Dynamical tidal response of neutron stars via scattering amplitudes

Author(s)

Saketh, M.V.S., Ghosh, Suprovo, Andersson, Nils

Abstract

A key challenge of gravitational-wave physics is distinguishing the nature of compact objects involved in binary coalescences, particularly whether they are black holes or neutron stars. Neutron stars are distinguished from black holes by a stronger tidal response, with both static and dynamical aspects directly linked to their rich internal physics. Measurements of the tidal response through gravitational observations constrains the neutron-star equation of state and provides insight into the physics of high-density matter. However, defining the tidal response of neutron stars in general relativity is challenging due to coordinate ambiguities and the complexity of connecting the star's response to binary dynamics and the associated gravitational waveforms. In this paper, we show how the dynamical tidal response of a neutron star can be systematically defined within the worldline effective field theory (EFT) framework, connecting the problem to gravitational-wave scattering off an isolated neutron star. These scattering amplitudes are computed both within the EFT, using standard quantum field-theory techniques, and within stellar perturbation theory (the corresponding ultraviolet theory), where the coupled metric and matter perturbation equations are solved in the stellar interior within general relativity and matched to the analytical Mano-Suzuki-Takasugi (MST) solutions in the exterior. We match the scattering amplitude between effective theory and the ultraviolet theory to obtain the dynamical tidal response. We show the result to be consistent with known expectations, such as the static limit and the behaviour near the neutron star's resonant modes, while also recovering the imaginary part of the dominant oscillation mode induced by gravitational-wave dissipation. We conclude with a discussion of potential future improvements within both the EFT and the perturbation theory.

Figures

Some of the Feynman rules for WEFT in momentum space.
Caption Some of the Feynman rules for WEFT in momentum space.
Tree level, tidal contribution to Raman scattering
Caption Tree level, tidal contribution to Raman scattering
Tree level, non-tidal contributions to Raman scattering. The $\otimes$ vertex is due to the recoil term in Eq.~\eqref{eq:action_expanded}, and vanishes in traceless transverse gauge. However, more generally, it is essential to preserve gauge invariance.
Caption Tree level, non-tidal contributions to Raman scattering. The $\otimes$ vertex is due to the recoil term in Eq.~\eqref{eq:action_expanded}, and vanishes in traceless transverse gauge. However, more generally, it is essential to preserve gauge invariance.
Additional tidal contributions to the scattering process that do not involve mass insertions such as those in Fig.~\ref{fig:treeMQQ}. These diagrams effectively resum the result of Fig.~\ref{fig:treeQQ}, and are required to account for backreaction due to gravitational radiation.
Caption Additional tidal contributions to the scattering process that do not involve mass insertions such as those in Fig.~\ref{fig:treeMQQ}. These diagrams effectively resum the result of Fig.~\ref{fig:treeQQ}, and are required to account for backreaction due to gravitational radiation.
The simplest diagrams at linear order in $M$. The loop leads to an infrared divergence as well, which does not show up in any observables. The labels for incident and outgoing momenta have been suppressed to avoid cluttering.
Caption The simplest diagrams at linear order in $M$. The loop leads to an infrared divergence as well, which does not show up in any observables. The labels for incident and outgoing momenta have been suppressed to avoid cluttering.
Additional contributions to $T_{\rm tid,(1)}$ involving one mass insertion.
Caption Additional contributions to $T_{\rm tid,(1)}$ involving one mass insertion.
Illustrating the dynamical tidal response near resonant frequencies; in this case the highest frequency g-modes for the chosen BSk model. The tidal response obtained from Eq.~\eqref{eq:dynamical_tides} is labeled ``$k_2^{\rm EFT}(\omega)$'', while that used in Ref.~\cite{Andersson:2025iyd} is labeled ``Andersson. et. al''. Given that the two sets of results are very close, the considerably more advanced formalism we have brought appears to have only a modest effect on the final result near the g-modes.
Caption Illustrating the dynamical tidal response near resonant frequencies; in this case the highest frequency g-modes for the chosen BSk model. The tidal response obtained from Eq.~\eqref{eq:dynamical_tides} is labeled ``$k_2^{\rm EFT}(\omega)$'', while that used in Ref.~\cite{Andersson:2025iyd} is labeled ``Andersson. et. al''. Given that the two sets of results are very close, the considerably more advanced formalism we have brought appears to have only a modest effect on the final result near the g-modes.
