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Dark energy era with a resolution of Hubble tension in generalized entropic cosmology

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A four-parameter entropy of the apparent horizon is proposed as dark energy and, for a narrow parameter range, yields H0≈73 km/s/Mpc, easing the Hubble tension.

desk verdict A clean entropic-cosmology derivation that overreaches: the claimed Hubble-tension resolution is a fitted parameter, and the SH0ES-independent case rests on an unvalidated Planck likelihood. read the letter →

arxiv 2507.15273 v1 pith:IJAJUIXL submitted 2025-07-21 gr-qc astro-ph.COhep-th

Priyanka Adhikary , Sudipta Das , Sergei D. Odintsov , Tanmoy Paul This is my paper
classification gr-qcastro-ph.COhep-th MSC 83F05 PACS 98.80.-k95.36.+x
keywords generalizedentropyentropiccosmologydarkenergyHubbletensionapparenthorizonthermodynamicscosmologicalconstantFriedmannequationslate-time
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 proposes that dark energy is a consequence of the four-parameter generalized entropy of the apparent horizon in a spatially flat universe. This entropy function reduces to all previously known horizon entropies at suitable parameter values, and its modified Friedmann equations contain an entropic energy density and pressure that, together with a cosmological constant, drive the late-time matter-to-dark-energy transition. The paper's key quantitative claim is that with β=1 and σ0 near 0.83–0.93 the present Hubble parameter comes out around 72.6–73.3 km/s/Mpc, higher than the ΛCDM value, while fits to Cosmic Chronometer, supernova, DESI DR1 BAO, and compressed Planck data remain viable. That higher H0 is presented as a possible resolution of the Hubble tension.

What carries the argument

The central object is the generalized entropy $S_g$, a four-parameter function of the Bekenstein-Hawking variable $S$ constructed so that Tsallis, Rényi, Barrow, Sharma-Mittal, Kaniadakis, and loop-quantum-gravity entropies are all recovered by special parameter choices. The paper feeds $S_g$ into the apparent-horizon first law $T_h\,dS_h = -dE + W\,dV$, which produces modified Friedmann equations; in the late-time limit the whole correction collapses to a single parameter $\sigma_0$ that controls how much $H_0$ differs from its $\Lambda$CDM value. The analytic form of $\Omega_D(z)$, and hence $H(z)$, $\omega_D(z)$, and $q(z)$, carries the comparison with data. The mechanism that raises $H_0$ is the redshift-dependent matching: requiring the model's $H(z)$ to equal $\Lambda$CDM's near recombination while allowing $\sigma_0<1$ at $z=0$ forces $H_0$ upward through the relation in Eq. (42).

What would settle it

Recompute the sound horizon at decoupling and at the drag epoch using the model's modified $H(z)$ with the best-fit parameters $\beta=1$ and $\sigma_0\approx 0.87$; if the resulting shift parameters $\theta_*$ and $R$ disagree with the Planck compressed values beyond their quoted errors, the DESI+P18 value $H_0=72.56$ is not supported.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the four-parameter generalized entropy $S_g = \frac{1}{\gamma}\left[\left(1+\frac{\alpha_+}{\beta}S\right)^\beta-\left(1+\frac{\alpha_-}{\beta}S\right)^{-\beta}\right]$ of the apparent horizon produces a late-universe dark-energy sector characterized by an entropic energy density $\rho_g$ and pressure $p_g$. In the late-time limit $GH^2\ll 1$ the deviation from $\Lambda$CDM is governed by a single parameter $\sigma_0$; for $\beta=1$ and $\sigma_0<1$ the entropic dark-energy density stays positive. Markov Chain Monte Carlo fits to CC, PantheonPlus+SH0ES, DESI DR1, and compressed Planck data return $H_0\approx 69.9$ (CC), $\approx 73.3$ (PantheonPlus+SH0ES), $\approx 73.2$ (CC+PantheonPlus+SH0ES), $\approx 72.6$ (DESI+P18), and $\approx 73.0$ (DESI+P18+PantheonPlus+SH0ES), with $\sigma_0$ between about 0.83 and 0.93. The $\Lambda$CDM fits to the same data give $H_0\approx 67.9$–$69.4$, so the model's higher $H_0$ is the advertised resolution of the tension. The scenario also reproduces the standard thermal history: a deceleration-to-acceleration transition near $z\approx 0.5$ and a future de Sitter phase.

