Imponente, Giovanni (2003) Fingerprints of the Very Early Universe. [Tesi di dottorato] (Unpublished)

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During the work developed in the PhD Thesis, it has been faced a number of problems concerning theoretical cosmology especially with respect dynamical evolution of the Universe near the cosmological singularity. Wide space has been devoted to the study of the subtle question concerning the covariance chaoticity, which led to important issues favourable to the independence of the ‘’chaos’’ with respect to the choice of the temporal gauge. Such analysis found its basis either on the standard approach using the Jacobi metric (a scheme allowed by the existence of an energy like constant of motion), either by a statistical mechanics approach in which the mixmaster evolution is represented as a billiard on a Lobatchevski plane and therefore admitting a microcanonical ensemble associated to such energy-like constant. Furthermore, an important step consisted in searching a Physical link between the chaoticity characterizing the system at a classical level and the quantum indeterminism appearing in the Planckian era for such a model. More precisely it was constructed che canonical quantization of the model via a Schroedinger approach (equivalent to the Wheeler-DeWitt scheme) and then developed the WKB semiclassical limit to be compared with the classical dynamics. As an issue, it resulted a correspondence between the continuity equation of the microcanonical distribution function and that one describing the dynamics of the first order corrections in the wave function for h  0. It is remarkable the investigation performed about a quasi-isotropic inflationary solution, which allowed to confirm how there is non chance for classical inhomogeneous perturbations to survive after the DeSitter phase; such an analysis supports strongly the idea that only quantum fluctuations ofthe scalar field can provide a satisfactory esplanation for the observed spectrum of inhomogeneous perturbations. A detailed discussion was pursued in view of clarifying the peculiarity existing to characterize chaos in General Relativity; in particular, it has been provived a critical discussion on the predictivity allowed by the fractal basin boundary approach in qualifying the nature of the mixmaster dynamics; the main issue on this direction relies on the numerical approximations limits when treating iterations of irrational numbers and overall on the potential methods commonly adopted in the dynamical systems approach. With this respect it is emphasized the ambiguity of describing chaos in terms of geodesic deviation when the backgroun metric is a pseudo Riemannian one; a correct characterization of the Lyapunov exponents required a projection of the connecting vector toward a Fermi basis.

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