Gravitational waves driven by Holographic dark energy

Prabir Rudra; Sayani Maity

REVIEW 3 major objections 4 minor 59 references

Gravitational waves driven by Holographic dark energy

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that gravitational-wave perturbation evolution in a flat FRW universe differs characteristically across six holographic dark energy models, giving a new comparative probe of dark energy.

desk verdict The central ODE in Eq. (2.12) doesn't follow from Eq. (2.7), so the plotted waveforms can't support the paper's comparative claims. read the letter →

arxiv 2411.16780 v1 pith:6WF25OIR submitted 2024-11-25 gr-qc

Sayani Maity , Prabir Rudra This is my paper · ORCID

classification gr-qc MSC 83C3583F05 PACS 04.30.-w95.36.+x98.80.-k

keywords gravitationalwavesholographicdarkenergyFRWcosmologymodelsperturbationequationsredshiftevolutionBarrowentropycosmologicalperturbations

topics Dark Energy

open problems Dark 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 aims to show that the evolution of gravitational wave perturbations in a flat expanding universe carries a characteristic fingerprint of the dark energy model driving the expansion. It derives a second-order redshift equation for the perturbation eta(z), plugs in six holographic dark energy densities, and plots the resulting waveforms. The authors claim the waveforms differ in amplitude and time-dependence across models, so gravitational wave evolution can serve as a comparative tool for dark energy models. A secondary claim is that the Barrow holographic dark energy prediction barely depends on which infrared cutoff is chosen. If correct, this gives cosmologists a new observational handle on the otherwise hard-to-probe dark energy sector.

What carries the argument

The load-bearing object is the gravitational wave perturbation eta(z), whose evolution is governed by Eq. (2.12): eta''(z) + (1/(2(1+z)))(Y(z)/X(z) + 4) eta'(z) + (4/(Y(z)^2 $H_0^{4}$))( $xi^{2}$ + $H_0^{2}$ Y(z)/(1+z)^2 ) eta(z) = 0, with X(z) and Y(z) built from the energy density and pressure of matter plus the holographic dark energy component. The paper obtains explicit X(z) and Y(z) for each model by assuming a power-law scale factor a(t) = b_0 t^n and inserting the corresponding holographic energy density, converting the time-domain gravitational wave equation into a redshift-domain ODE whose numerical solutions are plotted.

What would settle it

Re-derive Eq. (2.12) directly from Eq. (2.7) by substituting z = 1/a - 1; the direct substitution yields friction coefficient (Y/X + 6)/(2(1+z)) and restoring coefficient (1/($H_0^{2}$ X))($xi^{2}$ + $H_0^{2}$ Y/(1+z)^2), which differ from the printed coefficients. Integrating the directly derived equation for, say, Ricci holographic dark energy and comparing to Fig. 1 would settle whether the claimed waveforms are artifacts of that algebra.

Watch

Extended reading notes

Core claim

The paper constructs the second-order differential equation governing gravitational wave perturbations eta(z) in redshift space for a flat FRW background, then evaluates its coefficients X(z) and Y(z) for six holographic dark energy models obtained from a power-law scale factor. Numerically solving the equation, it finds that each model leaves a distinct imprint: Ricci, Renyi, and Kaniadakis waveforms look similar; Tsallis produces larger amplitudes; Sharma-Mittal produces much smaller, more concentrated ripples; and Barrow holographic dark energy gives nearly identical waveforms for the Hubble-horizon and Granda-Oliveros cutoffs. The authors conclude that the features of gravitational wave evolution can serve as a significant tool for studying different dark energy models comparatively.

Load-bearing premise

The entire comparison rests on Eq. (2.7) being the correct equation for gravitational wave perturbations and on the algebra that converts it to Eq. (2.12); if either is wrong, the plotted waveforms do not represent gravitational waves.

Editorial extensions

If this is right

If the waveforms are reliable, the redshift dependence of gravitational wave perturbation amplitude offers a new, purely gravitational way to compare dark energy models.
All six models predict amplitude decay toward the present time, consistent with the known damping effect of dark energy; this makes the decay a model-independent signature rather than a discriminator.
For Barrow holographic dark energy, the Hubble-horizon and Granda-Oliveros cutoffs yield nearly identical gravitational wave evolution, so the prediction is insensitive to the cutoff choice within this model.
Distinct features, such as Tsallis producing larger amplitudes and Sharma-Mittal producing smaller, more concentrated ripples, could be used to rule out or favor models once gravitational wave data become available.

