to these SALT2 lu-minosity correlations, studies have shown a correlationbetween the standardised brightness of SNIa and proper-ties of their host-galaxy, such as mass (e.g. Sullivan et al.2010), Star Formation Rate (e.g. Uddin et al. 2017), aswell as other properties (Rose et al. 2020). While c and x are rigorously included in the SALT2 model, the cor- Email: [email protected] relation with host properties is typically included as anad-hoc correction to the supernova brightness. Addition-ally, correlations between c and host-properties as wellas x and host-properties have been ignored when thisad-hoc correction is applied. In this analysis, we focuson host-galaxy stellar mass ( M stellar ) as a representativehost-galaxy property.To predict and correct for biases in the measurement ofdistance modulus values from standardisation methods,analyses have increasingly relied on simulations (Kessleret al. 2009a; Betoule et al. 2014; Scolnic et al. 2018;Brout et al. 2019b). These simulations used the pub-licly available SuperNova ANAlysis package (SNANA:Kessler et al. 2009b); a detailed description of the simu-lations is given in Kessler et al. (2019). To simulate SNIasamples, simulations rely on measurements of underlyingpopulations of c and x . Early SNIa cosmology analy-ses tuned these populations based on visual comparisonsof fitted c and x distributions between data and sim-ulations and lacked rigorous procedure. This motivatedScolnic & Kessler 2016 (SK16) to formalise a methodfor determining the underlying populations. By utiliz-ing large simulations, SK16 determined the underlyingpopulations by tracking how selection effects, noise andintrinsic scatter cause populations to ‘migrate’ from un-derlying distributions to observed distributions. Here,we improve on SK16 by determining underlying c and x populations as a function of M stellar . Our popula-tion fitting code is publicly available , along with sampleinputs. . https://github.com/bap37/ParentPops T.B.D a r X i v : . [ a s t r o - ph . C O ] F e b Simulated samples can be used to determine redshift-dependent bias corrections, as done in the analysisfor SDSS-II (Kessler et al. 2009a) and for the JointLightcurve Analysis (JLA; Betoule et al. 2014). Sub-tle c and x biases were shown in SK16 for simu-lated distance moduli, suggesting that 1-dimensionalredshift-dependent bias corrections are not sufficient.This motivated Kessler & Scolnic (2017) to develop theBEAMS with Bias Corrections (BBC) method, whichcorrects for observed light-curve fit parameters basedon rigorous simulations with corrections computed in 5-dimensional parameter space (redshift, colour, stretch,colour-luminosity relationship, and stretch-luminosityrelationship). Hereafter, we refer to this method asBBC5D, which has been used in Dark Energy Survey 3year analysis (DES3YR; Abbott et al. 2019; Brout et al.2019b), analyses of the Pan-STARRs sample (Scolnicet al. 2018; Jones et al. 2018), and in a re-analysis of theSloan Digital Sky Survey-II (SDSS) (Popovic et al. 2019).We refer to the redshift-only correction as BBC1D. Asummary of these methods is presented in Table 1.Using SNANA to simulate DES3YR with a knowninput value of γ , Smith et al. (2020) showed that theBBC5D-recovery of γ from an input value is significantlybiased when correlations between c / x and M stellar areincluded in the simulation. To address the γ bias foundin Smith et al. (2020), we improve the BBC formalismfrom BBC5D to BBC7D by introducing two new dimen-sions in the bias corrections.Brout & Scolnic 2020, hereafter BS20, present a newexplanation for the mass step γ as a consequence of vary-ing extinction ratios for high and low mass host galax-ies. BS20 extended the SALT2 formalism to include twosources of colour variation: intrinsic and dust. Thisformalism, however, is incompatible with BBC5D andBBC7D; therefore we adapt BBC to work with BS20(BBC-BS20).Here, we address the concerns about corrections forthe mass step raised by Smith et al. (2020), and confirmthe origin of the mass step presented in BS20. Section 2provides an overview of the data used in this work. Sec-tion 3 is an overview of population-fitting techniques andselection effects. In Section 4 we review the methodol-ogy introduced in previous studies and expand on themto track evolution with host-galaxy properties. Section 5details changes to the BBC formalism, the efficacy ofwhich is presented in Section 6. Finally, discussion andconclusions are presented in Section 7. DATA
In this analysis, we compile light-curves from 6 SNIasamples that have been spectroscopically confirmed. Thesamples used here are from the Sloan Digital Sky Survey-II (SDSS; York et al. 2000; Sako et al. 2018), the Super-nova Legacy Survey (SNLS; Sullivan et al. 2010), Pan-STARRs (PS1; Jones et al. 2018; Scolnic et al. 2018), theFoundation Supernova Survey (Foley et al. 2018) and theDark Energy Survey (DES; Brout et al. 2019a). Thelow-redshift (Low-z) supernovae include samples fromthe Carnegie Supernova Project (CSP) (Stritzinger et al.2011) and Harvard-Smithsonian Center for Astrophysics(CfA3-4) (Hicken et al. 2009b,a, 2012).These samples are flux-calibrated to the SuperCal sys-tem (Scolnic et al. 2015). Host-galaxy masses are taken . . . . . . z C o un t Fig. 1.—
The redshift distribution of the combined Foundation,Low-z, SDSS, DES, SNLS, and PS1 data sample. from past analyses, notably we use the masses for SDSS,PS1, and SNLS provided in the Pantheon sample (Scol-nic et al. 2018). For Foundation, we use the masses pro-vided by Jones et al. (2018). For DES, we use updatedmasses provided by Smith et al. (2020) and Wisemanet al. (2020). These analyses derive host-galaxy massesfrom SED fitting to broadband photometry. To measureparent populations, we apply the following selection re-quirements (cuts): • light-curve fit probability P fit > .
01 that varies foreach survey • fitted SNIa colour | c | < . • fitted SNIa colour uncertainty σ c < . • fitted SNIa stretch | x | < • fitted SNIa stretch uncertainty σ x < • fitted SNIa time-of-peak-brightness error σ t < • at least five observations • at least one observation before peak brightness • at least one observation after peak brightnessHowever, for Low-z, we drop the c / x uncertainty cutsto increase statistics. A summary of the total SNIa foreach sample is given in Table 2. The resulting red-shift distribution is shown in Figure 1. The c , x , and M stellar distributions for our combined data sample arepresented in Figure 2, along with the mean values for c and x as a function of M stellar . SIMULATIONS AND ANALYSIS
Simulations
Simulations of supernovae are needed to correct for bi-ases arising from inefficiencies and from the light-curvefitting process. Bias corrections are needed to measurepopulations for stretch and colour, and to measure dis-tances. Therefore we generate simulations to assess theimpact of systematics on cosmological measurements. Tothis end, we use SNANA to simulate realistic samples ofSNIa with appropriate noise, observing conditions, ca-dence, and detection efficiencies. We generate these sim-ulations with the Flat Λ-CDM cosmology from Planck
TABLE 1Breakdown of different bias correcting methods.
