Implications of quantum ambiguities in $k=1$ loop quantum cosmology: Distinct quantum turnarounds and the super-Planckian regime

The spatially closed Friedmann-Lemaître-Robertson-Walker model in loop quantum cosmology admits two inequivalent consistent quantizations: one based on expressing the field strength in terms of the holonomies over closed loops and another using a connection operator and open holonomies. Using the effective dynamics, we investigate the phenomenological differences between the two quantizations for the single-fluid and the two-fluid scenarios with various equations of state, including the phantom matter. We show that a striking difference between the two quantizations is the existence of two distinct quantum turnarounds, either bounces or recollapses, in the connection quantization, in contrast to a single distinct quantum bounce or a recollapse in the holonomy quantization. These results generalize an earlier result on the existence of two distinct quantum bounces for stiff matter by Corichi and Karami. However, we find that in certain situations two distinct quantum turnarounds can become virtually indistinguishable. And depending on the initial conditions, a pure quantum cyclic universe can also exist undergoing a quantum bounce and a quantum recollapse. We show that for various equations of states, connection-based quantization leads to super-Planckian values of the energy density and the expansion scalar at quantum turnarounds. Interestingly, we find that very extreme energy densities can also occur for the holonomy quantization, breaching the maximum allowed density in the spatially flat loop quantized model. However, the expansion scalar in all these cases is bounded by a universal value.

Figure 1

A cartoon depiction of the distinct quantum turnarounds for the connection quantization is shown. In the left figure, the evolution is sourced by a matter content obeying the strong energy condition ($w>-1/3$). Such an evolution results in two distinct quantum bounces and a classical recollapse in each cycle. In the right figure, evolution is sourced by a phantom fluid with $w\ll -1$. In this case, quantum turnarounds are no longer bounces but recollapses. The classical turnaround is a bounce in the phantom case. Distinct quantum recollapses become indistinguishable for equations of state closer to negative unity. In the holonomy quantization, there is only a single quantum bounce and a quantum recollapse in the left and the right figures respectively.

Figure 2

Relative error in the effective Hamiltonian is plotted for representative holonomy and connection solutions. The solid curve corresponds to the holonomy solution, whereas the dashed curve represents connection solution. The initial conditions (in the Planck units) for this simulation are $V_0=1000$, ${\rho }_0=0.2$, with ${\beta }_0$ set from the Hamiltonian constraint. The equation of state is $w=1$.

Figure 3

Variation of volume vs time for a massless scalar field ($w=1$) is shown. The connection quantization is represented by the dashed line (bottom panel), while the holonomy quantization is represented by the solid line (top panel). The initial conditions for this particular solution were as follows: $V_0=10^6,{\rho }_0={10}^{-4}$ (in Planck units), with ${\beta }_0$ set by the Hamiltonian constraint. The energy density at the bounce in the holonomy case is $\rho =0.411{\rho }_{\mathrm{Pl}}$. In the connection case, distinct quantum bounces occur at $\rho =0.658{\rho }_{\mathrm{Pl}}$ and $\rho =0.247{\rho }_{\mathrm{Pl}}$.

Figure 4

This plot shows the comparison of GR evolution with the holonomy case in LQC for the early period corresponding to Fig. 3. For clarity, we only show the holonomy case. The connection case yields a very similar picture of the contrast between GR and the LQC. The solid curve corresponds to the LQC evolution, and the dashed curve depicts the evolution in GR. After agreement initially, two curves show departures. The GR curve ends in a big crunch singularity, whereas the LQC curve undergoes cyclic evolution. Bounces in the LQC evolution occur at $\rho =0.411{\rho }_{\mathrm{Pl}}$.

Figure 5

Behavior of volume of the universe vs time is shown for a radiation-dominated ($w=1/3$) universe. Initial conditions for volume and energy density and designations for these solutions are the same as in Fig. 3. The energy density at the bounce in the holonomy case is $\rho =0.423{\rho }_{\mathrm{Pl}}$. In the connection quantization, the distinct bounces occur at $\rho =1.095{\rho }_{\mathrm{Pl}}$ and at $\rho =0.157{\rho }_{\mathrm{Pl}}$.

