A hydrodynamical approach to CMB μ-distortion from primordial perturbations

Spectral distortion of the cosmic microwave background provides a unique opportunity to probe primordial perturbations on very small scales by performing large-scale measurements. We discuss in a systematic and pedagogic way all the relevant physical phenomena involved in the production and evolution of the mu-type spectral distortion. Our main results agree with previous estimates (in particular we show that a recently found factor of 3/4 arises from relativistic corrections to the wave energy). We also discuss several subleading corrections such as adiabatic cooling and the effects of bulk viscosity, baryon loading and photon heat conduction. Finally we provide formulae for the spatial dependence of mu-distortions and its transfer function between the end of the mu-era and now.