Covariant Formulation of refracted gravity

We propose a covariant formulation of refracted gravity (RG), a classical theory of gravity based on the introduction of the gravitational permittivity -- a monotonic function of the local mass density -- in the standard Poisson equation. The gravitational permittivity mimics the dark matter phenomenology. Our covariant formulation of RG (CRG) belongs to the class of scalar-tensor theories, where the scalar field $\varphi$ has a self-interaction potential $\mathcal{V}(\varphi)=-\Xi\varphi$, with $\Xi$ 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 $a_\Xi=(2\Xi-8\pi G\rho/\varphi)^{1/2}$, with $\rho=\rho_s+\rho_\mathrm{bg}$ and $\rho_\mathrm{bg}$ an isotropic and homogeneous background. In the limit $2\Xi\gg 8\pi G\rho/\varphi$, we obtain $a_\Xi\sim 10^{-10}$~m~s$^{-2}$. This acceleration is comparable to the acceleration $a_0$ originally introduced in Modified Newtonian Dynamics (MOND). From CRG, we also derive the modified Friedmann equations for an expanding, homogeneous, and isotropic universe. We find that the same scalar field that mimics dark matter also drives the accelerated expansion of the Universe. Since $\Xi$ plays a role roughly similar to the cosmological constant $\Lambda$ in the standard model and has a comparable value, CRG suggests a natural explanation of the known relation $a_0\sim \Lambda^{1/2}$. CRG thus appears to describe both the dynamics of cosmic structure and the expanding Universe with a single scalar field, and 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.