Echoes of compact objects in scalar-tensor theories of gravity

Scalar-tensor theory predicts solutions to the gravitational field equations which describe compact objects in the presence of a nonminimally coupled scalar field to the Einstein tensor. These objects are black holes with scalar hair and wormholes supporting scalar phantom matter. The evolution of test fields in fixed asymptotically-flat backgrounds of exotic compact objects leads to the formation of echoes in the ringdown signal, which designate the existence of trapping regions close to the event horizon. Here, we consider minimally coupled test scalar fields propagating on compact object solutions of the Horndeski action, which possess an effective cosmological constant, leading to anti-de Sitter asymptotics, and show that echoes can form in the ringdown waveform due to the entrapment of test fields between the photon sphere and the effective asymptotic boundary. Although the presence of an event horizon leads to the usual echoes with decaying amplitude, signifying modal stability of the scalarized black hole considered, we find that test scalar fields propagating on a scalarized wormhole solution give rise to echoes of constant and equal amplitude to that of the initial ringdown, indicating the existence of normal modes. Finally, we find that, near extremality, the test field exhibits a concatenation of echoes; the primary ones are associated with the trapping region between the photon sphere and the effective anti--de Sitter boundary while the secondary ones are linked to the existence of a potential well at the throat of the wormhole.