Superluminality in beyond Horndeski theory with extra scalar field

We study the superluminality issue in beyond Horndeski theory with additional scalar field, which is minimally coupled to gravity and has no second derivatives in the Lagrangian. We present the quadratic action for perturbations in cosmological backgrounds, stability conditions and expressions for sound speeds. We find that in the case of conventional additional scalar whose flat-space propagation speed is that of light, one of the modes in interacting theory is necessarily superluminal when this scalar rolls, even arbitrarily slowly. This result holds in any theory of the beyond Horndeski class (with 6 arbitrary functions in the Lagrangian) and for any stable rolling background. More generally, the requirement of the absence of superluminality imposes non-trivial constraints on the structure of the theory.