Constraints on parity and Lorentz violations from gravitational waves: a comparison between single-parameter and multi-parameter analysis

Hai-Tian Wang; Tao Zhu; Wei-Hua Guo; Yuan-Zhu Wang

REVIEW 4 major objections 5 minor 3 cited by

Constraints on parity and Lorentz violations from gravitational waves: a comparison between single-parameter and multi-parameter analysis

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper establishes that single-parameter tests of parity and Lorentz symmetry in gravitational-wave propagation give constraints of the same order of magnitude as multi-parameter tests, and therefore remain robust for current and…

desk verdict Useful first multi-parameter comparison in the [24,25] parity/Lorentz framework, but the robustness claim is undercut by event selection on single-parameter tightness and by unreported alpha values. read the letter →

arxiv 2507.09705 v2 pith:O6V4NAMD submitted 2025-07-13 gr-qc

Wei-Hua Guo , Yuan-Zhu Wang , Tao Zhu , Hai-Tian Wang This is my paper

classification gr-qc MSC 83C3583D0583B05 PACS 04.30.-w04.80.Cc

keywords gravitationalwavesparityviolationLorentzwavebirefringencedispersionBayesianparameterestimationGWTC-3testsofgeneralrelativity

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

This paper asks whether the standard way of testing parity and Lorentz symmetry with gravitational waves—varying one deformation parameter at a time while fixing others to their general-relativity values—remains trustworthy when several symmetry-violating effects are actually present at once. It constructs multi-parameter waveform models motivated by modified gravity theories, runs full Bayesian parameter estimation on selected LIGO-Virgo-KAGRA binary black hole signals, and compares the resulting 90% limits on the parity-violation scale $M_{\rm PV}$ and Lorentz-violation scale $M_{\rm LV}$ with single-parameter limits. For three of the four two-parameter models the bounds agree to the same order of magnitude; the exception is a model in which both parameters modify the waveform phase, where the two effects partially compensate and weaken the multi-parameter bound. The paper concludes that single-parameter tests are robust for current and future observations, while cautioning that when parameters act on the same aspect of the waveform (phase) degeneracies matter.

What carries the argument

The central object is the parametrized propagation equation for circularly polarized gravitational waves, $h''_A+(2+\bar\nu+\nu_A)Hh'_A+(1+\bar\mu+\mu_A)k^2h_A=0$, which encodes parity violation in $\nu_A,\mu_A$ and Lorentz violation in $\bar\nu,\bar\mu$. These parameters generate frequency-dependent amplitude corrections $\delta h_1=A_\nu(\pi f)^{\beta_\nu}$, $\delta h_2=-A_{\bar\nu}(\pi f)^{\beta_{\bar\nu}}$ and phase corrections $\delta\Psi_1=A_\mu(\pi f)^{\beta_\mu+1}$, $\delta\Psi_2=A_{\bar\mu}(\pi f)^{\beta_{\bar\mu}+1}$ to the GR waveform. The argument works by adding these corrections to the IMRPhenomXPHM waveform template, running nested-sampling Bayesian inference, and comparing posteriors of the coefficient pairs; the presence or absence of correlation between the two deformation parameters is what decides whether single- and multi-parameter analyses agree.

What would settle it

Run the same two-parameter Bayesian inference on the events that were excluded from the top-six selection in Ref. [25] (or on the full LVK catalog) and check whether any model-1, model-2, or model-3 event yields a 90% $M_{\rm PV}$ or $M_{\rm LV}$ limit from the multi-parameter analysis more than ten times weaker than the single-parameter limit; one such event would contradict the robustness claim.

