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Future multi-band gravitational-wave observations can tighten constraints on parity and Lorentz violations by 1–3 orders of magnitude.

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2026-08-03 12:14 UTC pith:GEYDNCSL

load-bearing objection Competent FIM forecast with a plausible multiband story, but the headline M_LV/M_PV bounds are convention-dependent until the α_i normalization is stated. the 3 major comments →

arxiv 2601.03571 v2 pith:GEYDNCSL submitted 2026-01-07 gr-qc astro-ph.IM

Constraining Lorentz and parity violations in gravity with multiband gravitational wave observations

Zhi-Xin Jia , Tao Zhu , Zhoujian Cao This is my paper
classification gr-qc astro-ph.IM
keywords gravitational wavesLorentz violationparity violationmultiband astronomybinary black hole mergersparameter estimationmodified gravitysymmetry breaking tests
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.

Observing a single binary black hole both in the millihertz band (with space-based detectors) and in the high-frequency band (with third-generation ground-based detectors) could tighten limits on Lorentz and parity violation in gravity by one to three orders of magnitude over current gravitational-wave constraints. This paper projects these gains using a parameterized waveform whose amplitude and phase corrections scale as a power law in frequency, with the power-law index determining whether ground- or space-based detectors are more sensitive. Two representative sources are compared: a high-signal-to-noise 'golden' merger and a massive binary, and the analysis shows they are complementary—the loud, lighter event wins for high-frequency modifications (positive power-law indices), while the massive event wins for low-frequency modifications (negative indices). If realized, these constraints would make multi-band gravitational-wave astronomy a leading probe of fundamental symmetry violations predicted by quantum-gravity frameworks.

Core claim

This paper establishes that a single well-chosen binary black hole merger, observed jointly by future space-based and ground-based detectors, can constrain the energy scales M_PV and M_LV of parity and Lorentz violation more tightly than current observational catalogs do today. The analysis injects GR waveforms with zero symmetry-breaking coefficients into a standard matched-filter covariance forecast for two detector networks (CE+ET+LISA+Taiji and CE+ET+LISA+TianQin) and eight power-law models (β_ν=1; β_μ=-1,1,3; β_ν̄=2; β_μ̄=-2,2,4). The results show improvements of 1–3 orders of magnitude over existing bounds, with the golden event GW250114-like best for β>0 and the massive event GW231123

What carries the argument

The central mechanism is the parametrized waveform correction: amplitude and phase modifications to the GR signal that scale as (π f)^β, with independent power-law indices and energy scales for parity (M_PV) and Lorentz (M_LV) violations. The sign and magnitude of β determine which frequency band is most sensitive: positive β amplifies corrections at high frequency (favoring ground-based detectors), negative β at low frequency (favoring space-based detectors). The analysis combines detectors by summing their covariance matrices—a standard statistical approximation that estimates parameter uncertainties from the noise-weighted waveform derivatives—to project 1σ bounds on M_PV and M_LV for two

Load-bearing premise

The projected energy-scale bounds assume a specific but unstated value for the dimensionless constants α multiplying each symmetry-violating term, so the quoted M_PV and M_LV limits are not uniquely defined until that convention is fixed.

What would settle it

Recompute the covariance-forecast projections with the dimensionless prefactors α explicitly set to 1 and compare the resulting bound on M_LV for the β_μ̄=−2 model with the paper's value of 7.89×10^-35 GeV for the GW231123-like event with Network A; if the bound changes by an order of magnitude or more, the quoted constraints are artifacts of the arbitrary convention rather than physical sensitivity.

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

If this is right

  • Joint space-plus-ground multiband networks tighten bounds on all eight Lorentz- and parity-violation models by 1–3 orders of magnitude over current constraints.
  • The high-SNR golden event GW250114-like yields the strongest limits for high-frequency modifications (β>0); the massive GW231123-like event yields the strongest limits for low-frequency modifications (β<0).
  • Ground-based detectors alone beat space-based detectors by 3–7 orders of magnitude for positive-β effects, while space-based detectors dominate negative-β effects, with longer-arm detectors outperforming shorter-arm ones.
  • A single well-selected binary observed in both bands can compete with or surpass constraints from large catalogs of events.
  • For the low-frequency dispersion model β_μ̄=−2, the massive event improves prior limits by nearly an order of magnitude.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves the dimensionless prefactors α in the waveform corrections unspecified; since the waveform depends only on α/M^β, the quoted energy-scale bounds are convention-dependent. A natural next step is to fix α (e.g., α=1) and report bounds on the combination, or to translate results into constraints on α for each β.
  • The complementarity between the two source types suggests that future observing strategies should intentionally target both a loud stellar-mass merger and a massive binary to cover both frequency regimes, rather than relying on a single 'golden' event.
  • The linearized statistical forecast assumes high signal-to-noise and Gaussian noise; full Bayesian analyses of simulated signals would test whether the projected gains survive realistic noise transients and waveform systematics.
  • If these constraints are realized, they would probe parity and Lorentz violation at energy scales that could discriminate among quantum-gravity models with different predicted β dependence, since each β maps to a specific class of theories.

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

3 major / 4 minor

Summary. This paper presents a Fisher-matrix forecast of future multiband gravitational-wave constraints on parity- and Lorentz-violating propagation effects. Using the parametrized waveform framework of refs. [38,40], it injects two representative BBH signals—a high-SNR GW250114-like event and a massive GW231123-like event—into two networks (CE+ET+LISA+Taiji and CE+ET+LISA+TianQin). The analysis samples five parameters per event (Mc, chi_eff, t0, phi0, and one symmetry-breaking scale C) and reports 1-sigma bounds on M_PV and M_LV for eight power-law models (beta_nu=1; beta_mu=-1,1,3; beta_nubar=2; beta_mubar=-2,2,4). The central claim is that multiband networks improve current GWTC-3 constraints by one to three orders of magnitude, with the golden event better for beta>0 and the massive event better for beta<0.

