= Solution
The integrating factor for the Fourier-space transport equation is $e^{ik\mu\eta}$. Therefore the <line-of-sight solution for free-streaming photons> between an initial time $\eta_i$ and observation at $\eta_0$ is
$$
\boxed{
\Theta_{\mathbf k}(\eta_0,\mu)
=e^{-ik\mu(\eta_0-\eta_i)}\Theta_{\mathbf k}(\eta_i,\mu)
-ik\mu\int_{\eta_i}^{\eta_0}d\eta\,
e^{-ik\mu(\eta_0-\eta)}
[\delta N_{\mathbf k}(\eta)+\psi_{\mathbf k}'(\eta)]}.
$$
The first term freely streams the initial angular distribution; the integral accumulates the lapse and shift source along the unperturbed photon path.
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