Method Bias-Correction Comment CitationDependenceBBC1D z Biascor grid of z : correct µ ( z ) . Marriner et al. (2011)Kessler & Scolnic (2017)BBC5D { z, x , c, α, β } Biascor grid of α, β, c, x , z : correct m B ( z ) , c and x Kessler & Scolnic (2017)BBC7D { z, x , c, α, β, θ, M stellar } Biascor grid of α, β, ,¸ x , z, γ, M stellar : correct m B ( z ) , c and x . This WorkBBC-BS20 { z, x , c, M stellar } Biascor based on BS20 model, correct µ ( z ) in grid of c, x , z, M stellar This Work
TABLE 2Summary of Data Statistics
Sample
Host Logmass − − S N I a S t r e t c h ( x ) Data meanData Host Galaxy Mass − . − . . . . S N I a C o l o u r ( c ) Host Galaxy Stellar Mass
Fig. 2.—
The distributions of c and x for the combined Low-z, DES, PS1, Foundation, SNLS, and SDSS spectroscopic samplesas a function of host-galaxy mass. Circles are the data; purplerepresent individual data points and black are the average valuefor that bin. The dash-dotted line is 0. Collaboration et al. (2016): Ω M = 0 .
315 (matter densityat z = 0) and w = − spec , for the SNLSsurvey to better fit the data (Appendix A1). These simulations rely on previously derived frame-works that describe brightness variations among theSNIa population, which we call intrinsic scatter. Here,we use the G10 (Guy et al. 2010) and C11 (Chotardet al. 2011) scatter models, translated into SED vari-ants in Kessler et al. (2013), along with the BS20 model.G10 and C11 are spectral variation models that differin the amount of variation ascribed to chromatic ver-sus achromatic scatter. For G10, roughly 70% of scatteris achromatic and 30% is chromatic, whereas for C11,roughly 25% of scatter is achromatic and 75% is chro-matic. The BS20 model does not have an explicit SEDvariation model; instead it includes dust parameter pop-ulations that are not part of the SALT2 framework.To accurately characterise our systematic and statisti-cal uncertainties, we simulate 100 data-sized samples. Analysis
We fit the light-curves using SNANA with the SALT2model as developed in Guy et al. (2010) and updatedin Betoule et al. (2014). For each SNIa, the SALT2fit gives four fitted parameters: m B , the log of the fit-ted light-curve amplitude x ; x , the stretch parametercorresponding to light-curve width,; c , the light-curvecolour; and t , the time of peak brightness. From the fit-ted SALT2 parameters, we infer distance modulus valueswith a modified version of the Tripp distance estimator(Tripp 1998). Following the BBC formalism in Kessler& Scolnic (2017), the distance modulus ( µ ) is defined as: µ = m B + αx − βc − M z i + δµ host + δµ bias (1)where α and β are global nuisance parameters relatingto stretch and colour respectively. M z i is the distanceoffset in discrete redshift bins denoted by z i . δµ host is theluminosity correction for the mass step, and is defined as: δµ host = γ × (1 + e ( X stellar − S ) /τ X ) − − γ , (2)where γ is the magnitude of the SNIa luminosity differ-ence between SNIa in high and low mass galaxies, X stellar is log M stellar , S ∼
10 is the step location, and τ X isthe width of the step. Finally, δµ bias is the distancebias correction. As summarised in Table 1, for BBC1D, δµ bias is determined in bins of redshift using simulationswith a fixed α and β . For higher-dimensional correc-tions (BBC5D/BBC7D/BBC-BS20), the corrections areapplied in bins of redshift along with other SNIa and hostgalaxy properties.Furthermore, the total distance modulus error σ isdescribed as σ = σ N + σ + σ σz + σ (3)where σ N includes the uncertainties from SALT2 param-eters and their covariances (see Equation 3 in Kessler &Scolnic (2017)), σ lens = 0 . z is the uncertainty fromweak gravitational lensing (Scolnic et al. 2018), σ σz is theuncertainty from peculiar velocity (Carrick et al. 2015),and σ int is the intrinsic scatter contribution determinedin the BBC fitting process.The BBC fit maximises a likelihood that includes sep-arate terms for SNIa and core collapse (CC) SNe. Be-cause our sample includes only spectroscopically con-firmed events, the CC term is excluded. The BBC fitdetermines α , β , γ , σ int over the entire redshift range,and a distance modulus in each redshift bin ( z i ). Follow-ing Brout, Hinton & Scolnic (2020), because we are notassessing the covariances of systematics, a z -binned Hub-ble diagram results in equivalent cosmological constraintsas that of an unbinned set. These binned distances arecombined with priors in cosmological fits. Here, Ω M and w are obtained with “ wfit ”, a χ minimisation programusing MINUIT (James & Roos 1975). We use a GaussianΩ M prior with a mean of 0.315 and width of 0 . MODELING SNIA POPULATIONS WITH HOST GALAXYCORRELATIONS
Here, we determine the underlying correlations be-tween SNIa parameters and host-galaxy properties givendifferent models of intrinsic scatter (G10, C11: Sec-tion 2; BS20: Section 4.4). In Section 4.1, we review themethodology from SK16, who developed an underlyingparent population method to ensure that resulting simu-lated distributions of c and x match the data. Naively,we would extend SK16 to a three dimensional populationthat includes M stellar , however, the SK16 method relieson a robust model of simulated uncertainties that hasbeen validated for c and x , but not for M stellar . There-fore, we use the SK16 method for c and x , and introducea different method for M stellar which does not rely onmodeling M stellar uncertainties.We fit our underlying parameter populations sepa-rately for Low-z, Foundation, and the combined high- z collection of DES, SDSS, SNLS, and PS1. As with SK16,we find notable differences between low and high redshiftsurveys. The difference in M stellar distributions and se-lection effects between Low-z and Foundation (Figure 3in Jones et al. 2019) motivate the separate fits.We derive the dependence of underlying parameterpopulations on host-galaxy mass for the G10 and C11models (4.1 and 4.2). In Section 4.3 and 4.4, we reviewthe BS20 framework and describe improvements to mod-eling the parent populations. Review of Method to Determine UncorrelatedParent Populations
