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Late-Time Tails in Nonlinear Evolutions of Merging Black Holes

Marina De Amicis ^1,*, Hannes R. Rüter², Gregorio Carullo ^1,3,†, Simone Albanesi^4,5, C. Melize Ferrus⁶, Keefe Mitman⁷, Leo C. Stein⁸, Vitor Cardoso^1,2, Sebastiano Bernuzzi⁴ et al.

Michael Boyle⁷, Nils Deppe^9,10,7, Lawrence E. Kidder⁷, Jordan Moxon¹¹, Alessandro Nagar^5,12, Kyle C. Nelli¹¹, Harald P. Pfeiffer¹³, Mark A. Scheel¹¹, William Throwe⁷, Nils L. Vu¹¹, and An𝚤l Zenginoğlu¹⁴

¹Center of Gravity, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
²CENTRA, Departamento de Física, 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
⁴Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, 07743 Jena, Germany
⁵INFN sezione di Torino, Torino 10125, Italy
⁶Theoretical Particle Physics and Cosmology Group, Physics Department, King’s College London, Strand, London WC2R 2LS, United Kingdom
⁷Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, New York 14853, USA
⁸Department of Physics and Astronomy, University of Mississippi, Oxford, Mississippi 38677, USA
⁹Laboratory for Elementary Particle Physics, Cornell University, Ithaca, New York 14853, USA
¹⁰Department of Physics, Cornell University, Ithaca, New York 14853, USA
¹¹Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA
¹²Institut des Hautes Etudes Scientifiques, 35 Route de Chartres, Bures-sur-Yvette 91440, France
¹³Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, D-14476 Potsdam, Germany
¹⁴Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742-2431, USA

^*Contact author: marina.de.amicis@nbi.ku.dk
^†Contact author: g.carullo@bham.ac.uk

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Phys. Rev. Lett. 135, 171401 – Published 21 October, 2025

DOI: https://doi.org/10.1103/2brx-xnyr

Abstract

We uncover late-time gravitational-wave tails in fully nonlinear $3 + 1$ dimensional numerical relativity simulations of merging black holes, using the highly accurate spec code. We achieve this result by exploiting the strong magnification of late-time tails due to binary eccentricity, recently observed in perturbative evolutions, and showcase here the tail presence in head-on configurations for several mass ratios close to unity. We validate the result through a large battery of numerical tests and detailed comparison with a perturbative evolution, which display striking agreement with full nonlinear ones in the ringdown regime, and very similar tail morphologies. Our results offer yet another confirmation of the highly predictive power of black hole perturbation theory in the presence of a source, even when applied to nonlinear solutions. The late-time tail signal is much more prominent than anticipated until recently, and possibly within reach of gravitational-wave detector measurements, unlocking observational investigations of an additional set of general relativistic predictions on the long-range gravitational dynamics.

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References (140)

