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Chaotic flavor evolution in an interacting neutrino gas

Neutrino-neutrino refraction can lead to non-periodic flavor oscillations in dense neutrino gases, and it has been hypothesized that some solutions are chaotic in nature. This is of particular interest in the case of neutrino emission from core-collapse supernovae where the measurement of the spectral shape for different flavors can provide crucial information about both neutrino physics and the physical conditions close to the proto-neutron star. Whether a system is chaotic or not can be assessed by the Lyapunov exponents which quantify the rate of divergence of nearby trajectories in the system. We have done a numerical case study for a simple toy model of two neutrino flavors with two momentum states traveling against each other which is known to exhibit flavor transition instabilities. We find the leading Lyapunov exponent to be positive in all cases, confirming the chaoticity of the system for both the normal and the inverted neutrino mass hierarchy. However, more Lyapunov exponents were approximately zero in the inverted hierarchy compared to the normal which has implications for the stability of the system. To investigate this, we have calculated a generalized set of normal modes, the so-called covariant Lyapunov vectors. The covariant Lyapunov vectors associated with vanishing Lyapunov exponents showed the existence of marginally stable directions in phase space for some cases. While our analysis was done for a toy model example, it should work equally well for more realistic cases of neutrinos streaming from a proto-neutron star and provide valuable insight into the nature of the flavor instability. We finally stress that our approach captures many more properties of the physical system than the linear stability analyses which have previously been performed.