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Analytic waveform derivatives make Fisher-matrix tests of gravity stable, fast, and able to map how well space- and ground-based detectors constrain non-GR effects by PN order.

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2026-07-11 20:42 UTC pith:SMQGRHC6

load-bearing objection Solid, usable analytic Fisher pipeline for ppE tests; the trends and complementarity claims hold under the stated assumptions. the 2 major comments →

arxiv 2607.04238 v1 pith:SMQGRHC6 submitted 2026-07-05 gr-qc

Tests of general relativity using analytic derivatives of parametrized post-Einsteinian gravitational waveforms within the Fisher-matrix framework

Jie Wu , Mengfei Sun , Jin Li This is my paper
classification gr-qc PACS 04.30.Db04.80.Nn04.50.Kd
keywords parametrized post-EinsteinianFisher matrixanalytic waveform derivativestests of general relativitybinary black holesspace-based detectorsmultiband observationspost-Newtonian order
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that fully analytic derivatives of parametrized post-Einsteinian inspiral waveforms let researchers run Fisher-matrix forecasts without the noise and cost of finite differences. The method is applied across space-based, ground-based, and multiband detector networks and across several binary black hole populations. The resulting forecasts display clear, systematic trends: how tightly a non-GR correction can be bounded depends on the post-Newtonian order at which it enters, on the detector band, and on the source class. Space- and ground-based detectors prove complementary, especially for effects that accumulate during the long low-frequency inspiral. The practical payoff is a cheaper, more stable tool for large-scale parameter studies and more reliable forecasts of what future detector networks can say about gravity beyond general relativity and about environmental influences on compact binaries.

Core claim

Fully analytic expressions for the derivatives of frequency-domain TaylorF2 waveforms with respect to both binary parameters and parametrized post-Einsteinian deformation parameters make Fisher-matrix calculations stable and efficient, and those calculations reveal systematic trends in the constraints on non-GR effects as functions of post-Newtonian order, detector type, and source population, together with clear complementarity between space- and ground-based detectors for low-frequency-accumulating effects.

What carries the argument

The amplitude-phase decomposition of the waveform derivative, ∂_θ h̃ = h̃ (X_θ + i Y_θ), with all X_θ and Y_θ written in closed form for the GR phase coefficients and for every ppE correction function Ξ and Υ; this identity turns every Fisher integral into a real-valued frequency integral free of finite-difference step-size choices.

Load-bearing premise

The calculation rests on a 2PN-truncated GR baseline and on the claim that neglected eccentricity and spin-precession effects stay smaller than the statistical errors for the parameters being studied; if those systematics are larger, the reported uncertainties and complementarity statements no longer hold.

What would settle it

Recompute a representative set of Fisher matrices with a higher-order or precessing/eccentric waveform model for the same detector bands and populations; if the recovered σ_δ values and the space-versus-ground ranking shift by more than the statistical width claimed in the paper, the central forecasts fail.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript derives fully analytic waveform derivatives for the frequency-domain TaylorF2 inspiral within the parametrized post-Einsteinian (ppE) framework and uses them to construct Fisher-matrix forecasts for non-GR effects. After writing the waveform as amplitude and phase factors (Eqs. 9–10), the authors obtain closed-form expressions for the logarithmic amplitude and phase derivatives X_θ and Y_θ with respect to the full parameter set (Eqs. 16–27 and Appendices A–B). These derivatives are applied to space-based (LISA, Taiji, TianQin), ground-based (LVK, ET) and multiband configurations, combined with several BBH population models, to map how constraints on a deformation parameter δ scale with PN order, chirp mass, observation time and detector band. The same machinery is then specialized to a suite of concrete non-GR and environmental effects (noncommutative gravity, dCS, EdGB, scalar–tensor, charge, graviton mass, Ġ, Ṁ, dynamical friction). The central claims are that the analytic approach is numerically stable and substantially faster than finite differences (Fig. 1), that the resulting constraints exhibit systematic PN-order and detector-type trends (Figs. 3–5), and that space- and ground-based detectors are complementary for low-frequency-accumulating corrections (Fig. 6 and Sec. 6).

