Creation of the CMB spectrum: precise analytic solutions for the blackbody photosphere

The blackbody spectrum of CMB was created in the blackbody photosphere at redshifts z greater than or similar to 2 x 10(6). At these early times, the Universe was dense and hot enough that complete thermal equilibrium between baryonic matter (electrons and ions) and photons could be established on time scales much shorter than the age of the Universe. Any perturbation away from the blackbody spectrum was suppressed exponentially. New physics, for example annihilation and decay of dark matter, can add energy and photons to CMB at redshifts z greater than or similar to 10(5) and result in a Bose-Einstein spectrum with a non-zero chemical potential (mu). Precise evolution of the CMB spectrum around the critical redshift of z similar or equal to 2 x 10(6) is required in order to calculate the mu-type spectral distortion and constrain the underlying new physics. Although numerical calculation of important processes involved (double Compton process, comptonization and bremsstrahlung) is not difficult with present day computers, analytic solutions are much faster and easier to calculate and provide valuable physical insights. We provide precise (better than 1%) analytic solutions for the decay of mu, created at an earlier epoch, including all three processes, double Compton, Compton scattering on thermal electrons and bremsstrahlung in the limit of small distortions. This is a significant improvement over the existing solutions with accuracy similar to 10% or worse. We also give a census of important sources of energy injection into CMB in standard cosmology. In particular, calculations of distortions from electron-positron annihilation and primordial nucleosynthesis illustrate in a dramatic way the strength of the equilibrium restoring processes in the early Universe. Finally, we point out the triple degeneracy in standard cosmology, i.e., the mu and y distortions from adiabatic cooling of baryons and electrons, Silk damping and annihilation of thermally produced WIMP dark matter are of similar order of magnitude (similar to 10(-8)-10(-10)).