Suppression of the multi-azimuthal-angle instability in dense neutrino gas during supernova accretion phase

It has been recently pointed out that removing the axial symmetry in the "multi-angle effects" associated with the neutrino-neutrino interactions for supernova (SN) neutrinos, a new multi-azimuthal-angle (MAA) instability would arise. In particular, for a flux ordering $F_{\nu_e} > F_{\bar\nu_e} > F_{\nu_x}$, as expected during the SN accretion phase, this instability occurs in the normal neutrino mass hierarchy. However, during this phase the ordinary matter density can be larger than the neutrino one, suppressing the self-induced conversions. At this regard, we investigate the matter suppression of the MAA effects, performing a linearized stability analysis of the neutrino equations of motion, in the presence of realistic SN density profiles. We compare these results with the numerical solution of the SN neutrino non-linear evolution equations. Assuming axially symmetric distributions of neutrino momenta we find that the large matter term strongly inhibits the MAA effects. In particular, the hindrance becomes stronger including realistic forward-peaked neutrino angular distributions. As a result, in our model for a $10.8$ $M_{\odot}$ iron-core SNe, MAA instability does not trigger any flavor conversion during the accretion phase. Instead, for a $8.8$ $M_{\odot}$ O-Ne-Mg core SN model, with lower matter density profile and less forward-peaked angular distributions, flavor conversions are possible also at early times.