Significance. If the forecasts hold under the stated assumptions, the paper supplies a transparent, reusable analytic toolkit that removes step-size systematics from large-scale ppE Fisher studies and makes the PN-order and detector-complementarity trends quantitatively accessible. The explicit derivative catalogue in Appendices A–B, the runtime comparison in Fig. 1, and the population-averaged constraints in Fig. 6 are concrete, reusable contributions that future forecast and MCMC pipelines can adopt directly. The work therefore has clear methodological value for precision tests of gravity with next-generation detector networks, even though the underlying Fisher and 2PN approximations are standard.

major comments (2)
  1. Sec. 2 (paragraphs after Eq. 4) asserts that systematic biases from neglecting eccentricity and spin-precession remain sub-dominant to statistical Fisher errors for the parameters considered, citing only the authors’ earlier work [43]. Because the reported complementarity statements and the layered PN-order trends rest on the absolute scale of σ_δ, a short quantitative check (or an explicit statement of the mass/spin/eccentricity regime in which the claim holds) is needed before those trends can be regarded as robust forecasts rather than purely statistical scalings.
  2. The GR baseline is truncated at 2PN while ppE corrections are varied from –5.5PN to +2PN (Sec. 2 and Sec. 5.1). For positive-PN corrections the missing higher-order GR phase terms can correlate with the ppE parameters; the paper should either recompute a representative subset of the Fig. 4/6 constraints with a 3.5PN (or higher) baseline or demonstrate that the relative detector rankings and the 0PN degeneracy feature are insensitive to that truncation.
minor comments (5)
  1. Fig. 1 caption and text: clarify whether the reported speed-up factors include the cost of assembling the full multi-parameter Fisher matrix or only the derivative evaluation itself.
  2. Eq. (35) and the subsequent scaling discussion introduce an index n that is easily confused with the PN order n_PN; a brief notational remark would help.
  3. Table 1 lists f_high = 1 Hz for all space-based detectors; a short justification relative to the actual transfer-function cut-offs would be useful.
  4. In Appendix B.7–B.9 the simplified constants F_α, F_β and the sound-speed dependence of γ_DF are stated without numerical values or ranges; adding them would improve reproducibility of Fig. 6.
  5. A few typographical inconsistencies appear (e.g., “Liuet al.”, “O’Beirnet al.” in the introduction; missing spaces before citations).

Circularity Check

1 steps flagged

No significant circularity: analytic ppE–TaylorF2 derivatives and Fisher forecasts are direct consequences of the stated waveform model; self-citations supply only background assumptions and noise curves.

specific steps
  1. self citation load bearing [Sec. 2 (paragraphs after Eq. 4) and citation [43]]
    "A related assessment examined how different PN waveform models affect parameter estimation for SBBHs observed by space-based GW detectors [43]. … In particular, the 2PN inspiral waveform is sufficient for the Fisher-matrix forecast and scaling analysis performed here. … A related assessment is presented in our previous work [43]."

    The justification that the 2PN truncation (and the neglect of eccentricity/spin-precession systematics) remains adequate for the present forecasts rests solely on the authors’ own prior paper. The assumption is not re-derived or independently validated here; it is imported by self-citation. The step is minor because it affects only the domain of validity of the forecasts, not the algebraic construction of the analytic derivatives themselves.

full rationale

The derivation chain is self-contained. Section 2 writes the standard TaylorF2 GR phase (Eqs. 1–4) plus the conventional ppE amplitude/phase corrections (Eqs. 5–10). Section 3 then obtains the Fisher integrands by elementary logarithmic differentiation (Eqs. 16–27), with all GR phase derivatives listed explicitly in Appendix A and the theory-specific Ξ, ϒ derivatives in Appendix B; none of these steps is defined in terms of the final constraints. The subsequent numerical results are ordinary Fisher forecasts under stated detector curves, population models and the 2PN truncation. Self-citations ([43], [53], [55]) justify the 2PN baseline, noise models and the neglect of TDI, but they are not used to force the analytic expressions or the reported PN-order/detector trends. There is no parameter fitting followed by “prediction,” no uniqueness theorem imported from the authors, and no renaming of a known empirical pattern. Circularity burden is therefore minimal.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The work rests on the standard SPA/TaylorF2 and ppE formalisms, the high-SNR Fisher-matrix approximation, published detector noise curves, and several astrophysical population models. No new free parameters are fitted to data; the only free choices are the conventional truncation order, sky-averaging, and the specific population realizations taken from the literature.

