# SALT3: An Improved Type Ia Supernova Model for Measuring Cosmic Distances

W. D. Kenworthy Department of Physics and Astronomy, The Johns Hopkins
University, Baltimore, MD 21218.  D. O. Jones NASA Einstein Fellow  Department
of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064,
USA  M. Dai Department of Physics and Astronomy, The Johns Hopkins University,
Baltimore, MD 21218.  Department of Physics and Astronomy, Rutgers, The State
University of New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854, USA
R. Kessler  University of Chicago, Kavli Institute for Cosmological Physics,
Chicago, IL, USA.  D. Scolnic  Department of Physics, Duke University, Durham
North Carolina 27708, USA  D. Brout  NASA Einstein Fellow  Harvard-Smithsonian
Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA  M. R.
Siebert  Department of Astronomy and Astrophysics, University of California,
Santa Cruz, CA 95064, USA  J. D. R. Pierel  Department of Physics and
Astronomy, University of South Carolina, 712 Main Street, Columbia, SC 29208,
USA  Now at Space Telescope Science Institute  K. G. Dettman Department of
Physics and Astronomy, Rutgers, The State University of New Jersey, 136
Frelinghuysen Road, Piscataway, NJ 08854, USA  G. Dimitriadis  Department of
Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064,
USA  R. J. Foley Department of Astronomy and Astrophysics, University of
California, Santa Cruz, CA 95064, USA  S. W. Jha Department of Physics and
Astronomy, Rutgers, The State University of New Jersey, 136 Frelinghuysen
Road, Piscataway, NJ 08854, USA  Y.-C. Pan Graduate Institute of Astronomy,
National Central University, 300 Zhongda Road, Zhongli, Taoyuan 32001, Taiwan
A. Riess  Department of Physics and Astronomy, The Johns Hopkins University,
Baltimore, MD 21218.  Space Telescope Science Institute, Baltimore, MD 21218.
S. Rodney  Department of Physics and Astronomy, University of South Carolina,
712 Main Street, Columbia, SC 29208, USA  C. Rojas-Bravo Department of
Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064,
USA W. D. Kenworthy<EMAIL_ADDRESS>D. O. Jones<EMAIL_ADDRESS>
###### Abstract

A spectral-energy distribution (SED) model for Type Ia supernovae (SNe Ia) is
a critical tool for measuring precise and accurate distances across a large
redshift range and constraining cosmological parameters. We present an
improved model framework, SALT3, which has several advantages over current
models including the leading SALT2 model (SALT2.4). While SALT3 has a similar
philosophy, it differs from SALT2 by having improved estimation of
uncertainties, better separation of color and light-curve stretch, and a
publicly available training code. We present the application of our training
method on a cross-calibrated compilation of 1083 SNe with 1207 spectra. Our
compilation is $2.5\times$ larger than the SALT2 training sample and has
greatly reduced calibration uncertainties. The resulting trained SALT3.K21
model has an extended wavelength range $2000$-$11000$ Å (1800 Å redder) and
reduced uncertainties compared to SALT2, enabling accurate use of low-$z$ $I$
and $iz$ photometric bands. Including these previously discarded bands,
SALT3.K21 reduces the Hubble scatter of the low-z Foundation and CfA3 samples
by 15% and 10%, respectively. To check for potential systematic uncertainties
we compare distances of low ($0.01<z<0.2$) and high ($0.4<z<0.6$) redshift SNe
in the training compilation, finding an insignificant $2\pm 14$ mmag shift
between SALT2.4 and SALT3.K21. While the SALT3.K21 model was trained on
optical data, our method can be used to build a model for rest-frame NIR
samples from the Roman Space Telescope. Our open-source training code, public
training data, model, and documentation are available at
https://saltshaker.readthedocs.io/en/latest/, and the model is integrated into
the sncosmo and SNANA software packages.

††software: AstroPy (Astropy Collaboration et al., 2013, 2018),Astroquery
(Ginsburg et al., 2019),extinction(Barbary, 2016),iMinuit (James & Roos, 1975;
Dembinski et al., 2020), Matplotlib (Hunter, 2007), NumPy (Harris et al.,
2020),SciPy (Virtanen et al., 2020), sncosmo (Barbary et al., 2015, 2016),
tqdm (da Costa-Luis et al., 2021)

tablenum SIXtablenum

##  1 Introduction

Type Ia supernovae (SNe Ia) have been used as cosmological distance indicators
for more than two decades, providing early evidence of the accelerating
expansion of the Universe (Riess et al., 1998; Perlmutter et al., 1999).
Today, SN Ia distances are used at low redshift ($z\leq 0.15$) for distance
ladder measurements of the Hubble constant (H0; Riess et al., 2021; Freedman
et al., 2019), currently the subject of a $4-6\sigma$ tension (see Verde et
al., 2019 for a review), as well as measurements of the dark energy equation-
of-state parameter, $w$ which incorporate SNe at $z\leq 2.2$ (currently
consistent with $w=-1$; Scolnic et al., 2018; Abbott et al., 2019; Jones et
al., 2019). Recent measurements of $H_{0}$ (Riess et al., 2016), as well as
most large studies of SNe across the observed redshift range for the last
decade (Guy et al., 2010; Conley et al., 2011; Sako et al., 2018; Betoule et
al., 2014; Riess et al., 2018; Scolnic et al., 2018; Brout et al., 2019a;
Jones et al., 2019), have relied upon the SALT2 light-curve model (Guy et al.,
2007, 2010) for the brightness standardization of SNe Ia in their analysis.

SN Ia distances are typically estimated by fitting their light curves with a
model to determine an overall flux, a color, and one (or more) light-curve
shape parameters. The apparent magnitude (as computed from the flux) is
standardized with a linear combination of color and light-curve parameters
(referred to as the Tripp estimator; Tripp, 1998) to produce a standardized
apparent magnitude relative to a fiducial SN Ia. The SALT2 ( Spectral Adaptive
Light-curve Template) model describes SN Ia light curves as a combination of
component spectral energy distributions (flux surfaces defined in wavelength
and time), multiplied by a color-dependent term described by a color law that
is similar to that of the Milky Way. These components are determined through a
“model training” process; the last trained model to be used in a published
cosmological analysis was SALT2.4 (which we hereafter refer to as SALT2.JLA111
JLA refers to ”Joint Lightcurve Analysis” that included the SDSS-SN and SNLS
teams, and produced cosmology results in Betoule et al. (2014) ), presented in
Betoule et al. (2014) and Mosher et al. (2014), although a retrained model has
recently been presented in Taylor et al. (2021). The ubiquity of the SALT2
model in cosmology analyses of the last decade can be attributed to:

  1. 1.

The spectrophotometric model can be integrated over filter bands using the
appropriate rest-frame wavelengths, which removes the need for explicit
$k$-corrections.

  2. 2.

The training process is cosmology independent, as the overall normalization of
a light curve is a fitted parameter.

  3. 3.

The training set incorporates photometric data from multiple surveys, reducing
the sensitivity of the model to the calibration of any one survey.

  4. 4.

The training sample incorporates high redshift photometric data, allowing the
use of well-calibrated observer-frame optical data to extend the model into
the rest-frame ultra-violet (UV).

  5. 5.

Publicly available analysis tools such as SNANA (Kessler et al., 2009b) and
sncosmo. (Barbary et al., 2015, 2016) include infrastructure to fit
lightcurves and generate simulated data using the SALT2 model.

  6. 6.

The model has been tested by many independent groups as a consequence of its
use in cosmology analyses.

Despite these advantages, Scolnic et al. (2018) found that the calibration of
the training sample used to create the SALT2.JLA model was the largest single
systematic uncertainty in their measurement of $w$ ($\sigma_{w}=0.014$, 30% of
the total systematic uncertainty), though a new analysis methodology could
somewhat reduce this uncertainty (Brout et al., 2020). Achieving the science
goals of the Vera C. Rubin Observatory’s Legacy Survey of Space and Time
(LSST) will require that systematic uncertainty in the calibration of the
light-curve model be decreased by a factor of 5 for the year-one analysis (The
LSST Dark Energy Science Collaboration et al., 2018).

The SALT2.JLA model does not fully reproduce observed spectral features such
as varying absorption line velocities (Foley & Kasen, 2011; Siebert et al.,
2020). Similarly, studies have found evidence that standardized distances
based on SALT2 with the Tripp estimator (SALT2+Tripp) are dependent on SN Ia
host galaxy properties (Kelly et al., 2010; Sullivan et al., 2010), although
the best way to characterize this effect is still in question (Rigault et al.,
2013; Betoule et al., 2014; Jones et al., 2018; Rigault et al., 2018; Brout &
Scolnic, 2020; Smith et al., 2020b). Since these astrophysical effects are not
explicitly included in the training process, differences between the training
sample and cosmology samples can lead to subtle biases in the cosmology
results. To characterize and correct for such untrained effects, it is
essential to perform training studies on simulated data that incorporates a
broad range of physical effects. Therefore, in parallel with the the SALT3
development our team has developed a more general SED-simulation framework
described in Pierel et al. (2020).

Extending the wavelength range of the SALT model shows promise in the
reduction of statistical uncertainty. Distance standardization based on
SALT2+Tripp results in scatter about the Hubble diagram that is
$\sim$0.1\text{\,}\mathrm{mag}$$ larger than expected from photometric
uncertainties (which we refer to as “intrinsic scatter”). Numerous studies
over the last two decades have found that NIR peak magnitudes show smaller
Hubble residuals than SALT2+Tripp distances (Krisciunas et al., 2004, 2007;
Wood-Vasey et al., 2008; Burns et al., 2011; Mandel et al., 2011; Dhawan et
al., 2018; Avelino et al., 2019; Mandel et al., 2020). However, the SALT2.JLA
SED model extends only to $9200$ Å (extrapolated further for use in
simulations in Pierel et al., 2018) with substantial model uncertainties past
$\sim 7000$ Å that preclude the use of existing low-redshift optical data. By
extending the wavelength range to reliably include existing rest-frame $i$ and
$z$ band photometry we can improve on cosmology constraints on current data
sets. Further extension of the model into the NIR would allow future SN Ia
cosmology programs to make use of a wavelength range in which SNe Ia are
intrinsically more precise.

As a step toward these goals, we have defined a SALT3 spectrophotometric model
formalism and developed SALTshaker, a flexible and open source Python-based
code for training a SALT3 model, accepting both spectra and photometry in the
training process. The SALT3 formalism has been defined similarly to SALT2 to
retain the compatibility of our model with existing analysis frameworks. As
part of an overall testing and validation framework (Dai et al. in prep),
SALTshaker enables new SN Ia light-curve models to be quickly trained for new
samples of cosmological SNe, allowing uncertainties from the modeling process
to improve as larger and more accurately calibrated samples are collected. In
Section 2 we define our SALT3 model, and describe the procedure of our
training code. In Section 3 we apply SALTshaker on training data similar to
that of the SALT2.JLA model, allowing a direct comparison of our training
process to that of SALT2. Next we add recalibrated data from past and recent
SN Ia surveys to our training sample, described in Section 4, to increase the
size of the training sample by a factor of $\sim 2.5$. Finally, we build the
SALT3.K21 model and present our results in Section 5.

