Which process(es) kept matter in equilibrium with radiation in the early universe? Which of the following process kept the matter in thermodynamic equilibrium with radiation? Is it $e^++e^-\leftrightarrow \gamma+\gamma$ or is it the scatterings like $e^-+\gamma\rightarrow e^-+\gamma$? Can one explain what happens to both these processes as the temperature of the universe falls?
 A: Both of these processes keep plasma containing electrons-positrons and photons in equilibrium, since of them distribute the energy. For example, the high-energy electron-positron pair create the photons pair due to annihilation (the first process). Then one of photons pair interact with the low-energy electron through Compton scattering (the second process) and give to it large part of the energy. And so on.
When the temperature decreases down to $T \sim m_{e}$, the averaged energy of photon pair is typically not sufficient to produce the electron-positron pair (with minimal energy being $2m_{e}$). Therefore the process
$$
e^{-}+e^{+} \to \gamma + \gamma
$$
is no-longer invertible. The electron-positron pair can still annihilate to photons, as long as there are positrons (note that there must be the matter-antimatter asymmetry in the Early Universe). This in particular explains why the temperature of relic photons observed today is larger in the factor $\left(\frac{11}{4}\right)^{\frac{1}{3}}$ than the relic neutrinos temperature. 
The process 2 is remained invertible for these temperatures.
Next, suppose that the temperature continues to decrease. The photons feel interaction
$$
e^{-} + \gamma \leftrightarrow \gamma + e^{-}
$$
as long as corresponding interaction rate $\Gamma (e\gamma \to e\gamma)$ is longer than the Hubble scale $H(t) = \frac{\dot{a}}{a}$. The rate $\Gamma (e\gamma\to e\gamma)$ is proportional to the electron number density $n_{e}$,
$$
\Gamma(e\gamma \to e\gamma) \simeq n_{e}\sigma_{e\gamma \to e\gamma}
$$
When temperatures are small, $T << m_{e}$, the only remained charged states in plasma are electron $e^{-}$ and proton $p^{+}$. The Universe is assumed to be electrically neutral, so when they tend to form bounded state (hydrogen atom), the electron density tends to zero. If the temperature (and hence averaged energies of photons) in plasma are significantly smaller than the Hydrogen ionization energy $E_{\text{ion}} = 13.6 \text{ eV}$, then the density $n_{e}$ begins to decrease quickly. Then $\Gamma$ begins to be smaller than $H$, and photons decouple (propagates as "frozen state"). The process 2 then almost disappears.
