Constraining Palatini-Horndeski theory with gravitational waves after GW170817

In this paper, we investigate the possible parameter space of Palatini-Horndeski theory with gravitational waves in a spatially flat Universe. We find that if the theory satisfies the following condition: in any spatially flat cosmological background, the tensor gravitational wave speed is the speed of light $c$, then only $S=\int d^{4}x\sqrt{-g}[K(\phi ,X)-G_3(\phi ,X)\tilde{\mathrm{\square }}\phi +G_4(\phi )\tilde{R}]$ is left as the possible action in Palatini-Horndeski theory.