Instability in the dense supernova neutrino gas with flavor-dependent angular distributions

The usual description of self-induced flavor conversions for neutrinos (\nu's) in supernovae is based on the simplified assumption that all the \nu's of the different species are emitted "half-isotropically" by a common neutrinosphere, in analogy to a blackbody emission. However, realistic supernova simulations show that \nu angular distributions at decoupling are far from being half-isotropic and, above all, are flavor-dependent. We show that flavor-dependent angular distributions may lead to crossing points in the angular spectra of different \nu species (where F_{\nu_e}=F_{\nu_x} and F_{{\bar\nu}_e}=F_{{\bar\nu}_x}) around which a new multi-angle instability can develop. To characterize this effect, we carry out a linearized flavor stability analysis for different SN neutrino angular distributions. We find that this instability can shift the onset of the flavor conversions toward low-radii and produce a smearing of the splitting features found with trivial $\nu$ emission models. As a result the spectral differences among \nu's of different flavors could be strongly reduced.