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.
