Probing Dark Energy Properties with Barrow Holographic Model in f(Q, C) Gravity

Anirudh Pradhan; N. Myrzakulov; S. H. Shekh

REVIEW 4 major objections 5 minor 1 cited by

Probing Dark Energy Properties with Barrow Holographic Model in f(Q, C) Gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that a Barrow holographic dark energy model in f(Q,C) gravity can reproduce the observed cosmic acceleration, with an equation of state that evolves from matter-like values in the past to -1 in the future.

desk verdict A routine transplant of the authors' f(Q,T) BHDE work into f(Q,C) gravity, where the assumed ΛCDM Hubble parameter does all the work and the new ingredients are never constrained. read the letter →

arxiv 2411.18911 v1 pith:NPCOZOT7 submitted 2024-11-28 gr-qc

N. Myrzakulov , S. H. Shekh , Anirudh Pradhan This is my paper · ORCID

classification gr-qc MSC 83F0583D05 PACS 98.80.-k04.50.Kd95.36.+x

keywords BarrowHolographicDarkEnergyf(QC)gravitynon-metricitycosmicaccelerationequationofstateLambda-CDMPantheonsupernovaeHubbleparameter

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 Barrow Holographic Dark Energy in f(Q,C) gravity, a modified theory built from the non-metricity scalar Q and its boundary term C, can account for the late-time acceleration of the universe without a cosmological constant. The authors adopt the standard Lambda-CDM expansion history H(z)=H0[Omega0m(1+z)^3+(1-Omega0m)]^{1/2}, add the Barrow entropy-corrected holographic energy density rho_bhde = C H^{2-$\Delta$}, and derive the pressure from the f(Q,C)=a1 Q^$\alpha$ + a2 C action. They then fit H0 and Omega0m to 57 OHD data points and the Pantheon SNe Ia sample, obtaining H0=70.01 and Omega0m=0.262, and find the equation of state goes from matter-like values at high redshift to -0.62 today and asymptotically to -1 at z=-1. The point of the paper is that this dynamic, quantum-gravity-inspired dark energy reproduces the same background diagnostics as Lambda-CDM while offering a geometric explanation.

What carries the argument

The central machinery is the ansatz H(z)=H0 $\sqrt$(Omega0m(1+z)^3 + (1-Omega0m)), equation (12), which fixes the expansion history to exactly Lambda-CDM, combined with the Barrow holographic energy density rho_bhde=C H^{2-$\Delta$}, where $\Delta$ measures the fractal deformation of the black-hole horizon. Feeding these into the f(Q,C) field equations with f(Q,C)=a1 Q^$\alpha$ + a2 C generates closed-form expressions for pressure, equation of state, sound speed, and energy conditions, so that every diagnostic plotted is an algebraic consequence of the assumed H(z), the Barrow density, and the chosen f(Q,C) form.

What would settle it

Compute the linear growth index from the f(Q,C) perturbation equations for the best-fit parameters; if the predicted f sigma_8(z) at z=0.5 disagrees with the combined Planck, BAO, and redshift-space-distortion measurements by more than the reported error, the model is ruled out despite its background fit.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that coupling Barrow's fractal black-hole entropy to the f(Q,C) gravitational action yields a holographic dark energy whose equation of state is dynamically evolving yet asymptotically Lambda-CDM. Specifically, with the Hubble rate fixed to the Lambda-CDM form, the derived EoS parameter omega_bhde is near zero at z>0, equals about -0.62 at the present epoch, and converges to -1 as z approaches -1, while the deceleration parameter switches from positive to negative at z about 0.74 and the statefinder pair is (r,s)=(1,0). The null and dominant energy conditions hold throughout, while the strong energy condition is violated at late times, matching the standard picture of acceleration. The authors read this as evidence that the BHDE model inside f(Q,C) gravity is a viable dynamical alternative to a static cosmological constant.

Load-bearing premise

The load-bearing premise is that the expansion history is already known to be Lambda-CDM: equation (12) fixes H(z) by hand, and all subsequent parameters, including the equation of state, deceleration parameter, and statefinder pair, are derived from that choice rather than from the f(Q,C) dynamics.

