Hints from non-commutative geometry
By Marco de Cesare, Mairi Sakellariadou, and Patrizia Vitale
It is often argued that modifications of general relativity can potentially explain the properties of the gravitational field on large scales without the need to postulate a (so far unobserved) dark sector. However, the theory space seems to be virtually unconstrained. One may then legitimately ask whether there is any guiding principle —such as symmetry— that can be invoked to build such a modified gravity theory and ground it in fundamental physics. We also know that the classical picture of spacetime as a Riemannian manifold must be abandoned at the Planck scale. The question then arises as to what kind of geometric structures may replace it, and if there are any novel gravitational degrees of freedom that they bring along. Importantly, one may ask whether there are any potentially observable effects away from the experimentally inaccessible Planck regime. These questions are crucial both from the point of view of quantum gravity and for model building in cosmology; trying to answer them will help us in the attempt to bridge the gap between the two fields, and could have far-reaching implications for our understanding of the quantum structure of spacetime.
CQG+ 
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