Renormalization group approach to relativistic cosmology

We discuss the averaging hypothesis tacitly assumed in standard cosmology. Our approach is implemented in a ``3+1'' formalism and invokes the coarse-graining arguments, provided and supported by the real-space renormalization group (RG) methods, in parallel with lattice models of statistical mechanics. Block variables are introduced and the recursion relations written down explicitly enabling us to characterize the corresponding RG flow. To leading order, the RG flow is provided by the Ricci-Hamilton equations studied in connection with the geometry of three-manifolds. The possible relevance of the Ricci-Hamilton flow in implementing the averaging in cosmology has been previously advocated, but the physical motivations behind this suggestion were not clear. The RG interpretation provides us with such physical motivations. The properties of the Ricci-Hamilton flow make it possible to study a critical behavior of cosmological models. This criticality is discussed and it is argued that it may be related to the formation of sheetlike structures in the universe. We provide an explicit expression for the renormalized Hubble constant and for the scale dependence of the matter distribution. It is shown that the Hubble constant is affected by nontrivial scale-dependent shear terms, while the spatial anisotropy of the metric influences significantly the scale dependence of the matter distribution.