Spectro-spatial evolution of the CMB II: generalised Boltzmann hierarchy

TL;DR

This paper develops a generalized photon Boltzmann hierarchy to model spectral distortion anisotropies in the early Universe, simplifying calculations and enabling detailed analysis of physical effects influencing the CMB.

Contribution

It introduces a new formalism extending the Boltzmann hierarchy with spectral parameters, reducing computational complexity and paving the way for future studies of distortion anisotropies.

Findings

Formalism describes Doppler and potential effects on distortions

Incorporates spectral evolution via Compton scattering

Provides a basis for computing CMB power spectra with distortions

Abstract

In this paper, we formulate a generalised photon Boltzmann hierarchy that allows us to model the evolution and creation of spectral distortion anisotropies in the early Universe. We directly build on our first paper in this series, extending the thermalisation Green's function treatment to the anisotropic case. We show that the problem can be described with the common Boltzmann hierarchy for the photon field extended by new spectral parameters -- a step that reduces the complexity of the calculation by at least two orders of magnitude. Our formalism describes the effects of i) Doppler and potential driving, ii) spectral evolution by Compton scattering, iii) perturbed thermalisation and iv) anisotropic heating on the distortion anisotropies. We highlight some of the main physical properties of the equations and also outline the steps for computing CMB power spectra including distortion anisotropies. Limitations and extensions of the formulation are also briefly discussed. The novel Boltzmann hierarchy given here is the basis for a series of companion papers studying how distortion anisotropies evolve in the perturbed Universe and which physical processes could be constrained using future CMB imaging techniques.