Consistent Loop Quantum Cosmology

A consistent combination of quantum geometry effects rules out a large class of models of loop quantum cosmology and their critical densities as they have been used in the recent literature. In particular, the critical density at which an isotropic universe filled with a free, massless scalar field would bounce must be well below the Planck density. In the presence of anisotropy, no model of the Schwarzschild black hole interior analyzed so far is consistent. Aside from detailed technical constructions, one of the main achievements of loop quantum gravity [1, 2, 3] is to provide a framework for discrete dynamical geometries. Loop quantum cosmology [4] takes these ingredients and applies them to expanding universes or black hole models. In this way, effects of quantum physics as well as quantum geometry can be explored in detail. This is a specific realization of a general issue which has been discussed recurrently in the context of discrete models of gravity [5, 6, 7, 8, 9]. Loop quantum gravity makes many of the older considerations more specific, but since neither its complete form nor the precise transition to loop quantum cosmology has been hammered out as of now, the dynamics of loop quantum cosmology cannot be unique. Current ignorance must be parameterized so that at least qualitative implications can be found. In particular, all corrections from quantum physics and quantum geometry must be analyzed before reliable and robust conclusions can be drawn. Here we consider quantum geometry in regimes where its implications are dominant over genuine quantum corrections, and provide a consistent combination of its two main effects: holonomy corrections but also inverse volume corrections which have often been ignored. As we will see, this combination leads to tight consistency conditions which rule out parameter choices made so far in cases where only holonomy corrections were considered. Quantum back-reaction effects, which still remain to be fully derived following the methods of [10, 11], will not be required for our analysis. We start with a discussion of isotropic models, which classically have only one invariant scale: the Hubble distance H−1 = a/ȧ (a dot meaning a derivative by proper time). If we ∗e-mail address: [email protected]