Effects of nonlinear inhomogeneity on the cosmic expansion with numerical relativity

We construct a three-dimensional, fully relativistic numerical model of a universe filled with an inhomogeneous pressureless fluid, starting from initial data that represent a perturbation of the Einstein-de Sitter model. We then measure the departure of the average expansion rate with respect to this homogeneous and isotropic reference model, comparing local quantities to the predictions of linear perturbation theory. We find that collapsing perturbations reach the turnaround point much earlier than expected from the reference spherical top-hat collapse model and that the local deviation of the expansion rate from the homogeneous one can be as high as $28\%$ at an underdensity, for an initial density contrast of $10^{-2}$. We then study, for the first time, the exact behavior of the backreaction term ${\cal Q}_{\cal D}$. We find that, for small values of the initial perturbations, this term exhibits a $1/a$ scaling, and that it is negative with a linearly growing absolute value for larger perturbation amplitudes, thereby contributing to an overall deceleration of the expansion. Its magnitude, on the other hand, remains very small even for relatively large perturbations.