Dark Matter in Fractional Gravity. I. Astrophysical Tests on Galactic Scales

Abstract

We explore the possibility that the dark matter (DM) component in galaxies may originate fractional gravity. In such a framework, the standard law of inertia continues to hold, but the gravitational potential associated with a given DM density distribution is determined by a modified Poisson equation including fractional derivatives (i.e., derivatives of noninteger type) that are meant to describe nonlocal effects. We analytically derive the expression of the potential that in fractional gravity corresponds to various spherically symmetric density profiles, including the Navarro–Frenk–White (NFW) distribution that is usually exploited to describe virialized halos of collisionless DM as extracted from N-body cosmological simulations. We show that in fractional gravity, the dynamics of a test particle moving in a cuspy NFW density distribution is substantially altered with respect to the Newtonian case, mirroring what in Newtonian gravity would instead be sourced by a density profile with an inner core. We test the fractional gravity framework on galactic scales, showing that (i) it can provide accurate fits to the stacked rotation curves of spiral galaxies with different properties, including dwarfs; (ii) it can reproduce to reasonable accuracy the observed shape and scatter of the radial acceleration relation over an extended range of galaxy accelerations; and (iii) it can properly account for the universal surface density and the core radius versus disk scale length scaling relations. Finally, we discuss the possible origin of the fractional gravity behavior as a fundamental or emerging property of the elusive DM component.

Dark Matter in Fractional Gravity II: Tests in Galaxy Clusters

“…In such a framework, the DM component exists, but the gravitational potential associated to its density distribution is determined by a modified Poisson equation including fractional derivatives (i.e., derivatives of noninteger type), which are meant to describe nonlocal effects; as such, this scenario is substantially different from the above theories where baryonic matter emulates DM-like effects via modifications of gravity. In [1], we showed that DM in fractional gravity worked very well for reproducing the kinematics of disk-dominated galaxies, especially dwarfs. In addition, we found preliminary evidence that the strength of fractional effects tends to weaken toward more massive systems; however, the latter finding is still subject to large uncertainties since the rotation curves of massive spirals were not probed out to radii large enough for the DM contribution to clearly emerge.…”

“…In fractional gravity, the potential Φ F (r) is instead derived from the modified Poisson equation [31] (−∆) s Φ F (r) = −4πG ℓ 2−2s ρ(r) (5) where (−∆) s is the fractional Laplacian operator (see [1,31] for details), s ∈ [1, 3/2] is the fractional index (this range of values for s is required to avoid divergences; see Appendix A in [1]), and ℓ is a fractional length scale that must be introduced for dimensional reasons. At variance with the standard case, the fractional Laplacian is inherently nonlocal; the index s measures the strength of this nonlocality, while the length scale ℓ can be interpreted as the typical size below which gravitational effects are somewhat reduced and above which they are instead amplified by nonlocality (around r ≈ ℓ, the dynamics is almost unaffected and indistinguishable from the standard case).…”

“…In [1], we solved the fractional Poisson equation sourced by the NFW density distribution. For s ∈ [1, 3/2), the solution reads…”

“…In the microwave band, such a tilt mimics a diminution of the CMB temperature proportional to the Comptonization parameter y ∝ dℓ p(r), which encompasses the line-of-sight integral of the pressure profile. Combining X-ray and SZ data allows one to reconstruct the ICM thermodynamic profiles throughout most of the cluster volume, from the center to a few times R 500 or even beyond the virial boundary 1 .…”

“…Recently, in [1], we suggested that the DM component itself may originate fractional gravity. In such a framework, the DM component exists, but the gravitational potential associated to its density distribution is determined by a modified Poisson equation including fractional derivatives (i.e., derivatives of noninteger type), which are meant to describe nonlocal effects; as such, this scenario is substantially different from the above theories where baryonic matter emulates DM-like effects via modifications of gravity.…”