Building non commutative spacetimes at the Planck length for Friedmann flat cosmologies

Abstract

We propose physically motivated spacetime uncertainty relations (STUR) for flat Friedmann-Lema\^{i}tre cosmologies. We show that the physical features of these STUR crucially depend on whether a particle horizon is present or not. In particular, when this is the case we deduce the existence of a maximal value for the Hubble rate (or equivalently for the matter density), thus providing an indication that quantum effects may rule out a pointlike big bang singularity. Finally, we costruct a concrete realisation of the corresponding quantum Friedmann spacetime in terms of operators on a Hilbert space.