Linear stability analysis of collective neutrino oscillations without spurious modes

Collective neutrino oscillations are induced by the presence of neutrinos themselves. As such, they are intrinsically nonlinear phenomena and are much more complex than linear counterparts such as the vacuum or Mikheyev-Smirnov-Wolfenstein oscillations. They obey integro-differential equations, for which it is also very challenging to obtain numerical solutions. If one focuses on the onset of collective oscillations, on the other hand, the equations can be linearized and the technique of linear analysis can be employed. Unfortunately, however, it is well known that such an analysis, when applied with discretizations of continuous angular distributions, suffers from the appearance of so-called spurious modes: unphysical eigenmodes of the discretized linear equations. In this paper, we analyze in detail the origin of these unphysical modes and present a simple solution to this annoying problem. We find that the spurious modes originate from the artificial production of pole singularities instead of a branch cut on the Riemann surface by the discretizations. The branching point singularities on the Riemann surface for the original nondiscretized equations can be recovered by approximating the angular distributions with polynomials and then performing the integrals analytically. We demonstrate for some examples that this simple prescription does remove the spurious modes. We also propose an even simpler method: a piecewise linear approximation to the angular distribution. It is shown that the same methodology is applicable to the multienergy case as well as to the dispersion relation approach that was proposed very recently.

Figure 1

A schematic picture of neutrino emissions from the neutrino sphere. The circle indicates the neutrino sphere, which sits slightly outside the PNS. Its radius is $R$. The thick straight line is one of the trajectories of neutrinos emitted from a point on the neutrino sphere. The emission angle is denoted by ${\theta }_R$ and is defined as the angle between the trajectory and the radial direction at the emission point, while $\theta$ is given at each point on the trajectory, as displayed in the figure.

Figure 2

The imaginary part of $\mathrm{\Omega }$, $\mathrm{Im}\mathrm{\Omega }$, as a function of $\mu$ for the exact (red dashed lines) and approximate (blue solid lines) solutions for different numbers of angular bins. From top to bottom the number of bins $N_a$ is 1, 2, 4, and 8. We employ $B(u)=1$ for these calculations. Note that only the leftmost branch of the approximate solutions (blue solid lines) approaches the true solutions (red dashed lines) as $N_a$ increases and is regarded as the (approximate) physical solution, whereas all of the other approximate solutions are spurious, moving rightwards away from the true solutions as $N_a$ increases.

Figure 3

Absolute values of $D(\mathrm{\Omega })$ in the complex plane of $\mathrm{\Omega }$. The integrals in $I_n$ are performed analytically. The plus indicates one of the complex zero points of $D(\mathrm{\Omega })$. The gray line is the branch cut of $D(\mathrm{\Omega })$ on the Riemann surface. Note that only one of the zero points approaches the true one as $N_a$ increases.

Figure 4

Same as Fig. 3 except that the integrals are approximately evaluated with the discretization. Clockwise from the top-left panel, the number of angular bins $N_a$ is 1, 2, 4, and 8. Plus signs indicate some of the zero points of $D(\mathrm{\Omega })$, whereas orange dots correspond to the exact root of $D(\mathrm{\Omega })$ given in Fig. 3.

Figure 5

The original function (red line) and the polynomial approximations (blue lines) of $B(u)$ (top panel) and the absolute values of the differences between them (bottom panel). The dotted, dot-dashed, short dashed, long dashed, and solid lines correspond, respectively, to the polynomial degrees $d$ of 0, 1, 2, 3, and 4.

Figure 6

Top: The imaginary part of $\mathrm{\Omega }$ as a function of $\mu$ for the solutions of Eq. (10) with $I_n$ given in Eq. (22). The line styles denote the polynomial degrees $d$ as in Fig. 5. Bottom: The absolute values of the differences between the exact and approximate solutions for $d=4$.

Figure 7

The absolute values of $D(\mathrm{\Omega })$ in the complex plane of $\mathrm{\Omega }$ for $B(u)$ given in Fig. 5 with $\mu =20$. The degree of the polynomial is set to $d=1$ and 4 in the top and bottom panels, respectively.

Figure 8

Absolute values of $D(\mathrm{\Omega })$ in the complex plane of $\mathrm{\Omega }$ for the simple multienergy and multiangle model given in the text. $D$ is evaluated in the discretization approximation with $N_{\omega }=16$, $N_a=2$ (top panel) or calculated in Eq. (24) with 16 subintervals in energy, and with $G_n(p)$ being evaluated in Eq. (25) with 16 angular bins (middle panel) or obtained with Eq. (27) for $N_{\omega }=32,d=4$ (bottom panel). Zero points are marked with red pluses and the branch cuts are indicated with the gray lines.

Figure 9

The dispersion relation for a toy model with $G_{\mathit{v}}\propto (\mu +1)\mu (\mu -0.65)$. The top panel displays the results obtained by the discretization of the $\mu$ integrals with four angular bins, while in the bottom panel we present the results of the analytical-integration method. The shaded regions are the zone of avoidance.

Figure 10

The absolute values of $D(\omega ,\mathit{k})$ in the complex plane of $\omega$ for $k=0.1$. The angular distribution $G_{\mathit{v}}$ is the same as in Fig. 9. The top panel displays the results obtained by the discretization of the $\mu$ integrals with four angular bins, while in the bottom panel we present the results of the analytical-integration method. Red pluses mark the positions of the zero points of $D$ in each method. The gray line in the bottom panel indicates the branch cut of the Riemann surface obtained in the analytical-integration method.

Figure 11

The absolute values of $D(\omega ,\mathit{k})$ in the complex plane of $k$ for $\omega =0.4$ for a toy model with $G_{\mathit{v}}\propto -(\mu +0.5)(\mu -1)$. The top panel displays the results obtained by the discretization of the $\mu$ integrals with 20 angular bins, while in the bottom panel we present the results of the analytical-integration method. Red pluses mark the positions of the zero points of $D$ in each method. The gray lines in the bottom panel indicate the branch cuts of the Riemann surface obtained in the analytical-integration method.

Figure 12

The dispersion relation obtained in the analytical-integration method for the same toy model as in Fig. 11. The shaded regions are the zone of avoidance.

Figure 13

The angular distributions of ${\nu }_e$ (green pluses) and ${\overline{\nu }}_e$ (red crosses) at a time of 200 ms post bounce and at a radius of 40 km obtained in a SN simulation. The blue points are the difference between them and the blue solid line is the polynomial approximation to it.

Figure 14

Dispersion relations for the angular distribution in Fig. 13. The top panel displays the results obtained by the discretization approximation, while in the bottom panel we present the results of the analytical-integration method. The shaded regions are the zone of avoidance.

Figure 15

The absolute values of $D(\omega ,\mathit{k})$ in the complex plane of $\omega$ for $k=10{\mathrm{cm}}^{-1}$ for the angular distribution given in Fig. 13. The top panel displays the results obtained by the discretization approximation, while in the bottom panel we present the results of the analytical-integration method. Red pluses mark the positions of the zero points of $D$ in each method. The gray line in the bottom panel indicates the branch cut of the Riemann surface obtained in the analytical-integration method.

Figure 16

The absolute values of $D(\omega ,\mathit{k})$ in the complex plane of $k$ for $\omega =5{\mathrm{cm}}^{-1}$ for the angular distribution given in Fig. 13. The top panel displays the results obtained by the discretization approximation, while in the bottom panel we present the results of the analytical-integration method. Red pluses mark the positions of the zero points of $D$ in each method. The gray lines in the bottom panel indicate the branch cuts of the Riemann surface obtained in the analytical-integration method.