# Training Custom Light Curve Models of SN Ia Sub-Populations Selected
According to Host Galaxy Properties

G. Taylor,1, C. Lidman,1 B. Popovic,2 H. J. Abbot1
1Research School of Astronomy and Astrophysics, Australian National
University, Canberra, Australia
2Univ Lyon, Univ Claude Bernard Lyon 1, CNRS, IP2I Lyon/IN2P3, IMR 5822, 69622
Villeurbanne, France E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Type Ia supernova (SN Ia) cosmology analyses include a luminosity step
function in their distance standardization process to account for an observed
yet unexplained difference in the post-standardization luminosities of SNe Ia
originating from different host galaxy populations (e.g., high-mass ($M\gtrsim
10^{10}M_{\odot}$) versus low-mass galaxies). We present a novel method for
including host-mass correlations in the SALT3 light curve model used for
standardising SN Ia distances. We split the SALT3 training sample according to
host-mass, training independent models for the low- and high-host-mass
samples. Our models indicate that there are different average Si II spectral
feature strengths between the two populations, and that the average SED of SNe
from low-mass galaxies is bluer than the high-mass counterpart. We then use
our trained models to perform a SN cosmology analysis on the 3-year
spectroscopically confirmed Dark Energy Survey SN sample, treating SNe from
low- and high-mass host galaxies as separate populations throughout. We find
that our mass-split models reduce the Hubble residual scatter in the sample,
albeit at a low statistical significance. We do find a reduction in the mass-
correlated luminosity step but conclude that this arises from the model-
dependent re-definition of the fiducial SN absolute magnitude rather than the
models themselves. Our results stress the importance of adopting a standard
definition of the SN parameters ($x_{0},x_{1},c$) in order to extract the most
value out of the light curve modelling tools that are currently available and
to correctly interpret results that are fit with different models.

###### keywords:

supernovae: general

††pubyear: 2023††pagerange: Training Custom Light Curve Models of SN Ia Sub-
Populations Selected According to Host Galaxy Properties–5

##  1 Introduction

Type Ia supernovae (SNe Ia) are standardisable candles whose distances can be
measured to within $\sim 5\%$ via the classical two-parameter standardisation
process (Tripp, 1998).111Some studies have suggested that SN observations made
in the near-infrared are consistent enough to yield distance measurements
without the need for standardisation parameters (Mandel et al., 2009;
Johansson et al., 2021); however, the scarcity of near-infrared data means
that this technique has not yet been widely adopted. These distance
measurements can be made out to $z~{}\sim~{}2.2$ with current observing
facilities (Jones et al., 2013; Graur et al., 2014; Rodney et al., 2014; Riess
et al., 2018; Hayden et al., 2021), making SNe Ia a uniquely useful class of
object for studying the history of the expansion of the Universe and phenomena
such as dark energy (Guy et al., 2010; Suzuki et al., 2012; Betoule et al.,
2014; Abbott et al., 2019; Scolnic et al., 2018; Jones et al., 2019; Brout et
al., 2022a). Low redshift SNe Ia are also useful for making local measurements
of the Hubble constant (Branch, 1998; Riess et al., 2016; Galbany et al.,
2022; Riess et al., 2022).

The SN parameters necessary for standardising distances are obtained by
fitting a light curve model to observed time-series photometry of a sample of
SNe Ia. For the last decade, SALT2 has been the most widely used light curve
model (Guy et al., 2007; Guy et al., 2010; Mosher et al., 2014; Betoule et
al., 2014; Taylor et al., 2021; Brout et al., 2022b; Taylor et al., 2023).
Recently, Kenworthy et al. (2021) introduced SALT3, a modern refactored
version of the SALT2 model. SALT3 has a number of advantages over SALT2
(Kenworthy et al., 2021; Pierel et al., 2022; Taylor et al., 2023). In
particular, SALT3’s actively maintained, well-documented, open-source training
procedure makes it straightforward for the SN community to use and update
SALT3 as we improve training data sets and gain a deeper understanding about
the relationship between the model inputs and cosmological constraints (e.g.,
Jones et al., 2023; Dai et al., 2022).

An open question in the measurement of current SN distances (and moreover,
cosmological constraints) is the true nature of the “mass step". The mass step
is an observed $\sim 0.06$ magnitude shift in standardised SN Ia distance
moduli as a function of host galaxy mass (Kelly et al., 2010; Lampeitl et al.,
2010; Sullivan et al., 2010; Smith et al., 2020; Kelsey et al., 2021). It is
observed to be more significant for redder SNe Ia (Brout & Scolnic, 2021;
Kelsey et al., 2023). The astrophysical origins of this effect are as yet
unconfirmed; the step in luminosity has also been observed to correlate with
other parameters, such as progenitor metallicity (D'Andrea et al., 2011;
Hayden et al., 2013; Rose et al., 2021), progenitor ages (Childress et al.
2014), specific star formation rate (i.e., the star formation rate per unit of
stellar mass, be it local: Rigault et al. 2013, 2015; Kim et al. 2018; Rigault
et al. 2020; Briday et al. 2022; Hand et al. 2022; or global: D'Andrea et al.
2011; Hand et al. 2022), age of the stellar population (Rose et al., 2019),
host galaxy morphology (Pruzhinskaya et al., 2020), local host colour or mass
(Roman et al., 2018; Jones et al., 2018; Kelsey et al., 2021), local dark
matter density (Steigerwald et al., 2022), or host galaxy dust (Brout &
Scolnic, 2021; Popovic et al., 2021; Wiseman et al., 2022). However, it is
still most common for SN cosmology analyses to standardise their distances
with a mass step (Conley et al., 2011; Betoule et al., 2014; Scolnic et al.,
2018; Abbott et al., 2019; Jones et al., 2019; Brout et al., 2022a) — although
(Brout & Scolnic, 2021) suggest that the observed step in SN luminosities is
actually caused by differences in galactic dust laws that happen to correlate
with host galaxy mass, and that by accounting for the diversity of dust
properties, the need for the mass-step disappears.

Currently, the mass step is accounted for in SN cosmology analyses by deriving
distances from a model (such as SALT2 or SALT3) that does not parameterise any
host galaxy relationship, and then measuring how the Hubble residuals
correlate with host properties in order to make a luminosity correction. For
example, in the Dark Energy Survey 3-Year Supernova Analysis (DES-SN3YR;
Abbott et al. 2019) the host-mass correction is added to the standard Tripp
equation (Tripp, 1998) to measure distance moduli:

$\mu=m_{b}-M+\alpha x_{1}-\beta c+\delta\mu_{\text{host }}+\delta_{\mu_{\rm
bias}},$ (1)

where $\mu$ is the distance modulus of a SN Ia in magnitude units; $M$ is the
absolute magnitude of a fiducial ($x_{0}=1,x_{1}=0,c=0$) SN Ia (and is
degenerate with the Hubble constant $H_{0}$);222Following (Marriner et al.,
2011), $M$ is offset from the absolute $B$-band magnitude by $\sim 10$ mag.
$\alpha$ and $\beta$ are global dimensionless nuisance parameters representing
the amplitude of the stretch-luminosity and colour-luminosity relationships;
$m_{b}$, $x_{1}$ and $c$ are respectively the apparent magnitude at peak,
stretch and colour parameters of an individual SN (recovered from the SALT3
light curve fit); and $\delta_{\mu_{\rm bias}}$ accounts for bias corrections.
The host-mass correction is:

$\delta\mu_{\mathrm{host}}=\gamma\times\left[1+e^{\left(M_{\log\mathrm{host}}-M_{\log\mathrm{step}}\right)/\tau_{M_{\star}}}\right]^{-1}-\frac{\gamma}{2},$
(2)

where $M_{\log\mathrm{host}}=\log_{10}{M_{\star}}/M_{\odot}$ represents the
stellar mass of the host galaxy, $M_{\log\mathrm{step}}$ is the chosen “split"
between high-mass and low-mass galaxies (typically set to be $10$),
$\tau_{M_{\star}}$ is the width of the mass split, and $\gamma$ is the mass
step which is fit (e.g., Brout et al. 2019b).

Regardless of its origins, it is now possible to explore the cosmological
impact of the mass step in the underlying light curve model. By leveraging its
training code accessibility and the large SNe Ia training samples that are now
available, SALT3 can offer a insight into the mass step by revealing
differences in the spectral energy distributions (SEDs) between high-host-mass
and low-host-mass populations of SNe Ia. A SALT3-based investigation into the
impact of the mass step was first made in Jones et al. (2023) (hereafter J23).
They added a mass step parameterization to the model, finding differences in
some spectral features; most notably in the equivalent widths of Ca H&K and Si
II (at $\sim 2\sigma$ significance). While using the J23 model
parameterisation for light curve fitting does reduce the mass step by
$0.021\pm 0.002$ mag, they find little change to the Hubble residual
dispersion. In order to ensure robust host galaxy masses, J23 only used 278
low-$z$ ($z<0.15$) SNe Ia to train its models — just 30% of the SALT3 training
sample used in Kenworthy et al. 2021 (hereafter K21).