Editorial extensions

If this is right

The EoS parameter evolves from matter-like values near zero at z>0 to -0.62 today and to -1 at z=-1, so the model behaves like a dynamical dark energy that settles into a cosmological-constant phase in the far future.
The deceleration parameter crosses zero at z approximately 0.74, giving the observed transition from deceleration to acceleration without invoking a cosmological constant.
The statefinder pair (r,s)=(1,0) matches Lambda-CDM, so on the background geometry the model is indistinguishable from Lambda-CDM.
The null and dominant energy conditions hold at all redshifts while the strong energy condition is violated at z<=0, consistent with a repulsive late-time acceleration.
The sound-speed parameter is negative for z>0 and approaches zero at z=-1, indicating transient instabilities that fade in the distant future.

Reading between the lines

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

The authors do not claim, but it follows from their setup, that because Eq. (12) is assumed rather than derived, the kinematical diagnostics (q, Om(z), statefinder) cannot distinguish f(Q,C) gravity from Lambda-CDM.
A testable extension the paper leaves implicit would be to derive H(z) from the f(Q,C) field equations with rho_bhde as the matter source, rather than inputting the Lambda-CDM expansion by hand.
Another extension: compute the growth rate f sigma_8 from linear perturbations of this action; since perturbation dynamics depend on the full f(Q,C) structure, that observable would test the model beyond its background fit.
The Barrow exponent Delta in rho_bhde just rescales the density as H^{2-Delta}; an independent constraint on Delta (for example, from black-hole thermodynamics or a joint background-plus-perturbation fit) would separate the Barrow contribution from the f(Q,C) parameters.

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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

You should know two things up front. First, this is not an independent test of Barrow entropy or f(Q,C) gravity: the Hubble parameter in Eq. (12) is the ΛCDM expansion history, adopted by hand with a citation to the authors' own earlier paper. Second, the paper does have a usable core—the field equations for f(Q,C) are written down correctly, and the algebra from those equations to the EoS, deceleration parameter, statefinder, and energy conditions is internally consistent, as far as I checked. That is real work, and the presentation is readable. But the central claim in the abstract and conclusion—that the model provides a validated alternative to ΛCDM—is not supported. Because H(z) is fixed by hand, the derived ω(z), q(z), and statefinder (1,0) are algebraic consequences of ΛCDM, not predictions of the Barrow or f(Q,C) ingredients. The parameters Δ, α, a1, a2, C never enter Eq. (12), so the χ² fit to OHD and Pantheon constrains only H0 and Ω0m, and the reported values (H0≈70, Ω0m≈0.262) are just the standard ΛCDM best fit. That is the soft spot, and it is load-bearing. There are also smaller problems. The squared sound speed is negative for z≥0, which the text admits; claiming this 'aligns' with Bellini, Alam, and DES is a stretch, since those works do not validate negative sound speeds in this setup. The Data Availability statement says no data were used, which directly contradicts the fitted OHD and Pantheon samples, making the fit unreproducible as described. The text also has errors (e.g., the EoS reference to 'equations (20) and (21)' when the relevant equations are (14) and (15), and the section header that mentions f(Q,T)). None of these are fatal to the algebra, but they add noise. The citation pattern is conventional and mostly relevant; self-citation to ref. [35] is legitimate here because the construction is indeed a transplant, which only reinforces the novelty concern. Who is this for? Someone cataloging f(Q,C) phenomenology might want it on record, and a referee could usefully force the authors to state clearly that their fit does not distinguish the model from ΛCDM. But it is not a serious competitor to ΛCDM and it does not deliver the validation the abstract promises. My recommendation: if you have to referee it, accept the assignment but mark it as a modest phenomenological exercise needing major revision; if you are deciding whether to engage on your own, skip it unless you work on f(Q,C) model classification.

Referee Report

4 major / 5 minor

Summary. The paper investigates a Barrow Holographic Dark Energy (BHDE) model within an f(Q,C) gravity framework, using the functional form f(Q,C)=a1 Q^alpha + a2 C and the Barrow energy density rho_bhde = C H^(2-Delta). After presenting the f(Q,C) field equations for a flat FRW metric, the authors adopt a specific Hubble parameter H(z) and compute the energy density, pressure, equation-of-state parameter, squared sound speed, energy conditions, deceleration parameter, Om(z) diagnostic, and statefinder pair. They report a chi-squared fit to 57 OHD points and 1048 Pantheon points, obtaining H0 ~ 70 and Omega0m ~ 0.262, and conclude that the model transitions from matter-like behavior to dark-energy-dominated acceleration and aligns with Lambda-CDM. The central claim is that this constitutes a validated alternative description of cosmic acceleration.

