= Solution
Let the photon <phase-space distribution> be a Planck distribution whose local temperature is $T(\eta)[1+\Theta(\eta,\mathbf x,\widehat{\mathbf p})]$. In the absence of collisions, <Liouville theorem> says that the distribution is constant along a photon <null geodesic>. Linearizing $df/d\eta=0$ about the homogeneous distribution and using $d\mathbf x/d\eta=\widehat{\mathbf p}$ gives
$$
\left(\partial_\eta+\widehat{\mathbf p}\mathbin\cdot\nabla\right)\Theta
-\frac{d\log\epsilon}{d\eta}=0.
$$
A spatial <Fourier transform> sends $\widehat{\mathbf p}\mathbin\cdot\nabla$ to $i\widehat{\mathbf p}\mathbin\cdot\mathbf k$, so
$$
\boxed{
\frac{\partial\Theta}{\partial\eta}
+i(\widehat{\mathbf p}\mathbin\cdot\mathbf k)\Theta
-\frac{d\log\epsilon}{d\eta}=0}.
$$
This is the <Free-streaming photon Boltzmann equation>: the second term transports angular structure, while the last term is the gravitational redshift source.
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