- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Excitation factors for horizonless compact objects: Long-lived modes, echoes, and greybody factors

Romeo Felice Rosato1, Shauvik Biswas2, Sumanta Chakraborty3, and Paolo Pani1

PDF

Phys. Rev. D 113, 084002 – Published 1 April, 2026

DOI: https://doi.org/10.1103/v7pv-y3qy

Abstract

We present an analytical and numerical investigation of the quasinormal excitation factors of ultracompact horizonless objects. These systems possess long-lived quasinormal modes with extremely small imaginary parts, originating from the effective cavity between the photon sphere and the object’s interior. We show that the excitation of such modes is strongly suppressed, scaling with the imaginary part of their frequency, and therefore they contribute to the waveform only at very late times. This hierarchy naturally explains the structure of echo signals; the prompt ringdown is dominated by standard light-ring modes, the early echoes arise from moderately damped cavity modes, and only the latest echoes are governed by long-lived modes. Building on this, we propose a practical ringdown waveform model that combines ordinary black-hole quasinormal modes with cavity modes, capturing the complexity of the ringdown of horizonless ultracompact objects. We further demonstrate that the combination of small excitation factors and weak damping enhances the robustness of long-lived modes against localized perturbations, in contrast to the spectral instabilities affecting standard black-hole quasinormal modes. Finally, we extend the analysis of greybody factors to exotic compact objects and wormholes, showing that they remain stable under small deformations of the effective potential and thus represent robust observables. Our results provide a unified framework for understanding excitation, stability, and echoes in ultracompact horizonless objects, with direct implications for their spectral properties and gravitational-wave signatures.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (134)

