Neutrino fast flavor instability in three dimensions

Neutrino flavor instabilities have the potential to shuffle neutrinos between electron, mu, and tau flavor states, possibly modifying the core-collapse supernova mechanism and the heavy elements ejected from neutron star mergers. Analytic methods indicate the presence of so-called fast flavor transformation instabilities, and numerical simulations can be used to probe the nonlinear evolution of the neutrinos. Simulations of the fast flavor instability to date have been performed assuming imposed symmetries. We perform simulations of the fast flavor instability that include all three spatial dimensions and all relevant momentum dimensions in order to probe the validity of these approximations. If the fastest growing mode has a wave number along a direction of imposed symmetry, then the instability can be suppressed. The late-time equilibrium distribution of flavor, however, seems to be little affected by the number of spatial dimensions. This is a promising hint that the results of lower-dimensionality simulations to date have predictions that are robust against their the number of spatial dimensions, though simulations of a wider variety of neutrino distributions need to be carried out to support this claim more generally.

Figure 1

Graphical representation of the three initial conditions considered in this work. The radial extent of the curves represents the differential number density of that species of neutrino in the corresponding direction. The real initial conditions are distributed in all three directions and we only show the part in the $\widehat{x}–\widehat{z}$ plane; the full distribution can be visualized by rotating each curve around its axis of symmetry.

Figure 2

Volume rendering of contours of constant phase of $n_{e\mu }$ for the Fiducial_1D (top row), Fiducial_2D (center row), and Fiducial_3D (bottom row) simulations. Phases of $-2\pi /3$, 0, and $2\pi /3$ are shown in blue, white, and red, respectively. The left column shows the results at $t=0.29\mathrm{ns}$ during the linear growth phase of the fast flavor instability, the center column shows the results at $t=0.77\mathrm{ns}$ after the instability saturates, and the right column shows the results at $t=2.2\mathrm{ns}$ as the distribution is building power on small scales. The phase of $n_{e\mu }$ demonstrates wavefronts of the fastest growing unstable mode. The Fiducial_1D data is copied into the $x$ and $y$ dimensions and the Fiducial_2D data is copied into the $x$ direction for visualization purposes. Although there is significant multidimensional structure, the 3D results are qualitatively and quantitatively similar to the 1D and 2D results.

Figure 3

Time evolution of fraction of electron flavor neutrinos for the Fiducial (top), 90Degree (center), and TwoThirds (bottom) simulations. The 1D, 2D, and 3D simulation data are plotted in gray, black, and blue, respectively. The green line marks the equilibrium fraction of electron neutrinos ($1/3$ for Fiducial and 90Degree, 0.7 for TwoThirds). Equilibrium abundances (at late times) of each neutrino species seems to be independent of simulation dimensionality. The instability growth rates show weak dimensionality dependence in the 90Degree simulations, indicated by a slight offset in the time of the drop. The offsets in the drop for the TwoThirds simulation reflects initial mode amplitude differences in 1D, 2D, and 3D.

Figure 4

Probability that a neutrino (top) or antineutrino (bottom) starting in the electron flavor ends up in the electron (left), muon (center), or tauon (right) flavor state as a function of polar angle. Blue lines, black crosses, and gray circles are from 3D, 2D, and 1D simulations, respectively. The vertical gray line shows the angle of the crossing; to the left of this line there are more antineutrinos than neutrinos. The green horizontal line is at $⟨{\rho }_{ee}⟩=1/3$, marking complete flavor mixing. Flavor content equilibrates between the flavors in the antineutrino-dominated region, but no flavor transformation occurs far from the crossing.

Figure 5

Time evolution of the Fourier power spectra as defined in Eq. (8) for the Fiducial_3D (left column), 90Degree_3D (center column), and TwoThirds_3D (right column) simulations. The color indicates the time of the snapshot for which the power spectrum was evaluated. Early times are characterized by a rising bump with a peak at the fastest growing wave number, and late times show an exponential distribution centered at $k=0$. The slow spread of the exponential tail is a numerical artifact (see the Appendix).