Load-bearing premise

The load-bearing premise is that the compressed Planck shift parameters, calibrated for $\Lambda$CDM, remain valid constraints for a model whose expansion near recombination can differ from $\Lambda$CDM through the $\sigma_0$-dependent terms; the paper does not recompute the sound horizon or drag epoch in the modified model.

Editorial extensions

If this is right

  • If the central claim holds, the same CC, PantheonPlus, DESI, and Planck data that favor $\Lambda$CDM with $H_0\approx 68$ also accommodate the entropic model with $H_0\approx 73$, so the Hubble tension can be eased without changing early-universe physics.
  • The model predicts a smooth deceleration-to-acceleration transition near redshift $z\approx 0.5$ and a future de Sitter phase with $\omega_D\to -1$, matching the standard sequence of matter and dark-energy eras.
  • Both the cosmological constant and the entropic energy density are needed: without $\Lambda$ the deceleration parameter is constant and the model cannot produce the late-time accelerating transition.
  • With $\beta=1$ and $\sigma_0<1$ the entropic dark-energy density is positive throughout the evolution; the alternative branch $\sigma_0=1$ with $\beta\neq 1$ gives a negative entropic energy density beyond $z\approx 3$, which the authors set aside.

Reading between the lines

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

  • A decisive check the paper leaves implicit is to recompute the CMB sound horizon and drag epoch in the modified expansion; the compressed Planck shift parameters used in the DESI+P18 fit were calibrated for $\Lambda$CDM, so a shift in $r_s$ or $r_d$ would directly change the reported $H_0=72.56$.
  • Because the model's Hubble function matches $\Lambda$CDM near recombination and deviates only at low redshift, it acts as a late-time resolution; future BAO and supernova surveys at $z\approx 1$–$2$ could distinguish the predicted $\omega_D(z)$ from $\Lambda$CDM.
  • The same four-parameter entropy can be applied to other entropy-sensitive systems, such as black-hole thermodynamics or the early universe, where the allowed $(\beta,\sigma_0)$ region could be constrained independently of late-time cosmology.
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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

4 major / 5 minor

Summary. The paper derives modified Friedmann equations from a four-parameter generalized entropy of the apparent horizon, introduces an entropic dark-energy component, and specializes to beta=1 with sigma0<1. It then fits H0, Omega_m0, and sigma0 to Cosmic Chronometer, PantheonPlus+SH0ES, DESI DR1, and compressed Planck likelihood data, reporting H0 values around 72.6-73.3 km/s/Mpc in several dataset combinations and claiming a possible resolution of the Hubble tension. It also reconstructs the deceleration parameter and dark-energy equation of state and performs an AIC/BIC comparison with LambdaCDM.

Significance. The formal derivation in Sections II-III is coherent, and the paper correctly notes that without the cosmological constant the entropic term alone would produce a constant deceleration parameter (Eq. 38), so the dark-energy epoch requires both Lambda and rho_g. The paper also makes use of standard public datasets and emcee/GetDist, which is good practice. However, the central claim of a Hubble-tension resolution is not supported by the analysis as presented: H0 is a fitted free parameter in every MCMC run, the theoretical matching relation Eq. (42) is never imposed, the compressed Planck likelihood is applied with shift parameters that are not model-independent, and the AIC/BIC comparison actually favors LambdaCDM for every dataset. These are load-bearing defects, not presentation issues.

