An Overview on the Nature of the Bounce in LQC and PQM

We present a review on basic aspects of quantum cosmology in the presence of cut-off physics. We first analyze how the Wheeler-DeWitt equation in a pure metric approach describes the quantum Universe, showing how the singularity is not removed. We then discuss basic features of loop quantum cosmology. For the isotropic Universe, we compare the original $\mu_0$ scheme and the most commonly accepted formulation, the $\bar{\mu}$ scheme. Some results concerning the Bianchi Universes are also discussed. Finally, we consider some relevant criticisms on the real link between full loop quantum gravity and its minisuperspace implementation. In the second part of the review we consider the isotropic Universe and the Bianchi models in the framework of polymer quantum mechanics. We first address the polymerization in terms of the Ashtekar-Barbero-Immirzi connection and show how the resulting dynamics is isomorphic to the $\mu_0$. Then we analyze the polymerization of volume-like variables, and we see the resulting dynamics is isomorphic to the $\bar{\mu}$ scheme with a fixed critical energy density. Finally, we consider the polymer quantum dynamics of the homogeneous and inhomogeneous Mixmaster model through a metric approach. We compare the results obtained by using the volume variable to the standard Misner variable. In the latter case we find a cosmology that is still singular, and its chaotic properties depend on the ratio between the lattice steps for the isotropic and anisotropic variables. We conclude the review with some considerations on changing variables in the polymer representation. In particular we consider how the dynamics can be mapped in two different sets of variables (at the price of using a coordinate-dependent lattice step), and we infer some possible implications on the equivalence of the $\mu_0$ and $\bar{\mu}$ scheme of loop quantum cosmology.