Nonsingular ekpyrotic/cyclic model in loop quantum cosmology

We study the role of nonperturbative quantum gravity effects in the ekpyrotic/cyclic model using the effective framework of loop quantum cosmology in the presence of anisotropies. We show that quantum geometric modifications to the dynamical equations near the Planck scale as understood in the quantization of Bianchi-I spacetime in loop quantum cosmology lead to the resolution of classical singularity and result in a nonsingular transition of the Universe from the contracting to the expanding branch. In the Planck regime, the Universe undergoes multiple small bounces and the anisotropic shear remains bounded throughout the evolution. A novel feature, which is absent for isotropic models, is a natural turn-around of the moduli field from the negative region of the potential leading to a cyclic phenomena as envisioned in the original paradigm. Our work suggests that incorporation of quantum gravitational effects in the ekpyrotic/cyclic model may lead to a viable scenario without any violation of the null energy condition.

Figure 1

The plot of the mean scale factor (in Planck units), $a=(a_1{a}_2{a}_3{)}^{1/3}$ is shown for the initial conditions (all values in Planck units) $\varphi =0.43$, $\dot{\varphi }=-0.038$, $p_1=64$, $p_2=72$, $p_3=68$, $c_1=-0.8$, $c_2=-0.7$, and ${\Sigma }^2=5.80407$. The mean scale factor experiences multiple small bounces as a result of bounces of the individual scale factors ($a_i$). Unlike the classical theory, there is a nonsingular transition from the contracting to the expanding branch.

Figure 2

The evolution of the moduli field in Planck units is shown for the initial conditions in Fig. 1. The moduli field starts from a positive region of the potential, rolls to the negative part, and turns around in the Planck regime. After the turn-around the moduli again reaches the positive part of the potential, ready for another cycle.

Figure 3

The plot shows the variation of the shear ${\Sigma }^2$ in the transition region from the contracting to the expanding branch for the initial conditions in Fig. 1. Considerable variation for a short period before and after the bounce is evident from the spikes in value of $\log ({\Sigma }^2)$. We also see that in the classical regimes at both small and large values of $t$, the shear approaches similar values.

Figure 4

The variation in the directional Hubble rate $H_1$ is plotted for the initial data in Fig. 1. The spikes reflect the bounces and recollapses of individual scale factors in the Planck regime during the transition from the contracting to the expanding branch.

Figure 5

The plot shows the evolution of the scale factor in the presence (solid curve) and absence (dashed curve) of anisotropy for initial conditions $\varphi =0.4$, $\dot{\varphi }=-0.03$, $p_1=64$, $p_2=72$, $p_3=68$, $c_1=-0.6$, $c_2=-0.5$, and ${\Sigma }^2=9.2365$. The classical singularity is avoided in both the cases.

Figure 6

The evolution of the moduli field is compared for the initial conditions in Fig. 5 for the anisotropic (blue solid curve) and isotropic (red dashed curve) model. As in Fig. 2 the moduli field turns around in the presence of anisotropies in the Planck regime and leads to a nonsingular cyclic model. It fails to turn around, and approaches $\varphi =-\infty$, in the absence of anisotropies in confirmation with results of Ref. 35.

Figure 7

The shear scalar $\Sigma$ is plotted for the numerical run in Fig. 5. We see that its value is preserved before and after the transition period. The shear term is bounded across the evolution with spikes corresponding to multiple bounces and recollapses of the individual scale factors.

Figure 8

The plot of the energy density of the moduli field in the transition regime is shown. Bounces of the individual scale factors result in spikes in its behavior. Because of the presence of nonvanishing anisotropy, the energy density at these bounces does not need to saturate to its maximum allowed value.

Figure 9

The plots of the mean scale factor $a$ and the scalar field $\varphi$ are shown for the initial conditions with very small anisotropies. These are $\varphi =0.4$, $\dot{\varphi }=-0.03$, $p_1=64$, $p_2=72$, $p_3=68$, $c_1=-0.6$, $c_2=-0.532$, and ${\Sigma }^2=0.8517$.

Figure 10

This plot shows the evolution of different scale factors for the simulation corresponding to Fig. 9. We have plotted the logarithm of directional scale factors. The darker curve depicts $a_1$ and lighter curve $a_3$. The behavior of $a_2$ (not shown in the figure) is very similar to that of $a_1$.

Figure 11

These plots depict the evolution of the scale factor and scalar field for the initial conditions: $\varphi =0.4$, $\dot{\varphi }=-0.03$, $p_1=64$, $p_2=72$, $p_3=68$, $c_1=-0.6$, $c_2=-0.1$, and ${\Sigma }^2=1449.4278$. Unlike the previous cases, we find that the asymmetry in the mean scale factor and the moduli field around the bounce point is enhanced. Note that the moduli field reaches the positive value for $t\sim 220$ before starting a new cycle.