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A Bayesian PINN Framework for Barrow-Tsallis Holographic Dark Energy with Neutrinos: Toward a Resolution of the Hubble Tension

T0 review · 5 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A Barrow–Tsallis holographic dark energy model with massive neutrinos places $H_0$ between Planck and SH0ES, reducing the Hubble tension to $1.3$–$2.1\sigma$.

desk verdict The paper's central derivation has an exponent error, so the fitted model is not BTHDE; the data analysis is real but the tension 'alleviation' is a fitted result, not a prediction. read the letter →

arxiv 2506.02235 v1 pith:IIR4MQWG submitted 2025-06-02 astro-ph.CO gr-qc

Muhammad Yarahmadi , Amin Salehi This is my paper
classification astro-ph.COgr-qc
keywords Barrow–TsallisentropyholographicdarkenergyHubbletensionBayesianphysics-informedneuralnetworkneutrinomasssumGranda–OliveroscutoffcosmicchronometersPantheon+supernovae
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 seeks to establish that a holographic dark energy model built from a combined Barrow–Tsallis entropy, a Granda–Oliveros cutoff, and massive neutrinos can fit a broad set of cosmological observations while putting the Hubble constant at an intermediate value between early- and late-universe measurements. Both standard MCMC sampling and a Bayesian physics-informed neural network give $H_0$ near 70–70.6 km/s/Mpc, so the Planck–SH0ES tension falls to roughly $1.3\sigma$–$2.1\sigma$ depending on the dataset combination. The authors further claim that the Bayesian PINN constrains the model more tightly than MCMC, especially for the Granda–Oliveros parameter $\beta$ and the neutrino mass sum, with the tightest bound $\Sigma m_\nu < 0.114$ eV. If these constraints hold, the model offers a modified-entropy dark energy framework that partially reconciles the two sides of the Hubble tension while remaining compatible with neutrino-mass limits.

What carries the argument

The object carrying the argument is the generalized Barrow–Tsallis entropy, $S = \gamma (A/A_0)^{(1+\Delta/2)(3-q)/2}$, a single entropy formula that combines Barrow's fractal horizon deformation $\Delta$ with Tsallis' nonextensive statistical index $q$. From the first law $dE = T\,dS$ with Gibbons–Hawking temperature, the paper derives a holographic dark-energy density and inserts the Granda–Oliveros cutoff $L^{-2} = \alpha H^2 + \beta \dot H$ to obtain a first-order nonlinear differential equation for the normalized Hubble parameter $E(z) = H(z)/H_0$, Eq. (43). That equation is the machinery both pipelines solve or enforce: it determines every reported constraint on $H_0$, $q$, $\Delta$, $\alpha$, $\beta$, and $\Sigma m_\nu$, and it is the physical prior embedded in the neural-network loss.

What would settle it

Compute $\rho_D$ from $S = \gamma (A/A_0)^\xi$ and the first law: with $A \propto L^2$ one gets $dS/dL \propto L^{2\xi-1}$ and therefore $\rho_D \propto L^{2\xi-4}$; inserting the Granda–Oliveros cutoff $L^{-2} = \alpha H^2 + \beta \dot H$ gives $\rho_D \propto (\alpha H^2 + \beta \dot H)^{2-\xi}$, not the fitted $(\alpha H^2 + \beta \dot H)^\xi$. Checking Eq. (21) against Eq. (43) is a direct calculation that settles whether the reported constraints describe the Barrow–Tsallis model or a different power-law dark energy.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that the Barrow–Tsallis holographic dark energy density—taken with the Granda–Oliveros infrared cutoff $L^{-2} = \alpha H^2 + \beta \dot H$ and the entropy-index exponent $\xi = (1+\Delta/2)(3-q)/2$—yields expansion histories consistent with CMB, BAO, CMB lensing, cosmic chronometer, and Pantheon+ data. Every dataset combination places $H_0$ between the Planck 2018 value and the SH0ES R22 value, and the full CMB+All combination gives $H_0 = 70.6 \pm 1.35$ km/s/Mpc with $\Sigma m_\nu < 0.114$ eV at 95% confidence. The recovered model parameters favor a small positive Barrow deformation $\Delta$ and a Tsallis index $q$ slightly above 1, meaning only mild departures from standard holographic dark energy and from extensive thermodynamics. A secondary claim is methodological: because the Bayesian PINN embeds the Friedmann equation as a differential constraint in its loss function, it returns tighter posteriors for $\beta$ and stronger upper bounds on $\Sigma m_\nu$ than the MCMC pipeline, while staying consistent with it within uncertainties.

