The Speed of Gravity

Andrew J. Tolley; Claudia de Rham

REVIEW 2 major objections 4 minor 3 cited by

The Speed of Gravity

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Gravitational waves slightly outrun light on cosmological backgrounds, this paper argues.

desk verdict A carefully derived EFT calculation of gravitational wave speed corrections on FLRW whose superluminality headline is honestly conditional on the unproven sign of C_W2; the finite dimension-6 part is the most durable piece. read the letter →

arxiv 1909.00881 v2 pith:ZWP7WXTF submitted 2019-09-02 hep-th astro-ph.COgr-qchep-ph

Claudia de Rham , Andrew J. Tolley This is my paper · ORCID

classification hep-thastro-ph.COgr-qchep-ph

keywords gravitationalwaveseffectivefieldtheoryspeedofsuperluminalitypositivityboundsnullenergyconditioncurvature-squaredoperatorscosmologicalbackgrounds

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

Within the standard effective field theory of General Relativity, this paper tries to establish that the low-energy speed of gravitational waves on a cosmological background is generally not equal to the speed of light. Integrating out massive fields generates curvature corrections, and on an FLRW spacetime the leading correction is fixed by the Weyl-squared coefficient $C_{W^2}$: the tensor speed becomes $c_s^2 = 1 + 16 C_{W^2}(-\dot H)/M_{\rm Pl}^2$, which exceeds one when $C_{W^2}>0$ and matter satisfies the null energy condition, so $\dot H<0$. The paper argues that S-matrix locality, unitarity and analyticity—through positivity bounds applied to matter scattering after the massless graviton t-channel pole is neglected—select this positive sign, so gravitational waves are superluminal relative to the metric to which photons and Standard Model fields are minimally coupled. If the curvature-squared terms are tuned to zero, finite one-loop contributions from known particles still shift the speed, making it epoch-dependent and sensitive to the spin of the lightest particles above the Hubble scale. A sympathetic reader should care because this changes how the causal structure of cosmological effective field theories is to be assessed.

What carries the argument

The central object is the low-energy effective action for gravity organized in curvature operators, truncated at the Weyl-squared (dimension-4) and curvature-cubed (dimension-6) level. The speed is defined through the lightcone of the second-order hyperbolic equation obtained after perturbatively reducing the higher-derivative terms; it is the coefficient of $k^2$ in the tensor dispersion relation. At leading order the Weyl-squared coefficient $C_{W^2}$, the coupling of the conformal curvature-squared operator, carries the whole effect via $c_s^2 = 1 + 16 C_{W^2}(-\dot H)/M_{\rm Pl}^2$. Positivity bounds on matter amplitudes, together with the Källén–Lehmann spectral representation of the two-point function of the stress tensor, are the machinery that fixes $C_{W^2}>0$; the assumption of neglecting the massless graviton t-channel pole is what lets those bounds go through. At next order the finite one-loop coefficients for fields of spin 0, 1/2, and 1 supply the species-dependent terms that make the speed epoch-dependent.

What would settle it

In an explicit weakly coupled UV completion containing an infinite tower of massive spin-2 states, compute the full two-to-two amplitude without dropping the massless graviton t-channel pole; if the twice-subtracted forward amplitude does not force the Weyl-squared coefficient to be positive, the sign conclusion fails. A complementary observation would be a cosmological multi-messenger event whose gravitational-wave and photon arrival times imply a subluminal tensor speed in the matter frame.

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Extended reading notes

Core claim

The paper's central claim is that in the effective field theory of gravity the sound speed of tensor modes on a spontaneously Lorentz-breaking background is modified by irrelevant curvature operators, and for the signs selected by positivity bounds the modification is generically superluminal. On FLRW spacetime the Weyl-squared interaction gives $c_s^2 = 1 + 16 C_{W^2}(-\dot H)/M_{\rm Pl}^2$; since ordinary matter obeys the null energy condition and hence $\dot H<0$, a positive $C_{W^2}$ makes gravitational waves outpace the lightcone of the metric to which Standard Model fields are minimally coupled. A positive $C_{W^2}$ is what follows from Källén–Lehmann spectral positivity of the stress-tensor two-point function and from weakly coupled tree-level completions with massive spin-2 states, provided the massless graviton t-channel pole in the relevant amplitudes can be neglected. When the leading curvature-squared terms are set to zero, the finite one-loop dimension-6 effective action produces a speed shift of order $H^4/(M_{\rm Pl}^2 M^2)$ whose sign depends on the spin and mass of the lightest integrated-out particles: scalar-dominated content gives superluminal gravitational waves for most of standard cosmological history, while the radiation era is subluminal for all spins.

Load-bearing premise

The argument assumes that standard positivity bounds apply to matter scattering once the massless graviton t-channel pole is neglected, and that the Källén–Lehmann spectral integrals need at most one subtraction; if either premise fails, the sign of $C_{W^2}$ is unconstrained and the generic superluminality conclusion does not follow.