major comments (4)
  1. [Section IV, Eq. (42), and Section V] The abstract and conclusion claim that the model 'provides a higher value' of H0 for certain entropic parameters, but in the data analysis H0 is a free parameter with a flat prior [40,120] in every MCMC run. The only place where H0 is related to sigma0 through the recombination-scale matching is Eq. (42), and this relation is never imposed as a constraint or used as a prediction. Moreover, Eq. (42) as printed appears inconsistent with Eq. (41): for beta != 1 the redshift scalings of H(z -> z_rec) and the LambdaCDM expression do not match, and for beta = 1 the printed relation would make H0 proportional to sigma0^2 (or sigma0^{1/2} after a straightforward derivation), which would lower H0 for sigma0 ~ 0.87 rather than raise it. Therefore the high H0 values in Table II are best-fit values of a free parameter, not predictions of the entropic model. The authors should either impose a corrected Eq. (42) in the fitting or explicitly withdraw the predictive claim.
  2. [Section V, compressed Planck likelihood] The DESI+P18 entry in Table II, H0 = 72.56 km/s/Mpc, is the only SH0ES-independent result supporting a high H0. The analysis uses the fixed compressed Planck values 100*omega_b = 2.237, theta* = 1.0411, and R = 1.74998, without recomputing the sound horizon, the drag epoch, or the angular diameter distance to decoupling in the modified model. Because theta* = r_s(z_dec)/D_A(z_dec) and R = sqrt(Omega_m0) H0 D_A(z_dec) depend on the expansion history, and because Eq. (41) changes the high-redshift expansion rate by roughly sigma0^{-1/2} ~ 1.07 at fixed (H0, Omega_m0), these LambdaCDM-derived compressed values are not transferable to the entropic cosmology. The manuscript never evaluates whether the compressed Planck likelihood is valid for Eqs. (15)-(18), so the DESI+P18 high-H0 result is unestablished as presented; the analysis must either recompute the shift parameters in this model or remove this dataset from the claimed SH0ES-independent evidence.
  3. [Section V.A, Table III] The AIC/BIC comparison actually disfavors the proposed model. For all datasets DeltaAIC > 0 and DeltaBIC > 0 (CC: +2.0 and +1.5; PantheonPlus+SH0ES: +1.9 and +3.1; CC+PantheonPlus+SH0ES: +1.9 and +3.2), meaning LambdaCDM is preferred over the three-parameter entropic model in every comparison. The statement in Section V.A that for CC data the proposed model is 'strongly favored' is the opposite of what the numbers show; the positive DeltaBIC values indicate weak evidence in favor of LambdaCDM, not in favor of the model. This contradicts the abstract's claim of 'phenomenological viability' and should be corrected and discussed honestly.
  4. [Table II, PantheonPlus+SH0ES row] The LambdaCDM entry for the PantheonPlus+SH0ES dataset in Table II reports H0 = 69.06^{+0.477}_{-0.181} km/s/Mpc. This is inconsistent with standard analyses of that dataset, which include the Cepheid distance anchors and yield H0 near 73 km/s/Mpc. This suggests that the SH0ES likelihood (or the H0 anchor) may not have been implemented as described in Section V. If the pipeline does not actually anchor H0 through the SH0ES Cepheids, then the model's H0 = 73.34 in the same row is also an artifact of the analysis setup rather than a meaningful agreement with SH0ES. The authors should clarify the exact likelihood terms used for the PantheonPlus+SH0ES dataset.
minor comments (5)
  1. [Section V] The model is called a 'four-parameter generalized entropy' model, but in the data analysis beta is fixed to 1 and only the combination sigma0 is constrained; the parameters alpha+, alpha-, and gamma are never varied or reported. The text should state explicitly that the fits actually test a one-parameter extension of LambdaCDM.
  2. [Section V] The restriction sigma0 < 1 is imposed ad hoc to keep the entropic energy density positive, and no physical prior or independent motivation is given for this range. This limitation should be acknowledged when the paper claims that a 'certain range of entropic parameters' resolves the Hubble tension.
  3. [Section V, figure cross-references] The text refers to 'figure 3(a)' and 'figure 3(b)' when discussing what is labeled Figure 4 in the manuscript; the cross-references should be corrected.
  4. [Section III and IV] The notation changes from the entropic parameters (alpha+, alpha-, gamma, beta) to the single parameter sigma0 without a clear statement of the domain and mass dimension of sigma0 immediately after Eq. (21); please make this explicit and use the notation consistently.
  5. [Section V.A, Table III] Table III would be much easier to read if it indicated explicitly which model is preferred for each criterion, because with DeltaAIC and DeltaBIC defined as model minus LambdaCDM, positive values mean LambdaCDM is preferred, which is easily misread.

