Steady state of fast-oscillating neutrinos in an inhomogeneous medium

TL;DR

This work investigates how neutrino flavor evolution in an inhomogeneous medium is shaped by advection in a one-dimensional, azimuthally symmetric setup. By analyzing an ensemble of single-crossed ELN-XLN angular distributions, deriving a linear dispersion relation and performing nonlinear evolution, the authors show that advection drives a cascade of flavor waves to ever smaller spatial and angular scales, which are subsequently damped, leaving a quasi-homogeneous steady state. They introduce a simple empirical forecast for the final angle-averaged survival probability ⟨P_ee(t_f)⟩ based on the initial ELN-XLN distribution, and identify regimes where flavor equipartition on one side of the spectrum is or is not generic. The findings challenge the notion of universal equipartition, reveal the sensitivity to neutrino-antineutrino asymmetry, and provide a path toward semi-analytic flavor-conversion prescriptions that can be incorporated into hydrodynamic simulations of core-collapse supernovae and neutron-star merger remnants. The work lays groundwork for efficient subgrid models of flavor instabilities in complex astrophysical environments, while noting limitations such as neglect of collisions and vacuum oscillations.

Abstract

The streaming of neutrinos in an inhomogeneous medium is known to affect the physics of flavor conversion. We employ an ensemble of single-crossed angular distributions and explore the physics of flavor conversion, while neutrinos propagate across a one-dimensional inhomogeneous medium. The advective term in the neutrino equations of motion is responsible for the cascade of flavor waves towards ever smaller spatial and angular scales. However, as the system evolves, perturbations with large wavenumbers are damped, with a resulting smearing of the flavor configuration. We provide a simple recipe that allows to forecast the steady-state flavor configuration achieved by neutrinos without solving their kinetic equations. In particular, we find that flavor equipartition on one side of the angular spectrum and the cancellation of the spectral crossing in the lepton number distributions, proposed in the literature as generic flavor outcome, is a special solution depending on the degree of neutrino-antineutrino asymmetry. This work constitutes a step forward towards the development of semi-analytic schemes to account for flavor conversion physics in hydrodynamic simulations of core-collapse supernovae and neutron-star merger remnants.

Figures (5)

• Figure 1: Left: Growth rate of flavor instabilities as a function of their effective wavenumber, $K=k+\langle D^z_1\rangle$, where $\langle D_1^z\rangle$ is the space-averaged ELN-XLN flux, for the angular distributions C1--C4 defined by Eqs. \ref{['eq:ini_conditions']} and Table \ref{['tab:1']}. The vertical dotted (dashed) lines signal the location of the homogeneous (fastest-growing) mode. The dots on each $\omega_I(k)$ curve mark the Fourier modes $k_m=2\pi m/L$ for $0\leq m\leq 30$; this highlights that periodic boundary conditions impose discrete spectra on the solutions of the equations of motion. Right: Maximum growth rate of the flavor instability for the single-crossed angular distributions defined by Eqs. \ref{['eq:ini_conditions']}, in the plane spanned by the space-averaged ELN-XLN ($\langle D_0^z\rangle \propto \alpha$, cf. Eq. \ref{['eq:D0z']}) and the width of the $\bar{\nu}_e$ distribution ($\sigma$). For all cases in this parameter space, the fastest-growing mode is inhomogeneous ($k\neq 0$).
• Figure 2: Left: Evolution of the space-averaged lepton-number flux, $\langle D_1^z\rangle$, for the ELN-XLN distributions C1--C4 highlighted in Fig. \ref{['fig:1']} (see also Table \ref{['tab:1']}). The system enters the non-linear regime at $t~\sim 100 \mu^{-1}$ for all our benchmark ELN-XLN distributions. In a homogeneous medium, the evolution would be periodic, but advection forces the polarization vectors away from the $z$-axis, breaking such periodicity. Right: Angle-integrated ELN-XLN number, $D_0^z$, as a function of time for the ELN-XLN configuration C2. The orange line shows the space-averaged $D_0^z$, which is conserved according to Eq. \ref{['eq:fourier-polarization-ev']}; the remaining lines show the evolution of the local lepton number extracted at different spatial locations in the box ($r = 349.6 \mu^{-1}$, $450 \mu^{-1}$, and $649.8 \mu^{-1}$, from light to dark gray, respectively). Close to the peak of the initial perturbation (located at $r = 450 \mu^{-1}$), $D_0^z$ initially shows signs of periodicity (cf. light gray curves); such periodicity, however, disappears at later times as inhomogeneities cascade to small scales. Farther away from $r = 450 \mu^{-1}$ (cf. black line), the onset of the non-linear phase is delayed and no periodic features appear.
• Figure 3: Snapshots of the Fourier ($x$-axis) and multipolar ($y$-axis, for $\hat{D}_l(t)$ and $0\leq l\leq 9$) decomposition for the neutrino ensemble C4 extracted at $t=0$, $54 \mu^{-1}$, $450 \mu^{-1}$, and $900 \mu^{-1}$, from left to right, respectively. The top (bottom) panels represent the (off-)diagonal components of $D_l(t,r)$. Flavor instabilities trigger a cascade towards small spatial and angular scales (cf. middle panels), which subsides as the ELN-XLN distribution approaches a steady state (cf. right panel). The white line at $k\sim 0.5\mu^{-1}$ marks the position of the fastest-growing Fourier mode.
• Figure 4: Left: Angle-integrated and space-averaged $\nu_e$ survival probability in the plane spanned by the average ELN-XLN ($\langle D_0^z\rangle$) and the width of the $\bar{\nu}_e$ distribution ($\sigma$). The white, red, and cyan contours mark the regions with small relative error between the numerical solutions and the criteria defined by Eqs. \ref{['eq:finalstate']}, \ref{['eq:equipartition']}, and \ref{['eq:no-crossing']}, respectively. As for the relative error, we use different confidence levels (dotted contours for $\Delta \lesssim 8\%$ and dashed contours for $\Delta \lesssim 15\%$). Equation \ref{['eq:finalstate']} reproduces numerical results significantly better than the remaining criteria, which model partial (and complementary) regions of our parameter space with a lower degree of accuracy. Our four benchmark ELN-XLN configurations, C1--C4 (cf. Table \ref{['tab:1']}), are marked with yellow stars. Right: Spatially averaged ELN-XLN distribution for the angular configurations C1--C4. The dotted (solid) lines show the distribution at the beginning (end) of each simulation. We note that the part of the angular distribution with $v>v_c$ is not driven to $0$, but changes its sign as a result of flavor conversion.
• Figure 5: Left: Average ELN$-$XLN distribution at $t=900\mu^{-1}$ for the flavor configuration C4 and three different initial seed perturbations. Independent of the perturbation, the $v>0$ part of the ELN-XLN distribution changes sign as a result of fast flavor conversion. The initial perturbation mildly affects the shape of the distribution in the angular range with $v<0$. Right: Space-averaged and angle-integrated $\nu_e$ survival probability as a function of time for the flavor configuration C4. The initial perturbation affects the onset of flavor conversion and the early evolution of the system, but the effect of the perturbation on the asymptotic $\langle P_{ee}\rangle$ is relatively small.