In linear perturbation theory, the ringdown of a gravitational wave (GW) signal is described by a linear combination of quasinormal modes (QNMs). Detecting QNMs from GW signals is a promising way to test GR, central to the developing field of black-hole spectroscopy. More robust black hole spectroscopy tests could also consider the ringdown amplitude-phase consistency. That requires an accurate understanding of the excitation and stability of the QNM expansion coefficients. In this paper, we investigate the robustness of the extracted $m=2$ QNM coefficients obtained from a high-accuracy numerical relativity waveform. We explore a framework to assess the robustness of QNM coefficients. Within this framework, we not only consider the traditional criterion related to the constancy of a QNM’s expansion coefficients over a window in time, but also emphasize the importance of consistency among fitting models. In addition, we implement an iterative greedy approach within which we fix certain QNM coefficients. We apply this approach to linear fitting, and to nonlinear fitting where the properties of the remnant black hole are treated as unknown variables. We find that the robustness of overtone coefficients is enhanced by our greedy approach, particularly for the $(2,2,2,+)$ overtone. Based on our robustness criteria applied to the $m=2$ signal modes, we find the $(2\sim 4,2,0,+)$ and $(2,2,1\sim 2,+)$ modes are robust, while the $(3,2,1,+)$ subdominant mode is only marginally robust. After we subtract the contributions of the $(2\sim 4,2,0,+)$ and $(2\sim 3,2,1,+)$ QNMs from signal mode (4, 2), we also find evidence for the quadratic QNM $(2,1,0,+)\times (2,1,0,+)$.

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• Classical black holes
• General relativity equations & solutions
• Gravitational wave sources
• Gravitational waves