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REVIEW 2 major objections 4 minor 105 references

Cosmological constraints and standard sirens forecasts for non-dynamical dark energy in Horndeski gravity

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

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2608.04079 v1 pith:VWFOMMD6 submitted 2026-08-04 astro-ph.CO gr-qc

Marcello Miranda , Ruchika , Ivan De Martino , Daniele Vernieri , Salvatore Capozziello This is my paper
classification astro-ph.COgr-qc PACS 98.80.-k04.50.Kd95.36.+x
keywords ExtendedCuscutonHorndeskigravitynon-dynamicaldarkenergystandardsirensgravitational-wavecosmologyEinsteinTelescopeHubbleconstanttensioncosmicchronometers
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Sec. VI, Eq. (58); Sec. II, Eq. (22)] 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.
  2. [Sec. II, Eq. (40); Sec. III; Sec. IV, Table III] 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.
minor comments (4)
  1. [Sec. IV, Table III; Sec. II, Table II] 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.
  2. [Sec. VI] 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.
  3. [Sec. III.B and Sec. VI] 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.
  4. [Table II vs Sec. VI] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the model constraints and forecasts follow from data and the stated likelihood; the tensor-sector overclaim is a correctness caveat, not a circular step.

full rationale

The paper's current-data constraints (e.g., ΩΛ = 0.73 ± 0.01 and H0 = 71.71+0.20/−0.31 for SM01, Sec. IV) are obtained by fitting the background equations (31)–(33) to CC, BAO, and SN data through the χ2 in Eq. (42); the parameters are not defined in terms of the fitted outputs, and the closeness to ΛCDM is a data-driven result within the stated priors and viability cuts. The 3G forecasts are standard mock exercises: catalogs are generated from a Planck-ΛCDM fiducial distance (Sec. V A), and the likelihood in Eq. (58) is then applied to the four Extended Cuscuton submodels; recovering the fiducial parameters in Tables VII–IX validates the pipeline rather than constituting a prediction forced by construction. Self-citations [41–43,64] supply the mock-generation methodology but are not load-bearing for any physical derivation, and the model framework rests on external references [32,33] and on LLR/BBN bounds. One non-circular caveat: the forecast likelihood Eq. (58) uses the electromagnetic luminosity distance dL(z,λ) from E(z), whereas Eq. (22) defines dGW_L = dEM_L sqrt(f4(φ0)/f4(φ(z))); therefore the Sec. VIII statement that the catalogues are sensitive to the 'standard tensor-amplitude damping' is not supported by the analysis as written. This is an overclaim/omission, not a circular derivation, and it does not affect the current-data constraints.

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

The central forecasts depend on four fitted cosmological/gravity parameters (H0, ΩΛ, ~c2, ~c4) plus the SN nuisance M in the current-data analysis. The model sector itself rests on the polynomial ansatz of Ref. [33], the luminal-tensor assumption, the branch choice, and the neglect of screening in the GW distance; the benchmark submodels add the de Sitter conditions (38)-(39).

free parameters (5)
  • H0 = 71.71+0.20/-0.31 (SM01, CC+BAO+SN)
    Hubble constant, sampled in both current-data fit and forecast with prior U(50,90).
  • ΩΛ = 0.73+0.01/-0.01 (SM01, CC+BAO+SN)
    Dark-energy density parameter, sampled with prior [0.1,0.9] in current data and U(0,1) in forecasts.
  • ~c2 = -0.09+0.04/-0.04 (SM01, CC+BAO+SN)
    Linear coefficient in the non-minimal coupling f4; shape parameter controlling departures from ΛCDM.
  • ~c4 = -0.22+0.19/-0.10 (SM01, CC+BAO+SN)
    Cuscuton coupling coefficient; shape parameter controlling departures from ΛCDM.
  • M = -19.32+0.00/-0.01 (SM01, CC+BAO+SN)
    Supernova absolute magnitude, nuisance parameter with Gaussian prior N(-19.214,0.037^2).
assumptions (5)
  • domain assumption Polynomial ansatz for f1..f4 (Eqs. 5-8) from Ref. [33] defines the analytically tractable Extended Cuscuton sector.
    The whole analysis is restricted to this ansatz; other Extended Cuscuton sectors are not considered.
  • domain assumption Tensor modes propagate luminally (c_T=1), so GW effects are confined to amplitude damping via running Planck mass.
    Imposed by post-GW170817 viability; used to justify Eq. (22).
  • domain assumption Branch choice ˙ϕ < 0 (footnote 1) selects the future-directed scalar gradient.
    Fixes sign of sqrt(2X) in the action and is used in deriving the background equations.
  • domain assumption Local screening effects in dGW_L are neglected; background FLRW f4 values are used at source and observer.
    Stated after Eq. (22); affects the tensor-sector distance ratio if screening is significant.
  • ad hoc to paper The four benchmark submodels with ~c3=~c5=0 and de Sitter conditions (38)-(39) span the phenomenologically relevant non-degenerate sector.
    Imposed to avoid prior-dominated degenerate directions; restricts the model space tested and the conclusions drawn.

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Cite this review

Pith. "Pith review of Cosmological constraints and standard sirens forecasts for non-dynamical dark energy in Horndeski gravity." pith.science (2026). https://pith.science/paper/VWFOMMD6

@misc{pith2026260804079,
  author       = {Pith},
  title        = {Pith review of: Cosmological constraints and standard sirens forecasts for non-dynamical dark energy in Horndeski gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VWFOMMD6}},
  note         = {Machine review of arXiv:2608.04079}
}
abstract

We investigate an analytically tractable sector of the Extended Cuscuton model, a non-dynamical dark-energy realization within the framework of viable Horndeski gravity. We focus on four benchmark submodels and constrain them with current background probes, namely cosmic chronometers, Type-Ia supernovae, and BAO, while imposing theoretical viability, Lunar Laser Ranging, and Big Bang Nucleosynthesis bounds. We then forecast third-generation bright-standard-siren constraints with Einstein Telescope and Cosmic Explorer networks, considering prompt-emission, afterglow, and kilonova counterparts. Current data already restrict the viable parameter space to small departures from $\Lambda$CDM and do not remove the calibration-driven offset between the CC+SN and CC+BAO determinations of $H_0$. In principle, future bright sirens substantially sharpen the constraints, especially for kilonova catalogues and extended detector networks. Across the forecast configurations, the relative uncertainty on $H_0$ remains below $13.18\%$ and can reach $0.21\%$ in the most constraining cases, while $\Omega_\Lambda$ is recovered at the percent level in the best cases. These results show that third-generation standard sirens can provide a precise complementary test of non-dynamical dark energy beyond $\Lambda$CDM.

Figures

Figures reproduced from arXiv: 2608.04079 by the authors.

Figure 1
Figure 1. Posterior distributions (68% and 99% credible intervals) of the sampled parameters ( [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Posterior distributions (68% and 95% credible intervals) of the sampled parameters ( [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Posterior distributions (68% and 95% credible intervals) of the sampled parameters ( [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Posterior distributions (68% and 95% credible intervals) of the sampled parameters ( [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]

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