Tunneling dynamics of an oscillating universe model

Abstract

Quasiclassical methods for non-adiabatic quantum dynamics can reveal new features of quantum effects, such as tunneling evolution, that are harder to analyze in standard treatments based on wave functions of stationary states. Here, these methods are applied to an oscillating universe model introduced recently. Our quasiclassical treatment correctly describes several expected features of tunneling states, in particular just before and after tunneling into a trapped region where a model universe may oscillate through many cycles of collapse and expansion. As a new result, the oscillating dynamics is found to be much less regular than in the classical description, revealing a succession of cycles with varying maximal volume even when the matter ingredients and their parameters do not change.

Chaos in a tunneling universe

“…The results presented here show that even the isotropic dynamics is chaotic if it is described quasiclassically by including quantum fluctuation terms, applied in the specific analysis to tunneling-type potentials as in oscillating models. We will use the same model and quasiclassical extensions as derived in [7], reviewed in the next section, and show proofs of chaos based on a numerical analysis of the fractal domension in a space of initial values. In our conclusions we will demonstrate which features of the specific potential are likely to be responsible for chaos.…”

“…Our results do not depend much on the specific features of the contributions from σ and p ϕ , other than the trapped potential region they form together with the curvature term. For the latter, we choose k = 1, but smaller values are also possible; see [7,14] for more details.…”

“…As in [7], following [29,30], we bring the fourth-order term closer to Gaussian form, where ∆(α 4 ) = 3s 4 rather than s 4 , by adding the final term in Our analysis of chaos will use mainly the small-s dynamics in the trapped region before much tunneling happens, in which case the quasiclassical approximation is expected to be reliable.…”

“…Since the wall on the left has a height lower than the classical barrier for sufficiently large s, tunneling is possible when a quasiclassical trajectory crosses over the wall into the unbounded region to the left, where it will then continue almost freely except for one possible reflection at the U/s 2 potential. A detailed analysis of trajectories, given in [7], showed that the potential (2.17) does not reliably describe tunneling because the uniform nature of the channel, seen in figure 2, makes it much more likely for trajectories to follow the channel, rather than crossing the wall to the left. The fourth-order modification in (2.18), motivated by a more Gaussian behavior of states, was found to improve the tunneling description by quasiclassical trajectories.…”

“…Early models [1] have recently been extended to oscillating versions [2,3] in which the evolving scale factor, classically trapped in a finite region, may escape by quantum tunneling and approach a singularity [4][5][6]. An application [7] of quasiclassical methods revealed non-trivial features of the tunneling dynamics that is not captured by the traditional derivation of tunneling coefficients from stationary states. In particular, some of the dynamics was found to depend sensitively on the choice of initial values in the trapped region, suggesting chaotic behavior.…”