Significance. If the model's parameters were actually constrained by the data and the background dynamics were derived from the f(Q,C) field equations, this would be a potentially interesting extension of holographic dark energy. The paper does present the field equations for f(Q,C) gravity and attempts to connect Barrow entropy with non-metricity and boundary-term gravity. However, the main significance is undercut because the adopted Hubble parameter is exactly the Lambda-CDM one, and the model-specific parameters (alpha, a1, a2, Delta, C) are never constrained. The validation reported is therefore a fit of the Lambda-CDM background, not a test of f(Q,C) gravity or Barrow entropy. The paper also contains a direct contradiction in its data availability statement, which prevents replication. Overall, the claimed comprehensive framework is not substantiated by the analysis presented.

major comments (4)

[Section II, Eq. (12)] The Hubble parameter is assumed, not derived: H(z)=H0 [Omega0m(1+z)^3 + (1-Omega0m)]^(1/2) is the Lambda-CDM expansion history, stated as 'we adopt the following expression for H(z) [35]'. It is not obtained from the field equations (6)-(7) together with the BHDE density (10) and the action (11). Because this assumed background fixes the dynamics, the subsequent equation-of-state parameter (Eq. (16)), deceleration parameter (Eq. (21)), Om(z) diagnostic, and statefinder pair (1,0) (Eq. (23)) are algebraic consequences of the input rather than predictions of the model. The Barrow parameter Delta, the action parameters a1, a2, alpha, and the holographic coefficient C do not appear in Eq. (12), so the reported chi-squared fit to OHD and Pantheon data constrains only H0 and Omega0m, reproducing the known Lambda-CDM best-fit values. This circularity invalidates the central claim of model validation.
[Sections III.3 and IV, Figures 2-8] The plotted curves and the reported numerical values, such as omega_BHDE(z=0) approx -0.62 and q(z=0) approx -0.60, depend on unspecified choices of the free parameters a1, a2, alpha, Delta, and C. The MCMC fit described in Section II yields only H0 and Omega0m; the paper does not report priors or best-fit values for the remaining five parameters, nor the expressions used to generate the figures. Consequently, the figures and the conclusions drawn from them are not reproducible from the information given.
[Section III.4, Eq. (17) and Section V] The squared sound speed in Eq. (17) is negative for z >= 0, as the text explicitly states ('potential instability in the BHDE model during the high-redshift phase' and 'remains negative' at z=0). Yet the concluding section claims that 'the model effectively resolves initial perturbations, ensuring long-term stability.' A negative pressure-gradient speed is a known instability for dark energy perturbations unless a specific physical mechanism or a full perturbative analysis is provided; none is supplied here. The statement in the conclusion is therefore unsupported and inconsistent with the stability analysis presented in Section III.4.
[Data Availability statement and Section II] The Data Availability statement reads 'No data was used for the research described in the article,' which directly contradicts the detailed description in Section II of using 57 OHD data points (0 <= z <= 2.36) and 1048 Pantheon SN Ia data points and performing a chi-squared minimization. This contradiction makes the reported constraints and figures unreproducible and calls into question the provenance of the observational results.

minor comments (5)

[Figures 2-8] All figure captions in Section III and IV say 'The behavior of statefinder parameters of the fluid,' but the panels show energy density, pressure, equation of state, sound speed, energy conditions, deceleration parameter, and Om(z); the captions should be corrected to match each plotted quantity.
[Equations (15)-(16)] Equation (15) uses the symbol gamma in the pressure expression while Equation (16) uses alpha for what appears to be the same exponent; the notation should be made consistent, or the relationship between gamma and alpha should be defined.
[Section III.3, derivation of omega] The text says 'By solving equations (20) and (21), we derived the equation of state parameter' but there are no numbered equations (20) and (21) at that point; the reference should be to the preceding energy density and pressure equations, such as (14) and (15).
[Abstract and Section II] The abstract claims predictions 'align well with observational datasets, including Type Ia supernovae, cosmic microwave background (CMB) radiation, and baryon acoustic oscillations (BAO),' but the analysis in Section II only uses OHD and Pantheon data; the CMB and BAO claims are not supported by any analysis in the manuscript.
[Section IV.2, Om(z) definition] The paper defines Om(z) = [E(z)]^2 - 1, omitting the denominator (1+z)^3 - 1 that appears in the standard Om diagnostic introduced by Sahni et al. (2008). As a result, the 'model-independence' attributed to Om(z) is inaccurate, and the interpretation of the plotted Om(z) behavior should be revisited.