F. Pretorius, Evolution of binary black hole spacetimes, Phys. Rev. Lett. 95, 121101 (2005).
M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Accurate evolutions of orbiting black-hole binaries without excision, Phys. Rev. Lett. 96, 111101 (2006).
J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Gravitational wave extraction from an inspiraling configuration of merging black holes, Phys. Rev. Lett. 96, 111102 (2006).
E. Berti, V. Cardoso, J. A. Gonzalez, U. Sperhake, M. Hannam, S. Husa, and B. Bruegmann, Inspiral, merger and ringdown of unequal mass black hole binaries: A multipolar analysis, Phys. Rev. D 76, 064034 (2007).
B. P. Abbott et al., Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016).
J. Aasi et al. (LIGO Scientific Collaboration), Advanced LIGO, Classical Quantum Gravity 32, 074001 (2015).
F. Acernese et al. (Virgo Collaboration), Advanced Virgo: A second-generation interferometric gravitational wave detector, Classical Quantum Gravity 32, 024001 (2015).
T. Akutsu et al. (KAGRA Collaboration), Overview of KAGRA: Detector design and construction history, Prog. Theor. Exp. Phys. 2021, 05A101 (2021).
B. S. DeWitt and R. W. Brehme, Radiation damping in a gravitational field, Ann. Phys. (N.Y.) 9, 220 (1960).
R. H. Price, Nonspherical perturbations of relativistic gravitational collapse. I. Scalar and gravitational perturbations, Phys. Rev. D 5, 2419 (1972).
R. H. Price, Nonspherical perturbations of relativistic gravitational collapse. II. Integer-spin, zero-rest-mass fields, Phys. Rev. D 5, 2439 (1972).
C. T. Cunningham, R. H. Price, and V. Moncrief, Radiation from collapsing relativistic stars. I—Linearized odd-parity radiation, Astrophys. J. 224, 643 (1978).
R. Gómez and J. Winicour, Asymptotics of gravitational collapse of scalar waves, J. Math. Phys. (N.Y.) 33, 1445 (1992).
C. Gundlach, R. H. Price, and J. Pullin, Late-time behavior of stellar collapse and explosions. I. Linearized perturbations, Phys. Rev. D 49, 883 (1994).
C. Gundlach, R. H. Price, and J. Pullin, Late-time behavior of stellar collapse and explosions. II. Nonlinear evolution, Phys. Rev. D 49, 890 (1994).
E. W. Leaver, Spectral decomposition of the perturbation response of the Schwarzschild geometry, Phys. Rev. D 34, 384 (1986).
E. S. C. Ching, P. T. Leung, W. M. Suen, and K. Young, Late time tail of wave propagation on curved space-time, Phys. Rev. Lett. 74, 2414 (1995).
N. Andersson, Evolving test fields in a black hole geometry, Phys. Rev. D 55, 468 (1997).
L. M. Burko and A. Ori, Late time evolution of nonlinear gravitational collapse, Phys. Rev. D 56, 7820 (1997).
L. Barack, Late time dynamics of scalar perturbations outside black holes. II. Schwarzschild geometry, Phys. Rev. D 59, 044017 (1999).
S. Bernuzzi, A. Nagar, and R. De Pietri, Dynamical excitation of space-time modes of compact objects, Phys. Rev. D 77, 044042 (2008).
S. Hod, How pure is the tail of gravitational collapse?, Classical Quantum Gravity 26, 028001 (2009).
W. Krivan, P. Laguna, and P. Papadopoulos, Dynamics of scalar fields in the background of rotating black holes, Phys. Rev. D 54, 4728 (1996).
W. Krivan, P. Laguna, P. Papadopoulos, and N. Andersson, Dynamics of perturbations of rotating black holes, Phys. Rev. D 56, 3395 (1997).
S. Hod, The radiative tail of realistic gravitational collapse, Phys. Rev. Lett. 84, 10 (2000).
S. Hod, Mode coupling in rotating gravitational collapse of a scalar field, Phys. Rev. D 61, 024033 (1999).
