Covariant formulation of refracted gravity

¹ Dipartimento di Fisica, Università di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy
e-mail: asanna@dsf.unica.it
² Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy
³ Dipartimento di Fisica, Università di Torino, Via P. Giuria 1, 10125 Torino, Italy
e-mail: titos.matsakos@gmail.com; antonaldo.diaferio@unito.it
⁴ Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy

Received: 15 March 2022
Accepted: 17 April 2023

Abstract

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 φ has a self-interaction potential 𝒱(φ) = − Ξφ, with Ξ 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 ρ_s(r), the transition between the Newtonian and the RG regime appears below the acceleration scale a_Ξ = (2Ξ − 8πGρ/φ)^1/2, with ρ = ρ_s + ρ_bg and ρ_bg being an isotropic and homogeneous background. In the limit 2Ξ ≫ 8πGρ/φ, we obtain a_Ξ ∼ 10⁻¹⁰ m s⁻². This acceleration is comparable to the acceleration a₀ 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 φ that mimics dark matter also drives the accelerated expansion of the Universe. From the stress-energy tensor of φ, we derived the equation of state of a redshift-dependent effective dark energy w_DE = p_DE/ρ_DE. Current observational constraints on w_DE and distance modulus data of type Ia supernovae suggest that Ξ has a comparable value to the cosmological constant Λ in the standard model. Since Ξ also plays the same role of Λ, CRG suggests a natural explanation of the known relation a₀ ∼ Λ^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.