  1. T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957).
  2. S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, Oxford, 1985).
  3. E. Berti et al., arXiv:2505.23895.
  4. G. Carullo, Gen. Relativ. Gravit. 57, 76 (2025).
  5. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 116, 061102 (2016).
  6. R. Abbott et al. (LIGO Scientific and Virgo Collaborations), Phys. Rev. Lett. 125, 101102 (2020).
  7. A. G. Abac et al. (LIGO Scientific, KAGRA, and Virgo Collaborations), Phys. Rev. Lett. 135, 111403 (2025).
  8. A. G. Abac et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Phys. Rev. Lett. 136, 041403 (2026).
  9. O. Dreyer, B. J. Kelly, B. Krishnan, L. S. Finn, D. Garrison, and R. Lopez-Aleman, Classical Quantum Gravity 21, 787 (2004).
  10. S. L. Detweiler, Astrophys. J. 239, 292 (1980).
  11. E. Berti, V. Cardoso, and C. M. Will, Phys. Rev. D 73, 064030 (2006).
  12. S. Gossan, J. Veitch, and B. S. Sathyaprakash, Phys. Rev. D 85, 124056 (2012).
  13. R. Abbott et al. (LIGO Scientific, Virgo, and KAGRA Collaborations), Phys. Rev. D 112, 084080 (2025).
  14. E. Berti et al., Classical Quantum Gravity 32, 243001 (2015).
  15. E. Berti, K. Yagi, H. Yang, and N. Yunes, Gen. Relativ. Gravit. 50, 49 (2018).
  16. V. Cardoso and P. Pani, Living Rev. Relativity 22, 4 (2019).
  17. M. Colpi et al. (LISA Collaboration), arXiv:2402.07571.
  18. M. Maggiore et al. (ET Collaboration), J. Cosmol. Astropart. Phys. 03 (2020) 050.
  19. V. Kalogera et al., arXiv:2111.06990.
  20. S. Hild et al., Classical Quantum Gravity 28, 094013 (2011).
  21. M. Branchesi et al., J. Cosmol. Astropart. Phys. 07 (2023) 068.
  22. A. Abac et al., arXiv:2503.12263.
  23. B. P. Abbott et al. (LIGO Scientific Collaboration), Classical Quantum Gravity 34, 044001 (2017).
  24. R. Essick, S. Vitale, and M. Evans, Phys. Rev. D 96, 084004 (2017).
  25. M. Evans et al., arXiv:2306.13745.
  26. E. Berti, A. Sesana, E. Barausse, V. Cardoso, and K. Belczynski, Phys. Rev. Lett. 117, 101102 (2016).
  27. S. Bhagwat, C. Pacilio, E. Barausse, and P. Pani, Phys. Rev. D 105, 124063 (2022).
  28. S. Bhagwat, C. Pacilio, P. Pani, and M. Mapelli, Phys. Rev. D 108, 043019 (2023).
  29. C. V. Vishveshwara, Nature (London) 227, 936 (1970).
  30. K. D. Kokkotas and B. G. Schmidt, Living Rev. Relativity 2, 2 (1999).
  31. E. Berti, V. Cardoso, and A. O. Starinets, Classical Quantum Gravity 26, 163001 (2009).
  32. R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys. 83, 793 (2011).
  33. M. Isi, M. Giesler, W. M. Farr, M. A. Scheel, and S. A. Teukolsky, Phys. Rev. Lett. 123, 111102 (2019).
  34. N. Franchini and S. H. Völkel, arXiv:2305.01696.
  35. E. Maggio, L. Buoninfante, A. Mazumdar, and P. Pani, Phys. Rev. D 102, 064053 (2020).
  36. E. Maggio, P. Pani, and G. Raposo, arXiv:2105.06410.
  37. E. Maggio, Lect. Notes Phys. 1017, 333 (2023).
  38. E. Barausse, V. Cardoso, and P. Pani, Phys. Rev. D 89, 104059 (2014).
  39. V. Cardoso, K. Destounis, F. Duque, R. P. Macedo, and A. Maselli, Phys. Rev. D 105, L061501 (2022).
  40. V. Cardoso, K. Destounis, F. Duque, R. Panosso Macedo, and A. Maselli, Phys. Rev. Lett. 129, 241103 (2022).
  41. K. Destounis, A. Kulathingal, K. D. Kokkotas, and G. O. Papadopoulos, Phys. Rev. D 107, 084027 (2023).
  42. E. Barausse, V. Cardoso, and P. Pani, J. Phys. Conf. Ser. 610, 012044 (2015).
  43. M. H.-Y. Cheung, K. Destounis, R. P. Macedo, E. Berti, and V. Cardoso, Phys. Rev. Lett. 128, 111103 (2022).
  44. E. Berti, V. Cardoso, M. H.-Y. Cheung, F. Di Filippo, F. Duque, P. Martens, and S. Mukohyama, Phys. Rev. D 106, 084011 (2022).
  45. S. Biswas, C. Singha, and S. Chakraborty, Phys. Rev. D 109, 064043 (2024).
  46. C. Singha and S. Biswas, Phys. Rev. D 109, 024043 (2024).
