Cosmological constraints on the gravitational constant

We study the variation of the gravitational constant on cosmological scales in scalar-tensor theories of gravity. We focus on the simplest models of scalar-tensor theories with a coupling to the Ricci scalar of the form F(sigma) = N-pl(2) + xi sigma(2), such as extended JordanBrans-Dicke (N-pl = 0), or a non-minimally coupled scalar field with N-pl = M-pl, which permits the gravitational constant to vary self-consistently in time and space. In addition, we allow the effective gravitational constant on cosmological scales to differ from the Newton's measured constant G, i.e. G(eff)(z = 0) = G (1 + Delta)(2). We study the impact of this imbalance Delta jointly with the coupling into anisotropies of the cosmic microwave background and matter power spectrum at low-redshift. Combining the information from Planck 2018 CMB temperature, polarization and lensing, together with a compilation of BAO measurements from the release DR12 of the Baryon Oscillation Spectroscopic Survey (BOSS), we constrain the imbalance to Delta = -0.022 +/- 0.023 (68% CL) and the coupling parameter to 10(3) xi < 0.82 (95% CL) for Jordan-Brans-Dicke and for a non-minimally coupled scalar field with F(sigma) = M-pl(2) + xi sigma(2) we constrain the imbalance to Delta > -0.018 (< 0.021) and the coupling parameter to xi < 0.089 (xi > -0.041) both at 95% CL. With current data, we observe that the degeneracy between Delta, the coupling to the Ricci scalar, and H-0 allows for a larger value of the Hubble constant increasing the consistency between the distance-ladder measurement of the Hubble constant from supernovae type Ia by the SHOES team and its value inferred by CMB data. We also study how future cosmological observations can constrain the gravitational Newton's constant. Future data such as the combination of CMB anisotropies from LiteBIRD and CMB-54, and large-scale structures galaxy clustering from DESI and galaxy shear from LSST reduce the uncertainty in Delta to sigma(Delta) similar or equal to 0.004.