Vector photon Boltzmann hierarchy (${\Theta }_{\ell }^{(\pm )}$)

With $B_i^{(\pm )}=i{B}^{(\pm )}{m}_i^{(\pm )}/\sqrt{2}$ and the same convention for baryon velocity, choose ${\Theta }^{(\pm )}={\sum }_{\ell \ge 1}(-i{)}^{\ell }\sqrt{2\pi /(2\ell +1)}{\Theta }_{\ell }^{(\pm )}{Y}_{\ell ,\pm 1}$. The spherical-harmonic streaming recurrence givesThis is the temperature-only Thomson scattering hierarchy without polarization feedback, with optical depth to the observer satisfying $\dot{\tau }<0$. A photon multipole normalization change changes the denominators and the dipole-source prefactor. The lower coupling vanishes at $\ell =1$, so no vector monopole occurs.