Significance. If the quoted constraints are robust, the paper makes a useful contribution to the growing literature on future GW tests of fundamental symmetries. Its strengths are the use of realistic PSDs for five planned detectors, the systematic treatment of eight frequency-dependence models, and the direct comparison with published GWTC-3 constraints. However, the headline energy scales are not uniquely defined until the normalization of the coupling functions alpha_i is specified, and the five-parameter FIM may overstate statistical power. Both issues are fixable, but they are central to the numerical claims.

major comments (3)
  1. [Sec. II, Eqs. (11)-(14) and Tables III/IV] The waveform coefficients A_nu, A_mu, A_nubar, A_mubar are products of alpha_i and a power of 1/M_i, but the text only says that 'we treat alpha_nu, alpha_mu, alpha_nubar, alpha_mubar as constants' and never assigns their numerical values. Consequently, the FIM constrains the combinations alpha_i/M_i^{beta_i}, not M_i alone. For example, Eq. (11) with beta_nu=1 gives A_nu = (1/2)(2/M_PV) alpha_nu [1-(1+z)]; rescaling alpha_nu by c is equivalent to rescaling M_PV by c. The values in Tables III/IV and Fig. 2 are therefore convention-dependent unless a normalization such as alpha_i=1 is explicitly adopted. The comparison with the GWTC-3 bounds of refs. [38,44,46,48] is only meaningful if those references use the same convention. Please state the convention and, if needed, quote all constraints in terms of the actually constrained combination alpha_i/M_i^{beta_i}.
  2. [Sec. III C, Eq. (20)] The Fisher matrix is computed with respect to only five parameters (Mc, chi_eff, t0, phi0, C), while the injected waveform also depends on luminosity distance, inclination, polarization angle, sky location, mass ratio, and individual spin components. These omitted parameters can be degenerate with the propagation-effect coefficients—especially distance and inclination with the amplitude corrections delta_h1 and delta_h2, and chirp-mass/phase with the phase corrections. Fixing them removes part of the parameter space over which the likelihood must be marginalized and can make the quoted improvements appear larger than a full parameter-estimation analysis would give. Please justify the reduction or include the most correlated nuisance parameters; a minimal test would add distance and inclination to the FIM for one or two representative rows of Table IV and report the effect on the headline
  3. [Sec. IV, Table IV and Fig. 2] The captions for Tables III/IV state '1-sigma constraints', while Fig. 2 and the text describe '90% credible bounds' and compare with GWTC-3 limits. If the plotted points are the same as the 1-sigma table entries, the comparison is not apples-to-apples; if a conversion was applied, it should be described. In addition, the direction of the quoted bound (upper vs lower limit on M) is not stated. Since the waveform coefficients scale as M^{-beta}, for beta<0 the effect grows with M, so the sign of the allowed region is reversed relative to beta>0; the reader cannot tell whether a smaller numerical value in the table is a stronger or weaker constraint. Please clarify the convention for each beta.
minor comments (4)
  1. [Abstract and Sec. III C] The abstract refers to a 'Bayesian analysis', but the method used throughout is a Fisher information matrix with no priors or sampling. Please reword the abstract or justify the use of 'Bayesian'.
  2. [Eq. (9)] In delta_h2 = -A_nubar (pi f)^{beta_nu}, the exponent should be beta_nubar, not beta_nu.
  3. [Sec. IV, text around Table III] There is a typo: 'the GW250114-like event is lighter than the GW250114-like event' should compare with GW231123. Also 'CT' appears instead of 'CE' in the same paragraph.
  4. [Fig. 2] The panel labels appear to be missing the power-law index values (e.g., 'PV Vel ( = 1)' and 'LV Disp ( = 2)'), making the figure hard to read. Please ensure all beta labels are printed.

Circularity Check

0 steps flagged

No circular derivation; multiband forecast is a self-contained FIM calculation, but unstated α_i normalization makes absolute M_LV/M_PV bounds convention-dependent.

full rationale

The claimed result is a Fisher-matrix forecast, not a fit renamed as a prediction: GR waveforms are injected (C=0), derivatives of the modified waveform in Eqs. (6)-(14) are computed, and the inverse FIM yields the parameter uncertainties in Tables III/IV. No output constraint is fed back as an input, so the order-of-magnitude improvements are not forced by construction. The modified-waveform framework is taken from refs. [38,40], which share an author, but those are prior published model-building papers; the present calculation is a new application, and the comparison limits in Fig. 2 are external published constraints. Thus self-citations are present but not load-bearing in a circular sense. The substantive caveat is Sec. II's statement 'we treat α_ν, α_μ, α_νbar, α_μbar as constants' with no numerical assignment: Eqs. (11)-(14) enter the waveform only through combinations α_i/M^β, so the absolute energy-scale numbers in Tables III/IV are convention-dependent unless the intended α_i=1 normalization is stated. That is an identifiability/interpretation issue, not a result-in-equals-input circularity, and relative improvements survive if a common convention is used.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The main unstated input is the normalization of the four α coupling functions, which are degenerate with the energy scales M_LV and M_PV in Eqs. (11)–(14). The paper also assumes the validity of the prior parameterized framework, the FIM high-SNR approximation, and the chosen GR waveform model. No new particles, forces, or fields are introduced.

