Cosmological Backreaction in Spherical and Plane Symmetric Dust-Filled
  Space-Times
  We examine the implementation of Buchert's and Green & Wald's averaging
formalisms in exact spherically symmetric and plane symmetric dust-filled
cosmological models. We find that, given a cosmological space-time, Buchert's
averaging scheme gives a faithful way of interpreting the large-scale expansion
of space, and explicit terms that precisely quantify deviations from the
behaviour expected from the Friedmann equations of homogeneous and isotropic
cosmological models. The Green & Wald formalism, on the other hand, does not
appear to yield any information about the large-scale properties of a given
inhomogeneous space-time. Instead, this formalism is designed to calculate the
back-reaction effects of short-wavelength fluctuations around a given
"background" geometry. We find that the inferred expansion of space in this
approach is entirely dependent on the choice of this background, which is not
uniquely specified for any given inhomogeneous space-time, and that the
"back-reaction" from small-scale structures vanishes in every case we study.
This would appear to limit the applicability of Green & Wald's formalism to the
study of large-scale expansion in the real Universe, which also has no
pre-defined background. Further study is required to enhance the evaluation and
comparison of these averaging formalisms, and determine whether the same
difficulties exist, in less idealized space-time geometries.
