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In core-collapse supernovae and neutron star mergers, the neutrino density isso large that neutrino-neutrino refraction can lead to flavor conversion, if azero-crossing is present in the neutrino flavor lepton number (FLN) angulardistribution and the neutrino self-interaction strength $\mu=\sqrt{2} G_Fn_\nu$ represents the characteristic timescale of the system. It has beenempirically realized that the vacuum frequency $\omega=\Delta m^2/2E$ affectsthe development of flavor conversion in the presence of zero-crossing even if$\omega \ll \mu$. Focusing on a homogeneous and axially symmetric neutrino gas,we explore the role of $\omega$ in the onset of flavor instabilities. We findthat a non-zero vacuum frequency can be responsible for inducing flavorinstabilities even when the neutrino self-interaction strength is much largerthan the vacuum frequency. Moreover, mapping a neutrino ensemble with $\omega\neq 0$ into an effective system with $\omega =0$, we find that a system withno FLN zero-crossing can effectively develop one for $\omega \neq 0$ becomingunstable.

Energy Dependence of Flavor Instabilities Stemming from Crossings in the Neutrino Flavor Lepton Number Angular Distribution