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REVIEW 2 major objections 5 minor 136 references

In Palatini f(R) gravity with dynamical Chern-Simons, left- and right-handed gravitational waves damp and travel differently, an effect the metric version of the same action lacks.

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2026-08-01 09:38 UTC pith:LSP26XHN

load-bearing objection Read it for the Palatini propagation equation and the f_R scaling; treat the polynomial-in-redshift claim as a toy-model illustration, not a generic prediction. the 2 major comments →

arxiv 2607.20689 v1 pith:LSP26XHN submitted 2026-07-22 gr-qc astro-ph.CO

Gravitational Wave Birefringence in generalized Palatini Chern Simons

Jose Perdiguero , Macarena Lagos This is my paper
classification gr-qc astro-ph.CO
keywords gravitational wavesbirefringenceparity violationChern-SimonsPalatini formalismf(R) gravitydark energymodified gravity
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.

The paper establishes a concrete difference between two ways of formulating the same modified-gravity action. In the Palatini formalism, where the metric and the affine connection are varied independently, the gravitational-wave propagation equation acquires polarization-dependent friction and frequency terms: left- and right-handed modes are damped by different amounts (amplitude birefringence) and travel at different, frequency-dependent speeds (velocity birefringence). In the metric formalism, only the amplitude difference survives. The f(R) part of the action, which can drive late-time cosmic acceleration without a cosmological constant, enters these terms through the factor f_R = df/dR, so models with 0 < f_R < 1, as preferred by current cosmological fits, enhance the birefringence signal. The paper also shows that, in a de Sitter background with a light scalar field, the resulting birefringence grows polynomially with source redshift, in contrast to the linear-distance scaling used in most phenomenological analyses.

Core claim

The paper's central result is the gravitational-wave propagation equation h''_{L,R} + Ξ_{L,R} h'_{L,R} + ω²_{L,R} h_{L,R} = 0, where the friction Ξ_{L,R} and angular frequency ω_{L,R} are left/right polarization dependent. In the Palatini formalism, the dynamical Chern-Simons coupling generates both amplitude birefringence (a polarization-dependent damping) and velocity birefringence (a parity-violating, frequency-dependent phase shift); in the metric formalism the velocity term is absent. The f(R) sector enters through inverse powers of f_R = df/dR, so birefringence is enhanced when 0 < f_R < 1 and suppressed when f_R > 1. For three common f(R) models fitted to current cosmological data, f_

What carries the argument

The load-bearing object is the propagation equation (69), h''_{L,R} + Ξ_{L,R} h'_{L,R} + ω²_{L,R} h_{L,R} = 0, written in the circular polarization basis. The Palatini formalism treats the metric and the affine connection as independent fields; the independent connection perturbations bring extra degrees of freedom that feed a parity-violating, frequency-cubed term into the dispersion relation, which is what produces velocity birefringence. The dCS term, a parity-violating coupling of the Pontryagin density to a scalar field, enters with opposite signs for left and right modes, while the f(R) sector enters through inverse powers of f_R = df/dR, controlling whether birefringence is amplified

Load-bearing premise

The clean polynomial growth of birefringence with redshift rests on treating the late universe as de Sitter with a light scalar field (m ≪ H0) and a quadratic potential, so the scalar moves monotonically; if the scalar is heavier, it oscillates and the monotonic growth is lost.

What would settle it

Search a catalog of binary black hole events for the left/right amplitude and phase asymmetries predicted by Eqs. (118) and (123). If the phase asymmetry (velocity birefringence) is measured to be zero at the level where Palatini predicts it, or if the amplitude ratio at z≳1 follows a linear distance law rather than the polynomial in z, the Palatini prediction is ruled out.

