Quasinormal mode families: Classification and competition

The perturbation spectra of black holes beyond standard vacuum black hole solutions within generalrelativity (GR) may exhibit complex structures with long-lived modes. This usually generates echolikemodulations on the ringdown signal, which typically originate from modified boundary conditionsassociated with exotic compact objects. Recent studies also reveal that they can instead arise from themultipeaked structure of the perturbation potential. However, while some case-by-case studies have beencarried out, a framework for understanding the internal structure of such spectra, the physical nature ofdifferent mode families, and their dynamical excitation remains to be fully systematized. In this paper,we address this issue by proposing a potential methodology that combines frequency-domainclassification with time-domain analysis, using a hairy Schwarzschild black hole that admits adouble-peak perturbative potential as a theoretical platform. Our analysis of the quasinormal modespectrum identifies two distinct families of modes: the photon sphere (PS) family, arising fromdelocalized scattering resonances, and the echo family, corresponding to highly localized quasiboundstates. We then develop a windowed energy analysis framework in the time domain, which discloses adynamic competition for dominance between these families. In particular, our results explicitly showthat 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 modein GR. This study establishes the dynamic evolution of this energy competition as a new observationalsignature for probing new physics and further motivates a supplemental framework for analyzing long-lived ringdown signals.