#Today I Read#

Method

Separate the modes of flavor isospins.

Results

Two Colliding Beams

1. Plus mode (summation of d modes for the two beams) is unstable in IH; Minus mode is unstable in NH.

With Axial Distribution

1. For some axial distribution of emission, axial part of spherical harmonic expansion of the modes shows that the equation of motions for the modes are decoupled. And the $\lvert m\rvert > 1$ modes do not depend on the number density of neutrinos and are the same.
2. The $m=0$ mode, aka axial symmetric mode, is just like the plus mode in two colliding beams model, which means it’s unstable in IH.
3. The $m=\pm 1$ modes, behaves like the minus mode in two colliding beams model, which means it is unstable in NH.

A More General Angular Distribution

1. Expand the modes using spherical harmonics shows that the $l$, $m$ modes are decoupled and $l=0$ works like $m=0$ in previous case and l=1 works like $m=\pm 1$.