  47. V. Cardoso, E. Franzin, and P. Pani, Phys. Rev. Lett. 116, 171101 (2016); 117, 089902(E) (2016).
  48. V. Cardoso, S. Hopper, C. F. B. Macedo, C. Palenzuela, and P. Pani, Phys. Rev. D 94, 084031 (2016).
  49. V. Cardoso and P. Pani, Nat. Astron. 1, 586 (2017).
  50. J. Abedi, N. Afshordi, N. Oshita, and Q. Wang, Universe 6, 43 (2020).
  51. H.-P. Nollert, Phys. Rev. D 53, 4397 (1996).
  52. R. G. Daghigh, M. D. Green, and J. C. Morey, Phys. Rev. D 101, 104009 (2020).
  53. J. L. Jaramillo, R. Panosso Macedo, and L. Al Sheikh, Phys. Rev. X 11, 031003 (2021).
  54. K. Destounis, R. P. Macedo, E. Berti, V. Cardoso, and J. L. Jaramillo, Phys. Rev. D 104, 084091 (2021).
  55. E. Gasperin and J. L. Jaramillo, Classical Quantum Gravity 39, 115010 (2022).
  56. V. Boyanov, K. Destounis, R. Panosso Macedo, V. Cardoso, and J. L. Jaramillo, Phys. Rev. D 107, 064012 (2023).
  57. J. L. Jaramillo, Classical Quantum Gravity 39, 217002 (2022).
  58. S. Sarkar, M. Rahman, and S. Chakraborty, Phys. Rev. D 108, 104002 (2023).
  59. K. Destounis, V. Boyanov, and R. Panosso Macedo, Phys. Rev. D 109, 044023 (2024).
  60. D. Areán, D. G. Fariña, and K. Landsteiner, J. High Energy Phys. 12 (2023) 187.
  61. B. Cownden, C. Pantelidou, and M. Zilhão, J. High Energy Phys. 05 (2024) 202.
  62. K. Destounis and F. Duque, arXiv:2308.16227.
  63. A. Courty, K. Destounis, and P. Pani, Phys. Rev. D 108, 104027 (2023).
  64. V. Boyanov, V. Cardoso, K. Destounis, J. L. Jaramillo, and R. Panosso Macedo, Phys. Rev. D 109, 064068 (2024).
  65. L.-M. Cao, J.-N. Chen, L.-B. Wu, L. Xie, and Y.-S. Zhou, Sci. China Phys. Mech. Astron. 67, 100412 (2024).
  66. V. Cardoso, S. Kastha, and R. Panosso Macedo, Phys. Rev. D 110, 024016 (2024).
  67. A. Ianniccari, A. J. Iovino, A. Kehagias, P. Pani, G. Perna, D. Perrone, and A. Riotto, Phys. Rev. Lett. 133, 211401 (2024).
  68. R.-G. Cai, L.-M. Cao, J.-N. Chen, Z.-K. Guo, L.-B. Wu, and Y.-S. Zhou, Phys. Rev. D 111, 084011 (2025).
  69. R. J. Gleiser, C. O. Nicasio, R. H. Price, and J. Pullin, Classical Quantum Gravity 13, L117 (1996).
  70. R. J. Gleiser, C. O. Nicasio, R. H. Price, and J. Pullin, Phys. Rep. 325, 41 (2000).
  71. K. Ioka and H. Nakano, Phys. Rev. D 76, 061503 (2007).
  72. H. Nakano and K. Ioka, Phys. Rev. D 76, 084007 (2007).
  73. D. Brizuela, J. M. Martin-Garcia, and M. Tiglio, Phys. Rev. D 80, 024021 (2009).
  74. E. Pazos, D. Brizuela, J. M. Martin-Garcia, and M. Tiglio, Phys. Rev. D 82, 104028 (2010).
  75. J. L. Ripley, N. Loutrel, E. Giorgi, and F. Pretorius, Phys. Rev. D 103, 104018 (2021).
  76. N. Loutrel, J. L. Ripley, E. Giorgi, and F. Pretorius, Phys. Rev. D 103, 104017 (2021).
  77. L. Sberna, P. Bosch, W. E. East, S. R. Green, and L. Lehner, Phys. Rev. D 105, 064046 (2022).
  78. M. H.-Y. Cheung et al., Phys. Rev. Lett. 130, 081401 (2023).
  79. K. Mitman et al., Phys. Rev. Lett. 130, 081402 (2023).
  80. A. Kehagias, D. Perrone, A. Riotto, and F. Riva, Phys. Rev. D 108, L021501 (2023).
  81. D. Perrone, T. Barreira, A. Kehagias, and A. Riotto, Nucl. Phys. B999, 116432 (2024).
  82. M. H.-Y. Cheung, E. Berti, V. Baibhav, and R. Cotesta, Phys. Rev. D 109, 044069 (2024); 110, 049902(E) (2024).
  83. J. Redondo-Yuste, D. Pereñiguez, and V. Cardoso, Phys. Rev. D 109, 044048 (2024).
  84. J. Redondo-Yuste, G. Carullo, J. L. Ripley, E. Berti, and V. Cardoso, Phys. Rev. D 109, L101503 (2024).
  85. S. Yi, A. Kuntz, E. Barausse, E. Berti, M. H.-Y. Cheung, K. Kritos, and A. Maselli, Phys. Rev. D 109, 124029 (2024).
  86. H. Zhu et al., Phys. Rev. D 110, 124028 (2024).
  87. H. Zhu et al., Phys. Rev. D 109, 104050 (2024).
  88. M. Giesler, M. Isi, M. A. Scheel, and S. Teukolsky, Phys. Rev. X 9, 041060 (2019).
  89. S. Bhagwat, X. J. Forteza, P. Pani, and V. Ferrari, Phys. Rev. D 101, 044033 (2020).
