Gravitational waves in massive Horndeski theory with a potential

We investigate gravitational waves with an arbitrary potential within the framework of linearized Horndeski theory. We show that the minimum of the potential can play the role of an effective cosmological constant in this theory, which is usually neglected in previous studies of this subject. We first determine the background geometry in this setup by solving the weak field scalar and tensorial equations of linearized Horndeski theory. The solutions of linearized weak-field wave equations, in an appropriate gauge, are then obtained perturbatively to study the propagation and interactions of gravitational waves in this background. We compare our results with different realizations of the cosmological constant in Horndeski theory to compare the role of an arbitrary scalar potential with those of vacuum energy density and a linear potential. The results show that the background curvature arising from the minimum of the scalar potential effectively mimics a cosmological constant, producing distinct redshifts in the frequency and wave number that distinguish the tensor waves from massive scalar ones. We also find that the way the cosmological constant is introduced directly influences the speed and polarization of the scalar wave.