{"id":"76590149-9541-4d04-8415-1bcae1888575","arxiv_id":"2504.17798","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"A four-parameter power-law FLRW model with a dynamic cosmological constant is fit to OHD, Union, Pantheon, and BAO data, giving dataset-dependent H0 values and a universe age around 21.7 billion years.","lead":"This paper fits a flat FLRW model with a time-dependent cosmological constant, written as a linear combination of the acceleration, the squared Hubble parameter, and the density, to supernovae, Hubble, and BAO data. The model yields very different Hubble constant values depending on the dataset and predicts a 21.7 billion year old universe, which is hard to reconcile with standard age limits.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The model's predicted H(z) is not well defined: the equation-of-state parameter w appears in Eq. (13) and in every fit, but is never fixed or estimated, while the text also uses γ as the EOS parameter.","rationale":"Both the reader and I identify the same soft spot: the power-law H(z) is the engine of the paper, and it depends on an equation-of-state parameter that the fits never fix. The paper's own text wavers between p=γρ and p=wρ, and no table includes w. This is not a stylistic complaint; the exponent changes by a factor of about 2.7 between the two natural readings of the model for the paper's own combined best fit, so the H0, Ωm, Λ0, ρ0 and age numbers are not model predictions but artifacts of an unstated convention. A single refit with w fixed explicitly would settle the issue. Other problems reinforce the concern but are secondary: the BAO χ² in Eqs. (28)-(30) is internally inconsistent (the data vector lists z=0.2 and z=0.35 while X uses z=0.57; the text defines dz=rs/Dv but X uses dA(z*)/Dv), and the final 'Data Availability' statement says no data was used despite Section III fitting four public datasets. These would also need correction, but the undefined w is the most load-bearing because it affects every numerical result. The verdict should remain REJECT unless the w/EOS ambiguity is resolved and a corrected refit reproduces the claimed constraints.","tokens_in":14987,"tokens_out":11718,"duration_ms":114551,"concrete_test":"Rerun the MCMC fits using Eq. (13) exactly as coded, first with w fixed to 0 and then with w set equal to γ, on the same OHD, Union, Pantheon, BAO and combined likelihoods as in Tables I-VI. Report the posterior means and 68% intervals for H0. If the two runs differ by more than the quoted uncertainties, the paper's parameter estimates are conditional on an unstated value of w and the tension-resolution claim is not well defined; if the runs agree, then w is degenerate with α, β, γ and the model's parameter identifiability and the four-parameter claim need to be re-stated. The published tables should also state which w was used in the χ² and MCMC codes.","verdict_should_be":"REJECT","load_bearing_attack":"Eq. (13), the relation actually fitted to OHD, Union, Pantheon and BAO, is H(z)=H0(1+z)^{(w+1)(α+β−3)/[−γ+α(w+1)−2]}. This depends on w. Yet Section III states 'There are four model parameters H0, α, β, γ', no table reports w, and no prior or range for w is given. The text is self-contradictory: in Section II it defines p=γρ and uses γ for dust/radiation/stiff-fluid cases, then immediately writes p=wρ in Eqs. (8)-(9) and keeps both symbols. If w is meant to equal γ, Eq. (13) is not the solution of Eqs. (4)-(6): the exponent should be written with γ in place of w, and since γ is already constrained as part of Λ, the EOS is not an independent parameter. If w is a separate free parameter, the fits are over five parameters and the quoted α, β, γ, H0 posteriors are marginals over an unspecified w. The two interpretations give materially different predictions: for the OHD+BAO+Union best fit (α=0.973, β=0.901, γ=0.676), the exponent in Eq. (13) is 0.661 for w=0 but 1.806 for w=γ, more than a factor of 2.5 different. Thus every H0 value, density, and age in Tables I-VIII, and the claimed 'resolution' of the Hubble tension, is not reproducible until w is fixed.