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Observational constraints on entropic cosmology

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A generalized entropy-area relation, fitted to cosmic chronometer, BAO, and strong-lensing data, yields H0 = 0.727 and reproduces cosmic acceleration without a cosmological constant.

desk verdict A clean entropic-gravity derivation undermined by a fragile fitting pipeline: the H0 result is a conditional hint, not a measurement. read the letter →

arxiv 2412.11260 v1 pith:SIHPB7PE submitted 2024-12-15 gr-qc astro-ph.CO

Javier Chagoya , I. Díaz-Saldaña , Mario H. Amante , J. C. López-Domínguez , M. Sabido This is my paper · ORCID
classification gr-qcastro-ph.CO
keywords DarkEnergyEntropicGravityCosmologymodifiedFriedmannequationHubbletensionentropy-arearelationcosmicchronometersstronglensing
topics Dark EnergyDark Matter
open problems Dark MatterDark EnergyQuantum Gravity
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

This paper sets out to show that the observed late-time acceleration of the universe can be derived from a single generalized entropy-area relation for the apparent horizon, without invoking a separate dark-energy component. Starting from $S = A/4G + \alpha \ln(A/4G) + \sum_j \sigma_j (A/4G)^{1+j/2}$, the authors apply the Clausius relation $\delta Q = T\,dS$ with the apparent-horizon temperature $T = 1/(2\pi r_A)$ to obtain a modified Friedmann equation with additional powers of the Hubble rate. Fitting the model to cosmic chronometer, baryon acoustic oscillation, and strong-lensing data, they report a best fit with $h_0 = 0.727$ that matches the local distance-ladder measurement and a reduced chi-square competitive with $\Lambda$CDM. A sympathetic reader would care because this is a concrete, data-tested route by which entropic gravity could replace both dark energy and part of the dark-matter budget, and it suggests a geometric resolution of the Hubble tension.

What carries the argument

The load-bearing object is the modified entropy-area relation $S = A/(4G) + \alpha \ln(A/(4G)) + \sum_{j=0}^{N} \sigma_j (A/(4G))^{1+j/2}$, where $A = 4\pi r_A^2$ is the apparent-horizon area. Plugging this entropy, together with the horizon temperature $T = 1/(2\pi r_A)$ and the energy flux $\delta Q = A(\rho + P)\,dt$, into the Clausius relation $\delta Q = T\,dS$, and integrating with the continuity equation, produces the modified Friedmann equation (4)\text{--}(5). In that equation the logarithmic term gives $H^4$, the linear term gives $H$, the volumetric term gives a DGP-like $H^2$ piece, and higher terms give $H^{3-j}$. The truncated quartic model MFE4 of Eq. (8) is the version compared with the data.

What would settle it

A direct test is to compute the apparent-horizon entropy from a candidate quantum-gravity theory and check whether the coefficients $\alpha$ and $\sigma_j$ take the nonzero values the fit requires ($\alpha$ such that $\Omega_{0\alpha} = 0.008$, $\beta$ such that $\Omega_{0\beta} = 0.143$, $\epsilon$ such that $\Omega_{0\epsilon} = 0.770$); a fundamental calculation giving zero for the linear term would remove the central mechanism. A purely observational falsifier is to extend the fit to high-redshift BAO and CMB data and test whether the model's $H(z)$ remains consistent; the paper itself flags that the high-redshift behavior is generally far from $\Lambda$CDM unless a stiff fluid is added.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the generalized entropy-area relation of Eq. (2), combined with the Clausius relation on the apparent horizon, yields the modified Friedmann equation (Eqs. (4)\text{--}(5)), and that this equation, truncated to the quartic model MFE4, fits the combined cosmic-chronometer, BAO, and strong-lensing data with best-fit parameters $h_0 = 0.727$, $\Omega_{0m} = 0.095$, $\Omega_{0\epsilon} = 0.770$, $\Omega_{0\beta} = 0.143$, $\Omega_{0\alpha} = 0.008$, and $\chi^2_{\rm red} = 1.508$, compared with $1.552$ for $\Lambda$CDM with the same data. The $\Omega_{0\epsilon}$ term, associated with the linear entropy contribution, plays the role of the agent of current acceleration, slightly exceeding the cosmological-constant contribution at present time. The model is compatible with the local value of $H_0$, while $\Lambda$CDM fitted to the same data differs by $1.8\sigma$ from that value. The paper also argues that the entropic modifications mimic dark components, effectively acting as a geometric dark energy and, in part, dark matter.

Load-bearing premise

The entire derivation rests on the assumed entropy-area relation $S = A/(4G) + \alpha \ln(A/4G) + \sum_j \sigma_j (A/4G)^{1+j/2}$, together with the Clausius relation and the apparent-horizon temperature $T = 1/(2\pi r_A)$; if the true horizon entropy differs, the modified Friedmann equation and its fitted parameters no longer represent entropic-gravity predictions.