Circularity Check

3 steps flagged · score 8.0 of 10

The validation chain rests on an adopted ΛCDM H(z): Eq. (12) is inserted by hand from the authors' own prior work, and the EoS, q, Om(z), and statefinder results are algebraic consequences of that input; the OHD/Pantheon fit constrains only H0 and Ω0m.

ansatz smuggled in via citation [Section II, Eq. (12); reference [35]]
"Specifically, we adopt the following expression for H(z) [35]: H(z) = H0 [ Ω0m (1 + z)^3 + (1 − Ω0m ) ]^(1/2) (12). ... [35] N. Myrzakulov, S. H. Shekh, A. Pradhan, and K Ghaderi Barrow Holographic Dark Energy in f (Q, T) gravity, arXiv:2408.03961 [gr-qc] (2024)."

Eq. (12) is the flat ΛCDM Hubble rate, taken verbatim from a citation to the authors' own f(Q,T) preprint rather than derived from the f(Q,C) field equations (6)-(7) together with the BHDE density (10). The model parameters α, a1, a2, Δ, and C do not appear in Eq. (12). Subsequent quantities are explicitly computed 'in the framework of considered Hubble's parameter in the equation (12)' (Eqs. 14, 16, 21-23), so the claimed predictions inherit exactly the assumed ΛCDM background.
self definitional [Section IV.3, Eq. (23)]
"In the framework of considered Hubble’s parameter in the equation (12), the expression or values of r and s is observed as, r = 1 and s = 0 (23). ... Physically, these values indicate dark energy dominance, contributing approximately 68% of the total energy density, constant acceleration, and ΛCDM consistency, confirming the ΛCDM model as a viable description of the universe’s evolution."

The statefinder pair (r,s) = (1,0) is the defining diagnostic of ΛCDM. Because Eq. (12) already is the ΛCDM Hubble rate, deriving r = 1 and s = 0 from it and then reporting 'ΛCDM consistency' as a finding is computing the output from the input by construction. The same reduction applies to q(z) in Eq. (21): q(0) = -0.60 and q(-1) = -1 are properties of the assumed H(z), not of Barrow entropy or f(Q,C) gravity.

1 more flagged steps

fitted input called prediction [Section II (MCMC fit paragraph) and Section V conclusion]
"After successfully minimizing the χ2 function, our analysis yields the estimated values for the model parameters as H0 = 70.01+0.057 −0.057 km s−1 Mpc−1 and Ω0m = 0.262+0.017 −0.017. ... our derived values of H0 ≈ 70 and Ω0m ≈ 0.262 show remarkable consistency with established cosmological constraints."

The χ2 fit is performed on Eq. (12), which contains only H0 and Ω0m; those are exactly the values reported and compared with Planck, WMAP, Riess, and Scolnic. Presenting these standard ΛCDM best-fit values as validation of the BHDE/f(Q,C) model, and concluding that 'the model demonstrates compatibility with observational datasets' and 'offers a robust alternative to static dark energy models,' attributes the fitted ΛCDM background to modified ingredients that did not enter the fitted expression. The fit is thus a fitted input renamed as a prediction.

full rationale

The central derivation chain is not self-contained: the Hubble expansion (Eq. 12) is an adopted ΛCDM ansatz sourced from the authors' own prior work (ref. [35]), not a solution of the f(Q,C) field equations with the Barrow density. Every derived diagnostic — energy density (14), EoS (16), deceleration (21), Om(z) (22), and statefinder (23) — is computed from that same assumed H(z), so the qualitative behavior (matter-like EoS at high z, ω ≈ -0.62 at z=0, ω → -1, q → -1, r=1, s=0) is fixed by the input rather than being an independent prediction. The observational comparison is likewise a two-parameter fit of H0 and Ω0m in the ΛCDM form; the f(Q,C) and Barrow parameters do not affect the fitted H(z), so the agreement with OHD and Pantheon cannot validate those ingredients. Separately, the Data Availability statement ('No data was used for the research described in the article') contradicts the reported use of 57 OHD and 1048 Pantheon data points, which is a reproducibility concern rather than a circularity step. Because the paper's headline validation reduces to the assumed background plus a standard fit, the circularity score is high, though not maximal: the paper does assemble the field equations and BHDE density from the literature, and the derived EoS expressions are algebraically non-trivial. Score 8 reflects that the central claim is forced by the adopted input and the self-citation chain, not that every equation is vacuous.

