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For a homogeneous canonical inflaton oscillating near $V=m^2{\varphi }^2/2$, use reduced-Planck units and require $m\gg H$. The slowly varying ansatz $\varphi =A(t)\cos mt$ gives $H_m^2\simeq m^2{A}^2/6$. The Nonincreasing Hubble rate for a canonical inflaton becomesCycle averaging gives ${\dot{H}}_m=-3H_m^2/2$, so $H_m=2/[3(t-t_{\ast })]$. Integrating the rapidly varying term while holding its coefficient approximately constant gives the displayed correction. It is smaller than the mean by $O(H_m/m)$. Although the correction oscillates, the total Hubble parameter remains nonincreasing. Equal average kinetic and potential energies give pressureless-matter-like expansion, with amplitude $A\propto a^{-3/2}$.

A radiation-dominated universe has expansion controlled mainly by radiation in cosmology. When spatial curvature and the cosmological constant are negligible, the Friedmann equation and density scaling $\rho \propto a^{-4}$ give the displayed scale factor, with Hubble parameter $H=1/(2t)$.

A particle species thermally decouples when its interaction rate falls below the Hubble parameter. Its collisionless relativistic distribution subsequently redshifts with temperature proportional to $a^{-1}$, while annihilations among still-coupled species can heat the remaining plasma.

The physical Thomson scattering rate equals the inverse photon mean scattering time. Comparing it with the Hubble parameter gives a local approximate decoupling criterion. A more precise last-scattering temperature is obtained from optical depth and the visibility maximum; these conventions need not give exactly the same temperature.

In natural units with unreduced Planck mass $M_P=G^{-1/2}$, the radiation-era Hubble parameter scales as $\sqrt{g_{\ast }}{T}^2/M_P$, while a low-energy weak interaction rate scales as $G_F^2{T}^5$. Their equality gives a decoupling scale of order one MeV. Distinct weak processes have distinct rate coefficients: neutrino decoupling and cosmological weak freeze-out of neutron-proton conversion are nearby events, not an identical precise temperature inferred from the power laws alone.

For one particle species in chemical equilibrium under $2\varphi \leftrightarrow 3\varphi$, chemical balance gives $2{\mu }_{\varphi }=3{\mu }_{\varphi }$, hence ${\mu }_{\varphi }=0$. Elastic $2\varphi \leftrightarrow 2\varphi$ imposes no such constraint. In an expanding universe the number-changing reaction must remain fast compared with the Hubble parameter; after chemical freeze-out a nonzero chemical potential can develop even if elastic collisions maintain kinetic equilibrium.