It's Dust: Solving the Mysteries of the Intrinsic Scatter and Host-Galaxy Dependence of Standardized Type Ia Supernova Brightnesses
SS UBMITTED TO T HE A STROPHYSICAL J OURNAL
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IT’S DUST: SOLVING THE MYSTERIES OF THE INTRINSIC SCATTER AND HOST-GALAXY DEPENDENCE OFSTANDARDIZED TYPE IA SUPERNOVA BRIGHTNESSES D ILLON B ROUT & D
ANIEL S COLNIC Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA NASA Einstein Fellow Department of Physics, Duke University, Durham, NC 27708, USA
Submitted to The Astrophysical Journal
ABSTRACTThe use of Type Ia Supernovae (SNe Ia) as cosmological tools has motivated significant effort to: understandwhat drives the intrinsic scatter of SN Ia distance modulus residuals after standardization, characterize thedistribution of SN Ia colors, and explain why properties of the host galaxies of the SNe correlate with SN Iadistance modulus residuals. We use a compiled sample of ∼ σ detection of the dependence of Hubble residual scatter on SN Ia color. We introduce a physicalmodel of color where intrinsic SN Ia colors with a relatively weak correlation with luminosity are combinedwith extrinsic dust-like colors ( E ( B − V )) with a wide range of extinction parameter values ( R V ). This modelcaptures the observed trends of Hubble residual scatter and indicates that the dominant component of SN Iaintrinsic scatter is from variation in R V . We also find that the recovered E ( B − V ) and R V distributions differbased on global host-galaxy stellar mass and this explains the observed correlation ( γ ) between mass andHubble residuals seen in past analyses as well as an observed 4 . σ dependence of γ on SN Ia color. Thisfinding removes any need to prescribe different intrinsic luminosities to different progenitor systems. Finallywe measure biases in the equation-of-state of dark energy ( w ) up to | ∆ w | = 0 .
04 by replacing previous modelsof SN color with our dust-based model; this bias is larger than any systematic uncertainty in previous SN Iacosmological analyses.
Subject headings: supernovae, cosmology INTRODUCTIONStudies in the last decade of research in cosmology withType Ia supernovae (SNe Ia) have forewarned that the mea-surements of the equation-of-state of dark energy w will soonhit a systematic floor. Yet, such measurements (B14: Betouleet al. 2014, S18: Scolnic et al. 2018, B19b: Brout et al. 2019,Jones et al. 2019) continually reach better levels of both statis-tical and systematic precision. This is due to the improvementof systematic uncertainties in survey and camera design, butalso due to the possibility afforded from significantly largersamples to understand systematics in the analysis. In the mostrecent analyses (S18, B19b), it has been found that system-atic uncertainties in understanding the intrinsic scatter of stan-dardized SN Ia brightnesses is of a similar level or larger thanuncertainties due to external, photometric calibration. As cal-ibration uncertainties have been dominant in past systematicerror budgets, this moment marks a transition from a need tounderstand external issues independent of the supernovae to aneed to also better understand SN Ia physics.With current cosmological analyses of SNe Ia requiringmmag-level control of systematics, uncertainty over how tounderstand the intrinsic scatter of standardized SN Ia bright-nesses, which is on the 0.1 mag level, is problematic. Prac-tically, intrinsic scatter is measured as the excess scatter ofSN Ia distance residuals to a best-fit cosmology after ac-counting for measurement noise. A holistic understandingof SN Ia intrinsic scatter and its underlying characterizationhas remained elusive, but its size has been found to depend [email protected]@duke.edu on a wide variety of measurement components: redshift (e.g.,B14), wavelength range of the photometric observations (e.g.,Mandel et al. 2011), host-galaxy properties (e.g., Uddin et al.2017), and spectroscopic features (e.g., Fakhouri et al. 2015).Furthermore, Scolnic & Kessler (2016) showed that the rela-tive amounts of chromatic versus achromatic components ofthe intrinsic scatter models were directly linked to the intrin-sic SN Ia color population and reddening law; however, thisstudy was unable to discriminate between different models.After the discovery of the accelerating universe (Riess et al.1998; Perlmutter et al. 1999), there were two commonly usedlight-curve fitters: MLCS2k2 (Jha et al. 2007) and SALT2(Guy et al. 2010), that diverged in their approach to color andintrinsic scatter. MLCS2k2 attempted to model color basedon dust with the possibility that each SN could have its ownextinction law, and assumed that a large amount of the in-trinsic scatter was in color. The SALT2 model, on the otherhand, was agnostic to any physical properties of the SN colorand its relation to the intrinsic scatter. Cosmological analyseshave since favored the SALT2 model due to its native spectral-model to account for k-corrections and updated calibration,and it has been used in most recent cosmology analyses in-cluding the Joint Light-Curve Analysis (JLA: B14), Pantheon(S18), the Dark Energy Survey 3 Year Sample (DES3YR:Brout et al. 2019, B19a), and the Foundation + Pan-STARRS1photometric analysis (Jones et al. 2019). However, despite thefact that MLCS2k2 has not been used in recent cosmologicalanalyses, papers such as Scolnic et al. (2014b, 2018); Mandelet al. (2017) have attempted to bridge the gap between SALT2and MLCS2k2 methods by modeling a connection betweenthe underlying population of color, dust, and reddening laws. a r X i v : . [ a s t r o - ph . C O ] A p r Brout and ScolnicStill, SN Ia analyses that attempt to model dust using acosmological sample have typically made the simplistic as-sumption that there is a single total-to-selective extinction pa-rameter, R V , that can be fixed at a single number. R V is de-fined as A V / ( A B − A V ), where A V is the extinction in the V( λ V ∼ A B is the extinction in the blue( λ V ∼ R V varies for different dust grainsizes and composition, and galaxies have different dust prop-erties, it is well known that different galaxies and differentregions within galaxies exhibit a wide range of R V values.In fact, while the Milky Way galaxy has an R V on average ∼ .
1, it has a distribution of at least σ R V = 0 . R V values with a range of R V ∼ − R V = 2 .
61 andstar-forming galaxies, which are lower-mass on average, havea mean R V = 3 . R V has also been measured through large SN sample statis-tics and detailed studies of individual SNe, though often withvarying sets of assumptions. Cikota et al. (2016) compiled13 various studies of SN Ia samples from the literature whichdetermined a range of R V values from ∼ ∼ .
5. Cikotaet al. (2016) itself determined R V from nearby SNe and for 21SNe Ia observed in Sab-Sbp galaxies and 34 SNe in Sbc-Scpthey find R V = 2 . ± .
58 and R V = 1 . ± .
38 respectively.While so many past analyses have recovered R V < E ( B − V ), and it waspostulated R V may decrease with E ( B − V ). However, Nobili& Goobar (2008) found from a sample of modestly reddened( E ( B − V ) < .
25 mag) SNe Ia, a small value of R V ∼ R V ’s ranging from ∼ . ∼ . R V ’s of ∼ . ∼ .
