SALT3 Light Curve Model for SNe Ia

• SALT3 Light Curve Model is a PCA-inspired, empirically trained SED framework that standardizes Type Ia supernova observations.
• It refines uncertainty modeling and orthogonalizes color and stretch effects while extending wavelength coverage into the near-infrared.
• Integration with SNCosmo and SNANA enables robust light-curve fitting and improved cosmological distance estimations with reduced systematic errors.

The SALT3 light curve model constitutes a principal component analysis (PCA)-inspired, empirically trained spectral energy distribution (SED) model for Type Ia supernovae (SNe Ia), optimized for cosmological distance determinations. SALT3 extends and refines the widely used SALT2 framework, achieving improved uncertainty modeling, orthogonal separation of color and light-curve width (“stretch”) effects, and expanded wavelength/phase coverage. The model has been further generalized to the near-infrared (“SALT3-NIR”) to meet the demands of current and forthcoming SN Ia surveys extending to wavelengths up to 2 μm, thus reducing systematic uncertainties in light curve standardization for cosmological applications (Pierel et al., 2022, Kenworthy et al., 2021, Taylor et al., 2023).

1. Core Mathematical Structure

The SALT3 light curve model parameterizes the rest-frame SED of a SN Ia as a function of phase ($p$, days since $B$-band maximum) and rest-frame wavelength ($\lambda$):

$$F(p,\lambda) = x_0 \left[ M_0(p,\lambda) + x_1 M_1(p,\lambda) \right] \exp[c\,CL(\lambda)]$$

where:

• $M_0(p,\lambda)$ is the empirical mean SED surface,
• $M_1(p,\lambda)$ is the first principal variation, capturing stretch-dependent SED changes,
• $CL(\lambda)$ is a wavelength-dependent color law incorporating both intrinsic and dust-driven color diversity,
• $x_0$ is the normalization parameter (proportional to peak apparent flux),
• $x_1$ is the dimensionless stretch parameter,
• $c$ is the color parameter (relative $B-V$ color).

The predicted observer-frame broadband flux for filter $X$ and epoch $t$ is computed as:

$$f_X^\mathrm{model}(t) = \int T_X\left(\frac{\lambda}{1+z}\right) F\left(p(t),\lambda\right) \lambda\,d\lambda$$

with $T_X(\lambda)$ being the filter transmission and $p(t)=(t-t_0)/(1+z)$.

The distance modulus is derived through the Tripp estimator:

$$\mu = -2.5 \log_{10} x_0 + \alpha x_1 - \beta c - M_B$$

with $\alpha$ and $\beta$ as empirically constrained stretch–luminosity and color–luminosity coefficients, and $M_B$ the fiducial absolute $B$-band magnitude.

2. Model Training and Regularization

SALT3 is trained via the open-source SALTShaker pipeline, which performs joint likelihood maximization over:

• The SED surfaces ($M_0$, $M_1$),
• The polynomial color law $CL(\lambda)$,
• Per-SN light-curve parameters ($x_0$, $x_1$, $c$),
• Per-spectrum smooth polynomial “warp” factors correcting relative spectrophotometry,
• Phase–wavelength-dependent “in-sample” flux variance,
• Wavelength-dependent color dispersion.

The basis surfaces $M_0$ and $M_1$ are represented on a $(p, \lambda)$ grid (typically via second-order or cubic B-splines) to ensure smoothness and facilitate regularization. Regularization terms penalize rapid or nonphysical fluctuations in both phase and wavelength and discourage non-separable ($\partial_p\partial_\lambda$-like) structure in the basis components, with the strength set by the local density of spectroscopic data.

Degeneracies among $x_0$, $x_1$, $c$, and their corresponding model surfaces are controlled via sample-anchored priors:

• $x_1$ is fixed to satisfy $\langle x_1 \rangle = 0$, $\sigma(x_1) = 1$,
• $\langle c \rangle = 0$,
• $x_1$ and $c$ are forced uncorrelated in the training sample,
• $M_0$ is normalized at $B$-band peak for a fiducial event,
• $CL(\lambda)$ is anchored at 4302.57 Å ($B$) and 5428.55 Å ($V$).

An alternating optimization loop (e.g., using Levenberg–Marquardt or Minuit backends) sequentially updates the basis surfaces, error model, color law, and per-SN parameters until convergence (Pierel et al., 2022, Kenworthy et al., 2021, Taylor et al., 2023). The error model constrains both in-sample variance (with binned error surfaces in phase and wavelength) and out-of-sample variance (from the inverse Hessian of the fit).

