The luminosity distance–redshift relation up to second order in the Poisson gauge with anisotropic stress

Abstract

We present the generalization of previously published results, about the perturbed redshift and the luminosity–redshift relation up to second order in perturbation theory, for the case of the Poisson gauge and in the presence of anisotropic stress. The results are therefore valid for general dark energy models and (most) modified gravity models. We use an innovative approach based on the recently proposed ‘geodesic light-cone’ gauge. We then compare our findings with other results, which recently appeared in the literature, for the particular case of vanishing anisotropic stress. Arriving at a common accepted expression for the nonlinear and relativistic corrections to the redshift and distance–redshift relation is of fundamental importance in view of future cosmological surveys. Thanks to these surveys the Universe will be further probed with high precision and at very different scales, where nonlinear and relativistic effects can play a key role.