Cosmological constraints on post-Newtonian parameters in effectively massless scalar-tensor theories of gravity

We study the cosmological constraints on the variation of the Newton's constant and on post-Newtonian parameters for simple models of scalar-tensor theory of gravity beyond the extended Jordan-Brans-Dicke theory. We restrict ourselves to an effectively massless scalar field with a potential $V \propto F^2$, where $F(\sigma)=N_{pl}^2+\xi\sigma^2$ is the coupling to the Ricci scalar considered. We derive the theoretical predictions for cosmic microwave background (CMB) anisotropies and matter power spectra by requiring that the effective gravitational strength at present is compatible with the one measured in a Cavendish-like experiment and by assuming adiabatic initial condition for scalar fluctuations. When comparing these models with $Planck$ 2015 and a compilation of baryonic acoustic oscilation (BAO) data, all these models accomodate a marginalized value for $H_0$ higher than in $\Lambda$CDM. We find no evidence for a statistically significant deviation from Einstein's general relativity. We find $\xi < 0.064$ ($|\xi| < 0.011$) at 95 % CL for $\xi > 0$ (for $\xi < 0$, $\xi \ne -1/6$). In terms of post-Newtonian parameters, we find $0.995 < \gamma_{\rm PN} < 1$ and $0.99987 < \beta_{\rm PN} < 1$ ($0.997 < \gamma_{\rm PN} < 1$ and $1 < \beta_{\rm PN} < 1.000011$) for $\xi >0$ (for $\xi < 0$). For the particular case of the conformal coupling, i.e. $\xi=-1/6$, we find constraints on the post-Newtonian parameters of similar precision to those within the Solar System.