Linear analysis of fast-pairwise collective neutrino oscillations in core-collapse supernovae based on the results of Boltzmann simulations

Neutrinos are densely populated deep inside the core of massive stars after their gravitational collapse to produce supernova explosions and form compact stars such as neutron stars and black holes. It has been considered that they may change their flavor identities through so-called fast-pairwise conversions induced by mutual forward scatterings. If that is really the case, the dynamics of supernova explosion will be influenced, since the conversion may occur near the neutrino sphere, from which neutrinos are effectively emitted. In this paper, we conduct a pilot study of such possibilities based on the results of fully self-consistent, realistic simulations of a core-collapse supernova explosion in two spatial dimensions under axisymmetry. As we solved the Boltzmann equations for neutrino transfer in the simulation not as a postprocess but in real time, the angular distributions of neutrinos in momentum space for all points in the core at all times are available, a distinct feature of our simulations. We employ some of these distributions extracted at a few selected points and times from the numerical data and apply linear analysis to assess the possibility of the conversion. We focus on the vicinity of the neutrino sphere, where different species of neutrinos move in different directions and have different angular distributions as a result. This is a pilot study for a more thorough survey that will follow soon. We find no positive sign of conversion unfortunately at least for the spatial points and times we studied in this particular model. We hence investigate rather in detail the condition for the conversion by modifying the neutrino distributions rather arbitrarily by hand.

Figure 1

Entropy distributions in the meridian section of the central part of the core at $t_{pb}=15.0\mathrm{ms}$ (top), $t_{pb}=190.4\mathrm{ms}$ (middle) and $t_{pb}=275.9\mathrm{ms}$ (bottom), respectively.

Figure 2

Color contours for the sine of the angle that the flux vectors make: $|F_{{\nu }_e}\times F_{{\overline{\nu }}_e}|/|F_{{\nu }_e}||F_{{\overline{\nu }}_e}|$. Red and green circles correspond to the densities of $10^{12}$ and $10^{11}{\mathrm{g}/\mathrm{cm}}^3$, respectively. Violet, light-blue and brown circles indicate, respectively, the neutrino spheres of ${\nu }_e$, ${\overline{\nu }}_e$ and ${\nu }_x$, or the radii that their mean free paths calculated from both absorption and scattering rates are equal to the density scale height $\rho /(d\rho /dr)$. Yellow and dark-blue circles, on the other hand, show the neutrino spheres of ${\nu }_e$ and ${\overline{\nu }}_e$, respectively, calculated from the scattering rate alone.

Figure 3

Energy-integrated fluxes of different neutrino species at the point indicated by the yellow radial lines ($r=44.8\mathrm{km}$, $\theta =2.36\mathrm{rad}$). Top, middle and bottom panels correspond to $t_{pb}=15.0\mathrm{ms}$, $t_{pb}=190.4\mathrm{ms}$ and $t_{pb}=275.9\mathrm{ms}$, respectively.

Figure 4

Angular distributions in momentum space of ${\nu }_e$, ${\overline{\nu }}_e$ and ${\nu }_x$ at $t_{pb}=15.0\mathrm{ms}$. Panels A, B and C correspond to the neutrino energy of $E=5\mathrm{MeV}$ whereas D, E and F have $E=8.5\mathrm{MeV}$ and G, H and I share $E=10\mathrm{MeV}$. Each column shows the meridian sections, which correspond to a certain pair of ${\varphi }_{\nu }$ values: ${\varphi }_{\nu }[\mathrm{radian}]=(0.35,3.49)$, (1.57, 4.71) and (2.78, 5.92) for the left, middle and right columns, respectively.

Figure 5

Same as Fig. 4 but at $t_{pb}=190.4\mathrm{ms}$.

Figure 6

Same as Figs. 4 and 5 but at $t_{pb}=275.9\mathrm{ms}$.

