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

TL;DR

This paper introduces a relativistic scalar fractional gravity model where dark matter interacts non-locally with gravity, reproducing key dark matter phenomenology and predicting deviations in post-Newtonian parameters while maintaining light-speed gravitational wave propagation.

Contribution

It presents the first relativistic extension of fractional gravity with a non-local scalar coupling, deriving field equations and analyzing weak field and post-Newtonian limits.

Findings

Reproduces dark matter phenomenology in a relativistic framework

Predicts deviations in the post-Newtonian parameter gamma

Shows gravitational waves propagate at light speed with an additional scalar mode

Abstract

In a series of recent papers we put forward a ``fractional gravity'' framework striking an intermediate course between a modified gravity theory and an exotic dark matter (DM) scenario, which envisages the DM component in virialized halos to feel a non-local interaction mediated by gravity. The remarkable success of this model in reproducing several aspects of DM phenomenology motivates us to look for a general relativistic extension. Specifically, we propose 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. We derive the field equations starting from an action principle, and then we investigate their weak field limit, demonstrating that in the Newtonian approximation the fractional gravity setup of our previous works is recovered. We compute the first-order post-Newtonian parameter $\gamma$ and its relation with weak lensing, showing that although in RSFG the former deviates from its GR values of unity, the latter is unaffected. We also perform a standard scalar-vector-tensor-decomposition of RSFG in the weak field limit, to highlight that gravitational waves propagate at the speed of light, though also an additional scalar mode becomes dynamical. Finally, we derive the modified conservation laws of the DM stress energy tensor in RSFG, showing that a new non-local force emerges, and hence that the DM fluid deviates from the geodesic solutions of the field equations.