free parameters (2)
  • PN truncation order of GR baseline
    Fixed by hand at 2PN (Sec. 2); higher orders are stated to be straightforward extensions but are not varied in the main results.
  • Observation-time cut-offs T_obs
    Chosen as 1 yr (MBHB), 4 yr (space-SBBH), 10 min (ground-SBBH); these are conventional mission/observing-run values, not fitted.
axioms (4)
  • domain assumption Stationary-phase approximation yields the frequency-domain TaylorF2 inspiral waveform truncated at 2PN
    Invoked from the outset (Eqs. 1–4) and justified by reference to prior waveform studies.
  • domain assumption Fisher-matrix covariance equals the inverse of the noise-weighted inner product of waveform derivatives (high-SNR Gaussian noise)
    Standard assumption of the entire analysis (Sec. 3, Eqs. 13–14).
  • domain assumption Sky- and polarization-averaged detector response can be absorbed into an effective sensitivity curve with C=1
    Stated explicitly after Eq. 2 and used for all numerical results.
  • domain assumption ppE amplitude and phase corrections of the form α u^a and β u^b capture the leading inspiral deviations of the theories considered
    Taken from the literature (Refs. 25, 30) and used throughout Secs. 5–6 and Appendix B.

pith-pipeline@v1.1.0-grok45 · 29570 in / 2633 out tokens · 26961 ms · 2026-07-11T20:42:58.542719+00:00 · methodology

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read the original abstract

Testing gravity beyond general relativity (GR) is essential for probing fundamental physics in the strong-field and highly dynamical regime accessed by gravitational-wave (GW) observations. In this work, we derive analytic expressions for waveform derivatives in the Fisher-matrix formalism within the parametrized post-Einsteinian framework, using the frequency-domain inspiral waveform. These analytic derivatives enable stable and efficient Fisher-matrix calculations without relying on finite-difference schemes. We apply this method to a wide range of detector configurations, including space-based, ground-based, and multiband observations, and combine it with different binary black hole population models. Our results reveal clear and systematic trends in the constraints on non-GR effects as functions of post-Newtonian order, detector type, and source population. They also demonstrate the complementarity between space- and ground-based detectors, particularly for effects that accumulate during the low-frequency inspiral. The analytic approach substantially reduces computational cost and avoids numerical systematics associated with step-size choices, making it well suited for large-scale parameter studies. These results provide robust forecasts for the capability of future GW observations to constrain a broad class of non-GR effects and environmental influences, highlighting the scientific potential of upcoming detector networks for precision tests of gravity.

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Works this paper leans on

79 extracted references · 27 canonical work pages

  1. [1]

    Baker, D

    T. Baker, D. Psaltis, C. Skordis, Astrophys. J.802, 63 (2015). DOI 10.1088/0004-637X/802/1/63

    doi:10.1088/0004-637x/802/1/63 2015

  2. [2]

    Will, Living Rev

    C.M. Will, Living Rev. Rel.17, 4 (2014). DOI 10. 12942/lrr-2014-4

    2014

  3. [3]

    Kramer, et al., Phys

    M. Kramer, et al., Phys. Rev. X11(4), 041050 (2021). DOI 10.1103/PhysRevX.11.041050

    doi:10.1103/physrevx.11.041050 2021

  4. [4]

    Bertotti, L

    B. Bertotti, L. Iess, P. Tortora, Nature425, 374 (2003). DOI 10.1038/nature01997

    doi:10.1038/nature01997 2003

  5. [5]