##  2 The SALT3 Model and SALTshaker

Figure 1: Construction of a SALT light curve for a given SN Ia, here shown
using the SALT2.JLA model. In panels a-c, we apply the $M_{1}$ component and
the color law to the base $M_{0}$ spectral flux at a phase of +10 days. In
panel d, the spectrum is redshifted, and integrated across a bandpass
transmission. In panel e, this process is repeated at every phase to produce a
lightcurve. Panel f) shows realizations of the model uncertainties. Color
scatter can be seen as the coherent offset within each band, clearly seen at
the peak of $V$ band. At later phases in $V$ the uncorrelated uncertainties
dominate the realizations.

Using the framework of SALT, the SALT3 model defines the spectral flux of a SN
Ia as a function of rest-frame wavelength $\lambda$ and phase $p$. Given three
light-curve parameters for a given SN $x_{0}$ (the overall flux
normalization), $x_{1}$ (associated with the light-curve “stretch” ), and $c$
(a fitted color parameter), the spectral flux222In all cases we use $F$ to
refer to a spectral flux and $f$ to refer to a flux integrated over an
appropriate bandpass (in energy units) is

$\displaystyle F(p,\lambda)=$ $\displaystyle
x_{0}[M_{0}(p,\lambda;\bm{m_{0}})+x_{1}M_{1}(p,\lambda;\bm{m_{1}})]$
$\displaystyle\cdot\exp(c\cdot CL(\lambda;\bm{cl})),$ (1)

where $M_{0}(p,\lambda;\bm{m_{0}})$ and $M_{1}(p,\lambda;\bm{m_{1}})$ are flux
surfaces similar to principal components, representing respectively the SED of
a fiducial SN Ia and a first order correction. The two flux surfaces are
defined on a basis of second-order B-splines, with $N_{\text{Phase}}\times
N_{\lambda}$ basis functions for each surface, with coefficients
$\bm{m_{0}},\bm{m_{1}}$ that are model parameters. The number of basis
functions along each axis are determined based on the desired wavelength and
phase resolution; in our work we have chosen wavelength resolution $70$ Å and
phase resolution of $3$ days. $CL(\lambda;\bm{cl})$ is a single color law
which combines the effects of intrinsic color variation and host galaxy dust
extinction. The color law is polynomial with $N_{CL}$ coefficients $\bm{cl}$
within a specified wavelength range $\lambda_{-}$ to $\lambda_{+}$ and
linearly extrapolated for the rest of the SED wavelength range of the model.
Thus the color law is defined as

$\displaystyle CL_{p}(\lambda)$
$\displaystyle=\sum_{i=0}^{N_{CL}}\bm{cl}_{i}\lambda^{i},$
$\displaystyle\lambda<\lambda_{-}$ $\displaystyle CL(\lambda)$
$\displaystyle=\smash{\left\\{\begin{array}[]{@{}l@{}}CL^{\prime}_{p}(\lambda_{-})(\lambda-\lambda_{-})+CL_{p}(\lambda_{-})\\\\[3.0pt]
CL^{\prime}_{p}(\lambda_{+})(\lambda-\lambda_{+})+CL_{p}(\lambda_{+})\\\\[3.0pt]
CL_{p}(\lambda)\end{array}\right.}$ $\displaystyle\lambda>\lambda_{+}$
otherwise

The object of the model training process is to determine the model parameters
$\\{\bm{m_{0}},\bm{m_{1}},\bm{cl}\\}$ and estimate un-modeled variability in
the flux surfaces; the model training is detailed in Section 2.5.

For a given photometric observation of a SN Ia at heliocentric redshift
$z_{\text{Hel}}$ in a filter $X$ with observer-frame transmission function
$T_{X}(\lambda)$ in photon units, the broadband flux as a function of phase is

$\displaystyle f_{X}(p)=\int
T_{X}(\lambda/(1+z_{\text{Hel}}))F(p,\lambda;x_{0},x_{1},c)\lambda d\lambda$
(2)

We illustrate this procedure for modeling a light curve in Figure 1. The fixed
configuration parameters used to specify the model are listed in Table 1,
while the fitted model parameters (
$\\{\mathbf{m_{0}},\mathbf{m_{1}},\mathbf{cl}\\}$) are listed in Table 2.

###  2.1 Model Definitions

The specification of the model given above is degenerate; for example, the
scale of the flux surfaces may be changed by reducing $M_{0}$ and $M_{1}$ and
increasing $x_{0}$ by the same factor. To remove degeneracies we apply further
model definitions, which are used as constraints during the training process.
These definitions are arbitrary, and have been chosen to define a “fiducial”
supernova whose SED is the $M_{0}$ flux surface at the mean of the observed
distributions of lightcurve parameters. The definitions are:

  1. 1.

The rest-frame synthetic $B$-band flux of the $M_{0}$ component at peak is
fixed such that $m_{B}^{\text{peak}}=10.5$ when $x_{0}=1$

  2. 2.

The rest-frame synthetic $B$-band flux of the $M_{1}$ component at peak is
defined to be 0

  3. 3.

The distribution of the light-curve parameter $x_{1}$ in the training sample
is defined to have 0 mean

  4. 4.

The distribution of $x_{1}$ has standard deviation 1

  5. 5.

The distribution of the light-curve parameter $c$ in the training sample is
defined to have 0 mean

  6. 6.

The color law is defined such that $CL(4300\textrm{ \AA})=0$ and
$CL(5430\textrm{ \AA})=-1$ , corresponding to central wavelengths for $B$ and
$V$ bandpasses

  7. 7.

The distributions of $x_{1}$ and $c$ have no correlation in the training
sample

The location and scale of the $x_{1}$ distribution inferred for any
cosmological sample are thus relative to the demographics of the training
sample for a particular model, while $c$ is fixed to correspond to a phase-
independent inferred $B-V$ color relative to the mean of the training sample.
Our model definitions and SALT2 differ only in the last definition, intended
to separate the phase-independent color from the phase-dependent $M_{1}$
component. The SALT2 training code does not constrain the correlation between
$x_{1}$ and $c$, and instead fixes the $V$-band flux (and $B$-band flux
following definition 2) of the $M_{1}$ component to be $0$ at peak brightness,
implying that SALT2 $x_{1}$ has no effect on observed $B-V$ color at peak. For
SALT3, removing the correlation between the parameters $x_{1}$ and $c$ has a
physically intuitive meaning; the dust-like color term does not depend on the
processes associated with light-curve stretch. Further, it is easier to make
inferences about the latent populations when the stretch and color parameters
are uncorrelated, as correlated parameters imply that there is redundant
information in the two parameters.

###  2.2 Photometric model uncertainties

As Rubin (2020) and Rose et al. (2020) suggest that SN Ia SEDs are determined
by 3-5 SED parameters and a color, a model with a single SED parameter
($x_{1}$ in our case) will not capture the full diversity of the population.
We refer to the variation unexplained by our model as “in-sample variance”.
This variation can be contrasted with the uncertainty in the model parameters
due to training with a finite sample of SNe with finite signal-to-noise
photometry, which we refer to as “out-of-sample variance”. For a photometric
observation in a filter with central wavelength $\lambda_{c}$ and phase $p$,
in-sample variance is addressed by a “model variance” composed of two terms.
The first is a diagonal uncertainty defined as

$\displaystyle\sigma^{2}_{f}(p,\lambda_{c})=$
$\displaystyle\left[x_{0}\exp(c\cdot CL(\lambda_{c}))\int
T\left(\frac{\lambda}{1+z_{\text{Hel}}}\right)\lambda d\lambda\right]^{2}$
$\displaystyle\times\left[\sigma^{2}_{M_{0}}(p,\lambda_{c})+x^{2}_{1}\sigma^{2}_{M_{1}}(p,\lambda_{c})+\right.$
$\displaystyle\left.2x_{1}C_{M_{0}M_{1}}(p,\lambda_{c})\sigma_{M_{0}}(p,\lambda_{c})\sigma_{M_{1}}(p,\lambda_{c})\right]$
(3)

where $\sigma_{M_{0}}(p,\lambda_{c};\mathbb{\sigma_{M_{0}}})$ and
$\sigma_{M_{1}}(p,\lambda_{c};\mathbb{\sigma_{M_{1}}})$ represent variability
in the flux surfaces and
$C_{M_{0}M_{1}}(p,\lambda_{c};\mathbb{C_{M_{0},M_{1}}})$ is the correlation
between flux surfaces. These variance terms are described by zeroth order
B-splines (equivalent to binning the data by phase and wavelength) with 8
basis functions in wavelength and 12 basis functions in phase. As detailed in
Section 2.5 we use a maximum likelihood estimator to determine the B-spline
coefficients $\bm{\sigma_{M_{0}}},\bm{\sigma_{M_{1}}},\bm{C_{M_{0},M_{1}}}$.
This is distinct from the approach of SALT2, which took the in-sample variance
to have the same form as that of the out-of-sample variance, scaling the
latter (evaluated by leave-one-out tests) using a smooth function or “error
snake” to fix the $\chi^{2}_{\nu}$ of the photometry of the training sample to
1. Because the in-sample variance is determined by variability of the
underlying population of SNe light curves while the out-of-sample variance is
determined by the distribution of available training data, we consider our
approach to be a better account of the remaining variance of the SN Ia
population.