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents a spatially flat FLRW model with a time-varying cosmological constant Λ(t) = α ä/a + β(ȧ/a)^2 + 4πGγρ, derives a power-law Hubble parameter H(z) = H0(1+z)^((w+1)(α+β−3)/(−γ+α(w+1)−2)), and fits H0, α, β, γ to the 46-point OHD sample, the 580-point Union 2.1 compilation, the 1048-point Pantheon sample, and a BAO distance-ratio sample, both individually and in various combinations. It then computes Ωm, ΩΛ, ρ0, ρΛ0, and the age of the universe from the fitted parameters, and claims that the combined fits produce H0 values close to both the early-universe (67.7 km/s/Mpc) and late-universe (73 km/s/Mpc) measurements, thereby resolving the Hubble tension.","tokens_in":15428,"tokens_out":15376,"duration_ms":145821,"significance":"If the model and fits were correct, the paper would offer a compact phenomenological parametrization of a dynamical cosmological term and a possible way to bridge the Hubble tension. The manuscript has some positive features: the analytic formulas for H(z), q, j, densities, and age are explicit, and the MCMC methodology using emcee is a standard and appropriate tool for parameter estimation. However, the analysis is not reproducible as written: the equation-of-state parameter w in the fitted Hubble law is never defined or fitted, the BAO dataset is internally inconsistent, the Pantheon analysis fixes a poorly justified absolute magnitude, and the Data Availability statement contradicts the use of four public datasets. These issues affect every reported parameter and the central conclusion, so the claimed resolution of the Hubble tension is not established.","major_comments":[{"comment":"The fitted Hubble law is not well defined because the equation-of-state parameter w is never specified. The text first sets p = γρ and calls γ the equation-of-state parameter, then writes p = wρ in Eqs. (8)–(9) without relating w to γ. Eq. (13), the relation used for every dataset, contains w, but Section III and Tables I–V report only H0, α, β, γ, with no prior or posterior for w. If w = 0 (dust) is intended, the exponent for the OHD+BAO+Union best fit (α = 0.973, β = 0.901, γ = 0.676) is 0.661; if w = γ, the exponent is 1.806, a factor of about 2.7 difference that changes H(z) and hence every H0, density, and age in Tables I–VIII. If w is instead a fifth free parameter, the quoted posteriors are marginal over an unspecified w. The manuscript must state which interpretation is used and rerun the fits accordingly. In addition, because H(z) depends only on H0 and the combination E = (w+1)(α+β−3)/(−γ+α(w+1)−2), the datasets used here cannot separately pin down α, β, and γ; the individual parameter constraints in Tables I–V are therefore not meaningful tests of the Λ ansatz.","section":"Section II, Eq. (13); Section III"},{"comment":"The BAO analysis is internally inconsistent. The text lists six data points z = (0.106, 0.2, 0.35, 0.44, 0.6, 0.73) with dz = (30.95, 17.55, 10.11, 8.44, 6.69, 5.45), but the vector X in Eq. (29) uses z = 0.57 with value 6.72, omits the z = 0.2 and z = 0.73 entries, and uses reference values 30.84, 10.33, 6.72, 8.41, 6.66, 5.43 that do not correspond to the listed dz values. Equation (27) defines dz as rs(z*)/Dv(z), while Eq. (29) uses dA(z*)/Dv(z). The abstract states that the BAO sample has 5 redshifts, but the text and the 6×6 covariance matrix use 6. These inconsistencies make Table IV and the combined fits unverifiable as written.","section":"Section III.D, Eqs. (27)–(30)"},{"comment":"The Pantheon fit fixes the absolute magnitude M = −19.09 and treats mb0 = 23.29 − 5 log h as a known constant. Pantheon supernova analyses normally marginalize over M because M and H0 are degenerate in the distance modulus. Fixing M to a single value without a systematic uncertainty can shift the inferred H0 by several km/s/Mpc, so the reported H0 = 78.116^{+0.480}_{−0.479} and the combined constraints involving Pantheon depend on an unjustified assumption. The authors should either marginalize over M with a suitable prior or demonstrate that the results are insensitive to M.","section":"Section III.C, Eqs. (25)–(26)"},{"comment":"The claim that the model resolves the Hubble tension is not supported by the analysis. H0 is a free parameter in every fit, so obtaining values near 67.7 or 73 is not a prediction of the model; the four single-dataset fits already span 61.5 to 78.1 km/s/Mpc, and the two combined fits give 67.4 and 75.0, which merely reproduce the input datasets rather than providing a single model value. Moreover, Ωm, ΩΛ, ρ0, Λ0, and the age are algebraic functions of the fitted α, β, γ, and H0 through Eqs. (18)–(21) and (32), so their stated agreement with observations is not an independent test of the model.","section":"Section IV, Tables VI–VIII; Eqs. (18)–(21)"},{"comment":"The statement 'No data was used for the research described in the article' directly contradicts Section III, which fits the 46-point OHD sample, the 580-point Union 2.1 sample, the 1048-point Pantheon sample, and a BAO sample, and presents numerical results derived from those fits. This statement must be corrected to identify the public data products and their sources; as written, it makes the analysis impossible to reproduce and raises a serious integrity concern.","section":"Data Availability