Full GR cosmological simulations and back-reaction

The numerical solution of Einstein field equations, numerical relativity, has been essential in computing the merger of two black holes, predicting the signal that has been observed by LIGO in 2015, the centennial year of General Relativity.  A century after the foundation of cosmology as a physical theory, while we are developing larger and larger sky surveys, it is timely to consider the full application of GR to the nonlinear process of structure formation in the Universe.  In this talk I will review the idea of back-reaction in cosmology  and I will illustrate the first results of simulations representing the full GR nonlinear evolution of initial perturbations in a Einstein de Sitter background.  Main results are:  1) back-reaction  effects on the overall expansion  are very small; 2) voids expansion rate is significantly higher than that of the background; 3) over-densities can reach turn-around much earlier than predicted by the standard top-hat model. To establish the  significance of these results is the goal of future work. As more interdisciplinary work between the gravitational waves and the cosmology communities will develop, in the next few years numerical relativity may become a fundamental tool for understanding the  extent to which we can trust standard Newtonian N-body simulations on the largest scales.