free parameters (4)
  • α_ν normalization = unspecified; implicitly fixed to unity or O(1)
    Eq. (2) and Eq. (11): A_ν ∝ (1/M_PV)^βν [α_ν(τ0) − α_ν(τe)(1+z)^βν]. With α_ν an arbitrary constant, M_PV is degenerate with α_ν^{1/βν}; no value is stated.
  • α_μ normalization = unspecified; implicitly fixed to unity or O(1)
    Eq. (3) and Eq. (12): A_μ ∝ (2/M_PV)^βμ ∫ α_μ/a^{βμ+1} dt. α_μ is an arbitrary constant and is not assigned.
  • α_{\barν} normalization = unspecified; implicitly fixed to unity or O(1)
    Eq. (4) and Eq. (13) for Lorentz-violating damping; same degeneracy between α_{\barν} and M_LV.
  • α_{\barμ} normalization = unspecified; implicitly fixed to unity or O(1)
    Eq. (5) and Eq. (14) for Lorentz-violating dispersion; same degeneracy between α_{\barμ} and M_LV.
axioms (4)
  • domain assumption The parametrized propagation equation (1) and its waveform solutions (6)–(14) from [38, 40] correctly capture all parity- and Lorentz-violating effects.
    The entire Fisher forecast is built on this framework; Eq. (1) is quoted from prior work and not derived here.
  • standard math The Cramér–Rao bound via the Fisher information matrix gives accurate 1σ uncertainties at the injected SNRs.
    Sec. III.C assumes the high-SNR Gaussian limit (Eqs. 23–25); this is standard but can fail at low SNR or with strong parameter correlations.
  • domain assumption IMRPhenomXPHM is an adequate GR baseline for both representative events.
    Sec. III.B selects this phenomenological approximant; waveform systematics are not propagated into the forecast.
  • ad hoc to paper The coupling functions α_i are constants over propagation (local-event approximation).
    Stated after Eq. (14) without justification; this collapses the time dependence but leaves the normalization free.

pith-pipeline@v1.3.0-alltime-deepseek · 18095 in / 15239 out tokens · 143445 ms · 2026-08-03T12:14:18.495837+00:00 · methodology

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read the original abstract

This study evaluates the capability of future multi-band observations of gravitational waves emitted from binary black hole coalescences, utilizing joint third-generation ground-based (CE, ET) and space-based (LISA, Taiji, TianQin) detector networks, to constrain parity and Lorentz symmetry violations in the gravitational sector. We model these effects through a parameterized waveform framework that incorporates a set of parameters that quantify potential deviations from general relativity. The frequency-dependence of their effects is described by power-law indices $\beta$ (i.e., $\beta_{\bar \nu}$, $\beta_{\bar \mu}$, $\beta_{\nu}$, and $\beta_{\mu}$). By analyzing events such as a high-signal noise ratio (SNR) "golden event" like GW250114 and a massive binary system like GW231123 (total mass $190-265 M_\odot$) using two networks of ground- and space-based detectors, we demonstrate that multi-band observations can significantly improve the current constraints on Lorentz and parity violations by several order of magnitude, for both high-frequency ($\beta > 0$) and low-frequency ($\beta < 0$) modifications. Our Bayesian analysis reveals that while the exceptional SNR of the GW250114-like event yields superior constraints for high-frequency modifications ($\beta > 0$), the massive nature of GW231123 provides more stringent limits for low-frequency effects ($\beta < 0$). This work highlights the critical value of future multi-band gravitational wave astronomy for conducting precision tests of general relativity across diverse binary populations.

Figures

Figures reproduced from arXiv: 2601.03571 by Tao Zhu, Zhi-Xin Jia, Zhoujian Cao.

Figure 1
Figure 1. Figure 1: FIG. 1: Sensitivity curves of various GW interferometers, plotted alongside the characteristic amplitude of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Comparison of the 90% credible bounds on the parity-violating energy scale [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

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Forward citations

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Reference graph

Works this paper leans on

97 extracted references · 80 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Obser- vation of Gravitational Waves from a Binary Black Hole Merger,

    B. P. Abbottet al.(LIGO Scientific, Virgo), “Obser- vation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett.116, 061102 (2016), arXiv:1602.03837 [gr-qc]

    Pith/arXiv arXiv 2016

  2. [2]

    GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run,

    R. Abbottet al.(KAGRA, VIRGO, LIGO Scientific), “GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run,” Phys. Rev. X13, 041039 (2023), arXiv:2111.03606 [gr-qc]

    Pith/arXiv arXiv 2023

  3. [3]

    GW190521: A Binary Black Hole Merger with a Total Mass of 150M⊙,

    R. Abbottet al.(LIGO Scientific, Virgo), “GW190521: A Binary Black Hole Merger with a Total Mass of 150M⊙,” Phys. Rev. Lett.125, 101102 (2020), arXiv:2009.01075 [gr-qc]

    arXiv 2020

  4. [4]

    GW150914: First results from the search for binary black hole coales- cence with Advanced LIGO,

    B. P. Abbottet al.(LIGO Scientific, Virgo), “GW150914: First results from the search for binary black hole coales- cence with Advanced LIGO,” Phys. Rev. D93, 122003 (2016), arXiv:1602.03839 [gr-qc]

    Pith/arXiv arXiv 2016

  5. [5]

    B. P. Abbottet al.(LIGO Scientific, Virgo), “Properties 9 10 24 10 23 10 22 10 21 10 20 10 19 Energy Scale MPV [GeV] This Work References PV Amp ( = 1) 10 45 10 44 10 43 10 42 10 41 This Work References PV Vel ( = 1) 10 6 10 4 10 2 100 102 This Work References PV Vel ( = 1) 10 19 10 18 10 17 10 16 10 15 10 14 10 13 This Work References PV Vel ( = 3) GW250...

