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.