A zoomed in plot providing a detailed comparison near the dominant f-mode. Both tidal responses roughly show resonant behaviour close to the QNM frequency. The new result seems to align slightly more closely with the f-mode, but this shift is quite negligible.
Caption A zoomed in plot providing a detailed comparison near the dominant f-mode. Both tidal responses roughly show resonant behaviour close to the QNM frequency. The new result seems to align slightly more closely with the f-mode, but this shift is quite negligible.
A zoomed in plot providing a detailed comparison near the dominant f-mode for $M=1.98M\odot$, and $R=12.59$~km. Note that $k_2(\omega)$ shows resonance behaviour very close to the expected f-mode frequency (indicated by the dashed, vertical red line) whereas the earlier result in Ref.~\cite{Andersson:2025iyd} shows a slight offset towards higher frequencies.
Caption A zoomed in plot providing a detailed comparison near the dominant f-mode for $M=1.98M\odot$, and $R=12.59$~km. Note that $k_2(\omega)$ shows resonance behaviour very close to the expected f-mode frequency (indicated by the dashed, vertical red line) whereas the earlier result in Ref.~\cite{Andersson:2025iyd} shows a slight offset towards higher frequencies.
Low-frequency behaviour of the tidal response for the BSk22 equation of state and two stellar models. The top panel shows results for a star with $M=1.4M_\odot$ and $R=13.04~\mathrm{km}$ while the bottom panel represents a model with $M=1.98M_\odot$ and $R=12.59~\mathrm{km}$. The results show that the new model has significantly improved on the systematic error discussed in~\cite{Andersson:2025iyd}. It is also worth noting the slight wiggles at the lowest frequencies in the top panel. These indicate the low-frequency region where numerically solving the neutron-star perturbation equations is difficult.
Caption Low-frequency behaviour of the tidal response for the BSk22 equation of state and two stellar models. The top panel shows results for a star with $M=1.4M_\odot$ and $R=13.04~\mathrm{km}$ while the bottom panel represents a model with $M=1.98M_\odot$ and $R=12.59~\mathrm{km}$. The results show that the new model has significantly improved on the systematic error discussed in~\cite{Andersson:2025iyd}. It is also worth noting the slight wiggles at the lowest frequencies in the top panel. These indicate the low-frequency region where numerically solving the neutron-star perturbation equations is difficult.
Low-frequency behaviour of the tidal response for the BSk22 equation of state and two stellar models. The top panel shows results for a star with $M=1.4M_\odot$ and $R=13.04~\mathrm{km}$ while the bottom panel represents a model with $M=1.98M_\odot$ and $R=12.59~\mathrm{km}$. The results show that the new model has significantly improved on the systematic error discussed in~\cite{Andersson:2025iyd}. It is also worth noting the slight wiggles at the lowest frequencies in the top panel. These indicate the low-frequency region where numerically solving the neutron-star perturbation equations is difficult.
Caption Low-frequency behaviour of the tidal response for the BSk22 equation of state and two stellar models. The top panel shows results for a star with $M=1.4M_\odot$ and $R=13.04~\mathrm{km}$ while the bottom panel represents a model with $M=1.98M_\odot$ and $R=12.59~\mathrm{km}$. The results show that the new model has significantly improved on the systematic error discussed in~\cite{Andersson:2025iyd}. It is also worth noting the slight wiggles at the lowest frequencies in the top panel. These indicate the low-frequency region where numerically solving the neutron-star perturbation equations is difficult.
References
  • [1] LIGO Scientific Collaboration and Virgo Collaboration. Properties of the binary neutron star merger gw170817. Physical Review X, 9(1), January 2019. Publisher Copyright: © 2019 authors. Published by the American Physical Society.
  • [2] B. P. Abbott et al. Gw170817: Observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett., 119:161101, Oct 2017.
  • [3] M. Soares-Santos et al. The Electromagnetic Counterpart of the Binary Neutron Star Merger LIGO/Virgo GW170817. I. Discovery of the Optical Counterpart Using the Dark Energy Camera. Astrophys. J. Lett., 848(2):L16, 2017.
  • [4] David Reitze et al. Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO. Bull. Am. Astron. Soc., 51(7):035, 2019.
  • [5] Adrian Abac et al. The Science of the Einstein Telescope. JCAP, 03:081, 2026.
  • [6] Tanja Hinderer. Tidal love numbers of neutron stars. The Astrophysical Journal, 677(2):1216–1220, April 2008.
