Optimal covariant fitting to a Robertson-Walker metric and smallness of
  backreaction
  We define a class of "optimal" coordinate systems by requiring that the
deviation from an exact Robertson-Walker metric is "as small as possible"
within a given four dimensional volume. The optimization is performed by
minimizing several volume integrals which would vanish for an exact
Robertson-Walker metric. Covariance is automatic. Foliation of space-time is
part of the optimization procedure. Only the metric is involved in the
procedure, no assumptions about the origin of the energy-momentum tensor are
needed. A scale factor does not show up during the optimization process, the
optimal scale factor is determined at the end. The general formulation is non
perturbative. An explicit perturbative treatment is possible. The shifts which
lead to the optimal coordinates obey Euler-Lagrange equations which are
formulated and solved in first order of the perturbation. The extension to
second order is sketched, but turns out to be unnecessary. The only freedom in
the choice of coordinates which finally remains are the rigid transformations
which keep the form of the Robertson-Walker metric intact, i.e. translations in
space and time, spatial rotations, and spatial scaling. Spatial averaging
becomes trivial. In first order of the perturbation there is no backreaction. A
simplified second order treatment results in a very small effect, excluding the
possibility to mimic dark energy from backreaction. This confirms (as well as
contradicts) statements in the literature.