Load-bearing premise

Everything reported depends on the assumed Hubble-evolution equation, Eq. (43), which treats the dark-energy density as a power of a combination of the expansion rate and its time derivative; if that equation is not what the entropy model actually predicts, the parameter constraints do not test the model.

Editorial extensions

If this is right

  • The BTHDE model would reduce the Planck–SH0ES Hubble tension from its current $>4\sigma$ discrepancy to roughly $1.3\sigma$–$2.1\sigma$ depending on which datasets are combined.
  • The most complete dataset combination would give $H_0 = 70.6 \pm 1.35$ km/s/Mpc and $\Sigma m_\nu < 0.114$ eV, keeping the model within current neutrino-mass bounds.
  • The favored parameters ($q$ slightly above 1, $\Delta$ around 0.03–0.17, $\alpha$ near 1) imply only mild entropy corrections, so the model is a modest extension of standard holographic dark energy rather than a radically different cosmology.
  • Bayesian PINN and MCMC would agree on parameter means within uncertainties, with the PINN reporting tighter errors on $\beta$ and on $\Sigma m_\nu$ for the same low-redshift datasets.

Reading between the lines

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

  • An implication the paper leaves implicit is that its reported constraints are only as meaningful as the assumed Friedmann equation; if the exponent in Eq. (43) does not follow from the entropy derivation, the $H_0$ and neutrino numbers characterize an ad hoc power-law dark energy rather than Barrow–Tsallis entropy.
  • A natural extension would be to redo the analysis with the density exponent that actually follows from the paper's own first-law derivation, $\rho_D \propto (\alpha H^2 + \beta \dot H)^{2-\xi}$, and see whether the intermediate $H_0$ and the Bayesian PINN's precision persist.
  • The same entropy construction could be tested with other infrared cutoffs, such as the Hubble horizon or future event horizon, to check whether the tension alleviation is specific to the Granda–Oliveros choice.
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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

5 major / 4 minor

Summary. The paper proposes a Barrow–Tsallis holographic dark energy (BTHDE) model, derives an entropy-based dark energy density with a Granda–Oliveros infrared cutoff, and fits the model to cosmic chronometers, Pantheon+ supernovae, BAO, CMB lensing, and Planck CMB data using both MCMC and a Bayesian physics-informed neural network. The headline results are intermediate values of H0 (about 70 km/s/Mpc) that are claimed to alleviate the Hubble tension to 1.3–2.1 sigma, and an upper bound on the sum of neutrino masses of Sigma m_nu < 0.114 eV from the CMB+All combination. The manuscript also claims that the Bayesian PINN gives tighter constraints than MCMC, especially for the parameter beta and for Sigma m_nu.

Significance. If the derivation and numerical results were correct, the paper would be a useful contribution to the modified-entropy dark energy literature and would demonstrate a concrete methodological comparison between MCMC and Bayesian PINNs in cosmology. The use of several modern datasets and the explicit embedding of the Friedmann equation in the PINN loss are positive features. However, the central derivation contains exponent errors that change the model being fitted, and the printed equation of state is algebraically wrong. Because all of the reported cosmological constraints and tension numbers come from the incorrectly derived and unspecified model equations, the significance of the paper is not established in its current form. No code or reproducible implementation is provided, which further weakens the methodological contribution.

