Tensor cosmological perturbation ($h_{ij}$)

A tensor cosmological perturbation is the transverse, trace-free part of a metric perturbation about a Friedmann-Lemaitre-Robertson-Walker metric. In conformal coordinates, its spatial metric has the form $g_{ij}=a^2({\delta }_{ij}+h_{ij})$, with ${\partial }_i{h}_{ij}=0$ and $h_{ii}=0$. Its two independent polarizations describe a gravitational wave. Neutrino tensor anisotropic stress can damp its evolution through neutrino free streaming.

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.