The second term of the in-sample variance is a covariant “color scatter” that
allows light curves of the same supernova in different bands to be coherently
offset relative to one another. The color scatter term between photometric
measurements using the same bandpass is correlated, with no correlation
between measurements in different bandpasses. This model is similar to
chromatic models of intrinsic scatter like those of Guy et al. (2010) and the
diagonal terms of the covariance matrix from Chotard et al. (2011). The
relative covariance $k^{2}(\lambda_{c};\mathbf{a})$ is described by the
exponential of a fourth order polynomial in wavelength, where the polynomial
coefficients $\mathbf{a}$ are fitted parameters. Thus, the model covariance
matrix for two photometric observations $i$ and $j$ in broadband filters
$X_{i}$,$X_{j}$ of a given supernova with model fluxes
$\vec{f}_{\text{Model}}$ is

$\displaystyle(\Sigma_{\text{Model}})_{ij}=$
$\displaystyle\delta_{ij}\sigma^{2}_{f}(p_{i},\lambda_{c(i)})$
$\displaystyle+$
$\displaystyle\begin{cases}k^{2}([\lambda_{c}])(\vec{f}_{\text{Model}})_{i}(\vec{f}_{\text{Model}})_{j}&X_{i}=X_{j}\\\
0&\text{otherwise}\end{cases}$ (4)

###  2.3 Modeling of Spectral Data

Although the SALT3 model is intended for use as a photometric light-curve
model, spectral data is included in the training process to better constrain
the shape of spectral features. However, given that spectral data have larger
calibration uncertainties compared to broadband fluxes, we follow the SALT2
training code in “recalibrating” spectral data, modulating the model by a
smooth function to match the continuum of the observed spectrum. We modify the
spectral flux equation (Equation 2) for use with spectra during the training
by removing the color term and replacing it with a recalibration term of
similar form,

$F_{\text{spec}}(p,\lambda)=x_{0}[M_{0}(p,\lambda)+x_{1}M_{1}(p,\lambda)]\exp(\sum^{n}_{i}y_{i}\lambda^{i}/i!),$
(5)

where the spectral recalibration nuisance parameters $y_{i}$ are fitted during
the training procedure. In the limit of perfectly calibrated spectra, the
fitted recalibration term will reproduce the effect of the color parameter on
the spectrum; by removing the color law entirely from the spectral flux
equation, we mitigate the impact of mis-calibrated spectra on the color law.
The quantity $n$ controls how many recalibration parameters are allowed for
each spectrum, and is determined based on the wavelength extent of the
spectrum and the number of filter bands available for that SN (additional
filter bands better constrain the recalibration term, allowing for more free
parameters). The model variances are defined similarly to the photometry,
without the contribution of the color scatter:

$\displaystyle\sigma^{2}_{F}(p,\lambda)=$
$\displaystyle\left[x_{0}\exp(\sum^{n}_{i}y_{i}\lambda^{i}/i!))\right]^{2}\times$
$\displaystyle[\sigma^{2}_{M_{0}}(p,\lambda)+x^{2}_{1}\sigma^{2}_{M_{1}}(p,\lambda)$
$\displaystyle+$ $\displaystyle
2x_{1}C_{M_{0}M_{1}}(p,\lambda)\sigma_{M_{0}}(p,\lambda)\sigma_{M_{1}}(p,\lambda)].$
(6)

Table 1: Fixed Model Parameters Parameter | Description | JLA Training | SALT3 Training
---|---|---|---
$A_{\text{Spec}}$ | Spectral Suppression |  $\cdots$ | 0.75
$N_{\text{Phase}}$ | Number of phase basis functions for flux surfaces | 11 | 20
$N_{\lambda}$ | Number of basis functions on wavelength axis for flux surfaces | 97 | 127
$N_{\sigma,\text{Phase}}$ | Number of phase basis functions for flux uncertainty surfaces |  $\cdots$ | 9
$N_{\sigma,\lambda}$ | Number of basis functions on wavelength axis for flux uncertainty surfaces |  $\cdots$ | 5
$A_{\text{Phase}}$ | Phase gradient regularization weight | 0 | 1000
$A_{\lambda}$ | Wave gradient regularization weight | 10 | 10000
$A_{\text{Dyadic}}$ | Dyadic regularization weight | 1000 | 1000
Model parameters for SALT2.JLA training and SALT3.K21 training. We note that
given our adjusted regularization scheme, regularization weights are not
directly comparable. We were unable to determine the exact spectral
suppression in the JLA training, but from Mosher et al. (2014) we expect that
this is fine-tuned for a given training sample.
Figure 2: Activity diagram of the SALTshaker training process, with variable names in the model and model error fitting sections corresponding to the descriptions in Table 2. Table 2: Model and Model Uncertainty Parameters Parameter | Category | Number | Description
---|---|---|---
$\mathbb{m_{0}}$ | Flux Model | $N_{\text{Phase}}\times N_{\lambda}$ |  $M_{0}$ B-spline coefficients
$\mathbb{m_{1}}$ | Flux Model | $N_{\text{Phase}}\times N_{\lambda}$ |  $M_{1}$ B-spline coefficients
$\mathbb{cl}$ | Flux Model | $N_{CL}$ | Color law
$x_{0}$ | Nuisance | $N_{\text{SN}}$ | Overall flux normalization for each SN
$x_{1}$ | Nuisance | $N_{\text{SN}}$ | Stretch of each SN
$c$ | Nuisance | $N_{\text{SN}}$ | Color for each SN
$\mathbb{y}$ | Nuisance | $\propto N_{\text{spec}}$ | Spectral recalibration
$\mathbb{\sigma_{M_{0}}}$ | Uncertainty | 72 | Uncertainty in $M_{0}$
$\mathbb{\sigma_{M_{1}}}$ | Uncertainty | 72 | Uncertainty in $M_{1}$
$\mathbb{C_{M_{0},M_{1}}}$ | Uncertainty | 72 | Correlation between $M_{0}$ and $M_{1}$
$\mathbb{a}$ | Color Scatter | 4 | Color scatter

###  2.4  Regularization

In regions of phase-wavelength space that are poorly constrained by spectra,
the $M_{0}$ and $M_{1}$ components can acquire artifacts such as high-
frequency ringing, or deconvolution noise. To reduce these artifacts, we use a
“regularization” procedure that penalizes large derivatives in the model where
there is an absence of spectroscopic data, as parameterized by a binned
spectral density function $N_{\text{eff}}(p,\lambda)$, with every spectrum
assigned equal weight. We implement three kinds of regularization: phase
gradient, wave gradient, and dyadic. These are applied to the two flux
surfaces $M_{0}(p,\lambda)$ and $M_{1}(p,\lambda)$. For a flux surface
$S(p,\lambda)$ the regularization terms are defined as

  1. 1.

Phase gradient regularization penalizes large derivatives with respect to
phase in less-constrained regions

$\displaystyle\chi^{2}_{\text{Phase}}$
$\displaystyle[S(p,\lambda)]=A_{\text{Phase}}\sum^{2N_{\text{Phase}}}_{i}\sum^{2N_{\lambda}}_{j}\frac{{\Big{(}}\left.\frac{\partial
S}{\partial p}\right|_{\begin{subarray}{c}p=p_{i}\\\
\lambda=\lambda_{j}\end{subarray}}{\Big{)}}^{2}}{{N_{\text{eff}}}(p_{i},\lambda_{j})}$
(7)

  2. 2.

Wave gradient regularization similarly penalizes large derivatives with
respect to wavelength in less-constrained regions

$\displaystyle\chi^{2}_{\text{Phase}}$
$\displaystyle[S(p,\lambda)]=A_{\text{Wave}}\sum^{2N_{\text{Phase}}}_{i}\sum^{2N_{\lambda}}_{j}\frac{{\Big{(}}\left.\frac{\partial
S}{\partial\lambda}\right|_{\begin{subarray}{c}p=p_{i}\\\
\lambda=\lambda_{j}\end{subarray}}{\Big{)}}^{2}}{{N_{\text{eff}}}(p_{i},\lambda_{j})}$
(8)

  3. 3.

Dyadic regularization encourages the flux surfaces to be separable in phase
and wavelength, and is 0 when a flux surface takes the form
$S(p,\lambda)=g(p)\times h(\lambda)$

$\displaystyle\chi^{2}$
${}_{\text{Dyadic}}[S(p,\lambda)]=A_{\text{Dyadic}}\sum^{2N_{\text{Phase}}}_{i}\sum^{2N_{\lambda}}_{j}\frac{1}{{N_{\text{eff}}}(p_{i},\lambda_{j})}$
$\displaystyle\times{\Big{[}}\left.\frac{\partial S}{\partial
p}\right|_{\begin{subarray}{c}p=p_{i}\\\
\lambda=\lambda_{j}\end{subarray}}\left.\frac{\partial
S}{\partial\lambda}\right|_{\begin{subarray}{c}p=p_{i}\\\
\lambda=\lambda_{j}\end{subarray}}-S(p_{i},\lambda_{j})\left.\frac{\partial^{2}S}{\partial
p\partial\lambda}\right|_{\begin{subarray}{c}p=p_{i}\\\
\lambda=\lambda_{j}\end{subarray}}{\Big{]}}^{2}$ (9)

The relative strength of the three regularization terms is determined by the
weights $\\{A_{\text{Phase}},A_{\text{Wave}},A_{\text{Dyadic}}\\}$, which are
inputs to the SALTshaker code. The summations over phase and wavelength points
are evenly spaced over the model phase and wavelength ranges, with twice as
many points as basis functions along each axis. As regularization can bias the
model surfaces by over-smoothing them (Mosher et al., 2014), we tune the
weights to ensure that the regularization terms do not contribute
significantly to the total $\chi^{2}$. Further work to determine how this
regularization scheme affects the model and to choose optimal model
configurations will require applying our training and analysis framework to
simulations (Dai et al. in prep; Pierel et al., 2020).

###  2.5 Training Procedure

Based on the model definitions above, we construct the training $\chi^{2}$ for
the model as

$\displaystyle\chi^{2}_{\text{Total}}=$
$\displaystyle\chi^{2}_{\text{Phot}}+A_{\text{Spec}}\chi^{2}_{\text{Spec}}+\chi^{2}_{\text{Constraints}}+\chi^{2}_{\text{Reg}}.$
(10)

The photometric $\chi^{2}$ term is

$\displaystyle\chi^{2}_{\text{Phot}}=$
$\displaystyle\sum^{N_{\text{SN}}}_{n}(\vec{f}_{\text{obs}}^{(n)}-\vec{f}_{\text{Model}}^{(n)})^{T}(\Sigma_{\text{Total}}^{(n)})^{-1}(\vec{f}^{(n)}_{\text{obs}}-\vec{f}^{(n)}_{\text{Model}})$
$\displaystyle\Sigma_{\text{Total}}^{(n)}$
$\displaystyle=\text{diag}((\sigma^{(n)}_{Phot})^{2})+\Sigma_{\text{Model}}^{(n)}$
(11)

where for the $n$th supernova $\vec{f}^{(n)}_{\text{obs}}$ is the vector of
observed photometric fluxes, and $\vec{f}^{(n)}_{\text{Model}}$ is the vector
of the model fluxes integrated over the photometric bandpass. The covariance
$\Sigma_{\text{Total}}^{(n)}$ combines photometric uncertainties and the model
uncertainty covariance described in Equation 4. The factor $A_{\text{Spec}}$
is a constant “spectral suppression” term that downweights the contribution of
spectra to the training $\chi^{2}$ to reduce the sensitivity of the training
to unknown systematic errors in the spectral data (Guy et al., 2007). We set
this term such that the spectral and photometric data have roughly equal
contributions to the total training $\chi^{2}$. The spectral $\chi^{2}$ term
is then defined

