Physical evolution in loop quantum cosmology: The example of the vacuum Bianchi I model

We use the vacuum Bianchi I model as an example to investigate the concept of physical evolution in loop quantum cosmology (LQC) in the absence of the massless scalar field which has been used so far in the literature as an internal time. In order to retrieve the system dynamics when no such a suitable clock field is present, we explore different constructions of families of unitarily related partial observables. These observables are parametrized, respectively, by: (i) one of the components of the densitized triad, and (ii) its conjugate momentum; each of them playing the role of an evolution parameter. Exploiting the properties of the considered example, we investigate in detail the domains of applicability of each construction. In both cases the observables possess a neat physical interpretation only in an approximate sense. However, whereas in case (i) such interpretation is reasonably accurate only for a portion of the evolution of the universe, in case (ii) it remains so during all the evolution (at least in the physically interesting cases). The constructed families of observables are next used to describe the evolution of the Bianchi I universe. The performed analysis confirms the robustness of the bounces, also in absence of matter fields, as well as the preservation of the semiclassicality through them. The concept of evolution studied here and the presented construction of observables are applicable to a wide class of models in LQC, including quantizations of the Bianchi I model obtained with other prescriptions for the improved dynamics.

Figure 1

An example of eigenfunction of the operator ${\widehat{\Theta }}_i$, corresponding to the eigenvalue ${\omega }_i=100$ and the superselection sector ${\epsilon }_i=2$. The blue line (located on the imaginary plane) shows the part supported on the subsemilattice ${}^{(4)}\mathcal{L}_{{\tilde{\epsilon }}_i=4}^{+}$, whereas the red line (real plane) is the part supported on ${}^{(4)}\mathcal{L}_{{\tilde{\epsilon }}_i=2}^{+}$.

Figure 2

For Gaussian states [with profile (4.14)], the expectation values and dispersions of the observables $\ln ({\widehat{v}}_a{)}_{v_1}^{+}$ and $\ln ({\widehat{v}}_a{)}_{v_1}^{-}$ [corresponding to epochs when the wave packet is moving forward (green error bars) and backward (red error bars) in time, respectively] are compared with classical (pink lines) and effective (blue dotted line) trajectories. The states are peaked at ${\omega }_2^{\star }={\omega }_3^{\star }={10}^3$ with the relative dispersions $\Delta {\Theta }_2/{\Theta }_2=\Delta {\Theta }_3/{\Theta }_3=0.05$. The phases are, respectively, ${\beta }^2={\beta }^3=0$ for (a), and ${\beta }^2={\beta }^3=0.1$ for (b). The expectation values follow the effective trajectory till the point of bounce in $v_1$, where the dynamics “freezes”. On the other hand away from the bounce the quantum trajectory approaches the classical ones. In particular, in (a) the trajectories before and after the bounce coincide on the plane $v_1\mathrm{\text{−}}{v}_2$.

Figure 3

Comparison of the expectation values of the two families of observables $\ln ({\widehat{v}}_2{)}_{v_1}^{+}$ (blue) and $\ln ({\widehat{v}}_1{)}_{v_2}^{+}$ (red), corresponding to two different choices of emergent time, calculated on the state considered in Fig. 2a.

Figure 4

Expectation values and dispersions of the observables $\ln ({\widehat{v}}_i{)}_{b_1}$ on the Gaussian states [with profile (4.14)] presented in the $v_2\mathrm{\text{−}}{v}_1$ plane and compared with classical (green lines) and effective (blue dotted line) trajectories. The states are peaked at ${\omega }_2^{\star }={\omega }_3^{\star }={10}^3$ with relative dispersions $\Delta {\Theta }_2/{\Theta }_2=\Delta {\Theta }_3/{\Theta }_3=0.05$. The phases are, respectively, ${\beta }^2={\beta }^3=0$ for (a) and ${\beta }^2={\beta }^3=0.1$ for (b). The expectation values follow the effective trajectory through all the evolution. In particular, the trajectories before and after the bounce coincide in (a).