Quasinormal modes of hot, cold and bald Einstein-Maxwell-scalar black holes

Einstein-Maxwell-scalar models allow for different classes of black hole solutions, depending on the non-minimal coupling function \(f(\phi)\) employed, between the scalar field and the Maxwell invariant. Here, we address the linear mode stability of the black hole solutions obtained recently for a quartic coupling function, \(f(\phi)=1+\alpha\phi^4\) [1]. Besides the bald Reissner-Nordstr\"om solutions, this coupling allows for two branches of scalarized black holes, termed cold and hot, respectively. For these three branches of black holes we calculate the spectrum of quasinormal modes. It consists of polar scalar-led modes, polar and axial electromagnetic-led modes, and polar and axial gravitational-led modes. We demonstrate that the only unstable mode present is the radial scalar-led mode of the cold branch. Consequently, the bald Reissner-Nordstr\"om branch and the hot scalarized branch are both mode-stable. The non-trivial scalar field in the scalarized background solutions leads to the breaking of the degeneracy between axial and polar modes present for Reissner-Nordstr\"om solutions. This isospectrality is only slightly broken on the cold branch, but it is strongly broken on the hot branch.