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REVIEW 4 major objections 6 minor 62 references

Growth of Cosmic Structures in generalized mass-to-horizon relation Entropic Cosmology

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper argues that a thermodynamically consistent entropic dark energy, built on a generalized mass-to-horizon relation with Bekenstein entropy, reproduces observed cosmic growth as well as ΛCDM does without extra free parameters.

desk verdict The paper's growth analysis is new and clearly framed, but Eq. (1.19) contradicts the model's own Friedmann equation, so the ΛCDM-like match is an artifact rather than a prediction. read the letter →

arxiv 2507.08647 v1 pith:ZA343RXK submitted 2025-07-11 astro-ph.CO gr-qchep-ph

Shafqat Ali , Tomasz Denkiewicz This is my paper
classification astro-ph.COgr-qchep-ph
keywords entropiccosmologymass-to-horizonrelationBekensteinentropycosmicstructuregrowthfσ8linearperturbationsdarkenergyClausius
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

Earlier entropic models of cosmic acceleration routinely failed a critical test: they under-predict how fast cosmic structures grow. This paper argues that a thermodynamically consistent version, the generalized mass-to-horizon entropic cosmology (MHEC) with Bekenstein entropy $(n=1)$, passes that test. In this model an entropic energy density $\rho_e = \gamma H^2/(4\pi G)$ enters the Friedmann equations while the Hawking temperature is left intact and the Clausius relation is satisfied. Solving the linear perturbation equations with the interaction consistently perturbed ("Case A"), the authors find that the matter density contrast and the growth observable $f\sigma_8(z)$ track the ΛCDM histories within current uncertainties for $\gamma \lesssim 10^{-2}$, with the best fit at the ΛCDM limit $\gamma=0$. The paper concludes that entropic dark energy, with no extra free parameters, is observationally indistinguishable from ΛCDM in both background and growth probes.

What carries the argument

The load-bearing object is the generalized mass-to-horizon relation $M = \gamma c^2 L^n/G$, whose $n=1$ limit yields Bekenstein entropy $S \propto L^2$ and the entropic energy density $\rho_e = \gamma H^2/(4\pi G)$ with pressure $p_e = -c^2\rho_e$. Thermodynamic closure comes from the Clausius relation $dE = c^2 dM = T_H dS$ with the unmodified Hawking temperature $T_H = \hbar c/(2\pi k_B L)$, which the paper argues makes this the first thermodynamically consistent entropic framework. The mechanism that carries the growth argument is the cancellation in Case A: the first-order part of the interaction source, $bQ_i = 2\gamma H(1+w_i)\bar{\rho}_i\delta_i$, cancels the explicit drag term in the linearized continuity equation, leaving $\gamma$ present only through $H(a)$ and $\bar{\rho}_i(a)$.

What would settle it

Re-run the Case-A growth calculation with the background constructed from $\rho_e = \gamma H^2/(4\pi G)$ alone, so that $\Omega_{e,0} = 2\gamma/3$, and compare $f\sigma_8(z)$ to the same data; if the ΛCDM-like match disappears for $\gamma = 10^{-5}$, the paper's central claim is wrong.

Watch

Extended reading notes

Core claim

The central discovery is that in the $n=1$ MHEC model the entropic interaction term, when treated consistently at first order, cancels out of the perturbation equations: in Case A the first-order part of the interaction four-vector $Q^\mu$ exactly removes the explicit drag terms, so $\gamma$ affects growth only through the background expansion rate $H(a)$ and background densities. Consequently the coupled matter-radiation contrast system takes the standard ΛCDM form, and the numerical $f\sigma_8$ evolution follows ΛCDM within the current redshift-space-distortion uncertainties even for $\gamma$ as large as $10^{-2}$. By contrast, if the perturbation is neglected (Case B), extra friction and mass terms suppress $\delta_m$ by about a factor of four at $\gamma=10^{-1}$, which would clash with observations. The paper therefore asserts that the thermodynamically consistent, fully perturbed MHEC model reproduces both background and growth probes as well as ΛCDM does, with no extra free parameters, and refutes earlier claims that entropic-force cosmologies cannot match structure formation.