8. Importantly, Amanullah et al. (2015) stressed thatthe observed diversity in R V is not accounted for in analysesthat measure the cosmological expansion of the universe.Since the low R V values ( <
2) are not found in studies ofthe Milky Way, this has motivated various SN Ia studies toascribe the dust to circumstellar dust around the progenitor atthe time of the explosion (Wang 2005; Goobar 2008). How-ever, an alternative interpretation could be that the low R V values are caused by dust in the interstellar medium (Phillipset al. 2013). This understanding has been supported by (Bullaet al. 2018,a), which constrained the location of the dust thatcaused the reddening in the SN Ia spectra to be, for the major-ity of the SNe that they observed, on scales of the interstellarmedium, rather than circumstellar surroundings. This couldbe due to cloud-cloud collisions induced by the SN radiationpressure (Hoang 2017) which produce small dust grains (Gaoet al. 2015; Nozawa 2016).While accounting for dust remains a challenge for cur-rent and future photometric cosmology analyses, this pur-suit has often been done in parallel to the search for correla-tions between measured supernova luminosity after standard-ization and host-galaxy properties. Global and local proper-ties of SN Ia host galaxies such as stellar mass, star forma- tion rate (SFR), stellar population age, and metallicity haveall been shown to correlate with the distance modulus resid-uals after standardization (Hicken et al. 2009a; Sullivan et al.2010; Lampeitl et al. 2010; Childress et al. 2013; Rose et al.2019). This correlation is often parameterized as a step func-tion in host-galaxy stellar mass and is now commonplace inSN Ia cosmology analyses despite the lack of understandingof its physical underpinning or convincing evidence for ex-actly which host-galaxy property is most influential on SN Ialuminosity (e.g. Jones et al. 2018a; Scolnic et al. 2020). Toexplain this correlation, recent studies have suggested a poten-tial relation between the luminosity of the SN and the progen-itor, which can be related to the age of the galaxy, or the localenvironment of the galaxy (Childress et al. 2013; Rigault et al.2013; Roman et al. 2018). However, as the aforementionedgalaxy properties are all directly linked to dust properties, itis likely that the lack of dust modeling in SN Ia cosmology isrelated to the correlations between host galaxy properties andstandardized luminosities.In this analysis, we show that there are clear limitations inSN Ia standardization techniques with a single color luminos-ity correlation, but that these limitations can be addressed byinclusion of dust modeling with variation in R V . This paperrelies heavily on the work of Mandel et al. (2017), which fol-lows closer to the framework of MLCS2k2 and developed ahierarchical Bayesian model to build a more rich understand-ing of SN color. Mandel et al. (2017) only used low-redshiftdata, did not account for selection effects, and assumed a fixed R V extinction parameter; here we use a much larger datasetacross a wide redshift range and use survey simulations toforward-model what is done in Mandel et al. (2011), thoughwith additional features to explain discrepancies seen betweensimulations and data.In Section 2, we present the data compilation, light-curvefitting and discrepancies between the data and a simple un-derstanding of SN color. In Section 3, we discuss how to dif-ferentiate between past models of SN color and our new dust-based color model. In Section 4, we show how the new modelcan explain the commonly seen correlation between distancemodulus residuals and host-galaxy properties. In Section 5,we assess the impact on recovered cosmological parameters,and in Sections 6 & 7, we discuss further studies and conclu-sions. DATA SAMPLE, DISTANCE MODULI, ANDDESCRIPTION OF SN IA COLORS2.1.
Data
We use a compilation of publicly available, spectroscopi-cally classified, photometric light curves of SNe Ia that havebeen used in past cosmological analyses and that have beencalibrated to the SuperCal system (Scolnic et al. 2015). Thelow-redshift (low-z) SNe used here are made up of, in part,by those used in B19b which are from CSP (Stritzinger et al.2010) and CfA3-4 (Hicken et al. 2009b,a, 2012). At low-z, we also include the recently released 180 low-z SNe fromthe Foundation sample (Foley et al. 2018). At high-z, we in-clude SNe from PS1 (Rest et al. 2014; Scolnic et al. 2018),SDSS (Sako et al. 2011) and SNLS (B14) as was done inthe Pantheon analysis. Finally, we include data from the re-cently released DES 3-year sample (Brout et al. 2019), here-after DES3YR. The redshift distribution of SNe Ia used in thiswork can be found in the top panel of Figure 1.This analysis relies largely on the host galaxy mass esti-rout and Scolnic 2020 3 F IG . 1.— Top: Stacked redshift histograms of each of the samples analyzed.Second: Hubble Diagram residuals relative to flat Λ CDM cosmology with w = −
1. Third: Mean and 68% intervals for the measured SALT2 color fromthe data, shown in blue points. Predictions from survey simulations shown inpurple for simulations with the C11+SK16 model and orange for simulationswith the G10+SK16 scatter model. Bottom: Same as third panel, but for themeasured SALT2 stretch x . mates provided by past analyses. We adopt the same massesreleased in the Pantheon sample, and references therein, forSDSS, PS1, SNLS, CSPDR2, and CfA. For DES3YR masses,we use the updated masses provided by Smith et al. (2020);Wiseman et al. (2020). For the Foundation sample, we utilizemasses derived in Jones et al. (2018b).2.2. Light-curve fits and Distance Modulus Determination
We fit the SNe with the SALT2 model as presented in Guyet al. (2010) and updated in B14. In SALT2, the SN Ia flux atphase ( p ) and wavelength ( λ ) is given as F (SN , p , λ ) = x × [ M ( p , λ ) + x M ( p , λ ) + . . . ] × exp[ cCL ( λ )] , (1)where the parameter x describes the overall amplitude of thelight-curve, x describes the observed light-curve stretch, and c describes the observed color of each SN. M , M , CL areglobal model parameters of all SNe Ia: M represents the av-erage spectral sequence (SED); M is the SED variability; and CL is the average color correction law. The light-curve fits as-sume Fitzpatrick (1999) for Milky Way reddening. The meanobserved c and x for the data, binned over redshift, is shownin the bottom panels of Figure 1.Distances are inferred following the Tripp estimator (Tripp 1998). The distance modulus ( µ ) to each candidate SN Ia isobtained by: µ = m B + α SALT2 x − β SALT2 c − M (2)where m B is peak-brightness based off of the light-curve am-plitude (log ( x )) and where M is the absolute magnitude ofa SN Ia with x = c = 0. α SALT2 and β SALT2 are the correlationcoefficients that standardize the SNe Ia and are determinedfollowing Marriner et al. (2011), in a similar process to whatis done in B14. In recent analyses with the Tripp estimator(S18, B19b), there is often additional additive terms δ bias , thecorrection for distance biases calculated from survey simula-tions and δ γ , the correction due to the host-galaxy mass cor-relation; these additional corrections are not applied becausenew treatments for both of these terms are introduced in fol-lowing sections.Distance uncertainties are computed from the uncertaintiesin the light-curve fit parameters and their covariance ( C ): σ µ = C m B , m B + α C x , x + β C c , c + α SALT2 C m B , x − β SALT2 C m B , c − α SALT2 β SALT2 C x , c + σ + σ z + σ + σ , (3)where σ vpec is the distance modulus uncertainty due to pecu-liar velocities (250 km/s), σ z is the distance modulus uncer-tainty due to the measured redshift uncertainty, σ lens is the ad-ditional uncertainty from weak gravitational lensing (0 . z ),and σ int is determined such that the reduced χ relative to abest fit cosmology is 1.Typical selection cuts are applied on the observed datasample as was done in B19b: we require fitted color uncer-tainty < .
