The late-time heating Green's function and improvements to distortion frequency hierarchy treatment

Early energy injection leaves an imprint on the observed blackbody spectrum of the CMB, allowing us to study the thermal history of the Universe. For small energy release, the distortion can be efficiently computed using the quasi-exact Green's function method. For pre-recombination injections, the Green's function has already been studied previously. Here we reconsider the pre- and post-recombination periods, showcasing both the spectral distortion intensity and the relative temperature difference, which encrypt precious information about physical processes such as free-free interactions and thermal decoupling. We present the associated distortion visibility function, investigating the impact of various physical effects. We then study improvements to the so-called frequency hierarchy (FH) treatment, a method that was developed for the modelling of anisotropic distortions, which like the average distortion signals encode valuable cosmological information. Specifically, the FH treatment has shortcomings even in the $\mu$ era, that in principle should be easy to overcome. In this paper, we introduce a new approach to reduce the mismatch, concluding with a redefinition of the $\mu$ spectral shape using CosmoTherm. This solution takes into account double Compton and Bremsstrahlung effects in the low tail, which can be included in the FH. This opens the path towards a refined modeling of spectral distortion anisotropies.