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Observational viability of fractional holographic dark energy in LRS Bianchi type-I cosmological model

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims a fractional holographic dark-energy model with one free parameter reproduces ΛCDM's late-time expansion when fitted to Hubble, BAO, and supernova data.

desk verdict A competent wCDM fit wearing an anisotropic fractional-HDE costume: the Bianchi geometry drops out of the observable and the model contradicts its own pressure equations. read the letter →

arxiv 2506.16004 v1 pith:PMRQTDWZ submitted 2025-06-19 gr-qc

Shivangi Rathore , Elangbam Chingkheinganba Meetei , S. Surendra Singh This is my paper · ORCID
classification gr-qc PACS 98.80.-k95.36.+x
keywords fractionalholographicdarkenergyLRSBianchitype-ImodelobservationalconstraintsMarkovChainMonteCarlostatefinderdiagnosticsdecelerationparameterΛCDMcomparison
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

This paper argues that a new 'fractional holographic dark energy' — an energy density obtained from the holographic principle together with fractional-calculus corrections to black-hole entropy — can drive the universe's late-time acceleration as effectively as the cosmological constant. The authors place this density in a locally rotationally symmetric Bianchi type-I universe, derive a parameterized Hubble law, and fit its three parameters ($H_0$, the fractional index $\alpha$, and the matter density $\Omega_{m0}$) to Hubble, BAO, and Pantheon+SH0ES data using Markov Chain Monte Carlo. Across four dataset combinations the fits return $H_0 \approx 67.7$–$67.8$ km/s/Mpc, $\Omega_{m0} \approx 0.26$–$0.27$, and $\alpha \approx 0.86$–$0.89$. Taking $\alpha = 1.01$, the model produces a deceleration-parameter transition at $z_t = 0.55$, an equation-of-state parameter that approaches $-1$, and statefinder values $(r, s) = (0.74, 0.07)$ heading toward the $\Lambda$CDM fixed point, which the authors read as demonstrating the observational viability of fractional holographic dark energy.

What carries the argument

The load-bearing object is the fractional holographic dark-energy density $\rho_d = 3c^2 L^{-(3\alpha-2)/\alpha}$, formed by inserting the fractional entropy law $S_h \propto A^{(\alpha+2)/2\alpha}$ into the holographic inequality and taking the Hubble horizon $L = H^{-1}$ as the infrared cutoff; the fractional order $\alpha \in (1, 2]$ is the single new parameter, with $\alpha = 2$ recovering ordinary holographic dark energy. Combined with the LRS Bianchi type-I metric $ds^2 = -dt^2 + A_1^2 dx^2 + A_2^2 (dy^2 + dz^2)$ and the assumption $A_1 \propto A_2^n$ with $n > 2$, this density yields the field equations, the dark-energy equation of state $\omega_d = -1 + (3\alpha - 2)(1 - \Omega_d)/(2\alpha - (3\alpha - 2)\Omega_d)$, and, treating dark matter as pressureless and $\omega_d$ as constant, the fitted Hubble law $H(z) = H_0[\Omega_{m0}(1+z)^3 + (1-\Omega_{m0})(1+z)^{3(1+\omega_d)}]$. The deceleration parameter $q$ and the statefinder pair $(r, s)$ derived from the same equations supply the diagnostics used to compare the model with $\Lambda$CDM.

What would settle it

Evaluate the two pressure equations of the LRS Bianchi field equations, Eqs. (14) and (15), at the paper's fitted parameters $\alpha \approx 0.87$, $\Omega_{d0} \approx 0.73$, $\omega_d \approx -0.77$: consistency of the metric with a single isotropic pressure forces $\omega_d \Omega_d = 1$, whereas the fitted values give $\omega_d \Omega_d \approx -0.56$. That direct substitution settles whether the claimed Bianchi solution exists at all, independently of the goodness of the data fits.

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Extended reading notes

Core claim

The central claim is that fractional holographic dark energy, with density $\rho_d = 3c^2 H^{(3\alpha-2)/\alpha}$ under the Hubble-horizon cutoff, is an observationally viable alternative to the cosmological constant. The fractional index $\alpha$ enters through the entropy law $S_h \propto A^{(\alpha+2)/2\alpha}$ obtained from the fractional Wheeler-DeWitt equation, and $\rho_d$ reduces to the standard holographic dark energy density when $\alpha = 2$. Fitted to the combined data, the model gives a present dark-energy density parameter $\Omega_{d0} \approx 0.73$, an equation-of-state parameter $\omega_d \approx -0.77$ that drifts toward $-1$ at late times, and a deceleration parameter that crosses from deceleration to acceleration at $z_t \approx 0.55$, with the statefinder pair $(r, s) = (0.74, 0.07)$ approaching the $\Lambda$CDM point $(1, 0)$. The authors conclude that the LRS Bianchi type-I fractional holographic dark energy model is observationally viable, that its fitted $H_0$ agrees with early-universe CMB measurements, and that it closely reproduces the $\Lambda$CDM expansion history.