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed Hubble-tension resolution is a fitted result: H0 and sigma0 are free parameters, and the datasets used include SH0ES, so the high H0 is an output of the fit rather than a prediction from the entropic parameters.

  1. fitted input called prediction [Sec. V, Data analysis and results, text around Table II; cf. Sec. IV Eq. (42) which is not used in the MCMC.]
    "We perform the analysis considering H0, Ωm0 and σ0 as free parameters keeping β fixed at β = 1. ... As evident from table II, for the PantheonPlus + SH0ES dataset, the best-fit value of H0 comes out to be H0 = 73.34+0.119−0.119 km/s/Mpc which is compatible with the SH0ES result ... Thus the results obtained indicate that the proposed generalized entropic dark energy model is capable of providing a possible resolution to the Hubble tension problem."

    The paper presents the high H0 as a consequence of the generalized entropy, but H0 is itself a free parameter fitted by MCMC. The PantheonPlus+SH0ES likelihood includes the SH0ES Cepheid-calibrated distance-ladder data, so a fitted H0 near 73 km/s/Mpc is statistically forced by the input calibration rather than independently predicted by the entropic parameters. The relation in Eq. (42), which would predict H0 from sigma0 by matching the recombination-era expansion, is never imposed in the analysis. The 'resolution of Hubble tension' is therefore a restatement of the best-fit value of the very parameter that the model is claimed to explain.

full rationale

The formal derivation from the generalized entropy to the modified Friedmann equations is internally self-contained: Eq. (12) is an assumed entropy function and Eqs. (13)-(18) follow by substitution into the horizon thermodynamic law, so no circularity occurs at that level. The circularity enters at the phenomenological level. The paper fits H0, Omega_m0, and sigma0 jointly as free parameters. When the SH0ES-containing PantheonPlus likelihood is used, the fitted H0 about 73 km/s/Mpc is essentially the SH0ES input reflected back, so calling this a 'resolution' is a fitted-input-as-prediction. The DESI+P18 result, which is the only SH0ES-independent-looking support, has a separate, non-circular consistency problem: the compressed Planck shift parameters (omega_b, theta*, R) are LambdaCDM-derived and are not recomputed for the modified expansion history, even though Eq. (41) shows H(z_rec) differs from LambdaCDM for sigma0 not equal to 1. That issue is a correctness risk rather than a circularity, but it removes the independent confirmation. Refs. [23]-[25] are self-citations for the specific entropy ansatz, but they are used as an assumed form rather than as a uniqueness theorem forbidding alternatives, so they do not add circularity by themselves. Because the central H0 claim reduces to a fit (or to a LambdaCDM-derived likelihood), the overall circularity score is 6.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The central model depends on the entropic-cosmology framework, a late-time approximation, and a restriction to beta=1 and sigma0<1. The only genuinely new fitted parameter is sigma0; H0 and Omega_m0 are standard free parameters. The compressed Planck likelihood is used as an external anchor without recomputing early-universe quantities in the modified model.

