A Spectroscopic Model of the Type Ia Supernova–Host-galaxy Mass Correlation from SALT3

Fundamentally, each of these host-galaxy step measurements have similar methodologies: they generate Hubble residuals using a model that does not include host-galaxy properties and then subsequently explore the ways in which those Hubble residuals correlate with host properties. Recent work to build the SALT3 model for SN Ia standardization (Kenworthy et al. 2021, hereafter K21) has created a new opportunity to study the fine-grained correlations between SN Ia spectra with their host galaxies by incorporating host properties in the generation of the model itself.

To enable this work, K21 created an open-source training code called SALTshaker and added several improvements to the SALT2 model framework (Guy et al. 2007, 2010; Betoule et al. 2014; Taylor et al. 2021) that has been used to measure SN Ia light-curve parameters for nearly all measurements of dark energy properties in the past decade. SALTshaker constructs a SALT3 model by finding flux surfaces as a function of phase and wavelength that are associated with the zeroth and first principal components of SN Ia variation, where the zeroth component is a mean SN Ia SED and the first component correlates strongly with the SN Ia light-curve stretch. Simultaneously, it infers a phase-independent color law to model the effects of reddening. Dai et al. (2022) validated the model training framework, Taylor et al. (2023) tested the model's performance against SALT2, and the model was extended to the NIR in Pierel et al. (2022).

Here, we retrain the SALT3 model after adding a "host-galaxy mass" flux component as a function of phase and wavelength to the model framework. We train two models: first, we train nominal SALT3 model surfaces and subsequently add a host-galaxy component while keeping the baseline parameters and surfaces fixed; in this case, the host-galaxy component is a model of the "missing" features from our current SN Ia standardization paradigm that vary with host-galaxy mass. Second, we train all model surfaces simultaneously, and the host-galaxy component is defined to be the difference between the mean SN Ia SED in a high- versus low-mass galaxy. The goal of this work is to understand the properties of the SNe Ia that could drive the host-galaxy mass step, and develop a framework for building a wavelength- and phase-dependent understanding of the ways in which the SN–host connection could bias cosmological parameter measurements.

In Section 2 we describe the revised model framework, including alterations to the training sample and our host-galaxy mass measurements. In Section 3, we present the trained model surfaces as well as the resulting Hubble residuals and mass-step measurements. In Section 4, we discuss future steps, and in Section 5 we conclude.

Here, we build a new version of SALT3 that includes an explicit dependence on host-galaxy mass. To include host-galaxy properties in the model framework, we add a host-galaxy parameter, x_host, and model surface, M_host:

$$\begin{eqnarray}\begin{array}{rcl}F(p,\lambda ) & = & {x}_{0}[{M}_{0}(p,\lambda )+{x}_{1}{M}_{1}(p,\lambda )\\ & & +\,{x}_{\mathrm{host}}{M}_{\mathrm{host}}(p,\lambda )]\cdot \exp (c\cdot \mathrm{CL}(\lambda )).\end{array}\end{eqnarray} \tag{ 2 }$$

In the current analysis, the host-galaxy parameter is simply chosen to be +1/2 for a SN in a host galaxy with $\mathrm{log}({M}_{* }/{M}_{\odot })\gt 10$ and −1/2 for a SN in a host with $\mathrm{log}({M}_{* }/{M}_{\odot })\lt 10$. This choice allows the M_host surface to be interpretable as the difference between a mean high-mass-hosted SN versus a mean low-mass-hosted SN. We use the canonical location of the mass step of $\mathrm{log}({M}_{* }/{M}_{\odot })=10$ (e.g., Sullivan et al. 2010 and subsequent studies), even though the median host-galaxy mass from the data presented in Section 2.2 is approximately 10.3 dex; this choice reduces the signal-to-noise ratio (S/N) of our resulting mass-step component, but it ensures that our results will have direct relevance to the mass step measured in cosmological analyses.