Here, we build on the investigation of J23 by splitting the complete K21 SALT3
training sample of 1083 SNe Ia on host galaxy mass. We take a different
approach to J23, choosing to use the traditional two-parameter SALT3 model
(without any host galaxy considerations) and simply train separate SALT3
surfaces (i.e., model iterations) from a “high-host-mass" ($M_{\rm
host}>10^{10}M_{\odot}$) training sample and “low-host-mass" sample. In this
way, we can see how all SALT3 components change with host-mass, rather than
parameterising the impact of host-mass within a single SED component. We focus
on comparing the results on a cosmological level by performing a reanalysis of
the spectroscopically confirmed Dark Energy Survey 3 Year SN data (Brout et
al. 2019a; Brout et al. 2019b; Abbott et al. 2019).

The SALT3 model training procedure underlying this work is outlined in $\S$2.
Our methods for training SALT3 models on samples with different host-mass
distributions are given in $\S$3. We present and discuss all results in $\S$4.
Our conclusions are presented in $\S$5.

##  2 SALT3 Model Training

The SALT3 model spectral flux density at phase $p$ and wavelength $\lambda$ is
defined as:

$\displaystyle F_{\lambda}=x_{0}$
$\displaystyle\times\left[M_{0}(p,\lambda)+x_{1}M_{1}(p,\lambda)\right]\times\exp[cCL(\lambda)],$
(3)

where $x_{0},x_{1}$ and $c$ are the parameters of a particular supernova and
$M_{0}$, $M_{1}$, and $C_{L}$ are the global model components. These
components are iteratively fit using a $\chi^{2}$ minimization process during
the model training.

The SALT3 training process is outlined in K21. This process requires an input
training sample of photometric and spectroscopic data from well-measured SNe
Ia, along with information about the systems on which the photometry was
observed, the redshifts, and a set of starting guesses for the SALT3 model
components and training sample light curve parameters. We use the the public
`SALTShaker` training code with the default public K21 training configuration
in this work, and select our training subsamples from the K21 training
sample.333During the preparation of this manuscript, a new version of
SALTShaker that utilises JAX gradient descent optimisation was released
(https://github.com/djones1040/SALTShaker/releases/tag/v1.1. This version is
faster and has fewer convergence issues, but has not been publicly validated
yet (Kenworthy, 2023). Following Taylor et al. (2023), we use the photometric
calibration presented in Brout et al. (2022b).444Following Taylor et al.
(2023), some of the photometry is also updated from the original K21 release.
We make the following modifications to the default SALT3 configuration:

  * •

The number of component fitting iterations between each error fitting
iteration is changed from 5 to 10, for computational speed. The error
estimation is the most computationally-intensive part of the training process,
and this choice halves the time spent in that training stage. This is not
anticipated to yield any noticeable difference to the model surfaces (Jones &
Kessler, 2018).

  * •

Following Dai et al. (2022), we reduce the maximum number of spectral
calibration parameters from 10 to 5. During the training, spectra are
iteratively recalibrated to match the best-fit model. This accounts for any
poor-quality calibration in the spectral training sample, which often has
large uncertainties. However, using too high an order for the recalibration
polynomial may fit out real spectral features.

  * •

The spectral $\chi^{2}$ scaling, which ensures roughly equal photometric and
spectroscopic contributions to the final model reduced $\chi^{2}$, is fine
tuned for each surface such that the contributions are within 10%. This
procedure, which follows Guy et al. (2007); Kenworthy et al. (2021), ensures
that the spectroscopic and photometric training data are considered with equal
weighting, despite the spectroscopic uncertainties often being much larger.

  * •

SALTShaker defines the training convergence to occur when the change in
$\chi^{2}$ between consecutive iterations is less than 1. We find that for
some of the models trained in this work, this criteria is never reached. In
this case, we consider a model to have “converged" if the change in $\chi^{2}$
is less than 0.01%.

##  3 Investigating Host Galaxy Mass in the SALT3 Model

Here, we investigate the possibility that separate intrinsic populations of
SNe Ia exist, and are correlated with host galaxy mass. We perform separate
light curve modelling for SNe Ia in “low-mass" and “high-mass" host galaxies.
We split our populations using host stellar mass, as this is the most commonly
used proxy for the host galaxy-luminosity relationship in SN cosmology
analyses — SNe in high-mass hosts are observed to have brighter post-
standardization luminosities than those in low-mass hosts.555This trend is
reversed before standardization, where SNe in high-mass hosts (with less
active star formation) tend to be dimmer (Childress et al., 2014; Hand et al.,
2022). The methods developed in this work could be applied to other host
galaxy parameters of interest (e.g., host galaxy colour, age, morphology,
specific star formation rate), provided that there exists sufficient data to
train the models.

###  3.1 SALT3 Training Samples and Resultant Surfaces

Figure 1: Data density plot showing the phase-wavelength coverage of the full K21, SALT3.HIGHMASS, and SALT3.LOWMASS training samples, applying a consistent heatmap scaling. The upper panels show the coverage of the photometric data, and the lower panel shows the spectral coverage. The relative distribution of data between samples is broadly similar, with slightly higher spectroscopic coverage in SALT3.HIGHMASS (which has a higher proportion of low-redshift SNe). These plots were generated using SALTShaker. Table 1: Host galaxy mass statistics for the mass-split SALT3 training samples used in this work. The low-redshift subsample tends to include more high-mass host galaxy SNe, while the untargetted high-redshift subsample favours low-mass hosts. Sample | Sub-sample | # SNe | mean $M_{\log}$ | median $M_{\log}$
---|---|---|---|---
K21 | Full | 1083 | 9.843 | 10.227
K21 | $z<0.1$ | 337 | 10.075 | 10.452
K21 | $z>0.1$ | 748 | 9.729 | 10.081
TS-HIGHMASS | Full | 617 | 10.709 | 10.667
TS-LOWMASS | Full | 446 | 8.645 | 9.189
Figure 2: The combined $M_{0}$ and $M_{1}$ components of SALT3 at various
$x_{1}$ values for the trained SALT3.LOWMASS and SALT3.HIGHMASS surfaces, as
well SALT3.FRAG (Taylor et al., 2023). There is clear evidence of ringing
(rapid fluctuations in phase or wavelength space) in the UV region of the
SALT3.LOWMASS $M_{1}$ component, which limits its usefulness in this analysis.
Figure 3: SN parameter distributions of the mass split training samples used
in this work. Here, we plot $m_{B}=-2.5\ln(x_{0})$ as a measure of business.
The parameters are all obtained from light curve fits with the same SALT3
model, and are used as initial guesses in the SALTShaker training procedure.
While the colour distributions are similar between the high- and low-host-mass
training samples, the amplitude and stretch distributions change
significantly.

We use host-masses from the public K21 data files, which were provided for 662
of the 1083 SNe. The missing masses were primarily from SNe in the Sloan
Digital Sky Survey (SDSS, Holtzman et al. 2008), Supernova Legacy Survey
(SNLS, Astier et al. 2006), and various low-$z$ surveys (Jha et al., 2006). We
supplement the K21 host-mass information with data from SNLS (M. Sullivan
2021, private communication) and the Joint Light Curve Analysis (JLA, Betoule
et al. 2014). This gives us mass information for all but 20 SNe in the K21
sample. We split the K21 sample to construct two training sub-samples: TS-
HIGHMASS, consisting of SNe in “high-mass" ($M_{\log}>10$) host galaxies, and
TS-LOWMASS, consisting of SNe in “low-mass" ($M_{\log}<10$) host galaxies. We
apply the mass-split at $M_{\log}=10$ (where
$M_{\log}=\log_{10}M_{\star}/M_{\odot}$) for consistency with the literature;
this is slightly higher than the mean $M_{\log}=9.483$ of the full sample.
Summary statistics of the training samples are given in Table 1.

Unlike J23, we have not derived our own host galaxy masses. It’s therefore
possible that using multiple sources of host galaxy mass could introduce some
errors. However, as we are only applying a broad cut at $M_{\log}$ = 10 and do
not use the host galaxy mass information in the actual training process, our
masses need only be accurate enough to assign each SN to the appropriate high-
or low-host-mass bin. The K21 training sample has a mean low-host-mass error
of $\sigma^{\rm low}_{M_{\rm log}}=0.661$ and a mean high-host-mass error of
$\sigma^{\rm high}_{M_{\rm log}}=0.469$.666J23 found that the uncertainty on
their host galaxy masses was small enough to be negligible, and so did not
propagate that uncertainty into their SALT3 model uncertainty.

