The Full Second-Order Radiation Transfer Function for Large-Scale CMB Anisotropies

TL;DR

The paper delivers the first complete calculation of the full second-order radiation transfer function for large-scale CMB anisotropies in a flat $\\Lambda$CDM universe, incorporating the second-order Sachs-Wolfe effect, both early and late ISW effects, and second-order tensor contributions, for generic primordial non-Gaussian initial conditions. It develops the evolution of second-order scalar potentials in the Poisson gauge, formulates initial conditions via the non-Gaussian parameter $a_{nl}$, and derives explicit second-order ISW transfer functions that connect primordial non-Gaussianity to observable CMB multipoles. A detailed angular decomposition yields a generalized second-order transfer framework with convolution terms $\\Delta_{L_1L_2}(k_1,k_2)$ and kernels $f_n$, enabling computation of the non-Gaussian part of $a_{\\ell m}$. The work provides a foundational step toward a full second-order radiation transfer treatment, setting the stage for incorporating non-linear small-scale effects and informing CMB bispectrum analyses and searches for primordial non-Gaussianity.

Abstract

We calculate the full second-order radiation transfer function for Cosmic Microwave Background anisotropies on large angular scales in a flat universe filled with matter and cosmological constant. It includes (i) the second-order generalization of the Sachs-Wolfe effect, and of (ii) both the early and late Integrated Sachs-Wolfe effects, (iii) the contribution of the second-order tensor modes, and is valid for a generic set of initial conditions specifying the level of primordial non-Gaussianity.