Following SK16, we extend their stretch population de-scription for high-redshift surveys (PS1, SDSS, DES, andSNLS) by replacing the asymmetric Gaussian with anasymmetric generalised normal distribution that dependson four parameters, P ( x ) = (cid:26) e ( −| x − x | n /nσ n − ) if x ≤ x e |−| x − x | n /nσ n + ) if x > x (4)where x is the value at peak probability of the asym-metric generalised normal distribution, n is the shape,and σ − and σ + are the width parameters for negativeand positive x values, respectively. The motivation for this distribution is discussed in Section 6.1. For n = 2,Equation 4 reduces to the three parameter asymmetricGaussian distribution used in SK16. This parametrisa-tion is also used to describe the SNIa colour c .For Low-z and Foundation, the stretch distributionis double-peaked. We therefore take an alternative ap-proach following Scolnic et al. (2018) and use a doubleGaussian model, P ( x ) = A × e ( −| x − x | / σ ) + A × e ( −| x − x | / σ ) (5)where A i is the weight, x i is the mean value, and σ i isthe standard deviation of the respective Gaussian.A simulation with a flat distribution of true colour andtrue stretch is generated, and SALT2 light-curve fits de-termine the measured colour and measured stretch forthe SNIa that pass light-curve quality cuts. The trueand measured values from these flat simulations are usedto compute a migration matrix ( X for stretch and C forcolour) that captures the migration from the input dis-tribution to the observed distribution through selectioneffects, measurement noise and intrinsic scatter. Eachcomponent of the matrix, X ij , describes the likelihoodthat a true value in an input stretch bin i ( x i ) migratesto a measured value in stretch bin j ( x j ), and similarlyfor colour.For an underlying stretch population, P ( x ), we definea binned distribution (cid:126)P x with components P xi . SK16defines (cid:126) ∆ x , the data-simulation difference vector, as (cid:126) ∆ x = o x o x ... o xn − X , X , · · · X ,n X , X , · · · X ,n ... ... . . . ... X d, X d, · · · X d,n × P x P x ... P xn (6)where the measured distribution vector (cid:126)o x has the samebinning as X and (cid:126)P x .For the χ calculation, the associated data error vectoris (cid:126)e x = [ e x , e x , ..., e xn ], where e x i = √ o x i for o x i > e x i = 1 for o x i = 0; while not technically correctfor a Poisson distribution (Baker & Cousins 1984), SK16has shown this this error approximation is sufficient. Thefour parameters that describe P ( x ) are determined byminimising the χ x defined as χ x = n (cid:88) i =1 (cid:18) ∆ xi e xi (cid:19) . (7)For the colour distribution, χ c is defined similarly using (cid:126) ∆ c and (cid:126)e c .SK16 performed a grid search for their parameters,where here we use a Monte Carlo minimisation proce-dure using the emcee python package (Foreman-Mackeyet al. 2013). We compare our population parameters tothose of SK16 and find that we replicate their resultsto within 1 σ for parameters describing the colour andstretch distributions (Equation 4 with n = 2). We findthat this assumption works well for the c population,and therefore we fix n = 2 (see Appendix A2). For x ,however, we find that fixing n = 3 works better and theresulting χ x is smaller by ∼ n = 2. Extending Parent Populations to Include MassDependence
The ideal approach for including mass-dependent cor-relations is to replace (cid:126)o x in Equation 6 with a 2-dimensional array of stretch and host-galaxy mass, (cid:126)o x ,M , where the subscript is M = M stellar . Similarly,the migration matrix X and probability vector (cid:126)P x wouldalso be extended to include M stellar .However, the M stellar measurements lack a well de-fined uncertainty and therefore the migration matrix isnot as well determined in the M stellar dimension as itis for c or x . We therefore assume that the measured M stellar is the true M stellar . To reduce the dependenceon M stellar uncertainties, we implement a 2 step process.First, we fit the parent populations by minimising χ x and χ c (Eq. 7) in M stellar bins. Ideally we would perform thisminimisation in small M stellar bins, however, the statis-tics of SNIa per bin is insufficient. As a compromise, wefit in relatively large M stellar bins of 1 . × M (cid:12) butuse a small step size of 0 . × M (cid:12) . Although the M stellar bins are strongly correlated, we have used simu-lated data samples to validate our method. In the secondstep, we re-weight the simulated M stellar distribution tomatch the data. REVIEW OF BS20
BS20 observed a significant ( > σ ) colour-dependentHubble scatter, and a 5 σ colour-dependence on the massstep ( γ ). They model this effect with simulations usingthe SALT2 model combined with host galaxy dust. Theydefine 3 contributions to the observed colour, c obs = c int + E dust + (cid:15) noise (8)where c int is the intrinsic SNIa colour, E dust is the dustextinction, and (cid:15) noise is measurement noise. These pa-rameters contribute to a total change in true SNIa bright-ness of ∆ m B, obs = β SN × c int + R V × E dust (9)where β SN is introduced as the correlation coefficient be-tween the intrinsic colour and SNIa luminosity, and R V is the dust extinction ratio. Components of the BS20model that are intrinsic to the SNIa, i.e. c int and β SN ,are assumed to be independent of M stellar . Thus, BS20determine the following parameters describing the distri-butions of dust and colour: Gaussian distribution of in-trinsic color (¯ c int , σ c int ), Gaussian distribution of intrin-sic color-luminosity coefficient ( ¯ β SN & σ β SN ), and Gaus-sian distribution of dust extinction ratios ( ¯ R V & σ Rv )for low and high mass, and exponential distribution ofthe dust extinction ( τ E ) for low and high mass.The BS20 parameters are determined in a forward-modeling fitting process by minimising the χ in Eq.8 of BS20. The χ includes constraints enforcing con-sistency between data and simulations for: the colourdistribution, the Hubble diagram scatter versus colour,the BBC-fitted Hubble residual versus colour, and BBC-fitted β . BS20 fixes the α in the simulations to be thatobtained from the dataset using BBC1D, and fix γ = 0. Upgrades to Parent Populations for Dust BasedScatter Models
For the BS20 model, we derive the x populations fol-lowing the method discussed in Section 4.4. However, wekeep the same colour distribution presented in BS20 be-cause their dust model parameterisation is incompatiblewith the SK16 method. While this upgrade to the BS20population parameters is minimal, there are significantupgrades to the BBC formalism presented in Section 5.3.These upgrades to the BBC formalism are essential be-cause BS20 is incompatible with higher dimensional BBCprocedures. A new and significantly improved formalismfor determining the underlying population parameters forBS20 is under development and will be presented in a fu-ture work. CORRECTING DISTANCE BIASES
Here we review the BBC methodology for correctingdistance biases (Section 5.1) and describe improvements.In Section 5.2, we detail the improvements to accountfor correlations between SNIa brightness and host-galaxyproperties. Finally, in Section 5.3, we describe improve-ments to the BBC formalism required for bias correctionsusing the BS20 model.
