Loop quantum cosmology of k=1 FRW models

TL;DR

This paper demonstrates that loop quantum cosmology applied to closed k=1 FRW models successfully resolves the big-bang singularity, predicts a quantum bounce, and leads to a cyclic universe consistent with classical recollapse.

Contribution

It extends loop quantum cosmology to closed k=1 models, showing singularity resolution and cyclic behavior while maintaining classical recollapse.

Findings

Big-bang replaced by a quantum bounce

Recollapse occurs at classical scale factor values

Quantum dynamics remains deterministic across Planck regime

Abstract

The closed, k=1, FRW model coupled to a massless scalar field is investigated in the framework of loop quantum cosmology using analytical and numerical methods. As in the k=0 case, the scalar field can be again used as emergent time to construct the physical Hilbert space and introduce Dirac observables. The resulting framework is then used to address a major challenge of quantum cosmology: resolving the big-bang singularity while retaining agreement with general relativity at large scales. It is shown that the framework fulfills this task. In particular, for states which are semi-classical at some late time, the big-bang is replaced by a quantum bounce and a recollapse occurs at the value of the scale factor predicted by classical general relativity. Thus, the `difficulties' pointed out by Green and Unruh in the k=1 case do not arise in a more systematic treatment. As in k=0 models, quantum dynamics is deterministic across the deep Planck regime. However, because it also retains the classical recollapse, in contrast to the k=0 case one is now led to a cyclic model. Finally, we clarify some issues raised by Laguna's recent work addressed to computational physicists.