L. Barack and A. Ori, Late time decay of scalar perturbations outside rotating black holes, Phys. Rev. Lett. 82, 4388 (1999).
A. Ori, Evolution of linear gravitational and electromagnetic perturbations inside a Kerr black hole, Phys. Rev. D 61, 024001 (1999).
W. Krivan, Late time dynamics of scalar fields on rotating black hole backgrounds, Phys. Rev. D 60, 101501(R) (1999).
E. Poisson, Radiative falloff of a scalar field in a weakly curved space-time without symmetries, Phys. Rev. D 66, 044008 (2002).
L. M. Burko and G. Khanna, Radiative falloff in the background of rotating black hole, Phys. Rev. D 67, 081502(R) (2003).
L. M. Burko and G. Khanna, Universality of massive scalar field late time tails in black hole space-times, Phys. Rev. D 70, 044018 (2004).
L. M. Burko and G. Khanna, Late-time Kerr tails revisited, Classical Quantum Gravity 26, 015014 (2009).
L. M. Burko and G. Khanna, Late-time Kerr tails: Generic and non-generic initial data sets, ’up’ modes, and superposition, Classical Quantum Gravity 28, 025012 (2011).
I. Racz and G. Z. Toth, Numerical investigation of the late-time Kerr tails, Classical Quantum Gravity 28, 195003 (2011).
A. Zenginoğlu, G. Khanna, and L. M. Burko, Intermediate behavior of Kerr tails, Gen. Relativ. Gravit. 46, 1672 (2014).
L. M. Burko and G. Khanna, Mode coupling mechanism for late-time Kerr tails, Phys. Rev. D 89, 044037 (2014).
E. Harms, S. Bernuzzi, and B. Brügmann, Numerical solution of the $2 + 1$ Teukolsky equation on a hyperboloidal and horizon penetrating foliation of Kerr and application to late-time decays, Classical Quantum Gravity 30, 115013 (2013).
M. Maggiore, Gravitational Waves. Vol. 2 (Oxford University Press, New York, 2018).
M. De Amicis, S. Albanesi, and G. Carullo, Inspiral-inherited ringdown tails, Phys. Rev. D 110, 104005 (2024).
R. H. Price and J. Pullin, Colliding black holes: The close limit, Phys. Rev. Lett. 72, 3297 (1994).
S. Albanesi, S. Bernuzzi, T. Damour, A. Nagar, and A. Placidi, Faithful effective-one-body waveform of small-mass-ratio coalescing black hole binaries: The eccentric, nonspinning case, Phys. Rev. D 108, 084037 (2023).
T. Islam, G. Faggioli, G. Khanna, S. E. Field, M. van de Meent, and A. Buonanno, Phenomenology and origin of late-time tails in eccentric binary black hole mergers, Phys. Rev. D 112, 024061 (2025).
A. Buonanno and T. Damour, Effective one-body approach to general relativistic two-body dynamics, Phys. Rev. D 59, 084006 (1999).
A. Buonanno and T. Damour, Transition from inspiral to plunge in binary black hole coalescences, Phys. Rev. D 62, 064015 (2000).
T. Damour and A. Nagar, Faithful effective-one-body waveforms of small-mass-ratio coalescing black-hole binaries, Phys. Rev. D 76, 064028 (2007).
A. Nagar, T. Damour, and A. Tartaglia, Binary black hole merger in the extreme mass ratio limit, Classical Quantum Gravity 24, S109 (2007).
S. Bernuzzi, A. Nagar, and A. Zenginoglu, Horizon-absorption effects in coalescing black-hole binaries: An effective-one-body study of the non-spinning case, Phys. Rev. D 86, 104038 (2012).
E. W. Allen, E. Buckmiller, L. M. Burko, and R. H. Price, Radiation tails and boundary conditions for black hole evolutions, Phys. Rev. D 70, 044038 (2004).
M. Dafermos and I. Rodnianski, A note on boundary value problems for black hole evolutions, arXiv:gr-qc/0403034.
https://www.black-holes.org/SpEC.html
