Hubble expansion rate and reaction rates

Early universe is very dense and hot, in this condition protons can't capture electrons to form atoms. The mean time of interraction between 2 particles is negligeable than the expansion rate give approximatively by $1/H$.

For example interaction rate for neutrino with the $l$ species ($l=$ electron for example) is

$$Gamma_nu = <n_lsigma v>$$

where $n_l$ is the density of the species l, $sigma$ the cross-section of neutrino (=probability of interaction) and $v$ the relative velocity between the two particles. Neutrino interact with weak interaction with coupling constant $G_Fsim 10^{-5}$ (Fermi's constant). You can verify that $<sigma v> sim G_F^2 T^2$ and $n_l sim T^3 $ (with fermi-dirac statistic: https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics) so $Gamma_nu sim G_F^2 T^5$.

At early times Universe is dominated by radiation and the radiation density is such as $rho_r propto T^4$. Friedman equation give you $H propto rho_r^{1/2} propto T^2$

So the time between two interaction of neutrino is $tau propto frac{1}{Gamma_nu}propto T^{-5}$ and expansion time is $tsimfrac{1}{H}propto T^{-2}$. Comparing the two you can see that for high temperature $tau ll t$.

Note that with temperature decreasing $tau$ finally become larger than $t$, it's the decoupling of neutrino.

$T^{-2}$ vs $T^{-5}$" />

I hope this clarify your question !