statement (end of manuscript)"}],"minor_comments":[{"comment":"The text says the model is valid 'provided that the deceleration constant given by Eq. (16) is positive' and claims it will be seen to be positive, but an accelerating universe requires q < 0, and the reported fits give q ≈ −0.34. This should be corrected.","section":"Section II, text near Eq. (16)"},{"comment":"The section heading says 'Union 2.1 Compilation 680 data set' while the text and Table II use 580 data points, and Eq. (24) labels the fitted quantity mbth although the Union analysis is for distance moduli μth. These mismatches should be reconciled.","section":"Section III.B, Eq. (24)"},{"comment":"The MCMC value β = 0.083 ± 0.490 differs from the least-χ² value β = 0.880 ± 0.490 by almost 10σ, and the figure caption lists parameter values that do not match the table entries. This suggests either a typographical error or a convergence problem; it needs clarification.","section":"Table III and Fig. 5"},{"comment":"The caption of Fig. 7 reports H0 = 65 ± 1, α = 0.11 ± 0.020, β = 0.41 ± 0.020, γ = 0.91 ± 0.01, whereas Table IV reports H0 = 71.386, α = 1.456, β = 1.556, γ = 1.899. These are substantially different and should be reconciled.","section":"Table IV and Fig. 7"},{"comment":"The caption says 'Corner plots for 36 Hubble data set', but the text and Table I refer to the 46-point Hubble dataset.","section":"Fig. 2 caption"},{"comment":"The MCMC implementation is not described in enough detail: no prior ranges, chain lengths, burn-in, or convergence diagnostics are given, and no code or reproducibility instructions are provided. This should be added if the paper is resubmitted.","section":"Section III.A–III.E"}],"recommendation":"reject","confidential_remarks":"I recommend rejection rather than major revision because the central numerical results are not reproducible as written: the undefined w in the fitted Hubble law, the internal contradictions in the BAO analysis, and the fixed-M Pantheon treatment all require redoing the fits. The Data Availability statement is also disqualifying in its current form. If the authors clarify the equation of state, correct the BAO inconsistency, and re-run the analysis with reproducible data and code, a substantially revised manuscript could be reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nShort version: this one should not be published as is. The central fitting equation, Eq. (13), contains a parameter w that is never defined, fixed, or estimated. The text also calls γ the equation-of-state parameter, so the model is ambiguous. The stress-test note is right: for the OHD+BAO+Union best fit, the exponent changes from 0.66 (w=0) to 1.81 (w=γ), which changes every derived quantity in the paper. This is not a minor quibble; it makes the results unreproducible.\n\nWhat the paper does well: the authors run a genuine MCMC analysis on four public datasets, show corner plots and tables, and try to engage with the Hubble tension. The derivation from the field equations to a power-law H(z) is straightforward and checkable.\n\nThe soft spots are serious and plentiful. The BAO section lists six data points in the text (including z=0.2 with dz=17.55), but the X vector in Eq. (29) uses z=0.57 and value 6.72, and the abstract says five redshifts. The Data Availability statement says \"No data was used,\" which contradicts the entire fitting section. The model reduces to the well-known power-law cosmology H(z) ∝ (1+z)^n; the three parameters α, β, γ collapse into one exponent, so calling it a new model is a stretch. The derived age of 21.7 Gyr conflicts with stellar and CMB constraints. And the \"resolution\" of the Hubble tension is just choosing dataset combinations that give either 67 or 75; that is tuning, not a mechanism.\n\nMy verdict: reject. The paper needs a major revision where w is either set to 0, eliminated, or estimated, and where the BAO data and covariance matrix are made consistent. As it stands, the analysis is not reproducible.\n\nWho is this for? Someone working on phenomenological dark energy might want to see the MCMC pipeline, but they would have to redo all the fits. I would not cite it, and I would not send it to a serious referee until the w issue is fixed.\n\nRegards,","headline":"A power-law dark energy model that is not well-defined because the equation-of-state parameter w is never fixed; the fits and BAO analysis are internally inconsistent, so reject.","tokens_in":15920,"tokens_out":3409,"would_cite":false,"duration_ms":31759,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["98.80.Cq","98.80.