    Pith/arXiv arXiv 2016

  6. [6]

    Population of Merging Compact Binaries Inferred Using Gravitational Waves through GWTC-3,

    R. Abbottet al.(KAGRA, VIRGO, LIGO Scientific), “Population of Merging Compact Binaries Inferred Using Gravitational Waves through GWTC-3,” Phys. Rev. X 13, 011048 (2023), arXiv:2111.03634 [astro-ph.HE]

    Pith/arXiv arXiv 2023

  7. [7]

    Laser Interferome- ter Space Antenna,

    Pau Amaro-Seoaneet al.(LISA), “Laser Interferome- ter Space Antenna,” (2017), arXiv:1702.00786 [astro- ph.IM]

    Pith/arXiv arXiv 2017

  8. [8]

    Taiji program: Gravitational-wave sources,

    Wen-Hong Ruan, Zong-Kuan Guo, Rong-Gen Cai, and Yuan-Zhong Zhang, “Taiji program: Gravitational-wave sources,” Int. J. Mod. Phys. A35, 2050075 (2020), arXiv:1807.09495 [gr-qc]

    Pith/arXiv arXiv 2020

  9. [9]

    The Taiji Program in Space for gravitational wave physics and the nature of gravity,

    Wen-Rui Hu and Yue-Liang Wu, “The Taiji Program in Space for gravitational wave physics and the nature of gravity,” Natl. Sci. Rev.4, 685–686 (2017)

    2017

  10. [10]

    TianQin: a space-borne grav- itational wave detector,

    Jun Luoet al.(TianQin), “TianQin: a space-borne grav- itational wave detector,” Class. Quant. Grav.33, 035010 10 (2016), arXiv:1512.02076 [astro-ph.IM]

    Pith/arXiv arXiv 2016

  11. [11]

    Science with the TianQin observatory: Preliminary results on stellar-mass binary black holes,

    Shuai Liu, Yi-Ming Hu, Jian-dong Zhang, and Jianwei Mei, “Science with the TianQin observatory: Preliminary results on stellar-mass binary black holes,” Phys. Rev. D 101, 103027 (2020), arXiv:2004.14242 [astro-ph.HE]

    Pith/arXiv arXiv 2020

  12. [12]

    Science with the TianQin obser- vatory: Preliminary results on massive black hole bina- ries,

    Hai-Tian Wanget al., “Science with the TianQin obser- vatory: Preliminary results on massive black hole bina- ries,” Phys. Rev. D100, 043003 (2019), arXiv:1902.04423 [astro-ph.HE]

    Pith/arXiv arXiv 2019

  13. [13]

    The first round result from the TianQin- 1 satellite,

    Jun Luoet al., “The first round result from the TianQin- 1 satellite,” Class. Quant. Grav.37, 185013 (2020), arXiv:2008.09534 [physics.ins-det]

    Pith/arXiv arXiv 2020

  14. [14]

    The Japanese space gravitational wave antenna DECIGO,

    S. Kawamuraet al., “The Japanese space gravitational wave antenna DECIGO,” Class. Quant. Grav.23, S125– S132 (2006)

    2006

  15. [15]

    Parametrized and inspiral- merger-ringdown consistency tests of gravity with multi- band gravitational wave observations,

    Zack Carson and Kent Yagi, “Parametrized and inspiral- merger-ringdown consistency tests of gravity with multi- band gravitational wave observations,” Phys. Rev. D 101, 044047 (2020), arXiv:1911.05258 [gr-qc]

    Pith/arXiv arXiv 2020

  16. [16]

    A Horizon Study for Cosmic Ex- plorer: Science, Observatories, and Community,

    Matthew Evanset al., “A Horizon Study for Cosmic Ex- plorer: Science, Observatories, and Community,” (2021), arXiv:2109.09882 [astro-ph.IM]

    Pith/arXiv arXiv 2021

  17. [17]

    Science with the Einstein Tele- scope: a comparison of different designs,

    Marica Branchesiet al., “Science with the Einstein Tele- scope: a comparison of different designs,” JCAP07, 068 (2023), arXiv:2303.15923 [gr-qc]

    Pith/arXiv arXiv 2023

  18. [18]

    Multiband Gravitational-Wave As- tronomy: Parameter Estimation and Tests of General Relativity with Space- and Ground-Based Detectors,

    Salvatore Vitale, “Multiband Gravitational-Wave As- tronomy: Parameter Estimation and Tests of General Relativity with Space- and Ground-Based Detectors,” Phys. Rev. Lett.117, 051102 (2016), arXiv:1605.01037 [gr-qc]

    Pith/arXiv arXiv 2016

  19. [19]

    Multiband gravitational- wave event rates and stellar physics,

    Davide Gerosa, Sizheng Ma, Kaze W. K. Wong, Emanuele Berti, Richard O’Shaughnessy, Yanbei Chen, and Krzysztof Belczynski, “Multiband gravitational- wave event rates and stellar physics,” Phys. Rev. D99, 103004 (2019), arXiv:1902.00021 [astro-ph.HE]

    Pith/arXiv arXiv 2019

  20. [20]

    Prospects for Multiband Gravitational- Wave Astronomy after GW150914,

    Alberto Sesana, “Prospects for Multiband Gravitational- Wave Astronomy after GW150914,” Phys. Rev. Lett. 116, 231102 (2016), arXiv:1602.06951 [gr-qc]

    Pith/arXiv arXiv 2016

  21. [21]

    Multiband gravitational- wave parameter estimation: A study of future detectors,

    Stefan Grimm and Jan Harms, “Multiband gravitational- wave parameter estimation: A study of future detectors,” Phys. Rev. D102, 022007 (2020), arXiv:2004.01434 [gr- qc]

    Pith/arXiv arXiv 2020

  22. [22]

    Bound- ing alternative theories of gravity with multiband GW observations,

    Giuseppe Gnocchi, Andrea Maselli, Tiziano Abdelsal- hin, Nicola Giacobbo, and Michela Mapelli, “Bound- ing alternative theories of gravity with multiband GW observations,” Phys. Rev. D100, 064024 (2019), arXiv:1905.13460 [gr-qc]

