{"id":"fb85e800-1e2c-45e2-956c-e77437787873","arxiv_id":"2412.13460","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A class of scalar-tensor dark energy theories with photon couplings can have luminal gravitational waves and suppressed gravitational wave decay into dark energy.","lead":"The paper shows that dark energy scalar fields can couple to photons in a few specific ways so that gravitational waves and light travel at the same speed, keeping many modified gravity theories alive. It also constructs one such theory where gravitational waves do not decay into dark energy, which is needed for them to reach Earth.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The no-decay condition Eq. (29) is imported from Creminelli et al. [26] without re-derivation in the new photon-coupled theory; until the h->chi chi amplitude is recomputed, the F4 solution (30) does not establish suppressed decay 'in any cosmological background'.","rationale":"Good-faith reading: the luminality computation (Eqs. (17)-(24)) is transparent and likely correct; b3=0 is a valid sufficient condition, and the explicit counterexamples show care with DHOST degeneracy classes. The weakest link is not the speed calculation but the decay-suppression claim. The imported Eq. (29) comes from a theory without the new DE-photon couplings, and the paper's statement that the scalar mode is unchanged only guarantees the quadratic action; it does not by itself guarantee that the cubic h->chi chi amplitude is identical. Because the abstract explicitly promises suppression in any cosmological background, this missing re-derivation is load-bearing. The sign-convention footnote compounds the risk: if Eq. (29) was not translated from [26]'s F4 convention, Eq. (30) may solve a differently signed equation. This does not invalidate the rest of the paper, so the reader's CONDITIONAL verdict is appropriate; I recommend no change to the verdict pending the concrete check.","tokens_in":10662,"tokens_out":11700,"duration_ms":115115,"concrete_test":"Re-derive the no-decay condition from the cubic action of (26) on a generic FLRW background in the F4 sign convention of [21,24], including all LDHOSTA vertices, and compare term-by-term with Eq. (29) and Eq. (87) of [26]. Then substitute Eq. (30) and verify directly that the on-shell h->chi chi amplitude vanishes. If the sign-translated condition differs, or if any photon-sector vertex contributes, the suppressed-decay claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest sentence in the abstract is that a subclass of theory (ii) has GW decay into DE suppressed in any cosmological background. That claim rests entirely on Eq. (29), taken from [26] (their Eq. (87)), which was derived for pure beyond-Horndeski gravity without the new DE-photon couplings in LDHOSTA. The quadratic scalar action is indeed unchanged because the vector background vanishes on FLRW, but the GW decay amplitude is a cubic on-shell quantity. LDHOSTA contains cubic vertices such as F^2 box(chi) and F^2 pi_{mu nu} (Eqs. (12)-(15)), and unless the full h->chi chi (and h->chi A, h->AA) vertices are recomputed for action (26), the transfer of Eq. (29) is an assumption rather than a proof. The sign issue in footnote 3 makes this acute: [10,26,35] use the opposite sign for F4, while Eq. (30) solves Eq. (29) only in the stated convention; the paper does not show the conversion from [26]. The authors concede additional checks are needed for theory (i), but the same caveat applies to the decay-suppression part of the headline for theory (ii).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10967,"tokens_out":6565,"duration_ms":64647,"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":[{"comment":"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'.","section":"Eq. (29) and surrounding text"},{"comment":"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.","section":"Footnote 3 and Eq. (29)/(30)"},{"comment":"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).","section":"Conclusions, theory (ii) claim"}],"minor_comments":[{"comment":"The affiliation text contains typos: 'Sci ences' and 'Russ ia' should be 'Sciences' and 'Russia'.","section":"Title page affiliations"},{"comment":"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.","section":"Eq. (12)"},{"comment":"Footnote 4 contains the typo 'the GW s decay' and could be reworded for readability.","section":"Footnote 4"},{"comment":"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.","section":"Section on degenerate and luminal DHOST"}],"recommendation":"major_revision","confidential_remarks":"The paper is appropriate for a hep-th/HEP phenomenology journal and contains a clean central result. The main risk is the unverified transfer of Eq. (29) together with the unresolved sign convention; these are fixable with an explicit calculation or a clear derivation, so major revision rather than rejection is appropriate. The authors should also ensure the abstract's strong statement about suppressed decay is backed by the actual demonstrated result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nThe 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.\n\nAlso 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).