Load-bearing premise

The argument stands on the assumption that the LRS Bianchi type-I field equations are a consistent basis for the fitted expansion history, but equating the two pressure equations forces $\omega_d \Omega_d = 1$ while the fitted values ($\omega_d \approx -0.77$, $\Omega_d \approx 0.73$) give $\omega_d \Omega_d \approx -0.56$, so the Bianchi structure drops out of the dynamics.

Editorial extensions

If this is right

  • If the model is correct, the fitted $H_0 \approx 67.7$–$67.8$ km/s/Mpc matches early-universe CMB estimates, meaning this dark-energy model does not by itself resolve the Hubble tension.
  • The deceleration-to-acceleration transition at $z_t \approx 0.55$ falls inside the observationally favored range $0.5 \le z_t \le 0.8$, consistent with supernova and BAO evidence.
  • The dark-energy equation of state that starts near zero at high redshift, reaches about $-0.77$ today, and approaches $-1$ at late times reproduces the quintessence-to-$\Lambda$CDM trajectory without introducing a cosmological constant.
  • The consistent parameter estimates across all four dataset combinations indicate that the $\Lambda$CDM-like behavior is not an artifact of any single probe.

Reading between the lines

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

  • Because the fitted Hubble law is algebraically a flat wCDM model with a constant dark-energy equation of state, current distance data cannot distinguish fractional holographic dark energy from a simpler two-parameter dark-energy model; matching $\Lambda$CDM is a consistency check rather than a detection of fractional physics.
  • The paper's dynamical plots use $\alpha = 1.01$, yet the fitted values cluster near $\alpha \approx 0.87$, outside the model's stated range $1 < \alpha \le 2$; recomputing the deceleration and statefinder trajectories at the central fitted value is a direct check of whether the $\Lambda$CDM-like conclusions survive.
  • The paper treats $\omega_d$ as constant in deriving $H(z)$ even though its own Eq. (26) defines a redshift-dependent $\omega_d$; redoing the fit with an evolving $\omega_d(z)$ is a natural extension that could shift the best-fit parameters and the transition redshift.
  • The Bianchi geometry carries a shear that the fitting procedure ignores, so the same model could be tested against cosmic-microwave-background isotropy bounds on anisotropy, an independent check the paper does not pursue.
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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 proposes a locally rotationally symmetric (LRS) Bianchi type-I cosmological model filled with fractional holographic dark energy, derives a Hubble parameter H(z), and fits it to Hubble, BAO, and Pantheon+SH0ES data using MCMC. The authors report constraints on H0, the fractional parameter α, and Ω_m0, and use these to study the density parameters, deceleration parameter, equation of state, and statefinder diagnostics. The central claim is that the model is observationally viable and closely mimics ΛCDM, with a deceleration transition at z_t ≈ 0.55 and a statefinder pair (r,s) near (0.74,0.07). My assessment is that the central derivation is inconsistent with the paper's own field equations and that the fitted Hubble law is a standard wCDM proxy rather than the proposed Bianchi fractional-HDE model.

Significance. If the central claim were sound, the paper would be a useful test of whether a fractional-calculus holographic dark energy density can be embedded in an anisotropic Bianchi spacetime while remaining consistent with late-time cosmological data. The use of multiple public data sets and a standard MCMC pipeline is appropriate, and the reported H0 and Ω_m0 values are plausible. However, the load-bearing derivation is not a solution of the Einstein equations stated in Section 3, and the fitted H(z) is exactly the isotropic flat wCDM expression with no dependence on the Bianchi index n or on the fractional density evolution. The paper therefore does not establish the observational viability of the proposed model; it establishes only that a wCDM-like expansion history fits the data.