We found that the uncertainties on the host-galaxy masses are sufficiently small that the effect of including them would be negligible, but a future framework that includes the uncertainty on host-galaxy parameters could be necessary for incorporating additional host-galaxy properties into the model.

We use this formalism to train a SALT3 model that includes host-galaxy mass information in two ways, described below:

• 1.  
  We use our low-z training data (Section 2.2) to determine M₀, M₁, and CL assuming the original SALT model formalism (Equation 1). Then, while keeping M₀, M₁, and the color law fixed, we add the host component to the SALT model formalism and train again. The derived M_host component contains only spectroscopic and photometric features that depend on host mass and are "missing" from existing SALT models. We refer to this model as SALT3.HostResid.
• 2.  
  We train the full model, allowing M₀, M₁, and CL to vary simultaneously with M_host. In this case, the derived M_host component is the change in the mean spectrum of a SN Ia between low- and high-mass host galaxies. Additional host-independent variability is captured by a redefined x₁ parameter and M₁ surface. We refer to this model as SALT3.Host.

These models are compared to the SALT3.HostIgnore model, which is a model trained using the traditional SALT formalism (Equation 1) with the exact same training data as SALT3.Host and SALT3.HostResid.

For the SALT3.Host model, to separate the effects of host-galaxy mass from the component of intrinsic variability that is independent of the host properties, we include a constraint to set the correlation between x₁ and x_host to zero. ⁸ This is similar to the existing SALT3 constraint that the correlation between x₁ and c is zero in the training sample; the x₁, c constraint has the physically intuitive meaning that if the color is related to dust extinction, it should not depend on the stretch.

This x₁, x_host constraint redefines the x₁ parameter as the phase-dependent SN variability that is uncorrelated with the host-galaxy mass. Correspondingly, M_host can be interpreted as the difference in the mean host-galaxy spectrum between a SN Ia in a high-mass versus a low-mass host galaxy.

As the SALT3 training is cosmology independent, due to including no prior on the amplitude parameter x₀, the overall amplitude of M_host is degenerate with x₀. Therefore, we add offsets to both M₁ and M_host such that they have zero flux when integrated over the B bandpass at maximum light. We note that without this definition the size of the measured mass step from this model would be directly correlated with the B-band luminosity of the mass component; the M_host luminosity can be defined such that it removes the mass step entirely for a given data set, but the M_host luminosity cannot be determined by SALTshaker itself due to the degeneracy between x_host and x₀ for each SN.

Finally, the nominal SALT3 error model includes the in-sample and out-of-sample variances for M₀ and M₁ as well as the diagonal covariance between M₀ and M₁. We add to this model the variance of M_host and the diagonal covariance between M₀ and M_host (M_host and M₁ are defined as described above to have negligible covariance with each other). Because we found some evidence that SALTshaker overestimates the model uncertainties when trained on a small sample, perhaps due to its regularization assumptions, we used a simplified approach for uncertainty computation for the out-of-sample uncertainties in this model: we bootstrap resampled the input SNe 50 times and retrained the model on each new sample after reducing the regularization terms by a factor of 100.

To build the SALT3.HostResid and SALT3.Host models, we restrict to training data at redshifts z < 0.15 to ensure that host-galaxy masses can be measured robustly. This allows us to include Galaxy Evolution Explorer (GALEX) and Two Micron All Sky Survey (2MASS) photometry for the SN host galaxies, which are not deep enough to detect many higher-redshift galaxies, and it avoids potential concerns of biases in high-z mass estimates (e.g., Paulino-Afonso et al. 2022). This choice adds significant noise to the model training below ∼3500 Å due to the lack of well-calibrated near-UV data from high-redshift surveys—and perhaps avoids systematic redshift-dependent differences at those wavelengths (Foley et al. 2012)—but allows more robust mass estimates.

