Neutrino fast flavor oscillations with moments: Linear stability analysis and application to neutron star mergers

Providing an accurate modeling of neutrino physics in dense astrophysical environments such as binary neutron star mergers presents a challenge for hydrodynamic simulations. Nevertheless, understanding how flavor transformation can occur and affect the dynamics, the mass ejection, and the nucleosynthesis will need to be achieved in the future. Computationally expensive, large-scale simulations frequently evolve the first classical angular moments of the neutrino distributions. By promoting these quantities to matrices in flavor space, we develop a linear stability analysis of fast flavor oscillations using only the first two “quantum” moments, which notably requires generalizing the classical closure relations that appropriately truncate the hierarchy of moment equations in order to treat quantum flavor coherence. After showing the efficiency of this method on a well-understood test situation, we perform a systematic search of the occurrence of fast flavor instabilities in a neutron star merger simulation. We discuss the successes and shortcomings of moment linear stability analysis, as this framework provides a time-efficient way to design and study better closure prescriptions in the future.

Figure 1

Instability growth rate for different wave numbers in the Fiducial test case. The coordinates of the overall maximum are given in Eq. (39).

Figure 2

Numerical solution of three-dimensional neutrino flavor transformation in the Fiducial test case, for particle-in-cell (Emu) and moment (FLASH) codes, adapted from [96]. For comparison purposes, the time origin on the left and middle plots is taken at the saturation time $t_{\mathrm{sat}}$ (for which $N_{ex}$ is maximal), which occurs at different times in the Emu and FLASH simulations. Left: evolution of the normalized electron neutrino density averaged over the simulation box, which shows fast flavor conversion. Middle: evolution of the domain-averaged off-diagonal component of $N$, showing the exponential growth of the unstable mode. The brown line shows an exponential growth at the rate given by the linear stability analysis prediction (39). Right: magnitude of the spatial Fourier transform of the off-diagonal component (${\tilde{N}}_{ex}$) against wave number $k$, taken 0.1 ns before saturation. The dash-dotted line is the linear stability analysis prediction for $k_{\max }$.

Figure 3

Predicted characteristics of the FFI fastest growing mode on the slice $\{Y=0\mathrm{km}\}$ of the NSM simulation, obtained from moment LSA. Top: instability growth rate. Contours of equal flux factor $\widehat{f}=0.2$ are represented for ${\nu }_e$ (solid green line), ${\overline{\nu }}_e$ (dashed red line) and ${\nu }_x$ (dash-dotted blue line). Bottom: normalized vertical component of ${\mathbf{k}}_{\max }$, showing the predominance of modes aligned with the ELN flux direction. Regions with ${\mathbf{k}}_{\max }\nparallel \mathbf{z}$ are not expected to be unstable (see Sec. 5b), and are thus expected to be false LSA positives.

Figure 4

Volume rendering of the fast flavor instability growth rate in the NSM simulation from [6, 7], taken 5 ms post merger. Three colored contours are centered respectively around the growth rate values $7\times {10}^9{\mathrm{s}}^{-1}$, $2\times {10}^{10}{\mathrm{s}}^{-1}$, and $7\times {10}^{10}{\mathrm{s}}^{-1}$.

Figure 5

Instability growth rate from moment linear stability analysis, for various slices of the NSM simulation. Left: slice across the equatorial plane at $Y=0\mathrm{km}$. Middle: off-center slice across the equatorial plane, at $Y\simeq 40\mathrm{km}$. Right: centered slice orthogonal to the first one, at $X=0\mathrm{km}$. We identify the three points studied in [96] and discussed in Sec. 5a3.

Figure 6

Instability growth rate obtained from moment linear stability analysis in the equatorial plane of the NSM simulation snapshot ($Z=0\mathrm{km}$). Regions of FFI follow the spiral density wave structure of the nascent accretion disk, as indicated by the $\widehat{f}=0.2$ flux factor contours, which are represented for ${\nu }_e$ (solid green line), ${\overline{\nu }}_e$ (dashed red line), and ${\nu }_x$ (dash-dotted blue line).

Figure 7

Wave number corresponding to the fastest growing mode, across the same polar slice as the left panel of Fig. 5. These values show that the typical lengthscale associated with FFI in a classical neutron star merger simulation is on the scale of centimeters.

Figure 8

Fast flavor instability growth rate for the NSM slice $\{Y=0\mathrm{km}\}$, obtained from moment LSA restricted to the “zero mode” $\mathbf{k}=\sqrt{2}{G}_F({\mathbf{F}}_{ee}-{\mathbf{F}}_{xx}-{\overline{\mathbf{F}}}_{ee}+{\overline{\mathbf{F}}}_{xx})$ (Top), or to the homogeneous mode $\mathbf{k}=\mathbf{0}$ (Bottom), to be compared with the left panel of Fig. 5.

Figure 9

Comparison of linear stability analysis predictions and three-dimensional numerical solutions of neutrino flavor transformation (adapted from [93]) for the NSM 1 point. Left: instability growth rate obtained by linear stability analysis for different modes, similarly to Fig. 1. The values corresponding to the zero mode are represented in dotted green lines. Middle: evolution of the domain-averaged magnitude of the off-diagonal component of $N$ in Emu and FLASH. Right: Fourier transform of the off-diagonal component against wavenumber $k$, taken 0.1 ns before saturation. Plotting conventions are the same as in Fig. 2. The dash-dotted brown line is the linear stability analysis prediction (see Table 2).

Figure 10

On top of the growth rate (cf. Fig. 5, left panel), the blue regions indicate where an ELN crossing is determined from Eq. (43).

Figure 11

Fast flavor instability growth rate for the NSM slice $\{Y=0\mathrm{km}\}$, for a 3D search in $\mathbf{k}$ space, varying the cutoff of the consistency test. The large value ($\delta =20$) overestimates the possibility of FFI, notably in the polar regions (cf. Fig. 10 where the regions with an ELN crossing are represented), while the low value ($\delta =1.5$) rejects too many points.

Figure 12

Collisional instability growth rate [Eq. (c2)], on the same slice as the left panel of Fig. 5. It is generally several orders of magnitude smaller than the FFI growth rate. For reference, the $\widehat{f}=0.2$ flux factor contours, which indicate the “neutrinospheres,” are represented for ${\nu }_e$ (solid purple line), ${\overline{\nu }}_e$ (dashed red line), and ${\nu }_x$ (dash-dotted blue line).

Figure 13

Volume rendering of the collisional instability growth rate across the NSM simulation, predicted using Eq. (c2). Three colored contours are centered respectively around the growth rate values ${10}^3{\mathrm{s}}^{-1}$, ${10}^5{\mathrm{s}}^{-1}$, and ${10}^7{\mathrm{s}}^{-1}$.