Constraining Palatini–Horndeski theory with gravitational waves after GW170817

Abstract

Abstract In this paper, we investigate the possible parameter space of Palatini–Horndeski theory with gravitational waves in a spatially flat Universe. We develop a general method for obtaining the speed of gravitational waves in the Palatini formalism in the cosmological background and 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 $$S = \int d^4x \sqrt{-g} \big [K(\phi ,X)-G_{3}(\phi ,X){{\tilde{\Box }}}\phi +G_{4}(\phi ){\tilde{R}}\big ]$$ S = ∫ d 4 x - g [ K ( ϕ , X ) - G 3 ( ϕ , X ) □ ~ ϕ + G 4 ( ϕ ) R ~ ] is left as the possible action in Palatini–Horndeski theory. We also find that when $$G_{5}(\phi ,X)\ne 0$$ G 5 ( ϕ , X ) ≠ 0 , the tensor part of the connection will propagate and there are two different tensor gravitational wave speeds.