  • [7] B. P. Abbott et al. GW170817: Measurements of neutron star radii and equation of state. Phys. Rev. Lett., 121(16):161101, 2018.
  • [8] Dong Lai. Resonant oscillations and tidal heating in coalescing binary neutron stars. Mon. Not. Roy. Astron. Soc., 270:611, 1994.
  • [9] Tanja Hinderer, Andrea Taracchini, Francois Foucart, Alessandra Buonanno, Jan Steinhoff, Matthew D. Duez, Lawrence E. Kidder, Harald P. Pfeiffer, Mark A. Scheel, Béla Szilágyi, Kenta Hotokezaka, Koutarou Kyutoku, Masaru Shibata, and Cory W Carpenter. Effects of neutron-star dynamic tides on gravitational waveforms within the effective-one-body approach. Physical review letters, 116 18:181101, 2016.
  • [10] Adrian Abac, Tim Dietrich, Alessandra Buonanno, Jan Steinhoff, and Maximiliano Ujevic. New and robust gravitational-waveform model for high-mass-ratio binary neutron star systems with dynamical tidal effects. Phys. Rev. D, 109:024062, Jan 2024.
  • [11] Geraint Pratten, Patricia Schmidt, and Natalie Williams. Impact of dynamical tides on the reconstruction of the neutron star equation of state. Phys. Rev. Lett., 129:081102, Aug 2022.
  • [12] A.R. Counsell, F. Gittins, N. Andersson, and I. Tews. Interface modes in inspiralling neutron stars: A gravitational-wave probe of first-order phase transitions. Physical Review Letters, 135(8), August 2025.
  • [13] Hang Yu and Nevin N. Weinberg. Dynamical tides in coalescing superfluid neutron star binaries with hyperon cores and their detectability with third-generation gravitational-wave detectors. Monthly Notices of the Royal Astronomical Society, 470(1):350–360, May 2017.
  • [14] Fabian Gittins, Harsh Narola, Thibeau Wouters, Peter T. H. Pang, Tanja Hinderer, and Chris Van Den Broeck. Detecting Tidal Resonances in Binary Neutron Stars. 6 2026.
  • [15] Abhishek Hegade K. R., Justin L. Ripley, and Nicolás Yunes. Dynamical tidal response of nonrotating relativistic stars. Phys. Rev. D, 109(10):104064, 2024.
  • [16] Abhishek Hegade K. R., Yumu Yang, Mauricio Hippert, Jacquelyn Noronha-Hostler, Jorge Noronha, and Nicolás Yunes. Dynamical tidal response of neutron stars as a probe of dense-matter properties. 3 2026.
  • [17] Abhishek Hegade K. R., K. J. Kwon, Tejaswi Venumadhav, Hang Yu, and Nicolas Yunes. The Good, the Bad, and the Subtle: Relativistic mode sums for neutron-star tidal response. 5 2026.
  • [18] Nils Andersson, Rhys Counsell, Fabian Gittins, and Suprovo Ghosh. Tidal response of a relativistic star. Phys. Rev. D, 113(6):064051, 2026.
  • [19] Michèle Levi. Effective Field Theories of Post-Newtonian Gravity: A comprehensive review. Rept. Prog. Phys., 83(7):075901, 2020.
  • [20] Rafael A. Porto. The effective field theorist’s approach to gravitational dynamics. Phys. Rept., 633:1–104, 2016.
  • [21] Walter D. Goldberger. Effective field theories of gravity and compact binary dynamics: A snowmass 2021 whitepaper. 2022.
  • [22] Michele Levi and Jan Steinhoff. Spinning gravitating objects in the effective field theory in the post-Newtonian scheme. JHEP, 09:219, 2015.
  • [23] Zhengwen Liu, Rafael A. Porto, and Zixin Yang. Spin Effects in the Effective Field Theory Approach to PostMinkowskian Conservative Dynamics. JHEP, 06:012, 2021.
  • [24] Gihyuk Cho, Rafael A. Porto, and Zixin Yang. Gravitational radiation from inspiralling compact objects: Spin effects to the fourth post-Newtonian order. Phys. Rev. D, 106(10):L101501, 2022.
  • [25] Sylvain Marsat. Cubic order spin effects in the dynamics and gravitational wave energy flux of compact object binaries. Class. Quant. Grav., 32(8):085008, 2015.
