Towards cosmological dynamics from loop quantum gravity

We present a systematic study of the cosmological dynamics resulting from an effective Hamiltonian, recently derived in loop quantum gravity using Thiemann’s regularization and earlier obtained in loop quantum cosmology (LQC) by keeping the Lorentzian term explicit in the Hamiltonian constraint. We show that quantum geometric effects result in higher than quadratic corrections in energy density in comparison to LQC, causing a nonsingular bounce. Dynamics can be described by the Hamilton or Friedmann-Raychaudhuri equations, but the map between the two descriptions is not one to one. A careful analysis resolves the tension on symmetric versus asymmetric bounce in this model, showing that the bounce must be asymmetric and symmetric bounce is physically inconsistent, in contrast to the standard LQC. In addition, the current observations only allow a scenario where the prebounce branch is asymptotically de Sitter, similar to a quantization of the Schwarzschild interior in LQC, and the postbounce branch yields the classical general relativity. For a quadratic potential, we find that a slow-roll inflation generically happens after the bounce, which is quite similar to what happens in LQC.

Figure 1

In this figure, we plot Eq. (2.13). The top dashed line represents the $b_{+}$ branch while the bottom solid curve represents the $b_{-}$ branch. The two branches converge at the critical density $\rho ={\rho }_c$. For any choice of the initial density $\rho ={\rho }_0<{\rho }_c$, which is represented by the solid vertical line in the graph, there are two solutions of ${\sin }^2(\lambda b)$ marked by $b_{+}$ and $b_{-}$, respectively. In the actual simulations, ${\rho }_0$ is chosen in the range of ${10}^{-4}$–${10}^{-6}$ in terms of the Planck units.

Figure 2

In this figure, the initial time of our numerical simulations is chosen at $t_0=11.5$ and the final time is set to $t_f=-10$. Hence, the bounce occurs at $t_B=-0.004$, and at $t_0=11.5$, we have $v_0\approx 70.71$ and $b_0^{-}\approx 0.00688$.

Figure 3

In this figure, the initial time of our numerical simulations is chosen at $t_0=-300$ with ${\rho }_0={10}^{-6}$ and ${\pi }_{\varphi }^0=1$. Hence, the bounce occurs at $t_B=-185$; the de Sitter space is now located in the postbounce phase; the evolution of the Universe in this figure is represented by the segment $O\to E\to F$ in Fig. 6.

Figure 4

In this figure, we compare the results from three different models. The black solid straight line is the result from the classical theory of GR, the red dotted line is from the full LQG cosmology and the blue dot-dashed curve is from LQC. The initial conditions are chosen the same as in Fig. 2.

Figure 5

With the same initial conditions as adopted in Fig. 4, we show the behavior of the Ricci scalar and inverse Hubble rate near the bounce in Planck units. The LQG cosmology results are shown by the red dotted curve, and the LQC results are depicted by the blue dot-dashed curve.

Figure 6

In this figure, we plot a complete cycle of $\sin (\lambda b)$ in the interval $\lambda b\in (-\pi ,\pi )$ where $\xi \equiv \rho /{\rho }_c$. The four straight lines $\xi =0$ and $\xi =1$ are where $\sin (\lambda b)=\pm \sqrt{1/({\gamma }^2+1)}$ and $\sin (\lambda b)=\pm \sqrt{1/[2({\gamma }^2+1)]}$, respectively. The black dots on the curve represent the starting points (G, O, B, J) or end points (C, O, F, I) of the evolution, which takes place in the direction indicated by the arrows. The segments without the arrows ($C\to B$ and $F\to G$) are the forbidden regions where the energy density exceeds the critical density ${\rho }_c$. In our numerical simulations, we focus on the line $B\to A\to O$.

Figure 7

In this figure, we compare the numerical simulations of Hamilton’s equations (represented by the dotted line) and the combination of the two branches $b_{-}$ and $b_{+}$ (represented by the dot-dashed line) in the full LQG cosmology, as explained in detail in Sec. 4. Our initial data are chosen at $t_0=115$ with ${\rho }_0={10}^{-6}$, ${\pi }_{\varphi }^0=1$. The bounce occurs at $t_B=-0.158$.

Figure 8

In this figure, we show the numerical simulations with a massive scalar field and the mass is set to $m=1.3\times {10}^{-6}$. Our initial conditions are chosen at $t_0=36.5$ with ${\rho }_0={10}^{-5}$, ${\pi }_{\varphi }^0=1000$ and $v_0=223607$ so that ${\varphi }_0=4.63$. The red dotted lines represent the results from LQG cosmology, and the blue dot-dashed lines are for LQC.

Figure 9

With the same initial conditions as adopted in Fig. 8, we show the behavior of the potential, equation of state and slow-roll parameter near the bounce. The LQG results are shown by the red dotted curve, and the LQC results are depicted by the blue dot-dashed curve.

Figure 10

The plots with the same initial conditions as those in Fig. 9 are shown for late times in the postbounce branch during the inflationary phase and its end. The slow-roll parameter increases to unity at $2.6\times {10}^7{t}_{\mathrm{Pl}}$.

Figure 11

With the same initial conditions as adopted in Fig. 8, we show the behavior of the energy density, Ricci scalar and inverse Hubble rate near the bounce. The LQG results are shown by the red dotted curve, and the LQC results are depicted by the blue dot-dashed curve. In the third subfigure, we also plot the Ricci scalar in a longer range after the bounce to show that there exists a negative curvature regime in the classical limit.