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Weyl quadratic gravity predicts gravitational waves with four polarization states: the two standard tensor modes plus two transverse vector modes from the Weyl gauge field, and no scalar modes.

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2026-08-02 06:04 UTC pith:NKVSOPQQ

load-bearing objection A solid mode-count calculation for the truncated R^2+F^2 action, but the WQG-level claim needs a real justification of the dropped C^2 term and a consistent mass identification before the LVK prediction can be trusted. the 2 major comments →

arxiv 2607.13146 v1 pith:NKVSOPQQ submitted 2026-07-14 gr-qc hep-phhep-th

Weyl gauge symmetry at LIGO-Virgo-KAGRA

D. M. Ghilencea , V. -M. Mandric This is my paper
classification gr-qc hep-phhep-th PACS 04.30.-w04.50.Kd11.15.-q
keywords Weyl quadratic gravitygravitational wave polarisationsWeyl gauge symmetryde Sitter backgroundvector modesgeodesic deviationWeyl geometrytensor polarisations
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.

Weyl quadratic gravity, a gauge theory of scale symmetry built on Weyl conformal geometry, is a candidate extension of general relativity that in its broken phase reduces to the Einstein–Hilbert action plus a positive cosmological constant. The paper claims that gravitational waves around a de Sitter background in this theory carry the two transverse-traceless tensor polarizations of general relativity, plus two additional transverse vector polarizations originating from fluctuations of the Weyl gauge field, with no scalar polarizations present. To reach this conclusion the authors translate the geodesic-deviation equation from Riemannian to Weyl geometry and show that the vector modes are visible only when the perturbation analysis is done on a curved de Sitter background, not on flat spacetime. If the claim is correct, detecting transverse vector modes in gravitational waves would be direct evidence for Weyl gauge symmetry, and current ground-based detector data already begin to constrain such an outcome.

Core claim

The central claim is that the linearized gravitational-wave spectrum of Weyl quadratic gravity around a de Sitter background contains exactly two propagating tensor modes h^TT_ij and two transverse vector modes rho_i, and no scalar modes. The tensor modes obey equation (5.43), the same wave equation as in Einstein gravity with a positive cosmological constant; the vector modes obey equation (5.42), a modified wave equation with a mass term m_omega^2/H_0^2 = -72 d_1/d_2 = 12 alpha_0^2/xi^2, matching the mass acquired by the Weyl gauge boson through spontaneous symmetry breaking. In the flat-space limit the two vector modes become massless plane waves. The paper also establishes that the geode

What carries the argument

The machinery is the gauge-invariant scalar-vector-tensor decomposition of metric and Weyl-field fluctuations on a de Sitter background, built from the Weyl gauge covariant formulation of Weyl geometry in which the connection is metric-compatible. Gauge-invariant quantities S1, S2, S3, Vi, rho_i and h^TT_ij separate physical from pure-gauge degrees of freedom; the new physical objects are the transverse vector polarizations rho_i, whose wave equation (5.42) carries the Weyl gauge field mass. The geodesic-deviation equation (3.12) connects these modes to the observable displacement of test masses in a detector.

Load-bearing premise

The claim rests on the assumption that the simplified action used for the calculation, the de Sitter background with zero background Weyl field, and the decoupling of the massive Weyl boson together capture the full gravitational-wave content of Weyl quadratic gravity—in particular, that the omitted Weyl-tensor-squared term does not alter the mode spectrum.

What would settle it

A linearized perturbation calculation starting from the full Weyl quadratic gravity action (2.9), including the Weyl-tensor-squared term, would settle the internal consistency of the claim: if the vector modes rho_i change their wave equation or become ghost-like, the stated mode content fails. Observationally, a loud binary-black-hole event resolved by six unaligned detectors that rules out transverse vector polarizations at the predicted amplitude would falsify the prediction.

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

If this is right

  • Weyl quadratic gravity predicts four gravitational-wave polarization degrees of freedom—two tensor like general relativity and two transverse vector—with no scalar mode.
  • Detection of transverse vector modes would be a direct signal of the Weyl gauge field, hence of gauged scale symmetry rather than a generic alternative to general relativity.
  • The mass relation m_omega^2/H_0^2 = 12 alpha_0^2/xi^2 ties an observed vector-mode frequency to the Weyl coupling alpha_0 and the R^2 coupling xi, making the modes a probe of the theory's parameters.
  • Because the vector modes can be massive on sub-Hubble scales, they may travel slower than light and spread a chirp signal; multi-messenger events with electromagnetic counterparts can bound this velocity difference.
  • A definitive test requires enough independent detectors to separate all polarization hypotheses; current data do not yet distinguish the possibilities, though the tensor-vector hypothesis remains viable.

