Brans-Dicke-like field for co-varying $G$ and $c$: observational constraints

Ref. [Symmetry 15 (2023) 709] introduced a Brans-Dicke-like framework wherein the scalar field $\phi$ is composed of both $G$ and $c$ which, for this reason, co-vary according to $c^{3}/G=\text{constant}$. In this paper, we use observational data to constrain the supposed co-varying $G$ and $c$, under the hypothesis of the validity of the standard Lemaitre formula $1+z\sim a^{-1}$. The datasets include SN Ia, BAO and the value of $\theta$ extracted from CMB data. A proxy function is demanded for the varying $c$ since the framework does not provide a closed set of equations for computing the functional form of either $G$ or $c$ uniquely. Accordingly, we choose three separate parameterizations for $c\left(z\right)$ inspired both by desirable properties of the varying speed of light (VSL) and by successful phenomenological models from the literature -- including the one by Gupta (CCC framework in e.g. Ref. [Mon. Not. R. Astron. Soc., 498 (2020) 4481-4491]. When combined with DESI, Pantheon+ data strongly favor a variable speed of light with more than $3\sigma$ confidence level for all parameterizations considered in this paper, whereas Union2.1 suggests no variation of the speed of light. As we shall demonstrate, this apparent discrepancy is due to a strong correlation that emerges between $H_0$ and VSL.