Review of 5D Bias Corrections
A brief overview of the BBC process is described inSection 3.2 and here we elaborate on the bias correc-tions component for BBC5D. Kessler & Scolnic (2017)expanded on the 1D method by defining a bias-correcteddistance where the individual Tripp components of m B , c , and x are corrected. They define a bias-correcteddistance, µ ∗ = m ∗ B + αx ∗ − βc ∗ − M z i = ( m B − δ m B ) + α ( x − δ x ) − β ( c − δ c ) − M z i = m B + αx − βc − M z i − δµ bias ( z, x , c, α, β )(10)where bias-corrected quantities are denoted with a starsuperscript and δµ bias ≡ ( δ m B + αδ x − βδ c ) (11)Measurement noise and intrinsic scatter preclude calcu-lating the exact bias correction for each event; therefore, δµ bias is interpolated in 5D cells of { z, x , c, α, β } . The δµ bias term is calculated with a large simulation desig-nated as a ‘BiasCor’, by comparing the observed valuesto the simulated ones for each SNIa parameter. The firstthree dimensions ( z , x , c ) are interpolated using the sim-ulated populations. However, the luminosity correlationcoefficients ( α , β ) are single valued and not describedwith a population model akin to z, x , c . Therefore, thebias corrections for α and β are determined on a 2 × α and β found in thedata. This grid enables for an interpolation of the Bias-Cor sample at the value of the proposed α and β in eachiteration of the BBC fit. Improving Bias Corrections to Account forSNIa-Host Correlations
Smith et al. (2020) find that the fitted γ from BBC5Dis biased when analysing simulated samples that includecorrelations between c/x and host-galaxy mass. To ac-count for these biases in the mass step, we introduce twonew dimensions to the δµ bias term in Equation 10: θ ,which is a magnitude shift, and M stellar .The new θ dimension is incorporated into the Bias-Cor by adding a magnitude shift of + θ to a randomhalf of the simulation, and − θ to the other half; thereis no correlation between θ and host properties, and thus θ (cid:54) = γ . Using BiasCor with θ , the BBC fit allows fora mag-shift ( δµ host ) as an arbitrary function of X =log M stellar and other parameters. While previous cos-mology analyses have used δµ host = − γ/ X > and δµ host = + γ/ X < , here we adopt themore general δµ host function with step location S andstep width τ M (Equation 2). At each step of the BBCfit, the value of δµ host for each event is used to interpolatethe BiasCor between ± θ such that δµ bias = δµ bias ( (cid:126)x , X, − θ )+ f × [ δµ bias ( (cid:126)x , X, + θ ) − δµ bias ( (cid:126)x , X, − θ )] (12)where f = ( δµ host + θ ) / θ and (cid:126)x = { z, x , c, α, β } . It isimportant that θ > δµ bias to ensure a valid interpolation.The addition of θ and M stellar results in changing the5D δµ bias term in Equation 10 to a 7D δµ bias as follows, δµ bias ( z, x , c, α, β ) → δµ bias ( z, x , c, α, β, θ, M stellar ) . (13)For the first six BiasCor dimensions, δµ bias is interpo-lated. For the M stellar dimension, δµ bias is evaluated indiscrete bins to avoid interpolating across the luminositystep at 10 M (cid:12) .The effect of θ on the bias corrections is shown in Fig-ure 3 as a function of redshift. Here we set θ = ± . z < . ∼ .
02 mag at high- z . Because θ is independent ofsupernova parameters, it can be used for investigatingcorrelations between any host galaxy property and SNIaluminosity, not just host-galaxy stellar mass. Changes to the BBC formalism for BS20
While the SALT2 model is an accurate description ofSNIa light curves, it is nonetheless an approximation thatignores the difference between intrinsic color variationand dust. Previous SNIa cosmology analyses have usedthe same SALT2 model in both light curve fitting andBiasCor; this means that the BiasCor corrects for selec-tion effects, but does not correct for biases in the SALT2model. BS20 attempts to provide a more accurate light-curve model that can be used for the BiasCor, and herewe update BBC to be compatible with BS20.The BBC5D formalism is not compatible with BS20 forthree reasons: 1) BBC intrinsic scatter is characterisedby a single colour-independent σ int while BS20 uses adust-dependent scatter that is poorly characterised bya single σ int , 2) BBC assumes the single SALT2 colour-luminosity relation β , whereas BS20 uses distributionsfor intrinsic β SN and dust R V (Eq. 9), and 3) c int and β SN in BS20 refer to intrinsic colour and intrinsic colour-luminosity relationship, while SALT2 c and β includeboth dust and intrinsic properties.Ideally, to incorporate BS20, we would decompose thefitted colour c into c int and c dust and compute a true β in the simulation. While we plan to address these is-sues in a future work, they require significant updates to − . − . − . − . − . . B i nn e d ∆ µ ( m a g ) . . . . . . z − . − . − . − . − . .
00 0 . . . . . . zαx Corrections βc Corrections m B CorrectionsTotal Corrections mag shift = + θ mag shift = − θ Fig. 3.—
Bias correction vs. redshift for θ = +0 .
06 (blue) and θ = − .
06 (orange). The biased quantity is indicated on eachpanel. both training and fitting code; instead, we make severalapproximations. Here we describe BBC updates to becompatible with the BS20 model:1. Replace the SALT2 model in the BiasCor with theBS20 model.2. Include M stellar as in Section 5.2, but not θ becausethe mass step is predicted by the BS20 model with-out the γ parameter.3. Remove grid-interpolation of α and β because theBS20 simulations are forward-modeled and resultin BBC-fitted α and β that are in agreement withthat of the data.4. δµ bias is computed for distances instead of bias-correcting each SALT2 parameter.These changes are referred to as BBC-BS20 and result ina δµ bias dimensionality of { z, x , c, M stellar } . For BBC-BS20, Equation 10 becomes: µ ∗ = m B + αx − βc − M z i − δµ bias (14)and Equation 11 becomes δµ bias = m B CB + αx B C − βc B C − M z i − µ true . (15)The true distance modulus for the BiasCor is µ true , andthe superscript BC denotes that these values are fromSALT2 light-curve fits to the BiasCor events. We notethat Equations 14 and 15 can be used with BBC5D andthe SALT2 model, and yield consistent results comparedto using Equations 10 and 11. Host Logmass − − S N I a S t r e t c h ( x ) χ N = 0 . x /c σ − σ + Host Galaxy Mass − . − . . . . S N I a C o l o u r ( c ) χ N = 1 . Data Data mean Sim mean
Host Galaxy Stellar Mass
Fig. 4.—
The distributions of c and x for the combined DES,PS1, SNLS, and SDSS spectroscopic samples as a function of host-galaxy mass. Circles are the data; purple represent individual datapoints and black are the average value for that bin. Solid lines arethe G10 parent population parameters: green is σ + , blue is thepeak value, and orange is σ − . The dash-dotted line is 0. The binaverages for the data are presented in red-lined black dots, for thesimulations, red-lined yellow triangles. The histograms on the sideshow data in circles and simulated results in dashed line. For both c and x we find good agreement between the mean value in dataand sims, with a χ /N = 1 . χ /N = 0 .
