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>2x10^6. 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. 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>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~2x10^6 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, 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 ~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 (~ 10^{-8}-10^{-10}).
