Luminal Scalar-Tensor theories for a not so dark Dark Energy

A. Shtennikova; M. Valencia-Villegas; S. Mironov

REVIEW 3 major objections 4 minor 46 references

Luminal Scalar-Tensor theories for a not so dark Dark Energy

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Adding up to five dark-energy–photon couplings makes gravitational waves luminal in the most general scalar-tensor theories, and at least one Beyond Horndeski subclass also suppresses gravitational wave decay into dark energy on any…

desk verdict Clean luminality result in DHOST, but the headline decay-suppression claim is imported rather than re-derived. read the letter →

arxiv 2412.13460 v2 pith:2FQWJ3DZ submitted 2024-12-18 hep-th astro-ph.COgr-qc

S. Mironov , A. Shtennikova , M. Valencia-Villegas This is my paper · ORCID

classification hep-thastro-ph.COgr-qc

keywords gravitationalwavesluminalityscalar-tensortheoriesDHOSTBeyondHorndeskidarkenergyphotoncouplingsGW170817

topics Dark Energy

open problems Dark Energy

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

Gravitational waves from GW170817 arrived almost simultaneously with gamma rays, forcing gravity's speed to match light's to within one part in $10^{16}$. The paper argues this does not rule out scalar-tensor dark energy theories, if the dark-energy scalar also couples to the photon in a specific way. It shows that up to five such scalar–photon couplings are enough to make gravitational waves luminal across the most general scalar-tensor theories that propagate a graviton (DHOST, including Horndeski and Beyond Horndeski), and exhibits at least one Beyond Horndeski theory in which gravitational wave decay into dark energy is suppressed on any cosmological background.

What carries the argument

The load-bearing object is the 4D action (11), $L_{\mathrm{DHOST}\pi} + L_{\mathrm{DHOST}A}$, where $L_{\mathrm{DHOST}A}$ contains up to five dark-energy–photon couplings tied to the potentials $f_2$, $f_3$, $a_1$, $b_2$, $b_3$, $b_6$. The central identity is Eq. (23), $c_g^2/c^2 = 1 - 3 \dot{\pi} X H b_3 / G_\tau$, which reduces the speed coincidence to the single condition $b_3(\pi,X)=0$. The machinery then checks this condition against the degeneracy conditions of each DHOST class, selects the classes with a graviton, and in theory (ii) uses the freed potential $F_4$ to solve the linear no-decay condition (29), yielding the explicit solution (30).

What would settle it

Compute the full quadratic action on an FLRW background for theory (ii) with $F_4$ given by Eq. (30), including all scalar–vector mixing terms; if a nonzero leading-order mixing between the photon perturbation $A_i$ and the scalar perturbation $\chi$ appears, gravitational waves can decay into dark energy and the suppressed-decay claim fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that the post-GW170817 constraint on the speed of gravitational waves does not close the door on scalar-tensor dark energy, because light can be made to travel at the same modified speed. Starting from a 5D Kaluza-Klein compactification of DHOST, the authors derive a 4D dark-energy–photon sector with five couplings. Computing graviton and photon speeds on a flat FLRW background yields the identity $c_g^2/c^2 = 1 - 3 \dot{\pi} X H b_3 / G_\tau$, so setting $b_3 = 0$ restores luminality. The paper then checks this condition against the DHOST degeneracy conditions and exhibits two successful classes: every quadratic DHOST with a graviton (including Horndeski and Beyond Horndeski BH4), and a mixed quadratic-BH plus cubic-Horndeski theory with $G_5(\pi)$ and $F_5=0$, in which $F_4$ remains free. Imposing the no-decay condition imported from the literature fixes $F_4$ via Eq. (30), producing at least one luminal Beyond Horndeski theory whose gravitational waves do not decay into dark energy on any cosmological background.

Load-bearing premise

The claim that gravitational wave decay is suppressed rests on applying the no-decay condition derived for Beyond Horndeski gravity without scalar–photon couplings to a theory with those couplings, assuming the photon sector does not alter the scalar mode on a cosmological background.