We also update the photometry of the training data relative to K21 to use the "SuperCal-Fragilistic" cross-calibration update to these photometry presented in Brout et al. (2022b); this model is published in Taylor et al. (2023). SuperCal-Fragilistic uses Pan-STARRS (PS1) observations of tertiary standard stars employed by different SN surveys, along with updates to the Hubble Space Telescope absolute calibration from Bohlin et al. (2020), to put individual SN surveys on a common photometric system. We also update the Carnegie Supernova Project (CSP) photometry to use the third CSP data release (DR3; Krisciunas et al. 2017); K21 used CSP Data Release 2 to match the cross-calibration analysis done by Scolnic et al. (2015), whereas Brout et al. (2022b) uses DR3. A full SALT3 model using the recalibrated K21 data is publicly available as part of the sncosmo package (Barbary et al. 2021) as well as SNANA (Kessler et al. 2010).

A summary of the SALT3.Host training data is given in Table 1. We do not attempt to extend this model to the NIR (e.g., Pierel et al. 2022) because one of our primary goals is to understand spectral variation and there are currently just 51 publicly available NIR spectra of SNe Ia that have well-sampled NIR light curves.

Table 1. The SALT3 Host-galaxy Training Sample (z < 0.15)

	$\mathrm{log}({M}_{* }/{M}_{\odot })\lt 10$		$\mathrm{log}({M}_{* }/{M}_{\odot })\gt 10$
Survey	NSN	Nspectra	NSN	Nspectra	Filters	References
Calan-Tololo	1	0	4	0	BV RI	Hamuy et al. (1996)
CfA1	1	7	7	59	U BV RI	Riess et al. (1999)
CfA2	4	70	9	96	U BV RI	Jha et al. (2006)
CfA3	12	135	39	399	U BV RIri	Hicken et al. (2009)
CfA4	9	0	21	0	BV ri	Hicken et al. (2012)
CSP	3	5	10	31	uBV gri	Krisciunas et al. (2017)
Foundation	70	52	81	60	griz	Foley et al. (2018)
Misc. low-z	2	9	23	207	U BV RI	Jha et al. (2007)
Total	102	278	194	852	⋯	⋯

Download table as:  ASCIITypeset image

We measure host-galaxy masses for SNe in our training sample by first using the Galaxies HOsting Supernova Transients (GHOST; Gagliano et al. 2021) software to determine the host galaxy for each SN. GHOST uses a gradient-ascent algorithm to match SN locations to their likely host galaxies using deep PS1 postage-stamp images at each SN location. Next, we visually inspect the GHOST matches, and alternatively use the SExtractor-based "directional light radius" (DLR) method (Sullivan et al. 2006; Gupta et al. 2016) on r-band images from PS1 (Chambers et al. 2016) to correct the few percent of erroneous matches. Finally, for SNe with no identifiable host either visually or with GHOST, we assume the true host is at the SN location; at z < 0.15, hosts that are undetected in PS1 imaging will all have $\mathrm{log}({M}_{* }/{M}_{\odot })\lt 10$, but we use their location for photometric measurements in case upper limits on the mass are needed for future work.

With these host-galaxy locations, we use SExtractor (Bertin & Arnouts 1996) to measure the isophotal elliptical radius of each galaxy, again using PS1 r-band imaging. We measure photometry within this radius using images from GALEX, the Sloan Digital Sky Survey (SDSS), PS1, and 2MASS, modifying the radius slightly to account for the changing size of the point-spread function (PSF) for each filter and instrument (Table 2).

Finally, we use LePHARE (Ilbert et al. 2006) to measure the host-galaxy masses of each SN. We used the Chabrier (2003) initial mass function and the Bruzual & Charlot (2003) SED template library. The extinction, E(B − V), is varied from 0 to 0.4 mag. The resulting host-galaxy mass distributions for each subsample in our analysis are shown in Figure 1.