3. Wavelength/Phase Coverage and Data Sets

SALT3 significantly extends the empirical wavelength and phase domain relative to SALT2. The canonical SALT3.K21 model covers $2000$–$11000$ Å and phases from $-15$ to $+60$ days, enabling native fits to observed $I$ and $z$ bands at low redshift and robust performance across $0.001 < z < 0.85$ (Kenworthy et al., 2021). Training employs a cross-calibrated, multi-survey photometric and spectroscopic compilation ($\sim 1080$ SNe, $> 1200$ spectra), tied to a consistent absolute photometric system via the “SuperCal” method.

For SALT3-NIR (Pierel et al., 2022), the basis is expanded to $20,000$ Å (2 μm) and the color law is fit as a polynomial up to 12,500 Å (linear extrapolation to 2 μm). The NIR extension leverages additional SNe from public CfA, CSP, HST (SIRAH, RAISIN), and UKIRT DEHVILS programs, adding $\sim166$ SNe with well-sampled NIR photometry ($Y, J, H$ bands) and 51 NIR spectra.

4. Model Validation and Quantitative Performance

SALT3 achieves improved empirical performance over SALT2 and other light curve models:

• Model photometric uncertainties are reduced by up to 50% over SALT2 in the optical, with color-law uncertainties in $7000$–$9000$ Å reduced from $\sim10$% to $\sim1$–2% (Kenworthy et al., 2021, Taylor et al., 2023).
• SALT3-NIR achieves 2–3% precision in $M_0$ and 1–2% in $M_1$ across the full phase range, with smooth, well-calibrated behavior in NIR bands (Pierel et al., 2022).
• Utilizing two NIR bands ($YJ$) plus full light-curve parameter fitting, the Hubble diagram residual RMS drops to $\sim0.11$ mag—a $\sim30$% improvement over optical-only fitting ($\sim0.16$ mag); combining optical and NIR tightens the scatter by an additional $\sim$10–20%.
• Fitting with NIR bands alone can outperform optical-only regimes in several observational scenarios at $>$95% confidence.

The stretch parameter $x_1$ measured from NIR data is tightly correlated (slope $\sim$1.03) with the optical $x_1$, establishing a consistent stretch–luminosity relation into the infrared domain. The color parameter $c$ is only weakly constrained in NIR, consistent with diminished color variability.

For forthcoming wide-field surveys (e.g., Roman Space Telescope), SALT3-NIR enhances the fraction of usable SN Ia events within the SALT model framework by $\sim20$\% at $z \lesssim 0.4$ and $\sim50$\% at $z \lesssim 0.15$ due to redder filter coverage and NIR model extension (Pierel et al., 2022).

5. Integration with Cosmology Toolkits and Community Practices

SALT3 and SALT3-NIR are natively implemented in both SNCosmo and SNANA, the leading SN Ia cosmology analysis packages. Typical use in SNCosmo involves:

import sncosmo model = sncosmo.Model(source='salt3_nir')

SNANA supports both simulation and light-curve fitting modes with the SALT3 and SALT3-NIR surfaces; the $M_0$, $M_1$, $CL$ basis are accessed via standard model format files, and algorithms match those from SALT2 workflows.

The open-source SALTshaker training infrastructure enables custom retraining or extension as new data (especially in the NIR) become available, ensuring transparency, reproducibility, and community updatability for all aspects of the model training, regularization, and systematic priors (Pierel et al., 2022, Kenworthy et al., 2021).

6. Implications for Cosmological Analyses

The design and performance of SALT3 address key sources of systematic uncertainty in SN Ia cosmology:

• The model reduces the calibration-induced distance modulus error (systematic uncertainty on $w$ from model choice $\Delta w = +0.001 \pm 0.005$ is negligible at current statistical precision) (Taylor et al., 2023).
• Extension to 2 μm and inclusion of NIR data are critical for next-generation $z<0.5$ SN surveys, directly improving leverage for dark energy equation-of-state measurements and reducing sensitivity to optical-band extinction systematics (Pierel et al., 2022).
• The orthogonalization and robust error propagation offered by SALT3-SALTshaker reduce biases due to parameter degeneracies and population drift.
• Integration of host-galaxy specific models and further basis component expansion (e.g., SALT3+) are logical next steps for further reduction of residual biases as survey statistical power increases.

7. Future Developments and Extensions

SALT3 forms the basis for multiple model generalizations:

• SALT3-NIR provides the current open standard for rest-frame optical plus NIR light curves through 2 μm (Pierel et al., 2022).
• Host-dependent retraining (SALT3.HIGHMASS/LOWMASS) enables systematic studies of SN Ia population effects on cosmological inferences (Taylor et al., 14 Jan 2024).
• Additional principal components as in “SALT3+” (Kenworthy et al., 13 Feb 2025) capture higher-order variability (e.g., correlated color–secondary maximum effects), further reducing color-degeneracy systematics.

A plausible implication is that the SALT3 family will underpin precision SN Ia cosmology analyses into the LSST and Roman era, with continual empirical updates to accommodate new populations, multiple light-curve parameters, and ongoing improvements in calibration and data quantity.