Photon Boltzmann hierarchy (${\Theta }_{\ell }$ hierarchy)

Expanding the direction-dependent photon temperature in Legendre polynomials turns its transport equation into an infinite hierarchy. Free streaming couples each multipole ${\Theta }_{\ell }$ only to its neighbours ${\Theta }_{\ell -1}$ and ${\Theta }_{\ell +1}$.

The dipole member of the photon Boltzmann hierarchy is the photon Euler equation. In Newtonian gauge and without collisions,