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Dynamical quasinormal mode excitation

Marina De Amicis1,*, Enrico Cannizzaro2, Gregorio Carullo1,3,†, and Laura Sberna4

  • 1Center of Gravity, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
  • 2CENTRA, Departamento de Fisica, Instituto Superior Técnico—IST, Universidade de Lisboa—UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal
  • 3School of Physics and Astronomy and Institute for Gravitational Wave Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
  • 4School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

  • *Contact author: mdeamicis@perimeterinstitute.ca
  • †Contact author: g.carullo@bham.ac.uk
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Phys. Rev. D 113, 024048 – Published 22 January, 2026

DOI: https://doi.org/10.1103/bgjv-lvyd

Abstract

We study the dynamical excitation of quasinormal modes (QNMs) through the plunge, merger and ringdown of an extreme-mass-ratio-inspiral into a Schwarzschild black hole, for generic orbital configurations. We work out the QNM causality condition, crucial to eliminate amplitude divergences and to incorporate horizon redshift effects. We then use it to derive a model of the time-dependent QNM excitation via a Green’s function approach, driven by the point-particle source on a given trajectory. Our model predicts that: (i) QNMs propagates along hyperboloidal slices in the minimal gauge; (ii) the signal is composed of an “activation” term, depending on the source past history, and a local “impulsive” term; (iii) amplitudes grow in time in an “activation function” fashion, and the waveform displays a stationary ringdown regime at times ∼10–20M after its peak; (iv) at these late times, an infinite tower of nonoscillatory, exponentially damped terms appear: the redshift terms. The model is in good agreement with numerical solutions, capturing the main waveform features after the peak. Additional components of the Green’s function are required to complement the QNM description and reproduce the plunge-merger waveform. We predict the late-time, stationary amplitude of the quadrupolar mode as a function of eccentricity, in agreement with accurate numerical solutions, marking the first time that QNM amplitudes are predicted for generic binary configurations. Our work provides a first solid step toward analytically modeling the inspiral’s imprint onto ringdown signals, generalizable to include higher orders in the mass ratio, black hole spin, nonvacuum configurations and corrections to the Einstein-Hilbert action.

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Phys. Rev. D 113, 024048– Published 22 January, 2026
  • Received 2 September 2025
  • Accepted 17 December 2025
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