Dark energy in multifractional spacetimes

Abstract

We study the possibility to obtain cosmological late-time acceleration from a geometry changing with the scale, in particular, in the so-called multifractional 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 structureformation 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. After arguing that this finding might not be a numerical coincidence, we conclude that present-day acceleration could be regarded as the effect of a new restoration law for spacetime geometry.

Newtonian Fractional-Dimension Gravity and MOND

“…This procedure is practically equivalent to the one used for scalar-tensor theories of gravity (see Ref. [47] for a general overview) and it has been used extensively by Calcagni in the context of multi-scale spacetimes and fractional gravity theories [7,8,10,11,15,17,18]. In the following subsections we will review these techniques and adapt them to our particular case.…”

“…Different multi-scale theories were then developed by Calcagni, with reference to the possible derivative operators ∂ being used: theory T 1 with ordinary derivatives, theory T v with weighted derivatives, and theory T q with q-derivatives (see [11,17] for full details). These models were then used in connection with the most general measure derived from first principles [13] and then applied to quantum field theories, quantum gravity, and cosmology [11,[15][16][17][18].…”

Relativistic Fractional-Dimension Gravity

“…This procedure is practically equivalent to the one used for scalar-tensor theories of gravity (see Ref. [52] for a general overview) and it has been used extensively by Calcagni in the context of multi-scale spacetimes and fractional gravity theories [7,8,10,11,15,17,18]. In the following subsections we will review these techniques and adapt them to our particular case.…”

“…Different multi-scale theories were then developed by Calcagni, with reference to the possible derivative operators ∂ being used: theory T 1 with ordinary derivatives, theory T v with weighted derivatives, and theory T q with q-derivatives (see [11,17] for full details). These models were then used in connection with the most general measure derived from first principles [13] and then applied to quantum field theories, quantum gravity, and cosmology [11,[15][16][17][18].…”

“…In paper III, the use of a variable dimension D(r) as a function of the field point was also discussed and justified in terms of other similar existing studies. In all these three papers, we pointed out that NFDG is only loosely based on the methods of fractional calculus and fractional mechanics (see [6] and references therein), but is not a fractional theory in the sense used by other gravitational models [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. NFDG field equations are of integer order, therefore local, as opposed to non-local field equations based on fractional differential operators.…”