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

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.