    Stairs, Living Rev

    I.H. Stairs, Living Rev. Rel.6, 5 (2003). DOI 10.12942/ lrr-2003-5

    2003

  6. [6]

    Clifton, P.G

    T. Clifton, P.G. Ferreira, A. Padilla, C. Skordis, Phys. Rept.513, 1 (2012). DOI 10.1016/j.physrep.2012.01. 001

    doi:10.1016/j.physrep.2012.01 2012

  7. [7]

    Joyce, L

    A. Joyce, L. Lombriser, F. Schmidt, Ann. Rev. Nucl. Part. Sci.66, 95 (2016). DOI 10.1146/ annurev-nucl-102115-044553

    2016

  8. [8]

    Berti, et al., Class

    E. Berti, et al., Class. Quant. Grav.32, 243001 (2015). DOI 10.1088/0264-9381/32/24/243001

    doi:10.1088/0264-9381/32/24/243001 2015

  9. [9]

    Damour, G

    T. Damour, G. Esposito-Farese, Phys. Rev. Lett.70, 2220 (1993). DOI 10.1103/PhysRevLett.70.2220

    doi:10.1103/physrevlett.70.2220 1993

  10. [10]

    Abbott, et al., Phys

    B.P. Abbott, et al., Phys. Rev. Lett.116(6), 061102 (2016). DOI 10.1103/PhysRevLett.116.061102

    doi:10.1103/physrevlett.116.061102 2016

  11. [11]

    Postnov, L.R

    K.A. Postnov, L.R. Yungelson, Living Rev. Rel.17, 3 (2014). DOI 10.12942/lrr-2014-3

    doi:10.12942/lrr-2014-3 2014

  12. [12]

    Ghosh, et al., Phys

    A. Ghosh, et al., Phys. Rev. D94(2), 021101 (2016). DOI 10.1103/PhysRevD.94.021101

    doi:10.1103/physrevd.94.021101 2016

  13. [13]

    Ghosh, N.K

    A. Ghosh, N.K. Johnson-Mcdaniel, A. Ghosh, C.K. Mishra, P. Ajith, W. Del Pozzo, C.P.L. Berry, A.B. Nielsen, L. London, Class. Quant. Grav.35(1), 014002 (2018). DOI 10.1088/1361-6382/aa972e

    doi:10.1088/1361-6382/aa972e 2018

  14. [14]

    Abbott, et al., Phys

    B.P. Abbott, et al., Phys. Rev. Lett.116(22), 221101 (2016). DOI 10.1103/PhysRevLett.116.221101. [Er- ratum: Phys.Rev.Lett. 121, 129902 (2018)]

    doi:10.1103/physrevlett.116.221101 2016

  15. [15]

    Bhagwat, X.J

    S. Bhagwat, X.J. Forteza, P. Pani, V . Ferrari, Phys. Rev. D101(4), 044033 (2020). DOI 10.1103/PhysRevD.101. 044033

    doi:10.1103/physrevd.101 2020

  16. [16]

    Abac, et al., Phys

    A.G. Abac, et al., Phys. Rev. Lett.135(11), 111403 (2025). DOI 10.1103/kw5g-d732

    doi:10.1103/kw5g-d732 2025

  17. [17]

    Abbott, et al., Phys

    R. Abbott, et al., Phys. Rev. D112(8), 084080 (2025). DOI 10.1103/PhysRevD.112.084080

    doi:10.1103/physrevd.112.084080 2025

  18. [18]

    Collaboration, the Virgo Collaboration, the KAGRA Collaboration

    T.L.S. Collaboration, the Virgo Collaboration, the KAGRA Collaboration. Gwtc-4.0: Updating the gravitational-wave transient catalog with observations from the first part of the fourth ligo-virgo-kagra ob- serving run (2025). URLhttps://arxiv.org/abs/ 2508.18082

    Pith/arXiv arXiv 2025

  19. [19]

    Punturo, et al., Class

    M. Punturo, et al., Class. Quant. Grav.27, 194002 (2010). DOI 10.1088/0264-9381/27/19/194002

    doi:10.1088/0264-9381/27/19/194002 2010

  20. [20]

