A Generalized Kompaneets Formalism for Inelastic Neutrino-Nucleon Scattering in Supernova Simulations

TL;DR

This paper introduces a generalized Kompaneets formalism for inelastic neutrino-nucleon scattering in supernova simulations, ensuring conservation laws, scalability, and applicability to anisotropic distributions, enhancing spectral modeling accuracy.

Contribution

It develops a robust, scalable, and anisotropic-compatible formalism for neutrino-nucleon scattering, extending the classical Kompaneets equation for supernova simulations.

Findings

Conserves lepton number to machine precision.

Scales linearly with energy groups, improving computational efficiency.

Automatically reduces to zero energy transfer in thermal equilibrium.

Abstract

Based on the Kompaneets approximation, we develop a robust methodology to calculate spectral redistribution via inelastic neutrino-nucleon scattering in the context of core-collapse supernova simulations. The resulting equations conserve lepton number to machine precision and scale linearly, not quadratically, with number of energy groups. The formalism also provides an elegant means to derive the rate of energy transfer to matter which, as it must, automatically goes to zero when the neutrino radiation field is in thermal equilibrium. Furthermore, we derive the next-higher-order in {\epsilon}/mc2 correction to the neutrino Kompaneets equation. Unlike other Kompaneets schema, ours also generalizes to the case of anisotropic angular distributions, while retaining the conservative form that is a hallmark of the classical Kompaneets equation. Our formalism enables immediate incorporation into supernova codes that follow the spectral angular moments of the neutrino radiation fields.