  90. V. Baibhav, M. H.-Y. Cheung, E. Berti, V. Cardoso, G. Carullo, R. Cotesta, W. Del Pozzo, and F. Duque, Phys. Rev. D 108, 104020 (2023).
  91. R. H. Price, Phys. Rev. D 5, 2439 (1972).
  92. C. Gundlach, R. H. Price, and J. Pullin, Phys. Rev. D 49, 883 (1994).
  93. L. Barack, Phys. Rev. D 59, 044017 (1999).
  94. M. De Amicis, S. Albanesi, and G. Carullo, Phys. Rev. D 110, 104005 (2024).
  95. M. De Amicis et al., Phys. Rev. Lett. 135, 171401 (2025).
  96. R. F. Rosato and P. Pani, Phys. Rev. D 112, 024080 (2025).
  97. N. Oshita, J. Cosmol. Astropart. Phys. 04 (2023) 013.
  98. N. Oshita, Phys. Rev. D 109, 104028 (2024).
  99. K. Okabayashi and N. Oshita, Phys. Rev. D 110, 064086 (2024).
  100. S. W. Hawking, Commun. Math. Phys. 43, 199 (1975); 46, 206(E) (1976).
  101. R. F. Rosato, K. Destounis, and P. Pani, Phys. Rev. D 110, L121501 (2024).
  102. N. Oshita, K. Takahashi, and S. Mukohyama, Phys. Rev. D 110, 084070 (2024).
  103. C. Bambi et al., arXiv:2505.09014.
  104. Z. Mark, A. Zimmerman, S. M. Du, and Y. Chen, Phys. Rev. D 96, 084002 (2017).
  105. E. Maggio, P. Pani, and V. Ferrari, Phys. Rev. D 96, 104047 (2017).
  106. J. Abedi, H. Dykaar, and N. Afshordi, Phys. Rev. D 96, 082004 (2017).
  107. N. Siemonsen, Phys. Rev. Lett. 133, 031401 (2024).
  108. P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, Laser Photonics Rev. 12, 1700113 (2018).
  109. L. A. Sheikh, Scattering resonances and pseudospectrum: stability and completeness aspects in optical and gravitational systems, Ph.D. thesis, Institut de Mathématiques de Bourgogne [Dijon], France, 2022.
  110. M. Visser, Lorentzian Wormholes: From Einstein to Hawking (American Institute of Physics (AIP) Press, Woodbury, 1995).
  111. N. Oshita, Q. Wang, and N. Afshordi, J. Cosmol. Astropart. Phys. 04 (2020) 016.
  112. E. W. Leaver, Phys. Rev. D 34, 384 (1986).
  113. E. Berti and V. Cardoso, Phys. Rev. D 74, 104020 (2006).
  114. H. O. Silva, G. Tambalo, K. Glampedakis, K. Yagi, and J. Steinhoff, Phys. Rev. D 110, 024042 (2024).
  115. M. Mirbabayi, J. Cosmol. Astropart. Phys. 01 (2020) 052.
  116. K. Kyutoku, H. Motohashi, and T. Tanaka, Phys. Rev. D 107, 044012 (2023).
  117. K. Destounis, M. Malato Corrêa, C. F. B. Macedo, and R. Panosso Macedo, Phys. Rev. D 113, 044025 (2026).
  118. R. F. Rosato, S. Biswas, S. Chakraborty, and P. Pani, Phys. Rev. D 111, 084051 (2025).
  119. V. Cardoso, L. C. B. Crispino, C. F. B. Macedo, H. Okawa, and P. Pani, Phys. Rev. D 90, 044069 (2014).
  120. P. Bueno, P. A. Cano, F. Goelen, T. Hertog, and B. Vercnocke, Phys. Rev. D 97, 024040 (2018).
  121. S. Chakraborty, E. Maggio, A. Mazumdar, and P. Pani, Phys. Rev. D 106, 024041 (2022).
  122. A. A. Starobinskil and S. M. Churilov, Sov. Phys. JETP 65, 1 (1974), https://inspirehep.net/literature/1594792.
  123. E. Maggio, V. Cardoso, S. R. Dolan, and P. Pani, Phys. Rev. D 99, 064007 (2019).
  124. M. Zworski, Not. Am. Math. Soc. 46, 319 (1999).
  125. R. A. Konoplya and A. Zhidenko, J. Cosmol. Astropart. Phys. 12 (2016) 043.
  126. S. Nair, arXiv:2509.09986.
  127. N. Oshita, E. Berti, and V. Cardoso, Phys. Rev. Lett. 135, 031401 (2025).
  128. A. Maselli, S. H. Völkel, and K. D. Kokkotas, Phys. Rev. D 96, 064045 (2017).
  129. J. L. Jaramillo, R. Panosso Macedo, and L. A. Sheikh, Phys. Rev. Lett. 128, 211102 (2022).
  130. C. Warnick, arXiv:2407.19850.
  131. M. Hitrik, J. Sjöstrand, and M. Vogel, arXiv:2505.07292.
  132. T. Torres, Phys. Rev. Lett. 131, 111401 (2023).
  133. P. Pani, Int. J. Mod. Phys. A 28, 1340018 (2013).
  134. R. Brito, V. Cardoso, and P. Pani, Lect. Notes Phys. 906, 1 (2015).

Outline

Information

Phys. Rev. D 113, 084002– Published 1 April, 2026
  • Received 21 November 2025
  • Accepted 20 February 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation