# 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 $n_e$?

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.
• I am only given $n_b/n_{\gamma}=6*10^{-10}$ and from $$n_{\gamma}=2\int \frac{d^3q}{(2\pi)^3}\frac{q}{e^{q/T_{\gamma}}-e}=2\zeta (3)T_{\gamma}^3/\pi^2$$ I find the photon density which in turn gives me nucleon density but how do i get electron density from there? the abundance of the two species is not given do I assume all of them to be hydrogen – MSB Apr 20 '17 at 18:21
• Way to not answer the question. The gas was NOT completely ionized at the epoch of recombination. The $Z_{REC}$ is generally found by taking the maximum visibility function, but if you don't want to go to all that work, you are safe by taking $X_E = 0.1$ or at 90% ionized. – Quarkly Aug 23 '19 at 0:44