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Tsallis holographic dark energy in Fractal Universe

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A fractal-spacetime version of Tsallis holographic dark energy is shown to reproduce the current cosmic acceleration and to fit combined supernovae, BAO, CMB, and gamma-ray-burst observations.

desk verdict The algebra is coherent and the beta->0 limit recovers standard THDE, but the observational claims rest on LambdaCDM likelihoods that do not follow from the fractal THDE background, so the paper's main evidence for compatibility does not hold. read the letter →

arxiv 1908.10602 v1 pith:QQO633HA submitted 2019-08-28 gr-qc

S. Ghaffari , E. Sadri , A. H. Ziaie This is my paper · ORCID
classification gr-qc
keywords Tsallisholographicdarkenergyfractaluniverseinteractingsectorsdecelerationparameterstatefinderdiagnosticcosmologicalestimationlate-timecosmicacceleration
topics Dark Energy
open problems Dark MatterDark Energy
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

Tsallis holographic dark energy is a proposal in which the dark energy density follows a power law in the Hubble radius, motivated by generalized entropy. This paper embeds that proposal in a fractal universe, one whose spacetime is modified by a power-law fractal function, and asks whether the model can still produce the observed late-time acceleration and fit the available distance data. The authors derive the equation-of-state, deceleration, and jerk parameters, and show that in both interacting and non-interacting versions the universe transitions from deceleration to acceleration near z≈0.7, with a present deceleration of q0≈-0.55. They then fit the six free parameters to a combined Pantheon supernovae, BAO, CMB, and gamma-ray-burst data set, obtaining H0≈68.8 km/s/Mpc and Ω_D≈0.69 with a goodness-of-fit below one, and take this as evidence that the model describes the current accelerating universe.

What carries the argument

The load-bearing object is the fractal Friedmann equation H² + H\dotν/ν - (ω/6)\dotν² = (1/(3M_p²))(ρ_m+ρ_D), with ν=$a^{{-β}}$, together with the Tsallis holographic dark energy density ρ_D = (3B/8π)$H^{{4-2δ}}$. The central identity is Eq. (10), which expresses \dot H/H² purely in terms of Ω_D, the redshift z, the fractal parameters β and ω, and the interaction coupling b²; all later quantities, including the EoS ω_D, deceleration q, jerk j, and the statefinder pair, are obtained by inserting this identity into the definitions of those parameters. The fractal modification appears in the closure relation Ω_m+Ω_D = 1+γ with γ = -β - (β²ω/6)(1+z)^{2β}, which changes the standard flat-universe constraint and shifts the late-time dynamics.

What would settle it

Take the best-fit fractal THDE parameters and compute the model's own prediction for the CMB shift parameter, the acoustic scale, and the BAO distances directly from the fractal Friedmann expansion, without importing the ΛCDM formulas in Eqs. (28)-(35), then compare those predictions to the Planck 2015 and BOSS DR12 measurements; a discrepancy of more than a few percent in the acoustic scale or the BAO distances would falsify the claimed observational compatibility.

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Extended reading notes

Core claim

The central claim is that the Tsallis holographic dark energy density ρ_D = (3B/8π)$H^{{4-2δ}}$, with the Hubble radius as IR cutoff, can serve as the dark energy in a flat fractal Friedmann universe whose fractal function is ν=$a^{{-β}}$. The derived equations show that the EoS parameter runs from larger values in the past toward -1 today, and in the interacting case crosses the phantom divide; the deceleration parameter q reaches about -0.55 at the present and crosses zero at z≈0.7, signalling the onset of acceleration. With the model's parameters fitted by MCMC to the combined data, the Hubble constant is H0=68.$783^{{+0.961}}$_{-0.761} km/s/Mpc, the dark energy density parameter is Ω_D=0.$687^{{+0.024}}$_{-0.028}, and the interaction coupling b²=0.$0423^{{+0.02}}$_{-0.02}. The statefinder pair (r,s) approaches the ΛCDM fixed point (1,0) at late times while tracing a quintessence-like path, which the authors interpret as the model remaining close to, but distinguishable from, a cosmological constant.

