Quasinormal modes of a scalar field in the Einstein--Gauss--Bonnet-AdS black hole background: Perturbative and non-perturbative branches

It has recently been found that quasinormal modes of asymptotically anti-de Sitter (AdS) black holes in theories with higher curvature corrections may help to describe the regime of intermediate 't Hooft coupling in the dual field theory. Here, we consider quasinormal modes of a scalar field in the background of spherical Gauss--Bonnet--anti-de Sitter (AdS) black holes. In general, the eigenvalues of wave equations are found here numerically, but at a fixed Gauss-Bonnet constant $\alpha = R^2/2$ (where $R$ is the AdS radius), an exact solution of the scalar field equation has been obtained. Remarkably, the purely imaginary modes, which are usually appropriate only to some gravitational perturbations, were found here even for a test scalar field. These purely imaginary modes of the Einstein--Gauss--Bonnet theory do not have the Einsteinian limits, because their damping rates grow, when $\alpha$ is decreasing. Thus, these modes are nonperturbative in $\alpha$. The real oscillation frequencies of the perturbative branch are linearly related to their Schwarzschild-AdS limits $Re (\omega_{GB}) = Re (\omega_{SAdS}) (1+ K(D) (\alpha/R^2))$, where $D$ is the number of spacetime dimensions. Comparison of the analytical formula with the frequencies found by the shooting method allows us to test the latter. In addition, we found exact solutions to the master equations for gravitational perturbations at $\alpha=R^2/2$ and observed that for the scalar type of gravitational perturbations an eikonal instability develops.