65 respectively. RESULTS
Here we evaluate the accuracy of our population mod-elling and bias correction approaches. Section 6.1 dis-cusses the results of the parent population modelling andhow well our simulations match the data. Section 6.2presents a comparison of BBC7D to previous approachesfor a variety of metrics. Section 6.3 contains the resultsof the new BBC-BS20 method for dust-scatter models.A discussion of these results is in Section 7.
Parent Populations
Using the simulations described in Section 3.1, we fol-low the process presented in Section 4.2 to determine theunderlying populations of stretch and colour when as-suming the G10 or C11 scatter models. For both scattermodels, the population parameters for the generalisednormal distribution, σ + , x , σ − , are presented in Ap-pendix A2 for each individual survey. These populationparameters are provided in steps of 0 . × M stellar .In Figure 4, we show the comparison between sim anddata for c vs. M stellar and for x vs. M stellar . We alsoshow the M stellar dependence of the underlying combinedSDSS, DES, SNLS, and PS1 parent population. TheFoundation and Low-z parent populations are shown sep-arately in Figures 10 and 11 for colour and stretch re-spectively. There are three notable results. First, thereis excellent agreement between the observed mean valuesfor data and simulation. In the combined SDSS, DES,SNLS, and PS1 sample, this parent population also char-acterises the constituent individual surveys. Second, wefind that both the intrinsic x and c populations depend − . . . S N I a C o l o u r ( c ) − . . . S N I a S t r e t c h ( x ) SK16 average This work Data . . . . . . Redshift ( z )891011 S t e ll a r M a ss χ /N = 4 . χ /N = 1 . χ /N = 3 . Fig. 5.—
The redshift evolution of c , x , and host-galaxy stellarmass for the combined sample. The data is presented in purplecircles, and averaged bin values for the data are shown in red-linedblack circles. The averaged bin values for the SK16 simulationsare shown in red-lined blue squares, red-lined yellow triangles rep-resent the average bin values for this work. Individual simulatedevents are not shown for clarity. We present the χ /N values foreach parameter. SK16 assumes a flat distribution of M stellar withredshift. on M stellar .We evaluate the significance of M stellar dependence bycomparing each curve in Figure 4 to a null model that hasno M stellar dependence. For the x distribution, σ + isconsistent with no M stellar dependence with a confidenceof 99.9%. The probability that σ − has no M stellar depen-dence, however, is 1E-8. This means that the observed x dependence on mass is driven by σ − increasing withincreasing masses. Similarly, x has a low probability ofbeing independent of M stellar at only 0.01%, but does notcorrelate with the observed distribution and is thereforeunlikely to be the primary driver. For the colour dis-tribution, σ − and c are consistent with no M stellar de-pendence with a confidence of 90%. Similarly to x , theprobability that the faint-side σ + has no M stellar depen-dence is 99.9%. Taken together, this constitutes the thirdnotable result: the M stellar dependence of the observed c and x distributions is driven by increasing faint-sidewidths ( σ − for x and σ + for c ) rather than a shift inthe mean values.We also investigate the observed redshift dependenceof our c and x distributions. Figure 5 shows that theobserved colour and stretch distributions depend on red-shift for two reasons: 1) separate populations for low andhigh z (Section 4), and 2) selection effects. The data-simulation agreement is reasonable, with χ /N = 4.8,1.5, and 3.5 for stretch, colour, and M stellar , respectively.To model the double-peaked x distribution in Low-zand Foundation, we found it necessary to implement aprior requiring that x < x > x ∼ − − − − − C o un t Foundation
SimData − − − − x Low-z
Fig. 6.—
The x distributions for Foundation and Low-z surveys.Data is presented in black circles and simulation is presented in bluehistogram. Impact on Cosmology Using Bias Corrections withHost Properties
With our simulations from Section 3.1 and the parentpopulations shown in Figure 4, we create 100 simulateddata sets and a large BiasCor to test the accuracy of ourbias correction methods. We compare four different biascorrection methods for the G10 and C11 models. Twomethods use BBC1D, with and without M stellar informa-tion included in the BiasCor (designated ‘No Mass’, simi-lar to that of Betoule et al. (2014)). The other two meth-ods use the BBC5D and new BBC7D method. Over-all, we find our new method significantly improves uponBBC5D. In Table 3, we show the fitted nuisance param-eters, Hubble scatter (RMS of µ fit − µ true ), w and w -biasdefined as ∆ w = ( w fit − w true ), averaged over the 100simulations.For the G10 scatter model, we find an α bias below1% for all BBC methods. This broad agreement holdswell for the recovery of β as well. The BBC5D approachresults in a significant γ bias of ∆ γ = 0 . ± .
001 mag.BBC7D reduces this bias by a factor of ∼
5, althougha 2 millimag γ -bias remains with 2 σ significance. Thesetrends in α , β , and γ are qualitatively similar in the C11results. However, it is worth noting that the BBC1Drecovery of β and γ for the C11 scatter model is signifi-cantly biased.For both G10 and C11, we see a general decreasein Hubble scatter with increasing dimensionality of theBBC methodology. BBC5D and BBC7D have compara-ble Hubble scatter with each other for both G10 and C11,and both are ∼
10% smaller compared to using BBC1D.The presence of host-galaxy correlations in the BBC1DBiasCor does not make a significant impact on recovered w values for G10 or C11. In the case with and withouthost-galaxy correlations, the w -bias is ∼ . ± . ∼ . ± .
005 for C11. BBC5D doesnot significantly effect this BBC1D w -bias, recovering − z − . − . − . . . . . µ BB C − µ tr u e G10 Simulations with G10 BiasCor w = − . ± . w = − . ± . w = − . ± . Fig. 7.—
The binned µ residuals for the G10 scatter model usingthree different BBC methods on the same set of simulations. Un-certainties are shown for BBC7D only but are representative forthe other methods. − . ± . − . ± . w -bias for both G10 and C11is a factor of 2 smaller than their 5D counterparts: w -bias= 0 . ± . . ± . ∼ σ significance. Over-all, BBC7D has the smallest w -bias as well as the small-est biases for other nuisance parameters and scatter.In Figure 7, we show the binned distance modulusresiduals as a function of redshift for BBC1D, BBC5D,and BBC7D. BBC7D has smaller µ residuals (mostly ∼ .
005 across the z range) than the BBC5D ap-proach. All BBC µ residuals have a significant excess, µ BBC − µ true = 0 . ± .
