We propose physically motivated spacetime uncertainty relations (STUR) for flat Friedmann-Lemaître 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.

We study the metric aspect of the Moyal plane from Connes' noncommutative geometry point of view. First, we compute Connes' spectral distance associated with the natural isometric action of R 2 on the algebra of the Moyal plane A. We show that the distance between any state of A and any of its translated is precisely the amplitude of the translation. As a consequence, we obtain the spectral distance between coherent states of the quantum harmonic oscillator as the Euclidean distance on the plane. We investigate the classical limit, showing that the set of coherent states equipped with Connes' spectral distance tends towards the Euclidean plane as the parameter of deformation goes to zero. The extension of these results to the action of the symplectic group is also discussed, with particular emphasize on the orbits of coherent states under rotations. Second, we compute the spectral distance in the double Moyal plane, intended as the product of (the minimal unitization of) A by C 2 . We show that on the set of states obtained by translation of an arbitrary state of A, this distance is given by Pythagoras theorem. On the way, we prove some Pythagoras inequalities for the product of arbitrary unital & non-degenerate spectral triples. Applied to the Doplicher-Fredenhagen-Roberts model of quantum spacetime [DFR], these two theorems show that Connes' spectral distance and the DFR quantum length coincide on the set of states of optimal localization.

We derive new spacetime uncertainty relations (STUR) at the fundamental Planck length L P from quantum mechanics and general relativity (GR), both in flat and curved backgrounds. Contrary to claims present in the literature, our approach suggests that no minimal uncertainty appears for lengths, but instead for minimal space and fourvolumes. Moreover, we derive a maximal absolute value for the energy density. Finally, some considerations on possible commutators among quantum operators implying our STUR are done.

We question the emergence of a minimal length in quantum spacetime, comparing two notions that appeared at various points in the literature: on the one side, the quantum length as the spectrum of an operator L in the Doplicher Fredenhagen Roberts (DFR) quantum spacetime, as well as in the canonical noncommutative spacetime (θ-Minkowski); on the other side, Connes' spectral distance in noncommutative geometry. Although on the Euclidean space the two notions merge into the one of geodesic distance, they yield distinct results in the noncommutative framework. In particular on the Moyal plane, the quantum length is bounded above from zero while the spectral distance can take any real positive value, including infinity. We show how to solve this discrepancy by doubling the spectral triple. This leads us to introduce a modified quantum length d L , which coincides exactly with the spectral distance d D on the set of states of optimal localization. On the set of eigenstates of the quantum harmonic oscillator -together with their translations -d L and d D coincide asymptotically, both in the high energy and large translation limits. At small energy, we interpret the discrepancy between d L and d D as two distinct ways of integrating the line element on a quantum space. This leads us to propose an equation for a geodesic on the Moyal plane.

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