An exact quantification of backreaction in relativistic cosmology

TL;DR

This paper uses exact solutions of Einstein's equations with discrete masses arranged in symmetric topologies to quantify how inhomogeneities affect large-scale cosmological models, showing that backreaction diminishes with increasing number of masses.

Contribution

It provides the first exact, unambiguous demonstration of backreaction effects in relativistic cosmology using discrete, symmetric n-body solutions without averaging procedures.

Findings

Large backreaction (O(1)) occurs with fewer than 10 masses.

Backreaction becomes small (<1%) with more than 100 masses.

Results are independent of averaging, providing a clear demonstration of backreaction effects.

Abstract

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 behaviour 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 modelled. 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 (<10) then the amount of backreaction 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 (>100), however, then our measures of backreaction become small (<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 backreaction in general relativistic cosmological modelling. Discrete models such as these can therefore be used as laboratories to test ideas about backreaction that could be applied in more complicated and realistic settings.