Photon decoupling occurred when the universe expands enough that photons become unlikely to
scatter off electrons. This occurred abruptly when the rate of
Compton scattering of photons \Gamma was approximately equal to the rate of
expansion of the universe H, or alternatively when the
mean free path of the photons \lambda was approximately equal to the
horizon size of the universe H^{-1}. After this photons were able to
stream freely, producing the cosmic microwave background as we know it, and the universe became transparent. The event coincides with
recombination, the time when atoms form from free electrons and protons, emitting photons, but the physics of the two events differ. There were very many more photons than electrons. The interaction rate of the photons is given by :\Gamma = \frac{c}{\lambda} = n_e \sigma_t c where n_e is the
number density of free electrons, \sigma_t is the electron
Thomson scattering area, and c is the
speed of light. In the
matter-dominated era (when recombination takes place), :H \approx H_0 a^{-{3/2}} where a is the
cosmic scale factor and H0 is the
Hubble constant. \Gamma also decreases as a more complicated function of a, at a faster rate than H. By working out the precise dependence of H and \Gamma on the scale factor and equating \Gamma=H, it is possible to show that photon decoupling occurred approximately 380,000 years after the
Big Bang, at a
redshift of z = 1100 when the universe was at a temperature around 3000 K. ==Neutrino decoupling==