major comments (5)
  1. [§IV, Eqs. (15)–(21)] The power counting in Eq. (19) is wrong. With S = gamma (A/A0)^xi and A = 4 pi L^2, Eq. (15) gives rho_D proportional to L^(2 xi - 4), not L^(2 xi - 2) as printed in Eq. (20). Consequently, after substituting the Granda–Oliveros cutoff L^-2 = alpha H^2 + beta Hdot, the density is rho_D proportional to (alpha H^2 + beta Hdot)^(2 - xi), not (alpha H^2 + beta Hdot)^xi as used in Eq. (37). The standard limit also fails: for q = 1 and Delta = 0, Eq. (21) yields rho_D proportional to L^0, while the text claims rho_D proportional to L^-2. Since Eq. (43) and all subsequent constraints use the incorrect exponent xi, the model that is actually fitted is an ad hoc power-law dark energy whose connection to the Barrow–Tsallis entropy derivation is not established.
  2. [§VIII, Eq. (40)] Equation (40) does not follow from Eq. (39). For rho_D proportional to X^xi with X = alpha H^2 + beta Hdot, one obtains d ln rho_D/dt = xi (2 alpha H Hdot + beta Hddot)/X, so w_D = -1 - xi (2 alpha H Hdot + beta Hddot)/(3 H X). The printed expression has the opposite sign, a prefactor 2 xi/3 instead of xi/3, and no factor of 1/H, making it dimensionally and algebraically inconsistent. This matters because w_D enters the continuity equation (35) and the redshift integral in Eq. (31) that defines rho_D(z).
  3. [§VIII, Eq. (43)] Equation (43) is dimensionally inconsistent for xi different from unity. Starting from Eq. (37) and dividing by 3 M_p^2 H0^2, the bracket obtains an overall factor H0^(2 xi - 2), which is absent from Eq. (43). If this factor is absorbed into c^2, that redefinition is not stated. In addition, f_nu(z) is introduced in Eq. (31) but never specified anywhere in the paper, so the neutrino term Omega_nu f_nu(z) in Eq. (43) is undefined and the model is not fully specified.
  4. [§IX, Eqs. (45)–(60); Tables III and VI] The claim that the Bayesian PINN yields more precise constraints is not validated. The variational posterior is a mean-field Gaussian, but the paper provides no convergence diagnostics, no posterior predictive checks, and no direct comparison with a full MCMC posterior for the same model. The reported improvement in beta from +/- 0.18 (MCMC, Table VI) to +/- 0.01 (Bayesian PINN, Table III for CC) is a factor of about 18; such dramatic tightening can be an artifact of the mean-field approximation or of the physics-loss weight lambda, and it requires explicit justification. The absence of any code or implementation details also makes the numerical results unreproducible.
  5. [§XII, Tables IV–VIII] The claim of alleviating the Hubble tension is an interpretation of fitted values rather than a prediction. Since H0 is a free parameter fitted to the same combined likelihoods, obtaining an intermediate value is partly a restatement of the fit. A meaningful test would be to predict H0 from the CMB-only combination within the BTHDE model and compare directly with R22; the paper's CMB+Lensing result H0 = 69.52 +/- 2.1 km/s/Mpc (Table VII) is still about 1.4 sigma below R22. The reported 1.3–2.1 sigma tensions conflate fitting freedom with a resolution of the tension.
minor comments (4)
  1. [§IX, Eqs. (63) and (71)] The conversion from Omega_nu to the sum of neutrino masses uses 93.14 eV in Eq. (63) and 94 eV in Eq. (71); the factor should be consistent.
  2. [References] Several references are incomplete or duplicated: [35] is only an arXiv identifier, [88] appears twice, and citations numbered [84], [90], and [96]-[100] are missing from the list.
  3. [§XI, Tables III–VI] The text refers to the Bayesian PINN results as 'Table VI' in the discussion of alpha and beta, but Table VI is the MCMC table; the correct reference is Table III, which makes the comparison confusing.
  4. [Throughout] There are numerous typographical and formatting errors, including 'forβ', 'F eature', 'V ariational', ' T ension quantification', 'we constraints', and 'In Fig. 2, we constraints'; the figures appear to be low-resolution screenshots with unclear labels in the preprint version.

