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In an environment with high-density neutrinos formed in a core-collapsesupernova (CCSN), the neutrinos exhibit nonlinear and complex oscillationbehaviors due to their self-interactions. The onset of this nonlinearoscillation can be investigated by linearizing the evolution equation for smallperturbations around the flavor eigenstates. While the condition under whichthe flavor eigenstates are unstable has been investigated in many studies, howthe perturbations evolve in spacetime has yet to be elucidated. In this paper,we analytically and correctly derive the asymptotic behaviors of the linearperturbations in 4-dimensional spacetime in the linear regime for a 2-beamneutrino model using the recently proposed Lefschetz thimble formulation. Theresult suggests that the perturbations grow in the directions between the twoneutrino beams. We also briefly discuss the possible effects of neutrino flavorconversion on the explosion mechanism of a CCSN. In particular, the resultimplies that the flavor instability in the preshock region may propagate intothe postshock region, contrary to the previous study focusing on the groupvelocity in 1-dimensional space. How to treat the case of a more realisticcontinuous spectrum is also discussed.

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