Examples of backreaction of small scale inhomogeneities in cosmology

In previous work, we introduced a new framework to treat large scale backreaction effects due to small scale inhomogeneities in general relativity. We considered one-parameter families of spacetimes for which such backreaction effects can occur, and we proved that, provided the weak energy condition on matter is satisfied, the leading effect of small scale inhomogeneities on large scale dynamics is to produce a traceless effective stress-energy tensor that itself satisfies the weak energy condition. In this work, we illustrate the nature of our framework by providing two explicit examples of one-parameter families with backreaction. The first, based on previous work of Berger, is a family of polarized vacuum Gowdy spacetimes on a torus, which satisfies all of the assumptions of our framework. As the parameter approaches its limiting value, the metric uniformly approaches a smooth background metric, but spacetime derivatives of the deviation of the metric from the background metric do not converge uniformly to zero. The limiting metric has nontrivial backreaction from the small scale inhomogeneities, with an effective stress-energy that is traceless and satisfies the weak energy condition, in accord with our theorems. Our second one-parameter family consists of metrics which have a uniform Friedmann-Lemaitre-Robertson-Walker limit. This family satisfies all of our assumptions with the exception of the weak energy condition for matter. In this case, the limiting metric has an effective stress-energy tensor which is not traceless. We emphasize the importance of imposing energy conditions on matter in studies of backreaction.