Where Pith is reading between the lines

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

  • Editorial inference: the prediction likely survives in the full action only if the omitted Weyl-tensor-squared term is genuinely inert; a direct linearized calculation of the complete action (2.9) is the fastest check, since that term, identical to the Riemannian Weyl tensor, could in principle couple to the vector modes.
  • Editorial inference: the two-vector-mode pattern is probably a generic signature of any theory with a massive gauge field for local scale invariance, and the de Sitter-based SVT method used here is a template for deriving the mode content of such theories.
  • Editorial inference: because vector polarizations carry a preferred spatial orientation, null-stream and network-correlation searches optimized for vector modes could probe this prediction with existing data before a full six-detector polarization breakdown becomes available.
  • Editorial inference: in the strong-field WDBI generalization, of which this action is the leading order, the vector modes may be sourced more efficiently or acquire nonlinear corrections, giving distinct waveform predictions for binary mergers.

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 paper studies gravitational-wave polarizations in Weyl quadratic gravity (WQG). Section 2 reviews the Weyl-geometry action; Section 3 derives the geodesic deviation equation in Weyl geometry; Section 4 develops the SVT/gauge-invariant formalism for Einstein-Hilbert theory on de Sitter; Section 5 repeats the analysis for the truncated action (5.1) containing only the \hat R^2 and \hat F^2 terms. The main result is that, around a de Sitter background, the linearized theory has the usual two transverse-traceless tensor modes plus two transverse vector modes from the Weyl gauge field, with no scalar modes. These vector modes are then compared with LVK polarization and time-of-flight tests.

Significance. If the claim is correct, it is observationally important: future polarization-resolved gravitational-wave observations could distinguish Weyl quadratic gravity from Einstein-Hilbert gravity. The paper has clear strengths: it gives a self-contained SVT decomposition, writes explicit linearized equations, constructs gauge-invariant variables, and derives the Weyl-geometric correction to the geodesic deviation equation. However, the central WQG claim is not yet established, because the calculation is performed for a truncated action and because the identification of the vector-mode mass with the broken-phase Proca mass contains a missing factor and no derivation. The result is therefore interesting and potentially valuable, but the physical interpretation and the direct comparison to LVK data require substantial additional work.

major comments (2)
  1. [Section 5, footnote 4; Conclusions] The full WQG action (2.9) contains the Weyl-tensor-squared term proportional to \hat C^2. The truncated action (5.1) omits this term, with the justification that, with appropriate boundary conditions, it is equivalent to Einstein-dS [44,45] and that the authors 'expect' it not to change the fluctuation equations. This is load-bearing: the paper's conclusion is about WQG, not about the truncated R^2+F^2 model. The cited equivalence is not shown at the level of the linearized fluctuation equations, and the \hat C^2 term may contribute to the very modes (5.42)-(5.43) that are claimed to be the full mode content. Please either include the \hat C^2 term in the linearized SVT analysis or prove rigorously that it decouples from the vector and tensor equations under the stated boundary conditions.
  2. [Eq. (5.42), (5.44), (5.48); Sections 5.3-5.4] There is a serious inconsistency between the mass obtained in the linearized equation and the mass used in the phenomenological discussion. Eq. (5.42) gives (\partial_0^2-\partial_m^2)\rho_i - 72 H^2 d1/d2 \rho_i = 0, so the mass-like term is m^2 = 12 \alpha_0^2 H^2/\xi^2, with no factor of M_p. Section 5.3 claims this agrees with the broken-phase Proca mass m_\omega \sim \alpha_0 M_p from [21], but that agreement would require \xi \sim H_0/M_p, which is not stated and is inconsistent with treating \xi as an O(1) perturbative coupling. Section 5.4 then sets A = sqrt(72 d1/|d2|) \sim \alpha_0 M_p/H_0, inserting an M_p/H_0 factor that is absent from (5.42). Moreover, the calculation is performed with \bar\omega_\mu = 0 and no Stueckelberg compensator, so \rho_i is not the massive broken-phase Weyl gauge boson; in the flat limit (5.46) it becomes exactly massless. The no-scalar-mode conclus
minor comments (5)
  1. [Appendix B, after Eq. (B-3)] Typo: '\partial^2 x^\lambda/\partial s \partial \lambda' should presumably read '\partial^2 x^\lambda/\partial s \partial \tau'.
  2. [Eq. (5.17)] The notation '\rho_i \equiv \rho_i' is redundant; also the sign conventions for the decomposition of \omega^\mu should be checked against (5.18)-(5.19), where \sigma appears with opposite signs in different invariants.
  3. [Section 5.4, Eq. (5.48)] The definition of A should be stated unambiguously: A = sqrt(72 d1/|d2|) has dimensions of M_p/H_0 only after an assumed identification; as written it is dimensionless. This is related to the major comment above, but the notation should be clarified.
  4. [Reference [47]] Citing 'work in progress' is not a standard reference; it should either be replaced by a published/preprint citation or described as private communication.
  5. [Throughout] There are several typographical/formatting issues: 'L VK' should be 'LVK', 'ω µ' should be '\omega_\mu', and the action (2.9) is typeset with broken math delimiters.