\n\nWho 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.","headline":"Clean luminality result in DHOST, but the headline decay-suppression claim is imported rather than re-derived.","tokens_in":11490,"tokens_out":1721,"would_cite":true,"duration_ms":16922,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["gravitational waves","luminality","scalar-tensor theories","DHOST","Beyond Horndeski","dark energy","photon couplings","GW170817"],"falsifier":"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.","tokens_in":10462,"feed_emoji":"🌌","tokens_out":7129,"duration_ms":61117,"temperature":0.7,"pith_summary":"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.","feed_headline":"Five scalar-photon couplings can make gravity waves luminal","feed_subtitle":"A Beyond Horndeski theory with these couplings also suppresses gravitational wave decay into dark energy.","key_machinery":"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).","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Kaluza-Klein compactification method and the starting Luminal Horndeski theory that the present work generalizes by adding $F_4$.","marker":"[5]"},{"why":"Provides the gravitational wave decay condition (Eq. 87 of that paper) that the present theory must satisfy, and the disformal invariance argument that the new coupling evades.","marker":"[26]"},{"why":"Gives the classification of DHOST theories up to cubic order and their degeneracy conditions, used to check when $b_3=0$ is compatible with a propagating graviton.","marker":"[22]"},{"why":"Classifies which DHOST classes propagate a graviton, used to identify the phenomenologically relevant luminal cases and the counterexamples.","marker":"[23]"},{"why":"Review of DHOST theories supplying the degeneracy relations for quadratic Beyond Horndeski and the standard notation for $F_4$ and $F_5$.","marker":"[24]"},{"why":"Establishes the degeneracy conditions for degenerate higher-derivative theories, including the BH4 relations used in theory (i).","marker":"[21]"},{"why":"The competing scalar-photon coupling Lagrangian whose quadratic action is matter-dependent, ruling it out for a background-independent solution and highlighting the distinction from $L_{H5A}$.","marker":"[35]"},{"why":"The GW170817 observation that supplies the speed constraint the paper works to satisfy.","marker":"[1]"}],"fun_headline_variants":["Five photon couplings make gravity waves luminal","Luminal gravity waves from five scalar-photon couplings","How five couplings keep gravitational waves luminal","Luminality for gravity waves via five dark-energy couplings"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Five photon couplings make gravity waves luminal","Luminal gravity waves from five scalar-photon couplings","How five couplings keep gravitational waves luminal","Luminality for gravity waves via five dark-energy couplings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000628,"raw_usage":{"total_tokens":2898,"prompt_tokens":935,"completion_tokens":1963,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":1910}},"tokens_in":551,"tokens_out":1963,"duration_ms":14803,"temperature":1.0,"reasoning_tokens":1910,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:06:37.563286+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Reviving Horndeski after GW170817 by Kaluza-Klein compactifications,","cited_arxiv_id":null,"evidence_quote":"Supplies the Kaluza-Klein compactification method and the starting Luminal Horndeski theory that the present work generalizes by adding $F_4$."},{"cited_title":"Naturally, experimental constraints would be neces- sary on the DE—Photon couplings proposed in this letter LBH4A, LH5A Eqn","cited_arxiv_id":null,"evidence_quote":"Provides the gravitational wave decay condition (Eq. 87 of that paper) that the present theory must satisfy, and the disformal invariance argument that the new coupling evades."},{"cited_title":"Does DESI 2024 Confirm Λ CDM?,","cited_arxiv_id":null,"evidence_quote":"Gives the classification of DHOST theories up to cubic order and their degeneracy conditions, used to check when $b_3=0$ is compatible with a propagating graviton."},{"cited_title":"Effective Description of Higher-Order Scalar-Tensor Theories,","cited_arxiv_id":null,"evidence_quote":"Classifies which DHOST classes propagate a graviton, used to identify the phenomenologically relevant luminal cases and the counterexamples."},{"cited_title":"From the comprehen- sive classification in [23] Table 1 and [22] it is clear that there are many 2 Scalar-Tensor theories with a graviton that can be made Luminal with Eqn","cited_arxiv_id":null,"evidence_quote":"Review of DHOST theories supplying the degeneracy relations for quadratic Beyond Horndeski and the standard notation for $F_4$ and $F_5$."},{"cited_title":"DESI and SNe: Dynamical Dark Energy, Ω m Tension or Systematics?,","cited_arxiv_id":null,"evidence_quote":"Establishes the degeneracy conditions for degenerate higher-derivative theories, including the BH4 relations used in theory (i)."},{"cited_title":"From k-essence to generalised Galileons,","cited_arxiv_id":null,"evidence_quote":"The competing scalar-photon coupling Lagrangian whose quadratic action is matter-dependent, ruling it out for a background-independent solution and highlighting the distinction from $L_{H5A}$."}],"review_version":1}