Covariant formulation of refracted gravity

We propose a covariant formulation of refracted gravity (RG), which is a classical theory of gravity based on the introduction of gravitational permittivity - a monotonic function of the local mass density - in the standard Poisson equation. Gravitational permittivity mimics dark matter phenomenology. The covariant formulation of RG (CRG) that we propose belongs to the class of scalar-tensor theories, where the scalar field phi has a self-interaction potential V(phi) = -Xi phi with Xi being a normalization constant. We show that the scalar field is twice the gravitational permittivity in the weak-field limit. Far from a spherical source of density rho s(r), the transition between the Newtonian and the RG regime appears below the acceleration scale a(Xi) = (2(Xi) 8 pi G rho/phi)1/2, with rho = rho(s) + rho(bg) and rho(bg) being an isotropic and homogeneous background. In the limit 2(Xi) >> 8 pi G rho/phi, we obtain a(Xi) similar to 10(-10) ms(-2). This acceleration is comparable to the acceleration a(0) originally introduced in MOdified Newtonian Dynamics (MOND). From CRG, we also derived the modified Friedmann equations for an expanding, homogeneous, and isotropic universe. We find that the same scalar field phi that mimics dark matter also drives the accelerated expansion of the Universe. From the stress-energy tensor of phi, we derived the equation of state of a redshift-dependent e ffective dark energy w(DE) = p(DE)/rho DE. Current observational constraints on w DE and distance modulus data of type Ia supernovae suggest that Xi has a comparable value to the cosmological constant Lambda in the standard model. Since Xi also plays the same role of Lambda, CRG suggests a natural explanation of the known relation a(0) similar to Lambda(1/2). CRG thus appears to describe both the dynamics of cosmic structure and the expanding Universe with a single scalar field, and it falls within the family of models that unify the two dark sectors, highlighting a possible deep connection between phenomena currently attributed to dark matter and dark energy separately.