Abstract
We analyze the dynamics of the Bianchi I model in the presence of stiff matter, an ultrarelativistic component and a small negative cosmological constant. We quantize this model in the framework of the polymer quantum mechanics, in order to introduce cut-off features in the minisuperspace dynamics.We then apply to the polymer Wheeler–DeWitt equation, emerging from the Dirac constraint, an adiabatic approximation a la Vilenkin, which treats the Universe volume as a quasi-classical variable, becoming de facto the dynamical clock for the pure quantum degrees of freedom, here identified in the Universe anisotropies.The main issue of the present analysis consists of determining a cyclical evolution for the Bianchi I model, oscillating between the Big-Bounce induced by the cut-off physics and the turning point due to the small cosmological constant. Furthermore, the mean value of the Universe anisotropy variables remains finite during the whole evolution, including the phase across the Big-Bounce. Such a feature, according to a suitable choice of the initial conditions makes the present cosmological paradigm, a viable scenario for the description of a possible primordial and late phases of the actual Universe.
Highlights
The existence of a singularity in the dynamics of the cosmological models is a very general feature of the Einstein equations, see [1] for a characterization of the generic cosmological solution and [2] for general theorems on this topic.The idea that the non-physical feature of the singularity in the cosmological problem could be removed by quantum effects has been reliably proposed since very long time [3]
The idea that the so-called Big-Bounce could replace the Big-Bang singularity emerged in a more general perspective when it became clear that the implementation of Loop Quantum Gravity [9],[14] to the cosmological problem could provide a reliable cut-off on the physics of the singularity
The main aim of the present analysis was to demonstrate that such a model constitute a good paradigm for a cyclical Universe, whose anisotropy degree of freedom are always finite and, in principle, they can be controlled via suitable initial conditions
Summary
The existence of a singularity in the dynamics of the cosmological models is a very general feature of the Einstein equations, see [1] for a characterization of the generic cosmological solution and [2] for general theorems on this topic. The anisotropy variables, described via the standard Misner variables β±, remain instead real quantum degrees of freedom and their dynamics is analyzed via the Ehrenfest theorem, compared to the wave packet behavior By this scenario, we are able to demonstrate that the Bianchi I dynamics is reduced to a cyclical Universe, oscillating between the bounce, ensured via the polymer cutoff physics and the later turning point due to the negative cosmological constant. By choosing the Vilenkin interpretation for the wave function of the Universe, we analyze from the semiclassical point of view the evolution of the isotropic variable (related to the volume of the Universe) towards the singularity and from the quantum point of view the behavior of the quantum degrees of freedom: the anisotropies.
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