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

No new particles or forces are introduced; the Barrow exponent Delta and the constant C are parameters, not entities. The model's free parameters are numerous and mostly unreported, while the central assumption H(z) is the standard Lambda-CDM background.

free parameters (7)

H0 = 70.01 ± 0.057 km/s/Mpc
Fitted to OHD and Pantheon data; controls the overall expansion scale.
Omega0m = 0.262 ± 0.017
Fitted to OHD and Pantheon data; sets the matter density parameter.
alpha = not reported
Exponent in the f(Q,C) action; controls the modified gravity deviation but its value is never given.
a1 = not reported
Coefficient of Q^alpha in f(Q,C); required to compute pressure and EoS but not constrained.
a2 = not reported
Coefficient of C in f(Q,C); affects the field equations but its value is not specified.
C = not reported
Constant in rho_bhde = C H^(2-Delta); sets the BHDE energy density scale but is not fitted.
Delta = not reported
Barrow entropy exponent; interpolates between standard and fractal horizons but is not constrained.

assumptions (5)

domain assumption Flat, homogeneous, isotropic FRW metric
Section II, Eq. (3); standard cosmological assumption but unvalidated in this context.
ad hoc to paper The Hubble parameter takes the Lambda-CDM form H(z)=H0 sqrt(Omega0m(1+z)^3+1-Omega0m)
Equation (12); assumed, not derived from the field equations, and it pre-determines the main results.
ad hoc to paper Action f(Q,C)=a1 Q^alpha + a2 C
Equation (11); chosen without a selection principle; also its alpha, a1, a2 are unconstrained.
domain assumption Barrow entropy with apparent horizon IR cutoff L=H^{-1}
Equations (8)-(10); borrowed from Barrow and HDE literature; the choice of cutoff is a modeling assumption.
ad hoc to paper Energy density of BHDE equals C H^(2-Delta)
Equation (10); this identification fixes dark energy density to the Hubble scale, tying the model to the assumed background.

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Cite this review

Pith. "Pith review of Probing Dark Energy Properties with Barrow Holographic Model in f(Q, C) Gravity." pith.science (2026). https://pith.science/paper/NPCOZOT7

@misc{pith2026241118911,
  author       = {Pith},
  title        = {Pith review of: Probing Dark Energy Properties with Barrow Holographic Model in f(Q, C) Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NPCOZOT7}},
  note         = {Machine review of arXiv:2411.18911}
}

abstract

Understanding the accelerating expansion of the universe remains one of the foremost challenges in modern cosmology. This study investigates Barrow Holographic Dark Energy (BHDE), a model inspired by quantum gravitational corrections, within the framework of $f(Q,C)$ gravity. This extension of symmetric teleparallel gravity incorporates the non-metricity scalar $Q$ and the boundary term $C$, enabling a deeper exploration of cosmic dynamics without relying on a cosmological constant or exotic matter. The BHDE model is analyzed under a flat Friedmann-Robertson-Walker (FRW) metric, focusing on key cosmological parameters such as energy density, isotropic pressure, the equation of state (EoS) parameter, stability conditions and the energy conditions. The results demonstrate that the EoS parameter transitions from matter-like behavior ($z > 0$) to negative values at $z = 0$, indicating the dominance of dark energy and its role in the universe's accelerated expansion. As $z$ approaches $-1$, the EoS parameter asymptotically converges to $-1$, aligning with the $\Lambda$CDM model. This work underscores the potential of the BHDE model in $f(Q,C)$ gravity as a comprehensive framework for studying cosmic acceleration. By incorporating Barrow entropy and addressing the interplay between non-metricity and boundary terms, the model provides a dynamic approach to explaining dark energy.

