Dispersion relation of the neutrino plasma: Unifying fast, slow, and collisional instabilities

TL;DR

This work develops a unified, perturbative dispersion-theory for a collisionless neutrino plasma that couples fast, slow, and collisional flavor instabilities. By treating vacuum-mass effects and collisions as small perturbations to the fast refraction scale, the authors derive algebraic, moment-based dispersion relations and classify modes into gapped and gapless, as well as fast, slow, and collisional families. They obtain orders-of-magnitude estimates for growth rates, highlight the near-luminal nature of slow and collisional instabilities, and show that these modes are typically slow-growing and nonlocal, challenging the common fast-mode local-evolution approximations. The framework also clarifies the thermodynamic aspects of collisional instabilities, proving an H-theorem extension and distinguishing irreversible collisional dynamics from conservative fast/s slow evolution. Overall, the results provide a transparent, quantitative lens for understanding linear flavor evolution across regimes and lay groundwork for nonlinear, multi-scale treatments in realistic astrophysical environments.

Abstract

In neutrino-dense astrophysical environments, these particles exchange flavor through a coherent weak field, forming a collisionless neutrino plasma with collective flavor dynamics. Instabilities, which grow and affect the environment, may arise from neutrino-neutrino refraction alone (fast limit), vacuum energy splittings caused by masses (slow limit), or neutrino-matter scattering (collisional limit). We present a comprehensive analytical description of the dispersion relation governing these unstable modes. Treating vacuum energy splittings and collision rates as small perturbations, we construct a unified framework for fast, slow, and collisional instabilities. We classify modes into gapped, where collective excitations are already present in the fast limit but rendered unstable by slow or collisional effects, and gapless, which are purely generated by these effects. For each class, we derive approximate dispersion relations for generic energy and angle distributions, which reveal the order of magnitude of the growth rates and the nature of the instabilities without solving directly the dispersion relation. This approach confirms that slow and collisionally unstable waves generally grow much more slowly than they oscillate. Consequently, the common fast-mode approximation of local evolution within small boxes is unjustified. Even for fast modes, neglecting large-distance propagation of growing waves, as usually done, may be a poor approximation. Our unified framework provides an intuitive understanding of the linear phase of flavor evolution across all regimes and paves the way for a quasi-linear treatment of the instability's nonlinear development.

Figures (7)

• Figure 1: Collective modes in the near-symmetric case ($\epsilon\ll1$), showing the real (top panels) and imaginary (bottom panels) part of the eigenfrequency as a function of wavenumber. Left panels: slow limit for IO, blue real-valued modes, green unstable ones. Right panels: collisional limit for $\Gamma_E>\overline{\Gamma}_E$, colors to differentiate among branches.
• Figure 2: Same as Fig. \ref{['fig1']}, switching to NO in the slow case and to $\Gamma_E<\overline{\Gamma}_E$ in the collisional case.
• Figure 3: Same as Fig. \ref{['fig1']}, now for the asymmetric case ($\epsilon\sim1$).
• Figure 4: Same as Fig. \ref{['fig2']}, now for the asymmetric case ($\epsilon\sim1$), and the same as Fig. \ref{['fig3']} after switching to NO and $\Gamma_E<\overline\Gamma_E$.
• Figure 5: Growth rate of collisionally unstable gapless modes as a function of the asymmetry $\epsilon$ of the distribution. Numerical solution (solid) compared with theoretical prediction (dotted) of Eq. \ref{['eq:gapless_near_symmetric']}, and expected position of disappearance of the gapless mode (dashed) from Eq. \ref{['eq:condition_disappearance_collisional']}.