A Relativistic Tensorial Model for Fractional Interaction between Dark Matter and Gravity

TL;DR

The paper addresses small-scale tensions in the standard CDM paradigm by proposing a relativistic, nonlocal interaction between dark matter and gravity. It extends Relativistic Scalar Fractional Gravity (RSFG) to include a tensorial nonlocal coupling via R_{mu nu} F2(Box) T^{mu nu}_{DM}, yielding a general action with two operators F0(Box) and F2(Box). The authors derive the full field equations, analyze the weak-field and Newtonian limits where dark matter sources acquire a nonlocal pressure and anisotropic stress, and compute the lensing potentials to compare with observations. They constrain two FG realizations (EFG and RTFG) using CLASH cluster convergence profiles, finding fits comparable to General Relativity with indications of nontrivial fractional index s in several clusters but no decisive preference over GR. The work demonstrates that nonlocal dark matter–gravity interactions can reproduce cluster-scale lensing while remaining consistent with large-scale cosmology, offering a framework to address cusp-core and related small-scale issues without abandoning the success of ΛCDM on large scales.

Abstract

In a series of recent papers it was shown that several aspects of Dark Matter (DM) phenomenology, such as the velocity profiles of individual dwarfs and spiral galaxies, the scaling relations observed in the latter, and the pressure and density profiles of galaxy clusters, can be explained by assuming the DM component in virialized halos to feel a non-local fractional interaction mediated by gravity. Motivated by the remarkable success of this model, in a recent work we have looked for a general relativistic extension, proposing a theory, dubbed Relativistic Scalar Fractional Gravity or RSFG, in which the trace of the DM stress-energy tensor couples to the scalar curvature via a non-local operator constructed with a fractional power of the d'Alembertian. In this work we construct an extension of that model in which also a non-local coupling between the Ricci tensor and the DM stress energy tensor is present. In the action we encode the normalization between these scalar and tensorial term into two operators $F_0(\Box)$ and $F_2(\Box)$, and we derive the general field equations. We then take the weak field limit of the latter, showing that they reduce to general relativity sourced by an effective stress energy tensor, featuring a non local isotropic pressure and anisotropic stress, even if one starts with the assumption of a pressureless DM fluid. Finally, after having worked out the lensing theory in our setup, we test particularly interesting realizations of our framework against the measured convergence profiles of the individual and stacked clusters of the CLASH sample, finding remarkable consistency with the data.

Figures (2)

• Figure 1: The effective density (normalized to $\rho_s\, (\ell/r_s)^{2-2s}$) vs the radius (normalized to $r_s$), for different values of the fractional index $s$ (color-coded). For reference, the dotted line refers to the maximal value $s = 3/2$.
• Figure 2: Fits to the lensing convergence profiles $\kappa_\infty(R)$ of the individual and of the stacked CLASH cluster sample according to the EFG (magenta) and RTFG (cyan) model (see Section \ref{['sec|lens']} for details). Solid lines with shaded areas illustrate the median and the $2\sigma$ credible interval from sampling the posterior distribution. Dashed black lines display, for comparison, the bestfits in general relativity. The vertical dotted line marks the Einstein radius in the lens plane, on assuming a reference source redshift $z_s\sim 2$.