05, fitted stretch uncertainty <
1, fitted light-curvepeak date uncertainty <
2, light-curve fit probability (fromSNANA) > .
01, and Chauvenaut’s criterion is applied to dis-tance modulus residuals, relative to the best fit cosmologicalmodel, at 3.5 σ . In total, after selection cuts, there are 1445SNe in this sample.2.3. Key Pillars of the Complexity of the Colors of SNe Ia
The complexity of the SN Ia color model is readily appar-ent after a simple Tripp standardization. Here, three criticalfeatures are presented in the observed dataset that must be ex-plained by models of SN Ia color and intrinsic scatter. • The distribution of observed SN Ia colors is shown inthe top of Fig. 2. There is a clear asymmetry, with anexcess of red SNe in comparison with blue SNe, that isinconsistent with a symmetric Gaussian distribution. • The relation between the root-mean-square (RMS) scat-ter of distance modulus residuals (with mean residualremoved in each bin) as a function of SN Ia coloris shown in the middle panel of Fig. 2. There is a11 σ dependence relative to a flat line, where the redderSNe Ia ( c > .
1) exhibit nearly twice as much scatter( ∼ c < − . ∼ • The relation between Hubble residual binned distancebiases and SN Ia color is shown in the bottom panelof Fig. 2. There is a ∼ . σ dependence relative to astraight line. As shown in Fig 2, the recovered β SALT2 of the data is 3 . ± .
06. Brout and ScolnicThe relation of increased scatter as a function of color has notbeen analyzed in a previous analysis. This paper is motivatedby quantifying these observed features and building a modelthat can address all of them simultaneously.2.4.
Using Survey Simulations to Evaluate SN Ia Color andIntrinsic Scatter Models
For every model presented in this paper, 100 realizationsof dataset-sized simulations are run.
SNANA (Kessler et al.2009) is used to simulate realistic samples of SNe Ia. Thesesimulations account for observing cadence, observing condi-tions, noise properties, selection effects, cosmological effects,and astrophysical effects. A general description of the simu-lation methodology can be found in Kessler et al. (2019) andthe survey specific simulation details for SDSS and SNLS aredescribed in Kessler et al. (2013); PS1, CSP, and CfA are de-scribed in S18; DES3YR is described in B19b and Foundationis described in Jones et al. (2018b).We define three metrics based on the three panels of Fig. 2which are pseudo χ evaluations that assess agreement be-tween simulations that assume an SN Ia model and the data.The first metric is defined as χ for the agreement in color his-tograms of data (N data c ) and survey simulations (N sim c ) suchthat χ = (cid:88) j (N data c j − N sim c j ) / e n j , (4)and is determined in bins of color ( j ) where e n j is determinedby Poisson statistics.A second metric, the agreement in total Hubble diagramscatter (RMS) between data (RMS data ) and survey simulations(RMS sim ), is defined as χ over color bins such that χ = (cid:88) i (RMS data c i − RMS sim c i ) / e ci (5)and is determined in bins of color i and where e ci are the errorsdetermined from 100 realizations of the simulated dataset. Weuse RMS instead of intrinsic scatter as a metric because, forintrinsic scatter, the sensitivity of the different components ofthe error modeling is difficult to track.A third metric is the agreement in distance modulus resid-uals between data ( ∆ µ data ) and survey simulations ( ∆ µ sim )which can be expressed as χ ∆ µ over color bins such that χ ∆ µ = (cid:88) i ( ∆ µ data c i − ∆ µ sim c i ) / e µ i (6)and is determined in bins of color i and where e µ i are errorsderived from the data itself. EVALUATING MODELS OF TYPE IA SUPERNOVAECOLORS AND INTRINSIC SCATTER3.1.
Previous Models of Intrinsic Scatter and AssociatedIntrinsic Color Populations
Recent studies have focused on two models of intrinsic scat-ter, which to first order, can both be described by two pa-rameters: the magnitude of chromatic and achromatic scatter.The two models are the ‘G10’ scatter model (Guy et al. 2010)which prescribes 70% of the intrinsic scatter to coherent vari-ation and 30% to chromatic (wavelength dependent) variationand the ‘C11’ scatter model (Chotard 2011) which prescribesonly 25% of the intrinsic scatter to coherent variation but 75%to chromatic variation. Both of these models were trained on F IG . 2.— Top:
Observed color histogram from the full data sample, withsymmetric Gaussian overlaid.
Middle:
RMS of Hubble Diagram residualsas a function of color. The RMS is calculated after Tripp standardization andafter subtracting the mean Hubble residual bias.
Bottom:
Binned Hubblediagram residuals as a function of color, after Tripp standardization using thebest fit β SALT . In the bottom two panels, the significance of the deviationfrom a flat line is show in the bottom corner. data: for C11, it was trained on spectra from the SNFactory(Aldering et al. 2002) and for G10, it was trained during thecreation of the SALT2 model on a large subset of the lightcurves used in this analysis (Guy et al. 2010, B14).These scatter models cannot be used in survey simulationsto predict color distributions or the trends of Fig. 2 without anassociated color population and a β SALT as defined in Eq. 2.For both the G10 and C11 scatter model, Scolnic & Kessler(2016), hereafter SK16, determined the underlying color pop-ulation such that when it was combined with measurementnoise, the color scatter from the scatter model, and selectioneffects, the observed color distribution matched that seen forthe data in the top panel of Fig. 2. The underlying populationwas described by an asymmetric gaussian, with three free pa-rameters. The value of β SALT was determined by finding whatinput β SALT in the simulations would yield an output β SALT consistent with that found in the data from the methodologyoutlined in Section 2.2.The number of parameters that describe the framework forone of these scatter models is six: two parameters for thespectral and coherent scatter, three parameters for underlyingpopulation, and the value of β SALT . However, in order to ex-plain inconsistencies between the low-z targeted sample andthe high-z samples, SK16 determined the underlying popula-tion for each separately. Therefore, in total, a description ofthe full sample is described by 9 parameters.rout and Scolnic 2020 5For the simulations with G10 and C11, a single input β SALT2 value is used for each one: β SALT2 = 3 . β SALT2 = 3 . β SALT2 ∼ . β = 0) for the intrin-sic color distribution. The populations used for the samples inPantheon (Low-z, PS1, SNLS, SDSS) can be found in SK16,for Foundation in Jones et al. (2018b), and for DES3YR inB19b. 3.2. Evaluating Past SN Ia Scatter Models
As expected, because the SK16 populations were deter-mined so that simulations would reproduce the observed colordistribution of the data, simulations based on C11+SK16 andG10+SK16 show excellent agreement with the data (Figure4): χ c of 9.0 and 9.5 respectively (12 bins). The mean ob-served c and x for the simulations, binned over redshift, isshown in the bottom panels of Figure 1 and is in similarlygood agreement with the data. However, the agreement be-tween data and simulations for both the RMS (Fig. 5a) andmean Hubble residuals (Fig. 5b) is comparatively poor.For the RMS of Hubble residuals (Fig. 5a), it is clearthat neither G10+SK16 nor C11+SK16 produce the trend ob-served in the data. We do see non-linear behavior predictedfrom the simulations for the C11+SK16 model, which pre-scribes more scatter due to SN Ia chromatic variation andachieves a χ = 35, whereas G10+SK16, which prescribeslittle color variation, achieves a χ = 68. The relativelyflat dependence of the RMS on color as predicted from theG10+SK16 model shows that the trend in the data can not beexplained by lower signal-to-noise for SNe with redder col-ors.The agreement between data and simulations for meanHubble residuals (Fig. 5b) is somewhat better for G10+SK16( χ ∆ µ ∼
12) but worse for C11+SK16 ( χ ∆ µ ∼ Parameterization of a new dust-based color model
We discuss in Fig. 2 a simple and more physical under-standing of the trends seen: the redder colors can be explainedby dust extinction, the high RMS for red SNe Ia could be ex-plained by variations in the extinction parameter, and Hub-ble residual biases for the blue and red SNe can be explainedby different respective color-luminosity relations. Here, wefollow Mandel et al. (2011) and Mandel et al. (2017), whichbuild on the work of Jha et al. (2007) to create a model of SNcolor based on two components: 1) an intrinsic color com-ponent ( c int ) related to luminosity by a correlation coefficient β SN and 2) a dust-component ( E dust ) described by an expo- F IG . 3.— Explaining the BS20 model. Shown are input parameterizationsto simulations (dashed teal), the simulated values after measurement noiseand selection effects (solid teal), and the dataset (black points) for the samethree quantities (y-axes) as in Fig. 2. Left : A simulation based solely onan intrinsic color distribution, described by a symmetric Gaussian, withoutdust.
