{"id":"7730dc25-eefb-4d4a-9d02-f20690f694ed","arxiv_id":"2608.04079","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Standard sirens from third-generation detectors could measure H0 to 0.21% in Extended Cuscuton models, but the forecast omits the modified GW luminosity distance.","lead":"The authors test four variants of the Extended Cuscuton dark-energy model against today's supernova, chronometer, and BAO data, then model how future ET and Cosmic Explorer gravitational-wave networks would pin down H0 in these models. Their best-case forecast reaches 0.21% H0 precision, but the forecast uses only the background expansion, not the modified gravitational-wave distance the paper highlights.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Forecast likelihood uses the electromagnetic luminosity distance, not the GW luminosity distance of Eq. (22), so the advertised tensor-sector probe is absent from the quantitative forecasts.","rationale":"The reader's weakest_assumption is the most load-bearing. The central claim is that 3G standard sirens provide a precise complementary test of this non-dynamical dark energy sector, with tensor-sector amplitude damping a key advertised feature (Secs. I, II, VIII). The forecast likelihood is explicitly EM-only, so the quantitative reach of the tensor channel is never computed. I checked whether this is cosmetic: because the posteriors in Tables VIII/IX keep c2 and c4 within roughly 0.1-0.2 of zero, the ratio sqrt(f4(φ0)/f4(φ(z))) is typically within a few times 10^-4 of unity, so the H0/ΩΛ numbers are unlikely to change dramatically. The concern is therefore more about the validity of the 'modified amplitude damping' claim than about the headline precision. It is still load-bearing because the paper's novelty and conclusions lean on this channel, and the forecast as written cannot support statements about tensor-sector sensitivity. The radiation treatment in the early-time BAO analysis is also underspecified (Eq. 40 is introduced without the corresponding modified E' system), but that affects only the current-data part and is secondary. The conditional verdict is appropriate: the analysis is solid for what it actually computes, but the advertised tensor-sector forecast is missing. The concrete test above would settle whether the missing channel changes the numbers or only the interpretation.","tokens_in":34945,"tokens_out":10933,"duration_ms":112745,"concrete_test":"Re-run the SM02 ET+2CE kilonova forecast with Eq. (58) replaced by d_i - dGW_L(z,λ), where dGW_L = dEM_L sqrt(f4(φ0)/f4(φ(z))) with dEM_L from Eqs. (31)-(33) and f4 from Eq. (24), using the same ΛCDM mock catalogue and priors. Compare the recovered H0, ΩΛ, c2, c4 medians and 68% intervals to Tables VIII/IX. If the intervals shift by more than their quoted widths, the published forecasts are mis-specified; if they are unchanged, the issue is a claim/documentation gap rather than a numerical one.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The forecasting pipeline is internally inconsistent with the model's tensor-sector prediction. The likelihood in Eq. (58) compares data to d_L(z, λ), computed from the background expansion E(z) via Eqs. (31)-(33). That is the electromagnetic luminosity distance. The paper's own derivation in Eq. (22) gives the GW luminosity distance dGW_L(z) = dEM_L(z) sqrt(f4(φ0)/f4(φ(z))), and the introduction and conclusions repeatedly state that standard sirens probe both the background and the modified amplitude damping. The mock catalogues (Sec. V A) are also generated with an EM ΛCDM distance. Consequently, Tables VIII and IX and the quoted 0.21% H0 / 1.87% ΩΛ uncertainties are background-only forecasts; they do not exercise the tensor-sector channel. The statement in Sec. VIII that the mock catalogues are sensitive to 'the standard tensor-amplitude damping' is unsupported by the analysis as written. This does not invalidate the current-data constraints, but it means the advertised complementarity is overclaimed unless the likelihood is replaced by dGW_L or the claims are recast as background-only.