Gravitational potential and galaxy rotation curves in multi-fractional spacetimes

TL;DR

This paper explores how multi-fractional spacetime theories can produce gravitational potentials that explain galaxy rotation curves without dark matter, revealing a logarithmic growth at large radii linked to quantum gravity-inspired geometries.

Contribution

It derives the Newtonian potential in multi-fractional theories and demonstrates their potential to account for galaxy rotation curves as geometric effects, offering an alternative to dark matter.

Findings

Potential grows logarithmically at large radii for certain fractional exponents

Rotation curves of specific galaxies can be explained without dark matter

The model reproduces medium-to-large radius rotation curves but not the Tully-Fisher relation

Abstract

Multi-fractional theories with integer-order derivatives are models of gravitational and matter fields living in spacetimes with variable Hausdorff and spectral dimension, originally proposed as descriptions of geometries arising in quantum gravity. We derive the Poisson equation and the Newtonian potential of these theories starting from their covariant modified Einstein's equations. In particular, in the case of the theory $T_v$ with weighted derivatives with small fractional corrections, we find a gravitational potential that grows logarithmically at large radii when the fractional exponent takes the special value $\alpha=4/3$. This behaviour is associated with a restoration law for the Hausdorff dimension of spacetime independently found in the dark-energy sector of the same theory. As an application, we check whether this potential can serve as an alternative to dark matter for the galaxies NGC7814, NGC6503 and NGC3741 in the SPARC catalogue. We show that their rotation curves at medium-to-large radii can indeed be explained by purely geometric effects, although the Tully--Fisher relation is not reproduced well. We discuss how to fix the small-radius behaviour by lifting some approximations and how to test the model with other observables and an enlarged galaxy sample.