Making use of publicly available host-masses allows us to use nearly all of
the K21 training sample SNe, while J23 could only use a fraction of the
low-$z$ SNe. It is important to include as many SNe in the SALT3 training as
one reliably can. The training of SALT models will only converge if there is
adequate phase-wavelength coverage in the training sample. Even if the
training does converge, low-coverage regions of the phase-wavelength space may
be poorly constrained or suffer from training artefacts that require tailored
regularization. We do not adjust the regularization parameters in this work,
instead choosing to only focus on results obtained from well-constrained
regions of the trained models.

In order to assess the reliable regions of our trained models, we provide
phase-wavelength coverage plots for each of the training samples (Figure 1).
TS-HIGHMASS largely follows the phase-wavelength space footprint of the K21
sample, with only minor gaps in some late-phase spectral coverage. The density
of the data coverage is at least as good as that of the Joint Light curve
Analysis training sample from Betoule et al. (2014) (shown in Figure 10 of
K21) which was used to produce the SALT2.4 “JLA" model — the most widely used
model in the literature (e.g., Betoule et al., 2014; Abbott et al., 2019;
Scolnic et al., 2018). We are therefore confident that our high-host-mass
training sample is sufficient to produce a reliable SALT3 model
(“SALT3.HIGHMASS").

While the relative phase and wavelength coverage of TS-lowmass broadly matches
that of the K21 sample, the number of SNe is lower. It is even lower than the
JLA sample, with the maximum density bins reduced by $\sim$50%. This follows
directly from the $\sim$50% reduction in the number of SNe in TS-LOWMASS. The
spectral coverage is also slightly poorer: there are gaps at multiple phases
in the rest-frame ultraviolet (UV, $\lambda\lesssim 4000$Å) and near-infrared
(NIR, $\lambda\gtrsim 9000$Å) regions — though the spectroscopic coverage
matches JLA reasonably well for the other regions.777We note that the original
JLA sample also did not include spectra at NIR wavelengths, which are not
typically used for light curve fitting anyway. Given this limited data, we
carefully inspect the resultant surface (“SALT3.LOWMASS") components for
ringing artefacts. We find evidence of significantly ringing in the $M_{1}$
component (and mild ringing in $M_{0}$) at many phases, including peak, for
$\lambda\lesssim 4000$Å (e.g., Figure 2). At longer wavelengths, our
SALT3.LOWMASS surface appears to be smoothly constrained (there is a peculiar
$M_{1}$ feature around 6000Å, but we find no evidence to suggest this is a
training artefact). Ameliorating the UV ringing artefact would require a
refined SALT3 regularization scheme or a supplementary training sample of SNe
in low-mass host galaxies, which are beyond the scope of this paper. Instead,
we quantify the impact of the UV ringing by performing two identical
cosmological analyses over different rest-frame wavelength ranges, that either
include or exclude the affected region of the SALT3.LOWMASS SED.

A further limitation in our analysis relates to scaling employed by
SALT3.888SALT2 also employs a scaling of SN parameters in its training
process. The SALT3 training performs a scaling of the SN parameters, such that
the training sample has $\bar{x_{1}}=0,\ \sigma_{x_{1}}=1,\ \bar{c}=0$ (K21,
Dai et al. 2022). Similarly, the SALT3 components are defined relative to the
training sample demographics: $M_{0}$ is the average SED for a $x_{1}=0,\ c=0$
SN, and $M_{1}$ scales with $\sigma_{x_{1}}$. Assuming our input training
sample distributions are representative of the true $x_{1}$ and $c$
populations of SNe in high- and low-mass host galaxies as they occur in the
Universe, any differences in the $M_{0}$ components should represent the
average astrophysical differences between SNe originating from high- and low-
mass host galaxies. However, any biases in the input training sample
distribution will complicate this interpretation. The end result is that the
fitted light curve parameters from each model cannot be meaningfully compared
without a scaling correction. We therefore compare the SN parameters that are
determined from fitting a single SALT3 surface to the entire training sample.
We plot these parameter distributions for each of our mass-split training
samples in Figure 3. While the $c$ distributions are largely consistent across
our training samples, there is (unsurprisingly) a significant change in the
$x_{1}$ distributions — SNe Ia that occur in more massive galaxies tend to be
faster declining, so have lower $x_{1}$ values (Hamuy et al., 1995; Branch et
al., 1996; Hamuy et al., 2000; Howell, 2001; Gallagher et al., 2005).999A
relationship with colour has also been found, though as we see in Figure 3,
this are weaker than the stretch relationships (Smith et al., 2020). Moreover,
the low-mass training sample has a narrower range of $x_{1}$ values than the
high-mass or full training samples.

###  3.2 Methods to Assess the Impact on Distances and Cosmology

We use the Dark Energy Survey’s 3-year spectroscopically confirmed SNe Ia
sample and its companion low-$z$ sample (DES-SN3YR Brout et al., 2019a) to
test the end-to-end cosmological impact of treating SNe Ia from different host
populations as intrinsically different. We fit the DES-SN3YR light curves with
our SALT3.HIGHMASS and SALT3.LOWMASS surfaces, as well as with a control
surface, SALT3.FRAG (Taylor et al., 2023), which was trained with the same
inputs but uses the entire K21 training sample. We split the DES-SN3YR sample
SNe according to the surface demographics — for example, we fit the entire
sample with SALT3.FRAG, but only the $M_{\log}>10M_{\odot}$ subsample with
SALT3.HIGHMASS. We use the published SVA1-Gold DES-SN3YR host-masses to select
these subsamples (Abbott et al., 2019), as these were the masses used in the
K21 training sample. The high-host-mass subsample contains 91 DES SNe and 98
low-$z$ SNe; the low-host-mass subsample contains 116 DES SNe and 24 low-$z$
SNe. There are 18 SNe in the DES sub-sample for which hosts could not be
adequately identified. Following Brout et al. (2019b), these SNe are assumed
to be in low-mass hosts (as high-mass hosts would likely be detectable). There
are 10 “high-host-mass" DES SNe within $1\sigma_{M_{\log}}$ of our chosen mass
boundary ($M_{\log}=10$), and 6 “low-host-mass" DES SNe within
$1\sigma_{M_{\log}}$ of the boundary. There are 28 high-host-mass, low-$z$ SNe
within $1\sigma_{M_{\log}}$ of our chosen mass boundary, and 11 low-host-mass,
low-$z$ SNe within $1\sigma_{M_{\log}}$ of the boundary.

We use SNANA (Kessler et al., 2009) to fit the light curves over
$3000<\lambda_{\rm eff}<8000$Å, where $\lambda_{\rm eff}$ is the rest-frame
effective mean wavelength of the photometric filter. In this range, $U$-band
observations in low-$z$ SNe (and $g$ band observations at $z\gtrsim 0.3$) will
be impacted by the UV ringing artefact seen in the SALT3.LOWMASS surface
(Figure 2). We explore the impact of this ringing by performing a similar
analysis over a truncated rest-frame wavelength range ($4000<\lambda_{\rm
eff}<8000$Å); this is presented in Appendix A.

We apply standard light curve cuts of $-0.3<c<0.3$, $-3<x_{1}<3$ following
Brout et al. (2019b); Jones et al. (2023). We use the `SALT2mu` procedure of
Marriner et al. (2011) to fit nuisance parameters and distance moduli for
three sets of light curve fit results — the full DES-SN3YR sample (including
low-$z$) fit with SALT3.FRAG, the high host-mass SNe fit with SALT3.HIGHMASS,
and the low host-mass SNe fit with SALT3.LOWMASS. For these three sets of
results, we include 1D bias corrections but do not include any host-mass
correction ($\delta\mu_{\text{host }}$, Equation 1). We fit an additional set
of distance moduli (labelled SALT3.FRAG$+$Mass Step) that uses light curve
parameters from the full DES-SN3YR sample fit with SALT3.FRAG and does include
the fit of a mass step term, with $M_{\log\mathrm{step}}=10$. These results
use the same set of bias correction simulations as the nominal SALT3.FRAG
results, and yield a $\gamma$ value (Equation 2) that is comparable to the 1D
mass step found in (Smith et al., 2020) ($\gamma_{\rm{Smith2020}}$ = 0.066
$\pm$ 0.020mag, $\gamma_{\rm{Taylor2023}}$ = 0.079 $\pm$ 0.021mag).

The 1D bias corrections are calculated using large sets of realistic
simulations of the DES-SN3YR data, which are generated following Kessler et
al. (2019). Separate simulations are created for the low-$z$ and DES SNe, for
each SALT3 surface. The simulated SN are drawn from parent populations that
are calculated according to predicted distributions from each individual
surface, following the methods of Popovic et al. (2021). While there are no
explicit mass-luminosity relationships included in our simulations, each set
selects SNe from parent populations that are calculated according to the
corresponding real distribution of fitted DES-SN3YR data. For example, the
SALT3.HIGHMASS simulations are generated from stretch and colour distributions
that are calculated using only the high host-mass DES-SN3YR SNe fit with the
SALT3.HIGHMASS model. Each set of bias correction simulations also uses the
corresponding SALT3 model as the input model to generate the rest-frame SED.
We use the G10 intrinsic scatter model of Kessler et al. (2013), based on Guy
et al. (2010), and assume an underlying flat $\Lambda$CDM cosmology with
$\Omega_{M}=0.315,\Omega_{\Lambda}=0.685,H_{0}=70$km/s/Mpc. These assumptions
introduce a small systematic uncertainty in the recovered cosmological
parameters (Table 8 of Brout et al. 2019b), though we only include statistical
uncertainties in our results here.