$\displaystyle\chi^{2}_{\text{Spec}}=\sum^{N_{SN}}_{n}\sum^{N_{\text{spec}}}_{i}\sum^{N_{\text{points}}}_{j}\frac{[F_{\text{spec}}(p^{(n,i)},\lambda^{(n,i)}_{j})-(\vec{f}^{(n,i)}_{\text{obs}})_{j}]^{2}}{(\sigma_{\text{obs}})_{j}^{2}+\sigma_{\text{Model}}(p^{(n,i)},\lambda^{(n,i)}_{j})^{2}},$
(12)

where for the $n$th supernova,
$F_{\text{spec}}(p^{(n,i)},\lambda^{(n,i)}_{j})$ is the model spectral flux at
the $j$th wavelength bin of the $i$th spectrum,
$(\vec{f}^{(n,i)}_{\text{obs}})_{j}$ is the observed spectral flux,
$(\sigma_{\text{obs}})_{j}$ is the photometric uncertainty, and
$\sigma_{\text{Model}}(p^{(n,i)},\lambda^{(n,i)}_{j})$ is the model
uncertainty in Equation 6. In $\chi^{2}_{\textrm{phot}}$ and
$\chi^{2}_{\textrm{spec}}$, we redden the model to account for the Milky Way
$E(B-V)$ along the line of sight to each SN, but do not include this term in
the equations above for simplicity. The $\chi^{2}_{\text{Constraints}}$ is
composed of penalty terms used to enforce the model definitions described in
Section 2.1, and the regularization term is defined as

$\displaystyle\chi^{2}_{\text{Reg}}=\sum^{\\{M_{0},M_{1}\\}}_{S(p,\lambda)}$
$\displaystyle\chi^{2}_{\text{Phase}}[S(p,\lambda)]+\chi^{2}_{\text{Wave}}[S(p,\lambda)]$
$\displaystyle+\chi^{2}_{\text{Dyadic}}[S(p,\lambda)].$ (13)

The SALTshaker code is initialized with the configuration parameters shown in
Table 1. SALTshaker then determines best-fit values of the model parameters.
These parameters are shown in Table 2. We define flux model parameters as
those which control the flux surfaces and color law, uncertainty model
parameters as those which control the output model uncertainties, and nuisance
parameters as those which describe individual supernovae.

SALTshaker alternates between simultaneously fitting the nuisance parameters
and model flux parameters while keeping uncertainties fixed and fitting the
model uncertainty parameters while keeping model fluxes fixed. The color
scatter is kept fixed at $k(\lambda_{c})=0$ during this stage of the training
process, as we find that allowing nonzero color scatter results in biased flux
surfaces due to the regularization procedure. The flux and nuisance parameters
are fit using an iterative Levenberg-Marquardt algorithm. To reduce the number
of (computationally expensive) Jacobian evaluations of the model residuals
required, we use Schubert’s method to perform a rank one update on the
Jacobian after each iteration while maintaining its sparsity structure
(Marwil, 1979; Schubert, 1970). However this technique is unsuitable for
determining the model uncertainties.

To fit the model uncertainty parameters defined in Equation 4 we define a log-
likelihood $(\mathcal{L}_{\text{Phot. Errs.}}$ given as

$\displaystyle-2\log(\mathcal{L}_{\text{Phot. Errs.}})=$
$\displaystyle\chi^{2}_{\text{Phot}}-\sum^{N_{SN}}_{n}\log(\lvert\Sigma^{(n)}_{\text{Total}}\rvert).$
(14)

The model uncertainty parameters are chosen to maximize the log-likelihood
using the optimizer iMinuit while the flux model and nuisance parameters are
kept fixed (James & Roos, 1975; Dembinski et al., 2020). After the training
has converged with $k$ fixed to 0, we use iMinuit to fit the parameters
controlling the color scatter; during this final fit, the color law is allowed
to vary (having previously been fit as a flux model parameter). The out-of-
sample variance is then estimated by inverting the Hessian matrix of the flux
parameters obtained from the model fitting process, with the regularization
terms suppressed by a factor of 100, and propagating those parameter
uncertainties into the flux surfaces. The model uncertainties and out-of-
sample variance are added together as the total model uncertainty surface.

The computation time required for the SALT3.JLA training sample with
SALTshaker is approximately three minutes per iteration, with $\sim$25
iterations required for convergence. On the larger SALT3.K21 sample,
iterations are significantly longer, with $\sim$25 minutes required per
iteration. We note that 29-Å spectral binning, which was used in the SALT2.JLA
training, improves speeds substantially by reducing the amount of data by up
to an order of magnitude. We include this as an option in SALTshaker but the
models in this work use the native binning of the input spectra. Finally, the
slowest component of the code is the iterative fitting of the error model,
which for SALT3.K21 requires approximately 4 hours to perform 80 iterations
and reach convergence through iMinuit. Error model iterations are performed
every five iterations by default but could likely be performed less often
without adversely affecting the final model. Faster methods of error model
fitting will be an important avenue for future improvement and should improve
speeds considerably.

An overview of the SALTshaker training procedure is given in Figure 2.

##  3 SALTshaker Validation

Here, we show that our method is capable of producing a trained model that
reproduces inferred distances from SALT2 by comparing trained SALT3 models
with SALT2.JLA. We show comparisons between synthetic photometry in multiple
rest-frame filters as well as the spectral models, and compare distances
between the models.We describe our metrics in Section 3.1, the JLA training
sample of Betoule et al. (2014) and Mosher et al. (2014) in 3.2, a simulated
sample that mimics the demographics of JLA in 3.3, then train on these samples
in 3.4, 3.5.

###  3.1 Validation Procedure and Metrics

To distinguish between models trained on different input data, we define
“SALT3.X” as the SALT3 model created with SALTshaker using training sample X.
We refer to the samples used to evaluate the performance of a given SALT3
model as “validation” samples, as it will be expedient to evaluate the trained
model in some cases on data that was not used in the model training. We use
the SNANA light curve fitting program to fit validation samples with both
SALT2.JLA and SALT3.X.

Given fitted SALT parameters $m_{B},x_{1},c$ and their uncertainties, the
Tripp estimator for distance modulus is

$\mu=m_{B}+\alpha\cdot x_{1}-\beta\cdot c-\mathcal{M}$ (15)

with distance uncertainties

$\displaystyle\sigma_{\mu}=$
$\displaystyle\sigma_{\textrm{int}}^{2}+\sigma_{\mu,z}^{2}+\sigma_{\textrm{lens}}^{2}+\sigma_{m_{B}}^{2}+(\alpha\sigma_{x_{1}})^{2}+(\beta\sigma_{c})^{2}$
$\displaystyle+2\alpha\beta\sigma_{c,x_{1}}+2\alpha\sigma_{m_{B},x_{1}}+2\beta\sigma_{m_{B},_{c}}$
(16)

where $\sigma_{\textrm{int}},\alpha,\beta$ and $\mathcal{M}$ are nuisance
parameters, $\sigma_{\mu,z}$ is computed from a peculiar velocity uncertainty
of $250\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$, and
$\sigma_{\textrm{lens}}=0.055z$. We use the SALT2mu method (Marriner et al.,
2011; Kessler & Scolnic, 2017) implemented in SNANA to estimate the nuisance
parameters as well as distances in 5 redshift bins. Allowing for a shift in
location and scale of the light-curve parameters, the observed distributions
are similar as compared to SALT2.JLA (see Section 5.2.2). We thus expect
selection biases to be common between the two models, and therefore we do not
use SALT2mu to correct for selection biases.

Given estimated distance moduli $\mu$ and Hubble residuals relative to a
nominal $\Lambda$CDM cosmology ($\Delta\mu$) for both models we define two
metrics. First $\textrm{RMS}(\Delta\mu)$ across the validation sample and the
relative distance difference between models defined as

$\displaystyle\textrm{Diff}(\Delta_{z}\mu)$
$\displaystyle=\Big{[}\mu(0.40<z<0.6|\textrm{SALT3.{X}})$
$\displaystyle-\mu(0.40<z<0.6|\textrm{SALT2.JLA})\Big{]}$
$\displaystyle-\Big{[}\mu(0.01<z<0.2|\textrm{SALT3.{X}})$
$\displaystyle-\mu(0.01<z<0.2|\textrm{SALT2.JLA})\Big{]},$ (17)

where $\mu(Z|M)$ indicates a weighted average distance across a redshift range
$Z$ given a model $M$. Additionally we will show binned Hubble residuals
across the redshift range. We discuss differences in the nuisance parameter
$\beta$, as this has physical implications for dust properties, however
$\alpha$ depends on both the demographics of the training sample and our
revised separation of stretch and color, and thus has no useful comparison
across models (see Section 2.1). Similarly we compare synthetic photometry,
but these comparisons are most relevant for simulated data when the truth
model is known. We train on multiple simulated and real data samples to assess
how our models perform; in Table 3 we summarize the abbreviations used for
these training samples.

Table 3: Data Abbreviations

Abbreviation | Size |  Description | Data | Trained Model
---|---|---|---|---
JLA | 420 |  Mixture of low-redshift SNe from many surveys and high-redshift SNe from SNLS and SDSS; see Mosher et al. (2014) | 3.2 | 3.5
simJLA | 420 |  Simulated data that emulates the JLA training sample | 3.3 | 3.4
K21 | 1083 |  Compilation of the JLA training sample combined with new SNe from Foundation, Pan-STARRS, the Dark Energy Survey, CfA4, and CSP, with additional spectra from kaepora | 4 | 5.2
K21train | 541 |  Half of the SNe from the K21 compilation chosen at random for use as a training sample | 4 | 5.1
K21valid | 541 |  The other half of the K21 SNe, used as a validation sample to see how the SALT3 model performs on data that was not part of the training | 4 |  $\cdots$

Abbreviations we use to refer to compilations of cosmological SNe Ia in this
work, number of SNe in each, a brief description, the section in which we
discuss the data itself, and the section in which we discuss a SALT3 model
trained on each

Figure 3: SN 1992A light curves (left) and spectra (right) for real data
(black) and SNANA simulation (orange) generated with best-fit $x_{1}$, $c$,
$z$, and time of maximum light. Photometric data are shown on the left and
spectroscopic data on the right. Random offsets between data and simulations
are expected due to the C11 scatter model and spectral recalibration. The
simulated spectra at red wavelengths have better SNR than the data because we
model the average spectral SNR rather than its wavelength dependence.

###  3.2 JLA Training Sample

The JLA data used to train the SALT2.JLA model consist of 420 SNe with light
curves, 83 of which include spectroscopy, from a compilation of low-$z$
samples (see Table 4 for details and citations), the Sloan Digital Sky Survey
(SDSS; Holtzman et al., 2008; Kessler et al., 2009a; Sako et al., 2018), and
the Supernova Legacy Survey (Astier et al., 2006, with spectra from Walker et
al., 2011; Balland et al., 2018 and private communication with M. Betoule, C.
Balland). These data are summarized in Table 4, along with the new training
data discussed in Section 4, and are included in the data release at
http://saltshaker.readthedocs.io/.

