Semiclassical and Quantum Polymer Effects in the Flat Isotropic Universe

We analyze some relevant semiclassical and quantum features of the implementation of Polymer Quantum Mechanics to the phenomenology of the flat isotropic Universe. We firstly investigate a parallelism between the semiclassical polymer dynamics of the flat isotropic Universe, as reduced to the effect of a modified simplectic structure, and the so-called Generalized Uncertainty Principle. We show how the difference in the sign of the fundamental Poisson bracket is reflected in a sign of the modified source term in the Friedmann equation, responsible for the removal of the initial singularity in the polymer case and for the survival of a singular point in the Universe past, when the Generalized Uncertainty Principle is concerned. Then, we study the regularization of the vacuum energy of a free massless scalar field, by implementing a second quantization formalism in the context of Polymer Quantum Mechanics. We show that, from this reformulation, naturally emerges a Cosmological Constant term for the isotropic Universe, whose value depends directly on the polymer parameter of the regularization. Finally, we investigate the behaviour of gravitational waves on the background of a modified dynamics, according to the semiclassical Friedmann equation. We demonstrate that the presence of a Bounce in the Universe past, naturally removes the divergence of the gravitational wave amplitude and they can, in principle, propagate across the minimum volume turning point. This result offers the intriguing perspective for the detection of gravitational signals coming from the pre-Big Bounce collapsing Universe.