Evolution of low-frequency features in the CMB spectrum due to stimulated Compton scattering and Doppler broadening

We discuss a new analytic solution of the Kompaneets equation for physical situations in which low frequency photons, forming relatively narrow spectral details, are Compton scattered in an isotropic, infinite medium with an intense ambient blackbody field that is very close to full thermodynamic equilibrium with the free electrons. In this situation the background-induced stimulated Compton scattering slows down the motion of photons toward higher frequencies by a factor of 3 in comparison with the solution that only takes Doppler broadening and boosting into account. This new solution is important for detailed computations of cosmic microwave background spectral distortions arising from uncompensated atomic transitions of hydrogen and helium in the early Universe. It also clearly shows that the broadening of weak lines in this situation only depends on the Compton y-parameter defined by Te, even though the evolution of the ambient CMB blackbody spectrum itself is described by $y\propto T_{\rm e}-T_{\gamma}$. In addition, we derive another analytic solution that only includes the background-induced stimulated Compton scattering and is valid for power law ambient radiation fields. This solution might have interesting applications for radio lines arising inside of bright extra-galactic radio sources, where according to our estimates line shifts because of background-induced stimulated scattering could be amplified and even exceed the line broadening due to the Doppler effect