A few moments to diagnose fast flavor conversions of supernova neutrinos

Neutrinos emitted from a supernova may undergo flavor conversions almost immediately above the core, with possible consequences for supernova dynamics and nucleosynthesis. However, the precise conditions for such fast conversions can be difficult to compute and require knowledge of the full angular distribution of the flavor-dependent neutrino fluxes, that is not available in typical supernova simulations. In this paper, we show that the overall flavor evolution is qualitatively similar to the growth of a so-called `zero mode', determined by the background matter and neutrino densities, which can be reliably predicted using only the second angular moments of the electron lepton number distribution, i.e., the difference in the angular distributions of $\nu_e$ and $\bar{\nu}_e$ fluxes. We propose that this zero mode, which neither requires computing the full Green's function nor a detailed knowledge of the angular distributions, may be useful for a preliminary diagnosis of possible fast flavor conversions in supernova simulations with modestly resolved angular distributions