V. Cardoso, G. Carullo, M. De Amicis, F. Duque, T. Katagiri, D. Pereniguez, J. Redondo-Yuste, T. F. M. Spieksma, and Z. Zhong, Hushing black holes: Tails in dynamical spacetimes, Phys. Rev. D 109, L121502 (2024).
See Supplemental Material at http://link.aps.org/supplemental/10.1103/2brx-xnyr, which includes Refs. [54–61], for an investigation of the impact of initial separation on tails, resolution tests, details of the extrapolation procedure and its comparison with Cauchy–characteristic evolution results, as well as information on noise filtering.
A. Zenginoglu, A hyperboloidal study of tail decay rates for scalar and Yang-Mills fields, Classical Quantum Gravity 25, 175013 (2008).
N. Deppe, W. Throwe, L. E. Kidder, N. L. Vu, K. C. Nelli, C. Armaza, M. S. Bonilla, F. Hébert, Y. Kim, P. Kumar, G. Lovelace, A. Macedo, J. Moxon, E. O’Shea, H. P. Pfeiffer, M. A. Scheel, S. A. Teukolsky, N. A. Wittek et al., spectre v2024.09.29, 10.5281/zenodo.13858965 (2024).
J. Moxon, M. A. Scheel, and S. A. Teukolsky, Improved Cauchy-characteristic evolution system for high-precision numerical relativity waveforms, Phys. Rev. D 102, 044052 (2020).
J. Moxon, M. A. Scheel, S. A. Teukolsky, N. Deppe, N. Fischer, F. Hébert, L. E. Kidder, and W. Throwe, SpECTRE Cauchy-characteristic evolution system for rapid, precise waveform extraction, Phys. Rev. D 107, 064013 (2023).
K. Mitman et al., Fixing the BMS frame of numerical relativity waveforms, Phys. Rev. D 104, 024051 (2021).
L. Magaña Zertuche et al., High precision ringdown modeling: Multimode fits and BMS frames, Phys. Rev. D 105, 104015 (2022).
K. Mitman et al., Fixing the BMS frame of numerical relativity waveforms with BMS charges, Phys. Rev. D 106, 084029 (2022).
K. Mitman et al., A review of gravitational memory and BMS frame fixing in numerical relativity, Classical Quantum Gravity 41, 223001 (2024).
L. Gualtieri, E. Berti, V. Cardoso, and U. Sperhake, Transformation of the multipolar components of gravitational radiation under rotations and boosts, Phys. Rev. D 78, 044024 (2008).
D. Christodoulou and R. Ruffini, Reversible transformations of a charged black hole, Phys. Rev. D 4, 3552 (1971).
M. A. Scheel, M. Boyle, T. Chu, L. E. Kidder, K. D. Matthews, and H. P. Pfeiffer, High-accuracy waveforms for binary black hole inspiral, merger, and ringdown, Phys. Rev. D 79, 024003 (2009).
B. Szilagyi, L. Lindblom, and M. A. Scheel, Simulations of binary black hole mergers using spectral methods, Phys. Rev. D 80, 124010 (2009).
D. A. Hemberger, M. A. Scheel, L. E. Kidder, B. Szilágyi, G. Lovelace, N. W. Taylor, and S. A. Teukolsky, Dynamical excision boundaries in spectral evolutions of binary black hole spacetimes, Classical Quantum Gravity 30, 115001 (2013).
S. Ossokine, L. E. Kidder, and H. P. Pfeiffer, Precession-tracking coordinates for simulations of compact-object-binaries, Phys. Rev. D 88, 084031 (2013).
A. H. Mroué, M. A. Scheel, B. Szilágyi, H. P. Pfeiffer, M. Boyle, D. A. Hemberger, L. E. Kidder, G. Lovelace, S. Ossokine, N. W. Taylor, A. i. e. i. f. Zenginoğlu, L. T. Buchman, T. Chu, E. Foley, M. Giesler, R. Owen, and S. A. Teukolsky, Catalog of 174 binary black hole simulations for gravitational wave astronomy, Phys. Rev. Lett. 111, 241104 (2013).
M. Boyle et al., The SXS Collaboration catalog of binary black hole simulations, Classical Quantum Gravity 36, 195006 (2019).