Editorial extensions

If this is right

If the central claim survives, low-frequency gravitational waves on a cosmological background arrive slightly earlier than photons emitted from the same source, with the departure growing toward the infrared.
No field redefinition can make both sectors luminal at low energies: the ratio of tensor to matter sound speeds is frame invariant, so a frame with luminal gravity leaves matter fluctuations subluminal through gravitationally induced $T\bar T$-type interactions.
The front velocity is luminal in the high-frequency limit, so the low-energy superluminal group velocity does not imply propagation of information outside the lightcone.
Setting the curvature-squared corrections to zero does not restore luminality; the finite dimension-6 loop corrections still shift the speed, and scalar-dominated heavy spectra make gravitational waves superluminal through most of standard cosmological history.
Cosmological model builders should not impose subluminality of all fluctuations as a consistency criterion; the operative causality criterion becomes S-matrix analyticity, which for this system selects superluminal gravitational waves.

Reading between the lines

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

One testable extension: the predicted speed shift is frequency-dependent and grows in the IR, so a multi-messenger gravitational-wave event at cosmological distance could in principle measure the sign of the effect and thereby test $C_{W^2}>0$ in the matter frame.
If positivity bounds are invalidated by the t-channel pole, the leading-order sign is unknown, but the finite dimension-6 loop effect remains; the same measurement strategy could still extract the spin and mass content of the lightest states above the Hubble scale.
The lightcone ordering implied by the paper (matter cone inside gravity cone) reverses for NEC-violating sources, so these sign arguments could sharpen consistency conditions on phantom dark energy or other negative-energy cosmological models.

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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

Colleague,

Two things up front. The paper contains real, checkable physics: it derives the correction to the gravitational wave sound speed on FLRW from heavy loops and higher-spin tree effects, including the clean result c_s^2 = 1 + 16 C_W2 (−Hdot)/M_Pl^2, and—more durably—a finite, epoch- and spin-dependent dimension-6 contribution that survives even if the curvature-squared story collapses. But the headline claim, that GWs are in general superluminal, is conditional on the sign of C_W2, and the authors themselves concede the sign is not established. Section 4.3.3: if the Kallen-Lehmann integrals in (4.37) need subtractions, "it is hence not possible to conclude positivity of the LHS." The abstract states the conclusion flatly; the intro bullet carries the qualifier about the t-channel pole.

Credit where due. The speed derivation is careful, especially the discussion in Section 3.1 of why you cannot read the speed off a naive WKB dispersion relation. The frame-independence argument—the ratio of tensor to matter speed is invariant under metric field redefinitions—is clean and should reset how people phrase subluminality demands in cosmological EFTs. The dimension-6 one-loop coefficients are taken from Avramidi, reproduced in Appendix A, finite, and genuinely new in their FLRW application: the sign of the correction depends on the spin of the lightest integrated-out particle and on the equation of state. That part does not rely on positivity bounds and is the paper's most robust contribution.

The soft spots, in proportion. The load-bearing assumption is C_W2 > 0. Getting there requires discarding the massless graviton t-channel pole (following [27]) and assuming the spectral integrals converge. Neither is proven; the paper says so. If either fails, the sign of C_W2 is unconstrained and Eq. (3.36) does not enforce superluminality—indeed a negative C_W2 would give subluminal GWs on NEC-preserving backgrounds. So the general claim is a conditional statement, not a theorem. The authors are honest about this, which is why the paper still warrants refereeing, but the abstract and discussion should carry the same caveat as the intro bullet. The axion/fuzzy-dark-matter speculation is minor; it is hedged and observationally remote anyway.

Who benefits: anyone doing EFT model building in cosmology, especially work that assumes tensor modes must be subluminal. It deserves a serious referee. I would send it out, and instruct the referee to focus on the t-channel/subtraction question and on aligning the abstract with the qualifier.

Best,

[you]

Referee Report

2 major / 4 minor

Summary. The paper studies the low-energy speed of gravitational waves in the standard effective field theory of general relativity coupled to matter, on backgrounds that spontaneously break Lorentz invariance such as FLRW. The central technical result is Eq. (3.36): on NEC-preserving FLRW backgrounds, the tensor-mode sound speed is c_s^2 = 1 + 16 C_W2 (-$\dot H$)/M_Pl^2 to leading order in the curvature-squared EFT, so the sign of the Weyl-squared coefficient C_W2 controls whether gravitational waves are superluminal. The paper argues that positivity bounds, applied to matter scattering amplitudes after a field redefinition, select C_W2 > 0 and hence superluminal gravitational waves. It also analyzes dimension-6 and dimension-8 curvature operators: after tuning the IR curvature-squared coefficients to zero, finite one-loop corrections from massive scalars, spinors, and vectors produce an epoch- and species-dependent modification of the speed, given in Eq. (5.10), with superluminal behavior for scalar-dominated contributions over most of standard cosmological history and subluminal behavior in the radiation era. The appendices reproduce the one-loop effective action for a massive scalar and provide the detailed FLRW reduction for dimension-6 operators.

