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REVIEW 3 major objections 4 minor 133 references

Letting the entropy exponent run with cosmic scale yields H0 ≈ 69.3 and a sign flip near z ≈ 100.

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2026-08-01 02:20 UTC pith:S2PII3NL

load-bearing objection A clean derivation-plus-fit of a running Barrow entropy model with an honest BIC verdict, but the Hubble-tension claim is weaker than it looks and the BAO likelihood is inconsistent with the model's own sound horizon. the 3 major comments →

arxiv 2607.26105 v1 pith:S2PII3NL submitted 2026-07-28 gr-qc astro-ph.CO

Cosmological consequences of scale-dependent Barrow-Tsallis entropy

Giuseppe Gaetano Luciano , Orlando Luongo , Marco Muccino This is my paper
classification gr-qc astro-ph.CO MSC 83F0583D05 PACS 98.80.-k04.70.Dy
keywords Barrow entropyTsallis entropyscale-dependent entropyHubble tensiongravity-thermodynamics conjecturemodified Friedmann equationdark energycosmological parameter estimation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper argues that if the anomalous dimension of Barrow-Tsallis entropy varies with the Hubble scale, the resulting modified Friedmann equation describes cosmic expansion in a way that fits a broad combination of distance and expansion-rate data. The best-fit model yields a present-day entropy exponent around 10^-4 that changes sign near redshift 100, with negative values in the early universe and positive values today. This transition is interpreted as a physical imprint of the passage from a UV-dominated, microstate-suppressing phase to a late-time, fractal-like horizon regime. The inferred Hubble constant, H0 ≈ 69.3 km/s/Mpc, sits between early- and late-universe determinations, suggesting a partial alleviation of the Hubble tension. The standard ΛCDM cosmology is recovered when the running parameters vanish, and ΛCDM remains mildly preferred by the Bayesian Information Criterion.

Core claim

The central claim is that the entropy-area relation with a scale-dependent exponent Δ(x)=Δ0+Δ1/ln x, where x=(H1/H)^2, when fed into the gravity-thermodynamics first law on the apparent horizon, yields an exact analytic Hubble rate H(z). Combined fits to Hubble-rate measurements, type Ia supernova distances, baryon acoustic oscillations, and CMB shift parameters give H0 = 69.27^+0.62_-0.65 km/s/Mpc with Δ0 ≈ 2.4×10^-3 and Δ1 ≈ -0.65, implying Δ(0) ≈ 1.09×10^-4. The derived Δ(z) crosses zero at z_tr ≈ 100 (with broad uncertainties), connecting a negative-Δ early phase to a positive-Δ late phase. The model is fully consistent with the data but is not statistically preferred over ΛCDM (ΔBIC ≈ 7

What carries the argument

The central object is the Barrow-Tsallis entropy S=(A/A0)^(1+Δ/2) with a 'running' anomalous dimension Δ(x)=Δ0+Δ1/ln x. The gravitational side is the gravity-thermodynamics conjecture: applying the first law dE=T dS+W dV to the apparent horizon of a flat FRW universe converts the entropy functional into a modified first Friedmann equation and, ultimately, into a closed-form expression for H(z). The parameter Δ0 sets the amplitude of the entropy correction at the present epoch, Δ1 controls the rate of its scale dependence, and the ΛCDM limit is recovered exactly for Δ0=Δ1=0.

Load-bearing premise

The entire analysis stands on the assumed functional form Δ(x)=Δ0+Δ1/ln x, which is motivated by analogy to quantum-gravity log corrections but is not derived from any concrete theory; if the true scale dependence of the entropy exponent differs, the derived H(z), the sign-reversal redshift, and the claimed Hubble-tension relief are all artifacts.