major comments (3)
  1. [§3, Eqs. (13)–(15) and (24)] The full LRS Bianchi field equations are not satisfied by the fitted expansion history. Eq. (24) follows from the 00 equation (13) together with the continuity equations, but the pressure equations (14) and (15) impose an additional constraint. Substituting Eq. (24) into either (14) or (15) gives ω_d Ω_d = 1 for n > 2, which forces Ȟ/H² = −3 and hence q = 2. The fitted values (ω_d ≈ −0.78, Ω_d ≈ 0.73) give Ȟ/H² ≈ −0.54, so the H(z) used in the likelihood is not a solution of the Einstein equations presented in Section 3. Equivalently, subtracting (14) from (15) yields Ȟ/H² = −3n/(n+2), which for n > 2 gives q = −1 + 3n/(n+2) > 0.8, showing that an accelerating solution of the full system does not exist under the ansatz A1 ∝ A2^n with n > 2.
  2. [§3, Eqs. (26)–(29)] The fitted Hubble law is a wCDM proxy, not the fractional-HDE model. Eq. (26) gives ω_d as a function of Ω_d (and therefore of z), and it is equivalent to the conservation equation (16) combined with ρ_d = 3c² H^{(3α−2)/α}. Eq. (27), by contrast, integrates (16) with ω_d held constant, which is an additional assumption not implied by the model. The text then takes a single value of ω_d from Eq. (26) and inserts it into Eq. (29). Consequently α enters the likelihood only through a constant equation-of-state parameter, the fractional density (20) is never evolved, and the Bianchi parameter n cancels from Eq. (29). The MCMC constraints in Table 1 therefore characterize an isotropic flat wCDM model, not the LRS Bianchi fractional-HDE model claimed in the abstract.
  3. [§5, Figs. 7–11; Table 1] All physical predictions are made with α = 1.01, whereas the MCMC constraints in Table 1 give α ≈ 0.86–0.885 for every data combination. No derivation of the value 1.01 is provided, and it is not the best-fit value. The claimed transition redshift z_t = 0.55, present values q ≈ −0.35, ω_d ≈ −0.77, and statefinder (r,s) = (0.74,0.07) are therefore not the predictions of the fitted model. This disconnect undermines the central viability claim in the abstract, which is based on the α = 1.01 curves rather than on the observationally constrained parameters.
minor comments (4)
  1. [§3, Eq. (29)] Equation (29) is printed without the exponent 1/2, making it dimensionally inconsistent as a formula for H(z); it should read H(z) = H0 [Ω_m0(1+z)^3 + (1−Ω_m0)(1+z)^{3(1+ω_d)}]^{1/2}.
  2. [§4.6, Table 1] The text reports α = 0.885^{+0.121}_{-0.149} for the H(z) dataset, but Table 1 lists 0.885^{+0.141}_{-0.149}; these should be reconciled.
  3. [§5.1 and §5.2] The text states that α = 1.01 and Ω_d0 = 0.73 are 'based on observational analysis', but neither value appears in Table 1, and the uncertainty on Ω_d0 is not propagated into the plotted curves for q, ω_d, and the statefinder.
  4. [Data Availability] 'No data are associated with the manuscript' is inaccurate, since the analysis uses the 46-point Hubble sample, 15 BAO points, and the Pantheon+SH0ES catalog; the specific data versions or repository links should be cited.

Circularity Check

2 steps flagged · score 5.0 of 10

The fitted H(z) is the standard wCDM law with the Bianchi index and fractional z-dependence removed, so the ΛCDM-like diagnostics are outputs of the proxy fit rather than tests of the FHDE model.

  1. renaming known result [Section 3, Eqs. (27)-(29); Section 5]
    "By using the relation, d ln a = − dz/(1+z) in Eq. (16), and considering ω_d as a constant, we get ρ_d = ρ_d0(1+z)^{3(1+ω_d)} ... After incorporating Eq. (27) and Eq. (28) in Eq. (13), we get parameterized form of Hubble parameter is H(z) = H0[Ω_m0(1+z)^3 + (1−Ω_m0)(1+z)^{3(1+ω_d)}]"

    This H(z) is exactly the standard flat wCDM Hubble law. The LRS Bianchi anisotropy index n cancels in the normalized density sum Ω_d+Ω_m=1, and the fractional-HDE density (20) is replaced by a constant-EoS power law (27), with ω_d frozen from Eq. (26) at a fixed Ω_d. The MCMC constraints on H0, α, and Ω_m0 therefore constrain the wCDM proxy, not the anisotropic or fractional dynamics. The subsequent 'predictions' of q(z), ω_d(z), and the statefinder pair are evaluated with the same α and Ω_d0, so the ΛCDM-like transition and fixed-point behavior are consequences of the assumed constant-EoS form rather than independent tests of the proposed FHDE/Bianchi model.