Circularity Check

1 steps flagged

Vector-mode polarisation derivation is self-contained, but the LVK mass/TOF prediction imports mω∼α0Mp from self-cited [21], contradicting the paper's own Eq. (5.42).

specific steps
  1. self citation load bearing [Section 5.3–5.4, Eqs. (5.42)–(5.48), footnote 1 / Ref. [21]]
    "The last term is the mass (m ω) of the Weyl gauge field inH 0 units:m 2 ω/H 2 0 =−72d 1/d2 = 12α 2 0/ξ2 >0, which agrees with the result for the mass in [21] (forq= 2). ... The vector modes, on the other hand, behave as massive modes, whose mass term in (5.44),i.e.A= p 72d 1/|d2| ∼α0Mp/H0, is coupled to curvatureH ∼η−1."

    Eq. (5.42) yields mω²/H0² = 12α0²/ξ², i.e. mω ∼ α0 H0/ξ. Section 5.4 instead takes A = sqrt(72d1/|d2|) ∼ α0 Mp/H0, inserting the Stueckelberg mass mω ∼ α0 Mp from Refs. [18,19,21] (same author). The claimed agreement with [21] is asserted; no derivation in this paper connects the curvature-induced coefficient −72H²d1/d2 to α0Mp. The LVK/TOF claims (ω > A/|η| ∼ O(mω), subluminal group velocity) therefore rest on the self-cited mass value rather than on the linearized equations. The existence of the vector modes is nevertheless derived independently, so this is a partial, not total, circularity.

full rationale

The central derivation (Section 5) is self-contained: the SVT decomposition, gauge invariants, and linearized equations (5.22)-(5.28) are solved in the paper to obtain the two tensor modes (5.43), two vector modes (5.42), and no scalars. The dS background is forced by the background equations (5.6)-(5.8), not assumed for convenience. The vector-mode operator in (5.42) emerges from the d2 F^2 term of action (5.1); no parameter is fitted and the mode content does not depend on the self-cited mass formula. Self-citations appear in the review of Weyl geometry (Refs. [9,10,17]), the Stueckelberg breaking, and the mass comparison with [21]; these are background/consistency uses. The one load-bearing self-citation is the mass mω ∼ α0Mp used in Section 5.4 for the LVK/TOF discussion: it is taken from Refs. [18,19,21] (same authors) and is not derivable from the paper's own eq. (5.42), which gives α0 H0/ξ; the claimed agreement is asserted. Thus the detectability prediction partially reduces to self-cited input, while the central polarisation-mode claim remains independent. The omission of the C^2 term is explicitly an expectation ('we expect the inclusion of this term to not alter the structure of the equations of motion for fluctuations in dS background'), which is a missing-support/correctness risk, not a circular step.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The paper introduces no new particles or fields; it uses the existing Weyl gauge field. The central derivation rests on the truncated action, the dS background choice, boundary conditions, and the equivalence/truncation claims.

