A Core-Collapse Supernova Neutrino Parameterization with Enhanced Physical Interpretability

TL;DR

This work presents a diffusion-based parameterization of core-collapse supernova neutrino spectra centered on the physically interpretable quantity $\tau(t)=\int_{0}^{t} \kappa'(s)\,ds$, the integrated thermal-diffusion area. By solving a point-source diffusion problem and mapping to energy space, the authors derive a spectrum form that ties spectral evolution directly to the thermodynamic state of the explosion engine, offering an interpretable alternative to the KRJ parameterization. The model is validated against SN1987A data, yielding a statistically good fit with a tightly constrained $\tau$ and a broader $Q$, and is further applied to a suite of 3D CCSN simulations where $\tau(t)$ evolution discriminates exploding from failed models and correlates with gravitational-wave signals. A linear relation between the time-integrated diffusion quantity $\tau_{\rm int}$ and the integrated energy $Q_{\rm int}$ emerges for exploding models, and Smoothed Isotonic Regression constrains the SN1987A progenitor mass to about $19\,M_\odot$ within uncertainties; unsupervised clustering and UKF-based multi-messenger analyses reveal that $\tau(t)$ encodes the explosion dynamics and energetics, suggesting a promising diagnostic tool for decoding future galactic supernovae. While the framework is phenomenological and excludes detailed oscillation physics, it provides a concrete bridge between observable spectra and the core processes that drive explosions, offering a path toward richer, multi-messenger inferences with next-generation detectors.

Abstract

We introduce a novel parameterization of supernova neutrino energy spectra with a clear physical motivation. Its central parameter, $τ(t)$, quantifies the characteristic thermal-diffusion area during the explosion. When applied to the historic SN1987A data, this parameterization yields statistically significant fits and provides robust constraints on the unobserved low-energy portion of the spectrum. Beyond this specific application, we demonstrate the model's power on a suite of 3D core-collapse supernova simulations, finding that the temporal evolution of $τ(t)$ distinctly separates successful from failed explosions. Furthermore, we constrain the progenitor mass of SN 1987A to approximately 19 solar masses by applying Smoothed Isotonic Regression, while noting the sensitivity of this estimate to observational uncertainties. Moreover, in these simulations, $τ(t)$ and the gravitational-wave strain amplitude display a strong, synergistic co-evolution, directly linking the engine's energetic evolution to its geometric asymmetry. This implies that the thermodynamic state of the explosion is imprinted not only on the escaping neutrino flux, but also recorded in the shape of the energy spectrum. Our framework therefore offers a valuable tool for decoding the detailed core dynamics and multi-messenger processes of future galactic supernovae.

Figures (31)

• Figure 1: Left: SN1987A detected spectrum (black) with our best-fit model (red solid line), obtained using a Genetic Algorithm with Empirical Bayesian Guided Evolution to enable rapid convergence within a vast and largely unexplored parameter space (computational details are provided in the \ref{['PB']}). The fit shows good agreement with the data and is statistically significant. For comparison, we also plot the cold (green dashed) and hot (purple dashed) KRJ parameterized models from Fiorillo:2022cdq, as well as the KRJ model best fit curve obtained using the KRJ form and the same optimization method as the red line. Right: Corner plot of the posterior distributions of the parameters $\tau$ and $Q$, derived from the same optimization process. The red star marks the best-fit values, and the distributions indicate that $\tau$ is well constrained, while $Q$ has a relatively broader uncertainty range with a negative correlation between them.
• Figure 2: Horizontal axis: total neutrino energy released within 2 s of explosion, $Q_{\mathrm{int}} = \int_0^{2\,\mathrm{s}} Q(t)\,\mathrm{d}t$; vertical axis: time-integrated spectral parameter, $\tau_{\mathrm{int}} = \int_0^{2\,\mathrm{s}} \tau(t)\,\mathrm{d}t$. Numerals denote progenitor masses in $M_\odot$. (a) Shaded bands indicate the statistical uncertainty of the linear regression fit. (b) Dashed contours correspond to the 1$\sigma$, 3$\sigma$, and 5$\sigma$ credible regions in probability density, delineating the parameter space associated with Betelgeuse’s plausible progenitor mass range. Each contour represents the projected confidence boundary in its neutrino spectral parameters for a future explosion.
• Figure 3: Left: Spectral parameter versus progenitor mass, with the dashed line marking SN 1987A’s reference value. Models below and above $40\,M_\odot$ exhibit distinct distributions. Right: Smoothed Isotonic Regression applied to the low-mass models (red curve), with its intersection with SN 1987A’s value (red star) yielding an estimated progenitor mass of $\sim17\,M_\odot$. Shaded regions indicate 10%, 20%, and 30% parameter uncertainties.
• Figure 4: The temporal evolution of $\tau$ serves as a robust diagnostic of explosion outcome. (a) Unsupervised clustering based on engineered features of the $\tau(t)$ curves—specifically their volatility and rebound extent—separates successful (Pattern 0) and failed (Pattern 1) explosions into distinct clusters. The two KDE plots show the number distributions of Feature 1 and Feature 2 for different clustering patterns. (b) Representative $\tau(t)$ curves for each cluster, with blue shading indicating the time window of the first pronounced peak. Dashed curve segments are not physically meaningful, representing constant extrapolation beyond the simulation endpoint for visual clarity.
• Figure 5: (a) Figure demonstrates an example of model parameter evolution curves for a progenitor star with a mass of 18 solar masses. Additional models are presented in the \ref{['P:Parameter Evolution']}. (b) Figure illustrates the diff-corr parameters for $\tau$ and gravitational wave amplitude across various models. Notably, all of the models exhibit extremely high correlation.