Circularity Check

1 steps flagged · score 6.0 of 10

The fitted Friedmann equation uses exponent ξ, while the paper's own entropy derivation yields exponent 2−ξ; the reported Hubble-tension and neutrino constraints are therefore fits of an ad hoc power-law model, not of the claimed Barrow–Tsallis entropy density.

  1. other [Section IV, Eqs. (15)-(21); Section VII, Eq. (37); Section VIII, Eq. (43)]
    "Combining the powers of L, we get: ρD ∝ L(1+ ∆ 2 )(3−q)−2. Thus, the generalized Barrow–Tsallis holographic dark energy density is: ρD = B L(1+ ∆ 2 )(3−q)−2. ... The generalized holographic dark energy density constructed from the G-O cutoff is then given by ρD = 3c2M 2 p (αH 2 + β ˙H) ξ. ... E2(z) = (Ωb + Ωc) (1 +z)3 + Ωνfν(z) + c2 (αE2(z) − β(1 + z)E(z) dE(z) dz )ξ."

    The paper's own first-law derivation gives S=γ(4πL²/A0)^ξ, dS/dL∝L^{2ξ−1}, and ρD=(1/(8π²L³))dS/dL∝L^{2ξ−4}. With the G-O cutoff L^{-2}=αH²+βḢ, this is ρD∝(αH²+βḢ)^{2−ξ}. Instead Eq. (37), and the integrated ODE Eq. (43), use exponent ξ. The advertised standard limit also fails: inserting q=1, Δ=0 into the printed Eq. (21) gives ρD∝L^0=const, not L^{-2}. Since Eqs. (37)/(43) are the equations actually integrated and fitted, every reported constraint (H0, q, Δ, α, β, Σmν) and every tension number is a fit of an ad hoc power-law dark energy whose exponent is set by ξ without the entropy derivation. The 'BTHDE' label is attached to a model defined by the fitting equation rather than by the claimed Barrow–Tsallis input; the first-principles derivation is bypassed by construction.

full rationale

The central claimed derivation chain is broken by an internal exponent error. Starting from the paper's own Eq. (15), Eq. (17), and ξ=(1+Δ/2)(3−q)/2, one obtains ρD∝L^{2ξ−4}; substituting the Granda–Oliveros cutoff gives (αH²+βḢ)^{2−ξ}. The paper instead prints L^{2ξ−2} in Eq. (21), advertises ρD∝L^{-2} for q=1, Δ=0 (which its own Eq. (21) does not give), and then adopts (αH²+βḢ)^ξ in Eq. (37) and the fitted ODE Eq. (43). All H0, q, Δ, α, β, and Σmν results, and all 1.3σ–2.1σ tension claims, are outputs of fitting Eq. (43); they therefore constrain an unmotivated power-law density, not the entropy-derived BTHDE model claimed in the abstract. This is a definitional/fitted-model circularity rather than a self-citation issue: the self-citations to the authors' earlier papers [55–60] are used only for comparison and are not load-bearing, and the 'alleviated tension' is a posterior fit of a free H0 rather than an independent prediction. The tighter PINN uncertainties are a calibration concern (mean-field variational posterior plus dropout without validation against MCMC on identical inputs) but not circularity. The score of 6 reflects that the headline results reduce, by the paper's own equations, to fitting a model that does not follow from its stated first principles.

