Neutrino tensor anisotropic stress (${\pi }_{\nu }^{ij}$)

For massless neutrinos with a temperature perturbation $\Theta$ and background energy density ${\bar{\rho }}_{\nu }$, the physical trace-free spatial stress-energy tensor isSome conventions define ${\Pi }^{ij}=-{\pi }_{\nu }^{ij}$ and correspondingly reverse the sign of the stress source in the gravitational-wave equation.

Projecting free-streaming neutrinos onto the tensor spherical harmonic givesHere $j_2$ is a Spherical Bessel function. The continuous value at zero follows by integrating $(1-{\mu }^2{)}^2$, whose integral is $16/15$.

For either tensor helicity, neutrino free streaming produces the memory equationThe neutrino tensor free-streaming kernel encodes the angular propagation of the perturbation. Energy transferred to neutrino directional anisotropy damps the oscillation amplitude of a gravitational wave after horizon entry. This effect was calculated by Weinberg.