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies an analytically tractable sector of Extended Cuscuton gravity, a non-dynamical dark-energy model inside the viable Horndeski class, and selects four benchmark submodels fixed by c1=±1, c3=c5=0 and two asymptotic de Sitter conditions. The first part constrains these models with current cosmic-chronometer, Type-Ia supernova, and DESI DR1 BAO data, applying theoretical viability, Lunar Laser Ranging, and Big Bang Nucleosynthesis bounds; the second part forecasts constraints from third-generation bright standard sirens using mock BNS catalogs for ET, ET+CE, and ET+2CE networks and three electromagnetic counterpart channels (prompt emission, afterglow, kilonova). The current-data analysis finds that the models are already close to ΛCDM, with ΩΛ=0.73±0.01 and H0=71.71+0.20/−0.31 for SM01, and that the CC+SN versus CC+BAO H0 offset persists. The forecasts claim H0 relative uncertainties as low as 0.21% and ΩΛ uncertainties at the 1.87% level for the best configurations. The central advertised result is that third-generation standard sirens provide a complementary test of non-dynamical dark energy beyond the background expansion.","tokens_in":35128,"tokens_out":9743,"duration_ms":110512,"significance":"If the forecasts actually used the modified gravitational-wave luminosity distance derived in the paper, this would be a useful contribution: the model is analytically tractable within the post-GW170817 viable Horndeski sector, the four benchmark submodels are clearly defined, and the current-data analysis is carefully presented, including LLR and BBN priors that are often omitted in similar studies. The mock multi-messenger pipeline is detailed and follows an established methodology, and the ΛCDM recovery check is a sensible validation step. The main advertised tensor-sector probe, however, is not implemented in the forecast likelihood as written. The stated complementarity therefore is currently an overclaim, and the forecast tables must be recomputed with the GW luminosity distance or the conclusions must be explicitly restricted to the background channel. With that correction, the paper would offer a clean, falsifiable forecast target for 3G standard-siren cosmology.","major_comments":[{"comment":"The forecast likelihood in Eq. (58) compares each mock standard-siren distance with d_L(z,λ) computed from the background expansion rate E(z) through Eqs. (31)-(33). That quantity is the electromagnetic luminosity distance. The paper's own derivation gives d_GW^L(z)=d_em^L(z) sqrt(f4(φ0)/f4(φ(z))) in Eq. (22), and both the Introduction and Sec. VIII state that standard sirens probe the background and the modified tensor-amplitude damping. Since Eq. (58) never uses d_GW^L, and since the mock catalogs in Sec. V.A are generated with the electromagnetic ΛCDM distance, Tables VIII and IX and the quoted 0.21% H0 / 1.87% ΩΛ uncertainties are background-only forecasts. The ΛCDM recovery check in Sec. VI.A does not exercise the tensor channel because f4=1 along the fiducial. The closing statement in Sec. VIII that the mock catalogs are sensitive to 'the standard tensor-amplitude damping' is therefore unsupported by the analysis as written. This is load-bearing for the main advertised claim: the forecasts must be rerun with d_GW^L, or the conclusions must be explicitly recast as background-only forecasts.","section":"Sec. VI, Eq. (58); Sec. II, Eq. (22)"},{"comment":"The paper states in Sec. II that when early-time datasets are included, the radiation contribution must be restored in the background equation, giving Eq. (40). However, Sec. III presents the model for the current-data analysis as Eqs. (31)-(33) and does not state whether Eq. (40) was used. This matters because the adopted BAO dataset in Sec. III.A includes the DESI Lyman-α sample out to z=4.16. If Eq. (40) was not used, the high-redshift BAO constraints in Table III are evaluated with a matter-only background; if it was used, the text must say so explicitly for reproducibility. Please clarify which background equation was actually implemented in the CC+BAO+SN analysis and, if necessary, recompute the affected constraints.","section":"Sec. II, Eq. (40); Sec. III; Sec. IV, Table III"}],"minor_comments":[{"comment":"The recovered supernova absolute magnitude for the full combination is M=-19.32±0.01, which sits about 2.9σ away from the adopted Gaussian prior mean M_B=-19.214±0.037 in Table II. Please comment on this shift and verify that it is not caused by a sign or calibration convention in the distance-modulus implementation.","section":"Sec. IV, Table III; Sec. II, Table II"},{"comment":"The sentence defining the enlarged parameter space reads 'the parameter space is enlarged to λ={H0, ΩΛ, c2, c4}, while is derived from the closure condition'; the derived quantity is Ωm,0 and should be named explicitly.","section":"Sec. VI"},{"comment":"The current-data analysis is described as both an MCMC analysis in Sec. III.A and a χ2 minimisation in Sec. III.B, while Fig. 1 and Table III report credible intervals. Please clarify whether the current-data contours come from MCMC posterior sampling or from χ2 profiling, and specify the sampler and convergence criteria if the former.","section":"Sec. III.B and Sec. VI"},{"comment":"The forecast priors