Pith. sign in

REVIEW 5 major objections 5 minor 77 references

Interacting viscous generalized QCD ghost dark energy with a hybrid expansion law reconstructs a late-time-accelerating f(G) gravity that is thermodynamically consistent and compatible with cosmic-chronometer data.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 06:25 UTC pith:WB3ZQBPB

load-bearing objection The paper's central reconstruction equation has a sign error that forces ρ_eff = 0, so the main claim of a viable reconstructed f(G) framework collapses; the rest is a standard, well-referenced but modest model-building exercise. the 5 major comments →

arxiv 2607.13109 v1 pith:WB3ZQBPB submitted 2026-07-14 gr-qc

Reconstruction of f(G) Gravity from an Interacting Viscous Generalized QCD Ghost Dark Energy Model: Cosmology and Thermodynamics: Cosmology and Thermodynamics

Zhanar Umurzakhova , Aziza Altaibayeva , Ulbossyn Ualikhanova , Surajit Chattopadhyay This is my paper
classification gr-qc MSC 83F0583D05 PACS 98.80.-k04.50.Kd95.36.+x
keywords ghost dark energyf(G) gravityGauss-Bonnet gravityhybrid expansion lawbulk viscositydark sector interactionBarrow entropygeneralized second law
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to show that a modified Gauss–Bonnet (f(G)) theory of gravity can be reconstructed from a dark-energy model in which QCD ghost dark energy interacts with dark matter and has bulk viscosity, and that the resulting cosmology is both viable and thermodynamically consistent. The key move is to adopt a hybrid expansion law a(t)=t^m e^{λt} that interpolates between the early matter-dominated expansion and a late-time de Sitter phase, then to derive the ghost equation of state, numerically reconstruct f(G) from the field equations, and test the result through the effective equation of state, classical stability, horizon thermodynamics (including Barrow entropy), and a fit to 31 cosmic-chronometer Hubble measurements. If the reconstruction is right, the universe's late-time acceleration emerges naturally: w_eff tends to −1, the total entropy never decreases, and the model's H(z) matches the data with reduced chi-square 0.55. The paper does not claim this is the unique or fundamental theory, but rather a consistent, observationally acceptable framework.

Core claim

The central claim is that the combined system—interacting viscous generalized ghost dark energy embedded in f(G) gravity—admits a smooth reconstruction of the Gauss–Bonnet correction f(G) under the hybrid expansion law, and that this reconstructed theory behaves as a viable dark-energy model: it produces a late-time de Sitter attractor, is classically stable for suitable parameters, satisfies the generalized second law of thermodynamics (with both Bekenstein–Hawking and Barrow entropy), and is consistent with cosmic-chronometer Hubble data. The calculation equates the geometric f(G) terms in the modified Friedmann equations to the ghost dark-energy density, yielding a second-order differenti

What carries the argument

The load-bearing elements are (1) the hybrid expansion law a(t)=a0 t^m e^{λt}, which fixes H(t)=m/t+λ and smoothly joins the matter era (t^{2/3}) to the de Sitter era; (2) the master reconstruction equation f(G)−G f_G + 24 H^3 dot f_G = ρ_ghost = αH+βH^2, which converts the ghost dark-energy density into a second-order ODE for f(G); and (3) the power-law ansatz f(G)=μG^n, chosen to mimic the numerically reconstructed function and to make the effective fluid, sound speed, and entropy analytically tractable. Barrow entropy S_B ∝ (A/A0)^((1+Δ)/2) generalizes the horizon entropy and is used to check the generalized second law.

Load-bearing premise

The model's central assumption is that the scale factor is exactly a(t)=a0 t^m e^{λt}; this hybrid form is chosen because it reproduces the early matter-dominated and late de Sitter limits, but it is not derived from the interacting viscous dynamics, and all subsequent results—the reconstructed f(G), the effective equation of state, the entropy law, and the Hubble fit—inherit whatever error this assumption introduces.

