 Solution

In the absence of scalar anisotropic stress, the traceless spatial Einstein field equations giveThe photon mass-shell condition $g_{\mu \nu }{P}^{\mu }{P}^{\nu }=0$, together with $p^2=g_{ij}{P}^i{P}^j$ and $ϵ=ap$, gives to first orderUse the time component of the geodesic equation, divide it by $d\eta /d\lambda =P^0$, and substitute the stated Christoffel symbols. Keeping the perturbed ratios $P^i/P^0=(1+\Phi +\Psi ){\hat{p}}^i$ where they multiply the background $\mathcal{H}$ terms makes those terms cancel. Comparing the result with the total derivative of $P^0=ϵa^{-2}(1-\Psi )$ yieldsThe first term is the time-dependent gravitational redshift and the second is the gravitational frequency shift along the photon direction.