restrict c4 to (−5,0) for c1=+1 and (0,5) for c1=−1, justified as branch-consistency conditions, but Table II lists a symmetric [−5,5] prior for c4 in the current-data analysis. Please reconcile these two choices and state whether the branch-consistency condition was also imposed in the current-data fit.","section":"Table II vs Sec. VI"}],"recommendation":"major_revision","confidential_remarks":"The reader's circularity concern is not, in my assessment, supported by the manuscript: the mock recovery check is a pipeline validation, not a construction that forces the conclusions. The substantive problem is the mismatch between the advertised tensor-sector probe and the likelihood actually implemented in Eq. (58). If the forecast is redone with d_GW^L, the quantitative results may change, so this is not a minor wording fix. The current-data part of the paper is solid and should be preserved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe thing to know: this paper does a careful job constraining four new benchmark submodels of Extended Cuscuton with CC+BAO+SN data, and the current-data results are trustworthy. But the 3G forecast section has a structural mismatch: the likelihood in Eq. (58) compares mock data to the electromagnetic luminosity distance, not the GW luminosity distance of Eq. (22), so the advertised tensor-sector probe never enters the forecast numbers.\n\nWhat's new: the four submodels from the de Sitter branch conditions (38)-(39), with c3=c5=0, are original. The constraints show the model is already pinned close to LCDM, with Omega_Lambda=0.73±0.01 and H0=71.71±0.20 for SM01, and the mild CC+SN vs CC+BAO offset persists. That part is carefully done, with viability priors (BBN, LLR) implemented cleanly. The forecast pipeline is standard (following Refs. [41-43]) and the mock recovery of LCDM is a good validation.\n\nSoft spots: the forecast numbers are background-only. The paper says standard sirens probe both the expansion history and the modified amplitude damping, but the likelihood never uses dGW_L. So the quoted 0.21% H0 and 1.87% Omega_Lambda uncertainties do not include the tensor-sector channel. That's an overclaim unless fixed. Also, the radiation restoration for BAO is mentioned but not detailed; that's likely minor given the redshift range.\n\nThis is a solid new application, not a new framework. The gap is central but addressable. I'd send it to review, asking the authors to either use dGW_L in the forecast or explicitly state that the forecasts are background-only and adjust the abstract and conclusions accordingly.","headline":"Well-executed background constraints on a new Extended Cuscuton subclass, but the 3G forecast likelihood uses EM distances only, so the advertised tensor-sector probe is missing from the numbers.","tokens_in":35706,"tokens_out":2927,"would_cite":true,"duration_ms":29913,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["98.80.-k","04.50.Kd","95.36.+x"],"model":"deepseek-v4-flash","headline":"An analytically tractable Extended Cuscuton sector stays close to ΛCDM under current data, and third-generation bright sirens could measure H0 to 0.21% and ΩΛ to 1.87%, keeping the model testable beyond ΛCDM.","keywords":["Extended Cuscuton","Horndeski gravity","non-dynamical dark energy","standard sirens","gravitational-wave cosmology","Einstein Telescope","Hubble constant tension","cosmic chronometers"],"falsifier":"A concrete check: rebuild the mock-catalogue likelihood with the gravitational-wave luminosity distance of Eq. (22) in place of the background electromagnetic $d_L(z, \\lambda)$ and inspect whether the recovered $H_0$ and $\\Omega_\\Lambda$ shift by more than the quoted 68% credible intervals, and whether the 0.21% relative uncertainty on $H_0$ survives. If the shift exceeds the quoted bands, the headline forecast claim is not robust to the theory's own tensor-sector prediction.","tokens_in":34691,"feed_emoji":"🌌","tokens_out":13983,"duration_ms":124540,"temperature":0.7,"pith_summary":"The paper targets a question that matters for the Hubble tension and for modified gravity: can a viable dark-energy alternative to the cosmological constant be observationally distinguished from ΛCDM? It studies the Extended Cuscuton model, a Horndeski subclass in which the scalar field is non-dynamical, so only the two gravitational-wave polarizations propagate, yet the same non-minimal coupling that drives dark energy also changes the amplitude damping of gravitational waves. Using cosmic chronometers, Type-Ia supernovae, and BAO data, the authors find that four benchmark submodels are already confined to small departures from ΛCDM, with $\\Omega_\\Lambda = 0.73^{+0.01}_{-0.01}$ and $H_0 = 71.71^{+0.20}_{-0.31}$ km s$^{-1}$ Mpc$^{-1}$ for the full data combination, and that the calibration-driven offset between CC+SN and CC+BAO determinations of $H_0$ persists. Forecasting mock bright-siren catalogues for third-generation detector networks, they find that kilonova counterparts with an ET+2CE network could measure $H_0$ to 0.21% relative uncertainty and $\\Omega_\\Lambda$ to 1.87%, so future standard sirens can provide a precise complementary test of non-dynamical dark energy beyond $\\Lambda$CDM.","feed_headline":"Bright sirens could measure H0 to 0.21 percent","feed_subtitle":"Current data nearly erase the model's differences from ΛCDM; future detectors could still spot them.","key_machinery":"The load-bearing object is the polynomial Extended Cuscuton subclass ($f_4 = c_1 \\phi^2 + c_2 \\phi + 1$, $f_3 = 0$, $f_2 = c_3 \\phi + c_4$, $f_1 = 6c_5 \\phi^2 + 6c_6 \\phi + 2\\Lambda$) in which the scalar-field equation of motion reduces to a linear algebraic equation, giving the scalar as a rational function $\\varphi(E)$ of the expansion rate and reducing the dynamics to a first-order system for $\\varphi'(z)$ and $E'(z)$. The four benchmark submodels are fixed by setting $\\tilde c_1 = \\pm 1$ and $\\tilde c_3 = \\tilde c_5 = 0$ to remove a prior-dominated degenerate approach to $\\Lambda$CDM, and by imposing one of two asymptotic de Sitter conditions on the combination of $\\tilde c_2$, $\\tilde c_4$, and $\\tilde c_6$. The same function $f_4(\\varphi)$ then connects every observable sector: it controls the background via the Extended Cuscuton energy density $\\Omega_{\\rm ec}$, it sets the Lunar Laser Ranging and Big Bang Nucleosynthesis priors through $G_N \\propto 1/f_4$, and it enters the tensor sector through the gravitational-wave luminosity distance ratio $d^{\\rm GW}_L/d^{\\rm EM}_L = \\sqrt{f_4(\\varphi_0)/f_4(\\varphi(z))}$. This single non-minimal coupling is what lets background data and standard sirens test the same physical ingredient.","core_discovery":"On its own terms, the paper's central claim is that the analytically tractable Extended Cuscuton sector provides a controlled, minimally modified dark-energy target that remains close to $\\Lambda$CDM but is not observationally inert. The four benchmark submodels defined by the sign of $\\tilde c_1$ and by the two asymptotic de Sitter branches all fit the combined CC+BAO+SN data with $\\Omega_\\Lambda = 0.73 \\pm 0.01$ and $H_0 \\simeq 71.7$ km s$^{-1}$ Mpc$^{-1}$, with the model dependence absorbed entirely by the shape parameters $\\tilde c_2$ and $\\tilde c_4$; none of the models removes the $H_0$ offset between the supernova-calibrated and BAO-calibrated combinations (about 72.5 versus 69.2 km s$^{-1}$ Mpc$^{-1}$). The forecast claim is that third-generation bright-siren networks, especially the kilonova channel with two Cosmic Explorer detectors added to Einstein Telescope, can recover the fiducial $\\Lambda$CDM cosmology within the enlarged parameter space and measure $H_0$ with relative uncertainty as low as 0.21% and $\\Omega_\\Lambda$ at the 1.87% level, with all configurations staying below 13.18% on $H_0$. The paper concludes that standard sirens at third generation can provide a precise complementary probe of non-dynamical dark energy beyond $\\Lambda$CDM.","pith_inferences":["The forecast likelihood in Eq. (58) uses the electromagnetic luminosity distance from the background expansion rather than the gravitational-wave luminosity distance of Eq. (22), so the advertised tensor-sector probe is not actually exercised; re-running the forecasts with $d^{\\rm GW}_L$ could change the quoted precision and is the immediate test of the headline numbers.","If the same pipeline were applied to the full analytically tractable subclass with non-zero $\\tilde c_3$ and $\\tilde c_5$ (which the paper sets to zero to avoid a prior-dominated direction), bright-siren data might be the tool that breaks the $\\Lambda$CDM degeneracy that background probes cannot see.","The paper's hierarchy result suggests that for 3G standard-siren cosmology in modified gravity, the binding constraint is counterpart localization and count (the kilonova channel) rather than detector sensitivity per se; other Horndeski subclasses with a running Planck mass would likely show the same hierarchy.","A real detection of the ratio $d^{\\rm GW}_L/d^{\\rm EM}_L = \\sqrt{f_4(\\varphi_0)/f_4(\\varphi(z))}$, measurable in principle by comparing the gravitational-wave and electromagnetic distances of the same event, would directly test the non-minimal coupling that the current-data analysis constrains only indirectly through Lunar Laser Ranging and Big Bang Nucleosynthesis