Middle : A simulation based solely on a delta function in intrinsic colorand an exponential dust distribution.
Right : A simulation with both intrinsiccolor Gaussian and dust distribution combined. nential distribution of reddening values related to luminosityby the extinction ratio R V . The observed color c obs can beexpressed as c obs = c int + E dust + (cid:15) noise . (7)where (cid:15) noise is measurement noise. We expand on the modelfrom Mandel et al. (2011) by allowing R V to be described by aGaussian distribution to reflect that a range of values are seenin the literature, rather than a single value. In total, the modelhas seven fundamental parameters: • ¯ c : the mean of the intrinsic color distribution describedby a symmetric Gaussian. • σ c : the 1-sigma width of the intrinsic color distributiondescribed by a symmetric Gaussian. • ¯ β SN : the correlation between intrinsic color and lumi-nosity. • σ β SN : the 1-sigma width of the Gaussian distributionfrom which the correlation between intrinsic color andluminosity is drawn for each SN. • ¯ R V : the center of the Gaussian distribution from which R V values are drawn for each SN. • σ R V : the 1-sigma width of the parent Gaussian R V dis-tribution. • τ E : the parameter describing the exponential distribu-tion from which E dust reddening values are drawn.To set a ‘reddening-free’ color, it is assumed that the intrin-sic colors of SNe Ia can be determined by: P ( c int ) = 1 √ πσ c e − ( c int − ¯ c ) / σ c . (8) Brout and Scolnic F IG . 4.— A histogram of the observed color values from data (points) andsimulations (lines). As all models are fitted so that simulations match the datafor this metric, good agreement between data and simulation is expected forall the models. TABLE 1P ARAMETERS USED FOR
BS20
MODEL . Model a Sample ¯ c σ c ¯ β SN − σ β SN ¯ R V σ R V τ E No-Mass-split:Full Targeted -0.078 0.044 0.8 0.3 2.0 1.4 0.17Full Rolling -0.078 0.044 0.8 0.3 2.0 1.4 0.10Mass-split:High-mass b Targeted -0.078 0.044 0.8 0.3 1.5 0.8 0.18High-mass Rolling -0.078 0.044 0.8 0.3 1.5 0.8 0.11Low-mass c Targeted -0.078 0.044 0.8 0.3 2.5 2.2 0.16Low-mass Rolling -0.078 0.044 0.8 0.3 2.5 2.2 0.09 a Uncertainties on model parameters are not determined. b High mass: Host log( M ∗ / M sun ) > c Low mass: Host log( M ∗ / M sun ) < The reddening for each SN is described by E dust from Eq. 7and is related to the extinction of the SN by the standard equa-tion A V = R V ∗ E dust (9)where E dust corresponds to E ( B − V ).The reddening values E dust are drawn from an exponentialdistribution following Mandel et al. (2017) with probabilitydensity P ( E dust ) = (cid:26) τ − E e − E dust / τ E , E dust > , E dust ≤ τ E is a parameter in the model described above.In addition, we draw from distribution of possible valuesfor R V : P ( R V ) = 1 √ πσ R V e − ( R V − ¯ R V ) / σ RV (11)where ¯ R V is the center of the Gaussian distribution of R V , σ R V is the width, and where individual R V values below 0.5 are notallowed.Finally, similar to Eq. 11, values for β SN are drawn for eachSN using model parameters ¯ β SN and σ β SN such that: P ( β SN ) = 1 √ πσ β SN e − ( β SN − ¯ β SN ) / σ β SN . (12)In total, the change in observed peak brightness of a SN dueto color can be expressed as ∆ m B ∆ m B = β SN c int + ( R V + E dust + (cid:15) noise (13) where each observed parameter is unique to each SN. Thecoefficient R V + R V as in Eq. 9, becauseto measure the change in m B , the extinction parameter R B = R V + τ E for each survey design. Intotal, this makes 8 parameters. In contrast, as discussed previ-ously, the G10+SK16 or C11+SK16 models require 9 param-eters when ‘rolling’ and ‘targeted’ samples are accounted forseparately. Thus, the dust-based framework described herehas fewer free parameters than those used in past cosmologi-cal analyses.Grouping of rolling versus targeted subsamples is chosenbecause rolling surveys (DES, PS1, SNLS, SDSS and Foun-dation) have no preferential host-galaxy selection and Tar-geted surveys (CfA samples and CSP) preferentially targetedSNe in brighter galaxies. The split between rolling versustargeted surveys is similar to splitting between high versuslow-redshift, except for the Foundation Survey. As discussedin Foley et al. (2018) and Jones et al. (2019), Foundationfollows-up objects discovered by rolling surveys, and thesample properties look in-between a high-z rolling survey anda low-z targeted survey. As the Foundation sample is not largeenough to discriminate between the designations according toour metrics, we leave it as a rolling survey.3.4. Results for the New Color Model
The parameters described in Section 3.3 can be fit from thephotometric data itself using the three metrics (Eqs. 4, 5, &6) and the requirement that after running the fitting methodfrom Marriner et al. (2011) on simulations, we recover towithin 1 σ , a value consistent with the β data SALT observed fromthe data. To determine the model parameters properly withforward modeling would likely require both an ApproximateBayesian Computing technique (ABC, e.g. Jennings et al.2016) in combination with very large simulations of SNe Iawith flat parameter distributions to be later re-weighted viaimportance sampling. Since we do not have this computation-ally expensive infrastructure set up, we present results for aplausible set of model parameters (Table 1) achieved throughcoarse minimization with human supervision to find the low-est χ across the three metrics. Uncertainties on parametersare not computed, but we show the power of the constraintsfrom the three metrics to break degeneracies between modelparameters in Appendix Fig. 9.The parameters are shown in Table 1. We find a meanreddening-free color of ¯ c = − .