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Extended reading notes

Core claim

Within the parametrized framework of Refs. [24] and [25], the paper shows that for parity-violating models with $(\beta_\nu,\beta_\mu)=(1,-1)$ and $(1,1)$ and for the Lorentz-violating model with $(\beta_{\bar\nu},\beta_{\bar\mu})=(2,2)$, Bayesian two-parameter estimation gives 90% credible limits on $M_{\rm PV}$ and $M_{\rm LV}$ of the same order of magnitude as single-parameter estimation. The reason is that in these models one parameter changes the waveform amplitude and the other changes the phase, so their effects do not strongly correlate. In Model 4, where both parameters appear in the phase as $(\pi f)^3$ and $(\pi f)^5$ terms, the two parameters do correlate and the multi-parameter constraint is somewhat weaker, though still comparable. A four-parameter model combining parity and Lorentz effects, analyzed on the loudest event GW250114 082203, shows weaker multi-parameter limits because the signal is dominated by one circular polarization, causing amplitude parameters to degenerate.

Load-bearing premise

The robustness conclusion rests on four selected events per model being representative; if events with stronger mutual degeneracies were included, the multi-parameter bounds could fall more than an order of magnitude below the single-parameter ones.

Editorial extensions

If this is right

For the amplitude-plus-phase models, previously published single-parameter bounds on $M_{\rm PV}$ and $M_{\rm LV}$ from GWTC-3 events remain valid even if multiple parity- and Lorentz-violating effects coexist.
Phase-based constraints are orders of magnitude tighter than amplitude-based constraints, so future searches should prioritize phase modifications such as velocity birefringence and dispersion.
When two deformation parameters act on the same part of the waveform (both phase), a full multi-parameter run is needed to avoid overstating the bound; single-parameter limits should be reported with this caveat.
The same-order-of-magnitude agreement means the large computational cost of multi-parameter analyses may be unnecessary for models whose corrections separate into amplitude and phase channels.
For future ground- and space-based detectors with broader frequency coverage, the robustness conclusion implies that single-parameter forecasts remain indicative when the same model structure holds.

Reading between the lines

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

The paper's event selection (four per model from the six tightest in Ref. [25]) means the robustness claim is not yet a catalog-wide statement; applying the same comparison to all O1-O4 binary black holes, or to neutron-star binaries, could reveal models or events where the single/multi gap exceeds an order of magnitude.
The correlation pattern suggests a practical criterion: if a theory's deformation parameters act on different waveform channels (amplitude vs phase), single-parameter tests are safe; if they act on the same channel, a targeted multi-parameter run or a correlation-mitigating prior as in Ref. [55] is advisable.
The degeneracy with luminosity distance and effective spin seen for some events implies that joint multi-band observations (space-based plus ground-based) could break these correlations and sharpen both single- and multi-parameter bounds.
A testable extension would be to compute Bayesian evidence ratios between single- and multi-parameter models on the same events; GR-favoring evidence would independently quantify how much complexity the data support.

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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

You should know two things about this one. It is the first multi-parameter Bayesian comparison done inside the [24,25] parity/Lorentz propagation framework, and the headline finding — that amplitude and phase corrections do not degenerate for Models 1–3 — looks genuinely solid. But the paper's broader claim that single-parameter tests are robust for 'current and future GW observations' is not supported by the event selection, because they picked the events that gave the tightest single-parameter limits in their earlier paper. That selection can bias the comparison toward the no-degeneracy outcome.

What the paper does well: the analysis is careful and transparent. The Bayesian setup is standard (Bilby, IMRPhenomXPHM), the posterior plots back up the claim that Aν and Aμ (or Aνbar and Aμbar) are not correlated in Models 1–3, and the Model 4 degeneracy between two phase terms is physically reasonable and clearly shown. The appendix on Model 5 is also honest in showing that a four-parameter fit on one loud event gives weaker limits and visible correlations. The writing is straightforward and the connection to specific gravity theories is useful.

The soft spots, in order of importance. First, the event selection. Section IV, item (1), says they chose, per model, four events from the six that gave the tightest constraints in [25]. That is selection on the very thing being tested. Tight single-parameter constraints usually mean high-SNR, clean signals, and those may also be the events where degeneracies are weakest. The appendix's Model 5 already shows factor 2–3 weaker multi-parameter limits and correlations for one loud event, so the phenomenon is not confined to Model 4. A full-catalog rerun, or at least a set including some typical events, is needed before generalizing.

