Neutrino Flavor Pendulum Reloaded: The Case of Fast Pairwise Conversion

In core-collapse supernovae or compact binary merger remnants, neutrino-neutrino refraction can spawn fast pair conversion of the type ν_{e}ν[over ¯]_{e}↔ν_{x}ν[over ¯]_{x} (with x=μ, τ), governed by the angle-dependent density matrices of flavor lepton number. In a homogeneous and axially symmetric two-flavor system, all angle modes evolve coherently, and we show that the nonlinear equations of motion are formally equivalent to those of a gyroscopic pendulum. Within this analogy, our main innovation is to identify the elusive characteristic of the lepton-number angle distribution that determines the depth of conversion with the "pendulum spin." The latter is given by the real part of the eigenfrequency resulting from the linear normal-mode analysis of the neutrino system. This simple analogy allows one to predict the depth of flavor conversion without solving the nonlinear evolution equations. Our approach provides a novel diagnostic tool to explore the physics of nonlinear systems.