  2. fitted input called prediction [Section 5.2-5.4, Eqs. (42)-(47), Figs. 9-11]
    "By using Eq. (25) and Eq. (26), the expression for q is q = −1 + 3α(1−Ω_d)/(2α−Ω_d(3α−2)) ... Based on the previous analysis, we get the value of α and Ω_d are 1.01 and 0.73 respectively ... From Fig. 9, the deceleration parameter q begins with a positive value ... At the transition redshift z_t = 0.55, the Universe enters a transition phase from deceleration to acceleration."

    The deceleration parameter q, the EoS parameter ω_d(z), and the statefinder pair (r,s) are derived algebraically from the same best-fit parameters H0, α, and Ω_m0 (equivalently Ω_d0=1−Ω_m0) that were used to construct the fitted H(z) of Eq. (29). These are reconstructions of the fitted model, not independent predictions. In particular, the transition redshift z_t=0.55 and the present statefinder values (r0,s0)=(0.74,0.07) are statistically forced by the fitted wCDM-like expansion history; any fit with the same Ω_m0 and ω_d0 would produce the same diagnostics. Presenting them as evidence of viability is the fitted-input-called-prediction pattern.

full rationale

The algebraic derivation from the assumed metric and energy-momentum tensors is self-contained, and there is no load-bearing self-citation: the authors' own prior LRS Bianchi papers are cited only as background, not as justification for the central result. However, the observational core of the paper is circular in a specific, quotable sense. The fitted Hubble parameter, Eq. (29), is the standard flat wCDM law: the Bianchi index n cancels, and Eq. (27) deliberately freezes ω_d to a constant even though the model's own Eq. (26) makes ω_d a function of Ω_d(z). Consequently, the MCMC constraints on H0, α, and Ω_m0 are constraints on a wCDM proxy, and the later 'predictions' of q, ω_d(z), and the statefinder pair are computed from the same fitted parameters and the same wCDM form. The claimed resemblance to ΛCDM—transition at z_t=0.55 and (r,s)=(0.74,0.07)—is therefore an output of the fitting setup rather than an independent test of the fractional holographic dark energy or of the LRS Bianchi geometry. Separate consistency problems (the pressure equations (14)-(15) are not satisfied by the fitted expansion, and the plotted α=1.01 is not the central fitted value) are correctness concerns, not circularity, and do not by themselves raise the circularity score.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The model adds several ad hoc assumptions: a power-law anisotropy with no consistency check, a constant equation of state in the fit that becomes varying in the interpretation, and an unconstrained holographic constant c. These, rather than a first-principles derivation, carry the central results.

free parameters (5)
  • alpha (fractional parameter) = 0.86 to 0.885 (best fit); 1.01 used for all physics plots
    Controls the fractional entropy power law and the dark energy equation of state via Eq. (26).
  • H0 = 67.688 to 67.80 km/s/Mpc
    Hubble constant, fitted with MCMC to all datasets.
  • Omega_m0 = 0.264 to 0.27
    Present matter density parameter, fitted with MCMC.
  • c (holographic constant) = not reported
    Appears in rho_d = 3c^2 H^{(3alpha-2)/alpha} and would set the dark energy scale; not constrained or reported.
  • n (anisotropy exponent) = assumed natural number greater than 2
    Defines A1 proportional to A2^n; never fit or constrained and cancels from H(z).
assumptions (7)
  • domain assumption Hubble horizon cutoff L = H^{-1}
    Used in Eq. (20) to convert the general FHDE density to a function of H. Known to fail for standard HDE but adopted here.
  • domain assumption Non-interacting dark sector
    Separate continuity equations (16) and (17) for dark energy and dark matter.
  • domain assumption Dark matter is pressureless (omega_m = 0)
    Standard assumption, stated before Eq. (17).
  • ad hoc to paper A1 proportional to A2^n with n greater than 2
    Power-law relation between scale factors; never derived and inconsistent with pressure equations for the fitted parameters.
  • ad hoc to paper omega_d treated as constant in deriving Eq. (27)
    Needed to integrate the continuity equation to (1+z)^{3(1+omega_d)}; later contradicted by Eq. (26) with z-dependent Omega_d.
  • domain assumption Fractional entropy S_h = C A^{(alpha+2)/(2alpha)} taken from previous work
    Borrowed from Jalalzadeh et al. [30]; power-law entropy forms are already known, but this specific fractional derivation is cited.
  • ad hoc to paper The 00 equation and total energy conservation determine the expansion while pressure equations are not enforced
    Eq. (24) is derived from this reduced system; Eqs. (14)-(15) are inconsistent with the assumed metric unless omega_d Omega_d = 1.

