Normal-mode Analysis for Collective Neutrino Oscillations

In an interacting neutrino gas, collective modes of flavor coherence emerge that can be propagating or unstable. We derive the general dispersion relation in the linear regime that depends on the neutrino energy and angle distribution. The essential scales are the vacuum oscillation frequency $\omega=\Delta m^2/(2E)$, the neutrino-neutrino interaction energy $\mu=\sqrt{2}G_{\rm F} n_\nu$, and the matter potential $\lambda=\sqrt{2}G_{\rm F} n_e$. Collective modes require non-vanishing $\mu$ and may be dynamical even for $\omega=0$ ('fast modes'), or they may require $\omega\not=0$ ('slow modes'). The growth rate of unstable fast modes can be fast itself (independent of $\omega$) or can be slow (suppressed by $\sqrt{|\omega/\mu|}$). We clarify the role of flavor mixing, which is ignored for the identification of collective modes, but necessary to trigger collective flavor motion. A large matter effect is needed to provide an approximate fixed point of flavor evolution, while spatial or temporal variations of matter and/or neutrinos are required as a trigger, i.e., to translate the disturbance provided by the mass term to seed stable or unstable flavor waves. We work out explicit examples to illustrate these points.