G. Lovelace, R. Owen, H. P. Pfeiffer, and T. Chu, Binary-black-hole initial data with nearly-extremal spins, Phys. Rev. D 78, 084017 (2008).
L. T. Buchman, H. P. Pfeiffer, M. A. Scheel, and B. Szilagyi, Simulations of non-equal mass black hole binaries with spectral methods, Phys. Rev. D 86, 084033 (2012).
G. B. Cook and H. P. Pfeiffer, Excision boundary conditions for black hole initial data, Phys. Rev. D 70, 104016 (2004).
H. P. Pfeiffer, L. E. Kidder, M. A. Scheel, and S. A. Teukolsky, A multidomain spectral method for solving elliptic equations, Comput. Phys. Commun. 152, 253 (2003).
H. P. Pfeiffer and J. W. York, Extrinsic curvature and the Einstein constraints, Phys. Rev. D 67, 044022 (2003).
L. Lindblom, M. A. Scheel, L. E. Kidder, R. Owen, and O. Rinne, A new generalized harmonic evolution system, Classical Quantum Gravity 23, S447 (2006).
L. Lindblom and B. Szilágyi, Improved gauge driver for the generalized harmonic Einstein system, Phys. Rev. D 80, 084019 (2009).
S. Bernuzzi and A. Nagar, Binary black hole merger in the extreme-mass-ratio limit: A multipolar analysis, Phys. Rev. D 81, 084056 (2010).
S. Bernuzzi, A. Nagar, and A. Zenginoglu, Binary black hole coalescence in the large-mass-ratio limit: The hyperboloidal layer method and waveforms at null infinity, Phys. Rev. D 84, 084026 (2011).
D. Bini and T. Damour, Gravitational radiation reaction along general orbits in the effective one-body formalism, Phys. Rev. D 86, 124012 (2012).
D. Chiaramello and A. Nagar, Faithful analytical effective-one-body waveform model for spin-aligned, moderately eccentric, coalescing black hole binaries, Phys. Rev. D 101, 101501(R) (2020).
S. Albanesi, A. Nagar, and S. Bernuzzi, Effective one-body model for extreme-mass-ratio spinning binaries on eccentric equatorial orbits: Testing radiation reaction and waveform, Phys. Rev. D 104, 024067 (2021).
A. Nagar and L. Rezzolla, Gauge-invariant non-spherical metric perturbations of Schwarzschild black-hole spacetimes, Classical Quantum Gravity 22, R167 (2005); Classical Quantum Gravity23, 4297(E) (2006).
O. Rinne, Stable radiation-controlling boundary conditions for the generalized harmonic Einstein equations, Classical Quantum Gravity 23, 6275 (2006).
O. Rinne, L. Lindblom, and M. A. Scheel, Testing outer boundary treatments for the Einstein equations, Classical Quantum Gravity 24, 4053 (2007).
L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D 46, 4304 (1992).
L. Blanchet and G. Schaefer, Gravitational wave tails and binary star systems, Classical Quantum Gravity 10, 2699 (1993).
E. Poisson, Gravitational radiation from a particle in circular orbit around a black hole. I. Analytical results for the nonrotating case, Phys. Rev. D 47, 1497 (1993).
E. Poisson and M. Sasaki, Gravitational radiation from a particle in circular orbit around a black hole. V. Black hole absorption and tail corrections, Phys. Rev. D 51, 5753 (1995).
sxs waveform catalog, https://data.black-holes.org/simulations/index.html.
SXS Collaboration, Binary black-hole simulation SXS:BBH:3991, 10.26138/SXS:BBH:3991v3.0 (2025).
SXS Collaboration, Binary black-hole simulation SXS:BBH:3997, 10.26138/SXS:BBH:3997v3.0 (2025).
SXS Collaboration, Binary black-hole simulation SXS:BBH:3998, 10.26138/SXS:BBH:3998v3.0 (2025).
SXS Collaboration, Binary black-hole simulation SXS:BBH:3994, 10.26138/SXS:BBH:3994v3.0 (2025).
SXS Collaboration, Binary black-hole simulation SXS:BBH:3995, 10.26138/SXS:BBH:3995v3.0 (2025).
