Anti-de Sitter universe dynamics in loop quantum cosmology

A model for a flat isotropic universe with a negative cosmological constant $\Lambda$ and a massless scalar field as sole matter content is studied within the framework of loop quantum cosmology. By application of the methods introduced for the model with $\Lambda =0$, the physical Hilbert space and the set of Dirac observables are constructed. As in that case, the scalar field plays here the role of an emergent time. The properties of the system are found to be similar to those of the $k=1$ Friedmann-Robertson-Walker (FRW) model: for small energy densities, the quantum dynamics reproduces the classical one, whereas, due to modifications at near-Planckian densities, the big bang and big crunch singularities are replaced by a quantum bounce connecting deterministically the large semiclassical epochs. Thus in loop quantum cosmology the evolution is qualitatively cyclic.

Figure 1

An example of a Wheeler-DeWitt Gaussian wave packet (3.14) generated for the parameter values $\Lambda =-0.01$, $p_{\varphi }^{\star }=5\times {10}^3$, $\Delta p_{\varphi }/p_{\varphi }^{\star }=0.02$, and ${\varphi }^{\star }=0$. Figure (a) shows the absolute value of the wave function. For the presentation clarity, only the points of $|\Psi (v,\varphi )|>{10}^{-6}$ were plotted. Figure (b) presents the expectation values and dispersions of $|\widehat{v}{|}_{\varphi }$ (red bars) compared against the classical trajectory $v(\varphi )$ (blue line). As we can see, the quantum trajectory agrees with the classical one (the difference being much smaller that the spread). Because of the large changes in magnitude of $v$ during the evolution, the trajectory was plotted in logarithmic scale.

Figure 2

Examples of eigenfunctions of $\Theta$ supported on the lattices ${\mathcal{L}}_{\epsilon }$ for $\epsilon =0$ (a) and $\epsilon =1$ (b). (a) shows a normalizable eigenfunction of $\omega \approx 300.45$ (red solid line) and two divergent ones of $\omega$ respectively smaller (green dashed line) and larger (blue dotted line) by 0.1. For clarity, only the positive $v$ part is shown. (b) presents the absolute value of a normalizable eigenfunction of $\omega \approx 52.85$ (red solid line) along with two divergent examples: generic (green dashed line) and left-converging (blue dotted line) generated for, respectively, $\omega \approx 53.35$ and 54.35. To show the behavior in a wide range of values, a logarithmic scale was used for the $y$-axis. Both figures correspond to $\Lambda =-0.01$.

Figure 3

The eigenfunctions $e_0$ to $e_4$ of $\Theta$, corresponding to $\Lambda =-0.01$ and supported on ${\mathcal{L}}_{\epsilon }$ with $\epsilon =0$ (a) and $\epsilon =1$ (b). For clarity, only the $v>0$ part was shown in (a) and only the part supported on ${\mathcal{L}}_{+|\epsilon |}$ was shown in (b).

Figure 4

(a) The lowest ($\omega <44$) elements of $\Theta$’s spectrum are shown as functions of $\pm |\epsilon |$. The eigenvalues are divided into three groups corresponding to the following eigenfuctions: (1) left-suppressed (red crosses), for which $>50\%$ of the norm is located on $v>0$, (2) right-suppressed (green x-es) defined analogously, and (3) singularity-peaked (blue stars), where $>50\%$ of the norm is located at the three points closest to $v=0$. (b) The large $\omega$ limit of the eigenvalue separation $\Delta \omega$, shown as a function of $\Lambda$ (red crosses). The blue line represents the small $\Lambda$ approximation given by (5.5).

Figure 5

The rescaled eigenvalue separation correction term $A(\omega )≔|\Delta {\omega }_n-\Delta \omega |({\omega }_n/\Delta \omega {)}^2$ is shown in (a) for several values of $\Lambda$. Its $\omega \to \infty$ limit is plotted in (b) as a function of $\Lambda$. The “wiggles” at small values of $|\Lambda |$ are the results of numerical errors due to the precision limitation of the applied calculation method.

Figure 6

Convergence test for the integration method of a Gaussian wave packet generated with $\Lambda =-0.1$ and peaked at $p_{\varphi }^{\star }={10}^3$, with relative $p_{\varphi }$ spread 0.05 and $v^{\star }=0.5v_R(p_{\varphi }^{\star })$. The initial data were specified at $\varphi =0$ and evolved till $\varphi =1$. The upper (red) curve shows the norm of the difference $\parallel {\Psi }_{(N)}-\Psi \parallel$ between the slice $\varphi =1$ of the solution ${\Psi }_{(N)}$ corresponding to the integration with $N$ steps and the same slice of its $N\to \infty$ limit $\Psi$ (found via 8th order polynomial extrapolation). The lower (green) curve shows the analogous difference taken with respect to the metric (5.9).

Figure 7

The absolute value of the wave function representing a Gaussian state (4.1) generated via backward integration of an initial profile corresponding to $\Lambda =-1$, $p_{\varphi }^{\star }=5\times {10}^3\hslash$, $\Delta p_{\varphi }/p_{\varphi }^{\star }=0.01$, $v^{\star }=0.6v_R(p_{\varphi }^{\star })$ and evaluated at ${\varphi }_o=0$. For presentation clarity, only values $>{10}^{-6}$ were shown on the plot.

Figure 8

The expectation values (red bars) of $|\widehat{v}{|}_{\varphi }$ (a) and ${\widehat{\rho }}_{\varphi }$ (b) are compared against the classical (red lines) and effective (blue lines) trajectories of $v(\varphi )$ and ${\rho }_{\varphi }(v)$, respectively. The data corresponds to a Gaussian wave packet (4.1) with $\Lambda =-0.1$, $p_{\varphi }^{\star }={10}^4\hslash$, $\Delta p_{\varphi }/p_{\varphi }^{\star }=0.012$, $v^{\star }=0.55v_R(p_{\varphi }^{\star })$ specified at ${\varphi }_o=0$ and evolved backwards. Because of the large changes in magnitude of $⟨{\rho }_{\varphi }⟩$, a logarithmic scale was used for the $y$-axis of Fig. (b).

Figure 9

A detailed picture of the comparison of $⟨|v{|}_{\varphi }⟩$ against the $v(\varphi )$ trajectories presented in Fig. 8a is shown near the bounce (a) and recollapse (b) points, respectively.

Figure 10

A detailed picture of the comparison of $⟨{\rho }_{\varphi }⟩$ against the ${\rho }_{\varphi }(\varphi )$ trajectories presented in Fig. 8b is shown near the bounce (a) and recollapse (b) points, respectively.

Figure 11

(a) A set of lowest ($\omega <44$) spectrum elements of $\Theta$ for the symmetric and antisymmetric sector (with $\Lambda =-1$) is shown with respect to $\pm |\epsilon |$. The green x-es represent the eigenvalues corresponding to the cases where the relation between symmetric and antisymmetric eigenfunctions is given by (A1, A2) (denoted as generic). The red crosses and blue stars represent the eigenvalues of, respectively, the symmetric and antisymmetric eigenfunctions on the lattice ${\mathcal{L}}_{\epsilon =0}$. The antisymmetric eigenfunctions $e_0^a$ to $e_4^a$ corresponding to the eigenvalues shown in (a) are presented in (b). For clarity, they are plotted on the $v>0$ semiaxis only.