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

Bo-Yang Zhang; Jian-Ming Yan; Jing-Fei Zhang; Tao Zhu; Xin Zhang

REVIEW 2 major objections 7 minor 7 cited by

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

T0 review · 2 major / 7 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper forecasts that next-generation ground- and space-based gravitational-wave observatories will tighten parity- and Lorentz-violation bounds by one to three orders of magnitude, with space-based detectors winning by roughly a…

desk verdict Solid forecast with a clear qualitative message, but the headline energy-scale bounds and the graviton-mass claim are convention-dependent because the α couplings are never fixed. read the letter →

arxiv 2502.04776 v2 pith:HENWUP6W submitted 2025-02-07 gr-qc hep-th

Bo-Yang Zhang , Tao Zhu , Jian-Ming Yan , Jing-Fei Zhang , Xin Zhang This is my paper · ORCID

classification gr-qchep-th PACS 04.30.-w04.80.Nn11.30.Cp11.30.Er

keywords gravitationalwavesparityviolationLorentzgravitational-wavedetectorsEinsteinTelescopeLISABayesianinferencegravitonmass

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Parity and Lorentz symmetry are exact in general relativity, and any observed breaking would signal new physics in the gravitational sector. This paper asks how tightly the next generation of gravitational-wave detectors could test that breaking. It injects simulated compact-binary signals into planned ground-based networks (the Einstein Telescope and Cosmic Explorer) and space-based networks (LISA with Taiji or TianQin), then recovers them with Bayesian inference using waveforms modified by parity- and Lorentz-violating propagation effects. The central forecast is that the combined ground-based detectors tighten the accessible energy-scale bounds $M_{\rm PV}$ and $M_{\rm LV}$ by one to three orders of magnitude relative to current observations, and that space-based detectors become the more powerful probe for one Lorentz-violating dispersion case ($\bar\beta_\mu=-2$), gaining roughly three orders of magnitude over the ground-based detectors and enabling a graviton-mass bound $m_g\lesssim10^{-35}$ GeV. The result matters because it identifies which planned observatories can best test fundamental spacetime symmetries.

What carries the argument

The central object is the modified waveform $\tilde h_A(f)=\tilde h^{\rm GR}_A(f)\,e^{\rho_A\delta h_1+\delta h_2}\,e^{i(\rho_A\delta\Psi_1+\delta\Psi_2)}$. Parity violation enters through amplitude birefringence (left- and right-handed polarizations damp at different rates, exponent $\beta_\nu$) and velocity birefringence (the two polarizations travel at different speeds, exponent $\beta_\mu$). Lorentz violation enters through frequency-dependent damping (exponent $\bar\beta_\nu$) and modified dispersion (exponent $\bar\beta_\mu$). The paper fixes the frequency dependence to eight benchmark cases—odd integers for parity, even integers for Lorentz—and maps the recovered waveform coefficients $A_\nu,A_\mu,A_{\bar\nu},A_{\bar\mu}$ to the energy scales $M_{\rm PV}$ and $M_{\rm LV}$ through Eqs. (2.8) and (2.12). Bayesian inference on the injected signals then turns the detector sensitivity curves into posterior bounds on those energy scales.

What would settle it

Re-run the same injection-and-recovery pipeline with explicit stated values of the four $\alpha$ coupling functions and compare the resulting 90% bounds to Tables III and IV; if varying the $\alpha$'s within their allowed range shifts the inferred $M_{\rm PV}$ and $M_{\rm LV}$ by more than the quoted improvements, the forecast is set by that unstated convention rather than by detector sensitivity. A direct empirical check would be an actual Einstein Telescope plus Cosmic Explorer detection of a GW170817-like event whose recovered $M_{\rm PV}$ upper bound for $\beta_\mu=-1$ is weaker than the forecast $2.0\times10^{-43}$ GeV.