What would settle it

Solve the coupled conservation equations for ρ_m and ρ_ghost numerically without imposing the hybrid expansion law and check whether the resulting H(t) has the form m/t+λ; alternatively, test the reconstructed H(z) against an independent dataset such as BAO. A significant deviation would falsify the model's reconstruction and its claimed consistency.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The reconstructed f(G) remains smooth and monotonic across both early- and late-time regimes, so a single Gauss–Bonnet modification can cover the full expansion history.
  • The effective equation of state starts in the quintessence region and asymptotically settles at (or just below) −1, giving a stable late-time de Sitter-like attractor with a possible phantom crossing.
  • Total entropy production is non-negative for the chosen parameters, so the generalized second law holds; the result persists when Bekenstein–Hawking entropy is replaced by Barrow entropy with Δ in [0,1].
  • The model fits 31 cosmic-chronometer H(z) measurements with reduced chi-square 0.55, suggesting current observational compatibility.
  • Interaction and viscosity shift the ghost equation of state away from the non-viscous result, so dissipative and dark-sector exchange effects leave observable signatures in the expansion history.

Where Pith is reading between the lines

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

  • Because the hybrid expansion law is assumed rather than derived from the interaction and viscosity, the reconstruction is essentially a consistency test of that ansatz; a different interpolation between the matter era and de Sitter would generally yield a different f(G) and could change the stability and entropy conclusions.
  • The power-law f(G)=μG^n is one of many possible fits to the numerically reconstructed curve; the specific predictions (such as the exact timing of the phantom crossing) are therefore not robust features of the underlying model.
  • The observational case rests on a single dataset (31 cosmic chronometers); combining with BAO, supernova, or CMB distance data would test whether the same parameters remain viable and would sharpen the reduced chi-square claim.
  • A direct check of the generalized second law over the full allowed parameter space (including extreme values of Δ, α, β, and the viscosity coefficients) would strengthen the thermodynamic conclusion beyond the plotted ranges.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The manuscript claims to reconstruct f(G) gravity from an interacting viscous generalized QCD ghost dark energy model. The reconstruction uses a hybrid expansion a(t)=a0 t^m e^{λt} that interpolates between a matter-dominated power law and late-time de Sitter behavior. The f(G) function is first obtained numerically from a master reconstruction equation and then replaced by a power-law ansatz f(G)=μG^n. The authors study the effective equation of state, squared sound speed, thermodynamics with Bekenstein-Hawking and Barrow entropies, and compare H(z) with 31 cosmic chronometer measurements, concluding that the model is viable and thermodynamically consistent.

Significance. If correct, the paper would provide a fairly comprehensive phenomenological package: a concrete f(G) for a viscous interacting ghost dark energy model that is classically stable, satisfies the generalized second law of thermodynamics, and is consistent with cosmic chronometer data. The paper is explicit about the phenomenological status of the bulk viscosity and about the effective nature of the power-law parameters, which is good practice. However, the central reconstruction equation has a sign error that makes the effective energy density vanish identically, and independent algebra errors appear in the stability and thermodynamic sections. The H(z) agreement is a fit with many free parameters rather than a predictive test. In its present form the manuscript does not support the main conclusion.

