Abstract
Recent works have suggested that nonlinear (quadratic) effects in black hole perturbation theory may be important for describing a black hole ringdown. We show that the technique of uniform approximations can be used to accurately compute 1) nonlinear amplitudes at large distances in terms of the linear ones, 2) linear (and nonlinear) quasi-normal mode frequencies, 3) the wavefunction for both linear and nonlinear modes. Our method can be seen as a generalization of the WKB approximation, with the advantages of not losing accuracy at large overtone number and not requiring matching conditions. To illustrate the effectiveness of this method we consider a simplified source for the second-order Zerilli equation, which we use to numerically compute the amplitude of nonlinear modes for a range of values of the angular momentum number.
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References
E. Berti, V. Cardoso and A.O. Starinets, Quasinormal modes of black holes and black branes, Class. Quant. Grav. 26 (2009) 163001 [arXiv:0905.2975] [INSPIRE].
S. Chandrasekhar and S.L. Detweiler, The quasi-normal modes of the Schwarzschild black hole, Proc. Roy. Soc. Lond. A 344 (1975) 441 [INSPIRE].
E.W. Leaver, Spectral decomposition of the perturbation response of the Schwarzschild geometry, Phys. Rev. D 34 (1986) 384 [INSPIRE].
A. Buonanno, G.B. Cook and F. Pretorius, Inspiral, merger and ring-down of equal-mass black-hole binaries, Phys. Rev. D 75 (2007) 124018 [gr-qc/0610122] [INSPIRE].
E. Berti, V. Cardoso, J.A. Gonzalez and U. Sperhake, Mining information from binary black hole mergers: A Comparison of estimation methods for complex exponentials in noise, Phys. Rev. D 75 (2007) 124017 [gr-qc/0701086] [INSPIRE].
E.W. Leaver, An Analytic representation for the quasi normal modes of Kerr black holes, Proc. Roy. Soc. Lond. A 402 (1985) 285 [INSPIRE].
S. Iyer and C.M. Will, Black Hole Normal Modes: A WKB Approach. 1. Foundations and Application of a Higher Order WKB Analysis of Potential Barrier Scattering, Phys. Rev. D 35 (1987) 3621 [INSPIRE].
S. Iyer and C.M. Will, Black hole normal modes: a semianalytic approach. 1. Foundations, Report number: Print-86-0935, Washington University, St. Louis, U.S.A. (1986).
B.F. Schutz and C.M. Will, Black hole normal modes: a semianalytic approach, Astrophys. J. Lett. 291 (1985) L33 [INSPIRE].
R.A. Konoplya, Quasinormal behavior of the d-dimensional Schwarzschild black hole and higher order WKB approach, Phys. Rev. D 68 (2003) 024018 [gr-qc/0303052] [INSPIRE].
J. Matyjasek and M. Opala, Quasinormal modes of black holes. The improved semianalytic approach, Phys. Rev. D 96 (2017) 024011 [arXiv:1704.00361] [INSPIRE].
G. Franciolini, L. Hui, R. Penco, L. Santoni and E. Trincherini, Effective Field Theory of Black Hole Quasinormal Modes in Scalar-Tensor Theories, JHEP 02 (2019) 127 [arXiv:1810.07706] [INSPIRE].
L. Hui, A. Podo, L. Santoni and E. Trincherini, An analytic approach to quasinormal modes for coupled linear systems, JHEP 03 (2023) 060 [arXiv:2210.10788] [INSPIRE].
Y. Hatsuda and M. Kimura, Perturbative quasinormal mode frequencies, arXiv:2307.16626 [INSPIRE].
L. Motl and A. Neitzke, Asymptotic black hole quasinormal frequencies, Adv. Theor. Math. Phys. 7 (2003) 307 [hep-th/0301173] [INSPIRE].
M. Ansorg and R. Panosso Macedo, Spectral decomposition of black-hole perturbations on hyperboloidal slices, Phys. Rev. D 93 (2016) 124016 [arXiv:1604.02261] [INSPIRE].
J.L. Ripley, Computing the quasinormal modes and eigenfunctions for the Teukolsky equation using horizon penetrating, hyperboloidally compactified coordinates, Class. Quant. Grav. 39 (2022) 145009 [arXiv:2202.03837] [INSPIRE].
G. Aminov, A. Grassi and Y. Hatsuda, Black Hole Quasinormal Modes and Seiberg-Witten Theory, Ann. Henri Poincare 23 (2022) 1951 [arXiv:2006.06111] [INSPIRE].