078 with an intrinsic color dis-tribution of σ c = 0 .
044 and a mean intrinsic color-luminositycorrelation coefficient of ¯ β SN =1.8 with small variation of σ β SN = 0 .
3. The recovered β SN is smaller than the tradi-tional β SALT2 ∼ R V distribution for the dust com-ponent is described by ¯ R V = 2 . σ R V = 1 .
4. The value of σ R V = 1 . R V , though with a set-floorof R V = 0 .
5. Because a single color-luminosity relation is ap-plied in the fit, but there is a range of R V values, the measured R V variation dominates the scatter of distance modulus resid-uals, contributing 0.093 to σ int , the majority of the total σ int (0.105). On the other hand, the measured variation in the in-trinsic color-luminosity relation ( σ β SN = 0 .
3) contributes 0.039rout and Scolnic 2020 7 (a) (b)F IG . 5.— a) The zero-mean RMS of the Hubble residuals relative to Λ CDM versus the observed color c of the SNe Ia. The data is shown in black points,and the predictions from simulations of G10+SK16 and C11+SK16 models are shown in orange and purple dotted lines respectively. The model created for thiswork, labeled BS20, is shown in green. Inset: sames as main figure but for intrinsic scatter term σ int instead of RMS. b) Binned Hubble Diagram residuals versuscolor. Biases are seen in the observed data (black points) and predicted by the scatter models (solid/dotted lines). For the BS20 model used here, there is no spliton host-mass. to the total σ int .The results of simulations with our model are presented inFig. 4 and Fig. 5. We find a β SALT2 = 3 . ± .
01 when an-alyzed identically to the observed dataset, which is consis-tent with that of the observed dataset ( β SALT2 = 3 . ± . χ c ∼ χ ∼ R V . This is further demonstratedby the inset of Fig. 5a, which derives the magnitude of theintrinsic scatter ( σ int ) after removing additional uncertaintyfrom the SALT2 error-snake, and shows that the distributionof dust and extinction laws can account for the majority ofthe observed intrinsic scatter. Finally, as shown in Fig. 5b,the BS20 model results in excellent agreement with observedHubble residual biases ( χ ∆ µ ∼ χ values between the different color scat-ter models for the three metrics in Table 2, the advancementof the BS20 model is clear, and with one less parameter, theimprovement cannot be simply attributed to additional modelcomplexity. THE DEPENDENCE OF THE HOST-MASSCORRELATION WITH SNE IA LUMINOSITY ONCOLOR4.1.
Observed trends of color metrics based on Host-GalaxyStellar Mass
Many studies have found correlations between the the Hub-ble residuals and various host-galaxy properties (Hicken et al.2009a; Lampeitl et al. 2010; Sullivan et al. 2010; Childresset al. 2013; Rigault et al. 2013; Roman et al. 2018; Roseet al. 2019). Here, we focus on the host-galaxy stellar massas it is the most commonly used, most accessible, and often
TABLE 2 χ FOR EACH EVALUATED MODEL . Scatter Model Color Model χ c a χ χ ∆ µ c d G10 SK16 9.0 68.1 12.3 9C11 SK16 9.5 34.7 28.6 9BS20 No-Mass-split e BS20 8.2 8.4 9.8 8 a
12 color bins. b c
10 color bins. d Note: e Note: Agreement as a function of mass, as shown in Fig. 6b, is not includedin χ ∆ µ and thus the BS20 Mass-split model is not listed in this table. yields some of the strongest correlations with Hubble resid-uals. In the top panel of Fig. 6a, the RMS versus SALT2color plot as shown in Fig. 5a is remade, but for the high andlow host-mass subsamples separately. For the ‘dust-free’ blueSNe ( c ∼ − . ∼ .
15 mag dif-ference in Hubble residuals. Overall, the subsamples are dis-crepant at greater than 5 σ ( χ / N bin = 57 /
10) relative to eachother.Pursuing this further, we follow recent works like B19b anddefine γ as the mean difference in Hubble residuals given a Brout and Scolnic (a) (b)F IG . 6.— a) (Upper) Hubble Diagram scatter binned by SALT2 observed color and compared for SNe in host galaxies with low and high mass. (Lower)Recovered values of γ for SNe Ia in high (log( M ∗ / M sun ) >
10) versus low (log( M ∗ / M sun ) <
10) mass hosts, binned by SALT2 observed color. Predictions fromthe BS20 Mass-split model is shown in green. Significance of the deviation from a constant γ of 0.06 is shown (4.5 σ ). b) Binned Hubble Diagram residualsversus color split on host-mass. Biases are shown for the observed data (points) and predicted using the scatter models (solid lines). The difference between thered and blue points has typically been found by marginalizing over color and finding a single step γ . The BS20 model parameters for the rolling surveys aregiven in the legend. split in host galaxy properties: δγ = γ × [1 + e ( M host − M step ) / . ] − − γ , (14)where M = log(M ∗ / M sun ) and a log host-mass step location( M step ) of 10 is assumed. We determine γ for the samplein discrete color bins. This is shown in the bottom panel ofFig. 6a. As expected from the observations in Fig 6b, for‘dust-free’ SNe Ia that are bluer than the intrinsic color ¯ c , γ = 0 . ± . γ = 0 . ± .
011 as well as a 4 . σ increasing trend where ∆ γ ∼ . ± . × c , showing thatthe typical γ values around 0.06 mag recovered in previousanalyses are driven by the red SNe in the sample.While many studies have shown that host-mass and SNcolor are weakly correlated if at all (e.g. Sullivan et al. 2010),the dependence of γ itself on color has not been studied. Asour model shows that redder colors can be described by dust,the difference between observed correlations between Hubbleresiduals and mass for different colors are all indicative of adust-based explanation. We note that the trend seen in thebottom of Fig. 6a is largely insensitive to whether distancebias corrections are applied. If we apply corrections based onKessler & Scolnic (2017), the γ recovered is 0 . − .
02 maglower per bin than that shown, which is discussed at length inSmith et al. (2020). The trend with RMS is not affected bythese corrections because the RMS measured per bin is calcu-lated after a mean offset is removed, thereby effectively doinga similar correction as Kessler & Scolnic (2017).4.2.
Dust Modeling Explains Mass Step
We repeat the process as described in Section 3 for deter-mining the underlying dust-based color model, except nowfor the low and high-mass host-galaxy subsamples separately.The fitted parameters are given in the ‘Mass-split’ groupingof Table 1. Parameters that are intrinsic to the SNe Ia are fixed for both host-galaxy subsamples while the dust distribu-tions are allowed to vary for each subsample. We find that forSNe in low-mass hosts, ¯ R V = 2 . σ R V = 2 .