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

If this is right

  • In Palatini f(R)+dCS gravity, a single gravitational-wave event can in principle show both handedness-dependent damping and handedness-dependent phase/group velocity; the metric version of the same action predicts only damping.
  • Models with f_R0 < 1, including three common f(R) models fitted to current cosmological data, amplify birefringence relative to f(R)=R by 1–10% (Palatini) and up to a factor of 2 for amplitude birefringence (metric).
  • Existing bounds on amplitude and velocity birefringence translate into constraints on the combination α H0 ϑ'_0/(κ f_R0), with the parity-violating velocity term giving the strongest current limit, of order 10^-14.
  • In the de Sitter plus light-scalar limit, amplitude birefringence scales as a cubic polynomial in source redshift and velocity birefringence as a higher-order polynomial, not as distance; phenomenological templates that assume linear distance scaling would mis-model high-redshift sources.
  • Because the effects are controlled by f_R, gravitational-wave birefringence offers an independent observational handle on the f(R) function responsible for dark energy.

Where Pith is reading between the lines

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

  • A direct detection of velocity birefringence with the predicted frequency-cubed, parity-odd phase would cleanly discriminate between the Palatini and metric formulations of f(R)+dCS, since the metric formalism predicts it to be exactly zero.
  • The 1–10% enhancement from f_R0 < 1 is small but potentially resolvable by stacking tens of binary-black-hole events; such a measurement could constrain f_R0 independently of cosmic-microwave-background and baryon-acoustic-oscillation fits.
  • If the scalar field is not light (m ≳ H0/2), the scalar oscillates and birefringence oscillates with redshift rather than growing monotonically; this oscillatory signature would be missed by standard linear-distance analyses and could masquerade as noise in parameter estimation.
  • The polynomial redshift law is explicitly conditional on a de Sitter background, a quadratic potential, and a light scalar; before high-redshift detectors are used to set bounds, model-specific templates should be computed beyond the de Sitter approximation.

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 cosmological gravitational-wave (GW) propagation in a generalized Palatini Chern–Simons theory, i.e., a parity-violating dynamical Chern–Simons (dCS) term coupled to a scalar field and to an arbitrary f(R) function, and compares the predictions with the metric formalism. The authors derive the background equations and the tensor perturbation equations, obtaining the compact GW propagation equation (69): h''_{L,R} + Ξ_{L,R} h'_{L,R} + ω²_{L,R} h_{L,R}=0. Their main claims are: (i) in the Palatini formalism both amplitude and velocity birefringence arise, whereas the metric formalism produces only amplitude birefringence; (ii) for late-time dark-energy–driven f(R) models, the birefringence is enhanced when f_R<1 and suppressed when f_R>1, with quantitative estimates for Hu–Sawicki, Exponential, and Hyperbolic gravity; (iii) current GW constraints can be translated into bounds on the dCS coupling; and (iv) the birefringence grows polynomially with source redshift, unlike the linear-distance dependence assumed in phenomenological templates. The derivation is long and internally consistent, and the GR and dCS limits are recovered.

Significance. If correct, the central distinction is observationally important: velocity birefringence in Palatini f(R)+dCS is a qualitatively new effect absent in the metric formalism (and in metric dCS), and it could be tested with current and next-generation GW detectors. The paper also connects the effect to realistic f(R) dark-energy models through external fits of f_R0, which is a useful phenomenological step. The strength of the manuscript is its explicit, self-contained computation of the perturbed connection and the resulting propagation equations, with the GR and GR+dCS limits checked. The redshift-scaling claim, however, is only an illustrative prediction under a restrictive approximation and is partly conceded in the final paragraph of Sec. V.C; this overreach in the abstract and introduction needs correction.