G. Bonelli, C. Iossa, D.P. Lichtig and A. Tanzini, Exact solution of Kerr black hole perturbations via CFT2 and instanton counting: Greybody factor, quasinormal modes, and Love numbers, Phys. Rev. D 105 (2022) 044047 [arXiv:2105.04483] [INSPIRE].
G. Bonelli, C. Iossa, D. Panea Lichtig and A. Tanzini, Irregular Liouville Correlators and Connection Formulae for Heun Functions, Commun. Math. Phys. 397 (2023) 635 [arXiv:2201.04491] [INSPIRE].
G. Aminov, P. Arnaudo, G. Bonelli, A. Grassi and A. Tanzini, Black hole perturbation theory and multiple polylogarithms, JHEP 11 (2023) 059 [arXiv:2307.10141] [INSPIRE].
Y. Hatsuda and M. Kimura, Spectral Problems for Quasinormal Modes of Black Holes, Universe 7 (2021) 476 [arXiv:2111.15197] [INSPIRE].
M.V. Berry and K.E. Mount, Semiclassical approximations in wave mechanics, Rept. Prog. Phys. 35 (1972) 315 [INSPIRE].
M. Mariño, Advanced Topics in Quantum Mechanics, Cambridge University Press (2021), https://doi.org/10.1017/9781108863384 [INSPIRE].
S. Ma et al., Quasinormal-mode filters: A new approach to analyze the gravitational-wave ringdown of binary black-hole mergers, Phys. Rev. D 106 (2022) 084036 [arXiv:2207.10870] [INSPIRE].
L. London, D. Shoemaker and J. Healy, Modeling ringdown: Beyond the fundamental quasinormal modes, Phys. Rev. D 90 (2014) 124032 [arXiv:1404.3197] [Erratum ibid. 94 (2016) 069902] [INSPIRE].
K. Mitman et al., Nonlinearities in Black Hole Ringdowns, Phys. Rev. Lett. 130 (2023) 081402 [arXiv:2208.07380] [INSPIRE].
M.H.-Y. Cheung et al., Nonlinear Effects in Black Hole Ringdown, Phys. Rev. Lett. 130 (2023) 081401 [arXiv:2208.07374] [INSPIRE].
H. Nakano and K. Ioka, Second Order Quasi-Normal Mode of the Schwarzschild Black Hole, Phys. Rev. D 76 (2007) 084007 [arXiv:0708.0450] [INSPIRE].
K. Ioka and H. Nakano, Second and higher-order quasi-normal modes in binary black hole mergers, Phys. Rev. D 76 (2007) 061503 [arXiv:0704.3467] [INSPIRE].
M. Lagos and L. Hui, Generation and propagation of nonlinear quasinormal modes of a Schwarzschild black hole, Phys. Rev. D 107 (2023) 044040 [arXiv:2208.07379] [INSPIRE].
D. Brizuela, J.M. Martin-Garcia and G.A. Mena Marugan, Second and higher-order perturbations of a spherical spacetime, Phys. Rev. D 74 (2006) 044039 [gr-qc/0607025] [INSPIRE].
D. Brizuela, J.M. Martin-Garcia and M. Tiglio, A Complete gauge-invariant formalism for arbitrary second-order perturbations of a Schwarzschild black hole, Phys. Rev. D 80 (2009) 024021 [arXiv:0903.1134] [INSPIRE].
D. Brizuela, J.M. Martin-Garcia and G.A.M. Marugan, High-order gauge-invariant perturbations of a spherical spacetime, Phys. Rev. D 76 (2007) 024004 [gr-qc/0703069] [INSPIRE].
N. Andersson, Evolving test fields in a black hole geometry, Phys. Rev. D 55 (1997) 468 [gr-qc/9607064] [INSPIRE].
S. Okuzumi, K. Ioka and M.-a. Sakagami, Possible Discovery of Nonlinear Tail and Quasinormal Modes in Black Hole Ringdown, Phys. Rev. D 77 (2008) 124018 [arXiv:0803.0501] [INSPIRE].
G. Carullo and M. De Amicis, Late-time tails in nonlinear evolutions of merging black hole binaries, arXiv:2310.12968 [INSPIRE].
E. Berti, V. Cardoso and C.M. Will, On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA, Phys. Rev. D 73 (2006) 064030 [gr-qc/0512160] [INSPIRE].
A. Kehagias, D. Perrone, A. Riotto and F. Riva, Explaining nonlinearities in black hole ringdowns from symmetries, Phys. Rev. D 108 (2023) L021501 [arXiv:2301.09345] [INSPIRE].
