Retarded Green's function of a Vainshtein system and Galileon waves

Motivated by the desire to test modified gravity theories exhibiting the Vainshtein mechanism, we solve in various physically relevant limits, the retarded Galileon Green's function (for the cubic theory) about a background sourced by a massive spherically symmetric static body. The static limit of our result will aid us, in a forthcoming paper, in understanding the impact of Galileon fields on the problem of motion in the solar system. In this paper, we employ this retarded Green's function to investigate the emission of Galileon radiation generated by the motion of matter lying deep within the Vainshtein radius r_v of the central object: acoustic waves vibrating on its surface, and the motion of compact bodies gravitationally bound to it. If λ is the typical wavelength of the emitted radiation, and r₀ is the typical distance of the source from the central mass, with r ₀âr_v, then, compared to its noninteracting massless scalar counterpart, we find that the Galileon radiation rate is suppressed by the ratio (r_v/λ)^-3/2 at the monopole and dipole orders at high frequencies r_v/λ≠1. However, at high enough multipole order, the radiation rate is enhanced by powers of r _v/r₀. At low frequencies r_v/λâ1, and when the motion is nonrelativistic, Galileon waves yield a comparable rate for the monopole and dipole terms, and are amplified by powers of the ratio r_v/r₀ for the higher multipoles.