Editorial extensions

If this is right

The condition $b_3=0$ becomes the luminality condition for DHOST, so the class of dark energy models compatible with the GW170817 bound is much larger than previously thought.
For any quadratic DHOST theory with a graviton, adding the corresponding dark-energy–photon couplings yields luminal gravitational waves; this includes quadratic Horndeski and Beyond Horndeski BH4.
The new Luminal Beyond Horndeski theory (ii) keeps $F_4$ free, allowing the no-decay condition to be solved explicitly as $F_4(\pi,X)$ from Eq. (30), suppressing gravitational wave decay to dark energy.
The essential coupling $f_3 F^2 \nabla^2 \pi$ cannot be removed by conformal or disformal transformations, so previously derived obstructions to suppressing decay do not apply.
Cosmological models built on $b_3=0$ scalar-tensor theories can be studied without the graviton-speed constraint, subject to new laboratory and astrophysical constraints on the dark-energy–photon couplings.

Reading between the lines

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

If the argument holds, future multi-messenger observations of gravitational wave amplitude and electromagnetic counterpart could directly probe the new scalar–photon couplings, rather than just the speed ratio.
The same 5D Kaluza-Klein construction that generates the couplings suggests the luminality and no-decay properties may extend to spherically symmetric backgrounds, where tests of the Vainshtein screening could be sharpened.
The no-decay condition (29) is independent of the Hubble rate and $\ddot{\pi}$, so a theory satisfying it suppresses decay throughout the cosmological trajectory of a wave, not just at one epoch; a dedicated calculation including scalar-vector mixing would test the imported condition.

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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

Main thing to know: this paper has a real, checkable result — the b3=0 condition for luminal GWs in DHOST theories with DE-photon couplings — and one genuinely new construction, theory (ii), where F4 is free and can be chosen to suppress GW decay. The quadratic-action computation leading to Eq. (23) is transparent, and the authors are careful to walk through degenerate classes, including explicit counterexamples. That part deserves credit.

What's actually new: extending their own KK construction from Horndeski to general DHOST, identifying b3=0 as the luminality condition, and showing that a Beyond Horndeski plus cubic Horndeski subclass automatically satisfies it with free F4. The F4 solution (30) is new relative to [5] and [35]. The paper also says plainly which DE-photon couplings cannot be removed by conformal/disformal transformations — that matters.

The soft spot is exactly where the reader and stress test put it. Eq. (29) is taken from Creminelli et al. [26], where it was derived for beyond Horndeski gravity without the new DE-photon couplings. The paper does not re-derive it for action (26). On FLRW the vector background vanishes, so the quadratic scalar action is unchanged, but the decay amplitude is cubic on shell, and the new photon sector contains F^2 box(chi) and F^2 pi_mu nu vertices. Unless the full h->chi chi amplitude is recomputed, calling the decay 'suppressed in any cosmological background' is stronger than what is shown. The sign-convention footnote makes this acute: Eq. (30) solves Eq. (29) in the authors' convention, but the conversion from [26] is not displayed. This is not a fatal flaw — the claim is plausible and may well be right — but it is a proof gap in the headline sentence.

Also fair: the authors themselves concede extra checks are needed for theory (i). The same caveat applies, quietly, to the decay-suppression part for theory (ii).

Who this is for: cosmologists and modified-gravity people working on DHOST after GW170817. A serious referee should get this. The central luminality condition is solid, and the decay question is checkable. My own verdict would be conditional: publish after the decay claim is either re-derived or softened.

Referee Report

3 major / 4 minor

Summary. The paper proposes that dark-energy scalar couplings to the photon, derived from a five-dimensional Kaluza-Klein compactification of DHOST theories, can make gravitational waves luminal on a cosmological background. The central derivation is the speed ratio c_g^2/c^2 = 1 - 3\dot\pi X H b_3/G_\tau (Eq. 23), so the luminality condition reduces to b_3 = 0 (Eq. 24). The authors identify two phenomenologically relevant classes: (i) quadratic DHOST with DE-photon couplings, and (ii) quadratic Beyond Horndeski plus cubic Horndeski with f_3 = G_5(\pi). For class (ii), the free function F_4(\pi,X) is used to satisfy a no-decay condition imported from Creminelli et al. [26], yielding the explicit solution (30), and the authors conclude that a subclass of luminal Beyond Horndeski theories has suppressed GW decay into dark energy in any cosmological background.