We use the “Supercal” procedure (Scolnic et al., 2015) to update the
calibration of the JLA training sample. Supercal uses the 3$\pi$ sky coverage
of the PS1 system and its observations of secondary standard stars to
precisely determine the offsets between different photometric systems. Scolnic
et al. (2015) found that applying these calibration corrections to the sample
of SN Ia used to measure the Hubble flow without updating the training
calibration could shift $w$ by 0.026. While we do not study the impact of the
calibration on the trained model, Taylor et al. (2021) examines this with a
full reanalysis of the DES-SNIa cosmology results using a retrained SALT2
model. We also use the Schlafly & Finkbeiner (2011) corrections to the
Schlegel et al. (1998) Milky Way dust maps.

###  3.3 Simulated JLA Training Sample

To test SALTshaker with a known input model and cosmology, we first simulate
the SALT2.JLA training data to produce our “simJLA” training sample. Training
on the actual JLA training sample is also a useful test (Section 3.5), but it
does not provide a known truth model to validate the training outputs. For
this simJLA test, every simulated SN that goes into the training sample has
SALT2.JLA as the “truth” model, allowing us to test the consistency of input
and output models.

We use the SNANA software (Kessler et al., 2009c) to generate Monte Carlo
realizations of SN photometry and spectroscopy mimicking the demographics of
the JLA training sample of 420 SNe. SNANA simulations are frequently used to
simulate a random realization for a sample, in order to explore SN distance
biases as a function of redshift.Our goal here is different: we aim to
accurately simulate every individual event in the JLA sample, including
cadence and signal-to-noise for both photometric and spectroscopic
observations. We therefore use the cadence, redshift, and best fit $c$ and
$x_{1}$ from each JLA light curve as input to the simulation. Each set of
measured SN properties ($z$, $t_{0}$, $x_{1}$,$c$) is used to simulate a rest-
frame SED with the SALT2.JLA model. Spectral variations (intrinsic scatter)
are added to the rest-frame SED using the method in K13 and the covariance
model from the Chotard et al. (2011, hereafter C11) scatter model333We
simulate $\alpha=0.14$ and $\beta=3.5$. Note that the fit value of $\beta$
will be lower than the simulated value by $\sim$0.6 due to the characteristics
of the C11 scatter model (Scolnic & Kessler, 2016).. We do not use the default
Guy et al. (2010, hereafter G10), as the G10 model has non-physical scatter at
extremely red wavelengths that does not match observations. As described in
Kessler et al. (2019) the simulation applies cosmological dimming, lensing,
peculiar velocity, galactic extinction, and redshifting to produce a
redshifted SED at the top of the atmosphere. The filter transmissions and
cadence are used to determine measured fluxes and uncertainties. Finally, the
noise properties of each measured spectrum are applied to the simulated SED.

This simulation does not exactly reproduce the data set because of random
fluctuations in photometric noise and intrinsic scatter, and also because the
underlying SALT2 model formalism is an approximation as discussed in Pierel et
al. (2020). Nonetheless the simulation is very similar to the data and is
therefore sufficient for testing SALTshaker, as illustrated in Figure 3 for a
representative low-$z$ SN Ia. Comparisons between the parameters of
simulations and data after fitting with the SALT2.JLA model are shown in
Figure 4, with only slight observed differences in average $m_{B}$
uncertainty.

Figure 4: Distributions of parameters for the simulated (red lines) versus
real (black dots) SALT2.JLA training samples. Comparisons are shown for
redshift (A), $m_{B}$ (B), uncertainty on $m_{B}$ (C), $x_{1}$ (D), $c$ (E),
and the maximum S/N in each SN light curve (F). All distributions are well-
matched except for the uncertainty on $m_{B}$, which is slightly higher in the
real data, perhaps due to the C11 model resulting in fainter SNe than the data
in some bandpasses.

###  3.4 Training on a Simulated SALT2.JLA Training Set

We use SALTshaker to train a model using our simulated JLA sample, producing a
model we call SALT3.simJLA. In Figure 5 we show the relative difference
between the input SALT2.JLA synthetic photometry and the corresponding
synthetic photometry recovered from the SALT3.simJLA model. There is
significant discrepancy in the ultraviolet where regularization strongly
impacts the recovered model spectra, but at central passband wavelengths 4000
Å $<\lambda<7000$ Å and
$$-10\text{\,}\mathrm{days}$<p<$30\text{\,}\mathrm{days}$$ the other three
light curves ($B$, $V$, and $R$) are consistent with the truth model at a
level better than 1%. To illustrate the impact of changes in the color law on
a “typical” light curve in units of magnitudes, we use the quantity
$\sigma_{c}\cdot\Delta CL(\lambda)$, where the standard deviation of the
distribution of the SALT color parameter $c$ is $\sim 0.1$. As can be seen
from Figure 6, the color law difference is $>$0.05\text{\,}\mathrm{mag}$$ when
$\lambda<3000$ Å, the regime where the color law is least constrained.

Figure 5: Relative difference between integrated model surfaces of the input
SALT2.JLA truth model and SALTshaker model trained on simulated data. Grey
shading shows $\pm 1\%$. Lightcurves are integrated over square $UBVR$-like
bandpasses $1000$ Å wide, centered at 1000 Å intervals from 3500 Å to 6500 Å.
Significant discrepancies are seen in the ultraviolet where there is sparse
data and regularization schemes have largest impact.  Figure 6: Change between
color law of input truth model and model trained on simulated data, multiplied
by $\sigma_{c}=0.1$ to show the impact on a typical SN Ia light curve in
magnitudes. Dashed lines show $\pm$0.01\text{\,}\mathrm{mag}$$. Grey shaded
regions indicate the wavelengths for which the color law is linear. The slopes
of the color law match closely at $>7000$ Å, leading to very small differences
across the wavelength range where the color law is linear.

We find that the RMS of $\sigma_{c}\Delta CL(\lambda)$ is
$0.01\text{\,}\mathrm{mag}$ between wavelengths $3500\AA<\lambda<7000$ Å,
central filter wavelengths for which the SALT2.JLA model is considered
reliable. We conclude that because few SNe constrain the SED at $\lambda<3000$
Å, the limited set of lightcurve parameters $c$,$x_{1}$ in this wavelength
region poorly constrain the color law and the spectral components. In Sections
4 and 5 we substantially expand the training data to address this issue.

Next we compare distances from the two models. To avoid the statistics of the
training sample limiting the precision of our validation, we simulate a
“large” JLA simulation for validation, with summary statistics for the Hubble
diagram fits shown in Table 5, row 1. Instead of simulating exact $x_{1}$ and
$c$ values for an “apples-to-apples” comparison with the SALT2.JLA training
sample, we generate $\sim$3000 SNe mimicking a combination of CfA3 (Hicken et
al., 2009a), SDSS, and SNLS by using $x_{1}$ and $c$ distributions from
Scolnic & Kessler (2016). We show an example of Hubble residuals in Figure 7.
We observe consistent measurements of $\beta$ and RMS$(\Delta\mu)$, with
$\sigma_{\textit{int}}$ higher by $0.005\text{\,}\mathrm{mag}$ due to our
treatment of uncertainties. We also observe a slightly higher value of
$\alpha$, which is attributed to the $x_{1}$/$M_{1}$ degeneracy. The distance
difference between the models (Eq. 17) is consistent
($\textrm{Diff}(\Delta_{z}\mu)=$7(11)\text{\,}\mathrm{mmag}$$) despite the
U-band discrepancies in the light curves seen in Figures 5 and 6. As the
validation sample here has similar demographics to the training sample, the
same sparsity of data that allows the observed ultraviolet divergences causes
those same divergences to have little effect on the inferred distances. We
conclude that for a SALT model to be reliable at these wavelengths, the
density of data in this region must be increased to match the density of data
where we recover the input model to $\sim 1\%$, at least a factor of two
increase (see Section 4, where we discuss the density of photometric data
across the JLA wavelength range).

Figure 7: Hubble residual comparison with SALT2.JLA and SALT3.simJLA fits to
simulated data. Each point represents a simulated SN light curve from the
simJLA sample, which has been fit with either the SALT2.JLA model (blue) or
the SALT3.simJLA model (red) with larger points showing the binned distances
from each model. We do not expect SALT3 to outperform SALT2 here because SALT2
is the “truth” model, but we find consistent results.

###  3.5 Training on the JLA Training Set

Next, we run SALTshaker on the real JLA training sample; we refer to this
trained model as SALT3.JLA. We follow the validation procedure of Section 3.1
using the JLA training set for our validation sample. As shown in row 2 of
Table 5, we find a similar $\alpha$, a slightly lower $\beta$, and consistent
RMS$(\Delta\mu)$. We find that
$\textrm{Diff}(\Delta_{z}\mu)=$32(28)\text{\,}\mathrm{mmag}$$ is consistent
with zero. $\sigma_{\textrm{int}}$ is slightly higher for SALT3.JLA,
attributable to greatly decreased model uncertainties. Given the equivalent
RMS scatter, this $\sigma_{\textrm{int}}$ difference does not result in
reduced distance precision. The distance moduli are consistent at the
1-$\sigma$ level between SALT2.JLA and the SALT3.JLA, and the model surfaces
are consistent within the uncertainties. We show the Hubble residuals of this
model in Figure 8 (along with our extended SALT3 models discussed in Section
5).

Figure 8: Comparison of the SALT2.JLA (blue) versus SALT3.K21 (red) Hubble
residuals for the full K21 compilation, with results from SALT3.JLA and
SALT3.K21train shown in orange and teal, respectively. The same number of
$3\sigma$ outliers are seen when using either SALT2.JLA and SALT3.K21a. A
small change in beta can be attributed to a modified separation of color and
stretch, along with improved parameter constraints on color. Additional
details regarding the nuisance parameters in different training sets are shown
in Table 5.

a SALT3 $3\sigma$ outliers are the SNe 1998ab, 05D2ci, 1995ac, 5635, 40166,
160214, and 370369. SALT2 outliers are the SNe 05D2ci, 40166, 90037, 160214,
and 2002hu.

##  4 Expanded Training Data: The K21 Compilation

Figure 9: Redshift distribution of the JLA training data for SALT2.JLA (black
dashed) and our training data for SALT3.K21 (red solid), which adds Foundation
(blue), PS1 (green) and DES (orange) to the JLA training sample as well as
small samples from CSP and CfA4. At the top, we label the rest-frame central
filter wavelengths of each added dataset, with the full filter widths in
conjunction with spectroscopy constraining the full SALT3.K21 wavelength
range. The combined dataset contains approximately three times as many spectra
as the previous SALT2 training sample and contains approximately 2.5 times as
many SNe.