Editorial extensions

If this is right

  • The best-fit quartic model reproduces the low-redshift Hubble evolution with $H_0 = 0.727$, removing the $1.8\sigma$ tension that $\Lambda$CDM fitted to the same OHD plus SLS data shows with the local distance-ladder value.
  • The entropic linear term with $\Omega_{0\epsilon} \approx 0.770$ acts as the effective dark-energy component driving the current acceleration, so no separate cosmological constant is needed at late times.
  • The entropic modifications also mimic part of the dark-matter budget: $\Omega_{0m} = 0.095$ is much smaller than the standard matter density, with the missing contribution effectively carried by the geometric terms.
  • Entropic terms with $N > 2$ (higher powers of the area) are not decisive for early-universe dynamics; the paper finds that adding a stiff fluid can make the high-redshift $H(z)$ resemble $\Lambda$CDM without spoiling the low-redshift fit.
  • The model offers a potential resolution of the Hubble tension, though the paper notes that early-universe observations are still needed to assess full viability.

Reading between the lines

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

  • If the entropy-area relation is later derived from a microphysical theory, the fitted parameters $\Omega_{0\alpha}$, $\Omega_{0\beta}$, and $\Omega_{0\epsilon}$ could become predictions of that theory rather than free parameters; the paper stops at the phenomenological level.
  • The apparent equivalence to a $\Lambda$CDM model with $\Omega_{0m} = 0.21$ and $\Omega_\Lambda = 0.79$ at $z < 1$ suggests a degeneracy that could be broken by combining low-redshift data with CMB-scale measurements, a combination the paper does not perform.
  • The cosmographic-contour method used to define the MFE4 parameter region is a practical device; if the true horizon entropy has a different functional form, the recovered parameters would shift, so the interpretation of $\Omega_{0\epsilon}$ as the dark-energy agent rests on the assumed entropy form.
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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

3 major / 4 minor

Summary. The paper derives a generalized modified Friedmann equation from an assumed entropy-area relation (Eq. 2) using the Clausius relation on the apparent horizon of an FRW universe, obtaining Eq. (4). It then specializes to two truncations, MFE3 (cubic in H) and MFE4 (quartic in H, including a logarithmic entropy term), and fits them to a joint dataset of Hubble parameter measurements (OHD) and strong lensing systems (SLS). The MFE4 fit is reported in Table 1 with h0=0.727, Omega0m=0.095, Omega0epsilon=0.770, Omega0beta=0.143, Omega0alpha=0.008 and chi2_red=1.508, compared to chi2_red=1.552 for LCDM. The paper claims that the entropic modifications can mimic dark energy and that the model is compatible with the SH0ES value of H0, offering a potential resolution to the Hubble tension.

Significance. The thermodynamic derivation in Sec. 2 is standard and clean: given the entropy-area ansatz, the modified Friedmann equation follows rigorously. The paper also usefully maps various known modified-gravity models (DGP, H^2+H^-2 dark energy) into particular cases of the generalized entropy-area relation. However, the main quantitative claim—that the MFE4 model is compatible with SH0ES—rests on a statistical analysis that is not valid as presented. The MCMC for MFE3 fails, the MFE4 search region is defined using the same data that are later used to evaluate the fit, Omega0alpha is fixed arbitrarily, and Table 1 reports no uncertainties. If the statistical analysis were redone properly with theory-motivated priors and a full posterior, the entropic cosmology framework could be a useful contribution to the literature, but the current manuscript does not support the headline claims.