2, whereas forSNe in high-mass hosts, ¯ R V = 1 . σ R V = 0 .
8. After ac-counting for selection effects, the distribution shifts such thatthe average observed R V for the detected SNe in the sampleis 2.94 and 1.55 for low-mass hosts and high-mass hosts re-spectively. In these simulations, 6% of all the detected SNehave simulated R V values greater than 5. The dust distributionfor SNe in high-mass hosts that are discovered in rolling sur-veys can be described with τ E = 0 .
11 whereas for low-masshosts it is τ E = 0 .
09 and similarly for targeted surveys the SNecan be described with τ E = 0 .
18 whereas for low-mass hostsit is τ E = 0 .
16. Therefore, we find that the amount of dust isslightly higher for high-mass hosts in comparison to low-masshosts, but that R V is significantly lower for high-mass hosts.We show in Fig. 6b that simulations with these separate dustmodels do indeed each recover the trends in Hubble residu-als, and consequentially the trend seen in the bottom panelof Fig. 6a. Therefore, we conclude that modeling differentdust properties for different galaxy populations can fully ex-plain the net γ ∼ .
06 mag offset seen in past analyses as wellas the γ dependence on observed SN Ia color. Furthermore,the wider R V distribution for low-mass hosts explains why inthe top panel of Fig. 6, the amount of scatter is significantlyhigher for low-mass hosts versus high-mass hosts.As shown from the data, applying a single offset ( γ ) as hasbeen done in past analyses, is incorrect. Furthermore, it hasbeen unclear in past analyses why there should be any ‘step’behavior (Sullivan et al. 2010). Here, it is shown that the paststep is an artifact of improper fitting, and arises because ofsignificantly different R V distributions for different types ofgalaxies.rout and Scolnic 2020 9 F IG . 7.— (Top) Hubble diagram residuals of the compiledDES3YR+Foundation+PS1+SNLS+SDSS+CSP+CfA dataset as a functionof redshift. The dataset is bias corrected with the three different modelsof SN Ia color. Error bars for G10+SK16 and ‘C11+SK16’ are not shownand are indistinguishable from those of BS20. (Middle) The impact of biascorrections on real data relative to distances computed using G10+SK16.(Bottom) The impact of bias corrections using simulated data relative todistances computed using G10+SK16. IMPACT ON RECOVERY OF COSMOLOGICALPARAMETERSTo understand the impact of these different models of SN Iacolor on the recovery of cosmological parameters, both dataand simulations are used. Before measuring cosmological pa-rameters, we apply bias corrections following the methodol-ogy of Marriner et al. (2011) and B14 using large simulationswith the three color models (G10+SK16, C11+SK16, BS20)to measure the dependence of distance biases with redshift,which are then applied as corrections to the dataset or a sim-ulated dataset. Bias corrections following Kessler & Scolnic(2017) are not used because they have so far been only beendesigned to work given a β SALT2 and a variation in c , but not R V , nor variation thereof. Therefore, we apply bias correc-tions that assume a single β SALT2 and follow the same for-malism that was used in the JLA analysis and we do not splitby host-mass. This is so a self-consistent comparison can bemade against the impact of the G10+SK16 and C11+SK16scatter models.The impact of the bias corrections on the data is shown inFig. 7. The most noticeable differences between the correc-tions of BS20 versus the other scatter models are at z < . z > .
8, where selection has the greatest influence. Here,the differences in recovered distance modulus can change byup to ∼ .
05 mag at low or high-z depending on which colormodel is used. This difference is larger than any other sys-tematic in past cosmology analyses (e.g., B19b). As shown in Fig. 7, we see the same effect with simulationsas we do for data when simulating a sample of 10,000 SNewith realistic proportions and distributions of SNe Ia fromeach survey. Here, the simulations of ‘datasets’ are based onthe BS20 model, but bias corrections are determined from theother models.To determine cosmological parameters, we use CosmoMC(Lewis & Bridle 2002) and combine with CMB (Planck Col-laboration et al. 2018) constraints. In Table 5, the biases incosmological parameters are given when simulated SNe Iadatasets use different models of SN Ia color than the modelused for the determination of distance bias correction. We findthat if the ‘true’ model of SN Ia color is the dust-based modelpresented in Section 3.3, but the bias corrections are based onthe G10+SK16 or C11+SK16 models, the propagated bias in w will be -0.025 and -0.040 respectively. Again, this bias islarger than any other systematic uncertainty reported in recentcosmological analyses.In Table 5 we also show the differences in w for the real datawhen we apply bias corrections based on simulations usingthe three separate models of color: G10+SK16, C11+SK16and BS20. Relative to BS20 bias corrections, there arechanges in recovered w for G10+SK16 and C11+SK16 of -0.033 and -0.041 respectively, which is consistent with sim-ulations. Interestingly, as shown in Fig. 5a, while C11 andBS20 better match the trend in the data, they produce thelargest differences in w of ∼ DISCUSSION6.1.
The Dependence Between R V and Host GalaxyProperties That the mass correlation can be explained by separate dustproperties is now the only direct explanation for the cor-relation between host-mass and distance modulus residuals.This possibility was briefly discussed in Mandel et al. (2017),which showed if one changed the dust distribution ( τ E ) forthe SNe in low and high-mass subsamples, one could remove1 / γ , but not the whole effect. We fol-low this idea from Mandel et al. (2017), but add that the R V distribution as well should be different for these subsamples.This can then explain the full γ as well as its color depen-dence. Our dust explanation aligns well with the observationsin Burns et al. (2018) that at low-z, the host-mass correlationwith SN Ia luminosity is larger in the optical than in the NIR,where the correlation is consistent with 0. This should be thecase if the correlation is tied to reddening, as the correspond-ing extinction ratio of R V in the NIR is smaller. Furthermore,the range of R V values is in good agreement with studies of R V from individual SNe like in Amanullah et al. (2015). Whilethe model shows that a fraction of SNe should have R V above5, we find that this is only 6% after accounting for selectioneffects.This analysis makes a strong prediction that SNe in lower-mass galaxies have on average, higher R V values with a widerdistribution than SNe in higher-mass galaxies. As there arevery few measurements of R V in the interstellar medium ofgalaxies beyond the Milky Way, LMC and SMC, it is diffi-cult to find evidence that this trend would hold for galaxiesthemselves. Salim et al. (2018), which measured the dust at-tenuation curves of 230,000 individual galaxies in the localuniverse, found that quiescent galaxies, which are typicallyhigh-mass, have a mean R V = 2 .
61 and star-forming galax-ies, which are lower-mass on average, have a mean R V = 3 . TABLE 3R
ESULTS FROM L ARGE Λ CDM S
IMULATIONS AND E I A SN Ia Color Model Host Dust Model SN Ia Color Model Host Dust Model SN Ia + Dust w CDM + Planck ’16Data a Data 1D BiasCor b
1D BiasCor β SALT ∆ w c BS20 BS20 C11 + SK16 Parent No Host Dust 3 . ± .
01 -0.040BS20 BS20 G10 + SK16 Parent No Host Dust 3 . ± .
