QNM families: classification and competition

The perturbation spectra of black hole (BH) beyond standard general relativity (GR) may exhibit complex structures with long-lived modes that generate echo-like modulations on the ringdown signal. However, a systematic framework for understanding the internal structure of such spectra, the physical nature of different mode families, and their dynamical excitation is still undeveloped. In this paper, we address this issue by proposing a potential methodology that combines frequency-domain classification with time-domain analysis, using a hairy Schwarzschild BH that admits a double-peak perturbative potential as a theoretical platform. Our analysis of the quasinormal mode (QNM) spectrum identifies two distinct families of modes: the photon sphere (PS) family, arising from delocalized scattering resonances, and the echo family, corresponding to highly localized quasi-bound states. We then develop a windowed energy analysis framework in the time domain, which discloses a dynamic competition for dominance between these families. In particular, our results explicitly show that this competition is sensitive to the properties of the initial perturbation source, and that higher-overtone echo modes can dominate in the observed signal, which are in contrast to the standard PS mode in GR. This study establishes the dynamic evolution of this energy competition as a new observational signature for probing new physics and further motivate a supplemental framework for analyzing long-lived ringdown signals.