Difficulties in reconciling non-negligible differences between the local and cosmological values of the gravitational coupling in extended Brans-Dicke theories

Abstract

Recent studies by Sol\`a Peracaula, G\'omez-Valent, de Cruz P\'erez and Moreno-Pulido (2019,2020) have pointed out the intriguing possibility that Brans-Dicke cosmology with constant vacuum energy density (BD-$\Lambda$CDM) may be able to alleviate the $H_0$ and $\sigma_8$ tensions that are found in the framework of the concordance cosmological model (GR-$\Lambda$CDM). The fitting analyses presented in these works indicate a preference for values of the effective gravitational coupling appearing in the Friedmann equation, $G$, about $4-9\%$ larger than Newton's constant (as measured on Earth), and mildy evolving with the expansion of the universe. The signal reaches the $\sim 3.5\sigma$ c.l. when the prior on $H_0$ from SH0ES and the angular diameter distances to strong gravitationally lensed quasars measured by H0LICOW are considered, and the $\sim 3\sigma$ c.l. when only the former is included. Thus, the improvement in the description of the cosmological datasets relies on the existence of a mechanism capable of screening the modified gravity effects at those scales where deviations from standard General Relativity (GR) are highly constrained, as in the Solar System. In this paper we explore several extensions of BD-$\Lambda$CDM that can leave the cosmological evolution basically unaltered at the background and linear perturbations level, while being able to screen the Brans-Dicke effects inside the regions of interest, leading to standard GR. We search for weak-field solutions around spherical static massive objects with no internal pressure and show that, unfortunately, these mechanisms can only explain very tiny departures of the effective cosmological gravitational coupling from the one measured locally. This might hinder the ability of BD-$\Lambda$CDM to alleviate the cosmological tensions.