Positive cosmological constant in loop quantum cosmology

The $k=0$ Friedmann-Lemaitre-Robertson-Walker model with a positive cosmological constant and a massless scalar field is analyzed in detail. If one uses the scalar field as relational time, new features arise already in the Hamiltonian framework of classical general relativity: In a finite interval of relational time, the universe expands out to infinite proper time and zero matter density. In the deparametrized quantum theory, the true Hamiltonian now fails to be essentially self-adjoint both in the Wheeler-DeWitt approach and in loop quantum cosmology. Irrespective of the choice of the self-adjoint extension, the big-bang singularity persists in the Wheeler-DeWitt theory while it is resolved and replaced by a big bounce in loop quantum cosmology. Furthermore, the quantum evolution is surprisingly insensitive to the choice of the self-adjoint extension. This may be a special case of a yet-to-be discovered general property of a certain class of symmetric operators that fail to be essentially self-adjoint.

Figure 1

Qualitative behavior of the quantum evolution can be inferred from the classical dynamics coupled with the form of eigenfunctions. Expanding classical trajectories start at ($x=\infty$, $\varphi =-\infty$) (the big bang) and end at ($x=\pi /2$, $\varphi ={\varphi }_o$) (where the matter density $\rho$ goes to zero), while the contracting trajectories start at ($x=\pi /2$, $\varphi ={\varphi }_o$) and at ($x=\infty$, $\varphi =\infty$) (the big crunch). For definiteness we have set ${\varphi }_o=0$ in the figure. The forms of ${\underline{\Theta }}_{\Lambda ,\beta }$ and its eigenfunctions in the $x$ representation suggest that incoming semiclassical wave functions would be largely reflected at $x=\pi /2$ and follow the contracting classical trajectory which is the analytical continuation of the original one. The figure shows only the part $x>0$ because the physical states $\underline{\Psi }(x,\varphi )$ are symmetric under reflection in $x$.

Figure 2

Quantum dynamics in the WDW theory. Evolution of the wave packet initially peaked at $p_{\varphi }^{\star }=5\times {10}^3$ and $|\theta {|}_{\varphi =0}^{\star }=\pi /2$ with dispersion $\Delta p_{\varphi }/p_{\varphi }=0.03$. Figures 2a, 2b show the evolution in the $v\mathrm{\text{−}}\varphi$ and $b\mathrm{\text{−}}\varphi$ plane, respectively. Figures2c, 2d show the expectation values of the observables $|\widehat{\theta }{|}_{\varphi }$, and ${\widehat{\rho }}_{\mathrm{tot}}{|}_{\varphi }$ in this state and compare them with the analytically extended classical trajectories. In these simulations, $\beta =0$, the constant $K$ in the definition of $\theta$ was set to $K=5\times {10}^3$, and $\Lambda =0.2{\ell }_{\mathrm{Pl}}^{-2}$.

Figure 3

(a) The dependence of the functions $x_o$ and $x_1≔x_M-x_o$ [of Eq. (4.12)] on the cosmological constant $\Lambda$. (b) The dependence of $y_o=\pi x_o/x_M$ on $\Lambda$. Here ${\Lambda }_c=3/{\gamma }^2{\lambda }^2$ is the critical value of $\Lambda$ at which the energy density in the cosmological constant equals the maximum energy density ${\rho }_{\max }$ allowed by LQC in this model.

Figure 4

(a) The lowest three eigenvalue ${\omega }_n$ are plotted as functions of the self-adjoint extension parameter $\beta$ and compared with their approximations ${\omega }_n^{\prime }$ obtained by neglecting the remnant $O(e^{-2\pi n(\pi -y_o)/y_o})$ in (4.27), for $\Lambda =0.01{\Lambda }_c$. (b) The frequency gap $\Delta \omega$ is plotted as a function of cosmological constant.

Figure 5

LQC dynamics. Figures 5a, 5b show the evolution of the (absolute value of the) physical wave function in the $v\mathrm{\text{−}}\varphi$ and $b\mathrm{\text{−}}\varphi$ planes, respectively. Figures 5c, 5d compare the expectation values of the Dirac observables $|\widehat{\theta }{|}_{\varphi }$ and ${\widehat{\rho }}_{\mathrm{tot}}{|}_{\varphi }$ with those in the effective and classical theories. Away from the Planck regime there is excellent agreement with (the analytical extensions of) solutions in general relativity. But in the Planck regime [left ends of Figs. 5d, 5d] there is a very large departure from the classical behavior because of quantum geometry effects. However, the effective trajectories capture the quantum evolution very well even in the Planck regime. In these simulations, the initial state was a Gaussian peaked at $p_{\varphi }^{\star }=5\times {10}^3$ and $|\widehat{\theta }{|}_{\varphi =0}^{\star }=\arctan (10)$ with the relative spread $\Delta p_{\varphi }/p_{\varphi }=0.03$. The self-adjoint extension corresponded to $\beta =0$, the constant $K$ in the definition of $\theta$ was set to $K=5\times {10}^3$ and the cosmological constant was $\Lambda =0.01{\Lambda }_c\approx 0.2{\ell }_{\mathrm{Pl}}^{-2}$.

Figure 6

The period $\Delta \varphi$ of the evolution predicted by the effective theory via (b4) is compared against the approximate period of the genuine quantum evolution (4.35).