Possibility of realizing weak gravity in redshift space distortion measurements

We study the possibility of realizing a growth rate of matter density perturbations lower than that in General Relativity. Using the approach of the effective field theory of modified gravity encompassing theories beyond Horndeski, we derive the effective gravitational coupling $G_{\rm eff}$ and the gravitational slip parameter $\eta$ for perturbations deep inside the Hubble radius. In Horndeski theories we derive a necessary condition for achieving weak gravity associated with tensor perturbations, but this is not a sufficient condition due to the presence of a scalar-matter interaction that always enhances $G_{\rm eff}$. Beyond the Horndeski domain it is possible to realize $G_{\rm eff}$ smaller than Newton's gravitational constant $G$, while the scalar and tensor perturbations satisfy no-ghost and stability conditions. We present a concrete dark energy scenario with varying $c_{\rm t}$ and numerically study the evolution of perturbations to confront the model with the observations of redshift-space distortions and weak lensing.