  • [26] Walter D. Goldberger and Ira Z. Rothstein. An Effective field theory of gravity for extended objects. Phys. Rev. D, 73:104029, 2006.
  • [27] Manoj Kumar Mandal, Pierpaolo Mastrolia, Hector O. Silva, Raj Patil, and Jan Steinhoff. Renormalizing love: tidal effects at the third post-newtonian order. Journal of High Energy Physics, 2024, 2023.
  • [28] Thibault Damour. Gravitational scattering, postminkowskian approximation, and effective-one-body theory. Phys. Rev. D, 94:104015, Nov 2016.
  • [29] Gregor Kälin and Rafael A. Porto. Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics. JHEP, 11:106, 2020.
  • [30] Rafael Aoude, Kays Haddad, and Andreas Helset. Onshell heavy particle effective theories. JHEP, 05:051, 2020.
  • [31] Mikhail M. Ivanov, Yue-Zhou Li, Julio Parra-Martinez, and Zihan Zhou. Gravitational Raman Scattering in Effective Field Theory: A Scalar Tidal Matching at O(G3). Phys. Rev. Lett., 132(13):131401, 2024. [Erratum: Phys.Rev.Lett. 134, 159901 (2025)].
  • [31] Mikhail M. Ivanov, Yue-Zhou Li, Julio Parra-Martinez, and Zihan Zhou. Gravitational Raman Scattering in Effective Field Theory: A Scalar Tidal Matching at O(G3). Phys. Rev. Lett., 132(13):131401, 2024. [Erratum: Phys.Rev.Lett. 134, 159901 (2025)].
  • [32] Mikhail M. Ivanov and Zihan Zhou. Vanishing of Black Hole Tidal Love Numbers from Scattering Amplitudes. Phys. Rev. Lett., 130(9):091403, 2023.
  • [33] M. V. S. Saketh, Jan Steinhoff, Justin Vines, and Alessandra Buonanno. Modeling horizon absorption in spinning binary black holes using effective worldline theory. Phys. Rev. D, 107(8):084006, 2023.
  • [34] M. V. S. Saketh, Zihan Zhou, and Mikhail M. Ivanov. Dynamical tidal response of Kerr black holes from scattering amplitudes. Phys. Rev. D, 109(6):064058, 2024.
  • [35] Mikhail M. Ivanov, Yue-Zhou Li, Julio Parra-Martinez, and Zihan Zhou. Gravitational Raman Scattering: a Systematic Toolkit for Tidal Effects in General Relativity. 2 2026.
  • [36] M. V. S. Saketh, Zihan Zhou, Suprovo Ghosh, Jan Steinhoff, and Debarati Chatterjee. Investigating tidal heating in neutron stars via gravitational Raman scattering. Phys. Rev. D, 110:103001, 2024.
  • [37] Sayan Chakrabarti, Térence Delsate, and Jan Steinhoff. New perspectives on neutron star and black hole spectroscopy and dynamic tides. 4 2013.
  • [38] S. Detweiler and L. Lindblom. On the nonradial pulsations of general relativistic stellar models. Astrophys. J., 292:12–15, May 1985.
  • [39] Suprovo Ghosh, Bikram Keshari Pradhan, and Debarati Chatterjee. Tidal heating as a direct probe of strangeness inside neutron stars. Phys. Rev. D, 109(10):103036, 2024.
  • [40] Raymond F. Sawyer. Bulk viscosity of hot neutron-star matter and the maximum rotation rates of neutron stars. Phys. Rev. D, 39:3804–3806, Jun 1989.
  • [41] A. R. Counsell, F. Gittins, and N. Andersson. The impact of nuclear reactions on the neutron-star g-mode spectrum. Mon. Not. Roy. Astron. Soc., 531(1):1721–1729, 2024.
  • [42] Rhys Counsell, Fabian Gittins, Nils Andersson, and Pantelis Pnigouras. Neutron star g modes in the relativistic Cowling approximation. Mon. Not. Roy. Astron. Soc., 536(2):1967–1979, 2024.
  • [43] Vinh Tran, Suprovo Ghosh, Nicholas Lozano, Debarati Chatterjee, and Prashanth Jaikumar. g-mode oscillations in neutron stars with hyperons. Phys. Rev. C, 108(1):015803, 2023.
  • [44] C.J. Krüger, W.C.G. Ho, and N. Andersson. Seismology of adolescent neutron stars: Accounting for thermal effects and crust elasticity. Physical Review D, 92(6), 2015.
  • [45] Tullio Regge and John A. Wheeler. Stability of a schwarzschild singularity. Phys. Rev., 108:1063–1069, Nov 1957.
  • [46] Shuhei Mano, Hisao Suzuki, and Eiichi Takasugi. Analytic solutions of the Regge-Wheeler equation and the postMinkowskian expansion. Prog. Theor. Phys., 96:549– 566, 1996.
  • [47] Marc Casals and Adrian C. Ottewill. High-order tail in Schwarzschild spacetime. Phys. Rev. D, 92(12):124055, 2015.
  • [48] Walter D. Goldberger and Ira Z. Rothstein. Dissipative effects in the worldline approach to black hole dynamics. Phys. Rev. D, 73:104030, 2006.
  • [49] Walter D. Goldberger, Jingping Li, and Ira Z. Rothstein. Non-conservative effects on spinning black holes from world-line effective field theory. JHEP, 06:053, 2021.
  • [50] Chih-Hao Chang, Chia-Hsien Shen, and Zihan Zhou. Gravitational Sommerfeld Effects: Formalism, Renormalization, and Perturbation to O(G10 ). 4 2026.
  • [51] Simon Caron-Huot, Miguel Correia, Giulia Isabella, and Mikhail Solon. Gravitational Wave Scattering via the Born Series: Scalar Tidal Matching to O(G7) and Beyond. Phys. Rev. Lett., 135(19):191601, 2025.
  • [52] Taylor Binnington and Eric Poisson. Relativistic theory of tidal love numbers. Phys. Rev. D, 80:084018, 2009.
  • [53] Thibault Damour and Alessandro Nagar. Relativistic tidal properties of neutron stars. Phys. Rev. D, 80:084035, 2009.
  • [54] Kent Yagi and Nicolás Yunes. Approximate universal relations for neutron stars and quark stars. Phys. Rept., 681:1–72, 2017.
  • [55] Andrea Maselli, Vitor Cardoso, Valeria Ferrari, Leonardo Gualtieri, and Paolo Pani. Equation-of-stateindependent relations in neutron stars. Phys. Rev. D, 88:023007, 2013.
  • [56] Walter D. Goldberger and Andreas Ross. Gravitational radiative corrections from effective field theory. Phys. Rev. D, 81:124015, 2010.
  • [57] S. Goriely, N. Chamel, and J. M. Pearson. Hartree-FockBogoliubov nuclear mass model with 0.50 MeV accuracy based on standard forms of Skyrme and pairing functionals. Phys. Rev. C, 88(6):061302, December 2013.
  • [58] J M Pearson, N Chamel, A Y Potekhin, A F Fantina, C Ducoin, A K Dutta, and S Goriely. Unified equations of state for cold non-accreting neutron stars with brussels–montreal functionals – i. role of symmetry energy. Monthly Notices of the Royal Astronomical Society, 481(3):2994–3026, 12 2018.
  • [59] N. N. Shchechilin, N. Chamel, and J. M. Pearson. Unified equations of state for cold nonaccreting neutron stars with brussels-montreal functionals. iv. role of the symmetry energy in pasta phases. Phys. Rev. C, 108:025805, Aug 2023.
  • [60] Lee Lindblom, Gregory Mendell, and James R. Ipser. Relativistic stellar pulsations with near-zone boundary conditions. Phys. Rev. D, 56:2118–2126, Aug 1997.
  • [61] Manoj K. Mandal, Pierpaolo Mastrolia, Hector O. Silva, Raj Patil, and Jan Steinhoff. Renormalizing Love: tidal effects at the third post-Newtonian order. JHEP, 02:188, 2024.
  • [62] Jan Steinhoff, Tanja Hinderer, Alessandra Buonanno, and Andrea Taracchini. Dynamical Tides in General Relativity: Effective Action and Effective-One-Body Hamiltonian. Phys. Rev. D, 94(10):104028, 2016.
  • [63] Gustav Uhre Jakobsen, Gustav Mogull, Jan Plefka, and Benjamin Sauer. Tidal effects and renormalization at fourth post-Minkowskian order. Phys. Rev. D, 109(4):L041504, 2024.
  • [64] Suprovo Ghosh, José Luis Hernández, Bikram Keshari Pradhan, Cristina Manuel, Debarati Chatterjee, and Laura Tolos. Tidal heating in binary inspiral of strange quark stars. Phys. Rev. D, 112(8):084072, 2025.