Black holes that carry a scalar “hair” can show a special kind of spectral degeneracy called an exceptional point. Exceptional points are non‑Hermitian degeneracies where two frequencies and their associated vibration patterns merge. The authors scanned the parameter space of a family of hairy black holes in the Einstein‑Maxwell‑scalar (EMS) theory and identified such an exceptional point in the spectrum of scalar quasinormal modes. They then studied how this feature changes the black hole’s ringdown signal and how well different fitting models recover the underlying modes.

Quasinormal modes (QNMs) are the damped oscillations that dominate the late time response when a black hole is disturbed. In the usual description the ringdown is written as a sum of independent exponentially damped sinusoids. Near an exceptional point that description breaks down. The paper explains and demonstrates that the QNM expansion gains an extra resonant contribution: an exponential factor multiplied by a piece that is linear in time. This extra term comes from the coalescence of both frequencies and eigenfunctions at the exceptional point.

To reach these conclusions the authors used two complementary numerical approaches. In the frequency domain they solved the wave equation for a test, massless scalar field on static, spherically symmetric hairy black hole backgrounds in the EMS model. They converted the radial problem to a compact coordinate and used a Chebyshev spectral expansion together with Newton–Raphson root finding to compute complex QNM frequencies. In the time domain they evolved the same wave equation with a leapfrog finite‑difference scheme, starting from a Gaussian pulse near the potential peak, and extracted ringdown waveforms for fitting.

They compared the standard fitting ansatz (a sum of damped modes) with an ansatz that includes the exceptional‑point resonant piece. Their results show that the exceptional‑point ansatz, which explicitly contains the linear‑in‑time contribution multiplied by an exponential decay, provides a more robust description of the ringdown near the exceptional point. In practice this ansatz captures the resonant imprint of mode coalescence more naturally and leads to more reliable extraction of the mode parameters from the time‑domain signal.