What would settle it

A high-precision, model-independent measurement of H(z) at z ≈ 1–3 (e.g., from gravitational-wave standard sirens or improved cosmic chronometers) that shows no departure from ΛCDM at the level predicted by the best-fit Δ0 and Δ1 would falsify the model, as would a direct early-universe constraint showing Δ > 0 at z ≈ 1000.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the running-entropy picture is correct, the Hubble expansion history acquires a small, calculable deviation from ΛCDM that is most visible at intermediate redshifts, where future standard-siren and cosmic-chronometer measurements can test it.
  • The model's negative Δ at z > 100 implies a horizon with fewer effective microstates in the early universe; this would show up as a slight suppression of the radiation-like entropy contribution and could alter the effective number of relativistic species by a small amount.
  • The claimed H0 ≈ 69.3 km/s/Mpc is a concrete prediction: it is ~2 km/s/Mpc below the local distance-ladder value and ~2 km/s/Mpc above Planck, so improved independent H0 measurements can decide whether the partial alleviation survives.
  • The sign-change redshift z_tr ≈ 100 is consistent, within uncertainties, with the photon-decoupling and neutrino-nonrelativistic epochs, offering a possible link between entropy running and well-known early-universe transitions.
  • Because the model remains disfavored by ΔBIC ≈ 7 relative to ΛCDM, the correct reading is that the data allow such an entropy correction but do not demand it.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A natural next test is to fit the same data with a more flexible Δ(z) (e.g., a two-parameter power law in log x) and see whether the preferred H0 and z_tr are stable; if they shift strongly, the specific ansatz is doing the work.
  • If the sign reversal is physical, one might expect related signatures in CMB spectral distortions or 21-cm global signal, since the effective horizon entropy affects the background expansion during decoupling.
  • The logarithmic-running ansatz is reminiscent of renormalization-group running of couplings in asymptotically safe gravity; a derivation from that framework would convert an ad hoc fit into a predictive statement about quantum gravity.
  • The apparent alignment of z_tr with the neutrino transition epoch is intriguing but could be a coincidence of the fit; comparing with a version where Δ1 is forced to zero would isolate whether the sign flip is really required by the data.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a Barrow-Tsallis entropy with a scale-dependent anomalous dimension, Δ(x) = Δ0 + Δ1/ln x with x = (H1/H)^2, motivated by logarithmic quantum-gravity corrections. Using the gravity-thermodynamics conjecture on the apparent horizon of a flat FRW universe, the authors derive a modified first Friedmann equation and an analytic expression for H(z). They then fit the four parameters {H0, Ωm, Δ0, Δ1} jointly to OHD, Pantheon+ SNe Ia, DESI DR2 BAO, and CMB shift parameters, obtaining H0 = 69.27 ± 0.65 km/s/Mpc, Δ(0) ≈ 1.09×10^-4, and a sign reversal of Δ(z) at z_tr ≈ 102. The BIC mildly favors ΛCDM. The paper interprets the best-fit H0 as a partial alleviation of the Hubble tension and derives local H0 constraints from low-redshift SNe Ia with two different calibrations.

Significance. If the numerical results were robust, the paper would provide a useful, testable example of an entropy-based extension of ΛCDM. The derivation from the first law is transparent, the model reduces to ΛCDM in the appropriate limit, and the MCMC/BIC comparison with ΛCDM is a standard and welcome feature. The inclusion of DESI DR2 BAO data is timely. However, the headline results are undermined by an internal inconsistency in the BAO likelihood: the sound horizon used in Eq. (42) is the standard ΛCDM-calibrated fit of Eq. (41), while Appendix B shows that the model itself predicts an r_d that is 2–3% larger. The reported H0, Δ0, Δ1, Δ(0), z_tr, and the claimed partial alleviation of the Hubble tension are therefore all based on a hybrid comparison, not on the model's actual predictions. Section VI's early-time 'constraint' also has a circular/mapping character that is presented as an independent result. These issues are fixable by a re-analysis, but the current manuscript does not establish its central quantitative claim.