For each set of results, we fit $\Omega_{m}$ and $w$ assuming a flat $w$CDM
model and applying a CMB prior, tuned such that the constraining power is
similar to that of Planck (Planck Collaboration et al., 2016). This is
performed with `SNANA`’s fast cosmology fitting program, `wfit`. We verify our
SALT3.HIGHMASS and SALT3.LOWMASS surfaces’ ability to recover accurate
cosmological parameters by using them to fit simulated DES-SN3YR light curves
that are generated with a known cosmology. In these test, we recover the input
$\Omega_{m}$ and $w$ values to within $2\sigma$ (when applying a CMB prior).

We also test how much of the difference in results from our three surfaces can
be attributed to the surfaces themselves, versus the use of different
subsamples of DES-SN3YR SNe. We do this by fitting the high/low-host-mass DES-
SN3YR subsample light curves with SALT3.FRAG, and using each set of results to
obtain separate distance measurements and cosmological fits. The results of
these tests (given in $\S$ 4.3) are labelled “SALT3.FRAG (HIGH)" and
“SALT3.FRAG (LOW)". There is no explicit mass-step applied in this test, as we
expect any luminosity difference between the subsamples to be absorbed into
the $M$ parameter of Equation 1.

##  4 Results and Discussion

Figure 4: The $M_{0}$ components (upper panel) and change in $M_{0}$ (lower
panel) for the trained SALT3.LOWMASS and SALT3.HIGHMASS surfaces. $\Delta
M_{0}=M_{0}$(SALT3.HIGHMASS) - $M_{0}$(SALT3.LOWMASS). The Ca H&K and Si II
features discussed in J23 are labelled. Figure 5: The colour law component of
our trained SALT3.LOWMASS and SALT3.HIGHMASS surfaces, as well as SALT3.FRAG,
for a $c=-0.1$ SN. For reference, we also plot the Milky Way laws from
(Cardelli et al., 1989) for different values of $R_{V}$. For much of the
wavelength range, the SALT3.HIGHMASS and SALT3.FRAG colour laws directly
overlap. The shaded regions indicate the wavelengths where the SALT3 colour
law is linearly extrapolated, rather than fit as a polynomial.

###  4.1 SALT3 Model Components

The combined $M_{0}$ and $M_{1}$ components of SALT3.HIGHMASS and
SALT3.LOWMASS at different $x_{1}$ values are shown in Figure 2. The
difference in the $M_{0}$ components is shown in Figure 4. These plots show
that fiducial SNe Ia in low-mass galaxies appear to be bluer than those in
high-mass galaxies. We observe this trend at all phases. As $M_{0}$ is by
definition the average SED of a $x_{1}=0$, $c=0$ SNe, this result confirms
that the definition of $c=0$ changes slightly between the two surfaces.

Apart from the difference in the colour, there are subtle differences in some
of the spectral features. The Si II features in SALT3.HIGHMASS (at 4000, 5770
and 6100Å) are slightly blueshifted compared to SALT3.LOWMASS. The blueshift
reflects the velocity of ejecta coming towards us along the line of sight;
higher ejecta velocities are associated with brighter SNe Ia (Benetti et al.,
2005). The Si II feature at 5770Åis noticeably deeper in SALT3.HIGHMASS. The
depth of this line relative to the stronger Si II feature at 6100Åcorrelates
with SN Ia luminosity. Lower ratios correspond to fainter SNe Ia (Nugent et
al., 1995). This is inconsistent with the blueshifts seen in the Si II lines
but consistent with the redder colours of SALT3.HIGHMASS.

The $c\times CL$ components (i.e. the exponential term from Equation 3) for a
$c=-0.1$ SN are shown in Figure 5. We find that SNe occurring in low-mass
galaxies have a steeper colour law in the UV. J23 do not train separate phase-
independent colour laws for low-host-mass or high-host-mass SNe; but they do
find that the phase-dependent colours exhibit a change with host-mass. Their
results disagree with our findings (though we reiterate that we use different
training samples); the J23 model of high-host-mass SNe is slightly bluer than
the low-host-mass model, both at peak phase and in the Lira law tail (Lira et
al., 1998; Phillips et al., 1999). They find that these phase-dependent colour
differences were previously captured by the $M_{1}$ component and encompassed
by the $x_{1}$ parameter.

Our results may indicate that SNe in low-mass galaxies are intrinsically more
luminous than those in high-mass galaxies. Alternatively, the change in
colours between SNe from high- and low-mass hosts may be caused by changes in
the extrinsic dust component. However, at the current level of modelling it is
not possible to disentangle the intrinsic SN colour from the effects of host
galaxy dust.

Figure 6: Comparison of the $x_{1}$ parameter for the DES-SN3YR sample when
fit with SALT3.FRAG versus SALT3.HIGHMASS (left panel, pink points) and
SALT3.LOWMASS (right panel, blue points).

Figure 7: As in Figure 6, for $c$.

Figure 8: As in Figure 6, for distance moduli. We plot the difference in distance moduli on the y-axis. Table 2: The best fit nuisance parameters (from Equation 1) and cosmology parameters for the DES-SN3YR sample when fit with different SALT3 surfaces. All results are obtained using the fast cosmology fitting program, wfit, and include a Planck Collaboration et al. (2016) cosmic microwave background (CMB) prior. As we have used a different set of surfaces to the original DES-SN3YR analysis (which used SALT2.JLA from Betoule et al. 2014), and apply 1D rather than 5D bias corrections, we do not expect our results to exactly match those presented in Brout et al. (2019b); Abbott et al. (2019). The SALT3.FRAG (HIGH/LOW) results that are split by host galaxy mass have this selection applied prior to the SALT2mu distance fitting stage. The cosmology fit $\chi^{2}$ values have been calculated without including contributions from the SALT3 model uncertainties, which change between surfaces. Sample | # SNe fit | $\alpha$ | $\beta$ | $\sigma_{\rm int}$ | $M$ | $\Omega_{M}$ | $\sigma_{\Omega_{M}}^{\rm stat}$ | $w$ | $\sigma_{w}^{\rm stat}$ | $\chi^{2}_{\rm cosmo}$
---|---|---|---|---|---|---|---|---|---|---
SALT3.FRAG | 322 | 0.14 $\pm$ 0.01 | 2.78 $\pm$ 0.11 | 0.11 | -29.99 | 0.34 | 0.02 | -0.91 | 0.05 | 11.1
SALT3.FRAG (HIGH) | 187 | 0.15 $\pm$ 0.01 | 2.63 $\pm$ 0.13 | 0.11 | -30.01 | 0.36 | 0.02 | -0.85 | 0.06 | 7.3
SALT3.FRAG (LOW) | 135 | 0.12 $\pm$ 0.02 | 2.99 $\pm$ 0.18 | 0.11 | -29.97 | 0.35 | 0.03 | -0.89 | 0.08 | 12.7
SALT3.FRAG + Mass Step | 322 | 0.15 $\pm$ 0.01 | 2.79 $\pm$ 0.10 | 0.11 | -29.99 | 0.36 | 0.02 | -0.85 | 0.05 | 9.3
SALT3.HIGHMASS | 129 | 0.11 $\pm$ 0.01 | 2.77 $\pm$ 0.16 | 0.09 | -29.94 | 0.33 | 0.02 | -0.96 | 0.07 | 7.9
SALT3.LOWMASS | 134 | 0.06 $\pm$ 0.02 | 2.89 $\pm$ 0.17 | 0.11 | -30.08 | 0.33 | 0.02 | -0.97 | 0.08 | 6.3

###  4.2 Light Curve Fits

As the SN parameters and nuisance parameters are defined on independent scales
for each SALT3 surface, we cannot directly compare these results. For example,
Figures 6-7 show that the definitions of $c$ and $x_{1}$ have systematically
changed between the three surfaces used in our analysis. We find that
$x_{1}({\rm SALT3.LOWMASS})$ is systematically lower than $x_{1}({\rm
SALT3.FRAG})$, and $x_{1}({\rm SALT3.HIGHMASS})$ is systematically higher.
This is consistent with the findings of J23, as shown in their Figure 8 (where
their $x_{1}$ parameter is redefined to only include host-mass independent
variability). We also find a (weaker) redefinition of $c$ such that for the
same SN, $c({\rm SALT3.LOWMASS})$ appears to be systematically higher than
$c({\rm SALT3.FRAG})$, and $c({\rm SALT3.HIGHMASS})$ appears to be
systematically lower. This follows directly from the $c$ rescaling expected
based on Figure 3, in which the mean (i.e. $c_{\rm model}=0$ point) is lower
for SALT.LOWMASS and higher for SALT3.HIGHMASS. In contrast, the results from
J23 (where $c$ is defined on the same scale for all models) indicate that in
reality, $c({\rm SALT3.LOWMASS})$ is systematically lower than $c({\rm
SALT3.K21})$, and $c({\rm SALT3.HIGHMASS})$ is slightly higher. This
highlights the complications introduced by the floating definitions of $x_{1}$
and $c$.