-k","04.20.Jb"],"model":"deepseek-v4-flash","headline":"The paper claims that a single power-law Hubble parameter, derived from a linear dynamical $\\Lambda(t)$ ansatz, fits both early- and late-universe $H_0$ measurements.","keywords":["FLRW cosmology","dynamical cosmological constant","Hubble parameter","Hubble tension","power-law solution","observational constraints","MCMC parameter estimation","dark energy"],"falsifier":"Measure the deceleration parameter $q(z)$ at several redshifts from a model-independent expansion history (for example, cosmic-chronometer $H(z)$ estimates or standard-siren distances) with errors small enough to detect variation in $q$: Eq. (16) demands that $q$ be constant in redshift, so any statistically significant drift of $q$ with $z$ would falsify the assumed $\\Lambda(t)$ form.","tokens_in":14808,"feed_emoji":"🔭","tokens_out":11879,"duration_ms":106268,"temperature":0.7,"pith_summary":"The paper claims that a flat FLRW universe with a dynamical cosmological term of the form $\\Lambda(t)=\\alpha\\,\\ddot{a}/a+\\beta\\,(\\dot{a}/a)^2+4\\pi G\\gamma\\rho$ closes the field equations and produces a Hubble parameter that is a pure power law in $(1+z)$. Fitting that power law to 46 Hubble measurements, Union 2.1 and Pantheon supernovae, and BAO distances gives best-fit $H_0$ values of $61.53^{+0.453}_{-0.456}$, $69.270^{+0.229}_{-0.228}$, $78.116^{+0.480}_{-0.479}$, and $71.318^{+2.473}_{-2.283}$ km/s/Mpc, with combined fits yielding $67.427^{+0.197}_{-0.199}$ and $74.997^{+0.143}_{-0.145}$. The authors take this as evidence that one model can reproduce both the early-universe value near 67.7 and the local distance-ladder value near 73, offering a simple phenomenological route to the Hubble tension. They also derive present densities and an age of about 21.7 billion years.","feed_headline":"One power-law Hubble model yields H0 values from both eras","feed_subtitle":"Combined MCMC fits to supernova, Hubble, and BAO data yield 67.4 and 75.0, straddling the Hubble tension.","key_machinery":"The load-bearing object is the functional form chosen for the dynamical cosmological term, $\\Lambda(t)=\\alpha\\,\\ddot{a}/a+\\beta\\,(\\dot{a}/a)^2+4\\pi G\\gamma\\rho$ (Eq. 6), with $\\alpha$, $\\beta$, $\\gamma$ constant. Because each of the three terms is linear in curvature and energy-density quantities, the field equations reduce to relations that integrate to simple powers of the scale factor; after imposing $H=H_0$ at $t=t_0$ and converting $a_0/a$ to $1+z$, every observable the paper reports - $H(z)$, $\\rho(z)$, $\\Lambda(z)$, $\\Omega_m$, $\\Omega_\\Lambda$, $q$, $j$, and the redshift-time integral - becomes an explicit power law in $(1+z)$. The ansatz therefore carries the argument: the fitted coefficients simultaneously determine the expansion history, the densities, and the age of the universe.","core_discovery":"On its own terms, the central discovery is that the linear ansatz for $\\Lambda(t)$ turns the Friedmann equations into an integrable system. Solving Eqs. (4)-(6) gives $H(z)=H_0(1+z)^{(w+1)(\\alpha+\\beta-3)/(-\\gamma+\\alpha(w+1)-2)}$ (Eq. 13), with $\\rho(z)$ and $\\Lambda(z)$ following the same redshift power law and $\\Omega_m+\\Omega_\\Lambda=1$ identically. The expansion has constant deceleration parameter $q$ and constant jerk $j$, and the paper describes the result as an accelerating Einstein-de Sitter-type universe. The observational payload is that MCMC fits to the four data sets and their combinations place $H_0$ at values matching both the CMB-calibrated early-universe estimate and the local distance-ladder estimate, so the model's claimed significance is a one-family phenomenological resolution of the Hubble tension without modifying gravity or adding a separate dark-energy component.","pith_inferences":["Because Eq. (16) makes $q$ a constant for any fitted coefficients, the model cannot describe a transition from decelerated to accelerated expansion; a future model-independent detection that $q(z)$ varies with redshift would rule it out even if the current $H_0$ fits look good.","Equation (6) is a linear parametrization of the vacuum energy; relating $\\alpha$, $\\beta$, $\\gamma$ to a scalar-field potential or a modified-gravity action would be needed to turn the empirical fit into a physical mechanism rather than a curve-fitting scheme.","The paper is not consistent about which symbol is the barotropic index - $w$ appears in the derived formulas while $\\gamma$ is sometimes also called the equation-of-state parameter - so the reported parameter bounds should be read as conditional on $w=0$ until a refit with $w$ free is done.","The predicted age near 22 Gyr is high enough that an independent lower bound on the age of the oldest stellar populations, or a direct low-redshift measurement of $H(z)$, would discriminate this model