Zoom In Zoom Out Reset image size

We first examine the SALT3.Host model, and find that spectra at maximum light have significantly different Ca H and K and Si ii (6355 Å) line profiles as a function of host-galaxy mass. In particular, SNe in low-mass hosts have broader absorption features on average (Figure 2); at maximum light, we measure larger Ca NIR triplet, Ca H and K, and Si ii equivalent widths at 2.7, 2.3, and 2.2σしぐま significance, respectively, which are correlated with higher velocities and explosion energies per unit of SN mass (e.g., Wang et al. 2009; Zhao et al. 2015). ⁹ The Ca H and K velocity is higher in low-mass hosts, but at just 1.3σしぐま significance, and there is no difference in the Si ii velocity. ¹⁰ While the significance of these results could be somewhat enhanced by the look-elsewhere effect, we note that previous studies have specifically investigated differences in Si ii and Ca H and K absorption features as potentially correlated with Hubble residuals (Foley & Kasen 2011; Siebert et al. 2020), so we have a priori reasons to suspect host-dependent changes in these line profiles.

Additional spectral differences are observed at later phases, particularly in the Si ii features (Figures 4 and 6). The differences in light-curve properties are not limited to places where spectral features are expected to vary (e.g., we observe qualitatively similar behavior at ∼4500–5000 Å), but variation in these Si ii and other absorption features could explain some of the variation present in the broadband light curves. We also see other differences between the high-mass host galaxy and low-mass host galaxy SN models at 25–35 days (Figures 3 and 6), near the time of NIR maximum light, which is an epoch that could be particularly sensitive to the metallicity of the SN progenitor (Kasen 2006).

In Figure 3, we also confirm previous results that SNe Ia in high-mass hosts have narrower shapes on average than those in low-mass hosts (e.g., Hamuy et al. 2000; Gallagher et al. 2005), as well as an earlier time of secondary maximum light (Figure 3). Significant phase-dependent color differences as a function of host mass can be seen (Figure 8), which were previously captured as part of the SALT3 M₁ component and represented by the SALT3 x₁ component in light-curve fits.

Lastly, we note that the updated M₁ component appears qualitatively similar to the SALT3.HostIgnore model, but with larger-amplitude x₁-dependent color changes, particularly near maximum light (these are phase-dependent colors and can therefore be separated from the phase-independent color law).

We compare these SALT3.Host results to light-curve residuals and spectra from the SALT3.HostResid model in Figures 3 and 4 (bottom panels) and Figure 7. Near maximum light, we see that previous SALT models have not fully accounted for variations in the Ca H and K and Si ii absorption features between different host-galaxy types. There is also a slight possibility of increased amplitude of the host-mass component in the I band, indicated by the persistent low-significance offset in the bottom-right panel of Figure 3. Overall, however, the pre-existing M₁ surface had encoded most of the host-dependent variation in SN properties, and the amplitudes of the spectral changes in Figure 7 are significantly smaller than those in Figure 6.

3.1.1. Colors

By using the Bayesian information criterion with the log-likelihood returned by SALTshaker, we find that the SALT3.Host model is heavily favored over the SALT3.HostIgnore model, which is trained using the original SALT formalism on the same data set. However, we find that the likelihood of the photometric data alone changes less significantly between SALT3.HostIgnore and SALT3.Host, as discussed in Section 3.3 below.

To ensure that host-galaxy-dependent differences are not due to systematic errors in the SALTshaker training procedure, we use randomly assigned masses to train "random" versions of the SALT3.Host model, finding that only low-significance changes in spectral features are recovered. Those results are presented in the Appendix.

Next, we use the SNANA software (Kessler et al. 2010) to fit the photometric training data with the SALT3.Host, SALT3.HostResid, and SALT3.HostIgnore models. ¹¹ We apply selection cuts on redshift, x₁, and c (z > 0.01, −3 < x₁ < 3, and −0.3 < c < 0.3, following Scolnic et al. 2018), after which 261 of the original 296 SNe remain for measuring distances.