Second, the alpha constants αν, αμ, etc. are treated as constants but their values are never given. Without them the reported MPV and MLV bounds in Table II cannot be reproduced. That is a clear reproducibility gap, though it does not affect the single-vs-multi comparison within a fixed pipeline.

Third, there are no injection/recovery tests. For a paper whose point is that multi-parameter fits behave a certain way, a few injections would strengthen the claim a lot.

Fourth, the abstract says 'one specific model showing a degeneracy', but Model 5 also shows degeneracies. The conclusion should acknowledge that.

None of this is fatal. The central observation — amplitude and phase parameters do not degenerate in Models 1–3 — is well supported by the posteriors, and the comparison is a legitimate contribution to the GW tests-of-GR literature. I would send it to peer review, but with a request for a full-catalog or representative-event analysis, stated alpha values, and ideally injection tests. The authors have done a useful thing here; they just need to not overclaim the generality.

Referee Report

4 major / 5 minor

Summary. The paper constructs a set of multi-parameter gravitational-wave waveform models that incorporate parity-violating and Lorentz-violating propagation effects, and compares the resulting Bayesian constraints on the energy scales M_PV and M_LV with those obtained from single-parameter analyses. Four two-parameter models are analyzed with selected LIGO-Virgo-KAGRA binary black hole events, plus a four-parameter model applied to one loud event in an appendix. The authors report that for Models 1–3 the multi-parameter and single-parameter 90% limits are of the same order of magnitude, while Model 4 (and Model 5 in the appendix) yield weaker multi-parameter constraints due to parameter degeneracies. From this they conclude that single-parameter tests of parity and Lorentz symmetry are robust for current and future gravitational-wave observations.

Significance. The question addressed—whether single-parameter constraints on parity and Lorentz violations remain reliable when multiple deformation parameters are present—is methodologically important for the interpretation of existing and future gravitational-wave tests of gravity. The paper uses a well-established parametrized waveform framework, real LVK data, and a standard Bayesian inference pipeline, and it presents the comparison in a transparent way through tables and posterior plots. The comparison is empirical rather than circular: the multi-parameter constraints come from separate fits, not from the single-parameter results. If the central claim survives scrutiny, the paper would provide useful justification for the common practice of reporting single-parameter limits. However, the event selection and a few missing technical details currently limit the strength of the conclusion.

major comments (4)

[Section IV, item (1); Section V, first paragraph] The event selection is based on the very quantity whose robustness is being tested. The paper selects, per model, four events from the six that gave the tightest single-parameter constraints in [25]. This is a post hoc selection on single-parameter sensitivity, and it may preferentially choose events in which the two deformation parameters have weak degeneracy. The appendix Model 5 already demonstrates that a loud event can show factor 2–3 weaker multi-parameter constraints and visible correlations between A_nu and A_barnu and between A_mu and A_barmu. The abstract's general claim that single-parameter tests are robust for 'current and future GW observations' is therefore not supported by the selected subset. The authors should either perform a full-catalog comparison, or explicitly reframe the conclusion as holding only for events with strong single-parameter sensitivity, or justify representativeness with quantitative evidence.
[Section II, Eqs. (2.11)–(2.14); Section V; Table II] The numerical values of the alpha coefficients (alpha_nu, alpha_mu, alpha_barnu, alpha_barmu, alpha_(2), alpha_(4)) are never specified. The paper states that these functions are treated as constants, but it does not give their values, nor does it state that they are set to unity or taken from [25]. Since the reported limits on M_PV and M_LV are obtained by converting posterior samples of A_nu, A_mu, etc. through Eqs. (2.11)–(2.14), the absolute constraints in Table II are not reproducible without this information. The relative single- versus multi-parameter comparison is less affected if the same alpha values are used in both analyses, but the claimed agreement with [25] and the numerical limits themselves cannot be verified. Please specify the values or the exact prescription used in the mapping.
[Section V; Appendix A, Table III] No injection–recovery tests or convergence diagnostics are provided for the multi-parameter runs. The models have up to four additional parameters, and the 90% limits could be sensitive to the chosen uniform prior bounds on the deformation parameters. For example, the prior row in Table III for Model 2, beta_mu = 1, single-parameter analysis of GW190727 060333 is inconsistent with the stated setup: it lists A_nu in [-0.005, 0.005] and A_mu fixed to 0, whereas the single-parameter analysis should fix A_nu = 0 and sample A_mu. At minimum, the authors should show that the posterior distributions do not pile up at the prior boundaries and should demonstrate, for representative events, that injected non-GR signals are recovered with the intended credible intervals.
[Abstract; Section VI; Appendix B, Table II] The abstract states that only 'one specific model' shows a degeneracy between deformation parameters, but the appendix Model 5 also yields multi-parameter constraints that are weaker than the single-parameter ones by factors of about 2–3 for all four parameters. This is more than one model, and the conclusion should be softened to reflect that the robustness is model-dependent. In addition, the claim about 'future GW observations' is not established by this analysis, which uses only current O1–O3 events and one additional loud event; future detectors probe different frequency bands and source populations, where degeneracies may behave differently.