major comments (3)
  1. [§4.3, Eqs. (18)–(19), Fig. 3] The MFE4 parameter search is circular. The same OHD+SLS data are first used to fit the y-redshift cosmographic expansions (Eqs. 18–19) and to produce the contours in Fig. 3; those contours then define the 3-sigma allowed region for Omega0m and Omega0epsilon; and the same data are used again to compute the chi2 values for the MFE4 curves. This triple use of the dataset means that the quoted chi2_min=286.455 and the best-fit h0=0.727 in Table 1 are not likelihood-based estimates with the usual statistical meaning. The paper even states that the MCMC for MFE3 fails unless priors are extremely narrow, and then replaces the MCMC with a hand-restricted grid search. Consequently, the 0.15-sigma agreement with SH0ES mentioned in Sec. 4.3 and the compatibility claimed in Sec. 5 are not supported by the analysis as it stands.
  2. [Table 1] The MFE4 best-fit parameters are reported without any uncertainties, and the text states that Omega0alpha=0.008 is arbitrarily fixed to a small value. A point estimate with one parameter fixed ad hoc cannot be used to claim compatibility with an external measurement such as SH0ES, because the sensitivity of h0 and the other parameters to the arbitrary choice is unknown. The paper should provide posterior distributions or, at minimum, profile-likelihood intervals over all five parameters, and it should explicitly discuss how the results change when Omega0alpha is varied within a physically motivated range.
  3. [§5, Table 1] The comparison of chi2_red=1.508 for MFE4 versus chi2_red=1.552 for LCDM is not a valid model comparison. chi2_red does not penalize the larger number of free parameters in MFE4, and the search procedure itself was tuned using the same data, so the chi2 values are not directly comparable. The paper should use a proper information criterion (e.g., AIC or BIC) or a Bayesian evidence calculation, and should report uncertainties for the LCDM fit in the same table if a comparison of central values is to be made. Without this, the statement that the entropic model provides 'competitive constraints' is not established.
minor comments (4)
  1. [§2] The text after Eq. (2) contains a typo: 'Bekeinstein-Hawking' should be 'Bekenstein-Hawking'.
  2. [Fig. 1] The caption states that the data points shown are representative of the compilation, but the figure would be more informative if error bars were included or if the full dataset were shown, since the visual comparison is used to motivate the N=2 truncation.
  3. [§4.3, Eq. (18)] The y-redshift expansion in Eq. (18) includes a term proportional to s0, but the text says only that q0 and j0 are the cosmographic parameters and Fig. 3 shows only h0, q0, and j0. The treatment of s0 (whether it is fixed, marginalized, or fitted) should be stated explicitly.
  4. [§4.3] The sentence 'By assumption, the best-fit parameters of the MFE3 are close to those of the MFE truncated at higher orders' is presented without justification. This assumption is load-bearing for the transfer of the cosmographic constraints to MFE4, and it should be either tested (e.g., by comparing best-fit parameters over a range of truncations) or softened to a clearly labeled prior assumption.

Circularity Check

1 steps flagged · score 6.0 of 10

The H0 compatibility claim is partly built in: the same OHD+SLS data are used first to set the MFE4 search region via cosmography and then to select the best fit, so Table 1's h0 = 0.727 is a refit inside a data-defined box rather than an independent prediction.

  1. fitted input called prediction [Sec. 4.3 (Methodology, Fig. 3, Table 1)]
    "Relating the parameters of the model with the cosmographic ones by Taylor expanding Eq.(14), the cosmographic parameters are constrained with the same data sets described above. ... we consider the MFE4 and refine the search for the best-fit parameters in a region that is within the 3σ cosmographic confidence levels. ... we reduce the interval of interest to Ω0m ∈ [0.085, 0.12], Ω0ϵ ∈ [0.76, 0.83], while Ω0α = 0.008 is arbitrarily fixed to a small value."

    The same OHD+SLS data are used twice in the same fitting pipeline. First, the y-redshift cosmographic expansions (Eqs. 18-19) are fitted to those data to build the contours of Fig. 3; then those contours define the 3σ box in which MFE4 parameters are searched; finally, the same datasets are used to evaluate χ2_joint and select the Table 1 minimum. Because the search region is itself determined by the data being fit, the resulting h0 = 0.727 and the paper's 0.15σ agreement with SH0ES are not an independent prediction: the model is refit inside a region already favored by that same data.

full rationale

The derivation of the modified Friedmann equation from the assumed entropy-area relation (Eq. 2) via the Clausius relation is not circular: Eqs. (3)-(5) follow algebraically from the stated entropy and temperature assumptions, and the entropy ansatz is openly assumed rather than disguised as a prediction. The genuine circularity is confined to the parameter-estimation stage in Sec. 4.3. The cosmographic parameters are constrained with the same OHD+SLS data used to define the MFE4 priors, and the final best fit is then selected by minimizing χ2 on those same data. Thus the central quantitative claim of compatibility with SH0ES is partly built into the search protocol, not derived from an independent first-principles prediction. I find no load-bearing self-citation chain or imported uniqueness theorem; the authors' prior work motivates the entropy form, but the cosmological consequences are computed from that explicit assumption. The score reflects the partial circularity in the statistical pipeline while acknowledging that the entropy-to-Friedmann derivation itself is self-contained.

