Angular and radial stabilities of spontaneously scalarized black holes in the presence of scalar-Gauss-Bonnet couplings

We study the linear stability of spontaneously scalarized black holes (BHs) induced by a scalar field φ coupled to a Gauss-Bonnet (GB) invariant RGB². For the scalar-GB coupling ξ(φ)=(η/8) (φ²+α φ⁴), where η and α are constants, we first show that there are no angular Laplacian instabilities of even-parity perturbations far away from the horizon for large multipoles l 1. The deviation of angular propagation speeds from the speed of light is largest on the horizon, whose property can be used to put constraints on the model parameters. For α -1, the region in which the scalarized BH is subject to angular Laplacian instabilities can emerge. Provided that α -1 and -1/2<α φ₀²<-0.1155, where φ₀ is the field value on the horizon with a unit of the reduced Planck mass MPl=1, there are scalarized BH solutions satisfying all the linear stability conditions throughout the horizon exterior. We also study the stability of spontaneously scalarized BHs in scalar-GB theories with a nonminimal coupling -β φ² R/16, where β is a positive constant and R is a Ricci scalar. As the amplitude of the field on the horizon approaches an upper limit |φ₀|=4/√β, one of the squared angular propagation speeds cΩ-² enters the instability region cΩ-²<0. So long as |φ₀| is smaller than a maximum value determined for each β in the range β>5, however, the scalarized BHs are linearly stable in both angular and radial directions.