major comments (3)
  1. [IV.A, Eq. (42) and Appendix B] The BAO likelihood in Eq. (42) uses r_d from Eq. (41), an empirical fit calibrated to the standard pre-recombination ΛCDM expansion history. Appendix B, however, computes the model's own sound horizon from Eq. (B1) and finds r_d ∈ [149.9, 151.7] Mpc for z_d ∈ [1050, 1070], i.e., 2–3% larger than the ~147 Mpc ΛCDM value. DESI DR2 BAO distance ratios are measured to roughly 0.3–1% precision, so this percent-level shift in every D_M/r_d, D_H/r_d, D_V/r_d model prediction is comparable to the claimed H0 increase (69.27 vs 67.36, about 2.8%). The quoted best-fit parameters therefore do not represent a direct comparison of the model to the BAO data. The authors acknowledge the deviation in Appendix B but do not propagate it through the likelihood. This is an internal inconsistency, not a minor calibration detail, and it directly affects the headline H0 and the derived quantities Δ(0), z_tr, an
  2. [VI.A, Eqs. (49)–(50)] The 'early-time constraint' is constructed by equating the Barrow model's CMB shift parameters to those of ΛCDM and solving for H_Λ^0. With Ωm fixed to the Barrow best fit, Eqs. (49a) and (49b) define a mapping from the model's parameters to the ΛCDM H0 that would mimic the same angular scales. The central value H_Λ^0 = 67.84 in Eq. (50b) is therefore a consequence of the matching procedure, not an independent measurement or prediction. The statement that 'Barrow's entropy model purports H0 = 69.27 whereas H_P^0 only results because of a low-redshift extrapolation' overinterprets this calculation. In addition, Eqs. (49a,b) are two equations for one unknown, and the origin of the lower bound in Eq. (50a) as a separate constraint is not explained. Please either remove this subsection or reframe it explicitly as a consistency check of angular scales, with a transparent account of how Eq. (5
  3. [IV.A, Eq. (42)] The BAO log-likelihood in Eq. (42) is a diagonal Gaussian: it sums independent contributions with variances σ_Yi^2. DESI DR2 BAO measurements are not independent; the published data are correlated across redshift bins and between tracers, with full covariance matrices provided by the DESI collaboration. Ignoring these correlations can bias both the best-fit values and the quoted uncertainties. If the full covariance matrix was used, the paper should state this explicitly and describe how it enters the likelihood. If only diagonal errors were used, the error bars on H0, Ωm, Δ0, Δ1 in Table I are likely underestimated, which again affects the significance of the claimed Hubble-tension alleviation.
minor comments (4)
  1. [III, Eq. (28)] The text repeatedly calls Eq. (28) a first-order Taylor expansion in inverse powers of the Hubble parameter, but the functional form is logarithmic in x (hence in H). The physical motivation via logarithmic corrections is clear, but the wording should be corrected to avoid implying a power-law expansion.
  2. [Table I and Sec. V] The BIC difference ΔBIC = 6.94 is usually interpreted as strong evidence against the higher-dimensional model on the Kass–Raftery scale, not merely 'mildly favored.' Please justify the wording or soften the conclusion.
  3. [V, Sec. VA and Fig. 2] The reported transition redshift z_tr = 102^{+1770}_{-90} has a very asymmetric and large upper error. As a result, the claimed compatibility with z* ≈ 1089.92 is very weak. Please quote the full posterior interval or a credible interval that makes the constraining power clear.
  4. [Appendix B] The title 'Impact on the scaler d' contains a typo ('scaler' should be 'sound horizon' or 'scale'). More substantively, Appendix B lists r_d ∈ [149.9, 151.7] Mpc but does not give the corresponding z_d posterior; please provide the z_d value(s) and the resulting r_d uncertainty for the best fit.

