Detectability of overtones and quadratic nonlinear quasinormal modes in binary-black-hole ringdowns

Isi et al. (arXiv:1905.00869) reported an overtone in GW150914 at 3.6σ; Cotesta et al. (arXiv:2201.00822) found no evidence for it after the peak and showed that noise injections can produce comparable spurious evidence, while Isi & Farr (arXiv:2107.05609) set out the time-domain formalism behind the original claim, so the result depends on start-time and noise-model choices. Mitman et al. (arXiv:2208.07380) and Cheung et al. (arXiv:2208.07374) found quadratic modes in numerical-relativity ringdowns: in the (ℓ,m) = (4,4) harmonic the quadratic (220×220) mode scales as the square of the 220 amplitude, with A₂₂₀ₓ₂₂₀/A₂₂₀² ≈ 0.16 (Cheung et al.) or 0.15–0.20 depending on fit start time (Mitman et al.), and is comparable to or larger than the linear 440 mode. Second-order Kerr perturbation theory shows the ratio is spin dependent, from about 0.16 at a/M = 0.2 to about 0.08 near extremality (Redondo-Yuste et al., arXiv:2308.14796). The LIGO–Virgo–KAGRA analysis of GW250114 (detected 14 January 2025, the loudest signal so far; arXiv:2509.08099, September 2025) found that at least two quasinormal modes are needed after the peak (the more damped one significant for about one cycle), matched the 220 mode and its first overtone to the Kerr pattern to tens of percent, and bounded the 220 frequency to within a few percent of general relativity. The GWTC-4.0 tests of the remnants (arXiv:2603.19021, 2026) test the ringdown on 42 events from early O4 and find overall consistency with general relativity. In one hierarchical analysis the GR value sits at the boundary of the 98.6% credible region for the catalog (up from 93.8% for GWTC-3.0), which the authors attribute largely to finite-catalog variance; including GW250114 lowers this to 92.2%, so there is no strong evidence for a deviation. Since then, two unrefereed preprints claim quadratic modes in GW250114: Yang et al. (arXiv:2510.16903) find a model with the 220×220 mode preferred over a linear-only 440 model, and Wang et al. (arXiv:2601.05734) report a Bayes factor of about 62 and a 3.4σ rejection of zero amplitude for quadratic couplings of the (2,2,n) tones starting 4 M_f after merger (values differ between preprint versions). An orthonormal-mode reanalysis raises support for the first overtone to 99.9% with no deviation from Kerr (Suzuki et al., arXiv:2605.03576). The GWTC-5.0 GR tests (O4b, 168 events; arXiv:2607.19293, July 2026) find no deviation. O4 ended on 18 November 2025; O4c results are expected around the end of 2026. No independent confirmation of a nonlinear mode at ≥5σ exists, and whether the extra post-peak content is an overtone, a nonlinear mode or a modelling artefact is still debated, so the state is IN_PROGRESS.

Ringdown spectroscopy tests the no-hair theorem and general relativity in the strong field, and its pipelines apply to future detectors (Cosmic Explorer, Einstein Telescope, LISA) where SNR is above 1,000. No practical application.

Resolved when a high-SNR event (or a stack) shows a second mode with an amplitude and frequency consistent with theory, identified as an overtone or nonlinear mode with a start-time-robust analysis at ≥5σ, and the Kerr no-hair relation is tested at the percent level on multiple events.