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arxiv: 2606.22431 · v1 · pith:EHWNCKLUnew · submitted 2026-06-21 · 🌀 gr-qc · astro-ph.CO· hep-th

An Interplay Between Fractional Calculus and Holographic Dark Energy

Ayush Bidlan This is my paper

Pith reviewed 2026-06-26 10:09 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COhep-th
keywords fractional holographic dark energyRiesz derivativeWheeler-DeWitt equationBekenstein-Hawking entropyCohen inequalityHubble cutofflate-time accelerationmodified gravity
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The pith

Inserting a Riesz fractional derivative into the Schwarzschild black hole Hamiltonian constraint produces fractionally corrected entropy that defines a new holographic dark energy density.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a fractional-calculus version of holographic dark energy by placing a Riesz fractional spatial derivative inside the Hamiltonian constraint of a Schwarzschild black hole in quantum geometrodynamics. This produces a Fractional Wheeler-DeWitt equation whose solutions supply a Lévy-index-dependent correction to the Bekenstein-Hawking entropy. The corrected entropy is then fed into Cohen's inequality to obtain the Fractional Holographic Dark Energy density. A sympathetic reader would care because the construction supplies a direct route from non-local effects in quantum gravity to late-time cosmic acceleration, yielding observable deviations from ordinary holographic dark energy inside the Hubble cutoff. The resulting model is reconstructed in effective field theory and extended to several modified-gravity frameworks to extract redshift- and α-dependent equation-of-state and deceleration parameters.

Core claim

A Riesz fractional spatial derivative inserted into the Hamiltonian constraint of a Schwarzschild black hole generates a Fractional Wheeler-DeWitt equation. Solutions of this equation yield a fractional Bekenstein-Hawking entropy governed by the Lévy index α (1 < α ≤ 2). Substituting the entropy into Cohen's inequality defines the Fractional Holographic Dark Energy density, whose late-time evolution is analyzed within the Hubble cutoff, reconstructed via effective field descriptions, and extended to Brans-Dicke, DGP, EPN and Horndeski theories; the same framework also determines the occurrence of big, little and pseudo-rip singularities.

What carries the argument

The Riesz fractional spatial derivative placed inside the Hamiltonian constraint of the Schwarzschild black hole, which produces the Fractional Wheeler-DeWitt equation and the α-dependent correction to Bekenstein-Hawking entropy.

If this is right

  • FHDE deviates from standard holographic dark energy and can mitigate some Hubble-cutoff inconsistencies.
  • Cosmological observables, equation-of-state and deceleration parameters become explicit functions of redshift z and the Lévy index α.
  • Effective-field reconstructions yield kinetic and potential terms for spin-0 and spin-1 candidates as functions of z and α.
  • The framework supplies explicit expressions for the equation of state and deceleration parameter inside Brans-Dicke, DGP, EPN and Horndeski theories.
  • The Granda-Oliveros cutoff version of FHDE predicts the occurrence of big-rip, little-rip and pseudo-rip singularities at late times.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Non-local memory effects encoded in the fractional derivative could source cosmic acceleration without introducing new scalar fields.
  • Observational constraints on the present-day value of the deceleration parameter might bound the allowed range of α independently of other cosmological parameters.
  • The same fractional modification of the Wheeler-DeWitt equation could be applied to other holographic setups or to the thermodynamics of different black-hole solutions.
  • Consistency checks between the fractional entropy and other quantum-gravity approaches, such as loop quantum gravity or causal sets, would test whether the Lévy-index correction survives beyond the present construction.

Load-bearing premise

That a Riesz fractional spatial derivative can be inserted into the Hamiltonian constraint of a Schwarzschild black hole so that the resulting entropy correction remains consistent when used in Cohen's inequality to define a dark-energy density.

What would settle it

A high-precision measurement of the dark-energy equation-of-state parameter at late times showing no dependence on any parameter α in the interval (1,2] that matches the fractional-entropy prediction, or an independent calculation demonstrating that the fractional entropy correction violates the area law or unitarity for the same range of α.

Figures

Figures reproduced from arXiv: 2606.22431 by Ayush Bidlan.

