The (in)stability of quasinormal modes of Boulware-Deser-Wheeler black hole in the hyperboloidal framework

We study the quasinormal modes of Boulware-Deser-Wheeler black hole in Einstein-Gauss-Bonnet gravity theory within the hyperboloidal framework. The effective potentials for the test Klein-Gordon field and gravitational perturbations of scalar, vector, and tensor types are thoroughly investigated and put into several typical classes. The effective potentials for the gravitational perturbations have more diverse behaviors than those in general relativity, such as double peaks, the existence of the negative region adjacent to or far away from the event horizon, etc. These lead to the existence of unstable modes ($\text{Im} \omega<0$), and the presence of gravitational wave echoes. These rich phenomenons are inherent in Einstein-Gauss-Bonnet theory, rather than artificially introduced by hand. What's more, the (in)stability of quasinormal modes is studied in frequency domain and time domain, respectively. For the frequency-domain, the pseudospectrum is used to account for the instability of the spectrum. For the time-domain, we add a small bump to the effective potential, and find that the new waveform does not differ significantly from the original one, where the comparison is characterized by the so-called mismatch functions. This means that quasinormal modes are stable in time-domain regardless of the shapes of the original effective potentials. In this way, our study reveals the non-equivalence of the stability of quasinormal modes in the frequency-domain and the time-domain. Besides, we also numerically investigate Price's law at both finite distances and infinity with the assistance of the hyperboloidal approach.