Significance. If the claims hold, the paper reopens a substantial class of scalar-tensor dark-energy models that were considered ruled out by GW170817, by relaxing the assumption that the photon remains minimally coupled at cosmological scales. The speed-ratio derivation (Eq. 23) is clean, direct, and supports the b_3 = 0 condition, and the paper carefully checks degenerate classes where b_3 is fixed by degeneracy conditions. The identification of the F_4 freedom in class (ii) is a useful observation, and the disformal-invariance argument distinguishing this theory from earlier work in [35] is valuable. The main weakness is that the headline no-decay result rests on an equation imported from a theory without the new photon couplings, and the sign-convention mismatch is not resolved; this requires explicit verification before the strongest claim can be accepted.

major comments (3)

[Eq. (29) and surrounding text] The no-decay condition (29) is quoted from Creminelli et al. [26], where it was derived for pure beyond-Horndeski gravity without the DE-photon couplings LDHOSTA in Eq. (12). The manuscript does not show that the new photon sector leaves the on-shell h -> chi chi amplitude unchanged. At tree level the couplings in Eq. (27) all contain two powers of F_{\mu\nu}, so they do not directly contribute to the h chi chi vertex on a vanishing vector background; nevertheless, this transfer should be demonstrated explicitly, especially because the abstract claims suppression 'in any cosmological background'.
[Footnote 3 and Eq. (29)/(30)] The sign convention for F_4 is stated as taken from [21,24], while [26,35] use the opposite sign. Equation (29) is imported from [26], and the solution (30) is cited from [35], both of which use the opposite convention. Without an explicit conversion between conventions, it is not established that Eq. (30) actually solves Eq. (29) under the authors' sign convention. This is load-bearing because the F_4 solution is the basis for the suppressed-decay claim.
[Conclusions, theory (ii) claim] The paper explicitly states that additional checks are needed for theory (i) but does not apply the same caveat to the decay-suppression part for theory (ii). Given that Eq. (29) is the only evidence for suppressed decay and that its derivation is not re-examined in the presence of LDHOSTA, the conclusion should be softened or supported by an explicit recomputation of the decay amplitude for the action (26).

minor comments (4)

[Title page affiliations] The affiliation text contains typos: 'Sci ences' and 'Russ ia' should be 'Sciences' and 'Russia'.
[Eq. (12)] The phrase 'three types of DE–Photon couplings' lists F^2 \nabla^2\pi, F^2 (\nabla\pi)^2, and F^2 \nabla^2\pi (\nabla\pi)^2, but the second type is more precisely (F_{\mu\nu}\pi^\mu)^2; the text should clarify the correspondence with l^{(2)}_1 and l^{(3)}_j.
[Footnote 4] Footnote 4 contains the typo 'the GW s decay' and could be reworded for readability.
[Section on degenerate and luminal DHOST] The discussion of the full mixed quadratic-plus-cubic BH counterexample is compressed; a brief explanation of why G_\tau = G_A = 0 follows from the stated branch would help the reader follow the degeneracy argument.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the luminality condition is derived in-paper from explicit quadratic actions, and the decay-suppression claim is an externally imported consistency constraint, not a fit or self-citation chain.

full rationale

The central derivation is self-contained. Equation (23), c_g^2/c^2 = 1 - 3 pi_dot X H b3 / G_tau, is computed from the displayed quadratic graviton and photon actions (17)-(22), so the luminality condition b3 = 0 is a calculated consequence rather than an assumed input. The DE-photon sector (12)-(15) is produced by an explicit Kaluza-Klein compactification of the 5D metric (10), not by fitting to the GW170817 limit. Degeneracy compatibility is checked using the stated conditions (25) and (28), and for theory (ii), b3 = 0 follows from G5_X = F5 = 0, leaving F4 free. The decay-suppressed subclass is obtained by solving the externally derived no-decay equation (29) from Creminelli et al. [26] for F4, giving (30); solving a consistency constraint for a free potential is not fitting a parameter and then renaming it a prediction. The paper notes a sign-convention caveat in footnote 3 and concedes that additional checks would be needed for theory (i), but these are transfer-of-result or correctness issues, not circularity. Self-citations to [5] and [34] are methodological continuations, and the equations supporting the conclusions are written out in this paper rather than reduced to those citations. No exhibited circular step is found; at most there are minor non-load-bearing self-citations and an imported endpoint condition whose external validity could be questioned.

