Retarded Green's Function Of A Vainshtein System And Galileon Waves
Yi-Zen Chu, Mark Trodden
TL;DR
This work derives the retarded Green's function for the cubic Galileon about a static, spherically symmetric mass to study Galileon radiation from matter within the Vainshtein radius. By decomposing the Green's function into angular modes and solving a radial ODE in multiple limits (static, WKB, radiative), the authors reveal how Vainshtein screening suppresses high-frequency, low multipole Galileon signals but can enhance higher multipole radiation, and how low-frequency waves can remain unscreened or even amplified. The curved-spacetime interpretation via an effective metric and Hadamard form provides a cohesive picture of signal propagation, including superluminal radial modes inside $r_v$ and flat-space behavior outside. Two radiative scenarios are analyzed—surface waves on the central body and $n$ orbiting masses within $r_v$—showing rich $oldsymbol{ extomega}$- and $ l$-dependent spectra, with potential implications for solar-system tests and gravitational-wave observations. Overall, the paper advances analytical control over Galileon radiation in Vainshtein-screened setups and clarifies how nonlinearities shape energy loss and signal propagation in modified gravity theories.
Abstract
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 λis the typical wavelength of the emitted radiation, and r_0 is the typical distance of the source from the central mass, with r_0 << r_v, then, compared to its non-interacting 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/r_0. At low frequencies r_v/λ<< 1, and when the motion is non-relativistic, Galileon waves yield a comparable rate for the monopole and dipole terms, and are amplified by powers of the ratio r_v/r_0 for the higher multipoles.