004 (2 . σ ), around z = 0 . σ µ , in comparison to Hubble Residual (HR) scatter is an-other important metric in determining the effectivenessof BBC. Figure 8 shows BBC1D and BBC7D σ µ andHR scatter as a function of SNIa colour, c . We see bet-ter agreement for BBC7D compared to BBC1D. Figure8 and Table 3 show that for G10 and C11, the BBC7Dmethods have the smallest scatter. Impact on Cosmology Using Bias Corrections forBS20
The bottom tier of Table 3 shows how well BBC1D andBBC-BS20 perform on simulations generated with theBS20 model. While simulations of the BS20 model do nothave an input γ like the aforementioned simulations G10and C11, the simulation includes host-mass vs colour-luminosity dependence that matches that of the data.As reported in BS20, we find that using BBC1D resultsin γ ∼ .
06, consistent with the value found in the data.BBC-BS20 results in γ = 0 . ± . α (∆ α ∼ . ± . α -bias (∆ α ∼ . ± . γ -reduction, we applythe BBC-BS20 method on real data and find γ = 0 . ± . TABLE 3Fitted values and uncertainties a averaged over 100 simulations Method Model input α = 0 .
145 input β = 3 . γ = 0 .
05 input w = − α Fitted β Fitted γ Hubble scatter b Fitted w ∆ w ( N σ w )BBC1D (No Mass) c G10 0 . ± . . ± . . ± . . ± . − . ± . .
017 (3 . σ )BBC1D G10 0 . ± . . ± . . ± . . ± . − . ± . .
022 (4 . σ )BBC5D G10 0 . ± . . ± . . ± . . ± . − . ± . − . . σ )BBC7D G10 0 . ± . . ± . . ± . . ± . − . ± . − . . σ )Method Model input α = 0 .
145 input β = 3 . γ = 0 .
05 input w = − α Fitted β Fitted γ Hubble scatter Fitted w ∆ w ( N σ w )BBC1D (No Mass) C11 0 . ± . . ± . . ± . . ± . − . ± . . . σ )BBC1D C11 0 . ± . . ± . . ± . . ± . − . ± . . . σ )BBC5D C11 0 . ± . . ± . . ± . . ± . − . ± . − . . σ )BBC7D C11 0 . ± . . ± . . ± . . ± . − . ± . − . . σ )Method Model input α = 0 .
145 input β = ND d input γ = 0 input w = − α Fitted β Fitted γ Hubble scatter Fitted w ∆ w ( N σ w )BBC1D (No Mass) BS20 0 . ± . . ± . . ± . . ± . − . ± . . . σ )BBC1D BS20 0 . ± . . ± . . ± . . ± . − . ± . . . σ )BBC-BS20 BS20 0 . ± . . ± . . ± . . ± . − . ± . . . σ )space space space space space space space a Uncertainties are the average uncertainty among 100 simulations, divided by √ b The error in the Hubble scatter is the R.M.S divided by √ c Does not include M stellar dependent parent populations in the BiasCor. d Not Determined: For the BS20 model, input SALT2 β ( β (cid:54) = β SN ) is not defined. . . . m a g s BBC1D Hubble ScatterBBC7D Hubble Scatter BBC1D σ µ BBC7D σ µ − . − . − .
05 0 .
00 0 .
05 0 .
10 0 . SNIa Colour ( c )0 . . m a g s BBC-BS20 Hubble ScatterBBC1D Hubble Scatter BBC-BS20 σ µ BBC1D σ µ . . m a g s C11BS20G10
Fig. 8.—
The Hubble Scatter (line) and the BBC-fitted distancemodulus uncertainties, σ µ , (triangle) are plotted for three differentscatter models. We expect good agreement between the two for aneffective BBC method. which is more than × β SN and R V , and thus we cannot determine an input β for the simulation. For this reason, the BS20 section ofTable 3 refers to the input β as Not Defined (ND), andwe are unable to compute β biases. BS20 found theirmodel parameters such that their BBC1D β from simu-lations matched that of the data ( β ∼ . ± . M stellar distributions, their fitted model parameters are approxi-mate. Here we update the M stellar distribution to matchthe data (Figure 2) but do not update the BS20 model parameters; we find BBC1D correction on BS20 simula-tions results in β ∼ . ± .
01, which no longer matchesthat of the data. However, when running BBC-BS20 onour simulations and real data we find good agreement: β = 3 . ± .
005 (sim) and β = 3 . ± .
045 (data)respectively. This BBC1D β discrepancy warrants theneed for further improvements to the BS20 modeling.BBC1D results in a w -bias of ∼ . ± . w -bias is significantly reduced and consistentwith 0, ∼ . ± . ∼
10% smaller than that ofG10 or C11, which is illustrated in Figure 8. DISCUSSION AND CONCLUSION
The utilisation of host galaxy information in SNIastandardisation analyses has become common in recentanalyses. Typically, host galaxy properties have been in-cluded in the one of the final stages of the analysis tomake an additive correction to SNIa luminosities beforemeasuring cosmological parameters from the set of SNIadistances. However, this approach is conceptually flawedbecause it does not account for subtle biases due to thecorrelations of SNIa properties and host galaxies. Here,we have determined the underlying populations of SNIaproperties and their correlations with host galaxies sothat we can trace and correct for these biases.Our approach for these corrections is to modify theBBC method to allow for higher dimensional bias cor-rections. If the intrinsic scatter model is known, BBCdetermines the input mass step ( γ ) with a bias of 0.004mag for all models, and determines w with biases consis-tent with 0 for G10 and BS20. For γ , the BBC7D bias isa factor of 5 smaller than for BBC5D; for w , the BBC7Dbias is a factor of 2 smaller.We do not address potential biases if the intrinsic scat-ter model is not known. Using BBC5D, Scolnic et al.(2018) and Brout et al. (2019b) showed that the differ-ence in w between assuming the G10 and C11 is ∼ . .
00 0 .
25 0 .
50 0 .