SXS Collaboration, Binary black-hole simulation SXS:BBH:3996, 10.26138/SXS:BBH:3996v3.0 (2025).
A. Zenginoglu, Hyperboloidal foliations and scri-fixing, Classical Quantum Gravity 25, 145002 (2008).
A. Zenginoglu, Asymptotics of black hole perturbations, Classical Quantum Gravity 27, 045015 (2010).
A. Zenginoglu, Hyperboloidal layers for hyperbolic equations on unbounded domains, J. Comput. Phys. 230, 2286 (2011).
D. A. B. Iozzo, M. Boyle, N. Deppe, J. Moxon, M. A. Scheel, L. E. Kidder, H. P. Pfeiffer, and S. A. Teukolsky, Extending gravitational wave extraction using weyl characteristic fields, Phys. Rev. D 103, 024039 (2021).
M. Boyle, D. Iozzo, and L. C. Stein, moble/scri: v1.2, 10.5281/zenodo.4041972 (2020).
The scri package, https://github.com/sxs-collaboration/scri.
M. Boyle, Angular velocity of gravitational radiation from precessing binaries and the corotating frame, Phys. Rev. D 87, 104006 (2013).
M. Boyle, L. E. Kidder, S. Ossokine, and H. P. Pfeiffer, Gravitational-wave modes from precessing black-hole binaries, arXiv:1409.4431.
M. Boyle, Transformations of asymptotic gravitational-wave data, Phys. Rev. D 93, 084031 (2016).
A. Savitzky and M. J. E. Golay, Smoothing and differentiation of data by simplified least squares procedures, Anal. Chem. 36, 1627 (1964).
E. Barausse, A. Buonanno, S. A. Hughes, G. Khanna, S. O’Sullivan, and Y. Pan, Modeling multipolar gravitational-wave emission from small mass-ratio mergers, Phys. Rev. D 85, 024046 (2012).
B. Wardell, A. Pound, N. Warburton, J. Miller, L. Durkan, and A. Le Tiec, Gravitational waveforms for compact binaries from second-order self-force theory, Phys. Rev. Lett. 130, 241402 (2023).
T. Islam and G. Khanna, Interplay between numerical relativity and perturbation theory: Finite size effects, Phys. Rev. D 108, 044012 (2023).
A. Nagar, J. Healy, C. O. Lousto, S. Bernuzzi, and A. Albertini, Numerical-relativity validation of effective-one-body waveforms in the intermediate-mass-ratio regime, Phys. Rev. D 105, 124061 (2022).
S. Okuzumi, K. Ioka, and M.-a. Sakagami, Possible discovery of nonlinear tail and quasinormal modes in black hole ringdown, Phys. Rev. D 77, 124018 (2008).
D. Perrone, A. Kehagias, and A. Riotto, Nonlinearities in Kerr black hole ringdown from the Penrose limit, arXiv:2507.01919.
A. Pound and B. Wardell, Black hole perturbation theory and gravitational self-force, in Handbook of Gravitational Wave Astronomy, edited by C. Bambi, S. Katsanevas, and K. D. Kokkotas (Springer Nature Singapore, Singapore, 2022), pp. 1411–1529.
L. Sberna, P. Bosch, W. E. East, S. R. Green, and L. Lehner, Nonlinear effects in the black hole ringdown: Absorption-induced mode excitation, Phys. Rev. D 105, 064046 (2022).
J. Redondo-Yuste, D. Pereñiguez, and V. Cardoso, Ringdown of a dynamical spacetime, Phys. Rev. D 109, 044048 (2024).
H. Zhu et al., Imprints of changing mass and spin on black hole ringdown, Phys. Rev. D 110, 124028 (2024).
T. May, S. Ma, J. L. Ripley, and W. E. East, Nonlinear effect of absorption on the ringdown of a spinning black hole, Phys. Rev. D 110, 084034 (2024).
L. Capuano, L. Santoni, and E. Barausse, Perturbations of the Vaidya metric in the frequency domain: Quasinormal modes and tidal response, Phys. Rev. D 110, 084081 (2024).
J. Luk and S.-J. Oh, Late time tail of waves on dynamic asymptotically flat spacetimes of odd space dimensions, arXiv:2404.02220.