Similarly, the nuisance parameters $\alpha$ and $\beta$ (which quantify the
global stretch-luminosity and colour-luminosity relationships of SNe Ia) are
defined on different scales for each surface — though this effect is very
slight for $\beta$. The nuisance parameters are presented in Table 2.

The intricacies of comparing fitted light curve parameter values that we have
demonstrated here will also apply when comparing any fitted light curve
parameters from models trained on different samples (e.g. the Dark Energy
Survey’s 3-year versus 5-year SN analyses, which use light curve models from
Betoule et al. 2014 and Taylor et al. 2023 respectively).

During light curve fitting, we lose four SNe to light curve fitting cuts
related to the $x_{1}$ and $c$ bounds across the LOWMASS and HIGHMASS samples.
A further five fail the requirement of having at least one observation before
peak $B$-band maximum (as their estimated peak MJD have shifted when fit with
our new surfaces). We lose 56 SNe due to the fit probability > 0.01 criterion,
the majority of which are from the HIGHMASS low-$z$ sample and have high
signal-to-noise ratios.101010The fit probability is the probability of finding
an equal or larger light curve data-model $\chi^{2}$, assuming Gaussian-
distributed flux uncertainties. Our SALT3.HIGHMASS model has lower model
uncertainties than SALT3.FRAG or SALT3.LOWMASS, which leads to an increased
reduced $\chi^{2}$ for this data, thus causing it to fail fit probability
cuts. This may be remedied in future work by increasing the number of light
curve fit iterations when using models with low levels of uncertainty.

###  4.3 Distances and Cosmology

We compute the distance moduli for each set of light curve and nuisance
parameter fits. In Figure 8, the distance modulus values do not display any
systematic offset between the SALT3.FRAG and SALT3.HIGHMASS surfaces (left
panel). However, the right panel shows a slight slope, such that more distant
SNe are measured to be (on average) further away with SALT3.LOWMASS than
SALT3.FRAG. This trend is not observed when repeating our analysis over a
reduced rest-frame wavelength range of $4000<\lambda_{\rm eff}<8000$Å (Figure
10). This abridged range restricts the use of the less reliable UV region of
SALT3.LOWMASS, which is more impactful for distant SNe.

The spread of the Hubble residuals (HR) represents the level of remaining
scatter in the standardised SNe Ia luminosities and measurement errors. We
calculate Hubble residuals as
$\textrm{HR}_{i}=\mu_{\textrm{fit,}i}-\mu_{\textrm{cosmology,}i}$ for each set
of results $i$ obtained from our different SALT3 surfaces. In Table 3, we
report the root-mean-squared (RMS, i.e., spread of the Hubble residuals) for
each set of results. We also report the weighted RMS. We define the weight of
a particular HR to be $w=1/\sigma_{\mu_{\rm fit}}^{2}$. We note that the mean
measured uncertainty in the distance modulus is lower for the SALT3.HIGHMASS
results ($\bar{\sigma}_{\mu_{\rm fit}}=0.146$ mag) than those from SALT3.FRAG
($\bar{\sigma}_{\mu_{\rm fit}}=0.156$ mag) or SALT3.FRAG + Mass Step
($\bar{\sigma}_{\mu_{\rm fit}}=0.155$ mag). The mean uncertainty for the
SALT3.LOWMASS results is slightly higher ($\bar{\sigma}_{\mu_{\rm fit}}=0.157$
mag). For each set of results, we estimate the uncertainty in RMSHR using
bootstrapping, resampling the HR values 1000 times and reporting the standard
deviation in the resulting distribution.

We split the SALT3.FRAG results into low- and high-host-mass subsamples and
calculate the HR results for each, using the best-fit cosmology of the full
sample. We also report the HR results for the low- and high-host-mass
subsamples fit with SALT3.FRAG (a.k.a., SALT3.FRAG (HIGH/LOW)), where the
best-fit cosmology parameters are calculated for each subsample independently.

We find measuring the HR of a subsample of SNe from high-mass hosts lowers the
RMSHR by 0.5-1.2$\sigma$ (depending on weighting, application of mass step,
and subsample used to calculate best-fit cosmology) compared to using the full
DES-SN3YR sample fit with SALT3.FRAG. When fitting the high-host-mass sample
with SALT3.HIGHMASS, we find a reduction of 0.1-0.9$\sigma$ in the RMSHR
relative to the same sample fit with SALT3.FRAG.

On the other hand, using a subsample of SNe from low-mass hosts increases the
RMSHR by 0.5-1.2$\sigma$, compared to using the full DES-SN3YR sample fit with
SALT3.FRAG, when both samples use the SALT3.FRAG best fit cosmology. When
fitting the low-host-mass sample with SALT3.LOWMASS, we intead find a
reduction of 0.5-0.7$\sigma$ in the RMSHR relative to the same sample with
SALT3.FRAG.

These results indicate that the DES-SN3YR SNe in high-mass hosts are better
standardisable candles than those in low-mass hosts, and that the
standardisability in the DES-SN3YR sample is slightly improved by using custom
SALT3.HIGHMASS and SALT3.LOWMASS surfaces (albeit at low statistical
significance). This is consistent with the results of J23, who also find
negligible changes in the HR scatter.

The SALT3.HIGHMASS and SALT3.LOWMASS surfaces yield similar cosmology results,
with the estimated value of $w$ shifting by an insignificant
0.6-1.3$\sigma_{w}^{\rm stat}$ compared to the SALT3.FRAG results (Table 2).
Combining the binned Hubble diagrams of the mass-split samples in order to fit
a single cosmological model from all the data is not possible with the current
fitting infrastructure, but would be necessary if our methods were to be
adopted by future cosmology analyses.

Table 3: Scatter in the Hubble residual (from best-fit cosmology) for each
SALT3 surface. All values are in units of magnitude. The best-fit cosmology is
unique for each entry in the “Result" column. The SALT3.FRAG (HIGH) and (LOW)
results are split by host galaxy mass prior to the SALT2mu fitting stage, to
isolate the effects of fitting cosmology with a subsample of DES-SN3YR SNe
versus the effects of changing surfaces. Hubble residuals from best-fit
cosmology
---
Result | Subsample | RMSHR | Weighted RMSHR
| Full | 0.155$\pm$0.008 | 0.147$\pm$0.008
| $M_{\log}>10$ | 0.141$\pm$0.009 | 0.137$\pm$0.009
SALT3.FRAG | $M_{\log}<10$ | 0.173$\pm$0.013 | 0.159$\pm$0.014
SALT3.FRAG (HIGH) | $M_{\log}>10$ | 0.143$\pm$0.009 | 0.139$\pm$0.010
SALT3.FRAG (LOW) | $M_{\log}<10$ | 0.172$\pm$0.013 | 0.156$\pm$0.013
| Full | 0.158$\pm$0.008 | 0.149$\pm$0.008
| $M_{\log}>10$ | 0.147$\pm$0.009 | 0.142$\pm$0.010
SALT3.FRAG \+ Mass Step | $M_{\log}<10$ | 0.173$\pm$0.013 | 0.157$\pm$0.014
SALT3.HIGHMASS | $M_{\log}>10$ | 0.139$\pm$0.011 | 0.128$\pm$0.011
SALT3.LOWMASS | $M_{\log}<10$ | 0.162$\pm$0.013 | 0.146$\pm$0.014
Figure 9: Correlations between host galaxy masses and Hubble residuals for the
DES-SN3YR sample when fit with the SALT3.FRAG (upper plot), SALT3.HIGHMASS,
and SALT3.LOWMASS surfaces. Here, the Hubble residuals are taken from the
best-fit cosmology for each model; a similar result is obtained when
calculating Hubble residuals from a constant reference cosmology (i.e.,
$\Lambda$CDM from Planck Collaboration et al. 2020). The mass step for each
set of results is approximated as the difference in weighted mean Hubble
residual between the high- and low-host-mass samples, with uncertainties
estimated using bootstrapping. We plot a truncated mass range for visual
clarity, but use the full range for the mass step calculations. The Hubble
residuals are offset on the y-axis for each set of results. In the lower plot,
the mass-step is $\sim 0$ by design; the change in luminosity is absorbed by
the SALT3 model and the $M$ term of Equation 1 (see Table 2).

