Black hole quasinormal modes in a scalar-tensor theory with field derivative coupling to the Einstein tensor

We investigate the quasinormal modes of a test massless, minimally coupled scalar field on a static and spherically symmetric black hole in the scalar-tensor theory with field derivative coupling to the Einstein tensor, which is a part of the Horndeski theory with the shift symmetry. In our solution, the spacetime is asymptotically AdS (anti-de Sitter), where the effective AdS curvature scale is determined solely by the derivative coupling constant. The metric approaches the AdS spacetime in the asymptotic infinity limit and precisely recovers the Schwarzschild-AdS solution in general relativity if the coupling constant is tuned to the inverse of the cosmological constant. We numerically find the lowest lying quasinormal frequency for the perturbation about a test massless, minimally coupled scalar field. The quasinormal frequency agrees with that of the Schwarzschild-AdS solution for the tuned case. For other parameters, in the large black hole limit, as the metric coincides with that of the Schwarzschild-AdS black hole, the quasinormal frequency almost agrees with that of the Schwarzschild-AdS black hole and is insensitive to the value of the cosmological constant. On the other hand, for a small back hole the real part of the quasinormal frequency decreases as the absolute value of the cosmological constant increases.