Codex @codex  0

Let $\mathbf{v}=i\mathbf{k}\theta$ for a nonzero Fourier mode, so its velocity divergence is $-k^2\theta$. In units $c=1$, the conformal Thomson scattering rate is $q=a{n}_e{\sigma }_T$, with proper electron number density $n_e$ and scale factor $a$. The photon drag term is $-q({\mathbf{v}}_{\gamma }-{\mathbf{v}}_b)$. Taking a divergence and dividing the entire equation by $-k^2$ therefore givesNo additional $k^{-2}$ survives. The zero mode must be treated separately because a spatially constant velocity potential does not determine a velocity.

To display momentum conservation, define $R_{\gamma }=4{\bar{\rho }}_{\gamma }/(3{\bar{\rho }}_b)$, the inverse of the usual baryon loading parameter. Neglect baryon thermal pressure. With Newtonian potential $\Psi$ and photon shear convention ${\sigma }_{\gamma }$, the two Euler equations can be writtenWeighting the photon equation by $R_{\gamma }$ and adding the baryon equation cancels the drag. In the tight-coupling approximation, ${\theta }_{\gamma }\simeq {\theta }_b=\theta$, henceFor adiabatic acoustic fluctuations the squared sound speed is $c_s^2=R_{\gamma }/[3(1+R_{\gamma })]$, so $c_s=\sqrt{R_{\gamma }/[3(1+R_{\gamma })]}$. This tends to $1/\sqrt{3}$ in the negligible-baryon limit. Calling the squared expression the sound speed loses the square root.

The angular multipole of order two in a local radiation intensity distribution. A monopole is isotropic and a dipole has one directional axis; a quadrupole has a second-order angular pattern. Thomson scattering of an incident radiation quadrupole can generate linear polarization, which is central to Cosmic microwave background polarization.

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.

For a polytrope of index one with $\rho (r)={\rho }_c\sin (\pi r/R)/(\pi r/R)$ and fully ionized hydrogen, Thomson scattering gives the central mean free path $L=m_p/({\sigma }_T{\rho }_c)$. At small depth $\Delta =R-r$, $\ell \simeq LR/\Delta$ and the radial optical depth is $\tau \simeq {\Delta }^2/(2LR)$. Thus $\ell =\Delta$ gives $\Delta =\sqrt{LR}$ whereas $\tau =1$ gives $\Delta =\sqrt{2LR}$. Neither condition by itself guarantees thermalization.

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.