bounds."],"forward_implications":["All four submodels pass current CC+BAO+SN constraints only within a narrow region around $\\Lambda$CDM, with $\\Omega_\\Lambda \\approx 0.73$ and shape parameters $|\\tilde c_2| \\lesssim 0.1$ and $|\\tilde c_4| \\lesssim 0.3$ that are stable across data combinations.","Because the CC+SN and CC+BAO combinations still prefer different $H_0$ values (about 72.5 versus 69.2 km s$^{-1}$ Mpc$^{-1}$), the Extended Cuscuton model does not by itself resolve the Hubble tension.","For the most informative forecast configuration (ET+2CE with kilonova counterparts), bright sirens recover $H_0$ to 0.21% relative uncertainty and $\\Omega_\\Lambda$ to 1.87%, nearly matching $\\Lambda$CDM-only precision even with two extra shape parameters in the fit.","The de Sitter branch with $\\varphi_{\\rm dS} = 0$ (SM01, SM03) constrains $\\tilde c_2$ to roughly 39% to 83% relative uncertainty, while the $\\varphi_{\\rm dS} \\neq 0$ branch (SM02, SM04) keeps $\\tilde c_2$ consistent with zero, so the two branches are observationally distinguishable mainly through $\\tilde c_2$.","Prompt-emission and afterglow channels are far less constraining, with $H_0$ uncertainties up to about 13%, so the projected reach is driven by the number of well-localized kilonova counterparts."],"supporting_citations":[{"why":"Defines the polynomial Extended Cuscuton subclass with an algebraically solvable scalar field; the four benchmark submodels are built from its parameter restrictions.","marker":"[33]"},{"why":"Introduces the Extended Cuscuton action in which the scalar is non-dynamical, the starting point for the whole analysis.","marker":"[32]"},{"why":"Derives the luminal scalar-tensor gravitational-wave luminosity distance with modified amplitude damping, the basis of Eq. (20).","marker":"[2, 3]"},{"why":"Supplies the relation $d^{\\rm GW}_L/d^{\\rm EM}_L = M_*(0)/M_*(z)$ for a running effective Planck mass used in Eq. (20).","marker":"[13]"},{"why":"Provide the Lunar Laser Ranging bound on the present-day variation of $G_N$ imposed as a viability prior.","marker":"[27, 28]"},{"why":"Provide the Big Bang Nucleosynthesis bound on the early-time gravitational coupling imposed as a viability prior.","marker":"[29–31]"},{"why":"Establish the bright-siren mock catalogue and combined GW+EM analysis methodology the forecasts follow.","marker":"[41–43]"},{"why":"Supply the structured-jet, afterglow, and kilonova emission models that produce the counterpart detection counts.","marker":"[81, 82]"},{"why":"Specifies the Einstein Telescope and Cosmic Explorer network configurations used in the forecasts.","marker":"[11]"}],"fun_headline_variants":["Horndeski dark energy: bright sirens target 0.21% H0","Sirens could pin H0 to 0.21% for non-dynamical dark energy","Bright sirens at 0.21%: a sharp test for Horndeski dark energy","Future sirens measure H0 to 0.21%: dark energy beyond ΛCDM?","Dark energy hidden at 0.21%? Sirens will expose it"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The forecast likelihood assumes each standard-siren event measures the electromagnetic luminosity distance from the background expansion and ignores the modified gravitational-wave amplitude damping (the $\\sqrt{f_4(\\varphi_0)/f_4(\\varphi(z))}$ factor) that the theory predicts, so if that factor is real the quoted forecast precision is being computed on the wrong observable.","fun_headline_variants_meta":{"raw":{"variants":["Horndeski dark energy: bright sirens target 0.21% H0","Sirens could pin H0 to 0.21% for non-dynamical dark energy","Bright sirens at 0.21%: a sharp test for Horndeski dark energy","Future sirens measure H0 to 0.21%: dark energy beyond ΛCDM?","Dark energy hidden at 0.21%? Sirens will expose it"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001065,"raw_usage":{"total_tokens":4545,"prompt_tokens":1109,"completion_tokens":3436,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":725,"completion_tokens_details":{"reasoning_tokens":3316}},"tokens_in":725,"tokens_out":3436,"duration_ms":27018,"temperature":1.0,"reasoning_tokens":3316,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T00:34:40.683155+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: rebuild the mock-catalogue likelihood with the gravitational-wave luminosity distance of Eq. (22) in place of the background electromagnetic $d_L(z, \\lambda)$ and inspect whether the recovered $H_0$ and $\\Omega_\\Lambda$ shift by more than the quoted 68% credible intervals, and whether the 0.21% relative uncertainty on $H_0$ survives. If the shift exceeds the quoted bands, the headline forecast claim is not robust to the theory's own tensor-sector prediction.","supporting_citations":[],"review_version":1}