Theory of neutrino fast flavor evolution. Part II. Solutions at the edge of instability

TL;DR

This paper develops a theoretical framework to analyze weakly unstable neutrino flavor modes in dense astrophysical environments, providing explicit formulas for growth rates and mode velocities, with implications for supernova and neutron-star merger modeling.

Contribution

It introduces a dispersion relation expansion for weakly unstable neutrino modes, deriving explicit expressions for growth rates and velocities, including axial-symmetry-breaking modes.

Findings

Unstable modes with small growth rates move at subluminal or near-light speeds.

Explicit formulas for growth rates of resonant neutrino modes.

Identification of instability ranges in wavenumber space for 1D systems.

Abstract

In dense neutrino environments, such as provided by core-collapse supernovae or neutron-star mergers, neutrino angular distributions may be unstable to collective flavor conversions, whose outcome remains to be fully understood. These conversions are much faster than hydrodynamical scales, suggesting that self-consistent configurations may never be strongly unstable. With this motivation in mind, we study weakly unstable modes, i.e., those with small growth rates. We show that our newly developed dispersion relation (Paper~I of this series) allows for an expansion in powers of the small growth rate. For weakly unstable distributions, we show that the unstable modes must either move with subluminal phase velocity, or very close to the speed of light. The instability is fed from neutrinos moving resonantly with the waves, allowing us to derive explicit expressions for the growth rate. For axisymmetric distributions, often assumed in the literature, numerical examples show the accuracy of these expressions. We also note that for the often-studied one-dimensional systems one should not forget the axial-symmetry-breaking modes, and we provide explicit expressions for the range of wavenumbers that exhibit instabilities.