Watch

Extended reading notes

Core claim

Using the parametrized propagation equation $h''_A+(2+\bar\nu+\nu_A)\mathcal{H}h'_A+(1+\bar\mu+\mu_A)k^2h_A=0$ and the resulting modified waveforms, the paper derives posterior constraints on the parity-violation energy scale $M_{\rm PV}$ and the Lorentz-violation energy scale $M_{\rm LV}$ for eight frequency-dependence cases. It finds that the Einstein Telescope and Cosmic Explorer jointly tighten the 90% credible bounds on these scales by factors of roughly 20 to 263 for individual events relative to current observations, with the largest gain in the velocity-birefringence case $\beta_\mu=-1$. For positive $\beta$ exponents, ground-based detectors outperform space-based ones because the amplitude and phase corrections grow with frequency. For $\bar\beta_\mu=-2$, however, the space-based networks LISA+Taiji and LISA+TianQin yield bounds on $M_{\rm LV}$ about three orders of magnitude tighter than the ground-based networks, because lower-frequency gravitational waves carry more constraining power for that dispersion law; the same case translates into a graviton-mass bound $m_g\lesssim10^{-35}$ GeV.

Load-bearing premise

The forecast depends on assigning fixed numerical values to the arbitrary coupling functions (the $\alpha$'s) in Eqs. (2.8) and (2.12), and the paper never states those values, so the absolute energy-scale bounds are reproducible only if one guesses the intended convention.

Editorial extensions

If this is right

The Einstein Telescope and Cosmic Explorer jointly improve the 90% credible bounds on $M_{\rm PV}$ and $M_{\rm LV}$ by roughly one to three orders of magnitude per event compared with current observations, with the largest gain (about 263 times) in the $\beta_\mu=-1$ velocity-birefringence case.
For positive $\beta$ exponents, where the corrections grow with frequency, ground-based detectors outperform space-based detectors; for $\beta_\mu=-1$ the two classes of networks become comparable.
For the $\bar\beta_\mu=-2$ Lorentz-violating dispersion case, LISA+Taiji and LISA+TianQin tighten $M_{\rm LV}$ by about three orders of magnitude relative to the ground-based networks, and the same case implies a graviton-mass bound $m_g\lesssim10^{-35}$ GeV.
Heavier, more distant sources give tighter constraints for the low-$\beta$ cases, while lighter nearby sources such as GW170817-like binaries give the tightest constraints for the large positive-$\beta$ cases.
Ground- and space-based detectors are complementary: neither class dominates all eight frequency-dependence cases, so a combined network would cover a wider range of symmetry-breaking theories.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the $\alpha$ coupling functions are never assigned numerical values in the paper, the absolute energy-scale numbers in Tables III and IV should be read as order-of-magnitude forecasts under an implicit convention; making that convention explicit would let other groups reproduce and extend the tables.
The complementarity seen here suggests that a combined ground-plus-space network, which the paper does not simulate, would likely dominate every case because it would cover both the high-frequency corrections (positive $\beta$) and the low-frequency dispersion effects (negative $\beta$) at once.
The same injection-and-recovery machinery could rank the planned detectors for other propagation anomalies, such as frequency-dependent damping or extra polarizations, by substituting the corresponding waveform corrections.
A graviton-mass bound near $10^{-35}$ GeV, if realized, would compete with bounds from other astrophysical and cosmological channels, but that comparison inherits the same coupling-function convention.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a competent, narrowly-scoped forecast paper, and the qualitative findings are sound. The absolute energy-scale bounds in the tables are not reproducible as written because the paper never fixes the α couplings that appear in Eqs. (2.8) and (2.12). That's the one thing to know.

What's new: the authors apply their own parametrization (Refs. [30,31]) to space-based networks (LISA+Taiji, LISA+TianQin) and to two cases not covered before (βμ=1 and β̄μ=-2). They do real injection-recovery with BILBY and dynesty, using design sensitivity curves. The physical message—that high-frequency ground-based detectors win for positive β's, while space-based detectors win for βμ=-1 and β̄μ=-2 because of the low-frequency sensitivity—follows directly from the frequency scaling, so I trust that part.