    Pith/arXiv arXiv 2019

  23. [23]

    Non-Spinning Black Holes in Alternative Theories of Gravity,

    Nicolas Yunes and Leo C. Stein, “Non-Spinning Black Holes in Alternative Theories of Gravity,” Phys. Rev. D 83, 104002 (2011), arXiv:1101.2921 [gr-qc]

    Pith/arXiv arXiv 2011

  24. [24]

    The Confrontation between General Relativity and Experiment,

    Clifford M. Will, “The Confrontation between General Relativity and Experiment,” Living Rev. Rel.17, 4 (2014), arXiv:1403.7377 [gr-qc]

    Pith/arXiv arXiv 2014

  25. [25]

    Post-Newtonian, Quasi-Circular Binary In- spirals in Quadratic Modified Gravity,

    Kent Yagi, Leo C. Stein, Nicol´ as Yunes, and Takahiro Tanaka, “Post-Newtonian, Quasi-Circular Binary In- spirals in Quadratic Modified Gravity,” Phys. Rev. D 85, 064022 (2012), [Erratum: Phys.Rev.D 93, 029902 (2016)], arXiv:1110.5950 [gr-qc]

    Pith/arXiv arXiv 2012

  26. [26]

    Testing General Relativity with Present and Future Astrophysical Observations,

    Emanuele Bertiet al., “Testing General Relativity with Present and Future Astrophysical Observations,” Class. Quant. Grav.32, 243001 (2015), arXiv:1501.07274 [gr- qc]

    Pith/arXiv arXiv 2015

  27. [27]

    Question of parity conser- vation in weak interactions,

    T. D. Lee and C. N. Yang, “Question of parity conser- vation in weak interactions,” Phys. Rev.104, 254–258 (1956)

    1956

  28. [28]

    Experimental Test of Parity Conservation inβDecay,

    C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, “Experimental Test of Parity Conservation inβDecay,” Phys. Rev.105, 1413–1414 (1957)

    1957

  29. [29]

    Modern tests of Lorentz invariance,

    David Mattingly, “Modern tests of Lorentz invariance,” Living Rev. Rel.8, 5 (2005), arXiv:gr-qc/0502097

    Pith/arXiv arXiv 2005

  30. [30]

    Data Tables for Lorentz and CPT Violation,

    V. Alan Kostelecky and Neil Russell, “Data Tables for Lorentz and CPT Violation,” Rev. Mod. Phys.83, 11– 31 (2011), arXiv:0801.0287 [hep-ph]

    arXiv 2011

  31. [31]

    Spontaneous Breaking of Lorentz Symmetry in String Theory,

    V. Alan Kostelecky and Stuart Samuel, “Spontaneous Breaking of Lorentz Symmetry in String Theory,” Phys. Rev. D39, 683 (1989)

    1989

  32. [32]

    CPT and strings,

    V. Alan Kostelecky and Robertus Potting, “CPT and strings,” Nucl. Phys. B359, 545–570 (1991)

    1991

  33. [33]

    Nonstandard op- tics from quantum space-time,

    Rodolfo Gambini and Jorge Pullin, “Nonstandard op- tics from quantum space-time,” Phys. Rev. D59, 124021 (1999), arXiv:gr-qc/9809038

    Pith/arXiv arXiv 1999

  34. [34]

    Loop generated bounds on changes to the graviton dispersion relation,

    C. P. Burgess, James M. Cline, E. Filotas, J. Matias, and G. D. Moore, “Loop generated bounds on changes to the graviton dispersion relation,” JHEP03, 043 (2002), arXiv:hep-ph/0201082

    Pith/arXiv arXiv 2002

  35. [35]

    Chern-Simons modification of general relativity,

    R. Jackiw and S. Y. Pi, “Chern-Simons modification of general relativity,” Phys. Rev. D68, 104012 (2003), arXiv:gr-qc/0308071

    Pith/arXiv arXiv 2003

  36. [36]

    Chern-Simons Modified General Relativity,

    Stephon Alexander and Nicolas Yunes, “Chern-Simons Modified General Relativity,” Phys. Rept.480, 1–55 (2009), arXiv:0907.2562 [hep-th]

    Pith/arXiv arXiv 2009

  37. [37]

    The- oretical Physics Implications of the Binary Black-Hole Mergers GW150914 and GW151226,

    Nicolas Yunes, Kent Yagi, and Frans Pretorius, “The- oretical Physics Implications of the Binary Black-Hole Mergers GW150914 and GW151226,” Phys. Rev. D94, 084002 (2016), arXiv:1603.08955 [gr-qc]

    Pith/arXiv arXiv 2016

  38. [38]

    Constraints on parity and Lorentz violations in gravity from GWTC- 3 through a parametrization of modified gravitational wave propagations,

    Tao Zhu, Wen Zhao, Jian-Ming Yan, Yuan-Zhu Wang, Cheng Gong, and Anzhong Wang, “Constraints on parity and Lorentz violations in gravity from GWTC- 3 through a parametrization of modified gravitational wave propagations,” Phys. Rev. D110, 064044 (2024), arXiv:2304.09025 [gr-qc]

    Pith/arXiv arXiv 2024

  39. [39]

    Waveform of gravitational waves in the ghost-free parity-violating gravities,

    Jin Qiao, Tao Zhu, Wen Zhao, and Anzhong Wang, “Waveform of gravitational waves in the ghost-free parity-violating gravities,” Phys. Rev. D100, 124058 (2019), arXiv:1909.03815 [gr-qc]

    Pith/arXiv arXiv 2019

  40. [40]