###  4.4 The Mass Step

In Figure 9, we plot the Hubble residuals as a function of $M_{\log}$ for each
surface. We also show the binned weighted average the Hubble residuals in the
two mass step bins, and approximate the mass step as the difference in
weighted averages between the high- and low-host-mass sets of SNe.111111As we
perform independent SALT2mu processes for the low- and high-host-mass samples,
we cannot simply fit $\gamma$ to obtain the mass step value. We bootstrap our
samples 1000 times to obtain the uncertainty on the mass step.

We find that our method of treating SNe from low- and high-mass host galaxies
as unique populations in the SALT3 framework removes the observed mass step as
efficiently as our explicit fitting for $\gamma$, from step = -0.044 $\pm$
0.019 mag (SALT3.FRAG) to step = 0.006 $\pm$ 0.019 mag (SALT3.LOWMASS and
SALT3.HIGHMASS). However, the fitted fiducial SN ($x_{0}=1,x_{1}=0,c=0$)
absolute magnitude $M$ from Equation 1 changes by a +0.14 magnitudes between
SALT3.LOWMASS and SALT3.HIGHMASS, such that SNe in high mass galaxies are
fainter. This $\Delta M$ arises from both the redefinition of a fiducial SN
between different surfaces and the absorption of the host-mass luminosity
step. For our SALT3.FRAG (HIGH) and SALT3.FRAG (LOW) results — which are based
on the same surface and so have a consistently defined fiducial SN — we find
$\Delta M=-0.04$. In this case, the observed mass step is also “removed" (step
= -0.0005). This result also holds when calculating HR with respect to a
reference $\Lambda$CDM cosmology from Planck Collaboration et al. (2020), as
was done in J23.

Without enforcing a consistent definition of the parameters across different
light curve models, we cannot claim that the treatment of the mass-step
benefits from adopting bespoke SALT3.HIGHMASS and SALT3.LOWMASS light curve
models for sub-populations of SNe in our analysis. The analysis performed by
J23 (in which the parameters are defined on a consistent scale, and the host-
mass contribution to the SN flux is modelled and then excluded from the
calculation of distances) suggest that improved SALT3 modelling should at
least partially reduce the mass-step (by 0.021$\pm$0.002 mag).

###  4.5 Future Work

Here, we have used host-masses from a variety of sources, relying on the
broad, binary nature of the observed mass step to mitigate any biases from
inconsistent mass estimates. While this has allowed us to use the full
constraining power of the K21 training sample without having to derive our own
masses for $\sim 1000$ host galaxies, this analysis should be repeated with
more consistently derived masses for both the training and cosmology samples,
in order to develop this method into a robust tool for current SN cosmology
analyses. For example, we use the DES-SN3YR SVA1-Gold catalogue host-masses
described in Brout et al. (2019b) for the DES-SN training and cosmology
samples, but more recent work by Wiseman et al. (2020); Smith et al. (2020)
updates these masses using the full 5-yr DES deep-stack photometry. 19 DES SNe
change “mass bins" with the rederived masses. Similarly, 107 of our K21
training sample are assigned to different mass bins in the Pantheon+ analysis
(Scolnic et al., 2022). Future work should also consider the distributions of
host galaxy mass in the training and cosmology samples; many of the SNe used
in this work were discovered by targeted surveys which biases our sample.
Adopting larger training and cosmology samples would reduce the statistical
uncertainties on SN distance moduli, providing more clarity on the statistical
significance of the potential reduction in HR scatter hinted at in this work.

An interesting extension to this work would be to apply our methods to other
host galaxy parameters, such as the star formation rate. Childress et al.
(2014) predicts that there are a higher number of young SNe in low-mass
galaxies — and that this relationship appears to evolve with redshift.
Childress et al. (2014) conclude that SNe Ia in low-mass, active star-forming
galaxies represent a more standardisable population for cosmology analyses. We
have already seen promising results for SN standardisability when splitting
the populations according to host galaxy mass, despite the limitations of our
SALT3.LOWMASS model (e.g., inconsistent sources of mass, poor coverage of the
training sample, rest-frame UV ringing artefacts). Modelling separate
populations of SNe according to parameters such as specific star formation
rate (sSFR, i.e. the star formation rate normalised by the stellar mass) may
not only help to identify any unique physical processes in the SN Ia
explosions, but reduce the intrinsic scatter remaining in SN cosmology
analyses (Rigault et al., 2015, 2020).

Our straightforward method could be extended to other models that include
additional SN Ia variability (e.g., SNEMO, Saunders et al. 2018, SUGAR, Léget
et al. 2019, BayeSN (Mandel et al., 2020)). Using models that include near
infra-red data (e.g., BayeSN or SALT3.NIR Pierel et al. 2022) may be a
promising avenue to break the degeneracy between host galaxy mass (or other
properties), intrinsic colour, and extrinsic dust effects. However, further
use of this method must develop a solution for the pervasive light curve and
nuisance parameter scaling problem. Without a standardised, surface-
independent definition of $x_{0},x_{1}$ and $c$, any results derived using
different SALT surfaces will be difficult to interpret. Extensions to this
work would also benefit from a program to combine the Hubble diagrams from
multiple, independently fit samples into one input for cosmological model
fitting (for example, an extension of the current `Pippin` pipeline for SN
cosmology, Hinton & Brout 2020). Without this infrastructure, the method of
J23 is currently the clearest way to incorporate spectro-photometric modelling
of SNe Ia sub-populations into cosmology analyses.

##  5 Summary and Conclusions

For decades, SN studies have observed relationships between the luminosity of
a SN Ia and its host galaxy properties. These relationships may indicate the
existence of intrinsically unique sub-populations within the class of SNe Ia.
If that is case, the overall distance measurements and standardisation (i.e.
the distance moduli dispersion/Hubble residual scatter) of SNe Ia could be
improved by modelling each intrinsic population separately.

To test this hypothesis, we have produced and tested separate SALT3 light
curve models for SNe Ia that explode in high- or low-mass host galaxies. This
straightforward method utilises publicly available training programs and data
samples, as an alternative and complement to the method of J23 (which alters
the SALT3 model to include a host galaxy mass component).

In either method, the resulting high-resolution model spectrum may provide
astrophysical clues as to the origins of the observed mass step that would not
be obvious when looking at the population photometry or single-object spectra.
We find evidence of minor differences in Si II spectral features between low
host-mass and high host-mass samples, in agreement with Foley & Kasen (2011);
Pan et al. (2015); Siebert et al. (2020); Pan (2020); Dettman et al. (2021);
J23.

While we find that fiducial SNe in low-mass hosts are intrinsically bluer than
those in high-mass hosts, J23 find the opposite. The inconsistency in the
fiducial SN SED (which is defined relative to the mean $x_{0},x_{1},c$ values
of a SALT3 training sample) highlights the sensitivity of the SALT3 model to
the training sample demographics. This simple finding reinforces the
importance of a quality, unbiased light curve training sample (which must be
comprehensive and representative of the natural SN populations) for SN Ia
analyses (Dai et al., 2022).

Here, the fitted SN parameters ${x_{0},x_{1},c}$ are defined independently for
each surface, leading to unique definitions of the fitted nuisance parameters
$m_{B}$, $\alpha$, $\beta$ and $M$ for each surface. We assume that the
standardised distance moduli are defined consistently across all surfaces, and
so can be fairly compared. We recover consistent cosmological parameters for a
flat $w$CDM model (with CMB priors from Planck Collaboration et al. 2016) from
our low host-mass and high host-mass analyses. We find that fitting the high
host-mass subsample of SNe Ia from DES-SN3YR is intrinsically more
standardisable than the low host-mass subsample when fit with the same
SALT3.FRAG surface, with a $\sim 2\sigma$ difference between the Hubble
residual RMS of these subsamples. Using our custom SALT3.LOWMASS and
SALT3.HIGHMASS surfaces to fit these subsamples reduces the Hubble residual
RMS of both by $<1\sigma$. Though J23 suggest that the observed mass-step in
the Hubble residuals should be reduced when including host mass considerations
in the SALT3 modelling, the varying definitions of the nuisance parameter $M$
mean we cannot conclusively claim this is the case in our analysis.
Standardising the definitions of $x_{0},x_{1}$ and $c$ (and therefore, the
nuisance parameters of Equation 1) across different light curve models is a
critical step in realising the full potential of our method. This is also
important for comparing any results generated using different light curve
models; the nuance of the redefined parameters may not be obvious, but it can
be impactful.

We have demonstrated a novel application of the SALT3 light curve model
training procedure, that has the potential to improve SN Ia standardisability.
Our straightforward method may be easily used to incorporate new, high-quality
training data or explore other host-galaxy relationships. Moreover, this is
proof of how far the light curve model training infrastructure has developed
in recent years: the `SALTShaker` training code we use is publicly available,
well documented (and tested), and actively maintained, making it a useful tool
for the entire SN community.

## Data Availability

All data and training software underlying this work is publicly available.