Figure 3.1
Figure 3.1. Figure 3.1: Redshift evolution of Ω (fr) DE for various values of α ∈ (1, 2] [PITH_FULL_IMAGE:figures/full_fig_p029_3_1.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Redshift evolution of the EoS parameter w (fr) DE for various values of α ∈ (1, 2] [PITH_FULL_IMAGE:figures/full_fig_p029_3_3.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: The w (fr) DE–w ′(fr) DE plane over the redshift range −1 ≤ z ≤ 2, for various values of α [PITH_FULL_IMAGE:figures/full_fig_p030_3_5.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Redshift evolution of the quintessence kinetic energy Xq [PITH_FULL_IMAGE:figures/full_fig_p033_3_7.png] view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: Redshift evolution of the quintessence EoS parameter wq. from this limit. Nevertheless, throughout the range α ∈ [1.2, 1.8], the EoS remains within the quintessence regime, −1 < wq < −1/3, over the redshift interval approaching z → −1. Overall, Figure (3.7) suggests that larger values of α are less favourable for quintessence reconstruction, since they induce a kinetic term that evolves too rapidly to re… view at source ↗
Figure 3.10
Figure 3.10. Figure 3.10: Redshift evolution of the K￾essence kinetic energy Xkq [PITH_FULL_IMAGE:figures/full_fig_p034_3_10.png] view at source ↗
Figure 3.12
Figure 3.12. Figure 3.12: Redshift evolution of the dila￾ton kinetic energy Xd [PITH_FULL_IMAGE:figures/full_fig_p036_3_12.png] view at source ↗
Figure 3.14
Figure 3.14. Figure 3.14: Redshift evolution of the DBI￾essence kinetic energy XDBI [PITH_FULL_IMAGE:figures/full_fig_p038_3_14.png] view at source ↗
Figure 3.16
Figure 3.16. Figure 3.16: Redshift evolution of the DBI￾essence EoS parameter wDBI. • For larger values of α, such as 1.8 and 1.6, the kinetic term XDBI evolves more rapidly with decreasing redshift, whereas smaller values such as 1.4 and 1.2 yield a slower evolution. • The potential VDBI(φ) behaves qualitatively like the quintessence potential in [PITH_FULL_IMAGE:figures/full_fig_p038_3_16.png] view at source ↗
Figure 3.17
Figure 3.17. Figure 3.17: Redshift evolution of the tachyon kinetic energy Xt [PITH_FULL_IMAGE:figures/full_fig_p040_3_17.png] view at source ↗
Figure 3.19
Figure 3.19. Figure 3.19: Redshift evolution of the YMC electric field squared E 2 [PITH_FULL_IMAGE:figures/full_fig_p042_3_19.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Redshift evolution of the EoS parameter wGO(t) for the big rip scenario within the Granda–Oliveros cutoff. The results are as follows: Linear interaction (Figures 4.1a and 4.2a). 1. For fractional values α = 1.1 and 1.3, the EoS parameter crosses the phantom divide (w < −1) and tends to −∞ in the late-time limit, consistent with the expected divergence of wGO as the Universe approaches a big rip singular… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Redshift evolution of the squared sound speed parameter [PITH_FULL_IMAGE:figures/full_fig_p052_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Redshift evolution of the EoS parameter wHH(t) for the little rip scenario within the Hubble cutoff. (a) Squared sound speed parameter v 2 HH against cosmic time t for the little rip ansatz (4.2) with the linear interaction term (4.6), for various values of the Lévy index α. (b) Squared sound speed parameter v 2 HH against cosmic time t for the little rip ansatz (4.2) with the non-linear interaction term… view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Redshift evolution of the squared sound speed parameter [PITH_FULL_IMAGE:figures/full_fig_p054_4_4.png] view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: Redshift evolution of the EoS parameter wHH(t) for the pseudo rip scenario within the Hubble cutoff. (a) Squared sound speed parameter v 2 HH against cosmic time t for the pseudo rip ansatz (4.3) with the linear interaction term (4.6), for various values of the Lévy index α. (b) Squared sound speed parameter v 2 HH against cosmic time t for the pseudo rip ansatz (4.3) with the non-linear interaction term… view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Redshift evolution of the squared sound speed parameter [PITH_FULL_IMAGE:figures/full_fig_p055_4_6.png] view at source ↗
read the original abstract

