How to find the density of electrons during the recombination era? I am supposed to find the value of the redshift during photon decoupling when $\Gamma=H$, where $\Gamma=n_e \sigma c$ is the rate of photon-electron scattering and $H$ is the Hubble parameter. I know that during photon decoupling the Universe is matter dominated, so I take $$H^2\approx H_0^2 \Omega_{m0}(1+z)^3$$
But now how do I find $\sigma$ and $n_e$? 
 A: The cross section $\sigma$ is easy enough, it's the Thompson scattering cross section of the electron, $\sigma_t=6.65\times10^{-29}\,{\rm m}^2$. You can check that Thompson scattering is the correct scattering process by comparing the typical photon energy and the rest mass of the electron (if the photons are energetic enough, you get Compton scattering instead).
The electron density is also not too bad. At the time of recombination you can calculate the (matter) density of the Universe, and given $\Omega_{\rm b}$ you can calculate the baryon density. At the time there was a primordial mixture of mostly hydrogen and helium, so if you know the abundances of these two species (usually denoted $X$ and $Y$, with the "metal abundance" being $Z$) you can work out how many electrons per unit mass you should have for a completely ionized gas, allowing you to convert the mass density to a number density of electrons.
Donald Airy helpfully points out in the comments that the gas was not fully ionized at the time when photons decouple. Indeed, the dropping ionization fraction is part of the reason the photon mean free path becomes long. To quote him, the ionization fraction "can be found by taking the maximum visibility function". Or as a rough approximation, an ionization fraction of about 10% can work.
