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 nonlocal
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 γ 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-tensordecomposition
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.