Figure 7

Dispersion relations between real $k$ and $\omega$ for the original data at $t_{pb}=15.0\mathrm{ms}$. It is assumed that $\mathbf{k}$ is oriented in the crossing direction, which is different from case to case and is shown by a black arrow for each case in Figs. 10, 16 and 18. Red shaded area indicates the gap between branches, which is open in $\omega$ in this case.

Figure 8

Solutions of Eq. (11) in $k$ for the realistic data at $t_{pb}=15.0\mathrm{ms}$. On the top row, the real part of $\omega$ is Re $\omega =10{\mathrm{cm}}^{-1}$ whereas its imaginary part is Im $\omega =0$, 5 and $10{\mathrm{cm}}^{-1}$ from left to right, respectively. The middle and bottom rows correspond to Re $\omega =100$ and $-100{\mathrm{cm}}^{-1}$, respectively, and the imaginary part is Im $\omega =0$, 50 and $100{\mathrm{cm}}^{-1}$ for the left, middle and right columns, respectively. In each panel, blue and orange lines are the solutions of $\mathrm{Re}[\det [\mathrm{\Pi }]]=0$ and $\mathrm{Im}[\det [\mathrm{\Pi }]]=0$, respectively, and their intersections marked in red give an actual solution of Eq. (11). Note that we multiply $k$ with $\mathrm{\Pi }$ in drawing these pictures and hence the origin is not a solution. Gray lines indicate the zone of avoidance.

Figure 9

Dispersion relations (first and third rows) and the angular distribution differences (second and fourth rows) between ${\nu }_e$ and ${\overline{\nu }}_e$ for modified data at $t_{pb}=15.0\mathrm{ms}$. We multiply the original distribution functions of ${\overline{\nu }}_e$ by a factor 25, 31 and 35, respectively, on the second row from left to right whereas the multiplication factor is 39.5, 45 and 49, respectively, on the fourth row from left to right. In these pictures, the angular distribution differences are expressed as colored surfaces. Red implies that ${\nu }_e$ is dominant while blue means otherwise. The distance from the origin to a point on the surface is equal to the absolute value of the difference for the direction to the point. The coordinates are the same as those deployed locally in the simulation with the $z$ axis being aligned with the local radial direction; the $x$ axis corresponds to ${\varphi }_{\nu }=0$. Black arrows indicate the crossing direction, which coincides with the direction of $\mathbf{k}$ in these models. Note that the breaks found in some branches are just artifacts in drawing these pictures.

Figure 10

Growth rate or Im $\omega$ as a function of $k$ for different time steps when the crossing occurs for the first time. Note that $\mathbf{k}$ is oriented in crossing direction for all cases.

Figure 11

Solutions of Eq. (11) in $k$ for the modified data at $t_{pb}=15.0\mathrm{ms}$. The distribution function of ${\overline{\nu }}_e$ is multiplied by a factor 35 (see the rightmost pictures on the first and second rows in Fig. 9). We set Re $\omega =-5.0{\mathrm{cm}}^{-1}$ and Im $\omega =0$, 1.0 and $2.0{\mathrm{cm}}^{-1}$ from left to right. The red circles mark one of the solutions of $\det [\mathrm{\Pi }]=0$ we focus on here.

Figure 12

Angular-distribution difference between ${\nu }_e$ and ${\overline{\nu }}_e$ as a function of $\cos {\theta }_{\nu }$ for different multiplication factors at $t_{pb}=15.0\mathrm{ms}$. Right and left halves correspond to ${\varphi }_{\nu }=0.35$ and 3.49 rad, or the first and fourth azimuthal-angle bins in the simulation, respectively. The crossing occurs for the first time between the multiplication factors 31 (orange line) and 35 (blue line).

Figure 13

Same as Fig. 9 but for $\mathbf{k}$ in the radial direction (or the positive $z$ direction).

