We write the averaged Einstein equations in a form suitable for use with Newtonian gauge linear perturbation theory and track the size of the modifications to standard Robertson-Walker evolution on the largest scales as a function of redshift for both Einstein de-Sitter and ΛCDM cosmologies. In both cases the effective energy density arising from linear perturbations is of the order of 10 −5 times the matter density, as would be expected, with an effective equation of state w eff ≈ −1/19. Employing a modified Halofit code to extend our results to quasilinear scales, we find that, while larger, the deviations from Robertson-Walker behaviour remain of the order of 10 −5 .PACS numbers: 04.25.Nx, 95.36.+x, 98.80.Jk

We present a formalism for spatial averaging in cosmology applicable to general spacetimes and coordinates, and allowing the easy incorporation of a wide variety of matter sources. We apply this formalism to a Friedmann-Lemaître-Robertson-Walker universe perturbed to second-order and present the corrections to the background in an unfixed gauge. We then present the corrections that arise in uniform curvature and conformal Newtonian gauges. * Electronic address: I.Brown@thphys.uni-heidelberg.de † Electronic address: Juliane.Behrend@uni-ulm.de ‡ Electronic address: K.Malik@qmul.ac.uk 1 The issue of gauges in backreaction was considered in a study with somewhat different aims in [22], while a formalism dealing with general coordinates was presented in [23] while the first version of this work was in final preparation.

The cosmological backreaction arises when one directly averages the Einstein equations to recover an effective Robertson-Walker cosmology, rather than assuming a background a priori. While usually discussed in the context of dark energy, strictly speaking any cosmological model should be recovered from such a procedure. We apply the scalar spatial averaging formalism for the first time to linear Robertson-Walker universes containing matter, radiation and dark energy. The formalism employed is general and incorporates systems of multiple fluids with ease, allowing us to consider quantitatively the universe from deep radiation domination up to the present day in a natural, unified manner. Employing modified Boltzmann codes we evaluate numerically the discrepancies between the assumed and the averaged behaviour arising from the quadratic terms, finding the largest deviations for an Einstein-de Sitter universe, increasing rapidly with Hubble rate to a 0.01% effect for h = 0.701. For the ΛCDM concordance model, the backreaction is of the order of Ω 0 eff ≈ 4 × 10 −6 , with those for dark energy models being within a factor of two or three. The impacts at recombination are of the order of 10 −8 and those in deep radiation domination asymptote to a constant value. While the effective equations of state of the backreactions in Einstein-de Sitter, concordance and quintessence models are generally dust-like, a backreaction with an equation of state w eff < −1/3 can be found for strongly phantom models.PACS numbers: 04.25. Nx, 95.36.+x, 98.80.Jk

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