http://arxiv.org/abs/2112.13103
Multi-fractional theories with integer-order derivatives are simple models of gravitational and matter fields living in spacetimes with variable Hausdorff and spectral dimension, originally proposed as descriptions of spacetimes arising in quantum gravity. In this paper, we pose the question of whether they can serve as an alternative to dark matter. We give a preliminary positive answer. We find the Poisson equation and the Newtonian potential of the multi-fractional theories with integer-order derivatives starting from their covariant modified Einstein’s equations. We show that neither the theory $T_v$ with weighted derivatives in the extreme fractional limit nor the theory $T_q$ with $q$-derivatives fit the rotation curve of any of the galaxies NGC7814, NGC6503 and NGC3741 in the SPARC catalogue. The rotation curve of all three galaxies can be explained by purely geometric effects in the theory $T_v$ with weighted derivatives with small fractional corrections when the fractional exponent takes the special value $\alpha=4/3$, but only at large radii.
G. Calcagni and G. Varieschi
Tue, 28 Dec 21
17/55
Comments: 37 pages + appendices, 12 figures
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