Figure 6

Snapshot of the power spectrum of $n_{e\mu }$ at $t=1\mathrm{ns}$ for the 1D, 2D, and 3D Fiducial simulations under variations in the number of particles per cell; 32d means 32 equatorial directions or 378 particles per cell, 128d means 6022 particles per cell, and 256 means 24088 particles per cell. Solid curves reproduce a single snapshot of the data shown in Fig. 5. The 2D and 3D results at similar resolutions have very similar peak behavior, and the resolution dependence is similar for 2D and 3D simulations (though 1D simulations behave differently). The four black curves show that the slope of the exponential tail on either side of the upward-propagating feature is robust, but the time at which that feature is launched is increasingly late with increasing resolution.

Figure 7

Mollweide projection of the electron neutrino abundance in the 90Degree simulation at $t=0.56\mathrm{ns}$, or shortly before the flavor instability saturates. The 1D (top) and 2D (center) show similar mode structure, though the mode in the 3D simulation (bottom) much more closely tracks the initial ELN crossing (gray curve).

Figure 8

Evolution of the angular power spectrum of $f_{e\mu }$ as defined in Eq. (11) for the 1D (top panel) and 3D (bottom panel) 90Degree simulations. Solid lines show data from the standard simulations listed in Table 1 with 1506 computational particles per cell, while dashed lines show data from otherwise identical simulations with 378 particles per cell. The standard simulations contain significant information up to $l=32$, while the low-resolution simulations only contain significant information up to $l=16$. The time of the snapshot used to make the curve is indicated by color in the same way as in Fig. 5. The “tail kick” in Fig. 5 corresponds to the saturation of low-$l$ angular power (dark orange curves) and not to power cascading to smaller scales than the angular resolution can support (light orange curves).

Figure 9

Violation of the conservation of integrated electron flavor content, computed as $(\delta n_{ee}-\delta {\overline{n}}_{ee})/\mathrm{Tr}(n+\overline{n})$ and $\delta n=n(t)-n(0)$. The symmetry of the neutrino self-interaction part of the Hamiltonian requires this to be zero, but this is not enforced by the code.

Figure 10

Evolution of the domain-averaged electron neutrino abundance considering various numerical choices. Solid curves reproduce data shown in Fig. 3. The top row shows results from 2D simulations while the bottom shows those from 3D simulations. Each column shows results from each of our three initial conditions. Labels with a “d” refer to equatorial directions to be compared to the standard resolution of 64: 32d means 32 equatorial directions or 378 particles per cell, 128 means 6022 particles per cell, and 256d means 24088 particles per cell. “nx” labels the number of grid cells on each extended dimension of the simulation, keeping the domain size constant, where “nx128” is the standard resolution for the multidimensional simulations. “cm” labels the domain size of each dimension of the cube in centimeters. The horizontal green line marks complete flavor mixing. “inplane” for the 2D 90Degree simulations means the computational domain extends over the $\widehat{x}-\widehat{z}$ plane instead of the $\widehat{y}-\widehat{z}$ plane, slightly increasing the instability growth rate. Overall, our numerical choices have little effect on the instability growth rates and the late-time electron neutrino abundances.

Figure 11

Time evolution of the power spectrum of $n_{e\mu }$ for the Fiducial_3D simulation (bottom panel) employing 64 equatorial directions (i.e., 1506 particles per cell) and then an otherwise identical simulation employing only 32 equatorial directions (top panel, 378 particles per cell). The spread of the exponential tail toward higher values of $k$ is more strongly affected by this angular resolution than any other numerical choice.

Figure 12

Time evolution of the power spectrum of $n_{e\mu }$ for the Fiducial_2D simulation (second panel) employing 64 equatorial directions (i.e., 1506 particles per cell). The other panels show the equivalent results from simulations employing only 32, 128, and 256 equatorial directions (378, 6022, and 24088 particles per cell, respectively). The spread of the exponential tail toward higher values of $k$ is more strongly affected by this angular resolution than any other numerical choice.

Figure 13

Evolution of the value of $k$ at the peak of the power spectrum of $n_{e\mu }$ for 1D, 2D, and 3D simulations of varying angular resolution. The low-resolution (32d) simulations stay tightly packed until about 1 ns, while the standard resolution results are very similar until about 2 ns. The highest-resolution 2D simulation predicts a peak very close to $k=0$ throughout the postsaturation phase.