Load-bearing premise

The claim's load-bearing premise is that the standard ΛCDM formulas for the CMB shift parameter, the acoustic scale, and the BAO sound horizon (Eqs. 28-35) remain valid in the fractal THDE background, even though the model omits radiation and separate baryon and cold dark matter components; if those formulas do not carry over, the claimed observational compatibility is not established.

Editorial extensions

If this is right

  • The model's best-fit Hubble constant lies in the range H0≈68-70 km/s/Mpc, consistent with the Planck-based value and with recent local measurements, so a fractal-Tsallis dark energy does not require a Hubble rate outside the observed window.
  • The transition redshift z_t≈0.7 falls inside the 0.4-0.8 interval reported by independent studies of the deceleration-acceleration transition, so the model reproduces the timing of the onset of cosmic acceleration without a cosmological constant.
  • The statefinder trajectories converge toward the ΛCDM fixed point (r,s)=(1,0) while remaining on the quintessence side, so future geometric measurements that can distinguish (r,s) pairs would be able to tell this model apart from a pure cosmological constant.
  • The small positive coupling b²≈0.042 implies a mild transfer of energy from dark energy to dark matter, which the authors connect to the coincidence problem; the interacting version crosses the phantom divide, while the non-interacting version asymptotes to ω_D→-1.

Reading between the lines

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

  • Editorial inference: if the same model were tested against the full Planck 2018 CMB temperature power spectrum and matter growth data rather than the compressed shift parameter and acoustic scale, the extra fractal and Tsallis parameters would face a much tighter test; the paper's compressed-data fit does not settle that question.
  • Editorial inference: the choice of interaction term Q=3Hb²ρ_D is imported from a study of a different dark energy model; other couplings would change the phantom-crossing redshift and the fitted value of b², so the reported observational compatibility is specific to this interaction form.
  • Editorial inference: because the fractal function is fixed as a power law ν=a^{-β}, a natural extension is to allow β to vary with time or to use a different fractal measure; the derived equations and the fitted parameters would change, and comparing the resulting transition redshift with z_t≈0.7 would provide a direct robustness check.
  • Editorial inference: the same derivation machinery applies immediately to other generalized entropies, such as Rényi or Sharma-Mittal entropy, in the fractal background; differences in the predicted statefinder trajectory would let the same data set rank the entropy prescriptions.
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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. This paper studies a Tsallis holographic dark energy (THDE) model in a flat fractal Universe with the Hubble radius as the infrared cutoff. The authors derive the equation-of-state parameter, the deceleration parameter, and the evolution equation for the dark energy density parameter for both noninteracting and interacting cases, with interaction term Q = 3H b^2 rho_D. They also compute the statefinder pair and then fit the free parameters to Pantheon SNIa, GRB, BAO (BOSS DR12, 6dF, eBOSS), and Planck 2015 CMB data using an MCMC method. The paper claims that the model describes the current accelerated expansion in both scenarios and that a deceleration-to-acceleration transition occurs at late times (around z ~ 0.7), with a reported present deceleration parameter q0 ~ -0.55.

Significance. The analytic derivations in Secs. II-III are largely coherent and the beta -> 0 limit correctly recovers the standard THDE results, which is a useful consistency check. The paper is clearly organized in its formal parts. However, the observational claims depend on likelihoods that are not derived from the model's own background, and several implementation details are missing, so the empirical support for the model is not established. If the data analysis were redone with model-consistent formulas and full details, the model might still be viable, but the present evidence is inconclusive. The paper does present a nontrivial extension of THDE to fractal cosmology, but the strength of the claimed observational compatibility is not supported by the analysis as written.

