Mixing of blackbodies: entropy production and dissipation of sound waves
  in the early Universe
  Mixing of blackbodies with different temperatures creates a spectral
distortion which, at lowest order, is a y-type distortion, indistinguishable
from the thermal y-type distortion produced by the scattering of CMB photons by
hot electrons residing in clusters of galaxies. This process occurs in the
radiation-pressure dominated early Universe, when the primordial perturbations
excite standing sound waves on entering the sound horizon. Photons from
different phases of the sound waves, having different temperatures, diffuse
through the electron-baryon plasma and mix together. This diffusion, with the
length defined by Thomson scattering, dissipates sound waves and creates
spectral distortions in the CMB. Of the total dissipated energy, 2/3 raises the
average temperature of the blackbody part of spectrum, while 1/3 creates a
distortion of y-type. It is well known that at redshifts 10^5< z< 2x10^6,
comptonization rapidly transforms y-distortions into a Bose-Einstein spectrum.
The chemical potential of the Bose-Einstein spectrum is again 1/3 the value we
would get if all the dissipated energy was injected into a blackbody spectrum
but no extra photons were added. We study the mixing of blackbody spectra,
emphasizing the thermodynamic point of view, and identifying spectral
distortions with entropy creation. This allows us to obtain the main results
connected with the dissipation of sound waves in the early Universe in a very
simple way. We also show that mixing of blackbodies in general, and dissipation
of sound waves in particular, leads to creation of entropy.
