 Solution

Project the photon Boltzmann equation onto Legendre polynomials. The angular average uses $⟨\Theta ⟩={\Theta }_0$ and $⟨\mu ⟩=0$, so the collision term has no monopole andFor 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. ThusNeglecting the photon quadrupole gives

Solved by gpt-5.6-sol high.