Significance. If the sign assumption on C_W2 is eventually justified, the paper establishes a concrete, potentially observable consequence of a standard gravitational EFT: a tiny but nonzero difference between the gravitational-wave speed and the matter lightcone, scaling as |$\dot H$|/\Lambda^2. The derivation of the speed formula is careful, the field-frame invariance argument (Section 2.2) is a useful clarification, and the explicit one-loop coefficients from Avramidi are reproduced rather than fitted. The dimension-6 result is especially valuable because it is finite, calculable, and predicts a sign that depends on the spin of the lightest integrated-out field, including the interesting statement that radiation-era gravitational waves are subluminal for all three spin species considered. The paper is also honest in its main bullet by including the qualifier that the t-channel-pole contribution is ignored; the problem is that this qualifier is not consistently carried through the abstract and discussion.

major comments (2)

[§4.2, §4.3.3; Eq. (3.36)] The paper's headline claim that positivity bounds 'enforce' superluminal gravitational waves is not established, because the sign of C_W2 is never proven. The speed formula (3.36) gives c_s^2 = 1 + 16 C_W2(-\dot H)/M_Pl^2, so on NEC backgrounds (\dot H < 0) the conclusion c_s^2 > 1 requires C_W2 > 0. The arguments for C_W2 > 0 are explicitly conditional: Section 4.2 applies positivity bounds only 'provided we argue or assume that the contribution of the graviton exchange t-channel pole can be neglected', and Section 4.3.3 states that if the Kallen-Lehmann integrals in (4.37) require subtractions, 'it is hence not possible to conclude positivity of the LHS'. No proof of convergence of (4.37), and no independent justification of the t-channel-pole prescription beyond citation to [27], is supplied. The valid conclusion is therefore a conditional theorem, not the generic statement in the abstract and Section 6 that the speed is 'in general superluminal'. The abstract, introduction, and discussion should be revised so that the logical dependence on the unproven t-channel/subtraction assumption is explicit in every statement of the main result.
[§5.1.4; Eq. (5.11)] The phenomenological discussion of scalar dark matter in Section 5.1.4 applies the dimension-6 result to 'the whole standard cosmological history', but this application is only valid on the tuned subspace C_IR^{W2} = C_IR^{R2} = 0 introduced at the start of Section 5. If positivity bounds do select C_W2 > 0, the dimension-4 contribution generically dominates the dimension-6 one, and the epoch-dependent conclusions, including the subluminal radiation-era statement, would not hold. The paper partially acknowledges this in the 'Discriminator Redux' paragraph, but the abstract's claim that finite loop corrections 'lead to an epoch dependent modification' should be framed as a statement about a deliberately tuned EFT, not about the generic EFT. The section would benefit from a clear summary stating which conclusions are generic and which require the tuning.

minor comments (4)

[§4.3.5; Eq. (4.46)] Equation (4.46) contains an apparent typo: the inequality '|1-c_s^2(M_2)| < |1-c_s^2(M_2)|' for M_2 > M_1 is trivially false; the right-hand side should presumably refer to M_1. The surrounding text makes the intended RG monotonicity clear.
[Fig. 3] The axes of Figure 3 are not fully labeled; the approximate numerical boundaries (-1.8, 0.2, 1.2, -1.5) are given only in the caption. Adding explicit axis labels and marking the NEC boundary \omega = -1 directly on the figure would improve readability.
[§5.1.2, Eq. (5.2)] The effective numbers N_*^s weight fields by M^2/M_i^2, so the statement that the effect is 'determined by the lightest particle' is only true when there is no large multiplicity of heavier fields. This is stated implicitly, but a one-sentence caveat at Eq. (5.2) would prevent a common misreading.
[§6] The discussion says that the magnitude of the effect is of order |\dot H|/\Lambda^2, which is correct for the dimension-4 case, but the dimension-6 contributions scale as |\dot H| H^2/(M^2 M_Pl^2) and are suppressed by an additional H^2/M^2 factor. A sentence distinguishing the two parametric regimes would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the gravitational-wave speed is a direct EFT computation, and the positivity-sign input is an explicit external assumption rather than a recycled output.

