SALT3 Framework for SN Ia Cosmology

Updated 13 November 2025

SALT3 framework is an empirical model for SN Ia spectral energy distribution evolution, standardizing observations with enhanced uncertainty treatment and broader wavelength coverage.
It models SN flux using parametric surfaces (M0 and M1) on phase and wavelength grids, employing regularization techniques to prevent unphysical fluctuations.
Its refined calibration and training pipeline boost sample quality and reduce systematic biases, enabling robust cosmological distance estimations.

The SALT3 framework is an empirical model for the rest-frame spectral energy distribution (SED) evolution of Type Ia supernovae (SNe Ia), designed to standardize heterogeneous multi-wavelength SN Ia observations and enable precision cosmological distance measurements. As a direct successor to the widely-adopted SALT2 model, SALT3 introduces a more rigorous uncertainty treatment, enhanced separation of color and light-curve shape, expanded wavelength coverage, and public, reproducible training tools. The model is used extensively in current and planned cosmological analyses from DES, LSST, Roman, and other surveys.

1. Mathematical Formulation and Model Structure

SALT3 models the rest-frame spectral flux density of a SN Ia as a parametric surface in phase ( $p$ ; days from $B$ -band maximum) and wavelength ( $\lambda$ ), modulated by three event-specific parameters: $F(p,\lambda) = x_0 \Big[ M_0(p,\lambda) + x_1\,M_1(p,\lambda) \Big] \exp\left[c\,\mathrm{CL}(\lambda)\right].$

$M_0(p,\lambda)$ : Mean SED surface (zeroth component)
$M_1(p,\lambda)$ : First principal component (encodes stretch/width-luminosity variations)
$x_0$ : Amplitude (related to the distance to the SN)
$x_1$ : Stretch parameter (light-curve width/shape)
$c$ : Color parameter (intrinsic and host-dust reddening)
$\mathrm{CL}(\lambda)$ : Phase-independent color-law, representing wavelength-dependent attenuation, fitted as a polynomial with canonical normalization ( $\mathrm{CL}(\lambda_B)=0$ , $\mathrm{CL}(\lambda_V)=-1$ )

$M_0$ and $M_1$ are constructed on regular grids in phase (20 nodes, $\sim$ 3 day spacing, $-20< p < +50$ ) and wavelength (127 nodes, $2000< \lambda < 11000$ Å).

2. Model Parameters and Distance Standardization

After fitting observed SN photometry and/or spectra, one extracts best-fit $(x_0, x_1, c)$ per SN. The apparent magnitude in the $B$ -band, $m_B$ , is given by

$m_B = -2.5 \log_{10}(x_0).$

Distance estimation uses the "Tripp formula" for the bias-corrected distance modulus: $\mu = m_B - M + \alpha x_1 - \beta c - \delta\mu_\text{host} - \delta\mu_\text{bias},$ where $\alpha, \beta$ are empirically determined nuisance parameters, $M$ is the standardized absolute magnitude, $\delta\mu_\text{host}$ is a host-mass correction, and $\delta\mu_\text{bias}$ accounts for selection effects. This formula underpins cosmological inference using SALT3-fitted SN samples.

3. Model Training Methodology and Regularization

SALT3 surfaces and uncertainties are trained on large, cross-calibrated samples of SNe Ia (e.g., 1083 light curves with $\sim$ 1207 spectra), using the open-source Python package SALTShaker. The training pipeline cycles between:

Initial fitting of $M_0$ , $M_1$ , $\mathrm{CL}$ , with fixed variability parameters.
Estimation of phase- and wavelength-dependent in-sample variance using spline interpolation.
Regular updating of the surfaces and variance parameters until self-consistency.
Extraction of out-of-sample covariance via the Hessian of the final fit.

Regularization is imposed to control overfitting in sparsely sampled regions:

Phase gradient penalty (new in SALT3)
Wavelength gradient penalty
Dyadic (non-separability) penalty Unlike SALT2, which applied only the latter two, this three-pronged approach further damps unphysical fluctuations and negative-flux artifacts, especially in UV/blue wavelengths.