Load-bearing premise

The paper's growth match assumes that the entropic density parameter can be set to the large closure value $1 - \Omega_{m,0} - \Omega_{r,0}$, although the model's defining relation $\rho_e = \gamma H^2/(4\pi G)$ would force it to about $2\gamma/3$, and no derivation reconciles the two.

Editorial extensions

If this is right

  • If the claim is right, entropic dark energy survives the structure-growth test that disfavoured earlier entropic-force models, so it remains a live alternative to a cosmological constant.
  • A full MCMC fit to growth data would tighten the upper bound on $\gamma$ beyond the current cosmic-chronometer limit $\gamma < 0.02$, probing whether any departure from ΛCDM is present.
  • The Case-A/Case-B split establishes a consistency requirement: any interacting or entropic model must perturb the interaction four-vector; neglecting it artificially lowers growth and can produce false exclusions.
  • The same framework covers Tsallis-Cirto and Barrow entropies (via $n \neq 1$) and reduces to exact ΛCDM for $n = 3$, so the growth machinery developed here can be applied to discriminate among these entropy prescriptions.
  • A thermodynamically consistent entropic origin for the accelerating expansion could address fine-tuning and coincidence problems without modifying general relativity itself.

Reading between the lines

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

  • Because $\gamma$ drops out of the growth equations in Case A, the model is observationally a one-parameter family of ΛCDM-like histories; $f\sigma_8$ data alone cannot identify the entropic origin, so the viability claim depends entirely on the background solution being the true MHEC prediction.
  • Equation (1.19) fixes $\Omega_{e,0} = 1 - \Omega_{m,0} - \Omega_{r,0}$, but the model definition $\rho_e = \gamma H^2/(4\pi G)$ implies $\Omega_{e,0} = 2\gamma/3$; for the quoted $\gamma = 10^{-5}$ these differ by five orders of magnitude, so a self-consistent background may not reproduce the claimed growth match.
  • The same Case-A/Case-B decomposition could be applied to the $n \neq 1$ entropy branches, where the cancellation is not exact; those cases should produce distinctive growth signatures that would separate entropic models from ΛCDM.
  • A straightforward cross-check would be to fit the Case-B equations to the same $f\sigma_8$ data; the model's own prediction of a factor-4 suppression at $\gamma = 10^{-1}$ should be strongly excluded, providing a clean consistency test.
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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 / 6 minor

Summary. The manuscript extends the generalized mass-to-horizon entropic cosmology (MHEC) to linear structure growth. For the Bekenstein limit n=1, the entropic energy density is taken as ρe = γH^2/(4πG). The authors derive linear perturbation equations for matter and radiation, distinguishing a fully perturbed interaction (Case A) from a neglected one (Case B), and numerically integrate them. They claim that Case A reproduces the ΛCDM growth history and background expansion without extra free parameters, thereby providing a viable entropic explanation of cosmic acceleration. The paper includes fits to cosmic-chronometer H(z) data and to fσ8 measurements.

Significance. If the central derivation were sound, the paper would be a useful contribution: it lays out the perturbation equations for a thermodynamically consistent entropic energy density and explicitly contrasts the physically motivated Case A with the commonly used Case B. However, the manuscript's central claim is undermined by an internal inconsistency in the background solution. The model's own Friedmann equation forces Ωe,0 = 2γ/3, whereas Eq. (1.19) uses Ωe,0 ≈ 0.685. As a result, all numerical results are computed with a background that is not the MHEC model but effectively ΛCDM with a hand-inserted cosmological constant. The claimed agreement with growth data is therefore not a test of the model, and the conclusion that MHEC explains cosmic acceleration is not supported. The distinction between Case A and Case B may be useful for other entropic or interacting dark-energy models, but the present paper's quantitative results do not represent the model it proposes.

major comments (4)
  1. [Section I, Eq. (1.19)] Eq. (1.19) is not a solution of the model's own Friedmann equation. Substituting ρe = γH^2/(4πG) from Eq. (1.8) into f(t) = 8πGρe/3 from Eq. (1.12) and then into Eq. (1.9) gives (1 − 2γ/3)H^2 = (8πG/3)(ρm + ρr). Evaluating at a = 1 yields Ωe,0 = 2γ/3 ≈ 6.7 × 10^−6 for γ = 10^−5. Eq. (1.19) instead sets Ωe,0 = 1 − Ωm,0 − Ωr,0 ≈ 0.685, independent of γ. These two expressions are incompatible; the latter corresponds to an additive constant energy density, whereas the model's ρe tracks H^2 and cannot provide the constant term driving late-time acceleration. All numerical integrations (Figs. 1–8) therefore use a background that is not the MHEC model, and the claimed match to ΛCDM is an artifact.
  2. [Section I, derivation of Eq. (1.19)] The paper states that substituting Eq. (1.12) into Eq. (1.9) and using Eq. (1.16) 'we obtain' Eq. (1.19), but no derivation is shown. A consistent substitution instead yields ρe = [2γ/(3 − 2γ)](ρm + ρr), so the entropic energy density must scale with matter and radiation and cannot act as a cosmological constant. To produce late-time acceleration, ρe would need an additive constant component, which is absent from the definition (1.8). The background solution used in the paper is therefore an extraneous ansatz, not a consequence of the model. This is a load-bearing error that invalidates the quantitative claims in Sections III and IV.
  3. [Section III, Case A growth equations] The growth equations (2.13)–(2.14) for Case A contain no trace of γ once the background is specified, as the paper itself notes in Section II.D. Because the background used in the numerical solution is the inconsistent Eq. (1.19), which is ΛCDM-like with Ωe,0 ≈ 0.685, the fσ8 curves in Figs. 5–8 are effectively ΛCDM growth curves. The agreement with growth data therefore provides no independent support for MHEC; it is built into the chosen background. A correct background would give a different expansion history and, in particular, no late-time acceleration, so the central claim of the paper would fail.
  4. [Abstract and Section III] The abstract's claim that the model matches ΛCDM 'without extra free parameters' is not supported by the fitting procedure. In Section III, the parameters γ and σ8,0 are minimized over, so γ is a genuinely free parameter in addition to those of ΛCDM. Moreover, Ωe,0 in Eq. (1.19) is fixed by the closure relation independently of γ, meaning the background contains a separate input not derived from the model's entropic energy density. The parameter count should be stated accurately.
minor comments (6)
  1. [Section I] The citation in the thermodynamic-consistency paragraph appears as '[23, 28? ]', which contains an unresolved placeholder for a reference.
  2. [Section II.C] The word 'arive' should be 'arrive'.
  3. [Eq. (1.19)] The typesetting of Eq. (1.19) is garbled: the terms involving Ωm,0 and Ωr,0 are printed as negative fractions with unclear denominators, making the formula difficult to parse.
  4. [Section III] The discussion references Fig. 8 before Fig. 7, and the order of the figures could be adjusted to match the order of appearance in the text.
  5. [Section III, chi-square fit] The chi-square analysis for the cosmic-chronometer data does not specify the number of data points or the optimization algorithm, which hinders reproducibility.
  6. [Section IV] The quoted Ωe = 0.684 ± 0.012 from Ref. [28] appears to refer to the n = 3 case, while the paper focuses on n = 1; the distinction should be clarified to avoid confusion.

Circularity Check

3 steps flagged · score 8.0 of 10

Eq. (1.19) sets Omega_e,0 to the Lambda-CDM closure constant instead of the model value 2 gamma/3, so the reported growth match is computed with a Lambda-CDM background inserted by hand, not with the MHEC model as defined.

  1. self definitional [Section I, Eqs. (1.8), (1.12) and (1.19)]
    "rho_e = gamma H^2/(4 pi G) ... f(t) = 8 pi G/3 rho_e ... By substituting equation (1.12) into equation (1.9), and then using equation (1.16), we obtain H(a, gamma) = H0 [ Omega_m,0 a^{-3 alpha} + Omega_r,0 a^{-4 alpha} + Omega_e,0 - (2 gamma Omega_m,0/(-3 alpha))(a^{-3 alpha} - 1) - (8 gamma Omega_r,0/(-12 alpha))(a^{-4 alpha} - 1) ]^{1/2}, where alpha = (1 - 2 gamma/3), ..., and Omega_e,0 = 1 - Omega_m,0 - Omega_r,0."

    Substituting rho_e = gamma H^2/(4 pi G) and f = 8 pi G rho_e/3 into Friedmann gives (1 - 2 gamma/3) H^2 = (8 pi G/3)(rho_m + rho_r), so the model fixes Omega_e,0 = 2 gamma/3 ~ 6.7e-6 for gamma = 1e-5. Equation (1.19) instead defines Omega_e,0 = 1 - Omega_m,0 - Omega_r,0 ~ 0.685, which is the Lambda-CDM vacuum closure value. The constant entropic term used in all numerical work is therefore not the solution of the model as defined; it is the cosmological constant re-labelled. Any growth calculation using this H(a) is a Lambda-CDM background calculation by construction.

  2. fitted input called prediction [Section II.D and Section III growth analysis]
    "As a result, the coupled matter-radiation system written immediately above for Case A contains no trace of gamma beyond its effect on the background functions H(a) and rho-bar_i(a). Linear growth therefore follows the standard Lambda-CDM form, modified only indirectly through the altered expansion history."

    This statement shows that in Case A the entropic coupling enters the growth equations only through H(a) and rho-bar_i(a). Since Eq. (1.19) was forced to have Omega_e,0 = 1 - Omega_m,0 - Omega_r,0, the background is numerically indistinguishable from Lambda-CDM for gamma <= 1e-2. The reported f sigma_8 agreement is therefore inherited from the inserted Lambda-CDM background, not a prediction of the entropic energy density rho_e = gamma H^2/(4 pi G).

1 more flagged steps
  1. fitted input called prediction [Section III, chi^2 fit after Eq. (3.6)]
    "The global minimum of the chi^2 surface actually sits at gamma ~ 0, sigma_8,0 = 0.76, i.e. exactly the Lambda-CDM limit; however, a value as small as gamma = 10^-5 is already indistinguishable from that optimum within the current error bars, so we plot it as a representative entropic curve."

    The paper fits gamma and sigma_8,0 to the same f sigma_8 data and finds the best fit at gamma = 0, i.e. Lambda-CDM itself. The plotted entropic curve is an equally good fit only because gamma is so small that the model reduces to Lambda-CDM. Presenting this as 'MHEC matches growth without extra free parameters' converts a fitted parameter with best-fit value zero into a claimed prediction.

full rationale

The central circularity is algebraic rather than citational. The model defines rho_e = gamma H^2/(4 pi G) in Eq. (1.8) and inserts it into the Friedmann equation via f = 8 pi G rho_e/3 in Eq. (1.12). That combination forces Omega_e,0 = 2 gamma/3, about 10^-5 for the adopted gamma, not the 0.685 used in Eq. (1.19). By setting Omega_e,0 = 1 - Omega_m,0 - Omega_r,0, the paper reintroduces the cosmological constant as an independent closure density and then computes growth with that Lambda-CDM-like H(a). Since Case A growth equations contain gamma only through the background, the resulting f sigma_8 curves are Lambda-CDM curves by construction. Separately, the 'no extra free parameters' claim is contradicted by the paper's own fit of gamma and sigma_8,0, whose best fit is gamma = 0. No load-bearing self-citation is involved; the problem is that the model's defining relation is abandoned in the background solution used for the predictions.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central claim rests on the free parameters gamma, Omega_e,0, h, and sigma_8,0. The most consequential free parameter is Omega_e,0, because it is set to the Lambda-CDM closure value and decoupled from gamma, which is inconsistent with the model definition of rho_e. The thermodynamic axioms follow prior horizon-entropy literature.