This dissertation aims to put forth a systematic construction of a fractional-calculus extension of holographic dark energy (HDE). We show that linking late-time cosmic acceleration to non-local or memory effects encoded in a fractional (Riesz) derivative within black hole thermodynamics produces deviations from standard HDE and can address some challenges of the Hubble cutoff. In particular, a Riesz fractional spatial derivative is introduced into the Hamiltonian constraint of a Schwarzschild black hole in quantum geometrodynamics, leading to a Fractional Wheeler--DeWitt equation whose solutions yield fractionally corrected thermodynamic quantities, notably fractional Bekenstein--Hawking entropy governed by the L\'evy index \(\alpha\), with \(1<\alpha\leq2\). Using this entropy with Cohen's inequality, a new dark energy density is constructed, defining the Fractional Holographic Dark Energy (FHDE) framework. The cosmological implications of FHDE are then investigated. Within the Hubble cutoff, its late-time evolution is analysed through cosmological observables, and the model is reconstructed using effective field descriptions with spin-\(0\) and spin-\(1\) candidates, allowing kinetic and potential terms to be extracted as functions of redshift \(z\) and \(\alpha\). The framework is then extended to BD, DGP, EPN, and Horndeski theories to derive the equation-of-state and deceleration parameters in terms of \(z\) and \(\alpha\). In addition, the fate of the Universe is studied through late-time singularities, namely the big, little, and pseudo-rip, within the Granda--Oliveros FHDE setting. In short, this dissertation proposes FHDE as a theoretically motivated extension of HDE, bridging non-locality in quantum gravity with the late-time dynamics of the Universe, and offering a route toward understanding cosmic acceleration beyond \(\Lambda\)CDM.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs a fractional extension of holographic dark energy (FHDE) by inserting a Riesz fractional spatial derivative into the Hamiltonian constraint of a Schwarzschild black hole in quantum geometrodynamics. This produces a Fractional Wheeler-DeWitt equation whose solutions yield a fractional Bekenstein-Hawking entropy governed by the Lévy index α (1 < α ≤ 2). The fractional entropy is substituted into Cohen's inequality to define the FHDE density. Cosmological implications are then explored under the Hubble cutoff, including effective-field reconstructions with spin-0 and spin-1 candidates, extensions to Brans-Dicke, DGP, EPN and Horndeski theories, and analysis of late-time singularities (big rip, little rip, pseudo-rip) in the Granda-Oliveros setting.

Significance. If the insertion of the Riesz derivative can be placed on a firm footing, the work would supply a concrete route from non-local quantum-gravity effects to late-time acceleration, generating parameterized deviations from standard HDE that could be confronted with observational data on the equation-of-state and deceleration parameters. The explicit reconstructions and singularity classifications provide falsifiable outputs once α is fixed.

major comments (2)
  1. [Abstract and Fractional Wheeler-DeWitt construction] Abstract and the section describing the Fractional Wheeler-DeWitt equation: the Riesz fractional spatial derivative is stated to be 'introduced' into the Hamiltonian constraint without derivation from a fractional gravitational action, without verification that the constraint algebra is preserved, and without demonstration that the resulting solutions remain real and normalizable. This step is load-bearing; the fractional entropy, its insertion into Cohen's inequality, and all subsequent cosmological observables rest directly on it.
  2. [FHDE density definition] Construction of the FHDE density via Cohen's inequality: no consistency check or error estimate is supplied showing that the standard Cohen bound remains applicable once the entropy has been non-locally modified; the thermodynamic interpretation required for the dark-energy density therefore lacks supporting analysis.
minor comments (2)
  1. [Parameter definition] The range 1 < α ≤ 2 is stated without explicit justification for the lower bound or discussion of the α → 1 limit.
  2. [Notation] Notation for the Riesz derivative and the fractional Wheeler-DeWitt operator should be introduced with a brief reminder of the definition before first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments highlight important foundational aspects of the construction that require clarification. We address each major comment below and will revise the manuscript accordingly to improve transparency regarding the assumptions and limitations.

read point-by-point responses

  1. Referee: Abstract and the section describing the Fractional Wheeler-DeWitt equation: the Riesz fractional spatial derivative is stated to be 'introduced' into the Hamiltonian constraint of a Schwarzschild black hole in quantum geometrodynamics. This produces a Fractional Wheeler-DeWitt equation whose solutions yield a fractional Bekenstein-Hawking entropy governed by the Lévy index α (1 < α ≤ 2). The fractional entropy is substituted into Cohen's inequality to define the FHDE density. Cosmological implications are then explored under the Hubble cutoff, including effective-field reconstructions with spin-0 and spin-1 candidates, extensions to Brans-Dicke, DGP, EPN and Horndeski theories, and analysis of late-time singularities (big rip, little rip, pseudo-rip) in the Granda-Oliveros setting.

