Relativistic second-order perturbations of the Einstein-de Sitter Universe

TL;DR

<3-5 sentence high-level summary> The paper develops a comprehensive framework for relativistic second-order perturbations around the Einstein–de Sitter universe in both synchronous–comoving and Poisson gauges. It formulates exact and second-order gauge transformations, derives the evolution of second-order perturbations in the synchronous gauge including scalar and tensor initial conditions, and then transforms to the Poisson gauge to obtain physically interpretable results. A central finding is mode mixing: primordial density perturbations can source gravitational waves at second order, and primordial gravitational waves can seed second-order density fluctuations, with implications for mildly non-linear structure growth and secondary CMB anisotropies. The work provides new Poisson-gauge expressions and clarifies how second-order effects propagate across gauges, enabling more accurate modeling of non-linear relativistic effects in cosmology.

Abstract

We consider the evolution of relativistic perturbations in the Einstein-de Sitter cosmological model, including second-order effects. The perturbations are considered in two different settings: the widely used synchronous gauge and the Poisson (generalized longitudinal) one. Since, in general, perturbations are gauge dependent, we start by considering gauge transformations at second order. Next, we give the evolution of perturbations in the synchronous gauge, taking into account both scalar and tensor modes in the initial conditions. Using the second-order gauge transformation previously defined, we are then able to transform these perturbations to the Poisson gauge. The most important feature of second-order perturbation theory is mode-mixing, which here also means, for instance, that primordial density perturbations act as a source for gravitational waves, while primordial gravitational waves give rise to second-order density fluctuations. Possible applications of our formalism range from the study of the evolution of perturbations in the mildly non-linear regime to the analysis of secondary anisotropies of the Cosmic Microwave Background.