Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 312 / 4 / i / Solution

In a local orthonormal frame, a massless neutrino has $p^{\hat{a}}=E(1,\mathbf{e})$ and $E=ϵ/a$. With degeneracy $g_{\nu }$, kinetic theory gives the stress-energy tensorCoordinate components follow by inserting the orthonormal-frame vectors on the indices.

For the perturbation $\delta f=-\Theta ϵ{\bar{f}}^{\prime }(ϵ)$, integration by parts givesassuming the boundary term vanishes. Consequently the physical trace-free spatial stress isThe paper uses the opposite signed source, ${\Pi }^{\hat{i}\hat{j}}=-{\pi }_{\nu }^{\hat{i}\hat{j}}$, consistently with its tensor equation having $-16\pi G{a}^2{\Pi }_{ij}$ on the right. In that convention the neutrino tensor anisotropic stress is