Cosmological parameter constraints for Horndeski scalar-tensor gravity

We present new cosmological parameter constraints for general Horndeski scalar-tensor theories, using CMB, redshift space distortion, matter power spectrum and BAO measurements from the Planck, SDSS/BOSS and 6dF surveys. We focus on theories with cosmological gravitational waves propagating at the speed of light, $c_{\mathrm{G}\mathrm{W}}=c$, implementing and discussing several previously unaccounted for aspects in the constraint derivation for such theories, that qualitatively affect the resulting constraints. In order to ensure our conclusions are robust, we compare results for three different parametrisations of the free functions in Horndeski scalar-tensor theories, identifying several parametrisation-independent features of the constraints. We also consider models, where $c_{\mathrm{G}\mathrm{W}}\mathrm{\ne }c$ in cosmological settings (still allowed after GW170817 for frequency-dependent $c_{\mathrm{G}\mathrm{W}}$) and show how this affects cosmological parameter constraints.