Single-wave solutions of the neutrino fast flavor system. Part II. Weak instabilities and their resonant behavior

TL;DR

This paper analyzes nonlinear saturation of fast flavor instabilities in dense neutrino media by focusing on a single, axially symmetric wave mode. It derives a closed nonlinear set of equations for a fixed wavenumber $K$, showing that weak, nearly monochromatic instabilities lead to a resonant flavor pendulum where resonant neutrinos exchange lepton number with a coherently evolving flavor field. The study develops an adiabatic framework for resonant modes, derives a pendulum equation $\ddot\Phi_u = \gamma^2 \sin\Phi_u$, and demonstrates, via Rosen-Zener modeling, the robustness of this pendular dynamics over many cycles, with the collective field amplitude $\psi_0$ reaching $\psi_{0,\max} = 2\gamma/(1 - Uu)$. Complementary numerical results across varying crossing depths confirm the emergence, stability, and eventual damping of single-wave nonlinear solutions, highlighting the regime where early, narrow-band instabilities govern nonlinear evolution with potential astrophysical relevance.

Abstract

Flavor instabilities in dense neutrino media trigger exponential growth of flavor waves, yet their nonlinear saturation remains poorly understood. We examine a simple proxy for this effect in the form of a single-wave solution of an axially symmetric fast flavor system. When the angular crossing is shallow and the growth rate of the instability correspondingly small, the flavor wave primarily affects resonant neutrinos that move in phase with it. The evolution of these resonant neutrinos becomes periodic, undergoing cycles of full flavor reversal. They feed power into the unstable wave, and subsequently return to their initial state, draining power back out. This new flavor pendulum captures the dynamics of weak, nearly monochromatic fast flavor instabilities. Since weakly unstable distributions always exhibit a narrow range of unstable wavenumbers, our model likely describes the earliest development of a flavor instability when it first appears. When the instability is not weak, the linear phase of a single-wave excitation does not connect to a regular nonlinear solution, unless the angle distribution consists of only two beams.

Figures (3)

• Figure 1: Numerical evolution of the single-wave instability for the spectrum of Eq. \ref{['eq:reference-distribution']} with $a=0.001$. Left bottom: Spectrum $D_v(0)$ and its maximum excursion. Left top: Maximum swap factor, showing Lorentzian shape. Right bottom: Evolution of flavor field $|\Psi_0(t)|$, showing periodic collective motion of the entire ensemble. Right top: Contour of swap factor in the plane of $(v,t)$.
• Figure 2: Instability pattern for distributions of the type Eq. \ref{['eq:reference-distribution']}. Left: Crossed distributions for the reference values of $a$ listed in Eq. \ref{['eq:avalues']} and marked with asterisks in the right panel. Right: Growth rate of unstable modes, depending on the depth of the crossing $a$ and wavenumber $K$. There is no instability outside of the thin black contour.
• Figure 3: Time evolution for the $a$-cases listed in Eq. \ref{['eq:avalues']}, from top to bottom, corresponding to the angular distributions shown in Fig. \ref{['fig:benchmark']}. Left: Flavor fields $|\psi_0|$ (red) and $|\psi_1|$ (blue). Right: Swap factor as in Fig. \ref{['fig:pendulum']}. Horizontal lines denote the resonance velocity of the initial unstable mode (magenta) and the crossing velocity (green).