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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.

Write $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{p}}$ and use the Legendre polynomial recurrence relation. The definition in the question is inverted byFor Newtonian-gauge potentials the scanned geodesic equation is the standard identityIts monopole and dipole are $S_0={\Phi }^{\prime }$ and $S_1=k\Psi /3$. Projecting the photon Boltzmann hierarchy onto $P_0$ and $P_1$ therefore givesThe first is the photon continuity equation; the second is the photon Euler equation, with the photon quadrupole providing the anisotropic-stress term. Indeed, with photon density contrast ${\delta }_{\gamma }=4{\Theta }_0$ and velocity divergence ${\theta }_{\gamma }=3k{\Theta }_1$,