free parameters (4)
  • H0 = 69.94 to 73.34 km/s/Mpc depending on dataset
    Fitted in every MCMC run; the claim of a high Hubble constant depends on H0 being a free parameter.
  • Omega_m0 = 0.272 to 0.36 depending on dataset
    Fitted in every MCMC run; enters the Hubble function and the Planck shift parameters.
  • sigma0 = 0.83 to 0.93 depending on dataset
    The only entropic parameter actually constrained; for beta=1 it absorbs alpha_plus/gamma, and alpha_minus drops out entirely.
  • beta = 1 (fixed by hand)
    Chosen to keep the entropic dark energy density positive; not varied, so the advertised 4-parameter entropy is not explored.
assumptions (6)
  • domain assumption The apparent horizon satisfies Th dSh = -dE + W dV with temperature Th = H/(2 pi) |1 + Hdot/(2H^2)|
    Invoked in Sec. II, Eqs. (4)-(5); this is the entropic cosmology framework, not derived in the paper.
  • domain assumption Matter fields conserve separately from the dark energy sector
    Eq. (24) assumes no interaction between dust and the entropic dark energy component.
  • domain assumption The late-time expansion keeps only the leading term in the hypergeometric series (GH^2 << 1)
    Used to obtain Eq. (18) from Eqs. (16)-(17); higher-order terms are dropped without quantified error.
  • ad hoc to paper beta is restricted to 1 and sigma0 < 1 so that the entropic energy density stays positive
    Sec. V states this choice; it excludes the full 4-parameter model and pre-emptively avoids negative rho_D.
  • ad hoc to paper Compressed Planck likelihood values (100 omega_b, theta*, R) remain valid for the modified expansion history
    Sec. V takes P18 means from [113] without recomputing the sound horizon or drag epoch in the modified model.
  • domain assumption Spatially flat FLRW metric and dust-only late-time matter content
    Metric Eq. (1) and rho approximately rho_m in Sec. IV; no curvature or radiation at late times.
invented entities (1)
  • Entropic dark energy component (rho_g, p_g) from generalized horizon entropy
    purpose: Effective fluid that modifies the Friedmann equations and raises H0 relative to LambdaCDM
    It is defined by the chosen entropy function and constrained in this paper's fits; it has no falsifiable prediction outside the model. It is not a new fundamental entity, but a postulated effective contribution.

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Cite this review

Pith. "Pith review of Dark energy era with a resolution of Hubble tension in generalized entropic cosmology." pith.science (2026). https://pith.science/paper/IJAJUIXL

@misc{pith2026250715273,
  author       = {Pith},
  title        = {Pith review of: Dark energy era with a resolution of Hubble tension in generalized entropic cosmology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IJAJUIXL}},
  note         = {Machine review of arXiv:2507.15273}
}
abstract

We propose a new dark energy (DE) model from four parameter generalized entropy function of apparent horizon in a spatially flat universe. Such kind of generalized entropy is able to generalize all the known entropies proposed so far, for suitable representations of the entropic parameters. It turns out that the scenario can describe the correct thermal history of the universe, with the sequence of matter and dark energy epochs. Comparing with the $\Lambda$CDM model, the proposed generalized entropic DE model provides a higher value of present Hubble parameter for certain range of entropic parameter(s) leading to a possible resolution of Hubble tension issue. We confront the scenario with CC, PantheonPlus+SH0ES, DESI DR1 and compressed Planck likelihood datasets, which clearly depicts the phenomenological viability of the present model for some best fitted values of entropic parameter(s) that are indeed consistent with the resolution of Hubble tension.

Figures

Figures reproduced from arXiv: 2507.15273 by the authors.

Figure 1
Figure 1. FIG. 1. The 1 [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The 1 [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The 1 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of the generalized entropic dark energy model with the ΛCDM model for CC and [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Left Plot: Deceleration parameter, [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Early- and late-time constraints on Wald-Gauss-Bonnet topological dark energy and implications for the $H_0$ and $S_8$ tensions

    gr-qc 2026-07 conditional novelty 6.0 of 10

    A joint CMB, BAO, and supernova fit mildly prefers a non-zero Wald-Gauss-Bonnet dark-energy term (~3σ with SH0ES included), raising H0 from 68.5 to 69.8 km/s/Mpc and easing the Hubble tension by ~0.9σ at the cost of a...

  2. BAO miscalibration cannot rescue late-time solutions to the Hubble tension

    astro-ph.CO 2025-10 accept novelty 6.0 of 10

    Even after rescaling BAO data to prefer H0≈73 km/s/Mpc, none of six tested late-time dark-energy models can resolve the Hubble tension once unanchored SNeIa and CMB geometry are included.

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