Cosmic Microwave Background Anisotropies up to Second Order

TL;DR

The notes develop a comprehensive framework to compute cosmic microwave background anisotropies at second order by deriving the full Boltzmann system for photons, baryons, and cold dark matter in a flat FRW universe using the Poisson gauge. The approach combines the left-hand side of the Boltzmann equation, a detailed second-order collision term due to Thomson scattering including Vishniac and Kompaneets effects, and a line-of-sight integral solution for the second-order brightness function, together with energy and momentum conservation equations for baryons and CDM. Central to the work is the construction of the second-order photon multipole hierarchy and its coupling to lower-order perturbations, enabling the prediction of second-order radiation transfer functions and non-Gaussian signatures in the CMB. This formalism sets the foundation for disentangling primordial non-Gaussianity from secondary effects and provides a path toward accurate interpretation of Planck and future CMB observations.

Abstract

These lecture notes present the computation of the full system of Boltzmann equations describing the evolution of the photon, baryon and cold dark matter fluids up to second order in perturbation theory, as recently studied in (Bartolo, Matarrese & Riotto 2006, 2007). These equations allow to follow the time evolution of the cosmic microwave background anisotropies at all angular scales from the early epoch, when the cosmological perturbations were generated, to the present, through the recombination era. The inclusion of second-order contributions is mandatory when one is interested in studying possible deviations from Gaussianity of cosmological perturbations, either of primordial (e.g. inflationary) origin or due to their subsequent evolution. Most of the emphasis in these lectures notes will be given to the derivation of the relevant equations for the study of cosmic microwave background anisotropies and to their analytical solutions.