    Waveform of gravitational waves in the general parity- violating gravities,

    Wen Zhao, Tao Zhu, Jin Qiao, and Anzhong Wang, “Waveform of gravitational waves in the general parity- violating gravities,” Phys. Rev. D101, 024002 (2020), arXiv:1909.10887 [gr-qc]

    Pith/arXiv arXiv 2020

  41. [41]

    Grav- itational Wave Implications for the Parity Symmetry of Gravity in the High Energy Region,

    Yi-Fan Wang, Rui Niu, Tao Zhu, and Wen Zhao, “Grav- itational Wave Implications for the Parity Symmetry of Gravity in the High Energy Region,” Astrophys. J.908, 58 (2021), arXiv:2002.05668 [gr-qc]

    Pith/arXiv arXiv 2021

  42. [42]

    Constraints on the Nieh-Yan modified teleparal- lel gravity with gravitational waves,

    Qiang Wu, Tao Zhu, Rui Niu, Wen Zhao, and Anzhong Wang, “Constraints on the Nieh-Yan modified teleparal- lel gravity with gravitational waves,” Phys. Rev. D105, 024035 (2022), arXiv:2110.13870 [gr-qc]

    Pith/arXiv arXiv 2022

  43. [43]

    Gravita- tional wave constraints on Lorentz and parity violations in gravity: High-order spatial derivative cases,

    Cheng Gong, Tao Zhu, Rui Niu, Qiang Wu, Jing-Lei Cui, Xin Zhang, Wen Zhao, and Anzhong Wang, “Gravita- tional wave constraints on Lorentz and parity violations in gravity: High-order spatial derivative cases,” Phys. Rev. D105, 044034 (2022), arXiv:2112.06446 [gr-qc]

    Pith/arXiv arXiv 2022

  44. [44]

    Tests of gravitational-wave birefringence with the open gravitational-wave catalog,

    Yi-Fan Wang, Stephanie M. Brown, Lijing Shao, and Wen Zhao, “Tests of gravitational-wave birefringence with the open gravitational-wave catalog,” Phys. Rev. D106, 084005 (2022), arXiv:2109.09718 [astro-ph.HE]

    Pith/arXiv arXiv 2022

  45. [45]

    Search 11 for anisotropic, birefringent spacetime-symmetry break- ing in gravitational wave propagation from GWTC-3,

    Le ¨ ıla Haegel, Kellie O’Neal-Ault, Quentin G. Bailey, Jay D. Tasson, Malachy Bloom, and Lijing Shao, “Search 11 for anisotropic, birefringent spacetime-symmetry break- ing in gravitational wave propagation from GWTC-3,” Phys. Rev. D107, 064031 (2023), arXiv:2210.04481 [gr- qc]

    Pith/arXiv arXiv 2023

  46. [46]

    Gravi- tational wave constraints on nonbirefringent dispersions of gravitational waves due to Lorentz violations with GWTC-3 events,

    Cheng Gong, Tao Zhu, Rui Niu, Qiang Wu, Jing-Lei Cui, Xin Zhang, Wen Zhao, and Anzhong Wang, “Gravi- tational wave constraints on nonbirefringent dispersions of gravitational waves due to Lorentz violations with GWTC-3 events,” Phys. Rev. D107, 124015 (2023), arXiv:2302.05077 [gr-qc]

    Pith/arXiv arXiv 2023

  47. [47]

    Con- straints on parity and Lorentz violations from gravita- tional waves: a comparison between single-parameter and multi-parameter analysis,

    Wei-Hua Guo, Yuan-Zhu Wang, and Tao Zhu, “Con- straints on parity and Lorentz violations from gravita- tional waves: a comparison between single-parameter and multi-parameter analysis,” (2025), arXiv:2507.09705 [gr-qc]

    Pith/arXiv arXiv 2025

  48. [48]

    Modified gravitational wave propagations in linearized gravity with Lorentz and diffeomorphism violations and their gravitational wave constraints,

    Qiang Wang, Jian-Ming Yan, Tao Zhu, and Wen Zhao, “Modified gravitational wave propagations in linearized gravity with Lorentz and diffeomorphism violations and their gravitational wave constraints,” Phys. Rev. D111, 084064 (2025), arXiv:2501.11956 [gr-qc]

    Pith/arXiv arXiv 2025

  49. [49]

    Gravitational wave cosmology,

    Ligong Bianet al., “Gravitational wave cosmology,” Sci. China Phys. Mech. Astron.69, 210401 (2026), arXiv:2505.19747 [gr-qc]

    arXiv 2026

  50. [50]

    Gravitational wave constraints on spatial covariant gravities,

    Tao Zhu, Wen Zhao, and Anzhong Wang, “Gravitational wave constraints on spatial covariant gravities,” Phys. Rev. D107, 044051 (2023), arXiv:2211.04711 [gr-qc]

    Pith/arXiv arXiv 2023

  51. [51]

    Testing parity symmetry of gravity with gravitational waves,

    Jin Qiao, Zhao Li, Tao Zhu, Ran Ji, Guoliang Li, and Wen Zhao, “Testing parity symmetry of gravity with gravitational waves,” Front. Astron. Space Sci.9, 1109086 (2023), arXiv:2211.16825 [gr-qc]

    Pith/arXiv arXiv 2023

  52. [52]

    Polarized gravi- tational waves in the parity violating scalar-nonmetricity theory,

    Zheng Chen, Yang Yu, and Xian Gao, “Polarized gravi- tational waves in the parity violating scalar-nonmetricity theory,” JCAP06, 001 (2023), arXiv:2212.14362 [gr-qc]

    Pith/arXiv arXiv 2023

  53. [53]

