Exotic singularities and spatially curved loop quantum cosmology

We investigate the occurrence of various exotic spacelike singularities in the past and the future evolution of $k=\pm 1$ Friedmann-Robertson-Walker model and loop quantum cosmology using a sufficiently general phenomenological model for the equation of state. We highlight the nontrivial role played by the intrinsic curvature for these singularities and the new physics which emerges at the Planck scale. We show that quantum gravity effects generically resolve all strong curvature singularities including big rip and big freeze singularities. The weak singularities, which include sudden and big brake singularities, are ignored by quantum gravity when spatial curvature is negative, as was previously found for the spatially flat model. Interestingly, for the spatially closed model there exist cases where weak singularities may be resolved when they occur in the past evolution. The spatially closed model exhibits another novel feature. For a particular class of equation of state, this model also exhibits an additional physical branch in loop quantum cosmology, a baby universe separated from the parent branch. Our analysis generalizes previous results obtained on the resolution of strong curvature singularities in flat models to isotropic spacetimes with nonzero spatial curvature.

Figure 1

Type I future singularity: Hubble rates for $k=0$, $\pm 1$ models are shown. Dashed curves correspond to the classical theory and the solid ones to LQC. (Curves for different values of the curvature index coincide with each other for the resolution in the plot). In the classical theory, the Hubble rate diverges in a finite time with $a\to \infty$. We see that the effective LQC curves behave similarly to the classical ones for small values of the scale factor $a$ when the energy density is very small compared to the Planck scale. As the energy density increases, departures from classical trajectories become significant and we see a quantum recollapse. In this figure the parameters are $a_o=1000$, $A=0.1$, $B=1$ and $\alpha =0.8$.

Figure 2

Zoom of the Hubble rate around $a_o=1000$. The curves are, respectively, thin (blue) for $k=-1$ on the left, black for $k=0$, and thick (red) for $k=+1$ on the right. The parameters are the same as in Fig. 1. This zoom reveals the differences for evolution in Hubble rates in the region around $a_o$, which corresponds in this case to the “initial point” of cosmic evolution.

Figure 3

Type II future singularity: Hubble rate and Ricci scalar for $k=0$ in black, $k=+1$ in thick (red), and $k=-1$ in thin (blue) curves are shown. Classical curves are dashed and LQC curves are solid. The Hubble rate goes to zero both in the classical theory and LQC for all values of the curvature index at $a=a_o=100$. The Ricci scalar diverges at $a_o$ in the classical theory and LQC. (Since Ricci scalar is independent of the value of curvature index in the classical model, there is only curve for the classical theory). For $k=+1$ there exist two disjoint solutions of the Friedmann equation, and the further branch appearing there is bounded both in the Hubble rate and in the spacetime curvature. The parameters are $A=-0.1$, $B=10$ and $\alpha =1/4$.

Figure 4

Type II past singularity: Hubble rate and Ricci scalar for $k=0$ and $k=\pm 1$ are shown. Classical curves are dashed and LQC curves are solid. $k=0$ is in black, $k=+1$ is the thick (red) curve, and $k=-1$ in thin (blue) curve. The singularity is at $a_o=2$, where the Ricci scalar diverges. Note that for $k=+1$ the universe begins at $a>a_o$, and the Ricci scalar is always finite. In this figure the parameters are $A=0.1$, $B=-10$ and $\alpha =1/4$.

Figure 5

Type III past singularity: Classical (dashed) and effective LQC (solid) Hubble rate and Ricci scalar are shown. The curves are, respectively, thick (red) for $k=+1$ starting on the left, black for $k=0$ in the middle, and thin (blue) for $k=-1$ stating a bit more on the right. In the classical case there is a divergence of both $H$ and $R$ at $a_o=1$ while the curves for LQC remain bounded (in the figure the Ricci scalar for $k=-1$ is covered by the black line of the $k=0$ case). Here the parameters used are $A=-100$, $B=-1$ and $\alpha =2$.

Figure 6

This figure shows the additional branch occurring at small scale factors in the $k=-1$ model for equation of state leading to a type III past singularity in Fig. 5. The additional branch occurs for certain values of parameters both in the classical theory (dashed curve) and LQC (solid curve) when the scale factor is below the value $a_o$, where the big freeze occurs classically. Note that, also in this case, only LQC is immune from primordial singularity. The parameters used are $A=-100$, $B=-1$ and $\alpha =2$.

Figure 7

Type III future singularity: Classical (dashed) and effective LQC (solid) Hubble rate and Ricci scalar with $k=0$, $\pm 1$ are compared. In the classical case there is a divergence of both $H$ and $R$ at $a_o$ while the curves for LQC remain bounded. Notice that LQC cures also the initial singularity in the curve case, and, in particular, the thin (blue) solid curves ($k=-1$) shows a characteristic “tilt” for small $a$. The parameters are $a_o=1000$, $A=100$, $B=1$ and $\alpha =2$.

Figure 8

Type IV past singularity: Hubble rate and Ricci scalar for $k=0$, $\pm 1$. The singularity is at $a_o=2$, where the Ricci scalar diverges for $k=0$ and $k=-1$. Note that for $k=+1$ (dotted line for the classical solutions and thick red line for LQC effective solutions) the universe begins at $a>a_o$, thus the Ricci scalar is always finite. The parameters are $A=0.01$, $B=1$ and $\alpha =1/4$.

Figure 9

Type IV future singularity: Comparison of Hubble rates and $\dot{R}$ for the classical theory and LQC (in the figure LQC curves overlap). The singularity appears in the future at $a_o=1000$. The LQC universes (solid lines) do not have primordial singularity. Interestingly, for $k=1$ this happens also classically (dotted line). In this picture there are no baby universes, but they can be obtained for $k=1$ for a different choice of the parameters. Here the parameters are $A=-0.1$, $B=-1$ and $\alpha =1/4$.