Stability of black holes with non-minimally coupled scalar hair

General relativity admits a plethora of exact compact object solutions. The augmentation of Einstein's action with non-minimal coupling terms leads to modified theories with rich structure, which, in turn, provide non-trivial solutions with intriguing phenomenology. Thus, assessing their viability under external generic fluctuations is of utmost importance for gravity theories. We consider static and spherically-symmetric solutions of a Horndeski subclass which includes a massless scalar field non-minimally coupled to the Einstein tensor. Such theory possesses second-order field equations and admits an exact black hole solution with scalar hair. Here, we study the stability of such solution under axial perturbations and find that it is gravitationally stable at the linear level. The qualitative features of the ringdown waveform depend solely on ratio of the two available parameters of spacetime, namely the black hole mass $m$ and the non-minimal coupling strength $\ell_\eta$. Finally, we demonstrate that the gravitational-wave ringdown transitions from one which exhibits echoes to one with a typical quasinormal ringing phase followed by a late-time tail as the ratio $m/\ell_\eta$ increases.