    Amaro-Seoane, et al

    P. Amaro-Seoane, et al. Laser interferometer space an- tenna (2017). URLhttps://arxiv.org/abs/1702. 00786

    2017

  21. [21]

    Hu, Y .L

    W.R. Hu, Y .L. Wu, Natl. Sci. Rev.4(5), 685 (2017). DOI 10.1093/nsr/nwx116

    doi:10.1093/nsr/nwx116 2017

  22. [22]

    Luo, et al., Class

    J. Luo, et al., Class. Quant. Grav.33(3), 035010 (2016). DOI 10.1088/0264-9381/33/3/035010

    doi:10.1088/0264-9381/33/3/035010 2016

  23. [23]

    Cai, Z.K

    R.G. Cai, Z.K. Guo, B. Hu, C. Liu, Y . Lu, W.T. Ni, W.H. Ruan, N. Seto, G. Wang, Y .L. Wu, Fund. Res.4, 1072 (2024). DOI 10.1016/j.fmre.2023.10.007

    work page doi:10.1016/j.fmre.2023.10.007 2024

  24. [24]

    Mishra, K.G

    C.K. Mishra, K.G. Arun, B.R. Iyer, B.S. Sathyaprakash, Phys. Rev. D82, 064010 (2010). DOI 10.1103/ PhysRevD.82.064010

    2010

  25. [25]

    Yunes, F

    N. Yunes, F. Pretorius, Phys. Rev. D80, 122003 (2009). DOI 10.1103/PhysRevD.80.122003

    doi:10.1103/physrevd.80.122003 2009

  26. [26]

    Cornish, L

    N. Cornish, L. Sampson, N. Yunes, F. Pretorius, Phys. Rev. D84, 062003 (2011). DOI 10.1103/PhysRevD.84. 062003

    doi:10.1103/physrevd.84 2011

  27. [27]

    Chatziioannou, N

    K. Chatziioannou, N. Yunes, N. Cornish, Phys. Rev. D86, 022004 (2012). DOI 10.1103/PhysRevD.86. 022004. [Erratum: Phys.Rev.D 95, 129901 (2017)]

    doi:10.1103/physrevd.86 2012

  28. [28]

    Narikawa, H

    T. Narikawa, H. Tagoshi, PTEP2016(9), 093E02 (2016). DOI 10.1093/ptep/ptw126

    work page doi:10.1093/ptep/ptw126 2016

  29. [29]

    Yunes, X

    N. Yunes, X. Siemens, K. Yagi, Living Rev. Rel.28(1), 3 (2025). DOI 10.1007/s41114-024-00054-9

    work page doi:10.1007/s41114-024-00054-9 2025

  30. [30]

    Tahura, K

    S. Tahura, K. Yagi, Phys. Rev. D98(8), 084042 (2018). DOI 10.1103/PhysRevD.98.084042. [Erratum: Phys.Rev.D 101, 109902 (2020)]

    doi:10.1103/physrevd.98.084042 2018

  31. [31]

    Carson, K

    Z. Carson, K. Yagi, Phys. Rev. D101(4), 044047 (2020). DOI 10.1103/PhysRevD.101.044047

    doi:10.1103/physrevd.101.044047 2020

  32. [32]

    C. Liu, L. Shao, J. Zhao, Y . Gao, Mon. Not. Roy. As- tron. Soc.496(1), 182 (2020). DOI 10.1093/mnras/ staa1512

    doi:10.1093/mnras/ 2020

  33. [33]

    O’Beirne, N.J

    L. O’Beirne, N.J. Cornish, S.J. Vigeland, S.R. Taylor, Phys. Rev. D99(12), 124039 (2019). DOI 10.1103/ 21 PhysRevD.99.124039

    2019

  34. [34]

    C. Shi, M. Ji, J.d. Zhang, J. Mei, Phys. Rev. D108(2), 024030 (2023). DOI 10.1103/PhysRevD.108.024030

    work page doi:10.1103/physrevd.108.024030 2023

  35. [35]