We subsequently estimate nuisance parameters and distances from the Tripp estimator (Tripp 1998):

$$\begin{eqnarray}&&\mu ={m}_{B}+\alpha \cdot {x}_{1}-\beta \cdot c-{ \mathcal M }.\end{eqnarray} \tag{ 3 }$$

The αあるふぁ and βべーた variables are nuisance parameters that standardize the SN brightness and are determined in a global fit using SNANA's SALT2mu method (Marriner et al. 2011). ${ \mathcal M }$ is the absolute magnitude of a SN Ia, which is set to a nominal value in this work, and m_B is the log of the fitted light-curve amplitude. The mass step, Δでるた_M , can be included in this equation, but we treat it separately in this work. Additionally, bias corrections to correct the measured μみゅー for selection effects as a function of redshift are commonly included in the Tripp estimator but are not necessary here; we only examine the differential distance measurements between models and therefore this term cancels to first order.

We report the measured nuisance parameters αあるふぁ, βべーた, and Δでるた_M in Table 3. Each parameter is computed both for the complete sample and for SNe in high-mass and low-mass host galaxies separately. We also report the rms dispersion of Hubble residuals for each model. We see statistically insignificant differences of just ∼0.01 in the αあるふぁ parameter and ∼0.005 in the βべーた parameter between SALT3.Host and SALT3.HostIgnore. Between low-mass and high-mass hosts, however, we see marginally significant differences of ∼0.015–0.02 in αあるふぁ and ∼0.35 in βべーた, suggesting that fitting αあるふぁ and βべーた separately for these samples might reduce distance uncertainty and perhaps cosmological bias (Kelsey et al. 2023). After subtracting the ΛらむだCDM cosmological model prediction from Planck Collaboration et al. (2020), we see a consistent Hubble residual rms for all three models.

Table 3. SALT3 Light-curve Fitting χかい², Hubble Residual rms, and Nuisance Parameter Comparisons

Version	HR rms	LC χかい2/νにゅーa	αあるふぁ	βべーたb	ΔでるたM
	(mag)				(mag)
SALT3.HostIgnore	0.145	1.49	0.133 ± 0.009	2.675 ± 0.092	0.057 ± 0.018
− $\mathrm{log}({M}_{* }/{M}_{\odot })\lt 10$	0.140	1.49	0.133 ± 0.016	2.900 ± 0.140	⋯
− $\mathrm{log}({M}_{* }/{M}_{\odot })\gt 10$	0.147	1.49	0.149 ± 0.011	2.552 ± 0.119	⋯
SALT3.HostResid	0.148	1.48	0.137 ± 0.009	2.616 ± 0.093	0.032 ± 0.019
− $\mathrm{log}({M}_{* }/{M}_{\odot })\lt 10$	0.151	1.47	0.124 ± 0.015	2.828 ± 0.145	⋯
− $\mathrm{log}({M}_{* }/{M}_{\odot })\gt 10$	0.147	1.50	0.153 ± 0.011	2.482 ± 0.119	⋯
SALT3.Host	0.144	1.45	0.132 ± 0.009	2.659 ± 0.090	0.036 ± 0.019
− $\mathrm{log}({M}_{* }/{M}_{\odot })\lt 10$	0.141	1.44	0.125 ± 0.020	2.910 ± 0.134	⋯
− $\mathrm{log}({M}_{* }/{M}_{\odot })\gt 10$	0.146	1.49	0.137 ± 0.011	2.549 ± 0.119	⋯

Notes.

^a Neglecting model uncertainties and color dispersion terms, which are estimated such that each model has a reduced χかい² ≃ 1. To avoid biases to the χかい², we add a 0.1% error floor to the data uncertainties and omit z- and U/u-band data because they have significantly higher model uncertainties than the rest of the wavelength range. ^b With SALT2.4 (Betoule et al. 2014), we measure a consistent βべーた = 2.73 ± 0.099 from this sample, slightly lower than typical analyses including high-z SNe (βべーた ≃ 3; Scolnic et al. 2018).

