Relativistic perturbations in $\Lambda$CDM: Eulerian & Lagrangian approaches

We study the relativistic dynamics of a pressure-less and irrotational fluid of dark matter (CDM) with a cosmological constant ($\Lambda$), up to second order in cosmological perturbation theory. In our analysis we also account for vector and tensor perturbations and include primordial non-Gaussianity. We consider three gauges: the synchronous-comoving gauge, the Poisson gauge and the total matter gauge, where the first is the unique relativistic Lagrangian frame of reference, and the latters are convenient gauge choices for Eulerian frames. Our starting point is the metric and fluid variables in the Poisson gauge up to second order. We then perform the gauge transformations to the synchronous-comoving gauge and subsequently to the total matter gauge. Our expressions for the metrics, densities, velocities, and the gauge generators are novel and coincide with known results in the limit of a vanishing cosmological constant.