Approximate Energy-Integration Method for Identifying Collisional Neutrino Flavor Instabilities

Abstract

We present an approximate energy-integration method for identifying collisional neutrino flavor instabilities. Direct evaluation of the dispersion relation requires multi-dimensional integrals over neutrino phase space, making systematic searches for unstable modes in numerical models of core-collapse supernovae (CCSNe) and binary neutron star mergers (BNSMs) computationally expensive. In the literature there are some approximate schemes, but they are largely restricted to the homogeneous limit and can exhibit inaccuracies as reported in recent studies. In the current paper, we clarify the origin of the limitations in previous schemes and provide a better approximation method that robustly preserves the key physics of spectral asymmetries and collision rates. It yields a reduced dispersion relation that is inexpensive to evaluate. Comparison with exact solutions demonstrates that our new approximate method shows a good performance in computing both real frequencies and growth rates across a wide range of regimes, including isotropic and anisotropic neutrino distributions for both homogeneous and inhomogeneous modes. This provides a practical, accurate, and scalable framework for identifying collisional flavor instabilities in high-energy astrophysical simulations such as CCSNe and BNSMs.

Figures (11)

• Figure 1: Schematic decomposition of $E^{2}\Delta f(E)$ and $E^{2}\Delta\bar{f}(E)$ into four signed contributions (labeled $1$--$4$). The effective densities entering methods A/B and the present method are constructed from different combinations of these areas.
• Figure 2: Schematic locations of CFI solutions in the complex $\omega$-plane. In the non-resonance regime, the solutions separate into a near mode with $\mathrm{Re}\,\omega \approx 0$ and a far mode with $|\mathrm{Re}\,\omega| \sim 2|A|$. In the resonance regime, the pair of solutions is approximately symmetric about the origin with nonzero real and imaginary parts. The figure illustrates the IP branch; the IB branch behaves analogously.
• Figure 3: Effective coefficients $G^X$ and $A^X$ as functions of $g_{\nu_x}$ for the isotropic $k=0$ models. Model I (solid) corresponds to a non-resonance configuration, while Model II (dotted) is tuned to $A^X=0$.
• Figure 4: Effective collision rate parameters $\gamma^X$ (left) and $\alpha^X$ (right) for Model I. The shaded region indicates strong cancellation in the energy integrals. Method A exhibits divergences, while methods B and C remain finite.
• Figure 5: Real and imaginary parts of the collective eigenfrequencies for Model I. Left and right panels correspond to far and near modes, respectively. In the upper left panel, the blue and green dashed lines closely overlap with the black solid curve. Method A deviates strongly for the near mode in the strong-cancellation regime, while method C remains accurate across the full range.