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

The central claim rests on the assumed entropy-area relation and the thermodynamic derivation, plus several fitting parameters. No new physical entities are introduced; the stiff fluid mentioned in Sec. 5 is a known matter type used only in a discussion scenario.

free parameters (5)
  • h0 (dimensionless Hubble constant) = 0.727 (MFE4 best fit, Table 1); 0.727+0.020/-0.019 from cosmography
    H0 is a free parameter in the fit to OHD+SLS; the paper's compatibility claim with SH0ES rests on this fitted value.
  • Omega0m (matter density parameter) = 0.095 (MFE4 best fit)
    Fitted to the same low-redshift data; standard in cosmological fits.
  • Omega0epsilon (coefficient of the H term) = 0.770 (MFE4 best fit)
    Associated with the linear entropy term; the paper interprets it as mimicking dark energy and driving acceleration.
  • Omega0beta (coefficient of the H^3 term) = 0.143 (MFE4 best fit, derived from constraint Eq. (10))
    Associated with the volumetric entropy term; effectively the DGP-like contribution.
  • Omega0alpha (coefficient of the H^4 term, logarithmic entropy correction) = 0.008 (arbitrarily fixed, not fitted)
    Paper states this is arbitrarily fixed to a small value; it is a hand-chosen parameter affecting the quartic term.
assumptions (5)
  • domain assumption Clausius relation delta Q = T dS applies to the apparent horizon of a flat FRW universe.
    Used in Sec. 2 to convert the entropy-area relation into the modified Friedmann equation; this is standard in entropic gravity but not independently proven.
  • domain assumption Apparent horizon temperature is T = 1/(2 pi r_A) with r_A = H^{-1}.
    Assumed in Sec. 2 following [18]; if the temperature law differs, the derived MFE changes.
  • ad hoc to paper The generalized entropy-area relation Eq. (2) is the entropy of the apparent horizon.
    This is the central model assumption, motivated by black hole entropy corrections but not derived from a fundamental theory in this paper.
  • domain assumption The universe is spatially flat and contains pressureless dust (omega = 0).
    Stated in Sec. 2; the analysis restricts to dust, and extension to radiation or stiff fluid is only discussed qualitatively.
  • domain assumption The y-redshift cosmographic expansion, Eq. (18), converges for the redshift range used (z less than 2.5).
    Used to fit cosmographic parameters and to map them to MFE4 priors; convergence is assumed.

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Cite this review

Pith. "Pith review of Observational constraints on entropic cosmology." pith.science (2026). https://pith.science/paper/SIHPB7PE

@misc{pith2026241211260,
  author       = {Pith},
  title        = {Pith review of: Observational constraints on entropic cosmology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SIHPB7PE}},
  note         = {Machine review of arXiv:2412.11260}
}
abstract

In this work, we derive a generalized modified Friedmann equation based on an entropy-area relation that incorporates established modifications, such as volumetric, linear, and logarithmic terms, in addition to novel entropic modifications that might yield to relevant cosmological implications at different stages of the evolution of the Universe. Some of these modifications are capable of mimicking the effects of dark energy and describing the current state of accelerated expansion of the Universe. We study particular cases of the generalized Friedmann equation and constrain the free parameters using observational datasets, including Hubble parameter measurements, baryon acoustic oscillations, and strong lensing systems. Our findings indicate that the proposed models align well with current observational data, particularly in low-redshift regimes; furthermore, these models are compatible with the value of $H_0$ obtained by the SH0ES program.

Figures

Figures reproduced from arXiv: 2412.11260 by the authors.

Figure 1
Figure 1. Low-redshift behavior of H(z)/(1+z) for different MFEs arising from Eq.(5), compared to the ΛCDM model. Data points shown are representative of the compilation presented in [29]. reason, if we aim to focus on the redshift region, where the ma￾jority of available observational data are concentrated, we are motivated to truncate the series in Eq.(4) to N = 2. For this particular case, the resulting MFE is (1 + z) 3Ω0m… view at source ↗
Figure 2
Figure 2. Profiles of E(z), Eq.(14), for random values of Ω0m ∈ [0.08, 0.15] and Ω0ϵ ∈ [0.6, 0.8], and Ω0β = 1 − Ω0m − Ω0ϵ . 4.3. Methodology As already mentioned, if we consider the area term and the volumetric term in the entropy, the model corresponds to the DGP model. Adding the length term on the entropy gives a Friedmann equation that has a cubic term on the Hubble pa￾rameter. If we also include the logarithmic term, th… view at source ↗
Figure 4
Figure 4. Hubble parameter reconstruction for the MFE4 and [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (2 more)
Figure 3
Figure 3. Figure 3: Cosmographic contours from a joint analysis OHD [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]
Figure 5
Figure 5. Figure 5: Comparison of different cosmological models. A simplified ΛCDM with H0 = 67.34 Km s−1 Mpc−1 , Ω0m = 0.315 and ΩΛ = 1 − Ω0m is shown with dashed line. The other lines correspond to solutions of FE + FL + FGE with Ω0m = 0.120, Ω0β = 0.124, Ω0ϵ = 0.777, different amounts …

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