Damping the neutrino flavor pendulum by breaking homogeneity

The most general case of self-induced neutrino flavor evolution is described by a set of kinetic equations for a dense neutrino gas evolving in both space and time. Solutions of these equations have been typically worked out assuming that either the time (in the core-collapse supernova environment) or space (in the early Universe) homogeneity in the initial conditions is preserved through the evolution. In these cases, one can gauge away the homogeneous variable and reduce the dimensionality of the problem. In this paper, we investigate whether small deviations from an initial postulated homogeneity can be amplified by the interacting neutrino gas, leading to a new flavor instability. To this end, we consider a simple two-flavor isotropic neutrino gas evolving in time, and initially composed by only ${\nu }_e$ and ${\overline{\nu }}_e$ with equal densities. In the homogeneous case, this system shows a bimodal instability in the inverted mass hierarchy scheme, leading to the well-studied flavor pendulum behavior. This would lead to periodic pair conversions ${\nu }_e{\overline{\nu }}_e\leftrightarrow {\nu }_x{\overline{\nu }}_x$. To break space homogeneity, we introduce small amplitude space-dependent perturbations in the matter potential. By Fourier transforming the equations of motion with respect to the space coordinate, we then numerically solve a set of coupled equations for the different Fourier modes. We find that even for arbitrarily tiny inhomogeneities, the system evolution runs away from the stable pendulum behavior: the different modes are excited and the space-averaged ensemble evolves towards flavor equilibrium. We finally comment on the role of a time decaying neutrino background density in weakening these results.

Figure 1

Evolution of the component $P_{0,z}$ for $k_0=\kappa$. The continuous curve corresponds to a fluctuation seed $\epsilon =10^{-3}$, and the dashed one to $\epsilon =10^{-7}$, while the dotted one is for the homogeneous case with $\epsilon =0$.

Figure 2

Trajectory of the zero mode polarization vector ${\mathbf{P}}_0$ in the $x$-$z$ plane for the homogeneous case (left panel) and for an inhomogeneity with $\epsilon =10^{-3}$ and $k_0=\kappa$ (right panel).

Figure 3

Evolution of the component $P_{0,z}$ for $\epsilon =10^{-3}$ and different values of $k_0=c\kappa$: $c=1$ (continuous curve), $c=10^2$ (dashed curve), and $c=10^{-2}$ (dotted curve).

Figure 4

Evolution of the first four modes $|{\mathbf{P}}_n|$ for the case $c=1$ and $\epsilon =10^{-3}$. The continuous curve corresponds to $n=1$, the dashed one to $n=2$, the dash-dotted one to $n=3$, and the dotted one to $n=4$.

Figure 5

Flavor evolution with a declining neutrino density with $\mu ={\mu }_0\times \exp (-t/\tau )$. We take ${\mu }_0=50$, $\epsilon =10^{-3}$, and $k_0=\sqrt{2{\mu }_0\omega }$. In the left panels, we show the evolution of $P_{0,z}$ (dashed curves) and the modulus $|{\mathbf{P}}_0|$ (continuous curves) for different $\tau$, while in the right panels we show the evolution of $|{\mathbf{P}}_1|$ (continuous curves) and $|{\mathbf{P}}_2|$ (dashed curves).