N. T. Bishop, R. Gomez, L. Lehner, B. Szilagyi, J. Winicour, and R. A. Isaacson, Cauchy characteristic matching, in Black Holes, Gravitational Radiation and the Universe: Essays in Honor of C. V. Vishveshwara, edited by B. R. Iyer and B. Bhawal (Kluwer Academic, Dordrecht, Netherlands, 1999), pp. 383–408, .
B. Szilagyi, Cauchy characteristic matching in general relativity, Ph.D. Thesis, University of Pittsburgh, 2000, arXiv:gr-qc/0006091.
S. Ma et al., Fully relativistic three-dimensional Cauchy-characteristic matching for physical degrees of freedom, Phys. Rev. D 109, 124027 (2024).
A. Vañó-Viñuales, S. Husa, and D. Hilditch, Spherical symmetry as a test case for unconstrained hyperboloidal evolution, Classical Quantum Gravity 32, 175010 (2015).
A. Vañó-Viñuales and S. Husa, Spherical symmetry as a test case for unconstrained hyperboloidal evolution II: Gauge conditions, Classical Quantum Gravity 35, 045014 (2018).
A. Vañó-Viñuales, Spherically symmetric black hole spacetimes on hyperboloidal slices, Front. Appl. Math. Stat. 9, 1206017 (2023).
C. Peterson, S. Gautam, A. Vañó Viñuales, and D. Hilditch, Spherical evolution of the generalized harmonic gauge formulation of general relativity on compactified hyperboloidal slices, Phys. Rev. D 110, 124033 (2024).
T. F. M. Spieksma, V. Cardoso, G. Carullo, M. Della Rocca, and F. Duque, Black hole spectroscopy in environments: Detectability prospects, Phys. Rev. Lett. 134, 081402 (2025).
V. Cardoso, S. Yoshida, O. J. C. Dias, and J. P. S. Lemos, Late-time tails of wave propagation in higher dimensional spacetimes, Phys. Rev. D 68, 061503(R) (2003).
J. D. Hunter, matplotlib: A 2d graphics environment, Comput. Sci. Eng. 9, 90 (2007).
C. R. Harris et al., Array programming with numpy, Nature (London) 585, 357 (2020).
P. Virtanen et al., scipy 1.0: Fundamental algorithms for scientific computing in python, Nat. Methods 17, 261 (2020).
M. Boyle and M. Scheel, The sxs package, github.com/sxs-collaboration/sxs/ (2025).
S. Ma, M. A. Scheel, J. Moxon, K. C. Nelli, N. Deppe, L. E. Kidder, W. Throwe, and N. L. Vu, Merging black holes with Cauchy-characteristic matching: Computation of late-time tails, Phys. Rev. D 112, 024003 (2025).
sxs Collaboration, Binary black-hole simulation SXS:BBH:3985, 10.26138/SXS:BBH:3985v3.0 (2025).
sxs Collaboration, Binary black-hole simulation SXS:BBH:3986, 10.26138/SXS:BBH:3986v3.0 (2025).
sxs Collaboration, Binary black-hole simulation SXS:BBH:3987, 10.26138/SXS:BBH:3987v3.0 (2025).
sxs Collaboration, Binary black-hole simulation SXS:BBH:3988, 10.26138/SXS:BBH:3988v3.0 (2025).
sxs Collaboration, Binary black-hole simulation SXS:BBH:3989, 10.26138/SXS:BBH:3989v3.0 (2025).
sxs Collaboration, Binary black-hole simulation SXS:BBH:3990, 10.26138/SXS:BBH:3990v3.0 (2025).
sxs Collaboration, Binary black-hole simulation SXS:BBH:3992, 10.26138/SXS:BBH:3992v3.0 (2025).
sxs Collaboration, Binary black-hole simulation SXS:BBH:3993, 10.26138/SXS:BBH:3993v3.0 (2025).

Phys. Rev. Lett. 135, 171401– Published 21 October, 2025

Vol. 135, Iss. 17 — 24 October 2025

Received 23 December 2024
Revised 3 March 2025
Accepted 4 September 2025

DOI: https://doi.org/10.1103/2brx-xnyr

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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Late-Time Tails in Nonlinear Evolutions of Merging Black Holes

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