A 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. 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 properties of the Ricci-Hamilton flow make it possible to study a critical behaviour of cosmological models. This criticality is discussed and it is argued that it may be related to the formation of sheet-like 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 non-trivial scale dependent shear terms, while the spatial anisotropy of the metric influences significantly the scale-dependence of the matter distribution.Comment: 57 pages, LaTeX, 15 pictures available on request from the Author

Invariant Correlations in Simplicial Gravity

Some first results are presented regarding the behavior of invariant correlations in simplicial gravity, with an action containing both a bare cosmological term and a lattice higher derivative term. The determination of invariant correlations as a function of geodesic distance by numerical methods is a difficult task, since the geodesic distance between any two points is a function of the fluctuating background geometry, and correlation effects become rather small for large distances. Still, a strikingly different behavior is found for the volume and curvature correlation functions. While the first one is found to be negative definite at large geodesic distances, the second one is always positive for large distances. For both correlations the results are consistent in the smooth phase with an exponential decay, turning into a power law close to the critical point at $G_c$. Such a behavior is not completely unexpected, if the model is to reproduce the classical Einstein theory at distances much larger than the ultraviolet cutoff scale.Comment: 27 pages, conforms to published versio

Tomographic Representation of Minisuperspace Quantum Cosmology and Noether Symmetries

The probability representation, in which cosmological quantum states are described by a standard positive probability distribution, is constructed for minisuperspace models selected by Noether symmetries. In such a case, the tomographic probability distribution provides the classical evolution for the models and can be considered an approach to select "observable" universes. Some specific examples, derived from Extended Theories of Gravity, are worked out. We discuss also how to connect tomograms, symmetries and cosmological parameters.Comment: 15 page