Figures

Figures reproduced from arXiv: 2411.18911 by the authors.

**Figure 3.** FIG. 3. The behavior of statefinder parameters of the fluid with [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** FIG. 4. The behavior of statefinder parameters of the fluid with [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: FIG. 5. The behavior of statefinder parameters of the fluid with [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. The behavior of statefinder parameters of the fluid with [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. The behavior of statefinder parameters of the fluid with [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: FIG. 8. The behavior of statefinder parameters of the fluid with [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Hints Beyond $\Lambda$CDM from Barrow and Tsallis Holographic Dark Energy with GO cutoff
gr-qc 2026-01 conditional novelty 5.0 of 10

Barrow holographic dark energy with the Granda-Oliveros cutoff fits current background data as well as ΛCDM and shows a weak AIC preference only for the Union3-based dataset combination.

Reference graph

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Dark energy ∗ nmyrzakulov@gmail.com † da salim@rediff.com ‡ pradhan.anirudh@gmail.com The universe’s accelerating expansion is fueled by a mysterious entity known as dark energy, which makes up approximately 68% of its total energy density. This enigmatic component was first identified through groundbreaking observations of type-Ia supernovae, the cosmic ...

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Pith tools

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

[1] [35]

D. C. Maurya, Transit cosmological models in non- coincident gauge formulation of f(Q,C) gravity theory with observational constraints, Gravit. Cosmol. 30 (2024) 330

2024

[2] [1]

Dark energy ∗ nmyrzakulov@gmail.com † da salim@rediff.com ‡ pradhan.anirudh@gmail.com The universe’s accelerating expansion is fueled by a mysterious entity known as dark energy, which makes up approximately 68% of its total energy density. This enigmatic component was first identified through groundbreaking observations of type-Ia supernovae, the cosmic ...

arXiv 2024

[3] [2]

These modifications seek a deeper understanding of the universe’s nature and dynamics

Modified gravity Theoretical frameworks known as modified theories of gravity aim to enhance or extend the general theory of relativity, addressing phenomena that the original theory cannot fully explain. These modifications seek a deeper understanding of the universe’s nature and dynamics. To achieve this, researchers explore inno- vative mathematical fo...

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[4] [3]

It affects how the universe ex- pands, how matter clusters, and how galaxies form

Energy density As, the energy density is essential to understanding the universe’s evolution. It affects how the universe ex- pands, how matter clusters, and how galaxies form. In our study, energy density is key to explaining how the universe transformed into its current state. From the equations (10) and (12), it is observed as ρbhde = C H0 q Ω0mz(z(z +...

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[5] [4]

Specifically, isotropic pressure equal in all directions impacts the cos- mos

Isotropic pressure As, in cosmology, the pressure significantly influences the universe’s dynamics and evolution. Specifically, isotropic pressure equal in all directions impacts the cos- mos. Notably, dark energy, driving the universe’s accel- erated expansion, is typically associated with negative isotropic pressure. This negative pressure generates re-...

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[6] [5]

The equation of state parameter The equation of state parameter serves as a crucial tool for pinpointing pivotal moments in the cosmos’s evolution. Mathematically, it is expressed as (ωbhde = p ρ ) By solving equations (20) and (21), we derived the equa- tion of state parameter for Barrow holographic dark en- ergy, yielding: 7 ωbhde = 2α−13α(2α − 1)a1H2 0...

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[7] [6]

The stability parameter The squared velocity of sound, a crucial parameter derived from equations (14) and (15), gauges the uni- verse’s stability. Mathematically, it is expressed as (ϑ2 s = ∂p ∂ρ ) and it is obtained as ϑ2 s(bhde ) = a16α(α − 1)α(2α − 1)Ω0m(z + 1)3 C(∆ − 2)(H0Ω0mz(z(z + 3) +3) +H0)2 ! × −H2 0 (Ω0mz(z(z + 3) +3) +1) α H0 p Ω0mz(z(z + 3) +...

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[8] [7]

The energy conditions Next, the energy conditions provide valuable per- ception into gravitational systems’ behavior, enabling analysis without delving into matter specifics. The Raychaudhuri equation governs attractive gravity, categorizing the stress-energy-momentum tensor into four types: Strong Energy Condition (SEC: ρbhde + p ≥ 0 and ρbhde + 3p ≥ 0),...

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