To create a next-generation SALT model with extended wavelength range and
reduced uncertainties, we add high-quality data from the Dark Energy Survey
(Brout et al., 2019b), the Foundation Supernova Survey (Foley et al., 2018),
and the Pan-STARRS Medium Deep Survey (Scolnic et al., 2018). For SNe with
photometric data in the training sample, we add 693 low-$z$ spectra from the
Kaepora database (Siebert et al., 2019), the majority of which originate from
the Berkeley SN Ia Program (BSNIP; Silverman et al., 2012a; Stahl et al.,
2020). We show the density of photometric and spectral data in phase and
wavelength space with the original JLA training sample and the additional data
included in the K21 compilation in Figure 10. Our final training sample adds
data across the phase space, but is most impactful in the red and blue regions
of our wavelength range, where the JLA training data were limited. Wavelengths
$<3500$ Å are on average covered by $1.8\times$ more light curves, while
wavelengths $>7500$ Å are covered by an average of $2.1\times$ as many light
curves and $5.7\times$ as many spectra. Additional photometric data makes the
distribution across phase space more uniform, where the JLA data has
comparatively little data in the gap between low-redshift $B$ and $V$
bandpasses.

We initially characterize our performance with the extended set of data using
separate samples for training and validation. We define the “K21valid” and
“K21train” compilations by randomly assigning half of the supernovae to each.
Finally, to produce our best model, we combine all available data to create
the “K21” compilation, and summarize the data included below and in Table 4.
The redshift distributions of these data are shown in Figure 9. Trained models
using these compilations as training samples are discussed in Section 5.

Figure 10: Number of light curves or spectra constraining each bin in our
phase/wavelength space. Panels show coverage of photometric data (upper
panels) and spectral data (lower panels) in the JLA training sample (left
panels), the PS1, DES, and Foundation data added in the K21 compilation
(middle panels), and the complete K21 compilation (right panels). A bin is
considered “covered” by a given light curve if it is within the FWHM of the
rest-frame bandpass. Foundation photometry provides much more extensive
photometric coverage of red wavelengths, while PS1 and DES photometry provides
additional rest-frame blue photometry, and the new photometric data covers the
phase space more uniformly than JLA alone. Similarly, spectra from kaepora are
immensely impactful at wavelengths $\lambda>7500$ Å, greatly assisting the
deconvolution of $z$ band data.

###  4.1 The Pan-STARRS Medium Deep Survey

The Pan-STARRS medium deep survey covered 70 square degrees of sky over four
years, discovering approximately 5200 SNe (Jones et al., 2017; Villar et al.,
2020) and spectroscopically classifying $\sim$10% of these at a median
redshift of $\sim$0.35 (Rest et al., 2014). The Pantheon analysis, which
combined these data with JLA, includes 279 PS1-observed SNe Ia with an average
of approximately 6 observations per 10 days in $griz$ (Scolnic et al., 2018).
We use these SNe Ia in our training data.

###  4.2 The Foundation Supernova Survey

The Foundation Supernova Survey (Foley et al., 2018) followed SNe using the
Pan-STARRS1 telescope, and measured well-calibrated SN light curves in $griz$
filters with a five-day cadence near maximum light and an approximately 8-day
cadence beginning at $+$10 days after maximum light. To achieve reduced
selection effects compared to previous surveys that targeted bright, pre-
selected low-$z$ galaxies, Foundation primarily followed SNe discovered by
untargeted surveys such as the All-Sky Automated Survey for Supernovae
(Shappee et al., 2014), the Asteroid Terrestrial-impact Last Alert System
(Tonry et al., 2018), Gaia (Gaia Collaboration et al., 2016), and the Pan-
STARRS Survey for Transients (Huber et al., 2015) .

The Foundation first data release in Foley et al. (2018) contains 225 SNe Ia,
180 of which are cosmologically useful. These data have been used to measure
cosmological parameters in Jones et al. (2019) and the correlation of host
galaxy properties with SN distances in Jones et al. (2018). The $iz$ band
coverage of Foundation is particularly critical to creating a SALT3 model that
is trained to redder wavelengths than enabled by the JLA data alone. We
include spectra for 114 Foundation SNe from the Dettman et al. (2021) data
release.

###  4.3 The Dark Energy Survey

The Dark Energy Survey (DES) three-year spectroscopically classified SN sample
contains 207 SNe Ia at a median redshift of 0.36 (Abbott et al., 2019). These
SNe were discovered by imaging eight 2.7 deg2 “shallow” fields (depth $\approx
23.5$ mag) and two 2.7 deg2 “deep” fields (depth $\approx 24.5$ mag)
approximately once per week (Smith et al., 2020a). Transients were discovered
using a difference-imaging pipeline (Kessler et al., 2015) and final
photometry was performed with a “scene modeling” pipeline described in Brout
et al. (2019b). See Abbott et al. (2019) and Brout et al. (2019a) for
additional details regarding the DES SN Ia data and analyses. These data have
a maximum redshift of $\sim$0.85 and complement SNLS in probing rest-frame
near-UV wavelengths with well-calibrated (sub-percent) photometric data (Burke
et al., 2018).

Table 4: The K21 Compilation Survey | NSN | N${}_{\mathrm{spectra}}^{\rm a}$ | N${}_{\mathrm{spectra}}^{\rm a}$ | $z_{min}$ | $z_{med}$ | $z_{max}$ | Filters | Ref.
---|---|---|---|---|---|---|---|---
|  | (JLA) | (Total) |  |  |  |  |
Calan-Tololo | 5 | 0 | 0 | 0.015 | 0.020 | 0.051 | $BVRI^{\rm b}$ | Hamuy et al. (1996)
CfA1 | 8 | 46 | 66 | 0.004 | 0.011 | 0.050 | $UBVRI^{\rm b}$ | Riess et al. (1999)
CfA2 | 13 | 108 | 166 | 0.008 | 0.014 | 0.031 | $UBVRI^{\rm b}$ | Jha et al. (2006)
CfA3c | 51 | 31 | 534 | 0.004 | 0.023 | 0.041 | $UBVRIri$ | Hicken et al. (2009b)
SDSSc | 202 | 0 | 12 | 0.037 | 0.166 | 0.250 | $ugriz$ | Holtzman et al. (2008)
SNLS | 111 | 63 | 63 | 0.149 | 0.499 | 0.700 | $griz$ | Astier et al. (2006)
Misc. low-$z^{\rm c}$ | 25 | 152 | 216 | 0.001 | 0.009 | 0.077 | $UBVRI^{\rm b}$ | Jha et al. (2007)
SALT2.JLA Total | 415 | 400 | 1057 | 0.001 | 0.172 | 0.700 |  $\cdots$ |  $\cdots$
CfA4c | 30 | 0 | 0 | 0.009 | 0.029 | 0.070 | $BVri$ | Hicken et al. (2012)
CSP | 13 | 0 | 36 | 0.011 | 0.029 | 0.058 | $uBVgri$ | Krisciunas et al. (2017)
Foundation | 153 | 0 | 114 | 0.005 | 0.034 | 0.111 | $griz$ | Foley et al. (2018)
PS1 MDSc | 266 | 0 | 0 | 0.026 | 0.297 | 0.630 | $griz$ | Scolnic et al. (2018)
DES-spec | 206 | 0 | 0 | 0.078 | 0.362 | 0.850 | $griz$ | Abbott et al. (2019)
New Data Total | 668 | 0 | 150 | 0.005 | 0.244 | 0.850 |  $\cdots$ |  $\cdots$
SALT3 Total | 1083 | 400 | 1207 | 0.001 | 0.201 | 0.850 |  $\cdots$ |  $\cdots$
Note. SNe Ia in the K21 compilation, after cuts: the original JLA training
sample is reduced to 415 SNe (originally 420) and the new data adds 668 SNe,
for a total of 1083 SNe. a Total number of spectra, rather than number of SNe
with spectra, which is 83 for the JLA sample and an additional 297 from the
new data included here. Spectra are from Filippenko et al. (1992); Wells et
al. (1994); Patat et al. (1996); NAT (1997); Li et al. (2001); Salvo et al.
(2001); Valentini et al. (2003); Anupama et al. (2005); Benetti et al. (2004);
Kotak et al. (2005); Leonard et al. (2005); Garavini et al. (2007); Stanishev
et al. (2007); Thomas et al. (2007); Foley et al. (2008); Pignata et al.
(2008); Wang et al. (2009); Foley et al. (2010); Östman et al. (2011); Walker
et al. (2011); Blondin et al. (2012); Silverman et al. (2012b); Folatelli et
al. (2013); Balland et al. (2018) and from private communication with M.
Betoule and C. Balland. b We note that these filter transmission curves were
not provided for these samples. c 11 SNe in these samples have additional data
from the CSP (Contreras et al., 2010; Krisciunas et al., 2017), while SDSS SNe
2006oa, 2006ob, 2006on, and 2006nz also have data from CfA3 (Hicken et al.,
2009b). c See also Rest et al. (2014) and Jones et al. (2018).

###  4.4 Sample Selection Cuts

To ensure that SNe are suitable for inclusion in a training sample, we first
require that every SN in the compilation is a spectroscopically classified,
Branch-normal SN Ia (Branch et al., 1993; we include 1991T-like SNe Ia
following the original SALT2 training procedure; Guy et al., 2007). For the SN
light curves, we make the following selection requirements (cuts):

  * •

At least four epochs at phases between $-10<p<35$ days, where $p$ is the rest-
frame phase relative to time of B-band peak.

  * •

At least one measurement after peak brightness ($5<p<20$), to constrain the
shape.

  * •

At least one measurement in each of at least two filters at $-8<p<20$ to
constrain the color of the SN.

  * •

At least one measurement prior to peak brightness ($-10<p<-1$) to ensure a
well-measured time of maximum light. This is the only cut that was not
included in the original SALT2 training4445 of 420 SNe from the JLA sample are
removed with this cut..

For the spectra, we include the original SALT2.JLA training spectra in
addition to spectra taken from the kaepora database and spectra taken as part
of the Foundation Supernova Survey. A number of these spectra were taken by
the Foundation team, but most have been published on the Transient Name
Server555https://wis-tns.org/ as classification spectra.

To ensure minimal host galaxy contamination in the high-$z$ SN spectra used
for model training, Guy et al. (2007) fit the spectra with a combined model
including the predicted SN spectrum, the spectral recalibration parameters,
and a galaxy model (elliptical, S0, Sa, Sb, and Sc templates). They removed
spectra for which there was evidence for host galaxy contamination at the 68%
confidence level.

For Foundation and the additional low-$z$ SN spectra included here, low-$z$
SNe are much brighter relative to their host galaxies than at the redshifts
probed by SNLS and SDSS. We clip host galaxy lines, mask regions with
uncorrected telluric features, and remove excessively noisy or poorly
calibrated regions of each spectrum, but do not attempt to subtract a host
galaxy continuum. We remove a handful of spectra with poor quality from visual
inspection. After cuts there are 114 spectra from Foundation and 693 from
Kaepora, a subset of which are shown in Section 5. All but five Foundation SNe
have redshifts measured from host galaxy features. The complete training data
are available at https://saltshaker.readthedocs.io.