full rationale

The paper's central speed formula, Eq. (3.36), is obtained by starting from the EFT action (3.16), expanding tensor modes on an FLRW background, perturbatively reducing the fourth-order equation of motion to the second-order hyperbolic form (3.7)-(3.10), and reading off the coefficient beta_1 as the propagation speed. No parameter is fitted to data, and no quantity is defined in terms of the claimed prediction. The later conclusion that gravitational waves are generically superluminal depends on the sign of C_W2, which is argued from S-matrix and spectral-density positivity bounds. Those arguments are explicitly conditional in the paper itself: Section 4.2 says the bounds apply 'provided we argue or assume that the contribution of the graviton exchange t-channel pole can be neglected', and Section 4.3.3 concedes that if the Kallen-Lehmann integrals (4.37) require subtractions then 'it is hence not possible to conclude positivity of the LHS'. These are external assumptions and caveats, not circular reductions: the existence of a possible sign ambiguity or an unproven assumption does not make the derivation circular. The references used for the t-channel-pole treatment and spectral positivity are independent works, not author-uniqueness claims, and the paper's own self-citations are peripheral to the main derivation. Therefore no significant circularity is present.

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

The calculations rest on standard EFT input, but the sign conclusions require assumptions about UV subtraction constants and analyticity that are not derivable within the paper. The parameters listed are not fitted to data; they are unknown Wilson coefficients whose signs and values carry the central conclusions.

free parameters (4)

C_W2 (Weyl-squared coefficient)
Wilson coefficient of the dimension-4 operator W^2; not fitted to data but its positive sign is assumed via positivity bounds under the t-channel-pole-neglect assumption. Controls the sign of c_s^2 - 1 in Eq. (3.36).
C_R2 (Ricci-squared coefficient)
Renormalization-scheme dependent coefficient; set to zero in Section 5 to isolate dimension-6 effects. Not fixed by the paper.
UV subtraction constants C_UV^{W2}, C_UV^{R2}
Enter the spectral representation through Eqs. (4.29) to (4.33); assumed negligible or zero when concluding Delta C_W2 > 0. The paper notes positivity of the final C_IR cannot be concluded if subtractions are required.
C4 (non-minimal light-field curvature coupling)
Coupling of G_mu_nu partial_mu psi partial_nu psi introduced in Eq. (4.49); constrained by positivity to satisfy C4 + 4 C_W2 > 0 in Eq. (4.61).

assumptions (5)

domain assumption The massless graviton t-channel pole can be neglected when applying forward-limit positivity bounds to matter scattering amplitudes.
Invoked in Section 4.2 and relied on in Eqs. (4.19), (4.22) and (4.61); the paper notes this follows from [27] and is not proven here.
domain assumption The spectral densities rho_2(mu) and rho_0(mu) in the Kallen-Lehmann representation of the TT correlator are non-negative and the integrals in Eq. (4.37) converge without additional subtractions.
Needed for Eqs. (4.28) to (4.34) to conclude Delta C_W2 > 0; Section 4.3.3 states if subtractions are needed positivity cannot be concluded.
domain assumption Matter fields sourcing the background and light fields are minimally coupled to the metric and have spin less than 2.
Stated in Sections 2.1 and 3, Eq. (3.13); non-minimal couplings are treated separately in Section 4.4.
domain assumption Graviton loops are not included when integrating out heavy fields; only matter loops are included.
Section 2.1: "We will only be integrating out loops of matter fields no gravitons in the loops."
ad hoc to paper The IR curvature-squared coefficients are tuned to zero, C_IR^{W2} = C_IR^{R2} = 0, for the dimension-6 analysis.
Section 5 assumes this special tuning to expose next-order effects; it is a free choice, not a necessity.

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Pith. "Pith review of The Speed of Gravity." pith.science (2026). https://pith.science/paper/ZWP7WXTF

@misc{pith2026190900881,
  author       = {Pith},
  title        = {Pith review of: The Speed of Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZWP7WXTF}},
  note         = {Machine review of arXiv:1909.00881}
}

abstract

Within the standard effective field theory of General Relativity, we show that the speed of gravitational waves deviates, ever so slightly, from luminality on cosmological and other spontaneously Lorentz-breaking backgrounds. This effect results from loop contributions from massive fields of any spin, including Standard Model fields, or from tree level effects from massive higher spins $s \ge 2$. We show that for the choice of interaction signs implied by S-matrix and spectral density positivity bounds suggested by analyticity and causality, the speed of gravitational waves is in general superluminal at low-energies on NEC preserving backgrounds, meaning gravitational waves travel faster than allowed by the metric to which photons and Standard Model fields are minimally coupled. We show that departure of the speed from unity increases in the IR and argue that the speed inevitably returns to luminal at high energies as required by Lorentz invariance. Performing a special tuning of the EFT so that renormalization sensitive curvature-squared terms are set to zero, we find that finite loop corrections from Standard Model fields still lead to an epoch dependent modification of the speed of gravitational waves which is determined by the precise field content of the lightest particles with masses larger than the Hubble parameter today. Depending on interpretation, such considerations could potentially have far-reaching implications on light scalar models, such as axionic or fuzzy cold dark matter.

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