CMB Anisotropies at Second Order I

TL;DR

The paper addresses the challenge of predicting second-order CMB anisotropies by deriving the full Boltzmann system for photons, baryons, and CDM in a perturbed FRW universe. It presents the complete second-order Liouville operator and collision term, constructs the photon brightness function and its multipole hierarchy, and provides an integral line-of-sight solution, all within the Poisson gauge while neglecting polarization. These results establish the framework for the full second-order radiation transfer function under generic primordial non-Gaussian initial conditions and set the stage for a companion paper computing the CMB bispectrum. By detailing the energy-momentum tensors and the massive-particle Boltzmann equations, the work lays the groundwork for assessing non-Gaussianity arising from both primordial physics and late-time non-linear evolution.

Abstract

We present the computation of the full system of Boltzmann equations at second-order describing the evolution of the photon, baryon and cold dark matter fluids. These equations allow to follow the time evolution of the Cosmic Microwave Background (CMB) anisotropies at second-order at all angular scales from the early epoch, when the cosmological perturbations were generated, to the present through the recombination era. This paper sets the stage for the computation of the full second-order radiation transfer function at all scales and for a a generic set of initial conditions specifying the level of primordial non-Gaussianity. In a companion paper, we will present the computation of the three-point correlation function at recombination which is so relevant for the issue of non-Gaussianity in the CMB anisotropies.