Black hole spectroscopy and nonlinear echoes in Einstein-Maxwell-scalar theory

In the context of Einstein-Maxwell-scalar theory with a nonminimal coupling between the electromagnetic and scalar field, we study linear (non)radial perturbations and nonlinear radial dynamics of spherically symmetric black holes. In a certain region of the parameter space, this theory admits hairy black holes with a stable photon sphere. This has a counterpart in the effective potential of linear perturbations, featuring multiple maxima and minima. The corresponding quasinormal mode spectrum contains long-lived modes trapped in the potential cavity and the time-domain linear response displays echoes, as previously observed for horizonless compact objects. Interestingly, the black-hole dynamics in this theory can be studied at the nonlinear level. By performing fully fledged $1+1$ simulations, we show that echoes are present even when the nonlinearities are significant. To our knowledge, this is the first example of echoes appearing in a consistent theory beyond a linearized analysis. In a follow-up work we will study whether this feature is also present in the postmerger signal from black hole collisions in this theory.