major comments (2)
  1. [Abstract and Sec. V.C, Eqs. (118)–(123)] The abstract and introduction state that in this model birefringence grows polynomially with source redshift. This claim is derived under a stacked set of approximations: a pure de Sitter background, matter neglected via κ(ρ+3p)≪f (Eq. 105), a quadratic scalar potential, m≪H0, and Ω_ϑ0≪1. The authors themselves concede in the final paragraph of Sec. V.C that the result is 'highly sensitive to model assumptions'. More concretely, for the Hu–Sawicki-type parameters used in the paper, matter is not parametrically negligible at z≳1, so Eqs. (104)–(106) do not reliably describe the z∼1 events already observed. The polynomial scaling should be presented as a conditional toy-limit illustration, and the abstract/introduction should be revised accordingly, or the calculation should be extended to a background with matter. This does not threaten the central Palatini-vs-metric birefringence result,
  2. [Sec. V.B, Eqs. (100)–(102)] The translation of existing GW constraints into bounds on α uses, in Eq. (100), a low-redshift approximation z≪1 and then equates the integrated model expression to a phenomenological distance-proportional parametrization. Since LVK events extend to z∼1 and the paper itself argues that the model redshift dependence is not linear, this mapping introduces a systematic error that is not quantified. The statement that the constraint is O(10^-5) should be accompanied by an estimate of the error from using the low-z approximation, or the derivation should be repeated at the typical z of the events in the catalogs. This is secondary to the main result but directly affects one of the advertised quantitative outputs.
minor comments (5)
  1. [Fig. 1] The caption uses symbols like '10□2.3s', apparently intended to be negative exponents 10^{-2.3} s; the minus signs appear to be missing or corrupted. Please fix the typesetting.
  2. [Eq. (67) and Appendix B] The index ranges in Eq. (67) and in the coefficients A_{mn}, B_{mn}, C_{mn}, D_{mn} are not explicitly stated. It would help readers to specify the summation ranges and to state which coefficients vanish by parity.
  3. [Sec. V.A, Table II] The metric-formalism f_{R0} values are obtained by neglecting derivatives of f_R and using Eq. (96). For the Hyperbolic model with CC+PPS, f_{R0}=0.51 is a substantial deviation; the validity of neglecting R' and f_RR terms at z=0 deserves a brief quantitative check, especially because Eq. (95) contains explicitly the terms that are dropped.
  4. [Sec. V.A.1, Eq. (86)] The trace equation (86) neglects the scalar-field contribution to T. This is consistent with Ω_ϑ0≪1, but it would be useful to state this assumption in the same paragraph as Eq. (86), not only later, to avoid confusion.
  5. [Sec. V.C, final paragraph] The important caveat about model sensitivity appears only at the very end of Sec. V.C. It should be moved closer to Eqs. (118) and (123), and the abstract should carry the same qualification if Eqs. (118)–(123) are cited as a model prediction.

Circularity Check

0 steps flagged

No significant circularity: core GW equations are derived from the action; f_R0 enters from external cosmological fits; redshift scaling is explicitly conditional and not fitted.

full rationale

The derivation chain is self-contained: starting from the action (1), the field equations (9), (14), (15) are varied, the background is solved in Sec. III, and the GW propagation equation (69) is obtained from the perturbed metric and connection equations (62)-(66), with Ξ_L,R and ω^2_L,R defined in (70) as coefficient ratios. The Palatini-versus-metric difference (velocity birefringence present in Palatini, absent in metric) follows from the presence or absence of independent connection perturbations, not from an input assumption that already contains that conclusion. The f_R0 values used to claim enhancement (Tables I and II) come from external cosmological fits [125,39] and are not fitted to birefringence data; the enhancement is a computed consequence of the 1/f_R0 factors in Eqs. (84) and (85). The redshift-polynomial results (118) and (123) are derived, not fitted, from an explicit de Sitter + quadratic-potential + light-scalar stack, and the paper itself concedes in the final paragraph of Sec. V.C that this redshift evolution is 'highly sensitive to model assumptions'; that is a robustness caveat, not circularity. The author-overlapping citations [76,68,79] are used for consistency limits, prior dCS results, and translation of observational constraints, but the central f(R)+dCS derivation is independently performed here and does not reduce to those citations. No equation was found that equals an input by construction, and no fitted parameter is renamed as a prediction.

Axiom & Free-Parameter Ledger

7 free parameters · 8 axioms · 0 invented entities

The central equations depend on the action and on the symmetry/perturbation ansätze; no new particles or forces are introduced. The absolute birefringence magnitude is set by αϑ2 times powers of 1/f_R, with f_R0 taken from external fits and ϑ2 an integration constant, so the absolute size is not predicted. The enhancement percentages inherit no uncertainties from Tables I–II.

