Genericness of a Big Bounce in Isotropic Loop Quantum Cosmology

The absence of isotropic singularity in loop quantum cosmology can be understood in an effective classical description as the Universe exhibiting a big bounce. We show that with a scalar matter field, the big bounce is generic in the sense that it is independent of quantization ambiguities and the details of scalar field dynamics. The volume of the Universe at the bounce point is parametrized by a single parameter. It provides a minimum length scale which serves as a cutoff for computations of density perturbations thereby influencing their amplitudes.

Figure 1

For the small volume with nonperturbative corrections, the scalar matter Hamiltonian along a trajectory is given by $H_m\approx h{p}^{3/2}$. Thus nonperturbatively corrected scalar matter Hamiltonian vanishes at $p=0$. If there were no quantum geometry potential term, $W_{\mathrm{q}\mathrm{g}}$, in the effective Hamiltonian then for the spatially flat case ($\eta =0$) the $p=0$ point would have been accessible through the evolution. Although being a nonsingular evolution it would have taken an infinite coordinate time to reach $p=0$, nevertheless there would have been no minimum proper length for the given space-time. But once we incorporate the effect of quantum geometry potential $W_{\mathrm{q}\mathrm{g}}$ then we can see that the combined effect of $H_m$ and $W_{\mathrm{q}\mathrm{g}}$ will lead the extrinsic curvature to become zero at a nonzero value of $p$. Since for small $p$, $W_{\mathrm{q}\mathrm{g}}\sim -p$ and $H_m\sim p^{3/2}$ then there will always be a region where $W_{\mathrm{q}\mathrm{g}}$ dominates over $H_m$. Naturally there will always exist a classically inaccessible region leading to a generic big bounce. The shaded region represents the classically forbidden region.

Figure 2

The plot shows how the dimensionless parameter $h{p}_0$ varies as a function of the bounce scale $q_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{e}}$ as determined by Eqs. (2, 3). $p_0=\frac{1}{6}{\ell }_{\mathrm{P}}^2$ has been chosen for the plot.