##  5 The SALT3 Model: Extending the Wavelength Range and Training on
Pantheon, Foundation, and DES Data

Having demonstrated the effectiveness of SALTshaker on the original JLA
training data, we now train a SALT3 model using additional data, extending the
free model parameters in SALTshaker in three ways. First, we extend the SED
wavelength range to 11000 Å so that rest frame filters centered at wavelengths
up to $\sim 8500$ angstroms, such as the Foundation $z$-band, can be fit with
the model. Secondly, we extend the SED wavelength range over which the color
law is fit with a polynomial (Eq. 2) to 8000 Å , just below the central
wavelength of the PS1 $z$ band. We use a fourth-order polynomial, rather than
the third-order polynomial used for SALT2.JLA, to model the color law over the
increased wavelength range with $\times 2.5$ more training data. Third, the
color scatter model is changed from a third to a fourth-order polynomial to
allow additional flexibility over the increased wavelength range.

###  5.1 Validation on Extended Wavelength Range

Before presenting our model trained on all data in Section 5.2, we check the
performance on separate training and validation samples. We take the
“K21valid” compilation as our cosmology sample using our “SALT3.K21train”
model as described in Section 3.1, and show the resulting Hubble residuals in
Figure 8 and in row 3 of Table 5. We find that nuisance parameters are similar
between SALT2.JLA and the SALT3.K21train model, with $\sigma_{\textrm{int}}$
slightly higher by $0.01\text{\,}\mathrm{m}\mathrm{a}\mathrm{g}$ but with a
consistent $RMS(\Delta\mu)$ (the SALT3 RMS is negligibly smaller).
$\textrm{Diff}(\Delta_{z}\mu)$ is consistent with zero at the mmag level.

Figure 11: RMS of Hubble residuals (panel a) and quadratic means of
uncertainties in distances and light-curve parameters (panels b-e) of our
compilation using two SALT models in four logarithmically spaced redshift bins
from $0.015<z<1$. SALT3.K21 improves measured RMS Hubble scatter across our
entire redshift range. The smallest improvements appear to be in the
intermediate redshift range $z\sim 0.2$, with larger improvements at both
lower and higher redshifts where improvements to the red and blue regions of
the rest-frame model are most important.

###  5.2 Training on Complete K21 Compilation

Finally, we train our best SALT3 model, which we call SALT3.K21, using all of
the data described in previous sections as a training sample. The sample
includes 1083 SNe, a factor of 2.5 more SNe than the JLA training sample, and
1207 spectra, a factor of three increase in the number of spectra. Synthetic
light curves from SALT3.K21 and SALT2.JLA are compared in Figure 12 and the
model uncertainties are compared in Figure 13. We see good consistency between
SALT2.JLA and SALT3.K21, with modest differences in the $u$-band and some
additional differences in redder bands; at both wavelength ranges we have
substantially increased the training sample (see Figure 10). Similarly, as
shown in Figure 14, the color law is consistent with that of SALT2.JLA to
within 1% across the entire wavelength range.

Figure 12: Comparisons of synthetic light curves. Upper two panels show light
curves from both SALT2.JLA and SALT3.K21 in arbitrary units of flux.
Differences are most prominent in Bessell I band. Lower panels show a family
of SALT3 lightcurves created by varying the $x_{1}$ parameter. Figure 13:
Color scatter as a function of central filter wavelength (left) and example
light curves (right) with model uncertainties from SALT2.JLA (blue) and
SALT3.K21 (red). SALT3 has comparable errors in the $u$ and $g$ bands but much
smaller uncertainties in the $r$ and $i$ bands due to training data with much
better coverage at those wavelengths, especially at large or small $x_{1}$
(with errors in both models blowing up at very early phases). Only the SALT3
model covers the $z$ band (right-most panel). On the left, dashed lines
illustrate the wavelengths where the color scatter is unconstrained by data
from the JLA sample (blue) or the K21 compilation (red; grey shading). Figure
14: Comparison of SALT2.JLA versus SALT3.K21 color law, along with the
extinction curves of Cardelli et al. (1989) and Fitzpatrick (1999) for
comparison. For all four curves $R_{V}$ is fixed to 3.1.

We compare Hubble residuals of the SALT3.K21 and SALT2.JLA models in Figure 8.
Individual standardized distances are consistent to
$0.05\text{\,}\mathrm{mag}$ between the two models, and the effects on the
Hubble diagram are found in row 4 of Table 5. $\textrm{Diff}(\Delta_{z}\mu)$
is consistent with zero at $2(14)\text{\,}\mathrm{mmag}$. Finally, row 4 of
Table 5 shows nuisance parameters and Hubble diagram metrics, demonstrating
that SALTshaker produces a new SALT3 model with slightly lower total
dispersion, consistent $\sigma_{\textrm{int}}$, and consistent distances. We
note that the $\beta$ parameter is lower by 0.24 in the SALT3 model, likely
due to a reconsidered separation of color and stretch.

####  5.2.1 Model Uncertainties and Hubble Scatter of SALT3.K21

The uncertainties in Figure 13 show that both the color scatter and light-
curve uncertainties in redder bands are much lower in SALT3.K21 than
SALT2.JLA. The SALT2 color scatter was effectively unconstrained by the JLA
data past mean passband wavelengths $\sim 8000$ Å, and we find that while the
color scatter is significant at these wavelengths, it is much smaller than
implied by the SALT2.JLA model. Our additional data constrains this region up
to $\sim 8500$ Å, and redder data will be required to see how this effect
carries into the NIR. We also note lower color scatter by $\sim 1\%$ in the
blue, which we attribute to improved relative calibration of the training
sample.

Fitting the light curves using the SALT3.K21 model, we compare the
uncertainties in distances and light-curve parameters to those found using
SALT2.JLA. In Figure 11 we compare performance across the redshift range. Our
model shows reduced Hubble scatter and distance uncertainties over the
SALT2.JLA model at nearly every redshift, with the least improvement at
moderate redshifts $z\sim 0.2$, where signal to noise is high and the
SALT2.JLA model is already performing well. Light-curve parameters from
SALT3.K21train have smaller uncertainties across the redshift range, with the
exception of $x_{1}$ uncertainties. Our largest improvements, particularly in
color uncertainty, are found at low redshift, where the improved red
wavelength coverage of our model allows the use of additional light curve
bands in fitting these SNe, providing stronger constraints.

Figure 15: Upper panel: RMS$(\Delta\mu)$ versus the number of additional
filters used in SALT3.K21 (red) light-curve fits as compared to fits made
using SALT2.JLA (blue). The number of SNe and median redshift are printed
above each bin. Lower panel: RMS$(\Delta\mu)$ versus sample for SALT3.K21
(red) and SALT2.JLA (blue), for all samples which have more than 30 SNe Ia in
our compilation. As we move to surveys dedicated to searches at higher
redshift, fewer SNe have data that could not be fit with SALT2, and the
advantage of SALT3’s wavelength range becomes less important.

There is a slight improvement in RMS Hubble residuals, largely attributable to
decreased color uncertainties in the low redshift sample. Breaking this down
further, Figure 15 shows the Hubble scatter binned by the number of additional
filters used in the SALT3.K21 light-curve fit as compared to the SALT2.JLA
light-curve fit in Figure 15. Where two additional filters are available,
RMS$(\Delta\mu)$ improves by $\sim 10\%$. We conclude improvement in the SALT3
model is most noticeable when it allows us to fit SNe with existing light
curves in filters out of the SALT2.JLA wavelength range. At high redshift,
SALT3’s improved constraints on the NUV model reduce Hubble scatter by $\sim
0.01$ mag, although there is typically not sufficiently red data to take full
advantage of the extended wavelength range at these redshifts. Similarly, when
breaking down results by survey, as we show in Figure 15, the greatest
improvement is in the low-$z$ CFA3 and Foundation samples where we can make
use of $I$ and $iz$ band observations (respectively) previously unused in
SALT2.JLA-based analyses at low redshift. We are able to reduce the
RMS$(\Delta\mu)$ of the Foundation sample from $0.144\pm 0.001$ mag to 0.125
mag, an improvement of 15%.

####  5.2.2 Comparison of SALT3.K21 and SALT2.JLA Light-curve Parameters

The SALTshaker training procedure results in different distributions of light-
curve parameters compared to SALT2.JLA. Some differences are due to changes in
how the SALT3 model is defined, while others are from the demographics of the
training sample. For example, a $x_{1}$/$M_{1}$ degeneracy is broken by
setting a constraint on the standard deviation $\sigma_{x_{1}}=1$, both in our
procedure and in the original training procedure; therefore including
additional data with higher stretch increases the scale of the $M_{1}$
component.

Figure 16: Scatterplot of the $x_{1}$ and $c$ light-curve parameters measured
with SALT2.JLA and SALT3.K21. We observe a linear transformation of the
parameters, as expected from changes in the demographics of the training
sample and our procedure for separating stretch and color. Best fit
$c$-$x_{1}$ lines and correlation coefficients $r$ are shown to illustrate the
rotation of the distribution as SALT3 forces the independence of $x_{1}$ and
$c$.

In Figure 16 we compare the distributions of the $x_{1}$ and $c$ parameters
from both models. We note a slight rotation, shift, and scale change of the
color/stretch distribution relative to SALT2.JLA. These linear transformations
cancel in the Tripp standardization of distance, absorbed into the nuisance
parameters $\alpha,\beta,\mathcal{M}$. The offset in color in particular is
from including bluer high redshift SNe from PS1 and DES in the training
sample. The rotation is due to the distinct procedures for separating stretch
and color between the SALT2 training code and SALTshaker. SALT3.K21 $x_{1}$
and $c$ values can be approximated from the SALT2.JLA values by the
transformations

$\displaystyle x_{1}^{(\text{SALT3})}\approx$ $\displaystyle
1.028x_{1}^{(\text{SALT2})}+0.138c^{(\text{SALT2})}+0.005$ (18) $\displaystyle
c^{(\text{SALT3})}\approx$ $\displaystyle
0.002x_{1}^{(\text{SALT2})}+0.985c^{(\text{SALT2})}+0.013.$ (19)

To test the similarity of the shape of the distributions, we use the two
dimensional Kolmogorov-Smirnov -like test statistic of Press & Teukolsky
(1988). We modify the procedure by linearly transforming both sets of light-
curve parameters to have the same mean, standard deviation, and $x_{1}$-$c$
correlation before calculating the statistic. We find the test statistic for
the SALT2.JLA and SALT3.K21 $x_{1}$ and $c$ parameters, then bootstrap
resample the data and calculate the test statistic with the resampled data. We
derive a p-value$=0.895$, and conclude that we cannot distinguish the shape of
the underlying distributions.