free parameters (7)
  • α (dCS coupling)
    Controls all parity-violating effects; not fitted here, bounded via Eqs. (98)–(102).
  • β (scalar kinetic coupling)
    Appears in the action and background scalar dynamics but not in the leading GW equations; left free.
  • f_R0 for Palatini f(R) models = 0.90–0.99 (Table I)
    Taken from best fits in [125]; if f_R0<1, enhancement is claimed at 1–10%.
  • f_R0 for metric f(R) models = 0.51–0.99 (Table II)
    Taken from [39]; metric amplitude birefringence is enhanced up to a factor of 2 when f_R0≈0.51.
  • Scalar integration constant ϑ2
    Sets the amplitude of the scalar solution (113) and hence the magnitude of µA and µV; not constrained by the paper.
  • Scalar potential parameters (V0, m)
    Quadratic potential chosen ad hoc; the small-mass branch m≪H0 is selected to obtain the monotonic polynomial result.
  • f(R) model parameters (HS: Λ, µ²; Exp: ξ,σ; Hyp: ξ,R_T) = from [125,39]
    Enter only through f_R0 in the birefringence formulas; no error bars are shown.
axioms (8)
  • domain assumption The Palatini action (1) with a torsion-free connection is the correct starting point for f(R)+dCS gravity; Eq. (14) governs the connection.
    Central model input; if the connection variation is incomplete, all GW equations inherit the error.
  • domain assumption FRW symmetries reduce the connection to j(η), l(η), b(η) in Eq. (20), and the CS term vanishes at background level.
    Used to derive the background solution (27) and the modified Friedmann equation.
  • standard math SVT decomposition: scalar and vector perturbations decouple from tensor modes at linear order, and δϑ/matter perturbations can be ignored.
    Standard cosmological perturbation theory; stated in Sec. IV before Eq. (57).
  • domain assumption The perturbed connection has the four-tensor form (54) and the B_i=1 redefinition is complete.
    Needed to close the system (62)–(66); completeness against [76,119] is asserted, not proved.
  • domain assumption WKB approximation k≫H and small-α Taylor truncation up to order (αk)^3.
    Used to define Ξ, ω², µA, and µV in Eqs. (69)–(80).
  • ad hoc to paper Late-time universe is de Sitter, f(R) drives acceleration, Ω_ϑ0≪1, scalar potential quadratic with m≪H0 for the redshift-evolution section.
    Yields analytic solutions (108) and (113) and polynomial scaling; the paper concedes high model sensitivity.
  • domain assumption f_R0 values from external fits [39,125] are taken at face value without uncertainties.
    Enhancement percentages in Sec. V.A are directly controlled by these numbers.
  • domain assumption Existing GW birefringence constraints [66,68,77] can be translated by equating the model's µA and µV to the phenomenological parametrization.
    Basis of the coupling bounds in Sec. V.B; differs from the LVK anisotropic analysis, which the paper acknowledges.

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read the original abstract

The cosmological propagation of gravitational waves (GWs) can exhibit amplitude and phase polarization distortions when parity symmetry is broken, a phenomenon known as cosmological birefringence. In this paper, we investigate the phenomenology of GW birefringence in a gravitational model $f(R)$ coupled to a dynamical Chern-Simons (dCS) term, analyzed in the metric and Palatini formalisms. At the background level, this model can lead to dynamical dark energy, while for GW propagation we find that the Palatini formalism predicts both amplitude and velocity birefringence, whereas the metric formalism predicts amplitude birefringence only. We also find that the birefringence effects can either be suppressed or enhanced by the $f(R)$ interactions, depending on the specific form of $f(R)$. Considering three common $f(R)$ models (Hu-Sawicki, Exponential, and Hyperbolic gravity) fit to recent cosmological data, we find that birefringence is enhanced relative to the $f(R) = R$ case, by $1-10\%$ in the Palatini formalism and up to a factor of 2 in the metric formalism. We also translate current GW birefringence constraints to bounds on the dCS coupling within our model. Finally, we show that in our model the birefringence effect grows polynomially with source redshift, in contrast to the linear-distance scaling commonly assumed in phenomenological models of GW birefringence in the current literature.

Figures

Figures reproduced from arXiv: 2607.20689 by Jose Perdiguero, Macarena Lagos.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of amplitude (left) and velocity (right) birefringence for a GW signal with a purely left-handed polarization. In blue, it is [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

discussion (0)

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Reference graph

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