Quasi normal modes of hairy black holes at higher-order WKB approach

In this work, we implement the 13th order semi-analytical WKB method to explore the stability of hairy black holes obtained in the framework of Gravitational Decoupling. In particular, we perform a detailed analysis of the frequencies of the quasi-normal modes as a function of the primary hair of the solutions with the aim to bound their values. We explore a broad interval in a step of 0.1 of the hair parameters. We find that except for some cases where the method is expected to have poor accuracy, all the solutions seem to be stable and the role played by the primary hair is twofold: to modulate the damping factor of the perturbation and to decrease the frequency of its oscillation.