Figure 6

$V$ vs $t$ for a matter-dominated ($w=0$) universe is demonstrated. The choice of $V_0$ and ${\rho }_0$ and the labeling of solutions are the same as in Fig 3. The bounce in the holonomy case occurs at $\rho =0.629{\rho }_{\mathrm{Pl}}$, and the distinct bounces in the connection case occur at $\rho =4.073{\rho }_{\mathrm{Pl}}$ and at $\rho =0.0817{\rho }_{\mathrm{Pl}}$.

Figure 7

The volume of the universe is plotted vs time for a universe with equation of state $w=-0.3$. Initial conditions and designations for these solutions are the same as in Fig. 3. The energy density at the bounce in the holonomy case is $\rho =1.58\times {10}^{-3}{\rho }_{\mathrm{Pl}}$. In the connection case, the distinct bounces occur at $\rho =0.0525{\rho }_{\mathrm{Pl}}$ and $\rho =0.0253{\rho }_{\mathrm{Pl}}$. In contrast to simulations in previous figures, the energy density at bounce is smaller because for $w=-0.3$, $\rho \propto a^{-2.1}$ which is only slightly stronger than the way classical spatial curvature behaves ($a^{-2}$). The quantum turnarounds hence get significant help from the spatial curvature term, reducing the role of matter density in comparison to previous cases in causing the bounce.

Figure 8

$V$ vs $t$ for a universe in a two-fluid scenario is shown. The equations of state are $w_1=1/3$ and $w_2=-0.2$.

Figure 9

Variation of energy density $\rho$ and volume $V$ with respect to time is shown for a universe with phantom energy of $w=-1.2$ in the connection quantization approach (dashed curve) and the holonomy quantization approach (solid curve). The initial conditions for this universe are $V_0=10^6$ and ${\rho }_0={10}^{-4}$. The universe starts evolution from a classical regime and undergoes cyclic evolution with only a distinct quantum bounce in both the approaches.

Figure 10

Energy density and volume are plotted against time for connection quantization for the case of a universe with a strong phantom equation of state ($w=-10$). The universe starts from the initial conditions of $V_0=10^6$ and ${\rho }_0={10}^{-4}$. Due to the strong phantom dynamics and the quantum geometric effects, avoidance of big rip appears at smaller volumes in comparison to Fig. 9 and therefore are distinct quantum recollapses.

Figure 11

The variation of $\rho$ and $V$ in time is shown for a universe with initial conditions $V_0=1000$ and ${\rho }_0=0.2$ and with a phantom equation of state, $w=-1.1$. The black horizontal line represents ${\rho }_{\max }^{\text{flat}}$, and holonomy and connection designations are the same as in the earlier figures. Though the evolution in connection and holonomy quantization appears very similar, there are important differences in the nature of turnarounds as noted in the text.

Figure 12

For a two-fluid scenario with $w_1=1.0$ and $w_2=-1.25$, the behavior of energy density and volume are shown. Initial conditions are $V_0=10^6$, ${\rho }_0=0.0016$, such that ${\rho }_{0,\mathrm{tot}}=0.0032$. The horizontal black line represents ${\rho }_{\max }$. The solid curves correspond to holonomy quantization and dashed curves correspond to connection quantization.

Figure 13

Evolution of volume and energy density vs time for the case of $w=-1/3$ is shown for the holonomy (solid curve) and connection (dashed curve) quantization. Initial conditions in Planck units for this run were as follows: $V_0=1000$ and ${\rho }_0=0.16335$. The volume at the bounce in the case of holonomy quantization is $V_b\approx 64V_{\mathrm{Pl}}$.

Figure 14

$V$ vs $t$ for the same initial conditions as Fig. 13, for two fluids corresponding to $w_1=-1/3$ and $w_2=-1.25$.

Figure 15

Plot of energy density and expansion scalar vs time for a massless scalar field ($w=1$) with initial conditions ${\rho }_0=0.5>{\rho }_{\max }$ and $V_0=100$. The black horizontal line represents ${\rho }_{\max }^{\text{flat}}$ in the left figure and the maximum allowed value of expansion scalar in the holonomy quantization in the right figure. The holonomy solution is represented by the solid line, while the connection solution is represented by the dashed lines.