    Constraining gravitational wave am- plitude birefringence with GWTC-3,

    Thomas C. K. Ng, Maximiliano Isi, Kaze W. K. Wong, and Will M. Farr, “Constraining gravitational wave am- plitude birefringence with GWTC-3,” Phys. Rev. D108, 084068 (2023), arXiv:2305.05844 [gr-qc]

    Pith/arXiv arXiv 2023

  54. [54]

    Search for the Birefringence of Gravitational Waves with the Third Observing Run of Advanced LIGO-Virgo,

    Zhi-Chao Zhao, Zhoujian Cao, and Sai Wang, “Search for the Birefringence of Gravitational Waves with the Third Observing Run of Advanced LIGO-Virgo,” Astro- phys. J.930, 139 (2022), arXiv:2201.02813 [gr-qc]

    Pith/arXiv arXiv 2022

  55. [55]

    Parametrized test of parity-violating gravity using GWTC-1 events,

    Kei Yamada and Takahiro Tanaka, “Parametrized test of parity-violating gravity using GWTC-1 events,” PTEP 2020, 093E01 (2020), arXiv:2006.11086 [gr-qc]

    Pith/arXiv arXiv 2020

  56. [56]

    Constraining gravitational wave amplitude birefringence and Chern-Simons gravity with GWTC-2,

    Maria Okounkova, Will M. Farr, Maximiliano Isi, and Leo C. Stein, “Constraining gravitational wave amplitude birefringence and Chern-Simons gravity with GWTC-2,” Phys. Rev. D106, 044067 (2022), arXiv:2101.11153 [gr- qc]

    Pith/arXiv arXiv 2022

  57. [57]

    New Limits on the Lorentz/CPT Symmetry Through 50 Gravitational-wave Events,

    Ziming Wang, Lijing Shao, and Chang Liu, “New Limits on the Lorentz/CPT Symmetry Through 50 Gravitational-wave Events,” Astrophys. J.921, 158 (2021), arXiv:2108.02974 [gr-qc]

    Pith/arXiv arXiv 2021

  58. [58]

    Analysis of Birefringence and Dispersion Effects from Spacetime- Symmetry Breaking in Gravitational Waves,

    Kellie O’Neal-Ault, Quentin G. Bailey, Tyann Dumer- chat, Leila Haegel, and Jay Tasson, “Analysis of Birefringence and Dispersion Effects from Spacetime- Symmetry Breaking in Gravitational Waves,” Universe 7, 380 (2021), arXiv:2108.06298 [gr-qc]

    Pith/arXiv arXiv 2021

  59. [59]

    Testing Lorentz invariance of gravity in the Standard-Model Extension with GWTC-3,

    Rui Niu, Tao Zhu, and Wen Zhao, “Testing Lorentz invariance of gravity in the Standard-Model Extension with GWTC-3,” JCAP12, 011 (2022), arXiv:2202.05092 [gr-qc]

    Pith/arXiv arXiv 2022

  60. [60]

    Polarized primordial gravitational waves in spatial covariant gravi- ties,

    Tao Zhu, Wen Zhao, and Anzhong Wang, “Polarized primordial gravitational waves in spatial covariant gravi- ties,” Phys. Rev. D107, 024031 (2023), arXiv:2210.05259 [gr-qc]

    Pith/arXiv arXiv 2023

  61. [61]

    New probe of gravitational parity violation through nonobservation of the stochastic gravitational-wave background,

    Thomas Callister, Leah Jenks, Daniel E. Holz, and Nicol´ as Yunes, “New probe of gravitational parity violation through nonobservation of the stochastic gravitational-wave background,” Phys. Rev. D111, 044041 (2025), arXiv:2312.12532 [gr-qc]

    Pith/arXiv arXiv 2025

  62. [62]

    Birefringence tests of gravity with multimessenger bina- ries,

    Macarena Lagos, Leah Jenks, Maximiliano Isi, Kenta Ho- tokezaka, Brian D. Metzger, Eric Burns, Will M. Farr, Scott Perkins, Kaze W. K. Wong, and Nicolas Yunes, “Birefringence tests of gravity with multimessenger bina- ries,” Phys. Rev. D109, 124003 (2024), arXiv:2402.05316 [gr-qc]

    Pith/arXiv arXiv 2024

  63. [63]

    Gravitational Wave Birefringence from Fuzzy Dark Matter,

    Da Huang and Ze-Xuan Xiong, “Gravitational Wave Birefringence from Fuzzy Dark Matter,” (2024), arXiv:2406.13394 [gr-qc]

    Pith/arXiv arXiv 2024

  64. [64]

    Gravitational wave bire- fringence in symmetron cosmology,

    Ze-Xuan Xiong and Da Huang, “Gravitational wave bire- fringence in symmetron cosmology,” Phys. Rev. D111, 084020 (2025), arXiv:2409.09382 [gr-qc]

    Pith/arXiv arXiv 2025

  65. [65]

    Anisotropic Stress as a Signature of Nonstandard Propagation of Gravitational Waves,

    Ippocratis D. Saltas, Ignacy Sawicki, Luca Amendola, and Martin Kunz, “Anisotropic Stress as a Signature of Nonstandard Propagation of Gravitational Waves,” Phys. Rev. Lett.113, 191101 (2014), arXiv:1406.7139 [astro-ph.CO]

    Pith/arXiv arXiv 2014

  66. [66]

    Generalized framework for testing gravity with gravitational-wave propagation. I. Formula- tion,

    Atsushi Nishizawa, “Generalized framework for testing gravity with gravitational-wave propagation. I. Formula- tion,” Phys. Rev. D97, 104037 (2018), arXiv:1710.04825 [gr-qc]

    Pith/arXiv arXiv 2018

  67. [67]

    Parameterized Post- Einsteinian Gravitational Waveforms in Various Mod- ified Theories of Gravity,

