Covariant formulation of refracted gravity

TL;DR

This paper introduces a covariant scalar-tensor theory of gravity that unifies dark matter and dark energy phenomena through a scalar field, explaining galactic dynamics and cosmic acceleration with a single framework.

Contribution

It formulates a covariant version of refracted gravity, linking dark matter effects and cosmic acceleration via a scalar field with a specific self-interaction potential.

Findings

Scalar field mimics dark matter phenomenology.

The same scalar field drives cosmic acceleration.

The model's parameters align with observational data.

Abstract

We propose a covariant formulation of refracted gravity (RG), a classical theory of gravity based on the introduction of gravitational permittivity (GP), a monotonic function of the local mass density, in the standard Poisson equation. GP mimics dark matter (DM) phenomenology. The covariant formulation of RG (CRG) that we propose belongs to the class of scalar-tensor theories, where the scalar field $\varphi$ has a self-interaction potential $V(\varphi)=-\Xi\varphi$, with $\Xi$ a normalization constant. We show that $\varphi$ is twice the GP 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 $\rho=\rho_s+\rho_{bg}$, $\rho_{bg}$ being 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 is comparable to the acceleration $a_0$ originally introduced in MOND. From CRG, we also derived the modified Friedmann equations for an expanding, homogeneous, and isotropic universe. We find that the same scalar field that mimics DM also drives the accelerated expansion of the Universe. From the stress-energy tensor of $\varphi$, we derived the equation of state of a redshift-dependent effective dark energy (DE) $w_{DE}=p_{DE}/\rho_{DE}$. Current observational constraints on $w_{DE}$ and distance modulus data of SNIa suggest that $\Xi$ has a comparable value to the cosmological constant $\Lambda$ in the standard model. CRG, therefore, suggests a natural explanation of the known relation $a_0\sim \Lambda^{1/2}$ and appears to describe both the dynamics of cosmic structure and the expanding Universe with a single scalar field, highlighting a possible deep connection between phenomena currently attributed to DM and DE separately.