Regional averaging and scaling in relativistic cosmology

Averaged inhomogeneous cosmologies lie at the forefront of interest, since cosmological parameters like the rate of expansion or the mass density are to be considered as volume-averaged quantities and only these can be compared with observations. For this reason the relevant parameters are intrinsically scale-dependent and one wishes to control this dependence without restricting the cosmological model by unphysical assumptions. In the latter respect we contrast our way to approach the averaging problem in relativistic cosmology with shortcomings of averaged Newtonian models. Explicitly, we investigate the scale-dependence of Eulerian volume averages of scalar functions on Riemannian three-manifolds. We propose a complementary view of a Lagrangian smoothing of (tensorial) variables as opposed to their Eulerian averaging on spatial domains. This program is realized with the help of a global Ricci deformation flow for the metric. We explain rigorously the origin of the Ricci flow which, on heuristic grounds, has already been suggested as a possible candidate for smoothing the initial data set for cosmological spacetimes. The smoothing of geometry implies a renormalization of averaged spatial variables. We discuss the results in terms of effective cosmological parameters that would be assigned to the smoothed cosmological spacetime.Comment: LateX, IOPstyle, 48 pages, 11 figures; matches published version in C.Q.

Cosmological parameters are dressed

In the context of the averaging problem in relativistic cosmology, we provide a key to the interpretation of cosmological parameters by taking into account the actual inhomogeneous geometry of the Universe. We discuss the relation between `bare' cosmological parameters determining the cosmological model, and the parameters interpreted by observers with a ``Friedmannian bias'', which are `dressed' by the smoothed-out geometrical inhomogeneities of the surveyed spatial region.Comment: LateX, PRLstyle, 4 pages; submitted to PR

Entropy estimates for Simplicial Quantum Gravity

Through techniques of controlled topology we determine the entropy function characterizing the distribution of combinatorially inequivalent metric ball coverings of n-dimensional manifolds of bounded geometry for every n ≥ 2. Such functions control the asymptotic distribution of dynamical triangulations of the corresponding n-dimensional (pseudo)manifolds M of bounded geometry. They have an exponential leading behavior determined by the Reidemeister-Franz torsion associated with orthogonal representations of the fundamental group of the manifold. The subleading terms are instead controlled by the Euler characteristic of M. Such results are either consistent with the known asymptotics of dynamically triangulated two-dimensional surfaces, or with the numerical evidence supporting an exponential leading behavior for the number of inequivalent dynamical triangulations on three- and four-dimensional manifolds

Invariants of spin networks with boundary in Quantum Gravity and TQFT's

The search for classical or quantum combinatorial invariants of compact n-dimensional manifolds (n=3,4) plays a key role both in topological field theories and in lattice quantum gravity. We present here a generalization of the partition function proposed by Ponzano and Regge to the case of a compact 3-dimensional simplicial pair $(M^3, \partial M^3)$. The resulting state sum $Z[(M^3, \partial M^3)]$ contains both Racah-Wigner 6j symbols associated with tetrahedra and Wigner 3jm symbols associated with triangular faces lying in $\partial M^3$. The analysis of the algebraic identities associated with the combinatorial transformations involved in the proof of the topological invariance makes it manifest a common structure underlying the 3-dimensional models with empty and non empty boundaries respectively. The techniques developed in the 3-dimensional case can be further extended in order to deal with combinatorial models in n=2,4 and possibly to establish a hierarchy among such models. As an example we derive here a 2-dimensional closed state sum model including suitable sums of products of double 3jm symbols, each one of them being associated with a triangle in the surface.Comment: 9 page

Ricci Flow Conjugated Initial Data Sets for Einstein Equations

We discuss a natural form of Ricci--flow conjugation between two distinct general relativistic data sets given on a compact $n\geq 3$-dimensional manifold $\Sigma$. We establish the existence of the relevant entropy functionals for the matter and geometrical variables, their monotonicity properties, and the associated convergence in the appropriate sense. We show that in such a framework there is a natural mode expansion generated by the spectral resolution of the Ricci conjugate Hodge--DeRham operator. This mode expansion allows to compare the two distinct data sets and gives rise to a computable heat kernel expansion of the fluctuations among the fields defining the data. In particular this shows that Ricci flow conjugation entails a form of $L^2$ averaging of one data set with respect to the other with a number of desiderable properties: (i) It preserves the dominant energy condition; (ii) It is localized by a heat kernel whose support sets the scale of averaging; (iii) It is characterized by a set of balance functionals which allow the analysis of its entropic stability.Comment: 74 pages, 22 figures added, submitted version. The paper has been vastly expanded with a new detailed introduction and with added results on the asymptotics of the Ricci flow averaged data. This work is partly based on the lectures of the author at the GGI School "Coarse--Grained Cosmology", Florence January 26--29, 200

Renormalization Group and the Ricci flow

We discuss from a geometric point of view the connection between the renormalization group flow for non--linear sigma models and the Ricci flow. This offers new perspectives in providing a geometrical landscape for 2D quantum field theories. In particular we argue that the structure of Ricci flow singularities suggests a natural way for extending, beyond the weak coupling regime, the embedding of the Ricci flow into the renormalization group flow.Comment: 30 pages, 16 PNG figures, Conference talk at the Riemann International School of Mathematics: Advances in Number Theory and Geometry, Verbania April 19-24, 2009- Proceedings to appear in Milan Journal of Mathematics (Birkhauser

Quantum Tetrahedra

We discuss in details the role of Wigner 6j symbol as the basic building block unifying such different fields as state sum models for quantum geometry, topological quantum field theory, statistical lattice models and quantum computing. The apparent twofold nature of the 6j symbol displayed in quantum field theory and quantum computing -a quantum tetrahedron and a computational gate- is shown to merge together in a unified quantum-computational SU(2)-state sum framework

12j-symbols and four-dimensional quantum gravity

We propose a model which represents a four-dimensional version of Ponzano and Regge's three-dimensional euclidean quantum gravity. In particular we show that the exponential of the euclidean Einstein-Regge action for a $4d$-discretized block is given, in the semiclassical limit, by a gaussian integral of a suitable $12j$-symbol. Possible developments of this result are discussed.Comment: 12 pages, Late