Circularity Check

3 steps flagged

Headline results (Δ(0), z_tr, H0, early-time H0^Λ) are direct functions of the MCMC-fitted parameters or of the same data already used in the likelihood, so several 'predictions' are restatements of the fit rather than independent outputs of the entropy model.

specific steps
  1. other [Sec. V and V.A; ansatz Eq. (28); text around Δ(0) and z_tr]
    "Δ(x) = ∆0 + ∆1/ln x, where ∆0 and ∆1 are suitable constants. ... Remarkably, Δ(z) exhibits a sign change from negative to positive values as z decreases, with the transition occurring at z_tr = 102+1770−90."

    The sign reversal and its location are algebraic consequences of the assumed two-parameter ansatz. With the fitted values Δ0>0 and Δ1<0, z_tr is simply the root of the fitted equation Δ0 + Δ1/ln x(z) = 0, and Δ(0)=1.09×10^-4 is just the same fitted function evaluated at z=0. These are restatements of the Table I parameters, not independent predictions of Barrow-Tsallis entropy.

  2. fitted input called prediction [Sec. I (intro), Sec. IV.A (likelihood), Sec. VI (Table I and late-time results)]
    "predicting a viable dark energy evolution and enlarging the expected Hubble constant at late-times. ... we found the value H0 = 69.27+0.62−0.65 km/s/Mpc, suggesting a partial mitigation of the discrepancy between the Riess and Planck determinations."

    H0 is one of the four parameters {H0, Ωm, Δ0, Δ1} that the paper explicitly says are 'determined by maximizing the total log-likelihood function.' It is not derived from the entropy framework; it is a fitted normalization. Reporting the fitted H0 as the model's predicted 'enlarged' Hubble constant and as a partial resolution of the Hubble tension is therefore a restatement of the likelihood input, not a model-based prediction.

  3. fitted input called prediction [Sec. VI.A, Eqs. (49)-(50), CMB shift-parameter consistency]
    "we equate the CMB shift parameters given in Eqs. (43), computed using the best fit values from Barrow's entropy model in Table I ... Solving these relations for HΛ0 yields ... HΛ0 = 67.84+0.66−0.66 km/s/Mpc. Remarkably, the constraint from Eq. (50b) is in perfect agreement with HP0."

    The CMB shift parameters R and lA are themselves part of the fitted likelihood in Eq. (45). Equating the best-fit Barrow model's shift parameters to the ΛCDM expressions and solving for H0^Λ merely converts the same already-fitted CMB numbers into a ΛCDM parameter. The 'perfect agreement' with the Planck value is a consistency check of the fit, not an independent early-time prediction of the Barrow model.

full rationale

The central mathematical derivation — applying the gravity-thermodynamics conjecture to a scale-dependent Barrow entropy and obtaining the modified H(z) in Eq. (29) — is a genuine derivation from the stated ansatz, and the MCMC comparison to OHD, Pantheon+, DESI BAO, and CMB shift data is an honest parameter-estimation exercise. However, several quantities presented as 'physical results' or as supporting the Hubble-tension claim are pure functions of the fitted parameters. Δ(0) is the fitted curve evaluated at z=0; z_tr is the root of the fitted two-parameter curve; H0 is itself a fitted free parameter; and the early-time H0^Λ=67.84 is obtained by re-expressing the same CMB shift parameters that were already included in the likelihood. These are therefore fitted outcomes relabeled as predictions, which is the core circularity pattern. Separately, the BAO likelihood uses the empirical r_d of Eq. (41), calibrated to the standard ΛCDM expansion history, while Appendix B finds the model's own r_d from Eq. (B1) to be ~2-3% larger; this is a serious internal-consistency issue that can bias the reported H0 at roughly the level of the claimed tension alleviation, though it is more a correctness flaw than a definitional circularity. Because the central H(z) derivation and the fit themselves remain independent content, the overall circularity is partial, not total.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 1 invented entities

The central claim rests on the standard gravity-thermodynamics framework plus a hand-chosen running parametrization for Δ. The model adds two free parameters (Δ0, Δ1) on top of the usual cosmological parameters, and its physical content is effectively an added dark-energy-like component encoded in the entropy exponent. The BAO sound-horizon calculation assumes standard early-universe physics, with a small self-assessed bias.

