Early cosmology treats the universe as a rapidly expanding thermal plasma in approximate equilibrium. The key question is how interaction rates $\Gamma$ compare to the Hubble rate $H$; when $\Gamma \gg H$, species track equilibrium, and when $\Gamma \lesssim H$, freeze-out or decoupling occurs.

For relativistic species, $p=\rho c^2/3$ and the energy density scales as $$\rho_r \propto a^{-4}.$$ The extra $a^{-1}$ relative to matter comes from redshifting of photon energies. The entropy density scales as $$s \propto g_{*s} T^3,$$ and adiabatic expansion implies $s a^3 \approx$ constant.

Number densities evolve according to Boltzmann equations of the form $$\dot n + 3Hn = -\langle\sigma v\rangle\left(n^2-n_{\rm eq}^2\right),$$ or their more general phase-space versions. This framework underlies calculations of neutrino decoupling, dark matter freeze-out, and baryon relics.

Recombination is not a sudden event but a finite-width process governed by atomic transition rates and the expansion rate. The optical depth determines the visibility function, which peaks near the surface of last scattering and sets the origin of the cosmic microwave background anisotropies.