Reading between the lines

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

Editorial extension: the same redshift-space reduction could be applied to other dark energy constructions, such as quintessence, f(R) gravity, or interacting dark sectors, to build a library of gravitational wave fingerprints for model comparison.
Editorial extension: the claimed near-perfect agreement between the two Barrow cutoffs suggests that, within this framework, the infrared cutoff ambiguity—a known weakness of holographic dark energy—may not corrupt the gravitational wave prediction; if confirmed with the corrected equation, it would strengthen the testability of Barrow holographic dark energy.
Editorial extension: because the plots extend to z = 2000 with amplitudes set by the parameter xi, the quantitative distinctions between models may depend on the chosen model parameters; a sensitivity scan over n, delta, and other parameters would show whether the qualitative differences are robust.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper computes gravitational wave evolution for six holographic dark energy models and claims this can serve as a comparative tool. The new piece is the application of the perturbation framework to these six models, plus the observation that Barrow HDE waveforms are nearly cutoff-independent. That is a legitimate extension of earlier work, though a routine one.

The problem is the central equation. Eq. (2.12) does not follow from Eq. (2.7). Using z = 1/a - 1, a direct transformation gives an eta' coefficient of (Y/X + 6)/(2(1+z)) and an eta coefficient of (xi^2 + H0^2 Y/(1+z)^2)/(H0^2 X). The printed equation instead has (Y/X + 4)/(2(1+z)) and 4/(Y^2 H0^4)(xi^2 + H0^2 Y/(1+z)^2). The printed frequency coefficient is dimensionally inconsistent. Since every model-specific ODE in Section 3 is built from Eq. (2.12), all six numerical solutions and the comparative waveforms inherit this error.

Even if we grant Eq. (2.7) itself, which already looks non-standard because the friction term has the wrong sign and coefficient compared with the usual tensor perturbation equation, the transformation error alone is load-bearing. The paper's main claim, that GW evolution can distinguish HDE models, rests on a wrongly transformed ODE.

To its credit, the paper is honest about its limitations: no analytical solutions, no observational comparison, manually chosen parameters, no code or data. The Barrow cutoff-independence observation is interesting and could be true, but it depends on the same flawed equation.

What is useful here is the assembly of X(z) and Y(z) for six HDE models; someone could reuse that algebra as a starting point. But as it stands, the core math does not check out, and the plots are not trustworthy.

Recommendation: desk reject. The derivation error is central and easily caught; the paper would need major revision before it would warrant referee time. Not in current form.

Referee Report

3 major / 4 minor

Summary. The paper studies the evolution of gravitational-wave perturbations in a flat FRW background and compares the behaviour across six holographic dark energy models. The authors start from a second-order ODE for the perturbation η, convert it to redshift, define functions X(z) and Y(z) from the Friedmann equations, and insert each model's energy density into the resulting equation. Numerical solutions are plotted over a large redshift range, and the paper claims that gravitational-wave evolution can serve as a comparative probe of holographic dark energy models and that for Barrow HDE the result is nearly independent of the cutoff choice.

Significance. The comparative idea is potentially interesting: if the evolution of tensor perturbations were characteristically different for different holographic dark energy models, gravitational-wave observations or CMB polarization data could provide a new probe of dark energy. However, the paper contains no machine-checked proofs, no reproducible code or data, and the central conversion of the perturbation equation from cosmic time to redshift is algebraically incorrect. Because every model-specific equation and all figures are built on that conversion, the main quantitative results are not supported in the submitted form. The Barrow cutoff-independence claim is also not established quantitatively. A corrected rederivation from a justified gravitational-wave equation could be worth pursuing, but the present manuscript does not deliver it.

major comments (3)