free parameters (3)
  • d1 (or xi)
    d1 = 1/(4! xi^2) is the Rhat^2 coupling in the truncated action (5.1). It is not fitted here, but the vector-mode mass ratio in (5.42) and the detectability claims depend on it.
  • d2 (or alpha0)
    d2 = -1/(4 alpha0^2) is the F^2 coupling. The vector kinetic term and mass ratio both depend on d2; its value is not specified by the paper.
  • H0 (dS curvature scale)
    The dS background expansion rate enters the vector mass term in (5.42) and the flat-limit discussion in (5.46). It is not derived in this paper, and the broken-phase relation between H0 and the Proca mass is not established.
axioms (6)
  • domain assumption Weyl geometry is defined by the gauge transformations (2.1), the non-metricity condition (2.2), and the Weyl-covariant metric formulation (2.4).
    All curvature definitions, equations of motion, and the geodesic deviation equation rest on this formalism, which is taken from prior self-cited work [9,10,17,20].
  • domain assumption The only anomaly-free quantum gauge theory of a spacetime symmetry beyond Poincare is Weyl gauge theory.
    Motivates the framework and is cited to [20], but is not proven in this paper. It is not directly needed for the GW polarization algebra.
  • standard math Coordinate vector fields X^mu and S^mu commute in the Weyl metric formulation, X^mu nabla_mu S^lambda = S^mu nabla_mu X^lambda.
    Used to derive the Weyl geodesic deviation equation (3.12). The paper provides a proof in Appendix B.
  • domain assumption Boundary conditions at r -> infinity, stated below eq. (4.10), force S1, S2, V_i and related gradient modes to vanish.
    The scalar and vector metric invariants are eliminated using these boundary conditions; different boundary conditions would change the mode content.
  • ad hoc to paper The C^2 term in the full WQG action (2.9) can be omitted because, with appropriate boundary conditions, it is equivalent to Einstein-dS [44,45].
    This is load-bearing for the claim about WQG, but no linearized SVT computation including C^2 is presented. The paper itself says it 'expects' the term not to alter the structure.
  • domain assumption The flat-space limit H -> 0 at the end is consistent with LVK local experiments, and the massive phase of omega decouples as stated in the Introduction.
    Used to translate the dS equations into detector-language wave equations (5.46)-(5.47). The connection to the broken-phase Proca mass is not derived.

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read the original abstract

With current advances in gravitational wave (GW) detection made by the worldwide LIGO-Virgo-KAGRA (LVK) network of detectors, ever-more sensitive tests of gravity in the strong-field regime are now possible. This enables one to test gauge theories beyond Einstein-Hilbert action, such as Weyl gauge theories of gravity. The only anomaly-free (quantum) gauge theory of a space-time symmetry beyond Poincar\'e is based on Weyl gauge group (of dilatations and Poincar\'e symmetry) with Weyl conformal geometry as its natural underlying geometry. This gauge theory has spontaneous breaking of Weyl gauge symmetry to Einstein-Hilbert and Proca actions, plus a positive cosmological constant. We investigate the GW polarisation modes of Weyl (quadratic) gauge theory of gravity in Weyl geometry and compare our findings to the most recent experimental data. We show how the geodesic deviation equation from Riemannian geometry translates to Weyl geometry, and explain why it is crucial to perform the analysis around de Sitter background, which is the correct low-energy limit of Weyl quadratic gravity, to not alter the GW content, and then compute the polarisation modes. In addition to the two transverse-traceless tensor modes predicted by Einstein-Hilbert action, we find two additional vector modes induced by the transverse fluctuations of the Weyl gauge field. If detected, these vectors modes would be important evidence for Weyl gauge symmetry.

Figures

Figures reproduced from arXiv: 2607.13146 by D. M. Ghilencea, V. -M. Mandric.

Figure 3.1
Figure 3.1. Figure 3.1: The relative (space-like) displacement of two neighbouring geodesics in the initial [PITH_FULL_IMAGE:figures/full_fig_p008_3_1.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Comparison of likelihoods for different polarisation hypotheses against the purely [PITH_FULL_IMAGE:figures/full_fig_p019_5_1.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. World-Line Actions in Weyl Geometry

    hep-th 2026-07 accept novelty 6.0

    A dimensionless Weyl-invariant additive world-line action exists for time-like particles in Weyl geometry, but proper time cannot be defined until scale symmetry breaks.

Reference graph

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