Most gravitational-wave signals detected by the LIGO–Virgo–KAGRA network originate from binary black-hole coalescences. The newly formed black hole is highly distorted immediately after the merger and gradually settles into a stationary state by emitting gravitational waves. These waves exhibit a discrete set of exponentially decaying frequencies known as quasinormal modes. The corresponding phase, called the ringdown, encodes the unique fingerprint of the final black hole. Analyzing quasinormal modes provides a powerful avenue to probe fundamental physics, including stringent tests of general relativity in the strong-field regime. However, obtaining the quasinormal-mode spectrum of generic black holes presents severe mathematical challenges, as it requires solving a complex system of coupled partial differential equations. In this talk, I will introduce METRICS, Metric pErTuRbations wIth speCtral methodS, a spectral formalism that overcomes these difficulties and enables precise computation of quasinormal-mode spectra for general black holes. I will demonstrate how METRICS can be applied to a range of modified gravity theories motivated by high-energy physics, including axi-dilaton, dynamical Chern-Simons, and Einstein-scalar-Gauss-Bonnet theories, to conduct ringdown-only tests of gravity. These analyses yield, among other results, the first observational constraints on axi-dilaton gravity. If time permits, I will mention the results of the recent first accurate construction of nearly extremal black holes in modified gravity theories. I will conclude by outlining future applications of METRICS for extracting new insights into fundamental physics through black-hole ringdowns.
