Covarying-Bi-Scalar Theory: A model with covarying G and c and a natural screening mechanism

In this paper we consider a double-scalar-field extension to scalar–tensor theories of gravity. Our theory is based in the metric tensor $g_{\mu \nu }$ and two scalar fields. We keep the idea in Brans–Dicke theory that the gravitational coupling $G$ is an effective phenomenon from the dynamics of a scalar field $\varphi$. In addition to that, we raise the speed of light $c$ to the status of an independent scalar field. The dynamical analysis of $\varphi \left(x^{\mu }\right)$ and $c\left(x^{\mu }\right)$ is performed in the phase space and the conditions for the existence of stable equilibrium points are determined. These conditions accommodate the constraint $\left(c^3/G\right)=\text{constant}$ in accordance with the ansatz $\dot{G}/G=3\left(\dot{c}/c\right)$ that has recently received support from astrophysical and cosmological phenomenology. We analyze this possibility in the context of FLRW cosmology and a chameleon-like mechanism we dubbed $c$-meleon. We obtain specific solutions for the $c$-meleon field, including (i) an O’Hanlon and Tupper-like solution, (ii) dust-matter plus the varying $c$ (and $G$), and (iii) vacuum energy plus the varying $c$ (and $G$).