    Liu, W.H

    C. Liu, W.H. Ruan, Z.K. Guo, Phys. Rev. D102(12), 124050 (2020). DOI 10.1103/PhysRevD.102.124050

    work page doi:10.1103/physrevd.102.124050 2020

  36. [36]

    Wang, W.B

    G. Wang, W.B. Han, Phys. Rev. D103(6), 064021 (2021). DOI 10.1103/PhysRevD.103.064021

    work page doi:10.1103/physrevd.103.064021 2021

  37. [37]

    J. Wu, J. Li, Phys. Rev. D110(8), 084057 (2024). DOI 10.1103/PhysRevD.110.084057

    work page doi:10.1103/physrevd.110.084057 2024

  38. [38]

    Blanchet, Living Rev

    L. Blanchet, Living Rev. Rel.17, 2 (2014). DOI 10. 12942/lrr-2014-2

    2014

  39. [39]

    Buonanno, B

    A. Buonanno, B. Iyer, E. Ochsner, Y . Pan, B.S. Sathyaprakash, Phys. Rev. D80, 084043 (2009). DOI 10.1103/PhysRevD.80.084043

    doi:10.1103/physrevd.80.084043 2009

  40. [40]

    Sathyaprakash, B.F

    B.S. Sathyaprakash, B.F. Schutz, Living Rev. Rel.12, 2 (2009). DOI 10.12942/lrr-2009-2

    doi:10.12942/lrr-2009-2 2009

  41. [41]

    Robson, N.J

    T. Robson, N.J. Cornish, C. Liu, Class. Quant. Grav. 36(10), 105011 (2019). DOI 10.1088/1361-6382/ ab1101

    doi:10.1088/1361-6382/ 2019

  42. [42]

    S. Khan, S. Husa, M. Hannam, F. Ohme, M. Pür- rer, X. Jiménez Forteza, A. Bohé, Phys. Rev. D93(4), 044007 (2016). DOI 10.1103/PhysRevD.93.044007

    doi:10.1103/physrevd.93.044007 2016

  43. [43]

    J. Wu, J. Li, X. Liu, Z. Cao, Phys. Rev. D109(10), 104014 (2024). DOI 10.1103/PhysRevD.109.104014

    work page doi:10.1103/physrevd.109.104014 2024

  44. [44]

    Phurailatpam, G

    H. Phurailatpam, G. Achamveedu, M. Haney, T. Li, S. Tiwari. On constraining initial orbital eccentricity of purely inspiral events (2025). URLhttps://arxiv. org/abs/2508.12697

    Pith/arXiv arXiv 2025

  45. [45]

    X. Liu, Z. Cao, Z.H. Zhu, Class. Quant. Grav.41(19), 195019 (2024). DOI 10.1088/1361-6382/ad72ca

    work page doi:10.1088/1361-6382/ad72ca 2024

  46. [46]

    Owen, C.J

    C.B. Owen, C.J. Haster, S. Perkins, N.J. Cornish, N. Yunes, Phys. Rev. D108(4), 044018 (2023). DOI 10.1103/PhysRevD.108.044018

    work page doi:10.1103/physrevd.108.044018 2023

  47. [47]

    Kremer, et al., Phys

    K. Kremer, et al., Phys. Rev. D99(6), 063003 (2019). DOI 10.1103/PhysRevD.99.063003

    doi:10.1103/physrevd.99.063003 2019

  48. [49]

    Vallisneri, Phys

    M. Vallisneri, Phys. Rev. D77, 042001 (2008). DOI 10.1103/PhysRevD.77.042001

    doi:10.1103/physrevd.77.042001 2008

  49. [50]

    Cutler, E.E

    C. Cutler, E.E. Flanagan, Phys. Rev. D49, 2658 (1994). DOI 10.1103/PhysRevD.49.2658

    doi:10.1103/physrevd.49.2658 1994

  50. [51]

    Z. Luo, Z. Guo, G. Jin, Y . Wu, W. Hu, Results Phys.16, 102918 (2020). DOI 10.1016/j.rinp.2019.102918

    work page doi:10.1016/j.rinp.2019.102918 2020

  51. [52]