from $\\Lambda$CDM more sharply than the current $H_0$ scatter."],"forward_implications":["With the fitted coefficients, $H(z)$ stays a pure power law at every redshift, so the model predicts constant deceleration $q$ and jerk $j$; it has no transition epoch from deceleration to acceleration, only a fixed expansion class.","The combined OHD+BAO+Union and OHD+Pan+BAO+Union fits give $H_0=67.427^{+0.197}_{-0.199}$ and $74.997^{+0.143}_{-0.145}$ km/s/Mpc, so a single model lands close to both sides of the Hubble tension.","Matter and vacuum densities both scale as powers of $(1+z)$, and the present vacuum density is larger than the present matter density, matching the observed late-time acceleration regime.","The redshift-time integral (Eq. 32) gives a present age of 21.73 or 22.78 billion years depending on the combined data set, a value the paper notes is considerably higher than the $\\Lambda$CDM age."],"supporting_citations":[{"why":"Supplies the 46-point Hubble parameter sample (OHD) used for the first MCMC fit.","marker":"[35]"},{"why":"Supplies the Union 2.1 compilation of 580 SNIa distance moduli used to constrain the parameters.","marker":"[31]"},{"why":"Supplies the 1048-SN Pantheon apparent-magnitude sample used in the supernova fits.","marker":"[32, 33]"},{"why":"Supplies the BAO distance-ratio measurements and the inverse covariance matrix used to build the BAO chi-square.","marker":"[34]"},{"why":"Defines the Hubble-tension discrepancy between early- and late-universe H0 values that the model aims to reproduce.","marker":"[45]"},{"why":"Provides the Cepheid-SN distance-ladder H0 measurement (73 km/s/Mpc) used as a late-universe baseline.","marker":"[48]"},{"why":"Provides the Pantheon+ distance-ladder H0 measurement (73.4 km/s/Mpc) used as another late-universe baseline.","marker":"[46]"}],"fun_headline_variants":["Power-law Λ fits H0 from both cosmic eras","Single model fits both H0 measurements","Dynamic Λ yields H0=67.4 and 75.0","Power-law expansion matches both Hubble constants","One model, two H0 values: 67.4 and 75"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The model's results rest on Eq. (6) - a stipulated linear relation between the cosmological term, the acceleration, the square of the Hubble rate, and the matter density - which is assumed rather than derived; if the real vacuum dynamics are not of this exact form, the fitted parameters and all predictions derived from them do not survive.","fun_headline_variants_meta":{"raw":{"variants":["Power-law Λ fits H0 from both cosmic eras","Single model fits both H0 measurements","Dynamic Λ yields H0=67.4 and 75.0","Power-law expansion matches both Hubble constants","One model, two H0 values: 67.4 and 75"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000799,"raw_usage":{"total_tokens":3599,"prompt_tokens":1116,"completion_tokens":2483,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":732,"completion_tokens_details":{"reasoning_tokens":2404}},"tokens_in":732,"tokens_out":2483,"duration_ms":19432,"temperature":1.0,"reasoning_tokens":2404,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:27:28.991517+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the deceleration parameter $q(z)$ at several redshifts from a model-independent expansion history (for example, cosmic-chronometer $H(z)$ estimates or standard-siren distances) with errors small enough to detect variation in $q$: Eq. (16) demands that $q$ be constant in redshift, so any statistically significant drift of $q$ with $z$ would falsify the assumed $\\Lambda(t)$ form.","supporting_citations":[{"cited_title":"Goswami, and Anirudh Pradhan, An axially symmetric transi- tioning model with observational constraints, Chin J Phys, 80, 261–274(2022)","cited_arxiv_id":null,"evidence_quote":"Supplies the 46-point Hubble parameter sample (OHD) used for the first MCMC fit."},{"cited_title":"Suzuki et al., The Hubble space telescope cluster supernova survey V improving the dark energy constraints above z >1 and building an early-type-hosted supernova sample, Astrophys","cited_arxiv_id":null,"evidence_quote":"Supplies the Union 2.1 compilation of 580 SNIa distance moduli used to constrain the parameters."},{"cited_title":"Giostri, M","cited_arxiv_id":null,"evidence_quote":"Supplies the BAO distance-ratio measurements and the inverse covariance matrix used to build the BAO chi-square."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Cepheid-SN distance-ladder H0 measurement (73 km/s/Mpc) used as a late-universe baseline."},{"cited_title":"Brout, D","cited_arxiv_id":null,"evidence_quote":"Provides the Pantheon+ distance-ladder H0 measurement (73.4 km/s/Mpc) used as another late-universe baseline."}],"review_version":1}