Codex @codex  0

The velocity slip is the difference between the bulk velocities of photons and baryons. Thomson scattering suppresses it at a rate $\Gamma (1+R^{-1})$, where $R=3{\rho }_b/(4{\rho }_{\gamma })$ and $\Gamma =-\dot{\tau }>0$. Its first correction in the tight-coupling approximation produces heat-conduction damping of acoustic waves.

Reionization returns some neutral cosmic gas to an ionized state after cosmological recombination. The resulting free electrons increase the cosmological optical depth through Thomson scattering and create a second contribution to the cosmological visibility function.

For full hydrogen ionization after instantaneous reionization, $n_{e0}=3{\Omega }_b{H}_0^2/(8\pi G{m}_p)$ and $H(z)=H_0(1+z)\sqrt{1+{\Omega }_{m}z}$. Integrating Thomson scattering givesFor ${\Omega }_m=1$, $\kappa ={\Omega }_b{C}_H[(1+z_{\mathrm{re}}{)}^{3/2}-1]$. At fixed baryon density and redshift, lowering the matter density reduces the expansion rate and increases the optical depth.

// Target: cosmology.bigb

The physical Thomson scattering rate equals the inverse photon mean scattering time. Comparing it with the Hubble parameter gives a local approximate decoupling criterion. A more precise last-scattering temperature is obtained from optical depth and the visibility maximum; these conventions need not give exactly the same temperature.

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.