Assumptions & free parameters 1 free parameters · 4 assumptions · 1 invented entities

No observational data are fitted. The only hand-picked functional input is F4 via Eq (30). The central derivation relies on the standard DHOST degeneracy classification, linearized FLRW perturbation theory, and one imported decay condition from [26] whose validity for photon-coupled theories is not re-derived.

free parameters (1)

F4(pi,X), with integration function J4(pi) = Eq (30): F4 = (2G4 - X(4G4,X + G5,pi) + 4J4(pi)) / (2X^2 (2G4 + X G5,pi))
F4 is free in the luminal theory; the paper chooses the specific solution (30) to enforce the no-decay condition (29). This is a hand-picked functional tuning, not a fit to data.

assumptions (4)

domain assumption The DHOST degeneracy conditions classify ghost-free theories and determine which classes propagate a graviton.
Accepted from the DHOST classification literature [21-25]; used to identify which theory classes are compatible with the luminality condition b3=0.
standard math Linearized tensor and vector perturbations on a spatially flat FLRW background give the quadratic actions (17) and (20) and the speed ratio (23).
Standard cosmological perturbation theory applied to the action (11); no numerical verification is provided.
ad hoc to paper The Kaluza-Klein compactification with cylinder condition and constant dilaton produces the specific DE-photon couplings in Eq (12).
The 5D setup is a derivation tool; the paper suppresses dilaton dynamics and assumes no dependence on the fifth dimension.
domain assumption The GW decay suppression condition (29) from Creminelli et al. [26] remains valid for the new photon-coupled luminal theory.
The paper imports Eq (29) and asserts that L_DHOSTA does not modify the scalar mode on cosmological backgrounds; this is the key load-bearing premise and is not independently re-derived.

invented entities (1)

Fifth dimension in Kaluza-Klein compactification
purpose: Generate the DE-photon couplings from 5D diffeomorphisms
Introduced in Eqs (9)-(10), compactified with fixed cylinder condition and constant dilaton; explicitly treated as a mathematical tool, not as a physical extra dimension.

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Cite this review

Pith. "Pith review of Luminal Scalar-Tensor theories for a not so dark Dark Energy." pith.science (2026). https://pith.science/paper/2FQWJ3DZ

@misc{pith2026241213460,
  author       = {Pith},
  title        = {Pith review of: Luminal Scalar-Tensor theories for a not so dark Dark Energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2FQWJ3DZ}},
  note         = {Machine review of arXiv:2412.13460}
}

read the original abstract

In general the speed of Gravitational Waves (GWs) in Scalar-Tensor modifications of Einstein's gravity is different from the speed of Light. Nevertheless, it has been measured that their speeds are nearly the same. For the most general Scalar-Tensor theories classified to date that do propagate a graviton -- DHOST, {\it including Horndeski and Beyond Horndeski (BH) theories} -- we show that, remarkably, up to 5 self-consistent couplings of the scalar of Dark Energy (DE) to the Photon are enough to make their GWs luminal in a wide set of cases. We find at least one Luminal Beyond Horndeski theory for which the GW decay into Dark Energy is suppressed in any cosmological background.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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S. Mironov, A. Shtennikova, and M. Valencia-Villegas, “Higher derivative scalar-vector-tensor theories from Kaluza-Klein reductions of Horndeski theory,” Phys. Rev. D, vol. 111, no. 2, p. 024028, 2025

2025
[42]

Horndeski speed tests with scalar-photon couplings,

E. Babichev, C. Charmousis, B. Muntz, A. Padilla, and I. D. Saltas, “Horndeski speed tests with scalar-photon couplings,” 7 2024

2024
[43]

Disformal dark en- ergy at colliders,

P. Brax, C. Burrage, and C. Englert, “Disformal dark en- ergy at colliders,” Phys. Rev. D , vol. 92, no. 4, p. 044036, 2015