4. Covariance Treatment and Uncertainty Propagation

SALT3 provides an absolute, phase-and-wavelength-dependent covariance model, separating:

Component surface uncertainties (from the parameter Hessian)
In-sample variance (modeled as a spline surface) The resulting model–model covariance $\mathrm{Cov}_\mathrm{model}(p,\lambda;p',\lambda')$ is provided for each surface. During light-curve fitting, the total covariance used in the chi-square is the sum of the data (photometric) covariance and the model covariance. This supersedes SALT2's relative error "snake," which lacked a rigorous iterative variance fit.

5. Quantitative Comparison: SALT2 versus SALT3

SED Surfaces and Color Law

$M_0(p,\lambda)$ agrees with SALT2 to within 2–3% for $3500<\lambda<8500$ Å and $-5< p <+5$ days but exhibits fewer negative-flux regions in the UV due to stronger regularization.
$M_1$ displays up to 5% differences at extreme stretch values, with SALT3 enforcing greater independence between $x_1$ and $c$ .
The color law is a 6th-degree polynomial in reduced wavelength, closely matching SALT2 in the optical ( $3000<\lambda<7000$ Å), with minor deviations at the ends due to linear extrapolation.

Calibration Sensitivity and Field Performance

Systematic-shift surfaces, generated to probe the impact of photometric calibration uncertainties, demonstrate that SALT3's $M_0$ shifts are $\lesssim$ 2.5% across the core optical range, approximately half the instability observed in SALT2.JLA.
In DES-SN3YR+low-z samples, SALT3 recovers $\sim$ 13 more SNe passing quality cuts than SALT2.FRAG, due to more realistic and generous fit probabilities.
Mean uncertainties: $\sigma_{m_B}$ is unchanged (∼0.044 mag); $\sigma_{x_1}$ increases slightly (by ∼0.03 mag), and $\sigma_{c}$ decreases marginally (by ∼0.003 mag).
Cosmological impact: switching from SALT2 to SALT3 changes $w$ by $\Delta w = +0.001 \pm 0.005$ , negligible compared to the total statistical and systematic error budget ( $\sigma_w \sim 0.06$ ).

6. Recommendations and Implications for Cosmological Analyses

Analysts are encouraged to use SALT3 surfaces (e.g. SALT3.FRAG) trained on modern, globally cross-calibrated samples and to propagate the full model covariance and systematic-shift surfaces to capture calibration uncertainties robustly.
SALT3's more sophisticated error model and robust regularization yield increased sample sizes (especially at low-z), more stable fit probabilities, and a lower susceptibility to selection bias and calibration-driven systematics.
The open-source SALTShaker and integration with SNANA and SNCosmo facilitate reproducibility and ease of use in both training and application.
The negligible cosmological bias observed on current data sets ( $|\Delta w| \ll \sigma_w$ ) supports immediate adoption of SALT3 for current and forthcoming surveys (e.g., DES Y5, LSST, Roman).

7. Broader Context and Extensions

SALT3's framework enables further generalizations:

Expansion to NIR wavelengths (SALT3-NIR) for future rest-frame optical+NIR surveys.
Custom training on SN subsets (e.g., host-mass splitting) to explore systematic population-dependent effects.
Model extensions (e.g., SALT3+) with additional principal components to capture residual systematic trends in SN standardization, though the incremental gains in cosmological precision are presently modest.

In summary, SALT3 preserves the empirical, two-component plus color-law architecture of SALT2, while improving upon it in uncertainty modeling, wavelength coverage, and open, reproducible software infrastructure. It reduces both intrinsic and calibration-driven systematics in SN Ia cosmology and provides a robust, extensible basis for next-generation cosmological analyses (Taylor et al., 2023, Kenworthy et al., 2021, Pierel et al., 2022).