Sufficient and Necessary Conditions for Collective Neutrino Instability: Fast, Slow, and Mixed

TL;DR

This work establishes a rigorous, general set of sufficient (and largely necessary) conditions for collective neutrino instabilities in dense gases with distributions that depend on both energy and emission angle. By formulating and analyzing the dispersion relation ${\mathcal D}(\omega,{\bf k})=0$ and employing a Nyquist contour approach, the authors show that a zero-crossing of the flavor-difference distribution is not alone enough to guarantee instability; one must also satisfy a local positivity condition and global principal-value constraints. In fast, slow, and mixed regimes, these conditions simplify differently: fast instabilities reduce to the classic requirement that the velocity-space distribution change sign, while slow and mixed instabilities require additional global information and, at small $w_E$, can exhibit approximate $\lambda$-scaling with the collective potential $\mu$. The results clarify when simple instability diagnostics suffice and provide a robust framework to predict flavor evolution in environments like core-collapse supernovae and neutron-star mergers.

Abstract

Collective neutrino oscillations exhibit instabilities that induce appreciable flavor conversion, with crucial astrophysical implications. While the importance of initial phase-space distributions is well-established, a general instability criterion for distributions dependent on both energy and emission angle has been lacking. We identify and analyze the sufficient (and necessary) conditions for a generic collective neutrino flavor instability.

Figures (8)

• Figure 1: Sketch of contour mapping from ${\mathcal{C}_{\omega}}$ to ${\mathcal{C}_{\mathcal{D}}}$. The color gradient from blue to red indicates the direction of the contour, with only a finite range of $\omega \in (-\omega_C,+\omega_C)$ allowing a nonzero $\Im{\mathcal{D}}$. Crossing points of ${\mathcal{C}_{\mathcal{D}}}$ are denoted by ${\mathcal{D}}_i$, corresponding to $\omega_i$ where $\Im{\mathcal{I}}$ changes sign. The contour ${\mathcal{C}_{\mathcal{D}}}$ begins and terminates at ${\mathcal{D}}_\infty=1$, as the semicircular arc $|\omega|\to\infty$ on ${\mathcal{C}_{\omega}}$ gets mapped to that point indicated by the purple dot. Note the flat region near it, where for large $|\omega|>\omega_C$, the dispersion ${\mathcal{D}}$ is a real function.
• Figure 2: Sketch of ${\mathcal{C}_{\mathcal{D}}}$ contours for singly and multiply crossed $g_{\bf\Gamma}$, showing the relevance of Conditions 2a and 2b. For a singly crossed $g_{\bf\Gamma}$, if Condition 2a is satisfied, i.e., the principal value corresponding to ${\mathcal{I}}_{\rm PV}(\omega_0)<-1$, encirclement of origin is guaranteed (left panel). With multiple crossings, there may be more than one crossing to the left of the origin. If Condition 2b, viz. ${\mathcal{I}}_{\rm PV}(\omega_{i\neq0})>-1$, is not satisfied; there may not be an encirclement of the origin (middle panel); if 2b is satisfied as well, encirclement cannot be avoided (e.g., right panel).
• Figure 3: Illustrations of different types of crossing surfaces. The sign of the distribution $g_{\bf\Gamma}$ is shown as blue ($>0$) or red ($<0$), as a function of ${\bf p}=|\Delta m^2/(2{w_E})|{\bf v}$ for one half of the allowed range of ${w_E}$, say ${w_E}\in(0,\infty)$. Crossings in only ${\bf v}$ (left panel) are "fast" and unstable. Crossings in $E=|{\bf p}|$ (or ${w_E}$) but not ${\bf v}$, are "slow" (middle panel); these are unstable under further conditions, i.e., on sign of $\partial g/\partial {w_E}$ and on ${\mathcal{I}}_\text{PV}$. The general case of "mixed" crossings is shown on the right panel, which is unstable only under further conditions.
• Figure 4: Distribution function with no crossing results in no encirclement of origin for any value of $k$. Note how, for the fast case, increasing $k$ shrinks the contour without distortion of its shape.
• Figure 5: Distribution function with a single crossing at $v=0$, that leads to two ${\mathcal{C}_{\mathcal{D}}}$ crossings, at ${\mathcal{D}}_0<0$ and at crossing at ${\mathcal{D}}_\infty=1$, giving origin-encirclement. Instability exists in the range $\omega\leq|k|$ which is shown in the inset. For large values of $k$, ${\mathcal{C}_{\mathcal{D}}}$ shrinks and moves closer to ${\mathcal{D}}_\infty$, eventually resulting in no encirclement.