How to compute Coulomb scattering rate(in early universe)? I want to compare Coulomb scattering rate with Hubble's expansion rate. 
To compute Coulomb scattering rate, I know that
$$ \Gamma = n_p\sigma v_{rel} $$
Here, $\sigma = \frac{e^{4}}{(4\pi\epsilon_o)^{2}}\frac{1}{m_e^{2}v_{rel}^{4}}4\pi(1+\frac{m_e}{m_p}) ln\Lambda$
I can approximate all the terms here. But I do not know what should I substitute for $v_{rel}$ given that I am dealing with early universe.
Failed attempts- I thought I can substitute $$ v_{rel} = |v_{p,rms}-v_{e,rems}| \approx v_{e,rms}$$ Since, $m_p \gg m_e$
Here, I can get rems values by using the fact that electron and photons are tightly coupled until Decoupling and hence $$ T_e = T_\gamma = T_o*z $$ where, $T_o = 2.7^{o}C$ is the CMB temperature today and $Z$ is the redshift.
However I noticed that with smaller redshifts a.k.a later times temperature $T$ will fall and hence $v_{rel}$ will fall. But I see that $$\Gamma \propto \frac{1}{v_{rel}^{3}}$$
Thus interaction rate increases with time. This makes no sense at all.
So, what should $v_{rel}$ be taken as? In online PDF's I noticed that they take it as a constant along with masses during analysis. How? 
NOTE- After discussing with my professor and studying a bit more I came to know that this is actually a feature of Coloumb interaction. Its coupling increases actually increases over time. So the question is answered.
 A: You can look up the standard Coulomb collision rates in numerous places.  For example, a recent paper by Wilson et al. [2018] has some convenient formulas and numerical results for the solar wind.

Thus interaction rate increases with time. This makes no sense at all.

Let's ignore the extra factors and just assume the collision rate, $\nu$, goes as $\nu \propto n \ v^{-3}$ or if we replace the speeds with thermal speeds, then $\nu \propto n \ T^{-3/2}$.  At the time of the Big Bang nucleosynthesis, the temperature was ~$10^{9}$ K (or ~86200 eV) and the mass density was ~0.004 kg m-3.  If we assume all protons, then the corresponding number density is $n_{p} \sim 10^{24}$ m-3 (Note that even if we start getting detailed and use fractional densities for heavier ions, it would not change the results enough to matter.).
If we compare the solar wind now, we have $n_{p} \sim 10^{7}$ m-3 and $T_{p}$ ~ 10 eV [e.g., Wilson et al., 2018].  I realize that the solar wind is not characteristic of the cold interstellar medium (ISM) to which the evolution discussed implies, but it will not matter as you will see.
So if we take the old and current universe values we find that:
$$
\nu_{old} \sim 10^{31}  \\
\nu_{new} \sim 10^{8}
$$
or a decrease of ~23 orders of magnitude.
Suppose we use ISM averages for the warm ionized medium (i.e., $n_{p}$ ~ 0.2–0.5 cm-3 and $T_{p}$ ~ 8000 K ~ 0.7 eV), again assuming only protons, then we have $\nu_{new} \sim 10^{5}$, which is again much smaller than the early universe value.
Note:  I see that I said things backwards in my original comment.  I should have said, as the thermal speed decreases, the collision rate will increase for constant density.

In online PDF's I noticed that they take it as a constant along with masses during analysis. How?

I am not sure why the speeds used to estimate Coulomb collision rates would be considered constant over the age of the universe because that is not consistent with our current understanding.
References


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*Wilson, L.B., III, et al., "The Statistical Properties of Solar Wind Temperature Parameters Near 1 au," Astrophys. J. Suppl. 236(2), pp. 41, doi:10.3847/1538-4365/aab71c, 2018.