[§2, Eqs. (2.7), (2.8), (2.12)] The conversion from Eq. (2.7) to Eq. (2.12) is algebraically wrong. With a0=1, z=1/a−1 and d/dt=−H(1+z)d/dz, Eq. (2.7) together with H²=H0²X and 2ä/a=−H0²Y yields η'' + [(Y/X+6)/(2(1+z))] η′ + [ξ²/(H0²X) + Y/(X(1+z)²)] η = 0. The printed Eq. (2.12) instead has (Y/X+4)/(2(1+z)) for the friction coefficient and 4/(Y²H0⁴)(ξ² + H0²Y/(1+z)²) for the potential coefficient. The discrepancy already appears in Eq. (2.8), where the friction term should be (−a²ä/ȧ² + 3a)η′ with a0=1, not (−a²ä/ȧ² + 2a)η′. The printed potential coefficient is also not dimensionless if ξ carries the same units as H0. Since every model equation in Section 3 is obtained by substituting the model's X(z), Y(z) into Eq. (2.12), all six ODEs and all figures inherit this error. This is an internal inconsistency, not merely a different convention.
[§2, Eq. (2.7)] The starting equation itself is not the standard tensor-perturbation equation for a flat FRW background. If η is the metric perturbation h, the standard equation is h¨ + 3H ḣ + k²/a² h = 0, equivalently h″ + 2(a′/a)h′ + k²h = 0 in conformal time. Eq. (2.7) instead has −Hη̇ and an extra −2ä/a term. The authors do not state a variable redefinition or a derivation that would make Eq. (2.7) equivalent to the standard equation. They need to define what η represents and justify Eq. (2.7); otherwise the physical meaning of all subsequent solutions is unclear.
[§3, Eqs. (3.4)–(3.6) and related model equations] The assumed power-law scale factor is not checked against the Friedmann equation. The authors assume a(t)=b0 t^n and use this to compute the holographic dark energy density, but a power-law scale factor gives H(z)=n b0^{1/n}(1+z)^{1/n}, while Eq. (3.6) gives H(z)=H0 sqrt[Ωm0(1+z)^{3(1+ωm)} + ...]. For the parameters of Fig. 1 (n=5, b0=0.002, H0=70, Ωm0=0.25, α=0.5), these disagree by about a factor of 24 at z=0. The same inconsistency affects the other models. Unless the parameters are forced to satisfy the Friedmann equation, the X(z) and Y(z) used in the gravitational-wave equation do not describe the assumed scale factor, and the comparison across models is not a comparison of actual holographic dark energy cosmologies.

minor comments (4)

[Figures 1–7] The parameter ξ is quoted as 5×10^{11} and 2×10^{9} without units, while H0=70 is dimensional; please state the unit convention for ξ, for example by plotting against ξ/H0, so that the numerical solutions are reproducible.
[§3.4.2, Figs. 4 and 5] The claim of minimum dependence on the cutoff is supported only by visual comparison of two plots that use different parameter sets, including C1=2 in Fig. 4 versus additional α1, β1, and Mp in Fig. 5; a quantitative measure or a controlled comparison is needed.
[§3.4.1] The notation is inconsistent: the label 'Rhobhdeh' after Eq. (3.22) and the use of both C in Eqs. (3.21)–(3.26) and C1 in the Fig. 4 caption should be harmonized.
[§3 and §4] Statements such as 'the gravitational waves are concentrated (shorter time period) around z=0' describe a property in redshift, not in time; please rephrase to avoid conflating redshift with cosmic time.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: GW waveforms are derived from stated background equations and externally cited HDE densities; no fitted parameter is relabeled as a prediction.

full rationale

The derivation chain is self-contained: the paper adopts a flat FRW background, states the gravitational-wave perturbation equation (2.7), rewrites it using the definitions X(z) and Y(z) in (2.9)-(2.11) to obtain (2.12), and then substitutes the published energy-density expressions for six holographic dark energy models together with the assumed power-law scale factor (3.4). No parameter is fitted to the gravitational-wave output, no prediction is extracted from a subset of the same data, and no uniqueness claim is imported from the authors' prior work. The only self-citation, ref. [47], appears in an introductory list of studies showing that HDE models are effective; it is not load-bearing. The model-specific densities are attributed to external references [54]-[59], and the governing differential equation is stated as a background input rather than justified by a self-citation. The conclusion that the plotted waveforms are determined by the assumed model inputs is true of any model calculation and is not circularity. A possible algebraic inconsistency in the transformation from (2.7) to (2.12) would be a correctness or derivation-error concern, not a circularity concern, and is outside the scope of this circularity pass.