free parameters (4)
  • Δ0 = 0.00240 (+0.00130/-0.00130 at 1σ)
    Constant part of the running anomalous dimension, fitted to data (Table I). Drives the overall entropy modification.
  • Δ1 = -0.65 (+0.35/-0.38 at 1σ)
    Coefficient of the 1/ln x correction in Eq. (28), fitted to data (Table I). Controls the scale dependence of Δ.
  • H0 = 69.27 (+0.62/-0.65) km/s/Mpc
    Present-day Hubble constant, a standard cosmological free parameter, fit jointly with the model parameters.
  • Ωm = 0.310 (+0.009/-0.010)
    Matter density parameter, fitted as in standard cosmological analyses.
axioms (6)
  • domain assumption Gravity-thermodynamics conjecture: the Clausius-like relation dE = T dS + W dV holds on the apparent horizon.
    Adopted from Ref. [32] in Sec. II as the starting point for deriving modified Friedmann equations.
  • domain assumption Barrow-Tsallis entropy formula S = (A/A0)^(1+Δ/2) for the horizon entropy.
    Eq. (18) is taken as the entropy functional, without independent derivation.
  • ad hoc to paper The running form Δ(x) = Δ0 + Δ1/ln x with x = (H1/H)^2.
    Eq. (28) is a postulated ansatz, not derived from a specific quantum-gravity model.
  • domain assumption Spatially flat FRW background.
    Assumed in Eq. (4) and supported by CMB/BAO observations; used throughout the paper.
  • domain assumption Standard photon, baryon, and neutrino physics for computing the sound horizon r_d.
    Appendix B evaluates r_d using standard sound speed with the modified H(z); deviations up to ~3% are considered acceptable.
  • standard math The integration constant in the first Friedmann equation is interpreted as a cosmological constant Λ.
    Standard treatment following Eq. (17) and used in Eq. (31).
invented entities (1)
  • Scale-dependent anomalous dimension Δ(x) no independent evidence
    purpose: To modify the Barrow entropy exponent across cosmic epochs, producing a running entropy correction and an effective dark-energy component.
    Δ(x) is a phenomenological construct defined by Eq. (28) and fixed from fits to cosmological data. No independent, outside-the-paper prediction is provided to confirm the functional form.

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read the original abstract

We investigate an extended cosmological scenario based on the Barrow-Tsallis entropy, incorporating a varying (i.e., energy-scale-dependent) anomalous dimension. This behavior is reminiscent of quantum gravity and effective field theory settings, where the relevant couplings acquire a nontrivial scale dependence. By applying the gravity-thermodynamic conjecture on the apparent horizon of a flat Friedmann-Robertson-Walker Universe, we derive the corresponding modified cosmological equations. The standard dynamics is recovered as a limiting case when the Barrow-Tsallis entropy reduces to the conventional Bekenstein-Hawking form. The proposed model is tested against a combination of early- and late-time datasets, including observational Hubble parameter measurements, the Pantheon+ catalog of Type~Ia supernovae, the second data release of DESI baryon acoustic oscillations and cosmic microwave background constraints. Using the Bayesian Information Criterion, we finally compare the fitting performance of our framework with that of the $\Lambda$CDM paradigm. While the latter remains mildly favored, our model is shown to be fully compatible with current observations within suitable regions of the parameter space, unveiling a richer phenomenology that points towards a possible alleviation of the Hubble tension.

Figures

Figures reproduced from arXiv: 2607.26105 by Giuseppe Gaetano Luciano, Marco Muccino, Orlando Luongo.

Figure 1
Figure 1. Figure 1: FIG. 1. The ∆( [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Behavior of ∆( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. MCMC posteriors on the entropy model (left contours) and the ΛCDM paradigm (right contours). The dark (light) [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

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Reference graph

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