Soft spots, in order of importance. First, the α problem. The text says αν, αμ, ᾱν, ᾱμ are 'considered as constant' for local sources, but the constant values are never stated. Since the inference is on the A coefficients and the conversion to MPV/MLV goes through those α's, every number in Tables III and IV carries an implicit α convention. The mg≲10^-35 GeV bound inherits the same problem. This is fixable by stating the convention (e.g., α=1), but as it stands the headline numbers are not well-defined. Second, the injection details are too sparse: Table II lists masses and redshift only, with no sky position, inclination, spins, or distance, and the priors on the A parameters are described only as 'uniformly distributed' without ranges. That makes the numerical exercise hard to reproduce. Both issues are omissions, not conceptual errors.

The comparisons with LVK use the authors' own published results, which is fine. The improvement factors quoted are ratios, so they cancel a constant α and should survive the fix.

Verdict: worth refereeing. The paper is a legitimate extension of an established program, the methods are standard, and the problems are matters of missing detail, not wrong physics. A referee should ask for the α convention and full injection/prior specifications; with those added, this becomes a useful reference for the detector community.

Referee Report

2 major / 7 minor

Summary. Zhang et al. present a Bayesian injection-recovery forecast of how well future gravitational-wave detectors will constrain the energy scales M_PV and M_LV associated with parity and Lorentz violations in gravitational-wave propagation. The analysis uses the parametrized modified waveforms of Refs. [30,31], in which amplitude and phase corrections with power-law frequency dependence are controlled by exponents (beta_nu, beta_mu, beta_barnu, beta_barmu) and by coefficients A_nu, A_mu, A_barnu, A_barmu that are mapped to M_PV and M_LV through Eqs. (2.8) and (2.12). Three LVK-like events (GW170817-, GW150914-, and GW190521-like) are injected into an ET+CE network, and three massive black-hole binaries (m1 = 10^4, 10^5, 10^6 solar masses at z = 1, 5, 10) are injected into LISA+Taiji and LISA+TianQin networks; the resulting posteriors on M_PV and M_LV are compared with LVK constraints from Refs. [30,40,42]. The reported findings are that ET+CE improve on LVK by one to three orders of magnitude, that for beta_mu = -1 the space-based detectors become competitive with ET+CE, and that for beta_barmu = -2 the space-based detectors give the tightest M_LV bounds, from which a graviton-mass bound m_g less than about 10^-35 GeV is claimed.

Significance. If the quoted numbers are made well-defined, this paper will be a useful reference for planning tests of parity and Lorentz symmetry with third-generation and space-based detectors. Its strengths are a self-consistent Bayesian pipeline (BILBY/dynesty with externally specified sensitivity curves), systematic coverage of eight frequency-scaling cases including two cases (beta_mu = 1 and beta_barmu = -2) beyond the previous LVK analysis, and a clean qualitative separation between ground- and space-based detector capabilities that follows directly from the frequency dependence of the corrections and is robust to details of the analysis. The circularity concern raised in the stress-test note does not, on my reading, land: the forecast is an injection-recovery experiment against externally given noise curves, so the detector rankings are not imposed by construction. The main risks are the undisclosed conventions for the arbitrary functions alpha in Eqs. (2.8) and (2.12), on which all absolute bounds (and hence the graviton-mass claim) depend, and an inconsistency between the abstract's 'three orders of magnitude' statement and the tables for the beta_barmu = -2 case.

major comments (2)

