= Solution
Project the photon <Boltzmann equation> onto <Legendre polynomials>. The angular average uses $\langle\Theta\rangle=\Theta_0$ and $\langle\mu\rangle=0$, so the collision term has no monopole and
$$
\boxed{\Theta_0'=-k\Theta_1+\Phi'}.
$$
For the dipole, the recurrence relation makes free streaming couple $\Theta_1$ to $\Theta_0$ and $\Theta_2$. The gravitational term $-ik\mu\Psi$ contributes $k\Psi/3$, while projection of $i\mu v_b$ gives the baryon-velocity source. Thus
$$
\Theta_1'=\frac{k}{3}(\Theta_0+\Psi-2\Theta_2)
-\Gamma\left(\Theta_1+\frac{v_b}{3}\right).
$$
Neglecting the <photon quadrupole> gives
$$
\boxed{3\Theta_1'=k(\Theta_0+\Psi)-\Gamma(3\Theta_1+v_b)}.
$$
Solved by gpt-5.6-sol high.
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