Figure 14

Comparison of the growth rates as a function of $k$ between the crossing and radial directions at $t_{pb}=15.0\mathrm{ms}$. The multiplication factor is 39.5, which corresponds to the near equal populations of ${\nu }_e$ and ${\overline{\nu }}_e$.

Figure 15

Same as Fig. 11 but for $k$ in the radial direction. The distribution function of ${\overline{\nu }}_e$ is multiplied by a factor 35 (see the rightmost pictures on the first and second rows in Fig. 13). We set Re $\omega =0.1{\mathrm{cm}}^{-1}$ and Im $\omega =0$, 0.02 and $0.07{\mathrm{cm}}^{-1}$ from left to right. The red circle marks one of the solutions of $\det [\mathrm{\Pi }]=0$ that we focus on here.

Figure 16

Same as Fig. 9 but for modified data at $t_{pb}=190.4\mathrm{ms}$. The multiplication factor is 1.5, 1.6 and 1.7 for the top two rows while it is 1.95, 2.3 and 2.8 for the bottom two rows. Black arrows indicate the crossing direction in this case, to which we set the direction of $\mathbf{k}$.

Figure 17

Same as Fig. 11 but at $t_{pb}=190.4\mathrm{ms}$. The direction of $\mathbf{k}$ is set to the crossing direction. The multiplication factor is 1.7 and Re $\omega =-0.75{\mathrm{cm}}^{-1}$ and Im $\omega =0.0$, 0.05 and $0.1{\mathrm{cm}}^{-1}$ from left to right, respectively. The red circles mark one of the solutions of $\det [\mathrm{\Pi }]=0$ that we pay attention to here.

Figure 18

Same as Figs. 9 and 16 for modified data at $t_{pb}=275.9\mathrm{ms}$. The multiplication factor is 1.2, 1.3 and 1.4 for the top two rows while it is 1.5, 1.7 and 2.0 for the bottom two rows. from left to right. The direction of $\mathbf{k}$ is set to the crossing direction, which is indicated by black arrows.

Figure 19

Same as Fig. 18 but for $\mathbf{k}$ in the $y$ direction as shown by black arrows. The multiplication factor is 1.2, 1.3 and 1.4 for the top two rows while it is 1.5, 1.7 and 2.0 for the bottom two rows from left to right.

Figure 20

Comparison of the growth rates as a function of $k$ between the crossing and radial directions at $t_{pb}=275.9\mathrm{ms}$. The multiplication factor is 1.5, which corresponds to the near equal populations of ${\nu }_e$ and ${\overline{\nu }}_e$.

Figure 21

Neutrino fluxes of different species at different points both inside and outside the PNS at $t_{pb}=15.0\mathrm{ms}$. The spatial points are indicated by yellow arrows, which also show the local radial directions. Red, blue and green arrows display the fluxes of ${\nu }_e$, ${\overline{\nu }}_e$ and ${\nu }_x$, respectively. Note that they are normalized as shown in each panel.

Figure 22

Same as Fig. 21 but at $t_{pb}=190.4\mathrm{ms}$.

Figure 23

Same as Figs. 21 and 22 but at $t_{pb}=275.9\mathrm{ms}$.

Figure 24

Same as Fig. 7 but for different postbounce times. Top panel corresponds to the $t_{pb}=190.4$ whereas the bottom panel corresponds to $t_{pb}=275.9\mathrm{ms}$. Note that the scales are different from panel to panel.

Figure 25

Same as Fig. 8 but at $t_{pb}=190.4\mathrm{ms}$ (top panels) and for $t_{pb}=275.9\mathrm{ms}$ (bottom panels). In the former case, we take Re $\omega =0.5{\mathrm{cm}}^{-1}$ and Im $\omega =0$, 0.05 and $0.15{\mathrm{cm}}^{-1}$ from left to right whereas in the latter we set Re $\omega =0.2{\mathrm{cm}}^{-1}$ and Im $\omega =0$, 0.1 and $0.2{\mathrm{cm}}^{-1}$ from left to right.