Dynamical coherent states and physical solutions of quantum cosmological bounces

A new model is studied which describes the quantum behavior of transitions through an isotropic quantum cosmological bounce in loop quantum cosmology sourced by a free and massless scalar field. As an exactly solvable model even at the quantum level, it illustrates properties of dynamical coherent states and provides the basis for a systematic perturbation theory of loop quantum gravity. The detailed analysis is remarkably different from what is known for harmonic oscillator coherent states. Results are evaluated with regard to their implications in cosmology, including a demonstration that in general quantum fluctuations before and after the bounce are unrelated. Thus, even within this solvable model the condition of classicality at late times does not imply classicality at early times before the bounce without further assumptions. Nevertheless, the quantum state does evolve deterministically through the bounce.

Figure 1

Illustration of positive and negative energy solutions in an upside-down harmonic oscillator potential $V(q)=-\frac{1}{2}{q}^2$. The sign of the corresponding triad variable $p$ is indicated.

Figure 2

Two bouncing solutions for the expectation value of $\widehat{p}$ and the spread around it. Generic states have different spread before and after the bounce (dashed lines), while unsqueezed Gaussian initial states lead to solutions which are symmetric around the bounce not only in their expectation values but also in spreads (solid lines).

Figure 3

Sketch of two wave functions with the same peak position $p_0$ at $-{\varphi }_0$, where one function represents a collapsing branch, the other an expanding one. After the bounce of the first solution at $\varphi =0$, the wave packets deviate strongly at ${\varphi }_0$. Both wave packets are illustrated by their expectation values and spreads, assumed to be symmetric around the bounce for the sake of the argument.