major comments (4)
  1. [Sec. V.D, Eq. (35)] The CMB shift parameter is mis-specified: as printed, Eq. (35) defines R = sqrt(Omega_m0) H0/c rs(z*), which is proportional to the sound horizon, whereas the standard shift parameter is proportional to the comoving distance to last scattering. Consequently, the CMB chi2 in Eq. (30) is not evaluating the intended observable, and the reported Planck constraints on this model are not meaningful unless this is a typographical error corrected in a revised version.
  2. [Sec. V.C-D, Eqs. (28)-(35)] The BAO and CMB likelihoods use standard LambdaCDM formulas - the Hu-Sugiyama drag-epoch fitting formula (Eqs. 33-34), the baryon-photon sound speed cs(z) in Eq. (28), and the Planck 2015 compressed observables in Eq. (31) - that presuppose a radiation-dominated pre-recombination universe with baryons and photons. The fractal THDE background in Eqs. (3)-(13) contains only pressureless matter and THDE, with no radiation density entering H(z) and no separate baryon component. Using these LambdaCDM-based compressed likelihoods is therefore not a valid test of the model's background, and the fitted values in Table I cannot be taken as evidence of compatibility with CMB and BAO data.
  3. [Sec. V, Eqs. (22)-(36)] The statistical implementation is under-specified: the paper does not provide the explicit H(z) used in the fits, does not write down the chi2 expressions for eBOSS and 6dF, does not describe how the 109 GRB distance moduli are calibrated (GRBs are not self-calibrating distance indicators), and reports only chi_dof without chain convergence diagnostics or a breakdown of chi2 per dataset. Without these details, the joint chi2_min in Eq. (36) and the parameter uncertainties in Table I are not reproducible.
  4. [Sec. V, Table I and Sec. VI] The paper claims that the model 'can describe the current accelerating Universe' and that a transition occurs at late time, but this is a postdiction: the parameters delta, beta, omega, b^2, H0, Omega_D are fitted to the data, so the derived q0 ~ -0.55 and z_t ~ 0.7 are consequences of the fit, not independent predictions. The q0 value is quoted without an uncertainty, and the transition redshift is quoted with inconsistent ranges (0.5 < z < 0.9 in Sec. II and 0.6 < z < 0.8 in Sec. VI). The observational section should be reframed as parameter constraints rather than as model verification.
minor comments (5)
  1. [Table I] The table heading reads 'N ON - IN TERACT IN G' but the table includes b^2, and the text says the table gives both interacting and non-interacting values; only one column of numbers is shown. The heading and table need to be corrected to indicate which scenario is displayed.
  2. [Fig. 1 and Fig. 2 captions] The captions state omega = 0.263, whereas Table I lists omega = 0.201; the parameter values used in the figures should be reconciled with Table I.
  3. [Sec. V.C] The references for the 6dF and eBOSS BAO measurements appear to be swapped: Beutler et al. [79] is the 6dF survey, while Ata et al. [80] is eBOSS, but the text attributes z = 1.52 to [79] and z = 0.106 to [80].
  4. [Sec. III, after Eq. (18)] The statement that Planck gives q0 = -0.55 is misleading; Planck does not directly measure the deceleration parameter, and such a value is only inferred within a specific cosmological model.
  5. [General] There are numerous typographical and OCR artifacts (e.g., 'drive' for 'derive' in the abstract, garbled equation references such as Sec. II citing Eq. (12) twice, and inconsistent notation beta^3 omega vs beta^2 omega in Eqs. (12)-(13)). A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: model quantities are derived from the field equations and then evaluated at best-fit parameters.

full rationale

The derivation chain is not circular. Equations (6)-(13) and (18) follow algebraically from the fractal Friedmann equation (3), the THDE ansatz (8), and the conservation equations (4)-(5) with the interaction Q=3Hb^2ρD. The cosmological parameters H0, ΩD, δ, β, ω, and b^2 are free and fitted to SNIa, GRB, BAO, and CMB data; q0 ≈ −0.55 and zt ≈ 0.7 are then computed from Eq. (18) at the best-fit point. These are post-fit consistency checks, not inputs used to define the parameters, and the paper does not label them as independent predictions. The only self-citation that shapes a model choice is [65] for the linear interaction form, but that is a phenomenological ansatz supported by an external data-driven comparison, not an unverified self-cited theorem, so it is not load-bearing circularity. The use of standard LambdaCDM CMB/BAO compressed likelihoods in Sec. V may be a model-validity or correctness problem, but it is not a case of a result being equivalent to its own inputs by construction.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The model rests on the fractal-cosmology framework, the THDE density ansatz, and several ad hoc choices (Hubble cutoff, linear interaction). No new particles or fields are introduced. The fit uses seven free parameters, which is a large freedom for a phenomenological model.

