Loop quantum cosmology and thek=−1Robertson-Walker model

Abstract

The loop quantization of the negatively curved $k=\ensuremath{-}1$ Robertson-Walker model poses several technical challenges. We show that the issues can be overcome and a successful quantization is possible that extends the results of the $k=0$, $+1$ models in a natural fashion. We discuss the resulting dynamics and show that for a universe consisting of a massless scalar field, a bounce is predicted in the backward evolution in accordance with the results of the $k=0$, $+1$ models. We also show that the model predicts a vacuum repulsion in the high curvature regime that would lead to a bounce even for matter with vanishing energy density. We finally comment on the inverse volume modifications of loop quantum cosmology and show that, as in the $k=0$ model, the modifications depend sensitively on the introduction of a length scale which a priori is independent of the curvature scale or a matter energy scale.