minor comments (5)

[Section VI heading] The heading 'CONCULSION' should read 'CONCLUSION'.
[Throughout] There are several typos: 'E.q.' should be 'Eq.', 'briefringences' should be 'birefringences', 'Densitiy' in Figure 3 axis labels should be 'Density', and 'noise abstractions' in Section V should likely be 'noise realizations'.
[Appendix A, Table III] The Model 2 beta_mu = 1 single-parameter row for GW190727 060333 appears to be a typo: the prior should be A_nu = 0 and A_mu sampled, rather than A_nu in [-0.005, 0.005] and A_mu = 0. Please correct and check the corresponding entry.
[Section IV, paragraph after event selection] The sentence 'eight events are selected to constrain Lorentz violation and another eight to constrain both violations' is confusing; based on Table I, the second set appears to be for parity violation, not 'both violations'. Please clarify.
[Section II, Eq. (2.6)] The notation 'e^{rho_A delta h_1 + delta h_2} e^{i(...)}' is ambiguous; adding parentheses around the exponent arguments would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

Comparison between single- and multi-parameter fits is empirical and self-contained; self-citations to [25] set up the analysis but do not force the result.

full rationale

The central comparison in this paper is empirical and self-contained: for each model, two independent Bayesian fits (single-parameter with one deformation fixed to zero; multi-parameter with both free) are run on the same events, and the resulting 90% limits on M_PV and M_LV are compared. The multi-parameter limits are not obtained by transforming or re-weighting the single-parameter limits; they come from separate nested-sampling runs with independent priors, so the conclusion that the constraints are of the same order does not reduce to a fitted input. The main self-citations are to the waveform parametrization of [24,25] and to the event-selection rule of [25]. These are load-bearing for the setup, but they are not the quantity being tested and they do not by construction force the comparison: the same parametrization and event choices could have produced large discrepancies (as indeed happens in Model 4 and Model 5), which shows the comparison has discriminating content. The selection of the four tightest events from [25] is a potential selection-bias concern for the generality of the 'robustness' claim, but it is not a circular reduction: no prediction is defined in terms of the single-parameter fit, and the paper does not rename fitted values as predictions. Therefore no specific circular step can be identified with the required quote-and-reduction evidence, and the score is at the minor-self-citation level.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The analysis adds no new physical entities. Its adjustable content consists of the deformation amplitudes fitted to data and the unspecified constant alpha coefficients that set the scale of the converted mass bounds.

free parameters (6)