01 -0.025BS20 BS20 BS20 BS20 3 . ± .
01 0.000Real Data Real Data C11 + SK16 Parent No Host Dust 3 . ± .
06 -0.041Real Data Real Data G10 + SK16 Parent No Host Dust 3 . ± .
06 -0.033Real Data Real Data BS20 BS20 3 . ± .
06 0.000 a Datasets are based on large simulations of ∼ b Bias Correction samples are large simulations of > c ∆ w = w fit − w BS20 : this is relative to the last row (BS20) of each dataset grouping.
This trend is in general agreement with our prediction.The observation that global properties of the galaxy can im-pact the dust measured from the SNe is supported by Phillipset al. (2013) and Bulla et al. (2018b), which found the dust re-sponsible for the observed reddening of SNe Ia appears to bepredominantly located in the interstellar medium of the hostgalaxies and not in the circumstellar medium associated withthe progenitor system. It’s also supported by Childress et al.(2013) which showed that color of SNe Ia is strongly tied tothe metallicity of the host galaxy. For a future analysis, it isencouraged to repeat this same exercise but instead of usingstellar mass to use metallicity, specific star formation rate, orlocal color; improved estimates of the dust distribution param-eters would likely be obtained. For example, as shown in Sul-livan et al. (2010), when measuring a single color-luminositycoefficient β SALT2 for different samples, there is an even big-ger difference when splitting the sample for specific star for-mation rate than there is for mass. As our model constrainsboth the amount of dust and the properties of dust itself, it islikely that different galaxy properties (e.g., distance to hostand inclination, Holwerda et al. 2015; Galbany et al. 2012)will yield complementary insight about both of these compo-nents. We stress that our analysis does not limit the use ofhost-galaxy information in cosmological studies with SNe Ia,but rather, proposes a new path forward.Indirect explanations of γ have suggested that SNe fromdifferent progenitor systems have different luminosities, andthe progenitor system can be potentially linked to the age ofthe host galaxy (Childress et al. 2013). However, any modelthat assumes that the luminosity depends on progenitors doesnot predict the key observation in our analysis that the mag-nitude of γ depends on color. A progenitor-based explanationhas motivated studies by Rigault et al. (2013), Childress et al.(2014), Jones et al. (2015), Jones et al. (2018a), and Romanet al. (2018), which focus on the local specific star formationrate, local mass, and local color. Some of these studies seemto indicate that measuring the local color produces the high-est correlation with measured SN luminosity. In light of ourdust-based SN Ia color model, a simple explanation is that thelocal host color yields insight about the amount of dust and/ordust properties at the position of the SN. Our model does notdifferentiate whether the dust is in the circumstellar surround-ing which is still linked to the progenitor or in the interstellarmedium which is not linked to the progenitor, but we can ruleout a luminosity dependence on the progenitor system.Relatedly, many studies have found correlations betweenspectral features and Hubble residuals (Fakhouri et al. 2015;Siebert et al. 2020). Interestingly, Wang et al. (2009) split F IG . 8.— Simulations with the BS20 dust-based model predict a correlationbetween γ and observed intrinsic scatter. This correlation was originally seenfor real data in B19b. a sample of 158 SNe Ia based on whether their spectra in-dicate ‘normal velocity’ or ‘high velocity’ features, and find R V = 2 . ± .
07 and 1 . ± .
07 for the two subsamples re-spectively. Pan et al. (2015) show that the velocity of spec-tral features correlates with the mass of the host galaxies,such that high-mass host galaxies regularly have high-velocitySNe, so one would expect low R V to be found for high-masshosts. This is in great agreement with the results of our study,though we note that Foley & Kasen (2011) show that different R V from Wang et al. (2009) depend on using SNe with veryred colors E ( B − V ) > .
4. As velocity features has typicallybeen thought of indicative of properties of the progenitor andcircumstellar surrounding, it is unclear at what level things arecausally connected versus correlated.6.2.
Application of BS20 Model In Future Analyses
While we have shown that biases in w from our model rel-ative to previous models would have been the largest system-atic uncertainty of previous analyses, there is a clear path toutilize this model for future analyses. In order to do so op-timally, there are three necessary improvements. First, a fullABC fit to solve for the intrinsic and extrinsic parameters, bro-ken by survey, redshift range, or targeted versus un-targetedis needed. This could be facilitated by the recent advance-ments by Pippin (Hinton & Brout 2020). Future work canfully constrain and characterize systematic uncertainties onall 9 dust and hyper-parameters using a combination of therout and Scolnic 2020 11 χ metrics. Second, this model should be integrated into theSALT2 training, which currently only accounts for one com-ponent of the observed SN Ia color. Evidence of the benefit ofretraining is shown in the Appendix Fig. 11. It will be neces-sary in the future to attempt to train SALT2 based on intrinsicand extrinsic color components.Third, as discussed in Section 5, BBC5D from Kessler &Scolnic (2017) is not capable of bias correcting two effectivecolor-correlation coefficients ( β SN & R V ). Additionally futureanalysis should not be correcting for an observed γ . Rather,dust distributions should be fit to different subsamples of hostgalaxies and using this information, distance bias correctionscan be computed as a function of observables ( c , x , z and hostgalaxy properties). A future approach to such bias corrections(Popovic et al. in prep.) would be similar to that of Kessler& Scolnic (2017). If done properly, we predict that there willbe no residual γ in the distance modulus residuals. Doing sowill also improve the comparison of cosmological constraintsin Section 5, where we had to assume naive mass-independentbias corrections. Ultimately one should compare the impactof the bias corrections from the two-mass model to the biascorrections from the G10 and C11 model when a luminosity-correction due to host-mass is applied.The difference in RMS for ‘dust-free’ blue colors(RMS ∼ ∼ .
18) is striking.The statistical weight of these different SNe Ia when con-straining dark energy with our improved color model showsthat that a blue SN Ia is ∼ × more constraining than a redSN. The blue SNe Ia exhibit an RMS at the same level asNIR SN Ia standardized luminosities (Mandel et al. 2011).With tighter color measurement cuts and state of the art sam-ples (SNLS, DES), we have seen that the ‘dust-free’ RMScan even be as low as 0.08. In addition, as shown in Kessler& Scolnic (2017), bias corrections are much smaller for blueSNe Ia than red SNe Ia. Additionally, because γ is found tobe consistent with 0 for the blue SNe, it appears that there arenumerous advantages to using a sample of solely blue SNe.As LSST (Ivezi´c et al. 2019) and WFIRST (Spergel et al.2015; Hounsell et al. 2018) will discover thousands of SNe Iain this un-extincted regime ( c ∼ − . γ from that sample. They also re-marked that the σ int value of the low-z sample was more than3 σ discrepant from that of the DES3YR sample. However,we show in Fig. 8 that this behaviour arises naturally fromthe BS20 model: both σ int values are indeed consistent witha dust-based interpretation and that the relation between re-covered σ int and γ is a direct prediction of the BS20 model.This is because the different distributions of observed colorsfor each sample imply different amounts of dust, differentamounts of intrinsic scatter, as well as different magnitudesof γ .With our new model, we showed that that the bias in recov-ered w due to assuming the incorrect scatter model is ∼ σ w = 0 . w and for analyses of H as well. Dhawan et al. (2020) estimatesbiases due to scatter models to be on the level of 0 . − .