Table 5: Summary of SALT Model Nuisance Parameters and Relative Distance Difference for Different Training Sets Sample | Nuisance Parameters | Hubble Flow Dist.
---|---|---
Model | $N_{\text{Training}}$ | Validation Sample | $\alpha$ | $\beta$ | $\sigma_{int}$ | RMS$(\Delta\mu)$ | $\textrm{Diff}(\Delta_{z}\mu)$
| SALT2.JLA | 420 | large sim. JLAb | $0.151\pm 0.003$ | $3.101\pm 0.031$ | $0.098$ | 0.173 |  $\cdots$
1 | SALT3.simJLA | 420 | large sim. JLAb | $0.165\pm 0.003$ | $2.854\pm 0.028$ | $0.103$ | 0.182 |  $-0.007\pm 0.011$
| SALT2.JLA | 420 | JLA | $0.131\pm 0.009$ | $3.230\pm 0.114$ | $0.128$ | 0.153 |  $\cdots$
2 | SALT3.JLA | 420 | JLA | $0.142\pm 0.011$ | $2.952\pm 0.124$ | $0.145$ | 0.155 |  $0.032\pm 0.028$
| SALT2.JLA | 420 | K21 Valid | $0.151\pm 0.008$ | $3.064\pm 0.086$ | $0.103$ | 0.148 |  $\cdots$
3 | SALT3.K21train | 541 | K21 Valid | $0.115\pm 0.008$ | $2.967\pm 0.091$ | $0.118$ | 0.152 |  $0.013\pm 0.020$
| SALT2.JLA | 420 | K21 Full | $0.139\pm 0.005$ | $3.011\pm 0.060$ | $0.103$ | 0.146 |  $\cdots$
4 | SALT3.K21 | 1083 |  K21 Full |  $0.140\pm 0.005$ |  $2.833\pm 0.057$ |  $0.110$ | 0.143 |  $-0.002\pm 0.014$
Cosmological results from light-curve fits using SALT3 models created with
different training sets, compared to results when using the SALT2.JLA model.
Training sets include a simulated JLA training sample (simJLA, Row 1), the
real JLA training sample (JLA, Row 2), an expanded training sample using the
full JLA training sample but including only half the additional data we use,
with the remainder used for validation (K21train, Row 3), and the full K21
compilation (highlighted in Row 4), which is the training sample used to
create the SALT3 model published in this work. Our best model using the
complete K21 compilation results in nuisance parameters and distances
statistically consistent but slightly lower Hubble scatter as compared to
SALT2. a Relative to the equivalent SALT2 fitting results, the distance
between average Hubble residual at $0.01<z<0.2$ and the average Hubble
residual at $0.4<z<0.6$. b Large, combined simulations of CfA3, SDSS, and SNLS
with a total of $\sim$3000 SNe to measure distance biases more precisely than
is possible from a sample with the size of the JLA training sample

##  6 Conclusions

We have presented SALTshaker, a new Python-based training code to train a
phenomenologically motivated light-curve model using the SALT framework, in
addition to a retrained SALT model we call SALT3. SALTshaker is publicly
available at https://github.com/djones1040/SALTShaker with documentation at
https://saltshaker.readthedocs.io/en/latest/. The SALTshaker documentation
includes links to the training data and the SALT3.K21 model, and the SALT3.K21
model is compatible with and included in the latest versions of SNANA and
sncosmo.

The SALT3.K21 model itself includes updated calibration with Supercal, a
training sample with 1083 SNe $-$ 2.5 times larger than previous training
samples $-$ and extends to the rest-frame $iz$ bands. Due to its larger
wavelength range, we find that SALT3.K21 distances for both legacy low-$z$
data and Foundation data are approximately 15% more precise, equivalent to
increasing the low-$z$ sample size by 30%. The SALT3.K21 model is based on
updated calibration with Supercal (Scolnic et al., 2015) and revised MW E(B-V)
estimates from Schlafly & Finkbeiner (2011). As part of an upcoming cosmology
analysis we have employed SALT3.K21 within the PIPPIN (Hinton & Brout, 2020)
framework, generating simulations and performing bias corrections following
the methodologies of Kessler et al. (2019) and Kessler & Scolnic (2017).
Although this work is preliminary, we find that cosmological parameters found
using a larger light-curve dataset of $\sim 2000$ SNe Ia (to be released in
Brout+ in prep) are consistent between SALT3.K21 and SALT2.JLA. Looking to
future missions, we find that for the forecast SN Ia survey of the Roman Space
Telescope, assuming the ALL$z$ strategy (Hounsell et al., 2018), the extended
wavelength range in SALT3.K21 makes use of $\sim 20\%$ more observations
compared to SALT2.JLA; this increase is about a factor of 2 for redshifts
$z<0.5$, and falls to about a 10% increase at $z=1.5$.

Several light-curve models have been developed for cosmological supernovae,
including MLCS (Riess et al., 1996), MLCS2k2 (Jha et al., 2007), SiFTO (Conley
et al., 2008), SNooPy (Burns et al., 2011), SNEMO (Saunders et al., 2018),
SUGAR (Léget et al., 2019), and BayesSN (Mandel et al., 2011, 2020). In the
context of other modern light-curve models, SALT3 offers an approach to model
design and training process that prioritizes the use of heterogeneous spectral
and photometric data to provide extensive phase and wavelength coverage and
native k-corrections through cosmology-independent training. The model
framework is minimally changed from SALT2, so SNANA simulations, bias
corrections, and other analysis products are expected to require little
revision.

Over the coming years, we expect SALTshaker will continue to be developed and
improved as additional SN data becomes available and additional SN
standardization parameters (e.g., host mass) are discovered and explored.
Further development work could focus on the error model, which is currently
based on central filter wavelengths rather than integrated quantities. This is
a potential source of systematic uncertainty because observer-frame filter
functions are contracted in the rest frame. Additionally, SALTshaker enables a
more rigorous evaluation of systematic uncertainties such as those arising
from limited training data, photometric calibration uncertainties, or
treatment of SN spectra. These can be evaluated in a straightforward and
rigorous way by re-training the SALT3 model surfaces on simulated data.
Although we have demonstrated that SALTshaker can faithfully recover a truth
model at the $\sim$1% level, future work will also be able to fully validate
the model training process using an entire analysis chain that includes
training, bias corrections, and cosmology fitting (Dai et al in prep).
Similarly, while the SALT3 model surfaces presented in this work have been
trained on data recalibrated to the level of $1\%$ via the Supercal procedure
(Scolnic et al., 2015, 2018), we have left quantifying the reduced calibration
uncertainties as a topic for further work.

The BYOSED code (Pierel et al., 2020) implements a range of simulated effects
to a base SED template, such as perturbations to line velocity, multiple
sources of reddening with distinct effects, and correlations of host galaxy
properties with the SN Ia SED. Future work could perform the SALTshaker method
on a BYOSED-produced training sample with such underlying effects. By
propagating any biases introduced by the training code into cosmology, we may
quantify the potential impact of currently unmodeled supernova phenomenology
on cosmology.

Samples of SN Ia light curves will increase by orders of magnitude with the
Vera Rubin Telescope’s LSST (Ivezić et al., 2019) and the Roman Space
Telescope (Hounsell et al., 2018). For error budgets to continue improvement,
light-curve models should not be tied to outdated calibration standards, and
it is essential that the model training process be regarded as a key component
of an integrated cosmology analysis, as has been done in Betoule et al. (2014)
and Mosher et al. (2014).

We thank Julien Guy and the developers of the SALT and SALT2 models for their
contributions to the field, Julien Guy, Chris Lidman, and Georgie Taylor for
valuable assistance and discussion, and Julien Guy, Marc Betoule, and the SNLS
team for providing the original snpca code and JLA training sample. We are
grateful for all astronomers who acquired the photometry and spectroscopy used
to train the SALT3.K21 model. D.O.J. is supported by a Gordon and Betty Moore
Foundation postdoctoral fellowship at the University of California, Santa Cruz
and by NASA through the NASA Hubble Fellowship grant HF2-51462.001 awarded by
the Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., for NASA, under contract
NAS5-26555. Support for this work was provided by the STScI Director’s
Discretionary Fund. This research at Rutgers University (S.W.J., M.D.) was
supported by NASA contract NNG16PJ34C and DOE award DE-SC0011636. M.D. is also
supported by the Horizon Fellowship at the Johns Hopkins University. This work
at the University of South Carolina (S. Rodney, J. D. R. Pierel) was supported
by grant HST-AR-15808 from the Space Telescope Science Institute. This work
was supported by NASA contract No. NNG17PX03C issued through the Nancy G.
Roman Science Investigation Teams Program. This research is based on
observations made with the NASA/ESA Hubble Space Telescope obtained from the
Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555.
These observations are associated with programs GO-4016, GO-4252, GO-8611,
GO-9114, and GO-10182. This research is based on observations made with the
International Ultraviolet Explorer, obtained from the MAST data archive at the
Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555.
These observations are associated with programs METOO, SNMRK, SNNRK, STKRK,
and VILSP. This paper is based in part on observations made with the Southern
African Large Telescope (SALT) using the Robert Stobie Spectrograph (RSS),
through allocations made to Rutgers University via programs 2015-1-MLT-002,
2016-1-MLT-007, and 2017-1-MLT-002 (PI: S.W. Jha). Based in part on
observations obtained at the Southern Astrophysical Research (SOAR) telescope
(NOIRLab Prop. IDs 2015A-0253, 2015B-0313, 2017A-0306, 2017B-0169, 2018A-0277;
PI: R. Foley), which is a joint project of the Ministério da Ciência,
Tecnologia e Inovações do Brasil (MCTI/LNA), the US National Science
Foundation’s NOIRLab, the University of North Carolina at Chapel Hill (UNC),
and Michigan State University (MSU). Based in part on observations at Kitt
Peak National Observatory at NSF’s NOIRLab (NOIRLab Prop. IDs 2015A-0253,
2015B-0313, 2017A-0306, 2017B-0169, 2018A-0277; PI: R. Foley), which is
managed by the Association of Universities for Research in Astronomy (AURA)
under a cooperative agreement with the National Science Foundation. The
authors are honored to be permitted to conduct astronomical research on Iolkam
Du’ag (Kitt Peak), a mountain with particular significance to the Tohono
O’odham. A major upgrade of the Kast spectrograph on the Shane 3 m telescope
at Lick Observatory was made possible through generous gifts from the Heising-
Simons Foundation as well as William and Marina Kast. Research at Lick
Observatory is partially supported by a generous gift from Google. Some of the
data presented herein were obtained at the W. M. Keck Observatory, which is
operated as a scientific partnership among the California Institute of
Technology, the University of California and the National Aeronautics and
Space Administration. The Observatory was made possible by the generous
financial support of the W. M. Keck Foundation. The authors wish to recognize
and acknowledge the very significant cultural role and reverence that the
summit of Maunakea has always had within the indigenous Hawaiian community. We
are most fortunate to have the opportunity to conduct observations from this
mountain.

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