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Cite this review

Pith. "Pith review of Observational viability of fractional holographic dark energy in LRS Bianchi type-I cosmological model." pith.science (2026). https://pith.science/paper/PMRQTDWZ

@misc{pith2026250616004,
  author       = {Pith},
  title        = {Pith review of: Observational viability of fractional holographic dark energy in LRS Bianchi type-I cosmological model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PMRQTDWZ}},
  note         = {Machine review of arXiv:2506.16004}
}
abstract

We scrutinized the Locally Rotational Symmetry (LRS) Bianchi type-I cosmological model in the presence of fractional holographic dark energy. We calculate the value of the Hubble parameter $H(z)$ using the field equations. After that, we fit the model by employing the MCMC techniques to observational data, which includes Hubble, Hubble+BAO, Hubble+Pantheon+Shoes, and Hubble+BAO+Pantheon+Shoes datasets. With the help of these datasets, we calculate the model parameters, viz. $H_{0}(z), \alpha,$ and $\Omega_{m_{0}}$. The value of $H_{0}(z),\alpha$ and $\Omega_{m_{0}}$ lies in the range $67.688^{+1.246}_{-1.197}- 67.80^{+1.23}_{-1.23}, 0.86^{+0.14}_{-0.15}- 0.885^{+0141}_{-0.149}$ and $0.264^{+0.018}_{-0.018}- 0.27^{+0.02}_{-0.02}$ respectively. Our results indicate that the evolution of the density parameters corresponding to dark energy (DE) and dark matter (DM), particularly for $\alpha=1.01$, along with the transition at $z_{t} = 0.55$ of the deceleration parameter $q$ from positive values to $-1$, reflects a phase of accelerated expansion that closely resembles the $\Lambda$CDM model. The behavior of the equation of state (EoS) parameter further demonstrates that the Universe's evolution aligns well with the framework of the fractional holographic dark energy (FHDE) model, suggesting its compatibility with scenarios of late-time cosmology. Moreover, the analysis of the statefinder diagnostics ${(r,s)}$ and the present value of ${(r,s)} = (0.74,0.07)$ reveal characteristics that converge towards the $\Lambda$CDM fixed point. A comparison between the observational constraints on our model parameters and those of the $\Lambda$CDM model exhibits a strong degree of agreement, thereby reinforcing the physical plausibility and consistency of the proposed cosmological model.

Figures

Figures reproduced from arXiv: 2506.16004 by the authors.

Figure 1
Figure 1. Contour plot with 1 − σ and 2 − σ confidence regions for the parameter H0, α and Ωm0 along with the constraint values for Hubble datasets [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. H(z) is shown as a function of redshift z, with 46 observational data points and their corre￾sponding error bars, for the chosen cosmological model BAO data points, we selected 15 key representative points to minimise errors arising from dataset corre￾lations. We leverage transverse BAO experiments and the comoving angular diameter distance to probe the Uni￾verse’s expansion history and constrain cosmological models… view at source ↗
Figure 3
Figure 3. Contour plot with 1 − σ and 2 − σ confidence regions for the parameter H0, α and Ωm0 along with the constraint values for Hubble+BAO datasets. where dA(z) represents the comoving angular diameter distance, Dv(z) denotes the dilation scale, and c stands for the covariance matrix [34]. For the BAO dataset, the χ 2 function is defined as χ 2 BAO = X 15 i=1  Dth(zi) − Dobs ∆Di 2 (33) Within the redshift interval 0.24 … view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: H(z) +BAO vs. error bars plot, for the chosen cosmological model stringent constraints on the model’s parameters. The comoving distance DM(z) is a fundamental concept in cosmology, representing the proper distance between two points in the Universe at a given time, acc…
Figure 5
Figure 5. Figure 5: Contour plot with 1 − σ and 2 − σ confidence regions for the parameter H0, α and Ωm0 along with the constraint values for Hubble+Pantheon datasets. + Shoes datasets. This approach harnesses the complementary strengths of each dataset, yielding more robust and reliable …
Figure 6
Figure 6. Figure 6: Contour plot with 1 − σ and 2 − σ confidence regions for the parameter H0, α and Ωm0 along with the constraint values for Hubble+BAO+Pantheon datasets. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: The change in the dark energy density parameter Ω [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: The change in the dark matter density parameter Ω [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: The change in the deceleration parameter [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: Equation of State parameter ω with redshift z is shown for the value of α = 1.01 5.4 State-finder parameter The statefinder parameters r, s serve as valuable diagnostic tools for comparing the dynamical behaviour of various DE models in a model-independent way. Unlike…
Figure 11
Figure 11. Figure 11: State-finder parameter with redshift [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

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