Residual electron freeze-out (${\alpha }_{\mathrm{eff}}{n}_e\lesssim H$)

When the effective recombination rate per Electron becomes smaller than the expansion rate, the small remaining ionized fraction can no longer track chemical equilibrium. A nonzero Electron fraction remains, exceeding the exponentially small Saha equation prediction. The effective rate includes atomic radiative bottlenecks; chemical freeze-out is distinct from photon decoupling by scattering.