Download table as:  ASCIITypeset image

To compare SALT3.Host to the previously published SALT3 model (K21), we show the x₁ and c parameters measured with each model in Figure 9. The addition of a host-galaxy mass component changes the definition of x₁ such that there is a much lower mean difference in x₁ between high- and low-mass host galaxies for SALT3.Host (≃0.17) compared to the SALT3 model from K21 (≃0.91; hereafter SALT3.K21). ¹² The marginally significant host-dependent variation in the αあるふぁ parameter (0.015–0.02) seems to suggest that the x₁–luminosity relation is changing slightly between different host types.

Zoom In Zoom Out Reset image size

Interestingly, the mean color of SNe in high-mass hosts is slightly more red (≃0.05) in SALT3.Host compared to SALT3.K21 (≃0.03; Figure 9). This may mean that some variation in the dust color law or the intrinsic colors of SNe that was not previously modeled may have been captured by the new M_host component.

3.3.1. The Host-galaxy Mass Step

We measure the host-galaxy mass step by following the approach outlined in Jones et al. (2018) that incorporates both Hubble residual and host-galaxy mass uncertainties. Jones et al. (2018) construct a model to find the maximum-likelihood value of the step by assuming two Gaussian SN populations in Hubble residual space; the means and dispersions of these Gaussians are free parameters (see also Rigault et al. 2015 and Jones et al. 2015). The model takes into account that each SN has a probability of being in either a high-mass or low-mass host galaxy, which is given by the probability distribution function of the host-galaxy mass measurement.

From the SALT3.HostIgnore model, we measure a mass step of 0.057 ± 0.018 mag versus 0.036 ± 0.019 mag from the SALT3.Host model (Figure 10). After recomputing the mass step with 100 bootstrap iterations of the SN sample, which takes into account the fact that the same data are used for both mass-step measurements, we find that the mass step is reduced by a highly significant 0.021 ± 0.002 mag when the SALT3.Host model is applied.

Zoom In Zoom Out Reset image size

Even with the SALT3.Host model, the mass step remains significant at the ∼2σしぐま level. This implies a significant difference in the shape- and color-corrected B-band absolute magnitude of SNe Ia as a function of host mass even after explicit modeling of the SN Ia–host mass relation in a luminosity-independent way. Given the low uncertainties on the mass-step difference above, we consider it likely that larger samples would measure the mass step with high statistical significance even when using the SALT3.Host model.

In Figure 11, we show the difference in Hubble residuals between the SALT3.Host and SALT3.HostIgnore models. Both the SALT3.Host and SALT3.HostResid models change the Hubble residuals by shifting the measured color parameter c (Figure 9); this appears to be because some color differences previously attributed to the global, phase-independent color are now determined by SALTshaker to be dependent on host properties. Additionally, the SALT3.Host model changes the measurement of x₁ and therefore some x₁ differences are now modeled as part of the host component. The measured m_B does not change significantly in either model, which is unsurprising given that m_B is defined to be the same at maximum light for both models.

Zoom In Zoom Out Reset image size

These results show that our host-dependent SN Ia model alters the determinations of color and first-order variability, which in turn causes the mass step to shrink. These redefined terms appear to improve the Hubble residual scatter very slightly, and we find specifically that there is less Hubble residual scatter due to the αあるふぁ x₁ or βべーた c terms in the Tripp equation when applying the SALT3.Host framework (a small ∼1%–2% improvement). Our distance fitting and mass-step results are summarized in Figure 10 and Table 3.