[Sec. II, Eqs. (2.8) and (2.12); Tables III and IV] Equations (2.8) and (2.12) map the sampled waveform coefficients A_nu, A_mu, A_barnu, A_barmu to the energy scales M_PV and M_LV through the arbitrary functions alpha_nu(tau), alpha_mu(tau), alpha_barnu(tau), alpha_barmu(tau), but the paper never assigns values or functional forms to these functions, and the analysis treats them as constants. For constant alpha, each A is proportional to alpha, so a fixed posterior on A translates to M proportional to alpha^{1/beta} for positive beta and, for the two negative-exponent cases studied, to M_PV proportional to 1/alpha (beta_mu = -1) and M_LV proportional to alpha^{-1/2} (beta_barmu = -2). The absolute bounds in Tables III and IV, the improvement factors quoted in Section IV, and the graviton-mass bound m_g less than about 10^-35 GeV in the abstract are therefore not reproducible unless the alpha convention is stated; the LVK comparison values taken from Refs. [30,40,42] must likewise use the same convention for the quoted ratios to be meaningful. I ask the authors to state the adopted values of all four alpha functions (or to quote the results directly in terms of the A coefficients), to state the prior ranges for A_nu, A_mu, A_barnu, A_barmu in Sec. III.B (currently described only as 'uniformly distributed'), and to give the explicit formula connecting the beta_barmu = -2 bound on M_LV to m_g, since no such equation appears in the text.
[Abstract and Secs. IV.B and V] The abstract and the conclusion state that for beta_barmu = -2 the space-based detectors give bounds 'approximately three orders of magnitude tighter than those from ground-based GW detectors'. This is not what Tables III and IV show: the best ET+CE bound is 3.7 x 10^-33 GeV (GW190521-like) while the best LISA+Taiji bound is 5.1 x 10^-35 GeV (event3), a ratio of about 70, i.e., roughly two orders of magnitude. A factor of about 10^3 is instead obtained relative to the LVK combined bound of 8.3 x 10^-32 GeV quoted in Table III, which is the comparison actually made in Section IV.B. The wording in the abstract and conclusion should be corrected to match the tables, or the comparison basis should be stated explicitly.

minor comments (7)

[Sec. II.A and Sec. III.B] In the text following Eq. (2.7) the phase-correction exponent is printed as beta_nu + 1 ('delta Psi_1 = A_mu (pi f)^{beta_nu + 1}'), and the same typo is repeated in Sec. III.B in the description of the priors; both should read beta_mu + 1.
[Sec. IV] In the first paragraph of Section IV, the posteriors for the Lorentz-violating quantities are labeled M^{-beta_barnu}_PV and M^{-beta_barmu}_PV; the subscript should be LV for these two quantities, as in the figure captions.
[Table II] The label 'evnet2' in Table II is a typo for 'event2'.
[Sec. IV.A] The sentence reporting improvement factors 'about 20 times in binary neutron star systems and about 18 times in binary black hole systems' does not name the case it refers to; from Table III the numbers match the beta_barmu = -2 case (21 and 18 for GW170817-like and GW190521-like, while GW150914-like gives about 13), so the case label should be added and the event dependence clarified.
[Sec. II] The statement that Eqs. (2.5) and (2.6) 'can be considered as constant' is justified by the locality of the sources, but events 2 and 3 in Table II are at z = 5 and z = 10; the constant-alpha assumption should be stated explicitly as a modeling choice for these high-redshift events, since a time-varying alpha would change the integrals in Eqs. (2.8) and (2.12).
[Sec. III and Eq. (3.5)] The parity- and Lorentz-violating corrections are derived in the stationary-phase approximation and are strictly inspiral-phase results, yet they are appended to the full IMRPhenomPv2 templates that include merger and ringdown; a brief comment on the validity of this extension would be useful.
[Figs. 1-3] All posterior panels in Figs. 1-3 show the axis label 'Probability Densitiy', which should be 'Probability Density'.

Circularity Check

0 steps flagged · score 0.0 of 10

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities. Its forecast rests on a phenomenological parameterization with four coupling functions that are not specified, plus chosen injected events and adopted detector sensitivities.

free parameters (5)

alpha_nu (amplitude birefringence coupling function) = Not stated in the paper.
The conversion from the constrained coefficient A_nu to the energy scale M_PV in Eq. (2.8) depends on alpha_nu(tau0) and alpha_nu(tau_e); the paper never specifies these functions, so the M_PV bounds assume an implicit O(1) or other value.
alpha_mu (velocity birefringence coupling function) = Not stated in the paper.
The conversion from the constrained coefficient A_mu to M_PV in Eq. (2.8) depends on the integral of alpha_mu over redshift; without specifying alpha_mu, the reported M_PV bounds are not uniquely determined.
alpha_barnu (Lorentz-violating damping coupling function) = Not stated in the paper.
The conversion from A_barnu to M_LV in Eq. (2.12) requires alpha_barnu(tau0) and alpha_barnu(tau_e); the values are not given.
alpha_barmu (Lorentz-violating dispersion coupling function) = Not stated in the paper.
The conversion from A_barmu to M_LV in Eq. (2.12) depends on the integral of alpha_barmu; without specifying it, the M_LV bounds are not well-defined.
Simulated event parameters (masses, redshifts, spins, sky locations) = GW150914-like, GW170817-like, GW190521-like; event1, event2, event3 as in Table II.
The forecast constraints depend on the chosen source parameters. The paper selects three events per detector class, and the space-based events are equal-mass and non-spinning, which is a choice that affects the reported bounds.

