Dark energy in multi-fractional spacetimes.

We study the possibility to obtain cosmological late-time acceleration from a geometry changing with the scale, in particular, in the so-called multi-fractional theories with q-derivatives and with weighted derivatives. In the theory with q-derivatives, the luminosity distance is the same as in general relativity and, therefore, geometry cannot act as dark energy. In the theory with weighted derivatives, geometry alone is able to sustain a late-time acceleration phase without fine tuning, while being compatible with structure-formation and big-bang nucleosynthesis bounds. This suggests to extend the theory, in a natural way, from just small-scale to also large-scale modifications of gravity. Surprisingly, the Hausdorff dimension of spacetime is constrained to be close to the topological dimension 4. Promoting this finding to a principle, we conclude that present-day acceleration can be regarded as the effect of a new restoration law for spacetime geometry.