    Huang, Y .M

    S.J. Huang, Y .M. Hu, V . Korol, P.C. Li, Z.C. Liang, Y . Lu, H.T. Wang, S. Yu, J. Mei, Phys. Rev. D102(6), 063021 (2020). DOI 10.1103/PhysRevD.102.063021

    work page doi:10.1103/physrevd.102.063021 2020

  52. [53]

    J. Wu, J. Li, Phys. Rev. D108(12), 124047 (2023). DOI 10.1103/PhysRevD.108.124047

    work page doi:10.1103/physrevd.108.124047 2023

  53. [54]

    Vallisneri, J

    M. Vallisneri, J. Crowder, M. Tinto, Class. Quant. Grav. 25, 065005 (2008). DOI 10.1088/0264-9381/25/6/ 065005

    work page doi:10.1088/0264-9381/25/6/ 2008

  54. [55]

    J. Wu, M. Sun, J. Li, Class. Quant. Grav.43(6), 065017 (2026). DOI 10.1088/1361-6382/ae5144

    work page doi:10.1088/1361-6382/ae5144 2026

  55. [56]

    Hild, et al., Class

    S. Hild, et al., Class. Quant. Grav.28, 094013 (2011). DOI 10.1088/0264-9381/28/9/094013

    doi:10.1088/0264-9381/28/9/094013 2011

  56. [57]

    Poisson, Phys

    E. Poisson, Phys. Rev. D52, 5719 (1995). DOI 10.1103/PhysRevD.52.5719. [Addendum: Phys.Rev.D 55, 7980–7981 (1997)]

    work page doi:10.1103/physrevd.52.5719 1995

  57. [58]

    Colpi, A

    M. Colpi, A. Sesana,Gravitational Wave Sources in the Era of Multi-Band Gravitational Wave Astronomy (2017), pp. 43–140. DOI 10.1142/9789813141766_ 0002

    work page doi:10.1142/9789813141766_ 2017

  58. [59]

    Aghanim, et al., Astron

    N. Aghanim, et al., Astron. Astrophys.641, A6 (2020). DOI 10.1051/0004-6361/201833910. [Erratum: As- tron.Astrophys. 652, C4 (2021)]

    doi:10.1051/0004-6361/201833910 2020

  59. [60]

    Abbott, et al., Phys

    R. Abbott, et al., Phys. Rev. X13(1), 011048 (2023). DOI 10.1103/PhysRevX.13.011048

    doi:10.1103/physrevx.13.011048 2023

  60. [61]

    Abbott, et al., Astrophys

    R. Abbott, et al., Astrophys. J. Lett.913(1), L7 (2021). DOI 10.3847/2041-8213/abe949

    doi:10.3847/2041-8213/abe949 2021

  61. [62]

    Wang, et al., Phys

    H.T. Wang, et al., Phys. Rev. D100(4), 043003 (2019). DOI 10.1103/PhysRevD.100.043003

    work page doi:10.1103/physrevd.100.043003 2019

  62. [63]

    S.X. Yi, G. Nelemans, C. Brinkerink, Z. Kostrzewa- Rutkowska, S.T. Timmer, F. Stoppa, E.M. Rossi, S.F. Portegies Zwart, Astron. Astrophys.663, A155 (2022). DOI 10.1051/0004-6361/202141634

    work page doi:10.1051/0004-6361/202141634 2022

  63. [64]

    S.X. Yi, F. Stoppa, G. Nelemans, E. Cator, Astron. As- trophys.663, A156 (2022). DOI 10.1051/0004-6361/ 202243127

    doi:10.1051/0004-6361/ 2022

  64. [65]

    Z. Zhao, S. Shashank, D. Das, C. Bambi. Testing black hole metrics with binary black hole inspirals (2025). URLhttps://arxiv.org/abs/2510.04703

    Pith/arXiv arXiv 2025

  65. [66]