## Acknowledgements

This research was completed in part on Ngunnawal, Ngambri, and Whadjuk
Nyoongar country.
We thank the SALT3 development team, particularly David Jones and D’Arcy
Kenworthy, for their helpful discussions. We also thank Mark Sullivan for
providing us with the SNLS host mass data. This research was supported by an
Australian Government Research Training Program (RTP) Scholarship. This
project used public archival data from the Dark Energy Survey (DES). Funding
for the DES Projects has been provided by the U.S. Department of Energy, the
U.S. National Science Foundation, the Ministry of Science and Education of
Spain, the Science and Technology Facilities Council of the United Kingdom,
the Higher Education Funding Council for England, the National Center for
Supercomputing Applications at the University of Illinois at Urbana-Champaign,
the Kavli Institute of Cosmological Physics at the University of Chicago, the
Center for Cosmology and Astro-Particle Physics at the Ohio State University,
the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M
University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho
de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de
Desenvolvimento Científico e Tecnológico and the Ministério da Ciência,
Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft, and the
Collaborating Institutions in the Dark Energy Survey. The Collaborating
Institutions are Argonne National Laboratory, the University of California at
Santa Cruz, the University of Cambridge, Centro de Investigaciones
Energéticas, Medioambientales y Tecnológicas-Madrid, the University of
Chicago, University College London, the DES-Brazil Consortium, the University
of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi
National Accelerator Laboratory, the University of Illinois at Urbana-
Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the Institut de
Física d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-
Maximilians Universität München and the associated Excellence Cluster
Universe, the University of Michigan, the National Optical Astronomy
Observatory, the University of Nottingham, The Ohio State University, the
OzDES Membership Consortium, the University of Pennsylvania, the University of
Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the
University of Sussex, and Texas A&M University. Based in part on observations
at Cerro Tololo Inter-American Observatory, National Optical Astronomy
Observatory, which is operated by the Association of Universities for Research
in Astronomy (AURA) under a cooperative agreement with the National Science
Foundation.

## References

  * Abbott et al. (2019) Abbott T. M. C., et al., 2019, ApJ, 872, L30
  * Astier et al. (2006) Astier P., et al., 2006, A&A, 447, 31
  * Benetti et al. (2005) Benetti S., et al., 2005, The Astrophysical Journal, 623, 1011
  * Betoule et al. (2014) Betoule M., et al., 2014, A&A, 568, A22
  * Branch (1998) Branch D., 1998, ARA&A, 36, 17
  * Branch et al. (1996) Branch D., Romanishin W., Baron E., 1996, ApJ, 465, 73
  * Briday et al. (2022) Briday M., et al., 2022, A&A, 657, A22
  * Brout & Scolnic (2021) Brout D., Scolnic D., 2021, The Astrophysical Journal, 909, 26
  * Brout et al. (2019a) Brout D., et al., 2019a, The Astrophysical Journal, 874, 106
  * Brout et al. (2019b) Brout D., et al., 2019b, ApJ, 874, 150
  * Brout et al. (2022a) Brout D., et al., 2022a, The Pantheon+ Analysis: Cosmological Constraints, doi:10.48550/ARXIV.2202.04077, https://arxiv.org/abs/2202.04077
  * Brout et al. (2022b) Brout D., et al., 2022b, ApJ, 938, 111
  * Cardelli et al. (1989) Cardelli J. A., Clayton G. C., Mathis J. S., 1989, ApJ, 345, 245
  * Childress et al. (2014) Childress M. J., Wolf C., Zahid H. J., 2014, MNRAS, 445, 1898
  * Conley et al. (2011) Conley A., et al., 2011, ApJS, 192, 1
  * D'Andrea et al. (2011) D'Andrea C. B., et al., 2011, The Astrophysical Journal, 743, 172
  * Dai et al. (2022) Dai M., Jones D. O., Kenworthy W. D., Kessler R., Pierel J. D. R., Foley R. J., Jha S. W., Scolnic D. M., 2022, Propagating Uncertainties in the SALT3 Model Training Process to Cosmological Constraints, doi:10.48550/ARXIV.2212.06879, https://arxiv.org/abs/2212.06879
  * Dettman et al. (2021) Dettman K. G., et al., 2021, ApJ, 923, 267
  * Foley & Kasen (2011) Foley R. J., Kasen D., 2011, ApJ, 729, 55
  * Galbany et al. (2022) Galbany L., et al., 2022, arXiv e-prints, p. arXiv:2209.02546
  * Gallagher et al. (2005) Gallagher J. S., Garnavich P. M., Berlind P., Challis P., Jha S., Kirshner R. P., 2005, ApJ, 634, 210
  * Graur et al. (2014) Graur O., et al., 2014, The Astrophysical Journal, 783, 28
  * Guy et al. (2007) Guy J., et al., 2007, A&A, 466, 11
  * Guy et al. (2010) Guy J., et al., 2010, A&A, 523, A7
  * Hamuy et al. (1995) Hamuy M., Phillips M. M., Maza J., Suntzeff N. B., Schommer R. A., Aviles R., 1995, AJ, 109, 1
  * Hamuy et al. (2000) Hamuy M., Trager S. C., Pinto P. A., Phillips M. M., Schommer R. A., Ivanov V., Suntzeff N. B., 2000, AJ, 120, 1479
  * Hand et al. (2022) Hand J., Liu S., Galbany L., Perrefort D., Wood-Vasey W. M., Burns C., 2022, ApJ, 925, 115
  * Hayden et al. (2013) Hayden B. T., Gupta R. R., Garnavich P. M., Mannucci F., Nichol R. C., Sako M., 2013, ApJ, 764, 191
  * Hayden et al. (2021) Hayden B., et al., 2021, ApJ, 912, 87
  * Hinton & Brout (2020) Hinton S., Brout D., 2020, Journal of Open Source Software, 5, 2122
  * Holtzman et al. (2008) Holtzman J. A., et al., 2008, AJ, 136, 2306
  * Howell (2001) Howell D. A., 2001, ApJ, 554, L193
  * Jha et al. (2006) Jha S., et al., 2006, AJ, 131, 527
  * Johansson et al. (2021) Johansson J., et al., 2021, ApJ, 923, 237
  * Jones & Kessler (2018) Jones D., Kessler R., 2018, Saltshaker
  * Jones et al. (2013) Jones D. O., et al., 2013, The Astrophysical Journal, 768, 166
  * Jones et al. (2018) Jones D. O., et al., 2018, ApJ, 867, 108
  * Jones et al. (2019) Jones D. O., et al., 2019, The Astrophysical Journal, 881, 19
  * Jones et al. (2023) Jones D. O., Kenworthy W. D., Dai M., Foley R. J., Kessler R., Pierel J. D. R., Siebert M. R., 2023, ApJ, 951, 22
  * Kelly et al. (2010) Kelly P. L., Hicken M., Burke D. L., Mand el K. S., Kirshner R. P., 2010, ApJ, 715, 743
  * Kelsey et al. (2021) Kelsey L., et al., 2021, MNRAS, 501, 4861
  * Kelsey et al. (2023) Kelsey L., et al., 2023, MNRAS, 519, 3046
  * Kenworthy (2023) Kenworthy W. D. e. a., in prep. 2023
  * Kenworthy et al. (2021) Kenworthy W. D., et al., 2021, SALT3: An Improved Type Ia Supernova Model for Measuring Cosmic Distances (arXiv:2104.07795)
  * Kessler et al. (2009) Kessler R., et al., 2009, PASP, 121, 1028
  * Kessler et al. (2013) Kessler R., et al., 2013, The Astrophysical Journal, 764, 48
  * Kessler et al. (2019) Kessler R., et al., 2019, Monthly Notices of the Royal Astronomical Society, 485, 1171–1187
  * Kim et al. (2018) Kim Y.-L., Smith M., Sullivan M., Lee Y.-W., 2018, ApJ, 854, 24
  * Lampeitl et al. (2010) Lampeitl H., et al., 2010, ApJ, 722, 566
  * Léget et al. (2019) Léget P. F., et al., 2019, SUGAR: An improved empirical model of Type Ia Supernovae based on spectral features (arXiv:1909.11239)
  * Lira et al. (1998) Lira P., et al., 1998, AJ, 115, 234
  * Mandel et al. (2009) Mandel K. S., Wood-Vasey W. M., Friedman A. S., Kirshner R. P., 2009, ApJ, 704, 629
  * Mandel et al. (2020) Mandel K. S., Thorp S., Narayan G., Friedman A. S., Avelino A., 2020, A Hierarchical Bayesian SED Model for Type Ia Supernovae in the Optical to Near-Infrared (arXiv:2008.07538)
  * Marriner et al. (2011) Marriner J., et al., 2011, ApJ, 740, 72
  * Mosher et al. (2014) Mosher J., et al., 2014, ApJ, 793, 16
  * Nugent et al. (1995) Nugent P., Phillips M., Baron E., Branch D., Hauschildt P., 1995, ApJ, 455, L147
  * Pan (2020) Pan Y.-C., 2020, ApJ, 895, L5
  * Pan et al. (2015) Pan Y. C., Sullivan M., Maguire K., Gal-Yam A., Hook I. M., Howell D. A., Nugent P. E., Mazzali P. A., 2015, MNRAS, 446, 354
  * Phillips et al. (1999) Phillips M. M., Lira P., Suntzeff N. B., Schommer R. A., Hamuy M., Maza J., 1999, AJ, 118, 1766
  * Pierel et al. (2022) Pierel J. D. R., et al., 2022, SALT3-NIR: Taking the Open-Source Type Ia Supernova Model to Longer Wavelengths for Next-Generation Cosmological Measurements, doi:10.48550/ARXIV.2209.05594, https://arxiv.org/abs/2209.05594
  * Planck Collaboration et al. (2016) Planck Collaboration et al., 2016, A&A, 594, A13
  * Planck Collaboration et al. (2020) Planck Collaboration et al., 2020, A&A, 641, A6
  * Popovic et al. (2021) Popovic B., Brout D., Kessler R., Scolnic D., Lu L., 2021, The Astrophysical Journal, 913, 49
  * Pruzhinskaya et al. (2020) Pruzhinskaya M. V., Novinskaya A. K., Pauna N., Rosnet P., 2020, MNRAS, 499, 5121
  * Riess et al. (2016) Riess A. G., et al., 2016, ApJ, 826, 56
  * Riess et al. (2018) Riess A. G., et al., 2018, The Astrophysical Journal, 853, 126
  * Riess et al. (2022) Riess A. G., et al., 2022, ApJ, 934, L7
  * Rigault et al. (2013) Rigault M., et al., 2013, A&A, 560, A66
  * Rigault et al. (2015) Rigault M., et al., 2015, ApJ, 802, 20
  * Rigault et al. (2020) Rigault M., et al., 2020, A&A, 644, A176
  * Rodney et al. (2014) Rodney S. A., et al., 2014, The Astronomical Journal, 148, 13
  * Roman et al. (2018) Roman M., et al., 2018, A&A, 615, A68
  * Rose et al. (2019) Rose B. M., Garnavich P. M., Berg M. A., 2019, ApJ, 874, 32
  * Rose et al. (2021) Rose B. M., Rubin D., Strolger L., Garnavich P. M., 2021, ApJ, 909, 28
  * Saunders et al. (2018) Saunders C., et al., 2018, ApJ, 869, 167
  * Scolnic et al. (2018) Scolnic D. M., et al., 2018, ApJ, 859, 101
  * Scolnic et al. (2022) Scolnic D., et al., 2022, ApJ, 938, 113
  * Siebert et al. (2020) Siebert M. R., Foley R. J., Jones D. O., Davis K. W., 2020, MNRAS, 493, 5713
  * Smith et al. (2020) Smith M., et al., 2020, First Cosmology Results using Type Ia Supernovae from the Dark Energy Survey: The Effect of Host Galaxy Properties on Supernova Luminosity (arXiv:2001.11294)
  * Steigerwald et al. (2022) Steigerwald H., Rodrigues D., Profumo S., Marra V., 2022, MNRAS, 510, 4779
  * Sullivan et al. (2010) Sullivan M., et al., 2010, MNRAS, 406, 782
  * Suzuki et al. (2012) Suzuki N., et al., 2012, ApJ, 746, 85
  * Taylor et al. (2021) Taylor G., Lidman C., Tucker B. E., Brout D., Hinton S. R., Kessler R., 2021, Monthly Notices of the Royal Astronomical Society, 504, 4111
  * Taylor et al. (2023) Taylor G., et al., 2023, MNRAS,
  * Tripp (1998) Tripp R., 1998, A&A, 331, 815
  * Wiseman et al. (2020) Wiseman P., et al., 2020, MNRAS, 495, 4040
  * Wiseman et al. (2022) Wiseman P., et al., 2022, MNRAS, 515, 4587

