Refined approximations for the distortion visibility function and
  mu-type spectral distortions
  The physical processes affecting the thermalization of cosmic microwave
background spectral distortions are very simple and well understood. This
allows us to make precise predictions for the distortions signals caused by
various energy release scenarios, where the theoretical uncertainty is largely
dominated by the physical ingredients that are used for the calculation. Here,
we revisit various approximations for the distortion visibility function --
defined using the fraction of the released energy that does not thermalize --
and early $\mu$-type distortions. Our approach is based on a perturbative
expansion, which allows us to identify and clarify the origin of different
improvements over earlier approximations. It provides a better than ~0.1%-1%
description of our numerical results over a wide range of parameters. In
particular, we are able to capture the high-frequency part of the
mu-distortion, which directly depends on the time derivative of the electron
temperature. We also include lowest order double Compton and Compton scattering
relativistic corrections, finding that because of cancelation they increase the
thermalization efficiency in the tail of the distortion visibility function by
only ~10% (at $z\lesssim 6\times10^6$), although individually their effect can
reach ~20%-40%.
