Does Bose-Einstein condensation of CMB photons cancel μ distortions created by dissipation of sound waves in the early Universe?

Abstract

The difference in the adiabatic indices of photons and non-relativistic baryonic matter in the early Universe causes the electron temperature to be slightly lower than the radiation temperature. Thermalization of photons with a colder plasma results in the accumulation of photons in the Rayleigh-Jeans tail, aided by stimulated recoil, while the higher frequency spectrum tries to approach Planck spectrum at the electron temperature $T_γ^{final}=\Te<T_γ^{initial}$; i.e., Bose-Einstein condensation of photons occurs. We find new solutions of the Kompaneets equation describing this effect. No actual condensate is, in reality, possible since the process is very slow and photons drifting to low frequencies are efficiently absorbed by bremsstrahlung and double Compton processes. The spectral distortions created by Bose-Einstein condensation of photons are within an order of magnitude (for the present range of allowed cosmological parameters), with exactly the same spectrum but opposite in sign, of those created by diffusion damping of the acoustic waves on small scales corresponding to comoving wavenumbers $45< k< 10^4\, Mpc^{-1}$. The initial perturbations on these scales are completely unobservable today due to their being erased completely by Silk damping. There is partial cancellation of these two distortions, leading to suppression of $μ$ distortions expected in the standard model of cosmology. The net distortion depends on the scalar power index $n_S$ and its running $d n_S/d\ln k$, and may vanish for special values of parameters, for example, for a running spectrum with, $n_S=1,d n_S/d\ln k=-0.038$. We arrive at an intriguing conclusion: even a null result, non-detection of $μ$-type distortion at a sensitivity of $10^{-9}$, gives a quantitative measure of the primordial small-scale power spectrum.