Collective Neutrino Flavor Instability Requires a Crossing

Neutrinos in supernovae, neutron stars, and in the early Universe may change flavor collectively and unstably, due to neutrino-neutrino forward scattering. We prove that, for collective instability to occur, the difference of momentum distributions of two flavors must change sign, i.e., there is a zero crossing. This necessary criterion, which unifies slow and fast instabilities, is valid for Hamiltonian flavor evolution of ultrarelativistic standard model neutrino occupation matrices, including damping due to collisions in the relaxation approximation. It provides a simple but rigorous condition for collective flavor transformations that are believed to be important for stellar dynamics, nucleosynthesis, and neutrino phenomenology.