free parameters (4)
  • gamma (entropic coupling) = gamma < 0.02 (95% CL) from CC data; growth best fit gamma tending to 0
    Fitted to cosmic-chronometer H(z) data and to growth data. The paper's own equations require gamma = 1.5(1 - Omega_m,0 - Omega_r,0) for consistency, but this is not enforced.
  • Omega_e,0 (effective entropic/closure density) = 1 - Omega_m,0 - Omega_r,0, approximately 0.685
    Set independently of gamma in Eq. (1.19), effectively a cosmological constant. This is the key free parameter that makes the background Lambda-CDM-like.
  • h (dimensionless Hubble constant) = 0.68 +/- 0.02
    Fitted to cosmic-chronometer background data in Section I.A.
  • sigma_8,0 (present-day fluctuation normalization) = 0.76
    Fitted together with gamma in the f-sigma-8 analysis of Section III.
assumptions (5)
  • domain assumption Hawking temperature TH = hbar c / (2 pi k_B L) applies to the Hubble horizon.
    Used to integrate the Clausius relation and derive the generalized entropy (1.2) in Section I.
  • domain assumption Newtonian sub-Hubble perturbation theory with homogeneous dark energy (delta rho_e = 0).
    The perturbation equations (2.2)-(2.11) assume this framework, following Refs. [25, 29, 39-42].
  • domain assumption The entropic source functions satisfy f(t) = g(t) (the Lambda(t) ansatz).
    Assumed in Section I.A to write the modified continuity equation (1.13) and to derive the background H(a).
  • ad hoc to paper Closure relation Omega_m,0 + Omega_r,0 + Omega_e,0 = 1 with Omega_e,0 independent of gamma.
    Eq. (1.19) requires this, but it is not implied by rho_e = gamma H^2/(4 pi G), which gives Omega_e,0 = 2 gamma / 3. This is the load-bearing inconsistency.
  • domain assumption Spatially flat FLRW background.
    Used throughout the perturbation derivation, Eq. (2.1).
invented entities (1)
  • Entropic energy density rho_e = gamma H^2/(4 pi G)
    purpose: Postulated horizon-sourced component intended to drive late-time acceleration without a cosmological constant.
    No falsifiable handle outside the model; in the fits it is degenerate with a cosmological constant and gamma is not independently constrained.

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Cite this review

Pith. "Pith review of Growth of Cosmic Structures in generalized mass-to-horizon relation Entropic Cosmology." pith.science (2026). https://pith.science/paper/ZA343RXK

@misc{pith2026250708647,
  author       = {Pith},
  title        = {Pith review of: Growth of Cosmic Structures in generalized mass-to-horizon relation Entropic Cosmology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZA343RXK}},
  note         = {Machine review of arXiv:2507.08647}
}
read the original abstract

We investigate the growth of cosmic structures in the thermodynamically consistent generalised mass-to-horizon entropic cosmology (MHEC). For the Bekenstein case the entropic energy density augments the Friedmann equations without modifying the Hawking temperature and automatically satisfies the Clausius relation, thereby avoiding the inconsistencies that afflicted earlier entropy-force models. We then derive the linear perturbation equations, emphasising the distinction between a fully perturbed interaction term and the common approximation in which the perturbation is neglected. Numerical solutions show that fully perturbed follows the LCDM matter-growth history within the current growth uncertainties. Our results demonstrate that MHEC matches both background and growth probes as well as LCDM without extra free parameters, providing a viable entropic explanation for recent accelerated expansion of the Universe.

Figures

Figures reproduced from arXiv: 2507.08647 by the authors.

Figure 1
Figure 1. Evolution of the normalized density parameters [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Comparison of the deceleration parameter [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Evolution of the equation of state we(a) as a func￾tion of the scale factor a. diation component in the early Universe while fuelling late-time cosmic acceleration. Having thus established that the entropic compo￾nent evolves smoothly from radiation-like to acceleration￾driving behaviour, we now confront the same background solution with direct measurements of the expansion rate [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Hubble parameter data from cosmic chronome [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: displays the growth measurements alongside two theoretical curves: the reference ΛCDM prediction (γ = 0) and an entropic model with γ = 1 × 10−5 . The global minimum of the χ 2 surface actually sits at γ ≃ 0, σ8,0 = 0.76, i.e. exactly the ΛCDM limit; however, a value a…
Figure 6
Figure 6. Figure 6: The evolution of δm and δr density contrasts as functions of the scale factor a, for ΛCDM and MHEC. A: δm, γ = 10-5 A: δm, γ = 10-1 A: δr, γ = 10-5 A: δr, γ = 10-1 B: δm, γ = 10-5 B: δm, γ = 10-1 B: δr, γ = 10-5 B: δr, γ = 10-1 10-5 10-4 0.001 0.010 0.100 1 10-5 10-4 0…
Figure 7
Figure 7. Figure 7: The evolution δm and δr as functions of the scale factor a, for cases A and B. while the dashed curve repeats the calculation after set￾ting δr = 0. The feedback from δr boosts the growth of δm by over 100 times the value obtained by setting δr = 0. It demonstrates tha…

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