    Authors: We agree that the Riesz fractional derivative is introduced by construction into the Hamiltonian constraint without a derivation from a fractional gravitational action, without explicit verification that the constraint algebra remains closed, and without a demonstration that the resulting wave-function solutions are real and normalizable for the full range of α. The manuscript treats this insertion as a motivated ansatz to incorporate non-local effects, with solutions assumed to inherit the properties of the standard case. In the revised version we will expand the relevant section (and abstract if needed) to explicitly state these assumptions, note that preservation of the constraint algebra is an open question, and clarify that the construction is exploratory rather than derived from first principles. This will not alter the subsequent cosmological analysis but will better contextualize its foundational status. revision: yes

  2. Referee: Construction of the FHDE density via Cohen's inequality: no consistency check or error estimate is supplied showing that the standard Cohen bound remains applicable once the entropy has been non-locally modified; the thermodynamic interpretation required for the dark-energy density therefore lacks supporting analysis.

    Authors: We acknowledge that the manuscript applies Cohen's inequality in its standard form after substituting the fractional entropy, without providing a dedicated consistency check, error estimate, or extended thermodynamic analysis for the non-local modification. This step follows the conventional procedure used in the holographic dark energy literature. In the revision we will add a short discussion paragraph addressing the thermodynamic interpretation, noting the potential limitations introduced by the fractional entropy, and emphasizing the exploratory character of the bound in this setting. No quantitative error estimate will be added, as that would require further theoretical development beyond the present scope. revision: yes

Circularity Check

0 steps flagged

No significant circularity; model construction is self-contained

full rationale

The paper introduces a Riesz fractional derivative into the Schwarzschild Hamiltonian constraint by ansatz, obtains a Fractional Wheeler-DeWitt equation, and extracts a fractional Bekenstein-Hawking entropy parameterized by the Lévy index α (1 < α ≤ 2). This entropy is then inserted into Cohen's inequality to define the FHDE density. All subsequent cosmological observables, reconstructions, and extensions to other theories are expressed as functions of redshift z and the free parameter α. This constitutes a standard parameterized model-building exercise rather than any reduction of outputs to inputs by construction. No self-citations, fitted parameters renamed as predictions, or definitional loops are present in the abstract or described chain; the framework remains externally falsifiable against observational data for chosen α values.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 2 invented entities

The central claim rests on the introduction of the Levy index alpha as a free parameter, the assumption that fractional derivatives are physically appropriate in the Wheeler-DeWitt setting, and the direct application of Cohen's inequality to the resulting entropy. No independent evidence is supplied for these steps.

free parameters (1)
  • Levy index alpha = 1 < alpha <= 2
    Controls the order of the fractional derivative and therefore the form of the corrected entropy and dark energy density; range given as 1 < alpha <= 2.
axioms (2)
  • domain assumption Cohen's inequality remains valid when applied to the fractionally corrected Bekenstein-Hawking entropy.
    Invoked to construct the FHDE density from the fractional entropy.
  • ad hoc to paper The Riesz fractional derivative can be meaningfully substituted into the Hamiltonian constraint of the Schwarzschild black hole in quantum geometrodynamics.
    This substitution produces the Fractional Wheeler-DeWitt equation whose solutions yield the new entropy.
invented entities (2)
  • Fractional Holographic Dark Energy (FHDE) no independent evidence
    purpose: New dark energy density constructed from fractional entropy.
    Introduced as the central object of the dissertation.
  • Fractional Bekenstein-Hawking entropy no independent evidence
    purpose: Thermodynamic quantity obtained from solutions of the fractional Wheeler-DeWitt equation.
    Postulated as the bridge between fractional calculus and holographic dark energy.

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Reference graph

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