free parameters (7)
  • delta = 1.360+0.160-0.191
    Tsallis entropy exponent; fitted to SNIa+BAO+CMB+GRB data.
  • beta = 0.123+0.059-0.063
    Exponent in the fractal function nu=a^(-beta); fitted.
  • omega = 0.201+0.029-0.029
    Fractal parameter in the action; fitted. Note Fig. 2 caption uses 0.263.
  • b^2 = 0.0423+0.02-0.02
    Dark sector interaction coupling in Q=3Hb^2 rho_D; fitted.
  • H0 = 68.783+0.961-0.761
    Present-day Hubble constant; fitted.
  • OmegaD = 0.687+0.024-0.028
    Present-day dark energy density parameter; fitted.
  • M = -19.375+0.023-0.019
    Absolute magnitude nuisance parameter for Pantheon SNIa; fitted.
assumptions (6)
  • domain assumption The fractal Friedmann equation (3) with nu=a^(-beta) is the correct background.
    All equations follow from this modified Friedmann equation from Calcagni's fractal cosmology.
  • domain assumption The conservation equations for matter and dark energy include the factor (3-beta)H (Eqs 4-5).
    Derived from the fractal geometry; used to obtain the EoS parameter.
  • domain assumption The Tsallis holographic dark energy density is rho_D=(3/8pi) B L^(2delta-4) (Eq 1).
    Assumed from ref [37] without derivation.
  • ad hoc to paper The IR cutoff is the Hubble radius, L=H^(-1).
    A modeling choice; the paper selects it without justification and the results depend on it.
  • ad hoc to paper The interaction term is Q=3H b^2 rho_D.
    Chosen from ref [65]; the fit depends on this linear form.
  • domain assumption Standard CMB shift and BAO sound-horizon formulas (Eqs 28-35) apply to the fractal THDE background.
    The paper uses these standard formulas without noting that its background lacks radiation and separate baryons.

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Pith. "Pith review of Tsallis holographic dark energy in Fractal Universe." pith.science (2026). https://pith.science/paper/QQO633HA

@misc{pith2026190810602,
  author       = {Pith},
  title        = {Pith review of: Tsallis holographic dark energy in Fractal Universe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QQO633HA}},
  note         = {Machine review of arXiv:1908.10602}
}
read the original abstract

We study the cosmological consequences of interacting Tsallis holographic dark energy model in the framework of the fractal universe, in which, the Hubble radius is considered as the IR cut-off. We drive the equation of state (EoS) parameter, deceleration parameter and the evolution equation for the Tsallis holographic dark energy density parameter. Our study shows that this model can describe the current accelerating Universe in both noninteracting and interacting scenarios, and also a transition occurs from the deceleration phase to the accelerated phase, at the late time. Finally, we check the compatibility of free parameters of the model with the latest observational results by using the Pantheon supernovae data, eBOSS, 6df, BOSS DR12, CMB Planck 2015, Gamma-Ray Burst.

Figures

Figures reproduced from arXiv: 1908.10602 by the authors.

Figure 1
Figure 1. The evolution of density of dark energy versus [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The evolution of the EoS parameter ωD versus red￾shift parameter z. According to the best fitted values listed in Table I, we have taken β = 0 · 123, δ = 1 · 36, ω = 0 · 263 and some values of b 2 . THDE model can cross the phantom line (ωD < −1) at the late time. III. COSMOGRAPHY The scale factor as the important component of study￾ing the kinematics of the Universe is accountable for the dependency of spatial sepa… view at source ↗
Figure 3
Figure 3. The evolution of the cosmic jerk parameter in terms [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: The evolution of parameter s in terms of parameter r. According to the best fitted values listed in Table I, the different parameter values β = 0 · 123, δ = 1 · 36, ω = 0 · 263 are adopted. The star symbol denotes the ΛCDM model and dot symbols represent the present va…
Figure 6
Figure 6. Figure 6: The contour map of the interacting Tsallis holo [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Pith tools

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