75 1 . zCMB C o un t SNLS
DataNew EffOld Eff
Fig. 9.—
The SNLS redshift distribution for the data (solid black line with points), simulations with JLA spectroscopic efficiency (orangedash-dot line), and updated spectroscopic efficiency presented here (blue solid line). At higher redshifts, the new efficiency better matchesthe data. w -difference when doing 1D corrections between BS20,G10, and C11 was as much as much as 4%. However,in that study, they did not simultaneously refit for amass step, and it is unclear how this approximationwould affect the cosmological bias. This intrinsic scatteruncertainty will be quantified in an upcoming analysis(Popovic et al. in prep.), which will determine optimalparameters of the BS20 model and address the discrep-ancy in the fitted 1D β values between data and simula-tion.In determining the underlying populations, we followa purely empirical approach that does not rely on the-oretical models to relate SNIa properties to host galaxyproperties. Nicolas et al. (2020) use theoretical model-ing to predict an evolution of the relation between the x distribution with redshift due to the different sam-pling of progenitor systems at high- z compared to low- z .While we are unable to show whether this evolution isdue to progenitor systematics or survey selection effects,our analysis illustrates how the model from Nicolas et al.(2020) can be incorporated in a cosmological analysis.From our empirical approach, we find that peak prob-abilities of the parent distributions of color and stretchdo not evolve significantly with host-galaxy stellar mass,but rather that the observed dependence on M stellar isexplained by increasing asymmetries of the parent distri-butions. For c , this finding agrees with the model pro-posed in BS20, who suggested that dust is responsiblefor one-sided SNIa c scatter towards redder colors. Simi-larly, Rigault et al. (2018) propose that local specific starformation rate (lsSFR) is a tracer of the progenitor agewhich itself traces the SNIa x distribution. They find that the parent distributions of x for low lsSFR and highlsSFR are inconsistent. We find that the observed cor-relation between x and mass is driven by an increasingfaint side standard deviation ( σ − ) of the parent distribu-tion, which suggests a more subtle relationship between x and mass that can be described by a non-evolvingpeak probability and an evolving asymmetry.Finally, while we focused here on using host-galaxymass to correlate with SNIa properties, our methods canbe applied to other host galaxy properties. Furthermore,while we used a step function for the mass dependence,an arbitrary functional form can be used with BBC7D.Ultimately, our improved BBC method will be evaluatedby varying the SNIa model such that simulated distri-butions match those of the data, and propagating thesevariations to the BiasCor used by BBC. Acknowledgements
Dillon Brout acknowledges support for this work wasprovided by NASA through the NASA Hubble Fellow-ship grant HST-HF2-51430.001 awarded by the SpaceTelescope Science Institute, which is operated by Asso-ciation of Universities for Research in Astronomy, Inc.,for NASA, under contract NAS5-26555.Richard Kessler acknowledges that this work was sup-ported in part by the Kavli Institute for CosmologicalPhysics at the University of Chicago through an endow-ment from the Kavli Foundation and its founder FredKavli. R.K. is supported by DOE grant DE-SC0009924.Dan Scolnic is supported by DOE grant DE-SC0010007and the David and Lucile Packard Foundation.
APPENDIX
A1. EFFICIENCIES AND ERRATA
For the SNLS sample, we improve the spectroscopic efficiency function described as a function of peak i -band mag,Eff spec( i ) . Figure 9 shows that the JLA Eff spec , also used in BS20, underestimates the number of supernovae at redshiftsgreater than 0 .
75. Our update is to replace Eff spec( i ) with Eff spec( i − . . A2. PARENT POPULATION PARAMETERS
Here we show the parent populations as a function of mass for each survey, characterised by an asymmetric generalisednormal distribution with three parameters: a peak probability and two standard deviations. See Equation 4 for rigorousdefinition. For c , the n value is fixed at n = 2. For x , n = 3. For the Low-z and Foundation surveys, we present1 TABLE 4LOWZ parent colour populations
Mass mean (G10) σ − (G10) σ + (G10) mean (C11) σ − (C11) σ + (C11)9.8 − . ± .
042 0 . ± .
037 0 . ± . − . ± .
044 0 . ± .
038 0 . ± . − . ± .
02 0 . ± .
016 0 . ± . − . ± .
023 0 . ± .
019 0 . ± . − . ± .
017 0 . ± .
012 0 . ± . − . ± .
018 0 . ± .
013 0 . ± . − . ± .
015 0 . ± .
01 0 . ± . − . ± .
016 0 . ± .
011 0 . ± . − . ± .
014 0 . ± .
009 0 . ± . − . ± .
015 0 . ± .
01 0 . ± . − . ± .
02 0 . ± .
013 0 . ± . − . ± .
018 0 . ± .
012 0 . ± . − . ± .
023 0 . ± .
015 0 . ± . − . ± .
021 0 . ± .
014 0 . ± . − . ± .
034 0 . ± .
022 0 . ± . − . ± .
031 0 . ± .
021 0 . ± . − . ± .
039 0 . ± .
027 0 . ± . − . ± .
039 0 . ± .
025 0 . ± . . ± .
035 0 . ± .
03 0 . ± .
029 0 . ± .
033 0 . ± .
03 0 . ± . a double-Gaussian approach characterised in Equation 5. Population fit results for c are in Tables 4, 6, 9 for Low-z,Foundation, and the combined DES, SDSS, SNLS, and PS1 sample, respectively. The population fit results for x arein Tables 5 7, 8 for Low-z, Foundation, and the combined DES, SDSS, SNLS, and PS1 sample, respectively. Thesepopulations are shown in Figures 10 and 11 for c and x for the G10 scatter model. . . . Host Galaxy Mass − . − . . . . c HIZ . . . Host Galaxy Mass
LOWZ . . . Host Galaxy Mass
FOUND
Fig. 10.—
The G10 c parent populations as a function of host-galaxy mass for several spectroscopic surveys. The mean of the asymmetricdistribution is presented in blue, the peak in orange, the bright-side standard deviation in green, and the faint-side standard deviation inred. Errors are included for each parameter and the 68% confidence interval for each is shown in grey fill. . . . Host Galaxy Mass − x HIZ . . . Host Galaxy Mass
LOWZ . . . Host Galaxy Mass
FOUND
Fig. 11.—
The G10 x parent populations as a function of host-galaxy mass for several spectroscopic surveys. The mean of the asymmetricdistribution is presented in blue, the peak in orange, the faint-side standard deviation in green, and the bright-side standard deviation inred. Errors are included for each parameter and the 68% confidence interval for each is shown in grey fill. For Foundation and Low-z, twopeak values are shown by purple lines, and the line thickness denotes the relative weight of each peak. TABLE 5LOWZ parent stretch populations
Mass Weight (G10) x (G10) σ (G10) Weight (C11) x (C11) σ (C11)9.8 1 . ± . − . ± .
644 1 . ± .
454 3 . ± .
204 0 . ± .
181 0 . ± . . ± . − . ± .
342 0 . ± .
52 3 . ± .
169 0 . ± .
142 0 . ± . . ± . − . ± .
102 0 . ± .
107 2 . ± .
144 0 . ± .
635 0 . ± . . ± . − . ± .
163 0 . ± .
169 1 . ± .
579 0 . ± .
116 0 . ± . . ± . − . ± .
058 0 . ± .
043 1 . ± .
665 0 . ± .
603 0 . ± . . ± . − . ± .
055 0 . ± .
041 1 . ± .
539 0 . ± .
108 0 . ± . . ± . − . ± .
059 0 . ± .
046 1 . ± .
592 0 . ± .
101 0 . ± . . ± . − . ± .
064 0 . ± .
06 1 . ± .
544 0 . ± .
127 0 . ± . . ± . − . ± .
085 0 . ± .
111 0 . ± .
481 0 . ± .
163 0 . ± . . ± . − . ± .
377 1 . ± .
624 0 . ± .
61 1 . ± .