Assumptions & free parameters 11 free parameters · 5 assumptions · 0 invented entities

The results are driven by a fixed power-law background, imported holographic dark energy density formulas, and many hand-picked parameters. No free parameter is fit to data, and no observable constraints are used. The gravitational wave equation itself is adopted without derivation.

free parameters (11)

n (power-law index) = n=5 (Ricci, RHDE, SMHDE, KHDE), n=2 (THDE), n=5 (BHDE)
Controls the time dependence of H and rho_D; chosen per model, not fitted.
b0 (scale factor normalization) = 0.002, 20, 250
Sets amplitude in plots; arbitrary.
xi (GW wavenumber) = 5e11 and 2e9
Two hand-picked values; determines oscillation frequency.
alpha (Ricci HDE constant) = 0.5
Absorbed constant in rho_Ricci, chosen by hand.
beta and delta (Tsallis HDE) = beta=2.3, delta=2
Parameters in rho_THDE, hand-picked.
c and delta (Renyi HDE) = c=250, delta=1.8
Parameters in rho_RHDE, hand-picked.
C and Delta (Barrow HDE) = C=2, Delta=3.2
Parameters in rho_BHDE, hand-picked.
alpha1 and beta1 (GO cutoff) = alpha1=2, beta1=5
Parameters in GO cutoff for BHDE, hand-picked.
d, delta, C (Sharma-Mittal HDE) = d=200, delta=1.8, C=250
Parameters in rho_SMHDE, hand-picked.
C and kappa (Kaniadakis HDE) = C=250, kappa=0.2
Parameters in rho_KHDE, hand-picked.
H0, omega_m, Omega_m0 (cosmological parameters) = H0=70, omega_m=0.0001 or 0.001, Omega_m0=0.25 or 0.24
Fixed to conventional or arbitrary values; no uncertainties.

assumptions (5)

ad hoc to paper The gravitational wave perturbation equation (2.7), with friction term -(dot a/a)dot_eta and curvature term -2(ddot a/a)eta, is the correct evolution equation.
Introduced in Section 2 without derivation or citation; it differs from the standard tensor perturbation equation, so the paper's results depend on an unproved equation.
ad hoc to paper The scale factor follows a power law a(t)=b0 t^n for all holographic dark energy models.
Assumed in Section 3 (Eq. 3.4) and used to express H(z) and rho_D; not derived from the Friedmann equations with two fluids.
domain assumption Matter and dark energy are separately conserved with no interaction.
Continuity equations (2.4)-(2.5) assume separate conservation.
domain assumption Holographic dark energy density formulas from the literature (Ricci, Tsallis, Renyi, Barrow, Sharma-Mittal, Kaniadakis) are valid.
Equations (3.3), (3.9), (3.15), (3.21), (3.28), (3.33), (3.38) are imported from cited works without verification.
standard math Flat Friedmann-Robertson-Walker spacetime and General Relativity field equations with 8*pi*G = 1.
Background equations (2.1)-(2.3) assume these.

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Cite this review

Pith. "Pith review of Gravitational waves driven by Holographic dark energy." pith.science (2026). https://pith.science/paper/6WF25OIR

@misc{pith2026241116780,
  author       = {Pith},
  title        = {Pith review of: Gravitational waves driven by Holographic dark energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WF25OIR}},
  note         = {Machine review of arXiv:2411.16780}
}

read the original abstract

In this paper, we have studied the effects of holographic dark energy on the evolution of gravitational waves. The background evolution of gravitational waves in a flat FRW universe is considered and studied in the presence of various holographic dark energy models. The perturbation equations governing the evolution of the gravitational waves have been constructed and solutions are obtained. These solutions are studied in detail to get a proper understanding of the characteristics of the gravitational waves in the presence of holographic dark energy. The work can be a significant tool in studying different dark energy models comparatively using the features of the gravitational wave evolution.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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