Horndeski gravity and standard sirens

Standard sirens have been proposed as probes of alternative theories of gravity, such as Horndeski models. Hitherto, all studies have been conducted on a homogeneous-isotropic cosmological background, which is unable to consistently account for realistic distributions of matter, and for inhomogeneities in the Horndeski scalar field. Yet, the latter arc essential for screening mechanisms. In this article, we comprehensively analyze the propagation of Horndeski gravitational waves in an arbitrary background spacetime and scalar field. We restrict our study to the class of theories in which gravitational waves propagate at light speed, we work in the geometric-optics regime, and we neglect scalar radiation. We find that kinetic braiding produces a nonphysical longitudinal mode, whereas conformal coupling affects the amplitude of the standard transverse modes but not their polarization. We confirm that any observable deviation from general relativity depends on the local value of the effective Planck mass at emission and reception of the wave. This result is interpreted as the conservation of the number of gravitons.