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Recent studies by Sol\`a Peracaula, G\'omez-Valent, de Cruz P\'erez andMoreno-Pulido (2019,2020) have pointed out the intriguing possibility thatBrans-Dicke cosmology with constant vacuum energy density (BD-$\Lambda$CDM) maybe able to alleviate the $H_0$ and $\sigma_8$ tensions that are found in theframework of the concordance cosmological model (GR-$\Lambda$CDM). The fittinganalyses presented in these works indicate a preference for values of theeffective gravitational coupling appearing in the Friedmann equation, $G$,about $4-9\%$ larger than Newton's constant (as measured on Earth), and mildyevolving with the expansion of the universe. The signal reaches the $\sim3.5\sigma$ c.l. when the prior on $H_0$ from SH0ES and the angular diameterdistances to strong gravitationally lensed quasars measured by H0LICOW areconsidered, 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 theexistence of a mechanism capable of screening the modified gravity effects atthose scales where deviations from standard General Relativity (GR) are highlyconstrained, as in the Solar System. In this paper we explore severalextensions of BD-$\Lambda$CDM that can leave the cosmological evolutionbasically unaltered at the background and linear perturbations level, whilebeing able to screen the Brans-Dicke effects inside the regions of interest,leading to standard GR. We search for weak-field solutions around sphericalstatic massive objects with no internal pressure and show that, unfortunately,these mechanisms can only explain very tiny departures of the effectivecosmological gravitational coupling from the one measured locally. This mighthinder the ability of BD-$\Lambda$CDM to alleviate the cosmological tensions.

journal of cosmology and astroparticle physics

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