assumptions (4)

domain assumption The modified GW propagation equation (2.4) with parameters nu_A, mu_A, barnu, barmu correctly describes parity and Lorentz violations.
Adopted from Refs. [30,31]; the paper explicitly assumes linear perturbation, transverse-traceless modes, and subdominant modifications in Sec. II.
standard math The waveform models IMRPhenomPv2 and IMRPhenomPv2NRTidal accurately describe the GR baseline.
Used to generate injections and as the GR template; these are standard approximants in the field.
standard math The stationary phase approximation (SPA) is valid for the modified waveforms.
Used in deriving Eq. (2.7), following Refs. [30,31].
domain assumption The detector sensitivity curves (ET-D, CE, LISA, Taiji, TianQin) represent the actual future detectors.
Forecasts are only as good as these PSDs; systematic uncertainties in the curves are not propagated into the reported bounds.

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Cite this review

Pith. "Pith review of Constraining parity and Lorentz violations in gravity with future ground- and space-based gravitational wave detectors." pith.science (2026). https://pith.science/paper/HENWUP6W

@misc{pith2026250204776,
  author       = {Pith},
  title        = {Pith review of: Constraining parity and Lorentz violations in gravity with future ground- and space-based gravitational wave detectors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HENWUP6W}},
  note         = {Machine review of arXiv:2502.04776}
}

abstract

The future ground- and space-based gravitational wave (GW) detectors offer unprecedented opportunities to test general relativity (GR) with greater precision. In this work, we investigate the capability of future ground-based GW detectors, the Einstein Telescope (ET) and the Cosmic Explorer (CE), and space-based GW detectors, LISA, Taiji, and TianQin, for constraining parity and Lorentz violations in gravity. We inject several typical GW signals from compact binary systems into GW detectors and perform Bayesian inferences with the modified waveforms with parity and Lorentz-violating effects. These effects are modeled in the amplitude and phase corrections to the GW waveforms with their frequency-dependence described by factors $\beta_{\nu}$, $\beta_{\mu}$, $\beta_{\bar \nu}$, and $\beta_{\bar \mu}$. Our results show that the combined observations of ET and CE will impose significantly tighter bounds on the energy scale of parity and Lorentz violations ($M_{\rm PV}$ and $M_{\rm LV}$) compared to those given by LIGO-Virgo-KAGRA (LVK) detectors. For cases with positive values of $\beta_{\nu}$, $\beta_{\mu}$, $\beta_{\bar \nu}$, and $\beta_{\bar \mu}$, the constraints on $M_{\rm PV}$ and $M_{\rm LV}$ from ground-based detectors are tighter than those from the space-based detectors. For the $\beta_{\mu} = -1$ case, space-based GW detectors provide constraints on $M_{\rm PV}$ that are better than current LVK observations and comparable to those from ET and CE. Additionally, space-based detectors exhibit superior sensitivity in constraining $M_{\rm LV}$ for $\beta_{\bar \mu} = -2$ case, which is approximately three orders of magnitude tighter than those from ground-based GW detectors. This scenario also enables bounds on the graviton mass at $m_g \lesssim 10^{-35}\; {\rm GeV}$. These findings highlight the promising role of future GW observatories in probing fundamental physics beyond GR.

Figures

Figures reproduced from arXiv: 2502.04776 by the authors.

**Figure 2.** FIG. 2. The posterior distributions for [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3. The posterior distributions for [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

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