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

In core-collapse supernovae and neutron star mergers, the neutrino density is so large that neutrino-neutrino refraction can lead to flavor conversion, if a zero-crossing is present in the neutrino flavor lepton number (FLN) angular distribution and the neutrino self-interaction strength $\mu=\sqrt{2} G_F n_\nu$ represents the characteristic timescale of the system. It has been empirically realized that the vacuum frequency $\omega=\Delta m^2/2E$ affects the 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 find that a non-zero vacuum frequency can be responsible for inducing flavor instabilities even when the neutrino self-interaction strength is much larger than the vacuum frequency. Moreover, mapping a neutrino ensemble with $\omega \neq 0$ into an effective system with $\omega =0$, we find that a system with no FLN zero-crossing can effectively develop one for $\omega \neq 0$ becoming unstable.