Assumptions & free parameters 9 free parameters · 7 assumptions · 1 invented entities

The central claim depends on a large set of fitted parameters (q, Delta, alpha, beta, c, H0, Omega_nu, Omega_b, Omega_c) and on several unverified modeling choices: the ad hoc Barrow-Tsallis entropy, the Granda-Oliveros cutoff, an unspecified neutrino redshift function, and a mean-field variational posterior. The number of free parameters is large relative to the constraining power of the data, and the most important 'prediction', H0, is itself a fitted quantity.

free parameters (9)
  • q = 1.02 to 1.10 depending on dataset
    Tsallis nonextensive index; free parameter in the entropy and in Eq (43); marginalized posteriors in Tables III, VI, and VII.
  • Delta = 0.027 to 0.17
    Barrow deformation exponent; free parameter in the entropy and Eq (43); constrained in Tables III, VI, and VII.
  • alpha = 0.973 to 1.099
    Granda-Oliveros cutoff coefficient; free parameter in Eqs (36)-(43); reported in Tables III, VI, and VII.
  • beta = 0.45 to 0.59
    Granda-Oliveros cutoff coefficient; free parameter; the PINN is claimed to constrain it much more tightly than MCMC.
  • c = not reported
    Dimensionless holographic parameter multiplying the dark energy density in Eq (43); included in the parameter list but never reported in the results tables.
  • H0 = 69.5 to 70.7 km/s/Mpc
    Hubble constant; free parameter fitted to the same early and late datasets that define the tension.
  • Sigma_m_nu or Omega_nu = upper bounds 0.114 to 0.32 eV
    Total neutrino mass derived from fitted Omega_nu via Eq (63) or (71); the headline bound is a fitted posterior bound.
  • Omega_b, Omega_c = not reported
    Present-day baryon and CDM density parameters listed in the parameter vector but not reported in tables.
  • PINN loss weight lambda and prior widths = not reported
    Hand-chosen hyperparameters in loss functions Eq (60) and Eq (69) that balance data against physics; no values are given, and they directly affect the claimed precision.
assumptions (7)
  • domain assumption Flat FLRW metric and standard Friedmann equations with components rho_b, rho_c, rho_nu, rho_D
    Section V, Eqs (22)-(24); adopted as the cosmological framework without justification beyond the standard model.
  • domain assumption No interaction between dark energy and matter; separate continuity equations
    Section VI, Eqs (33)-(35); used to evolve matter and dark energy independently.
  • ad hoc to paper Generalized Barrow-Tsallis entropy S = gamma (A/A0)^[(1 + Delta/2)(3 - q)/2]
    Introduced in Section III, Eq (5); only consistency limits are checked, and the later density derivation does not reproduce the exponent used in the fitted Friedmann equation.
  • domain assumption Granda-Oliveros IR cutoff L^-2 = alpha H^2 + beta Hdot
    Section VII, Eq (36); taken from prior holographic dark energy literature without derivation.
  • domain assumption Massive neutrino density factor f_nu(z) exists with an unspecified functional form
    Eqs (31) and (43) require f_nu(z), but no explicit form is given anywhere; the model cannot be evaluated without this function.
  • standard math Initial condition E(0) = 1
    Eq (44); normalizes the dimensionless Hubble parameter to its present-day value.
  • domain assumption Mean-field Gaussian variational posterior for all cosmological parameters
    Section IX, Eqs (45)-(48); neglects parameter correlations and can underestimate uncertainty in the precision claims.
invented entities (1)
  • Generalized Barrow-Tsallis entropy S = gamma (A/A0)^xi
    purpose: To define a modified holographic dark energy density and hence the Friedmann equation used for all constraints.
    No falsifiable prediction outside the fitted expansion history is provided, and the derived density is algebraically inconsistent with the entropy as shown in Sections IV-VII.

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Pith. "Pith review of A Bayesian PINN Framework for Barrow-Tsallis Holographic Dark Energy with Neutrinos: Toward a Resolution of the Hubble Tension." pith.science (2026). https://pith.science/paper/IIR4MQWG

@misc{pith2026250602235,
  author       = {Pith},
  title        = {Pith review of: A Bayesian PINN Framework for Barrow-Tsallis Holographic Dark Energy with Neutrinos: Toward a Resolution of the Hubble Tension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IIR4MQWG}},
  note         = {Machine review of arXiv:2506.02235}
}
abstract