3.3.2. Light-curve Fitting Residuals

3.3.3. The Wavelength Dependence of the Mass Step

Finally, we explore the wavelength dependence of the mass step by performing a global mass-step fit to the SN data using the SALT3.K21 model, the SALT3.HostResid model, and the SALT3.Host model. Next, we use the global x₁ and c parameters measured using a fit to all bands for each SN, but generate amplitudes (x₀) from the V/g bands, the R/r bands, and the I/i bands separately. The goal is to see if the mass step changes significantly in different bands, with the different models explored here, and within the wavelengths covered by the SALT model. The results are summarized in Figure 12. ¹⁴

Zoom In Zoom Out Reset image size

For the K21 model, the mass step is consistent with no wavelength dependence, but we cannot rule out significant variation. Although well within the statistical errors, the mass step is very slightly larger in the R/r and I/i bands than it is in the V/g bands, as would be predicted from Figure 3. Counterintuitively, a larger mass step at red wavelengths is consistent with a lower R_V in high-mass hosts, as the amplitude of a SN in these redder bands is extrapolated to the B band assuming the host-independent SALT3.K21 model. We also see again that the addition of the host-galaxy mass component reduces the mass step by ∼0.02 mag; as the mass component is normalized to be equal to zero at B-band maximum light, this reduction cannot be due to luminosity information encoded in the mass step (e.g., dust extinction differences or other physical mechanisms that change the B-band SN luminosity).

We also investigate the dependence of the Hubble residual rms on the color parameter c to see if the trend of increased color scatter with c observed by Brout & Scolnic (2021) is altered by applying the SALT3.Host model. We see no significant change, with both sets of distance residuals from SALT3.HostIgnore and SALT3.Host having an rms that increases from ∼0.13 mag at c ≃ −0.05 to ∼0.16 mag at c ≃ 0.1, ¹⁵ indicating that adding a single host-mass component does not significantly affect the increased color scatter of red SNe. This result is consistent with the Brout & Scolnic (2021) model, as our correction to the mean of the flux would be incapable of substantially addressing the variance caused by a large diversity of extinction laws.

The measured βべーた parameter between low- and high-mass hosts differs by ∼0.35 across all models, with high-mass-hosted SNe preferring lower values of βべーた. This is again consistent with the prediction of lower R_V in high-mass hosts (Brout & Scolnic 2021).

Given the uncertainty in SN Ia physics and progenitor systems, a model that depends only on the host-galaxy mass is unlikely to be truly optimal for cosmological distance fitting. Alternative SALT3 models that depend on specific star formation rate, progenitor age, or other properties could be used within the SALT3.Host framework. However, all of these models will be subject to uncertainty regarding whether they remain valid as a function of redshift as SN Ia progenitors and host-galaxy dust properties evolve. A logical additional extension would be to incorporate the full host-galaxy parameter probability distribution functions as priors into the model training, allowing a model to be trained as a function of galaxy properties that have larger uncertainties.

Given that the SALT3.Host does not yield clear differences in Hubble residuals or improvements in dispersion, and only results in a small improvement in the quality of the light-curve fits themselves, it may be appropriate to continue using the simpler model formalism of the SALT3.HostIgnore model for cosmology. However, for measuring w, the leverage that the SALT3.Host model provides may be extremely useful as a systematic test that helps us understand the wavelength ranges, redshifts, and filter sets that are particularly sensitive to the observed spectral variations as a function of host-mass properties. As the model itself can be trained at any redshift, it is also possible to measure whether the SN Ia–host mass correlation is evolving with redshift as a function of both wavelength and phase.

Finally, the lack of a large difference in the goodness-of-fit statistic when comparing SALT3.Host to SALT3.HostIgnore is somewhat surprising given that both SN Ia spectra and light curves vary significantly as a function of their host-galaxy properties. Given that spectra are more homogeneous in particular host-galaxy types, this result may limit the precision one could expect from a "twinning" approach in which SNe with similar spectra are compared to yield more precise relative distance estimates (e.g., Fakhouri et al. 2015; Boone et al. 2021b). However, it is possible that modeling the SN spectra as a function of alternative global or local host-galaxy properties might result in more drastic improvements to Hubble residuals.

