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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.

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.