7.4. Where did the energy of the CMB come from?

Recombination occurs when the CMB temperature has dropped low enough such that there are no longer enough high energy photons to keep hydrogen ionized; + H e^- + p⁺. Although the ionization potential of hydrogen is 13.6 eV (T ~ 10⁵ K), recombination occurs at T 3000 K. This low temperature can be explained by the fact that there are a billion photons for every proton in the Universe. This allows the high energy tail of the Planck distribution of the photons to keep the comparatively small number of hydrogen atoms ionized until temperatures and energies much lower than 13.6 eV. The Saha equation (e.g. Lang 1980) describes this balance between the ionizing photons and the ionized and neutral hydrogen.

The energy in the CMB did not come from the recombination of electrons with protons to form hydrogen at the surface of last scattering. That contribution is negligible - only about one 10 eV photon for each baryon, while there are ~ 10¹⁰ times more CMB photons than baryons and each of those photons at recombination had an energy of ~ 0.3 eV: E_rec / E_CMB = (10 eV × 10^-10) / 0.3 eV ~ 10^-9. The energy in the CMB came from the annihilation of particle/anti-particle pairs during a very early epoch called baryogenesis and later when electrons and positrons annihilated at an energy of ~ 1 MeV.

As an example of energy injection, consider the thermal bath of neutrinos that fills the Universe. It decoupled from the rest of the Universe at an energy above an MeV. After decoupling the neutrinos and the photons, both being relativistic, cooled as T R^-1. If nothing had injected energy into the Universe below an MeV, the neutrinos and the photons would both have a temperature today of 1.95 K. However the photons have a temperature of 2.725 K. Where did this extra energy come from? It came from the annihilation of electrons and positrons when the temperature of the Universe fell below an MeV. This process injected energy into the Universe by heating up the residual electrons, which in turn heated up the CMB photons. The relationship between the CMB and neutrino temperatures is T_CMB = (11/4)^1/3 T. Derivation of this result using entropy conservation during electron/positron annihilation can be found in Wright (2003) or Peacock (2000). The bottom line: T_CMB = 2.7 K > T = 1.9 K because the photons were heated up by e^± annihilation while the neutrinos were not. This temperature for the neutrino background has not yet been confirmed observationally.