0% in H . As the H measurement has different systematic sensitiv-ity than w due to the comparison of SNe in calibrator galaxiesversus Hubble flow galaxies, we recommend these two sam-ples to have similar demographics of blue and red SNe. Afull systematics treatment, as done in Dhawan et al. (2020),should be done using the new dust-based SN Ia color modeldescribed in this paper. Furthermore, we note that past dis-cussions (e.g., Rigault et al. 2013; Jones et al. 2018a) aboutpotential biases in H should be reconsidered in light of thispaper’s findings. CONCLUSIONIn this paper, we introduced a new, physical, two-component color model of SNe Ia with an intrinsic compo-nent modeled as a simple symmetric Gaussian that correlateswith SN Ia luminosity and an extrinsic component that can bemodeled by a dust distribution that is tied to extinction by awide R V distribution. This model has fewer free parametersthan previous models of SN Ia color and a more physical mo-tivation that better matches the data. Our findings suggest thatthe dominant component of observed SN Ia intrinsic scatter isfrom R V variation of the dust around the SN. We also showthat there is a 4 . σ dependence on color of the correlation ofhost-mass with distance modulus residuals. Strikingly, thisshows that previously observed host-galaxy property corre-lations with SN Ia luminosity are driven by the redder SNeof the sample. This also suggests a dust-based explanationfor the host-galaxy property correlations. By allowing ourmodel to have different parameters for the dust distributionsof SNe in high-mass versus low-mass host-galaxies, we showthat the correlation between distance modulus residuals andhost-galaxy stellar mass can be attributed to correlations be-tween host-galaxy properties and R V .By finding that the previously seen host-galaxy correlationwith SN Ia luminosity after standardization is actually due todifferences of dust, and not due to possible variation in the lu-minosity based on progenitor systems, we find that that thereis a tremendous amount of leverage to continue to improvecosmological analyses by studies of larger samples, measure-ments covering larger wavelength ranges, more host galaxyproperties examined and improved dust models. Our studyshows that so many disparate analyses of SNe Ia are actuallyintricately connected, and unifying these studies will providetremendous improvements to measurements of the expansionof the universe. ACKNOWLEDGEMENTSWe thank Rick Kessler, Adam Riess, Saurabh Jha,The Goobar Research Group, David Jones, Mat Smith,Doug Finkbeiner, Eddie Schlafly, Charlie Conroy, AntonellaPalmese, and Sam Hinton for very useful discussions. We areappreciative of Rick Kessler for his ever-useful
SNANA pack-age. DB acknowledges support for this work was providedby NASA through the NASA Hubble Fellowship grant HST-HF2-51430.001 awarded by the Space Telescope ScienceInstitute, which is operated by Association of Universitiesfor Research in Astronomy, Inc., for NASA, under contract2 Brout and ScolnicNAS5-26555. DS is supported by DOE grant DE-SC0010007and the David and Lucile Packard Foundation. DS is sup-ported in part by NASA under Contract No. NNG17PX03C issued through the WFIRST Science Investigation Teams Pro-gramme.APPENDIX
A1. Model-data Agreement and Parameter Sensitivity
Here we review variants on parameters in the various color models to show what impact it has on the three metrics. We listthose variants here: • ‘BS20’ - the main model proposed in this work. • ‘No Dust’ - a model with a narrow intrinsic color distribution and a weak ( ¯ β SN = 2) correlation between color and luminosity. • ‘Only Dust’ - a model with only a dust distribution and a delta function for the intrinsic color distribution. • ‘G10+SK16’ - Described in Section 3.1 • ‘C11+SK16’- Described in Section 3.1 • ‘C11+SK16 + ¯ β SN variation’ - a model similar to the ‘C11+SK16’ one, except we allow the β SN to vary to reproduce theRMS for redder colors. • ‘BS20, σ β SN = 0’ - the nominal BS20 model, except β SN values are drawn from a delta function with value ¯ β SN . • ‘BS20, R V + .
5’ - the nominal BS20 model, except we shift our R V distribution by the full sample by 0 . • ‘BS20, τ E − .
5’ - the nominal BS20 model, except we reduce τ E to describe the dust distribution by 0 . • ‘BS20, ¯ β SN + .
5’ - the nominal BS20 model, except we increase ¯ β SN by 0.5. • ‘BS20, ¯ β SN = 0’ - the nominal BS20 model, except we set β SN to be 0. This effectively describes the intrinsic colordistribution as color scatter, similar to what is in the C11+SK16 model. • ‘BS20, No R V variation’ - the nominal BS20 model, except the variation in R V is removed.We show the results from using these different variants in Fig. 9. We include on the bottom panel the recovered β SALT for eachcase because as some variants may have a good χ in the three metrics, the recovered β SALT is far from that the data ( ∼ . A2. Observed Correlations with SALT2 x SN Ia cosmology analyses that measure correlations between SN light curve parameters and host galaxy mass regularly finda correlation between host-galaxy stellar mass and x (e.g., Sullivan et al. 2010; Scolnic et al. 2014a). This correlation is shownin Fig. 10a for our compiled dataset and from this, we expect similar trends with x that we observed with host stellar mass inSection 4. While there is no dependence of the RMS of distance modulus residuals on x (Fig. 10b) seen in the data or predictedfrom simulations, we do see similar trends with color when splitting on x (Fig. 10c) as we do when splitting on M host . We alsocompute a Hubble residual step when splitting on x : δκ = κ × [1 + e ( x − x ) / . ] − − κ , (1)at SN Ia stretch step location ( x step ) of -0.5 is assumed, albeit x step values between -0.5 and +0.5 provide good discriminationbetween sub-samples according to our three metrics. We determine κ for the sample in discrete color bins (Fig. 10d). Whenderiving δκ for the full sample with a single x step split, we find δκ = 0 . ± .
011 mag, roughly half the size of the step whensplitting by host stellar mass. As shown in Fig. 10d, similarly to host mass, the magnitude of κ depends on color; there is a 3 . σ deviation relative to a single step.When examining Hubble diagram residual biases in bins of color (Fig. 10e), simulations using the dust and R V distributions thatwere fit in Sec. 4.2 roughly predict the residuals when splitting on x . This indicates that x and M host yield similar informationabout the dust properties. However, upon studying the mean Hubble residual bias with color, as shown in Fig. 10f, we find theinformation from x and M host are complementary in potentially constraining R V as the difference in Hubble residuals from thesubsample of low x values and large host mass values (purple) in comparison to those from high x values and small mass values(orange) is larger than simply splitting on host mass (data points) as was done in Section 4. This finding is consistent with studieslike Rose et al. (2019), which argue that combinations of various host-galaxy properties and light-curve parameters could furtherimprove the standardizability of SNe Ia brightnesses, as well as with Galbany et al. (2012) who find that x is a good discriminatorof galaxy morphology.rout and Scolnic 2020 13 A3. The SALT2 Color Law