J. Redondo-Yuste, G. Carullo, J.L. Ripley, E. Berti and V. Cardoso, Spin dependence of black hole ringdown nonlinearities, arXiv:2308.14796 [INSPIRE].
A. Spiers, A. Pound and B. Wardell, Second-order perturbations of the Schwarzschild spacetime: practical, covariant and gauge-invariant formalisms, arXiv:2306.17847 [INSPIRE].
R.J. Gleiser, C.O. Nicasio, R.H. Price and J. Pullin, Second order perturbations of a Schwarzschild black hole, Class. Quant. Grav. 13 (1996) L117 [gr-qc/9510049] [INSPIRE].
R.J. Gleiser, C.O. Nicasio, R.H. Price and J. Pullin, Gravitational radiation from Schwarzschild black holes: The Second order perturbation formalism, Phys. Rept. 325 (2000) 41 [gr-qc/9807077] [INSPIRE].
D. Perrone, T. Barreira, A. Kehagias and A. Riotto, Non-linear Black Hole Ringdowns: an Analytical Approach, arXiv:2308.15886 [INSPIRE].
R. Pan et al., Uniform Asymptotic Approximation Method with Pöschl-Teller Potential, arXiv:2309.03327 [INSPIRE].
T. Regge and J.A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108 (1957) 1063 [INSPIRE].
F.J. Zerilli, Gravitational field of a particle falling in a Schwarzschild geometry analyzed in tensor harmonics, Phys. Rev. D 2 (1970) 2141 [INSPIRE].
S.C. Miller and R.H. Good, A wkb-type approximation to the Schrödinger equation, Phys. Rev. 91 (1953) 174.
R. Dingle, The method of comparison equations in the solution of linear second-order differential equations (generalized WKB method), Appl. Sci. Res. B 5 (1956) 345.
H. Moriguchi, An improvement of the wkb method in the presence of turning points and the asymptotic solutions of a class of hill equations, J. Phys. Soc. Jpn. 14 (1959) 1771.
P. Pechukas, Analysis of the miller-good method for approximating bound states, J. Chem. Phys. 54 (1971) 3864.
C.E. Hecht and J.E. Mayer, Extension of the WKB Equation, Phys. Rev. 106 (1957) 1156 [INSPIRE].
BHPToolkit Development Team, BHPToolkit: Black Hole Perturbation Toolkit, http://bhptoolkit.org/.
R.A. Konoplya, A. Zhidenko and A.F. Zinhailo, Higher order WKB formula for quasinormal modes and grey-body factors: recipes for quick and accurate calculations, Class. Quant. Grav. 36 (2019) 155002 [arXiv:1904.10333] [INSPIRE].
https://centra.tecnico.ulisboa.pt/network/grit/files/ringdown/.
P. Anninos, D. Hobill, E. Seidel, L. Smarr and W.-M. Suen, The Headon collision of two equal mass black holes, Phys. Rev. D 52 (1995) 2044 [gr-qc/9408041] [INSPIRE].
S.L. Detweiler, Black Holes and Gravitational Waves. I. Circular Orbits About a Rotating Hole, Astrophys. J. 225 (1978) 687 [INSPIRE].
E. Berti et al., Inspiral, merger and ringdown of unequal mass black hole binaries: A Multipolar analysis, Phys. Rev. D 76 (2007) 064034 [gr-qc/0703053] [INSPIRE].
V. Baibhav and E. Berti, Multimode black hole spectroscopy, Phys. Rev. D 99 (2019) 024005 [arXiv:1809.03500] [INSPIRE].
S.A. Hughes, A. Apte, G. Khanna and H. Lim, Learning about black hole binaries from their ringdown spectra, Phys. Rev. Lett. 123 (2019) 161101 [arXiv:1901.05900] [INSPIRE].
E. Berti, V. Cardoso and C.M. Will, On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA, Phys. Rev. D 73 (2006) 064030 [gr-qc/0512160] [INSPIRE].
T. Robson, N.J. Cornish and C. Liu, The construction and use of LISA sensitivity curves, Class. Quant. Grav. 36 (2019) 105011 [arXiv:1803.01944] [INSPIRE].
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Bucciotti, B., Kuntz, A., Serra, F. et al. Nonlinear quasi-normal modes: uniform approximation. J. High Energ. Phys. 2023, 48 (2023). https://doi.org/10.1007/JHEP12(2023)048
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DOI: https://doi.org/10.1007/JHEP12(2023)048