##  Appendix A The Cosmological Impact of UV Ringing in SALT3.LOWMASS

We repeat our cosmological analysis over an abridged rest-frame wavelength
range of $4000<\lambda_{\rm eff}<8000$Å, in order to subtract out any effects
of non-physical ringing in the UV region of our SALT3.LOWMASS surface. The
results reported here will also include the unavoidable effects of fitting
light curve models and distances on fewer data points, although that effect
should impact all surfaces to the same degree.

In Figure 10 we plot the change in fitted $\mu$ between the SALT3.FRAG and
SALT3.LOWMASS/HIGHMASS surfaces. We see that the slight distance-dependent
slope in the recovery of $\mu$ between SALT3.LOWMASS and SALT3.FRAG in our
nominal analysis (Figure 8) is no longer present when using the reduced
wavelength range. We do not observe any such change in slope in the recovery
of $\mu$ between SALT3.HIGHMASS and SALT3.FRAG.

In Table 4, we report the best fit parameters from Equation 1. There are no
statistically significant changes in the recovered parameters between this
analysis and our nominal analysis (Table 2), though we do lose four low-host-
mass SNe and five high-host-mass SNe from our samples due to fitting cuts.

In Table 5, we report the scatter in the Hubble residual from the best-fit
cosmology for each surface. All surfaces see an increase in Hubble residual
scatter compared to the nominal analysis (Table 3). However, the relationships
between the Hubble residual scatter across different surfaces has not changed
significantly from the nominal analysis. While in some cases the Hubble
residual scatter now appears to be worse with SALT3.LOWMASS than SALT3.FRAG
when considering only SNe from low-mass hosts, the statistical significance of
this is low; the Hubble residual scatter decreases by $0.6-0.7\sigma$ when
using SALT3.LOWMASS compared to SALT3.FRAG in the nominal analysis, and
similarly changes by $\pm 0.1\sigma$ in the analysis without UV data.

Figure 10: Comparison of the distance moduli for the DES-SN3YR sample when fit over an abridged range of $4000\leq\lambda_{\rm rest}\leq 8000$Å with SALT3.FRAG (including a mass step) versus SALT3.HIGHMASS (left panel, pink points) and SALT3.LOWMASS (right panel, blue points). We plot the difference in distance moduli on the y-axis. Table 4: The best fit nuisance parameters (from Equation 1) and cosmology parameters for the DES-SN3YR sample when fit with different SALT3 surfaces, when using a reduced wavelength range for light curve fitting. Sample | # SNe fit | $\alpha$ | $\beta$ | $\sigma_{\rm int}$ | $M$ | $\Omega_{M}$ | $\sigma_{\Omega_{M}}^{\rm stat}$ | $w$ | $\sigma_{w}^{\rm stat}$ | $\chi^{2}_{\rm cosmo}$
---|---|---|---|---|---|---|---|---|---|---
SALT3.FRAG | 314 | 0.14 $\pm$ 0.01 | 2.74 $\pm$ 0.12 | 0.12 | -29.98 | 0.33 | 0.02 | -0.96 | 0.05 | 8.3
SALT3.FRAG + Mass Step | 314 | 0.16 $\pm$ 0.01 | 2.76 $\pm$ 0.11 | 0.11 | -29.97 | 0.35 | 0.02 | -0.89 | 0.05 | 7.8
SALT3.HIGHMASS | 124 | 0.13 $\pm$ 0.01 | 2.63 $\pm$ 0.17 | 0.09 | -29.93 | 0.32 | 0.02 | -0.99 | 0.07 | 9.3
SALT3.LOWMASS | 130 | 0.08 $\pm$ 0.02 | 2.84 $\pm$ 0.18 | 0.11 | -30.09 | 0.35 | 0.02 | -0.88 | 0.09 | 11.8
Table 5: Scatter in the Hubble residual (from best-fit cosmology) for each
SALT3 surface, when using a reduced wavelength range for light curve fitting..
All values are in units of magnitude. The best-fit cosmology is unique for
each entry in the “Result" column Hubble residuals from best-fit cosmology
---
Result | Subsample | RMSHR | Weighted RMSHR
| Full | 0.169$\pm$0.008 | 0.158$\pm$0.008
| $M_{\log}>10$ | 0.154$\pm$0.009 | 0.145$\pm$0.010
SALT3.FRAG | $M_{\log}<10$ | 0.188$\pm$0.012 | 0.174$\pm$0.014
| Full | 0.171$\pm$0.008 | 0.158$\pm$0.008
| $M_{\log}>10$ | 0.157$\pm$0.010 | 0.147$\pm$0.010
SALT3.FRAG \+ Mass Step | $M_{\log}<10$ | 0.188$\pm$0.012 | 0.173$\pm$0.014
SALT3.HIGHMASS | $M_{\log}>10$ | 0.146$\pm$0.012 | 0.132$\pm$0.013
SALT3.LOWMASS | $M_{\log}<10$ | 0.19$\pm$0.013 | 0.172$\pm$0.014