Spatial averaging limit of covariant macroscopic gravity: Scalar corrections to the cosmological equations

It is known that any explicit averaging scheme of the type essential for describing the large scale behavior of the Universe must necessarily yield corrections to the Einstein equations applied in the cosmological setting. The question of whether or not the resulting corrections to the Einstein equations are significant is still a subject of debate, partly due to possible ambiguities in the averaging schemes available. In particular, it has been argued in the literature that the effects of averaging could be gauge artifacts. We apply the formalism of Zalaletdinov's macroscopic gravity (MG), which is a fully covariant and nonperturbative averaging scheme, in an attempt to construct gauge independent corrections to the standard Friedmann-Lema\^tre-Robertson-Walker (FLRW) equations. We find that whereas one cannot escape the problem of dependence on one gauge choice---which is inherent in the assumption of large scale homogeneity and isotropy---it is however possible to construct space-time scalar corrections to the standard FLRW equations. This partially removes the criticism concerning the corrections being gauge artifacts. For a particular initial choice of gauge which simplifies the formalism, we explicitly construct these scalars in terms of the underlying inhomogeneous geometry, and incidentally demonstrate that the formal structure of the corrections with this gauge choice is identical to that of analogous corrections derived by Buchert in the context of spatial averaging of scalars.