Paper detail

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.