    Sharaban Tahura and Kent Yagi, “Parameterized Post- Einsteinian Gravitational Waveforms in Various Mod- ified Theories of Gravity,” Phys. Rev. D98, 084042 (2018), [Erratum: Phys.Rev.D 101, 109902 (2020)], arXiv:1809.00259 [gr-qc]

    Pith/arXiv arXiv 2018

  68. [68]

    Gravitational wave propagation beyond general relativity: waveform distortions and echoes,

    Jose Maria Ezquiaga, Wayne Hu, Macarena Lagos, and Meng-Xiang Lin, “Gravitational wave propagation beyond general relativity: waveform distortions and echoes,” JCAP11, 048 (2021), arXiv:2108.10872 [astro- ph.CO]

    Pith/arXiv arXiv 2021

  69. [69]

    Parametrized parity violation in gravitational wave propagation,

    Leah Jenks, Lyla Choi, Macarena Lagos, and Nicol´ as Yunes, “Parametrized parity violation in gravitational wave propagation,” Phys. Rev. D108, 044023 (2023), arXiv:2305.10478 [gr-qc]

    Pith/arXiv arXiv 2023

  70. [70]

    Forecasts for constraining Lorentz-violating damping of gravitational waves from compact bi- nary inspirals,

    Bo-Yang Zhang, Tao Zhu, Jing-Fei Zhang, and Xin Zhang, “Forecasts for constraining Lorentz-violating damping of gravitational waves from compact bi- nary inspirals,” Phys. Rev. D109, 104022 (2024), arXiv:2402.08240 [gr-qc]

    Pith/arXiv arXiv 2024

  71. [71]

    Con- straining the modified friction in gravitational wave prop- agation with precessing black hole binaries,

    Chunbo Lin, Tao Zhu, Rui Niu, and Wen Zhao, “Con- straining the modified friction in gravitational wave prop- agation with precessing black hole binaries,” Phys. Rev. D112, 024010 (2025), arXiv:2404.11245 [gr-qc]

    Pith/arXiv arXiv 2025

  72. [72]

    Constraining parity and Lorentz violations in gravity with future ground- and space-based gravitational wave detectors,

    Bo-Yang Zhang, Tao Zhu, Jian-Ming Yan, Jing-Fei Zhang, and Xin Zhang, “Constraining parity and Lorentz violations in gravity with future ground- and space-based gravitational wave detectors,” Phys. Rev. D 111, 104012 (2025), arXiv:2502.04776 [gr-qc]

    Pith/arXiv arXiv 2025

  73. [73]

    Parity violation in gravitational waves and observational bounds from third-generation detectors,

    Matteo Califano, Rocco D’Agostino, and Daniele Vernieri, “Parity violation in gravitational waves and observational bounds from third-generation detectors,” Phys. Rev. D109, 104062 (2024), arXiv:2311.02161 [gr- qc]

    Pith/arXiv arXiv 2024

  74. [74]

    Nontensorial gravitational wave polarizations from the tensorial degrees of freedom: Linearized Lorentz- 12 violating theory of gravity,

    Shaoqi Hou, Xi-Long Fan, Tao Zhu, and Zong-Hong Zhu, “Nontensorial gravitational wave polarizations from the tensorial degrees of freedom: Linearized Lorentz- 12 violating theory of gravity,” Phys. Rev. D109, 084011 (2024), arXiv:2401.03474 [gr-qc]

    Pith/arXiv arXiv 2024

  75. [75]

    Constraining Generic Lorentz Violation and the Speed of the Graviton with Gravitational Waves,

    Saeed Mirshekari, Nicolas Yunes, and Clifford M. Will, “Constraining Generic Lorentz Violation and the Speed of the Graviton with Gravitational Waves,” Phys. Rev. D85, 024041 (2012), arXiv:1110.2720 [gr-qc]

    Pith/arXiv arXiv 2012

  76. [76]

    Probing parity violation in the stochastic gravitational wave background with astrometry,

    Qiuyue Liang, Meng-Xiang Lin, Mark Trodden, and Sam S. C. Wong, “Probing parity violation in the stochastic gravitational wave background with astrometry,” Phys. Rev. D109, 083028 (2024), arXiv:2309.16666 [astro- ph.CO]

    Pith/arXiv arXiv 2024

  77. [77]

    Joint Observations of Space-based Gravitational-wave Detec- tors: Source Localization and Implication for Parity- violating gravity,

    Qian Hu, Mingzheng Li, Rui Niu, and Wen Zhao, “Joint Observations of Space-based Gravitational-wave Detec- tors: Source Localization and Implication for Parity- violating gravity,” Phys. Rev. D103, 064057 (2021), arXiv:2006.05670 [gr-qc]

    Pith/arXiv arXiv 2021

  78. [78]

    Gravitational waves from merging compact binaries: How accurately can one extract the binary’s parameters from the in- spiral wave form?

    Curt Cutler and Eanna E. Flanagan, “Gravitational waves from merging compact binaries: How accurately can one extract the binary’s parameters from the in- spiral wave form?” Phys. Rev. D49, 2658–2697 (1994), arXiv:gr-qc/9402014

    Pith/arXiv arXiv 1994

  79. [79]

    Sensitivity Studies for Third-Generation Gravitational Wave Observatories,

    S. Hildet al., “Sensitivity Studies for Third-Generation Gravitational Wave Observatories,” Class. Quant. Grav. 28, 094013 (2011), arXiv:1012.0908 [gr-qc]

    Pith/arXiv arXiv 2011

  80. [80]

    Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy be- yond LIGO,

    David Reitzeet al., “Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy be- yond LIGO,” Bull. Am. Astron. Soc.51, 035 (2019), arXiv:1907.04833 [astro-ph.IM]

    Pith/arXiv arXiv 2019

Showing first 80 references.