Simple method of diagnosing fast flavor conversions of supernova neutrinos

Neutrinos emitted from a supernova may undergo flavor conversions almost immediately above the core, with possible consequences for supernova dynamics and nucleosynthesis. However, the precise conditions for such fast conversions can be difficult to compute and require knowledge of the full angular distribution of the flavor-dependent neutrino fluxes that is not available in typical supernova simulations. In this paper, we show that the overall flavor evolution is qualitatively similar to the growth of a so-called “zero mode” determined by the background matter and neutrino densities, which can be reliably predicted using only the second angular moments of the electron lepton number distribution, i.e., the difference in the angular distributions of the ${\nu }_e$ and ${\overline{\nu }}_e$ fluxes. We propose that this zero mode, which neither requires computing the full Green’s function nor detailed knowledge of the angular distributions, may be useful for a preliminary diagnosis of possible fast flavor conversions in supernova simulations with modestly resolved angular distributions.

Figure 1

Growth of absolute flavor instability for a toy model of neutrinos with a boxlike ELN distribution with a crossing at $v_{\mathrm{c}}=0$ given by $(G_{{\nu }_e},G_{{\overline{\nu }}_e})=(0.3,0.5)$ for $v<v_{\mathrm{c}}$ and $(G_{{\nu }_e},G_{{\overline{\nu }}_e})=(1.2,0.5)$ for $v>v_{\mathrm{c}}$. Left panel: The instability is absolute and spreads around its origin without drifting. Right panel: The numerically computed growth rate of the zero mode labeled by $k=0$ (continuous red line), the true homogeneous mode labeled by $k=-\mathrm{\Phi }$ (unbroken green line), and the mode with the largest growth rate labeled by $k=k_{\max }$ (unbroken blue line). The numerically observed growth rate for the zero mode matches the analytical prediction using the moments (dashed red line).

Figure 2

Growth of convective flavor instability for a toy model of neutrinos with a boxlike ELN distribution given by $(G_{{\nu }_e},G_{{\overline{\nu }}_e})=(0.6,1.0)$ for $0<v<v_{\mathrm{c}}$ and $(G_{{\nu }_e},G_{{\overline{\nu }}_e})=(2.4,1.0)$ for $v_{\mathrm{c}}<v<1.0$, with $v_{\mathrm{c}}=0.5$. Left panel: The instability grows along a particular direction, spreading around it, and advecting away from the original site of instability at $z=100$. Right panel: The numerically computed growth rate of the zero mode labeled by $k=0$ (continuous red line), the true homogeneous mode labeled by $k=-\mathrm{\Phi }$ (unbroken green line), and the mode with the largest growth rate labeled by $k=k_{\max }$ (unbroken blue line). The numerically observed growth rate for the zero mode agrees with the analytical prediction using the moments (dashed red line).

Figure 3

Left panel: Zenith-angle distributions of ${\nu }_e$ and ${\overline{\nu }}_e$ inspired by 1D SN models simulated by the Garching group. The relative weights of the fluxes have been changed to generate a smaller asymmetry that leads to a crossing at $v_{\mathrm{c}}=0.3$. Right panel: Difference of angular spectra of ${\nu }_e$ and ${\overline{\nu }}_e$ showing a crossing at $v_{\mathrm{c}}=0.3$.

Figure 4

Growth of absolute flavor instability for neutrino angular distributions following SN simulations, as shown in Fig. 3. Left panel: The instability is absolute and spreads over space, without drifting. Right panel: The numerically computed growth rate of the zero mode labeled by $k=0$ (continuous red line), the true homogeneous mode labeled by $k=-\mathrm{\Phi }$ (unbroken green line), and the mode with the largest growth rate labeled by $k=k_{\max }$ (unbroken blue line). The numerically observed growth rate for the zero mode matches the analytical prediction using the moments (dashed red line).