Title: Averaging procedure in variable-G cosmologies

Abstract: Previous work in the literature had built a formalism for spatially averaged equations for the scale factor, giving rise to an averaged Raychaudhuri equation and averaged Hamiltonian constraint, which involve a backreaction source term. The present paper extends these equations to include models with variable Newton parameter and variable cosmological term, motivated by the non-perturbative renormalization program for quantum gravity based upon the Einstein--Hilbert action. The coupling between backreaction and spatially averaged three-dimensional scalar curvature is found to survive, and all equations involving contributions of a variable Newton parameter are worked out in detail. Interestingly, under suitable assumptions, an approximate solution can be found where the universe tends to a FLRW model, while keeping track of the original inhomogeneities through two effective fluids.