We investigate the Barrow-Tsallis Holographic Dark Energy (BTHDE) model using both traditional Markov Chain Monte Carlo (MCMC) methods and a Bayesian Physics-Informed Neural Network (PINN) framework, employing a range of cosmological observations. Our analysis incorporates data from Cosmic Microwave Background (CMB), Baryon Acoustic Oscillations (BAO), CMB lensing, Cosmic Chronometers (CC), and the Pantheon+ Type Ia supernova compilation. We focus on constraining the Hubble constant $ H_0 $, the nonextensive entropy index $ q $, the Barrow exponent $ \Delta $, and the Granda-Oliveros parameters $ \alpha $ and $ \beta $, along with the total neutrino mass $ \Sigma m_\nu $. The Bayesian PINN approach yields more precise constraints than MCMC, particularly for $ \beta $, and tighter upper bounds on $ \Sigma m_\nu $. The inferred values of $ H_0 $ from both methods lie between those from Planck 2018 and SH$_0$ES (R22), alleviating the Hubble tension to within $ 1.3\sigma $-$2.1\sigma $ depending on the dataset combination. Notably, the Bayesian PINN achieves consistent results across CC and Pantheon+ datasets, while maintaining physical consistency via embedded differential constraints. The combination of CMB and late-time probes leads to the most stringent constraints, with $ \Sigma m_\nu < 0.114 $ eV and $ H_0 = 70.6 \pm 1.35 $ km/s/Mpc. These findings suggest that the BTHDE model provides a viable framework for addressing cosmological tensions and probing modified entropy scenarios, while highlighting the complementary strengths of machine learning and traditional Bayesian inference in cosmological modeling.

Figures

Figures reproduced from arXiv: 2506.02235 by the authors.

Figure 1
Figure 1. Constraints on the Barrow–Tsallis Holographic Dark Energy (BT-HDE) model parameters obtained using Cosmic [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Reconstructed Hubble parameter H0 as a function of redshift z from Cosmic Chronometer (CC) data within the Barrow–Tsallis Holographic Dark Energy (BT-HDE) model using the Bayesian Physics-Informed Neural Network (PINN) framework. combinations of datasets—Cosmic Chronometers (CC), Pantheon+, and their joint analysis—against the baseline values from Planck 2018 [73] and SH0ES R22 [51]. The vertical bands represent the… view at source ↗
Figure 3
Figure 3. Constraints on the Barrow–Tsallis Holographic Dark Energy (BT-HDE) model parameters obtained using the Pan [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Distance modulus µ(z) reconstruction from Pantheon+ data in the BT-HDE model using the Bayesian PINN approach. Table VI: Marginalized constraints on the free parameters of the Barrow–Tsallis Holographic Dark Energy (BTHDE) model using different combinations of observat…
Figure 5
Figure 5. Figure 5: Constraints on the Barrow–Tsallis Holographic Dark Energy (BT-HDE) model parameters obtained using the CC+ [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 6
Figure 6. Figure 6: The comparison of H0 measurement in baysian PINN model with Planck 2018 and R22 Results for different combi￾nation of data sets for BTHDE model. Table VII: Marginalized constraints on the free parameters of the Barrow–Tsallis Holographic Dark Energy (BTHDE) model using…
Figure 7
Figure 7. Figure 7: Comparison of the free parameters of the Barrow–Tsallis Holographic Dark Energy (BTHDE) model using different [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]
Figure 8
Figure 8. Figure 8: The comparison of H0 measurement in MCMC approach with Planck 2018 and R22 Results for different combination of data sets (CC+Pantheon+) for BTHDE model. with Planck and R22, respectively. These values are slightly more balanced than those from the MCMC method, suggest…
Figure 9
Figure 9. Figure 9: The comparison of parameters measurement in MCMC approach for different combination of data sets for BTHDE [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]
Figure 10
Figure 10. Figure 10: The comparison of H0 measurement in MCMC approach with Planck 2018 and R22 Results for different combination of data sets(CMB + All) for BTHDE model. Acknowledgments This work is based upon research funded by Iran National Science Foundation (INSF) under project No.40…

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