Emerging singularities in the bouncing loop cosmology

In this paper we calculate $\mathcal{O}({\mu }^4)$ corrections from holonomies in the loop quantum gravity, usually not taken into account. Allowance of the corrections of this kind is equivalent with the choice of the new quatization scheme. Quantization ambiguities in the loop quantum cosmology allow for this additional freedom and presented corrections are consistent with the standard approach. We apply these corrections to the flat Friedmann-Robertson-Walker cosmological model and calculate the modified Friedmann equation. We show that the bounce appears in the models with the standard $\mathcal{O}({\mu }^2)$ quantization scheme and is shifted to the higher energies ${\rho }_{\mathrm{bounce}}=3{\rho }_{\mathrm{c}}$. Also, a pole in the Hubble parameter appears for ${\rho }_{\mathrm{pole}}=\frac{3}{2}{\rho }_{\mathrm{c}}$ corresponding to hyperinflation/deflation phases. This pole represents a curvature singularity at which the scale factor is finite. In this scenario the singularity and bounce coexist. Moreover, we find that an ordinary bouncing solution appears only when quantum corrections in the lowest order are considered. Higher order corrections can lead to nonperturbative effects.

Figure 1

Upper: Evolution of $H_{\mathcal{O}({\mu }^2)}^2$ (red, bottom curve) and $H_{\mathcal{O}({\mu }^4)}^2$ (green, top curve) as a function of energy density $\rho$. The parts below zero value of $H^2$ are unphysical. The $\rho$ is in units of ${\rho }_{\mathrm{c}}$. Lower: Evolution of $H_{\mathcal{O}({\mu }^2)}^2$ (red, bottom curve) and $H_{\mathcal{O}({\mu }^4)}^2$ (green, top curve) as a function of $p$ for the model with radiation ($\rho \propto 1/p^2$). On both panels a physically admissible region corresponds to $H^2\ge 0$.

Figure 2

The phase portrait for all admissible initial conditions. The vertical (blue) line is representing points at which trajectories pass horizontally through the inflection point (hyperinflationary/deflationary phases).

Figure 3

The schematic picture of the evolution of the model. The top curve (green) represents the energy density of the free scalar field. It is worth noting that this energy density is finite during whole evolution, even during transitions through singularities. The bottom curve (blue) presents schematic evolution of the scale factor for the investigated model. The dashed curve represents the classical evolution which is not realized in the presented quantum model.