Relativistic second-order perturbations of nonzero-Λflat cosmological models and CMB anisotropies

TL;DR

This work develops a comprehensive treatment of second-order relativistic perturbations in spatially flat cosmologies with a nonzero cosmological constant, parameterized by an arbitrary spatial potential $F$ and analyzed in the synchronous gauge. By extending Lifshitz's approach and employing nonlinear gauge transformations, the author derives explicit second-order metric, density, and velocity perturbations, and tracks their evolution into the Poisson gauge. The study demonstrates that second-order effects generate tensor (gravitational-wave) and vector (shear) perturbations from first-order scalar modes, and it provides gauge-invariant expressions for the resulting CMB temperature anisotropies, including nonlinear integral Sachs-Wolfe contributions and gravitational-radiation terms. This framework enables analysis of nonlinear couplings between local inhomogeneities and CMB anisotropies, with potential implications for interpreting high-precision CMB data.

Abstract

First the second-order perturbations of nonzero-Λcosmological models are derived with an arbitrary potential function of spatial coordinates, using the nonlinear version of Lifshitz's method in the synchronous gauge. Their expression is the generalization (to the nonzero-Λcase) of second-order perturbations in the Einstein-de Sitter model which were derived previously by the present author. Next the second-order temperature anisotropies of Cosmic Micriwave Background radiation are derived using the gauge-invariant formula which was given by Mollerach and Matarrese. Moreover the corresponding perturbations in the Poisson gauge are derived using the second-order gauge transformations formulated by Bruni et al. In the second-order it is found in spite of gauges that tensor (gravitational-wave) perturbations and vector (shear) perturbations without vorticity are induced from the first-order scalar perturbations. These results will be useful to analyze the nonlinear effect of local inhomogeneities on Cosmic Microwave Background anisotropies.