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It has been recently pointed out that removing the axial symmetry in the"multi-angle effects" associated with the neutrino-neutrino interactions forsupernova (SN) neutrinos, a new multi-azimuthal-angle (MAA) instability wouldarise. 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 occursin the normal neutrino mass hierarchy. However, during this phase the ordinarymatter density can be larger than the neutrino one, suppressing theself-induced conversions. At this regard, we investigate the matter suppressionof the MAA effects, performing a linearized stability analysis of the neutrinoequations of motion, in the presence of realistic SN density profiles. Wecompare these results with the numerical solution of the SN neutrino non-linearevolution equations. Assuming axially symmetric distributions of neutrinomomenta we find that the large matter term strongly inhibits the MAA effects.In particular, the hindrance becomes stronger including realisticforward-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 flavorconversion during the accretion phase. Instead, for a $8.8$ $M_{\odot}$ O-Ne-Mgcore SN model, with lower matter density profile and less forward-peakedangular distributions, flavor conversions are possible also at early times.

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