Effective relational dynamics of a nonintegrable cosmological model

We apply the effective approach to evaluating semiclassical relational dynamics to the closed Friedman-Robertson-Walker cosmological model filled with a minimally coupled massive scalar field. This model is interesting for studying relational dynamics in a more general setting because (i) it features a nontrivial coupling of the relational clock to the evolving degrees of freedom, (ii) no temporally global clock variable exists, and (iii) it is nonintegrable, which is typical for generic dynamical systems. The effective approach is especially well geared for addressing the concept of relational evolution in this context, since it enables one to switch between different clocks and yields a consistent (temporally) local time evolution with transient observables so long as semiclassicality holds. We provide evidence that relational evolution in this model universe, while possible for sufficiently semiclassical states, generically breaks down in the region of maximal expansion. This is rooted in a defocusing of classical trajectories, which leads to a rapid spreading of states that are initially sharply peaked and to a mixing of internal time directions in this region. These results are qualitatively compared to previous work on this model, revisiting conceptual issues that have been raised earlier in the literature.

Figure 1

Two typical classical solutions to the closed FRW spacetime—both $\varphi$ and $a$ generically fail to be globally valid internal clock functions in this model. Here we used $\alpha =\ln (a)$ as appropriate for the canonical discussion following Eqs. (25, 26). Insets (a) and (c) show extended segments of (both the expanding and recontracting branches of) relational evolution up to the point of maximal expansion, ${\alpha }_{\max }=\ln (a_{\max })$. The (new) scale factor $\alpha$ oscillates between points of regular (nonglobal) maxima ${\alpha }_{\max ,k}=\ln (a_{\max ,k})$ and (nonglobal) minima ${\alpha }_{\min ,k}=\ln (a_{\min ,k})$; Inset (b) shows a close-up of the same configuration space trajectory as (a) near ${\alpha }_{\max }$, displaying the nonglobal extrema in a greater detail, while Inset (d) depicts a close-up on an intermediate section of the trajectory in (c).

Figure 2

Evolution of the canonical variables governed by Eq. (26) for a rather benign classical solution. Notice how $\alpha$ features quasi–turning points close to the turning points of $\varphi$ (also manifested in $p_{\alpha }$ having a local minimum close to the zeros of $p_{\varphi }$).

Figure 3

Defocusing of nearby trajectories and caustics developed along the extrema of $\varphi$ (see also Ref. 24).

Figure 4

(a) Classical trajectory (dotted) and patched up effective trajectory: $\alpha$ gauge (solid line), $\varphi$ gauge (dashed line). (b) Moments in $\alpha$ gauge on the incoming branch: $(\Delta \varphi {)}^2$ (thick dashed line), $(\Delta p_{\varphi }{)}^2$ (thin dashed line), $\Delta (\varphi p_{\varphi })$ (solid line). ${\alpha }_{Q_1}$ is the quasi–turning point of $\alpha$ on the incoming branch, where the clock becomes slow [see discussion and Fig. 6a].

Figure 5

(a) Moments in $\varphi$ gauge: $(\Delta \alpha {)}^2$ (thick dashed line), $(\Delta p_{\alpha }{)}^2$ (thin dashed line), $\Delta (\alpha p_{\alpha })$ (solid line). (b) Moments in $\alpha$ gauge on the outgoing branch: $(\Delta \varphi {)}^2$ (thick dashed line), $(\Delta p_{\varphi }{)}^2$ (thin dashed line), $\Delta (\varphi p_{\varphi })$ (solid line). ${\alpha }_{Q_2}$ is the quasi–turning point of $\alpha$ on the outgoing branch, where the clock becomes slow [see discussion and Fig. 6b].

Figure 6

(a) Classical momenta on the incoming branch with quasi–turning point ${\alpha }_{Q_1}$ of the clock $\alpha$: $p_{\varphi }$ (dashed line), $p_{\alpha }$ (solid line). (b) Classical momenta on the outgoing branch with quasi–turning point ${\alpha }_{Q_2}$: $p_{\varphi }$ (dashed line), $p_{\alpha }$ (solid line).

Figure 7

Top: a classical configuration space trajectory computed using the same model parameters as in Fig. 4a, but with different initial conditions: incoming branch (solid line), outgoing branch (dashed line). Bottom: a closeup of the same trajectory near $\alpha ={\alpha }_{\max }$; there are two other local extrema in $\alpha$ labeled by ${\alpha }^{\prime }$ (a maximum) and ${\alpha }^{\prime \prime }$ (a minimum); in addition, there $\varphi$ reaches a locally minimal value ${\varphi }_1$ very near $\alpha ={\alpha }^{\prime \prime }$.

Figure 8

Moments in $\alpha$ gauge on the incoming branch evolved effectively in a state initially peaked around the trajectory in Fig. 7: $(\Delta \varphi {)}^2$ (thick dashed line), $(\Delta p_{\varphi }{)}^2$ (thin dashed line), $\Delta (\varphi p_{\varphi })$ (solid) line.