= Solution
In the absence of <scalar anisotropic stress>, the traceless spatial <Einstein field equations> give
$$
\boxed{\Phi=\Psi}.
$$
The photon <mass-shell condition> $g_{\mu\nu}P^\mu P^\nu=0$, together with $p^2=g_{ij}P^iP^j$ and $\epsilon=ap$, gives to first order
$$
\boxed{P^\mu=\frac{\epsilon}{a^2}
\left[1-\Psi,(1+\Phi)\widehat p^i\right]}.
$$
Use 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 $P^i/P^0=(1+\Phi+\Psi)\widehat p^i$ where they multiply the background $\mathcal H$ terms makes those terms cancel. Comparing the result with the total derivative of $P^0=\epsilon a^{-2}(1-\Psi)$ yields
$$
\boxed{\frac{d\ln\epsilon}{d\eta}
=\Phi'-\widehat p^i\partial_i\Psi}.
$$
The first term is the time-dependent gravitational redshift and the second is the gravitational frequency shift along the photon direction.
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