    Harikumar, V .O

    E. Harikumar, V .O. Rivelles, Class. Quant. Grav.23, 7551 (2006). DOI 10.1088/0264-9381/23/24/024

    work page doi:10.1088/0264-9381/23/24/024 2006

  66. [67]

    H. Song, H. Li, Z. Lyu, J. Zhu, J.C. Wang, P. Ji. Con- straining noncommutative spacetime with gw150914 and gw190814 (2025). URLhttps://arxiv.org/ abs/2507.23442

    arXiv 2025

  67. [68]

    Jackiw, S.Y

    R. Jackiw, S.Y . Pi, Phys. Rev. D68, 104012 (2003). DOI 10.1103/PhysRevD.68.104012

    doi:10.1103/physrevd.68.104012 2003

  68. [69]

    Smith, A.L

    T.L. Smith, A.L. Erickcek, R.R. Caldwell, M. Kamionkowski, Phys. Rev. D77, 024015 (2008). DOI 10.1103/PhysRevD.77.024015

    doi:10.1103/physrevd.77.024015 2008

  69. [70]

    Wang, P.C

    H.T. Wang, P.C. Li, J.L. Jiang, G.W. Yuan, Y .M. Hu, Y .Z. Fan, Eur. Phys. J. C81(8), 769 (2021). DOI 10. 1140/epjc/s10052-021-09555-1

    2021

  70. [71]

    Jacobson, Phys

    T. Jacobson, Phys. Rev. Lett.83, 2699 (1999). DOI 10.1103/PhysRevLett.83.2699

    work page doi:10.1103/physrevlett.83.2699 1999

  71. [72]

    Horbatsch, C.P

    M.W. Horbatsch, C.P. Burgess, JCAP05, 010 (2012). DOI 10.1088/1475-7516/2012/05/010

    work page doi:10.1088/1475-7516/2012/05/010 2012

  72. [73]

    Maeda, N

    K.i. Maeda, N. Ohta, Y . Sasagawa, Phys. Rev. D80, 104032 (2009). DOI 10.1103/PhysRevD.80.104032

    work page doi:10.1103/physrevd.80.104032 2009

  73. [74]

    Gao, S.P

    B. Gao, S.P. Tang, H.T. Wang, J. Yan, Y .Z. Fan, Phys. Rev. D110(4), 044022 (2024). DOI 10.1103/ PhysRevD.110.044022 22

    2024

  74. [75]

    Zhang, S.Y

    J. Zhang, S.Y . Zhou, Phys. Rev. D97(8), 081501 (2018). DOI 10.1103/PhysRevD.97.081501

    work page doi:10.1103/physrevd.97.081501 2018

  75. [76]

    Yunes, F

    N. Yunes, F. Pretorius, D. Spergel, Phys. Rev. D81, 064018 (2010). DOI 10.1103/PhysRevD.81.064018

    work page doi:10.1103/physrevd.81.064018 2010

  76. [77]

    K. Yagi, N. Tanahashi, T. Tanaka, Phys. Rev. D83, 084036 (2011). DOI 10.1103/PhysRevD.83.084036

    work page doi:10.1103/physrevd.83.084036 2011

  77. [78]

    Genova, E

    A. Genova, E. Mazarico, S. Goossens, F.G. Lemoine, G.A. Neumann, D.E. Smith, M.T. Zu- ber, Nature Commun.9(1), 289 (2018). DOI 10.1038/s41467-017-02558-1

    work page doi:10.1038/s41467-017-02558-1 2018

  78. [79]

    Toubiana, et al., Phys

    A. Toubiana, et al., Phys. Rev. Lett.126(10), 101105 (2021). DOI 10.1103/PhysRevLett.126.101105

    doi:10.1103/physrevlett.126.101105 2021

  79. [80]

    Caneva Santoro, S

    G. Caneva Santoro, S. Roy, R. Vicente, M. Haney, O.J. Piccinni, W. Del Pozzo, M. Martinez, Phys. Rev. Lett. 132(25), 251401 (2024). DOI 10.1103/PhysRevLett. 132.251401

    doi:10.1103/physrevlett 2024