An exact quantification of backreaction in relativistic cosmology

An important open question in cosmology is the degree to which the Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions of Einstein's equations are able to model the large-scale behavior of the locally inhomogeneous observable universe. We investigate this problem by considering a range of exact n-body solutions of Einstein's constraint equations. These solutions contain discrete masses, and so allow arbitrarily large density contrasts to be modeled. We restrict our study to regularly arranged distributions of masses in topological 3-spheres. This has the benefit of allowing straightforward comparisons to be made with FLRW solutions, as both spacetimes admit a discrete group of symmetries. It also provides a time-symmetric hypersurface at the moment of maximum expansion that allows the constraint equations to be solved exactly. We find that when all the mass in the universe is condensed into a small number of objects (less than or similar to 10) then the amount of back-reaction in dust models can be large, with O(1) deviations from the predictions of the corresponding FLRW solutions. When the number of masses is large (greater than or similar to 100), however, then our measures of back-reaction become small (less than or similar to 1%). This result does not rely on any averaging procedures, which are notoriously hard to define uniquely in general relativity, and so provides (to the best of our knowledge) the first exact and unambiguous demonstration of back-reaction in general relativistic cosmological modelling. Discrete models such as these can therefore be used as laboratories to test ideas about back-reaction that could be applied in more complicated and realistic settings.