  Black hole (BH) spectroscopy has emerged as a powerful approach to extract
spacetime information from gravitational wave (GW) observed signals. Yet,
quasinormal mode (QNM) spectral instability under high wave-number
perturbations has been recently shown to be a common classical general
relativistic phenomenon [1]. This requires to assess its impact on the BH QNM
spectrum, in particular on BH QNM overtone frequencies. We conclude: i)
perturbed BH QNM overtones are indeed potentially observable in the GW
waveform, providing information on small-scale environment BH physics, and ii)
their detection poses a challenging data analysis problem of singular interest
for LISA astrophysics. We adopt a two-fold approach, combining theoretical
results from scattering theory with a fine-tuned data analysis on a
highly-accurate numerical GW ringdown signal. The former introduces a set of
effective parameters (partially lying on a BH Weyl law) to characterise QNM
instability physics. The latter provides a proof-of-principle demonstrating
that the QNM spectral instability is indeed accessible in the time-domain GW
waveform, though certainly requiring large signal-to-noise ratios. Particular
attention is devoted to discuss the patterns of isospectrality loss under QNM
instability, since the disentanglement between axial and polar GW parities may
already occur within the near-future detection range.

General Relativity and Quantum Cosmology