Axial perturbations of the scalarized Einstein-Gauss-Bonnet black holes

We study the axial perturbations of spontaneously scalarized black holes in Einstein-Gauss-Bonnet theories. We consider the nodeless solutions of the fundamental branch of the model studied by Doneva and Yazadjiev [Phys. Rev. Lett. 120, 131103 (2018)], which possesses a region of radially stable configurations, as shown by Blazquez-Salcedo et aL [Phys. Rcv. D 98, 084011 (2018)]. Here we show that almost all of the radially stable black holes are also stable under axial perturbations. When the axial potential is no longer strictly positive, we make use of the S-deformation method to show stability. As for the radial perturbations, hyperbolicity is lost below a certain critical horizon size for a fixed coupling constant. In the stable region, we determine the spectrum of the quasinormal modes by time evolution and by solving the associated time-independent eigenvalue problem.