A_nu (amplitude birefringence amplitude) = posterior distributions, not a single value; priors e.g. [-0.05, 0.05] in some runs (Table III)
Fitted deformation parameter in Models 1, 2, and 5.
A_mu (velocity birefringence phase amplitude) = posterior distributions; priors vary by event, e.g. [-2,2] or [-0.08,0.08] (Table III)
Fitted deformation parameter in Models 1, 2, and 5.
A_barnu (Lorentz-violating amplitude correction) = posterior distributions; priors e.g. [-0.002,0.002] (Table III)
Fitted in Models 3 and 5.
A_barmu (Lorentz-violating phase correction) = posterior distributions; priors e.g. [-5e-7,5e-7] (Table III)
Fitted in Models 3, 4, and 5.
A_barmu,(2) and A_barmu,(4) (two dispersion phase amplitudes) = posterior distributions; priors e.g. [-5e-7,5e-7] and [-5e-15,5e-15] (Table III)
Fitted in Model 4.
alpha_nu, alpha_mu, alpha_barnu, alpha_barmu, alpha_(2), alpha_(4) (constant coefficients of the alpha functions) = not specified; assumed constant
Their values are not stated in the paper. The reported M_PV and M_LV bounds in Table II depend on them through Eqs. (2.11)-(2.14).

assumptions (6)

domain assumption The parametrized wave equation (2.1) from [24,25] correctly describes parity and Lorentz violations in GW propagation.
The entire analysis assumes this parametrization; its validity is taken from prior work by the same group.
domain assumption The alpha functions vary slowly over z <~ 1 and can be treated as constants.
Stated in Section II; if alpha evolves with redshift, the mapping from fitted A parameters to M_PV/M_LV changes.
domain assumption The waveform distortion has the exponential form of Eq. (2.6) with the listed amplitude and phase corrections.
Derived in [24,25], not re-derived here.
domain assumption Detector noise is stationary and Gaussian with known power spectral density.
Standard assumption in GW parameter estimation, stated in Section IV.
domain assumption Planck 2018 cosmology is correct.
Used to compute the redshift-distance relation in the mapping from fitted amplitudes to mass scales.
domain assumption The four events selected per model are representative for testing the comparison.
Selection criteria in Section IV pick events with the tightest single-parameter constraints from [25]; representativeness is assumed.

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Cite this review

Pith. "Pith review of Constraints on parity and Lorentz violations from gravitational waves: a comparison between single-parameter and multi-parameter analysis." pith.science (2026). https://pith.science/paper/O6V4NAMD

@misc{pith2026250709705,
  author       = {Pith},
  title        = {Pith review of: Constraints on parity and Lorentz violations from gravitational waves: a comparison between single-parameter and multi-parameter analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O6V4NAMD}},
  note         = {Machine review of arXiv:2507.09705}
}

read the original abstract

The growing catalog of gravitational wave (GW) detections by the LIGO-Virgo-KAGRA Collaboration enables increasingly stringent tests of general relativity, particularly regarding possible violations of parity and Lorentz symmetry. Parity and Lorentz violations in gravity can modify both the damping rate and dispersion relation of GWs, leading to birefringence, frequency-dependent damping, and dispersion effects in the propagation of GWs. These effects result in amplitude and phase corrections of the waveforms of GWs produced by the coalescence of compact binaries, which enable us to constrain parity- and Lorentz-violating effects by analyzing GW signals detected by LIGO-Virgo-KAGRA detectors with the distorted waveforms. While most current analyses employ single-parameter methods-varying one deformation parameter at a time-modified gravity theories often predict multiple, coexisting deviations. In this work, we construct several specific multi-parameter GW waveform models incorporating parity- and Lorentz-violating effects and perform full Bayesian parameter estimation to compare multi-parameter and single-parameter constraints. We find that including multiple deformation parameters yields constraints on individual parameters that are generally comparable to those from single-parameter analyses, despite one specific model showing a degeneracy between the deformation parameters. Our results support the robustness of single-parameter tests for parity and Lorentz symmetry of gravity in current and future GW observations.

Figures

Figures reproduced from arXiv: 2507.09705 by the authors.

**Figure 2.** FIG. 2. Corner plots of posterior distributions for one gravitational-wave event chosen for each of the four non-GR models. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3. The posterior probability distributions of [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Corner plots of posterior distributions and correlations of parity- and Lorentz-violating amplitude parameters ( [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5. The same as Figure [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: FIG. 6. The same as Figure [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

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Forward citations

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