Spatiotemporal linear instability analysis of collective neutrino flavor conversion in 4-dimensional spacetime

In an environment with high-density neutrinos formed in a core-collapse supernova (CCSN), the neutrinos exhibit nonlinear and complex oscillation behaviors due to their self-interactions. The onset of this nonlinear oscillation can be investigated by linearizing the evolution equation for small perturbations around the flavor eigenstates. While the condition under which the flavor eigenstates are unstable has been investigated in many studies, how the perturbations evolve in spacetime has yet to be elucidated. In this paper, we analytically and correctly derive the asymptotic behaviors of the linear perturbations in 4-dimensional spacetime in the linear regime for a 2-beam neutrino model using the recently proposed Lefschetz thimble formulation. The result suggests that the perturbations grow in the directions between the two neutrino beams. We also briefly discuss the possible effects of neutrino flavor conversion on the explosion mechanism of a CCSN. In particular, the result implies that the flavor instability in the preshock region may propagate into the postshock region, contrary to the previous study focusing on the group velocity in 1-dimensional space. How to treat the case of a more realistic continuous spectrum is also discussed.