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 $n_e$?
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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.
answered Apr 20, 2017 at 17:01
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$\begingroup$ 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 $\endgroup$– MSBApr 20, 2017 at 18:21
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$\begingroup$ @MSB assuming al hydrogen will be reasonably close, but you could also pick typical values for the primordial abundances, which is something like X=.75, Y=.25, I think. $\endgroup$ Apr 20, 2017 at 23:58
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$\begingroup$ 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. $\endgroup$– user32023Aug 23, 2019 at 0:44
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$\begingroup$ @DonaldAirey you mean 10% ionized? If you mean 90%, that seems like a rather small correction to be accusing me of not answering the question. $\endgroup$ Aug 23, 2019 at 19:21