Analytic Estimates for Quasi-normal Mode Frequencies of Black Holes in scalar Gauss-Bonnet Gravity*

The ringdown part of a gravitational waveform from a binary black hole merger is characterized by quasi-normal mode (QNM) frequencies. Although one needs to rely on numerical calculations to find accurate QNM frequencies, there are several analytic techniques available to obtain physical implications such as the relation between QNM frequencies and properties of null geodesics. Here, we adopt an eikonal approximation to derive analytic estimates of QNM frequencies for black holes in scalar-Gauss-Bonnet gravity. In this theory, a scalar field is coupled to the metric through a quadratic curvature (Gauss-Bonnet invariant) and the theory is a generalization to Einstein-dilaton Gauss-Bonnet gravity motivated by string theory. We find that the eikonal machinery leads to axial perturbation modes deviating from the general relativistic results. We show that this result is in agreement with an analysis of unstable circular null orbits around black holes in this theory, allowing us to establish the geometrical optics-null geodesic correspondence for the axial modes. For the polar modes, the tensor and scalar perturbation equations are coupled and the scalar-Gauss-Bonnet corrections lift the general relativistic degeneracy between scalar and tensorial eikonal quasinormal modes. In general, our analytic, eikonal QNM frequencies agree with numerical results with an error of ~10% in the regime of a small coupling constant.