= Vector photon Boltzmann hierarchy
{title2=$\Theta_\ell^{(\pm)}$}
With $B_i^{(\pm)}=iB^{(\pm)}m_i^{(\pm)}/\sqrt2$ and the same convention for baryon velocity, choose $\Theta^{(\pm)}=\sum_{\ell\geq1}(-i)^\ell\sqrt{2\pi/(2\ell+1)}\Theta_\ell^{(\pm)}Y_{\ell,\pm1}$. The <spherical-harmonic streaming recurrence> gives
$$
\dot\Theta_\ell^{(\pm)}+k\left[\frac{\sqrt{(\ell+1)^2-1}}{2\ell+3}\Theta_{\ell+1}^{(\pm)}-\frac{\sqrt{\ell^2-1}}{2\ell-1}\Theta_{\ell-1}^{(\pm)}\right]=\dot\tau\left(1-\frac{\delta_{\ell2}}{10}\right)\Theta_\ell^{(\pm)}\mp(\dot B^{(\pm)}+\dot\tau v_b^{(\pm)})\delta_{\ell1}.
$$
This 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.
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