Mauro Carfora

On the curvature of the present-day Universe

We discuss the effect of curvature and matter inhomogeneities on the averaged scalar curvature of the present-day universe. Motivated by studies of averaged inhomogeneous cosmologies, we contemplate on the question of whether it is sensible to assume that curvature averages out on some scale of homogeneity, as implied by the standard concordance model of cosmology, or whether the averaged scalar curvature can be largely negative today, as required for an explanation of dark energy from inhomogeneities. We confront both conjectures with a detailed analysis of the kinematical backreaction term and estimate its strength for a multi-scale inhomogeneous matter and curvature distribution. Our main result is a formula for the spatially averaged scalar curvature involving quantities that are all measurable on regional (i.e. up to 100 Mpc) scales. We propose strategies to quantitatively evaluate the formula, and pinpoint the assumptions implied by the conjecture of a small or zero averaged curvature. We reach the conclusion that the standard concordance model needs fine tuning in the sense of an assumed equipartition law for curvature in order to reconcile it with the estimated properties of the averaged physical space, whereas a negative averaged curvature is favoured, independent of the prior on the value of the cosmological constant.

Is there proof that backreaction of inhomogeneities is irrelevant in cosmology

No. In a number of papers Green and Wald argue that the standard FLRW model approximates our Universe extremely well on all scales, except close to strong field astrophysical objects. In particular, they argue that the effect of inhomogeneities on average properties of the Universe (backreaction) is irrelevant. We show that this latter claim is not valid. Specifically, we demonstrate, referring to their recent review paper, that (i) their two-dimensional example used to illustrate the fitting problem differs from the actual problem in important respects, and it assumes what is to be proven; (ii) the proof of the trace-free property of backreaction is unphysical and the theorem about it fails to be a mathematically general statement; (iii) the scheme that underlies the trace-free theorem does not involve averaging and therefore does not capture crucial non-local effects; (iv) their arguments are to a large extent coordinate-dependent, and (v) many of their criticisms of backreaction frameworks do not apply to the published definitions of these frameworks. It is therefore incorrect to infer that Green and Wald have proven a general result that addresses the essential physical questions of backreaction in cosmology.

THE GEOMETRY OF CLASSICAL CHANGE OF SIGNATURE

The proposal of the possibility of change of signature in quantum cosmology has led to the study of this phenomenon in classical general relativity theory, where there has been some controversy about what is and is not possible. Here we present a new analysis of such a change of signature, based on previous studies of the initial value problem in general relativity. We emphasize that there are various continuity suppositions one can make at a classical change of signature, and consider more general assumptions than have been made up to now. We confirm that in general such a change can take place even when the second fundamental form of the surface of change does not vanish.The proposal of the possibility of change of signature in quantum cosmology has led to the study of this phenomenon in classical general relativity theory, where there has been some controversy about what is and is not possible. We here present a new analysis of such a change of signature, based on previous studies of the initial value problem in general relativity. We emphasize that there are various continuity suppositions one can make at a classical change of signature, and consider more general assumptions than made up to now. We confirm that in general such a change can take place even when the second fundamental form of the surface of change does not vanish.

Quantum states of elementary three-geometry

We introduce a quantum volume operator K—which could play a significant role in discretized quantum gravity models—by taking into account a symmetrical coupling scheme of three SU(2) angular momenta. The spectrum of K is discrete and defines a complete set of eigenvectors which is alternative with respect to the complete sets employed when the usual binary coupling schemes of angular momenta are considered. Each of these states, which we call quantum bubbles, represents an interference of three-dimensional geometrical configurations. We study the generalized recoupling coefficients connecting the symmetrical and the binary basis vectors, and provide an explicit recursive solution for such coefficients by analysing also their asymptotic limit.

RICCI FLOW DEFORMATION OF COSMOLOGICAL INITIAL DATA SETS

Ricci flow deformation of cosmological initial data sets in general relativity is a technique for generating families of initial data sets which potentially would allow to interpolate between distinct spacetimes. This idea has been around since the appearance of the Ricci flow on the scene, but it has been difficult to turn it into a sound mathematical procedure. In this expository talk we illustrate, how Perelmans recent results in Ricci flow theory can considerably improve on such a situation. From a physical point of view this analysis can be related to the issue of finding a constant-curvature template spacetime for the inhomogeneous Universe, relevant to the interpretation of observational data and, hence, bears relevance to the dark energy and dark matter debates. These techniques provide control on curvature fluctuations (intrinsic backreaction terms) in their relation to the averaged matter distribution.

Triangulated surfaces in twistor space: a kinematical set up for open/closed string duality

We exploit the properties of the hyperbolic space 3 to discuss a simplicial setting for open/closed string duality based on (random) Regge triangulations decorated with null twistorial fields. We explicitly show that the twistorial N-points function, describing Dirichlet correlations over the moduli space of open N-bordered genus g surfaces, is naturally mapped into the Witten-Kontsevich intersection theory over the moduli space of N-pointed closed Riemann surfaces of the same genus. We also discuss various aspects of the geometrical setting which connects this model to PSL(2,) Chern-Simons theory.

The modular geometry of random Regge triangulations

We show that the introduction of triangulations with variable connectivity and fluctuating egde-lengths (Random Regge Triangulations) allows for a relatively simple and direct analyisis of the modular properties of 2 dimensional simplicial quantum gravity. In particular, we discuss in detail an explicit bijection between the space of possible random Regge triangulations (of given genus g and with N vertices) and a suitable decorated version of the (compactified) moduli space of genus g Riemann surfaces with N punctures. Such an analysis allows us to associate a Weil-Petersson metric with the set of random Regge triangulations and prove that the corresponding volume provides the dynamical triangulation partition function for pure gravity.

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.

Implementing holographic projections in Ponzano-Regge gravity

We consider the path-sum of Ponzano-Regge with additional boundary contributions in the context of the holographic principle of Quantum Gravity. We calculate an holographic projection in which the bulk partition function goes to a semi-classical limit while the boundary state functional remains quantum-mechanical. The properties of the resulting boundary theory are discussed.

Wigner Symbols and Combinatorial Invariants of Three-Manifolds with Boundary

Abstract:In this paper we generalize the partition function proposed by Ponzano and Regge in 1968 to the case of a compact 3-dimensional simplicial pair (M, ∂M). The resulting state sum Z[(M, ∂M)] contains both Wigner 6j symbols associated with tetrahedra and Wigner 3jm symbols associated with triangular faces lying in ∂M. In order to show the invariance of Z[(M, ∂M)] under PL-homeomorphisms we exploit some results due to Pachner on the equivalence of n-dimensional PL-pairs both under bistellar moves on n-simplices in the interior of M and under elementary boundary operations (shellings and inverse shellings) acting on n-simplices which have some component in ∂M. We find, in particular, the algebraic identities – involving a suitable number of Wigner symbols – which realize the complete set of Pachners boundary operations in n=3.The results established for the classical SU(2)-invariant Z[(M, ∂M)] are further extended to the case of the quantum enveloping algebra Uq(sl(2,ℂ)) (q a root of unity). The corresponding quantum invariant, Mq[(M, ∂M)], turns out to be the counterpart of the Turaev–Viro invariant for a closed 3-dimensional PL-manifold. To Giorgio Ponzano and Tullio Regge