Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Big Bounce in Wheeler-DeWitt scattering theory: Ekpyrotic and LQC-like transitions

S. Lo Franco, G. Montani

Abstract

We present a rigorous formulation of the Quantum Big Bounce for the closed isotropic Universe, filled with a self-interacting scalar field, that emerges from the interaction with an ekpyrotic potential. Working in a covariant approach to the minisuperspace, we demonstrate the quantum equivalence between parametrizations in terms of the logarithmic scale factor and the volume variable. The analogy between the Wheeler-DeWitt equation and the Klein-Gordon equation, alongside a proper definition of asymptotic states, allows the identification of two different bouncing scenarios: one in which the transition occurs over a fixed direction of the internal time arrow, corresponding to a LQC-like scenario, and one involving a reversal of the internal time flow. The high-energy divergence in the former case shows the incompleteness of the WDW theory and the need for regularization. Therefore, the WDW theory is valid up to a given energy threshold. The latter transition, corresponding to an ekpyrotic scenario, is instead well-posed at any energy scale at the first perturbative order. While the Ashtekar school Big Bounce is expected to be recovered when high-energy corrections are included in this scheme, the WDW alone can avoid the cosmological singularity in a quantum mechanical fashion.

Quantum Big Bounce in Wheeler-DeWitt scattering theory: Ekpyrotic and LQC-like transitions

Abstract

We present a rigorous formulation of the Quantum Big Bounce for the closed isotropic Universe, filled with a self-interacting scalar field, that emerges from the interaction with an ekpyrotic potential. Working in a covariant approach to the minisuperspace, we demonstrate the quantum equivalence between parametrizations in terms of the logarithmic scale factor and the volume variable. The analogy between the Wheeler-DeWitt equation and the Klein-Gordon equation, alongside a proper definition of asymptotic states, allows the identification of two different bouncing scenarios: one in which the transition occurs over a fixed direction of the internal time arrow, corresponding to a LQC-like scenario, and one involving a reversal of the internal time flow. The high-energy divergence in the former case shows the incompleteness of the WDW theory and the need for regularization. Therefore, the WDW theory is valid up to a given energy threshold. The latter transition, corresponding to an ekpyrotic scenario, is instead well-posed at any energy scale at the first perturbative order. While the Ashtekar school Big Bounce is expected to be recovered when high-energy corrections are included in this scheme, the WDW alone can avoid the cosmological singularity in a quantum mechanical fashion.
Paper Structure (9 sections, 30 equations, 3 figures)

This paper contains 9 sections, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Covariant approach to the minisuperspace
  3. Universe's semiclassical states
  4. Scattering with the Ekpyrotic potential
  5. The Big Bounce transition
  6. Conclusions
  7. The Bogolyubov transformation
  8. Feynman propagator and S-matrix
  9. Transition amplitudes for semiclassical states

Figures (3)

  • Figure 1: Internal time evolution of $\overline{v}$ from the wavelets in Eq. (\ref{['eq:lognorm-weight']}) (dotted lines) vs. the classical law in Eq. (\ref{['eq:v_phi_cl']}) (solid lines). Both positive (blue dots) and negative (red dots) frequency states are considered, with arrows representing the internal time flow.
  • Figure 2: Plot of $O(\eta^2)$ probability densities versus the outgoing mean energy $\Omega'$ for ekpyrotic (left column) and LQC-like (right column) transitions. We choose different values for $\gamma, \Omega,\Phi$ and $\Phi'$, while $\sigma$ and $\eta$ are fixed.
  • Figure 3: Semiclassical evolution of an incoming state (solid line) with possible outgoing states representing bouncing scenarios (dashed/dash-dotted lines), together with $U_s(\phi)$ (red line). We consider the evolution of free states to be recovered as $U_s(\phi) \sim \eta \times10^{-3}$.