A relativistic scalar model for fractional interaction between dark matter and gravity

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.