CMB Anisotropies at Second-Order II: Analytical Approach

TL;DR

The paper develops an analytical framework to compute second-order CMB anisotropies generated by non-linear physics at recombination, extending the tight-coupling treatment of the photon-baryon fluid and generalizing the Meszaros effect to include second-order CDM perturbations. It derives the full set of second-order Boltzmann equations, including a non-negligible second-order photon quadrupole, and formulates the second-order radiation transfer function across relevant scales, incorporating primordial non-Gaussianity through aNL. By solving in two k-regimes and analyzing subhorizon evolution through equality and recombination, the work provides explicit expressions for the second-order monopole, dipole, and higher multipoles, and identifies surface- and gravity-induced contributions to the CMB bispectrum. The results establish a tractable analytical baseline for the second-order CMB, particularly useful for isolating recombination-era non-Gaussian signatures and guiding numerical implementations of the full second-order transfer function. This advances the ability to separate primordial NG from secondary non-Gaussianity in Planck-like data and informs future analyses of CMB statistics at second order.

Abstract

We provide an analytical approach to the second-order Cosmic Microwave Background (CMB) anisotropies generated by the non-linear dynamics taking place at last scattering. We study the acoustic oscillations of the photon-baryon fluid in the tight coupling limit and we extend at second-order the Meszaros effect. We allow for a generic set of initial conditions due to primordial non-Gaussianity and we compute all the additional contributions arising at recombination. Our results are useful to provide the full second-order radiation transfer function at all scales necessary for establishing the level of non-Gaussianity in the CMB.