827 1 . ± . TABLE 6Foundation parent colour populations
Mass mean (G10) σ − (G10) σ + (G10) mean (C11) σ − (C11) σ + (C11)10.0 0 . ± .
049 0 . ± .
03 0 . ± . − . ± .
05 0 . ± .
034 0 . ± . . ± .
041 0 . ± .
031 0 . ± . − . ± .
041 0 . ± .
029 0 . ± . − . ± .
04 0 . ± .
025 0 . ± . − . ± .
034 0 . ± .
023 0 . ± . . ± .
047 0 . ± .
027 0 . ± . − . ± .
039 0 . ± .
026 0 . ± . . ± .
051 0 . ± .
028 0 . ± . − . ± .
048 0 . ± .
031 0 . ± . . ± .
05 0 . ± .
029 0 . ± .
043 0 . ± .
049 0 . ± .
031 0 . ± . TABLE 7Foundation parent stretch populations
Mass Weight (G10) x (G10) σ (G10) Weight (C11) x (C11) σ (C11)10.0 2 . ± . − . ± .
506 0 . ± .
554 2 . ± .
367 0 . ± .
543 0 . ± . . ± . − . ± .
153 0 . ± .
192 1 . ± .
99 0 . ± .
28 0 . ± . . ± . − . ± .
097 0 . ± .
101 1 . ± .
743 0 . ± .
278 0 . ± . . ± . − . ± .
092 0 . ± .
103 1 . ± .
011 0 . ± .
189 0 . ± . . ± . − . ± .
097 0 . ± . . ± .
732 0 . ± .
276 0 . ± . . ± . − . ± .
152 0 . ± .
256 0 . ± .
644 0 . ± .
451 0 . ± . TABLE 8Combined SDSS, DES, SNLS, PS1 parent colour populations
Mass mean (G10) σ − (G10) σ + (G10) mean (C11) σ − (C11) σ + (C11)8.2 − . ± .
026 0 . ± .
016 0 . ± . − . ± .
029 0 . ± .
019 0 . ± . − . ± .
02 0 . ± .
013 0 . ± . − . ± .
03 0 . ± .
018 0 . ± . − . ± .
015 0 . ± .
01 0 . ± . − . ± .
021 0 . ± .
012 0 . ± . − . ± .
023 0 . ± .
015 0 . ± . − . ± .
022 0 . ± .
014 0 . ± . − . ± .
019 0 . ± .
013 0 . ± . − . ± .
018 0 . ± .
011 0 . ± . − . ± .
019 0 . ± .
013 0 . ± . − . ± .
025 0 . ± .
015 0 . ± . − . ± .
02 0 . ± .
013 0 . ± . − . ± .
021 0 . ± .
013 0 . ± . − . ± .
019 0 . ± .
013 0 . ± . − . ± .
02 0 . ± .
013 0 . ± . − . ± .
02 0 . ± .
014 0 . ± . − . ± .
026 0 . ± .
017 0 . ± . − . ± .
012 0 . ± .
009 0 . ± . − . ± .
017 0 . ± .
012 0 . ± . − . ± .
015 0 . ± .
011 0 . ± . − . ± .
014 0 . ± .
009 0 . ± . − . ± .
016 0 . ± .
011 0 . ± . − . ± .
015 0 . ± .
01 0 . ± . − . ± .
018 0 . ± .
013 0 . ± . − . ± .
016 0 . ± .
011 0 . ± . − . ± .
018 0 . ± .
012 0 . ± . − . ± .
017 0 . ± .
013 0 . ± . − . ± .
018 0 . ± .
013 0 . ± . − . ± .
014 0 . ± .
011 0 . ± . − . ± .
019 0 . ± .
013 0 . ± . − . ± .
015 0 . ± .
011 0 . ± . − . ± .
022 0 . ± .
016 0 . ± . − . ± .
019 0 . ± .
013 0 . ± . − . ± .
025 0 . ± .
017 0 . ± . − . ± .
021 0 . ± .
014 0 . ± . − . ± .
023 0 . ± .
016 0 . ± . − . ± .
026 0 . ± .
017 0 . ± . − . ± .
031 0 . ± .
023 0 . ± . − . ± .
04 0 . ± .
028 0 . ± . TABLE 9Combined SDSS, DES, SNLS, PS1 parent stretch populations
Mass mean (G10) σ − (G10) σ + (G10) mean (C11) σ − (C11) σ + (C11)8.2 0 . ± .
288 0 . ± .
241 0 . ± .
201 0 . ± .
227 0 . ± .
751 0 . ± . . ± .
349 0 . ± .
245 0 . ± .
244 0 . ± .
312 0 . ± .
611 0 . ± . . ± .
416 0 . ± .
305 0 . ± .
287 0 . ± .
281 0 . ± .
212 0 . ± . . ± .
365 0 . ± .
265 0 . ± .
244 0 . ± .
253 0 . ± .
182 0 . ± . . ± .
277 0 . ± .
202 0 . ± .
188 0 . ± .
273 0 . ± .
196 0 . ± . . ± .
253 0 . ± .
176 0 . ± .
171 0 . ± .
287 0 . ± .
201 0 . ± . . ± .
273 0 . ± .
206 0 . ± .
172 0 . ± .
315 0 . ± .
229 0 . ± . . ± .
31 0 . ± .
228 0 . ± .
21 0 . ± .
295 0 . ± .
21 0 . ± . . ± .
257 1 . ± .
177 0 . ± .
192 0 . ± .
246 0 . ± .
171 0 . ± . . ± .
225 1 . ± .
155 0 . ± .
169 0 . ± .
239 1 . ± .
166 0 . ± . . ± .
202 1 . ± .
142 0 . ± .
15 0 . ± .
262 1 . ± .
179 0 . ± . . ± .
26 1 . ± .
187 0 . ± .
189 0 . ± .
307 1 . ± .
212 0 . ± . . ± .
221 1 . ± .
162 0 . ± .
149 0 . ± .
309 1 . ± .
227 0 . ± . . ± .
26 1 . ± .
186 0 . ± .
176 0 . ± .
338 1 . ± .
246 0 . ± . . ± .
285 1 . ± .
207 0 . ± .
191 0 . ± .
35 1 . ± .
258 0 . ± . . ± .
334 1 . ± .
25 0 . ± .
227 0 . ± .
344 1 . ± .
257 0 . ± . . ± .
41 1 . ± .
301 0 . ± .
281 0 . ± .
243 1 . ± .
192 0 . ± . . ± .
512 1 . ± .
398 0 . ± .
325 0 . ± .
31 1 . ± .
258 0 . ± . . ± .
512 1 . ± .
428 0 . ± .
363 0 . ± .
392 1 . ± .
547 0 . ± . − . ± .
845 1 . ± .
69 1 . ± .
672 0 . ± .
605 2 . ± .
538 0 . ± .4