The SALT3.Host model gives a more precise understanding of the ways in which a mean SN Ia spectrum changes depending on its host galaxy. Though the dependence of SN Ia properties on host-galaxy properties is already well known, by constraining the host dependence at the same time as the principal-component-like M₀ and M₁ surfaces and phase-independent color law, our model is constructed in such a way that it separates the component of SN Ia intrinsic diversity uncorrelated with host mass from the change in mean properties as a function of host mass.

At 1.3σしぐま significance, we find evidence that the Ca H and K velocity is less blueshifted in higher-mass hosts, indicating at low significance—and contrary to results from composite spectra in Siebert et al. (2019)—that SNe in higher-mass host galaxies may have lower explosion energies per unit mass (the difference could be due to our more robust prescription of controlling for intrinsic variation when comparing high- to low-mass-hosted SNe). Similar evidence is seen from the larger observed equivalent widths in the Ca NIR triplet, Ca H and K, and Si ii line profiles, at 2.7, 2.3, and 2.2σしぐま significance. At late times, we note that Siebert et al. (2020) found evidence for larger equivalent widths in spectral features for SNe with positive Hubble residuals (more likely to be low-mass hosts), which aligns with our results from spectra nearer to peak. Siebert et al. (2019) also saw hints of increased Ca H and K and Si ii absorption in late-type galaxies, which is consistent with our findings. Finally, we see that the I-band secondary maximum may be slightly earlier relative to the peak flux, which would indicate a larger synthesized mass of electron-capture elements in high-mass hosts (Kasen 2006). As a caveat, we note that the Ca H and K line may be blended with Si ii at 3858 Å (Foley 2013), which may have some effect on the Ca H and K results.

Building this model as a function of other host-galaxy parameters may help to isolate additional physical processes in SN Ia explosions that depend on the metallicity or progenitor properties that are more prevalent in certain galaxy types. We hope that this approach will be effective at isolating subtle SN Ia physical behaviors with the use of large samples. We also hope that future investigations can continue to explore the ways in which differences in, for example, explosion energies, could affect the spectra, light curves, and luminosities of SNe in cosmological samples.

4.2.1. Constraining the Role of Dust in the SN Ia Mass Step

The training sample used here is subject to significant selection effects. In particular, approximately half the data are comprised of samples that were chosen from a preselected set of bright galaxies; controlling for the mass step alone may not sufficiently remove the biases in this sample as a function of host-galaxy properties. The way in which the demographics of those data could affect the recovered spectral model are difficult to fully understand from our limited data set; however, these data are the same as those used in other recent SALT model trainings (e.g., Betoule et al. 2014; K21; Taylor et al. 2021), meaning that these types of biases are already present in existing models used for SN Ia cosmology studies. ¹⁶

Despite these selection effects, spectroscopic data are important for models that explore the spectroscopic dependence of the SALT3 model on host-galaxy properties, and there exists no high-S/N sample of SN spectra comparable to the one that has been assembled over the last three decades at low redshift. Replacing this sample with future, less biased spectroscopic data sets will take considerable work.

SALTshaker also does not yet have a process to correct the data or model for observational biases, which could potentially be addressed by a fitting process that includes an iteratively updated, simulation-based bias-correction process to refine, for example, x₁ and c estimations and correct the resulting model surfaces for biases in those parameters. Understanding the role that such biases could play in the training process are an important consideration for future work (Dai et al. 2022).

Lastly, our data set remains limited redward of ∼7000–8000 Å. The Foundation sample alone has rest-frame z-band data to constrain, for example, the SN model in the vicinity of the Ca ii NIR triplet. However, ongoing surveys such as the Young Supernova Experiment (Jones et al. 2021) and future surveys including the Rubin Observatory's Legacy Survey of Space and Time will supplement these data and—with additional spectroscopic follow-up observations—allow a better understanding of spectroscopic behavior at longer wavelengths. Additionally, recent SN Ia NIR model extensions and spectroscopic data releases (Lu et al. 2023; Pierel et al. 2022) will help to constrain the model's dependence on host-galaxy properties at YJH wavelengths in the near future.