Fast oscillations, collisionless relaxation, and spurious evolution of supernova neutrino flavor

Mounting evidence indicates that neutrinos likely undergo fast flavor conversion (FFC) in at least some core-collapse supernovae. Outcomes of FFC, however, remain highly uncertain. Here we study the cascade of flavor-field power from large angular scales in momentum space down to small ones, showing that FFC enhances this process and thereby hastens relaxation. Cascade also poses a computational challenge, which is present even if the flavor field is stable: When power reaches the smallest angular scale of the calculation, error from truncating the angular-moment expansion propagates back to larger scales, to disastrous effect on the overall evolution. Essentially the same issue has prompted extensive work in the context of plasma kinetics. This link suggests new approaches to averting spurious evolution, a problem that presently puts severe limitations on the feasibility of realistic oscillation calculations.

Figure 1

Flavor evolution—$P_{0,z}$ (top), $S_0$ (middle), and $D_1$ (bottom)—with time $t$ in homogeneous, axially symmetric calculations, using the angular distributions and ratio $\mu /\omega$ from a spherically symmetric fornax simulation at a radius of 70 km and a postbounce time of 200 ms. The ${\overline{\nu }}_e$ angular distribution is rescaled to $\alpha =n_{{\overline{\nu }}_e}/n_{{\nu }_e}=0.90$ (black) and $\alpha =0.85$ (red), and the normalization is such that $|{\mathbf{P}}_0(0)|=0.5$. The normal hierarchy is shown on the left, inverted hierarchy on the right. Time is in units of ${\omega }^{-1}$. For typical supernova conditions and a neutrino energy of $\sim 10\mathrm{MeV}$, the horizontal axis spans about a microsecond. Gray curves represent the same data as the black curves but time-averaged over windows of duration $6\times {10}^{-3}{\omega }^{-1}$, spanning about three periods of fast collective oscillations. In the bottom panels, the dashed lines mark the initial values. At $\alpha =0.90$, the system is unstable to fast oscillations, which facilitate kinematic decoherence (decay of $S_0$) by a dephasing mechanism. See text for further discussion.

Figure 2

Momentum-space angular power spectra of neutrino flavor for the same set of calculations presented in Fig. 1. The power ${\mathrm{\Pi }}_l$ in angular moment $l$ is defined in Eq. (25). From darkest to lightest, the curves show ${\log }_{10}{\mathrm{\Pi }}_l$ at $t=0.2$, 0.4, 0.6, 0.8, and 1.0, in units of ${\omega }^{-1}$. FFC enhances cascade and hastens relaxation. All cases show interesting features at angular scales intermediate between the expanding front of nonzero power and the low-$l$ scales directly involved in fast and bipolar collective oscillations.

Figure 3

Flavor composition $P_{v,z}$ plotted as a function of propagation angle $v=\cos \theta$, for the $\alpha =0.90$, NH calculation of the previous figures. The curves are color coded by time, beginning at the top of a fast-oscillation dip (blue), lasting a duration of $\sim {10}^{-3}$, and ending at the bottom of the dip at time $t_f$ (red). As usual, times are given in units of ${\omega }^{-1}$. Finite-$\omega$ effects—the disruption of the fast pendulum and the cascade of power to smaller angular scales—become increasingly apparent at later times.

Figure 4

Angular power spectra in calculations with power seeded at $l=50$. In the plot labels, “Parallel” indicates that ${\mathbf{P}}_{50}$ and ${\overline{\mathbf{P}}}_{50}$ are set parallel to the flavor axis $\mathbf{z}$ at $t=0$, and “Perpendicular” indicates that they are perpendicular to the flavor axis and parallel to each other. From darkest to lightest, the curves are at times $1.1\times {10}^{-5}$, $2.2\times {10}^{-3}$, $4.4\times {10}^{-3}$, $6.6\times {10}^{-3}$, and $8.8\times {10}^{-3}$ in units of ${\omega }^{-1}$.

Figure 5

Same as the top four panels of Fig. 1, but the calculations here were done using $l_{\max }=199$. Significant discrepancies, including spurious decoherence, set in at late times due to truncation of the angular-moment expansion.

Figure 6

Angular power spectra, comparing a calculation (NH, $\alpha =0.85$) with $l_{\max }=199$ (light blue) to one that is converged in $l_{\max }$ (dark blue). In the former case, error accrues at $l_{\max }$ and propagates back to lower $l$. Power artificially builds up until backreaction incites spurious decoherence.

Figure 7

Same as Fig. 6, but with $\alpha =0.90$. Errors associated with truncation at $l_{\max }$ again cause highly spurious evolution. Note that the sampled times and the range of the vertical axis differ with respect to the previous figure.

Figure 8

Same as the top four panels of Fig. 1, but the calculations here were done using $l_{\max }=3$. Because $l\le 3$ are the most relevant for evolution of the slow and fast pendulums, important features of the evolution are decently well approximated. Relaxation is not captured, however, because momentum-space cascade is prohibited by the small number of angular moments.