CMB Temperature Anisotropies from Third Order Gravitational Perturbations

In this paper we present a complete computation of the cosmic microwave background (CMB) anisotropies up to third order from gravitational perturbations accounting for scalar, vector and tensor perturbations. We then specify our results to the large-scale limit, providing the evolution of the gravitational potentials in a flat universe filled with matter and cosmological constant which characterizes the integrated Sachs–Wolfe effect. As a by-product in the large scale approximation we are able to give non-perturbative solutions for the photon geodesic equations. Our results are the first step towards providing a complete theoretical prediction for cubic nonlinearities which are particularly relevant for characterizing the level of non-Gaussianity in the CMB through the detection of the four-point angular connected correlation function (trispectrum). For this purpose we also allow for generic initial conditions due to primordial non-Gaussianity.