  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.