2015
[44]

Beyond the Cosmological Standard Model,

A. Joyce, B. Jain, J. Khoury, and M. Trodden, “Beyond the Cosmological Standard Model,” Phys. Rept., vol. 568, pp. 1–98, 2015

2015
[45]

Eﬀective theory for the Vainshtein mechanism from the Horndeski ac- tion,

K. Koyama, G. Niz, and G. Tasinato, “Eﬀective theory for the Vainshtein mechanism from the Horndeski ac- tion,” Phys. Rev. D , vol. 88, p. 021502, 2013

2013
[46]

Break- ing of Vainshtein screening in scalar-tensor theories be- yond Horndeski,

T. Kobayashi, Y. Watanabe, and D. Yamauchi, “Break- ing of Vainshtein screening in scalar-tensor theories be- yond Horndeski,” Phys. Rev. D , vol. 91, no. 6, p. 064013, 2015

2015
[47]

Horndeski theory and beyond: a review,

T. Kobayashi, “Horndeski theory and beyond: a review,” Rept. Prog. Phys. , vol. 82, no. 8, p. 086901, 2019

2019

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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Reviving Horndeski after GW170817 by Kaluza-Klein compactiﬁcations,

S. Mironov, A. Shtennikova, and M. Valencia-Villegas, “Reviving Horndeski after GW170817 by Kaluza-Klein compactiﬁcations,” Phys. Lett. B , vol. 858, p. 139058, 2024

work page 2024

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From k-essence to generalised Galileons,

C. Deﬀayet, X. Gao, D. A. Steer, and G. Zahariade, “From k-essence to generalised Galileons,” Phys. Rev. D , vol. 84, p. 064039, 2011

work page 2011

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Naturally, experimental constraints would be neces- sary on the DE—Photon couplings proposed in this letter LBH4A, LH5A Eqn

or (i) — may be potentially used with minor consideration of the graviton speed, be- cause DE—Photon couplings exist that can take care of the luminality of GWs and the experimental bound ( 1). Naturally, experimental constraints would be neces- sary on the DE—Photon couplings proposed in this letter LBH4A, LH5A Eqn. ( 26). Indeed, laboratory and astro- p...

work page

[4] [2]

We ﬁnd that only two types of DE–Photon couplings are necessary

— with Horndeski and Beyond Horndeski as particular cases. We ﬁnd that only two types of DE–Photon couplings are necessary. One of them cannot be removed by a conformal/ disformal transformation of the metric. It is involved in a new Luminal Beyond Horndeski (BH) the- ory that we show below, for which the GW decay to DE is suppressed. Altogether passing t...

work page

[5] [3]

contain diﬀeomorphism invariant combinations of respectively, (up-to) linear, quadratic and cubic in ∇2π terms. Explicitly, LLinear = G2 + G3 □ π (4) LQuad = 5∑ i=1 ai(π, X) L(2) i , (5) where L(2) i are of order (∇2π)2, L(2) 1 = (πµν )2 , L (2) 2 = (□ π)2 , L (2) 3 = □ π (πµν πµ πν ) , L(2) 4 = (πµρ πµ )2 , L (2) 5 = (πµν πµ πν )2 , (6) and LCubic = 10∑ ...

work page

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Constraining Disformally Cou- pled Scalar Fields,

P. Brax and C. Burrage, “Constraining Disformally Cou- pled Scalar Fields,” Phys. Rev. D , vol. 90, no. 10, p. 104009, 2014

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Speed of Gravitational Waves and the Fate of Scalar-Tensor Gravity,

D. Bettoni, J. M. Ezquiaga, K. Hinterbichler, and M. Zu- malacárregui, “Speed of Gravitational Waves and the Fate of Scalar-Tensor Gravity,” Phys. Rev. D , vol. 95, no. 8, p. 084029, 2017

work page 2017

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Dark Energy After GW170817: Dead Ends and the Road Ahead,

J. M. Ezquiaga and M. Zumalacárregui, “Dark Energy After GW170817: Dead Ends and the Road Ahead,” Phys. Rev. Lett. , vol. 119, no. 25, p. 251304, 2017

work page 2017