 Solution

Insert the photon temperature multipole expansion into the Fourier-space transport equation. The Legendre polynomial recurrence relation turns multiplication by $\mu$ into nearest-neighbour multipole couplings, while Orthogonality of Legendre polynomials projects onto a fixed $\ell$. The monopole projection has no collision term because Thomson scattering conserves photon number:The dipole projection receives the electron-velocity source,For every $\ell \ge 2$, the collision term damps the anisotropic multipole and the photon Boltzmann hierarchy isThus the Free-streaming photon Boltzmann equation moves angular structure between neighbouring multipoles, whereas Thomson scattering suppresses all multipoles above the dipole in the tight-coupling approximation.