major comments (5)
  1. [§6, Eq. (36) vs Eqs. (8)/(35)] The master reconstruction equation is inconsistent with the paper's own definition of ρ_eff. Eq. (8) defines ρ_eff = ρ_ghost + Gf_G − f(G) − 24H^3 ẏf_G. Eq. (36) states f(G)−Gf_G+24H^3 ẏf_G = ρ_ghost, which is equivalent to Gf_G−f(G)−24H^3 ẏf_G = −ρ_ghost. Substituting this into Eq. (8)/(35) gives ρ_eff = 0. Thus the geometric terms cancel the ghost dark energy instead of reproducing it. The correct equation that equates the geometric part to ρ_ghost is f(G)−Gf_G+24H^3 ẏf_G = −ρ_ghost. Eq. (45) inherits the same sign error. Consequently, the numerical reconstruction of §6 describes a model with no effective dark-energy fluid, and the subsequent use of ρ_eff in the equation of state, stability, and thermodynamics is invalidated.
  2. [§8.1, Eq. (56) and §10] The power-law f(G)=μG^n is introduced as 'reconstruction-inspired' and its parameters are described as 'effective fitting parameters' (§8.1). It is not derived from the reconstruction equation, and the numerical reconstruction it is supposed to mimic is based on the incorrect Eq. (36). The comparison with cosmic chronometer data in §10 uses this power-law together with the hybrid expansion to fit 31 data points. With ten free parameters in the model, the reported χ^2_ν = 0.55 is a measure of the quality of a fit, not an independent test of the reconstructed f(G). It therefore cannot offset the reconstruction error.
  3. [§8, Eqs. (53)–(55)] The time derivatives of ρ_eff and p_eff are algebraically wrong. The derivative of G f_G − f(G) is G ̈f_G (the ẏf_G terms cancel), but Eqs. (53) and (54) contain G ẏf_G instead. The error propagates into the squared sound speed Eq. (55), so the classical stability conclusion is not supported. The same kind of derivative mistake is likely to affect the numerical stability analysis if it is based on these formulae.
  4. [§9, Eq. (75)] The expression for ẏG has an extra 2H^3ẏH term. From G=24H^2(ẏH+H^2), the correct derivative is ẏG = 24[4H^3ẏH + 2HẏH^2 + H^2̈H]. Eq. (75) gives 24[4H^3ẏH + 2HẏH(H^2+ẏH) + H^2̈H] = 24[6H^3ẏH+2HẏH^2+H^2̈H]. This error enters ρ_eff and p_eff via Eqs. (70)–(71) and hence the fluid entropy rate (74); the positivity of ẏS_total shown in Figs. 6–7 and the GSLT conclusion are therefore not reliable.
  5. [§5, Eq. (27)] The claim that the hybrid expansion law is 'derived' from the model is overstated. The asymptotic analysis only gives a∝t^{2/3} for t→0 and a∝e^{λt} for t→∞; the product form a(t)=a0 t^m e^{λt} is then 'chosen for further analysis' (end of §5). This is a phenomenological ansatz with two free parameters (m,λ), and all subsequent results depend on it. The paper should explicitly acknowledge this as an assumption rather than a reconstruction from the interacting viscous dynamics.
minor comments (5)
  1. [Title/heading] The title repeats 'Cosmology and Thermodynamics: Cosmology and Thermodynamics'.
  2. [§9.1] Unresolved citation placeholders '[?,?,?]' appear in the Barrow entropy discussion.
  3. [General] Typos and spacing issues: 'amooth', 'consder', 'negarive', 'pictirially', 'hgher', 'dstage', and inconsistent 'FR W' vs. FLRW.
  4. [Eq. (9)] The pressure expression has '16H2 ẏf_G' instead of '16H^3 ẏf_G' as in Eq. (5); presumably a typesetting error.
  5. [Eq. (34)] Eq. (34) is very long and unwieldy; the accompanying text (and later approximations such as Eq. (78)) would benefit from a clear identification of the limiting regime in which the interaction term is neglected.

Circularity Check

2 steps flagged

The central viability claim rests on a self-chosen expansion law whose late-time limit is presented as a prediction, and on a Hubble-parameter 'agreement' that is an in-sample χ² fit of an ansatz; the sign error in the master reconstruction equation is a separate consistency problem.

specific steps
  1. self definitional [Section 5, Eq. (27); Section 7, Eq. (50)]
    "In the opposite limit t → ∞ ... the effective equation of state goes to w_eff → −1 ... we can take Hdot→0 ... a(t)∝e^{λt}. Thus, combining the two regimes, the functional form that can reproduce both limits is a(t)=a0 t^m e^{λt}. ... the hybrid scale factor is chosen for further analysis. ... Equation (50) represents the asymptotic effective equation of state ... This behaviour indicates that the reconstructed modified Gauss–Bonnet gravity naturally admits a stable late-time cosmological attractor."

    The late-time de Sitter behavior (w_eff→−1) was inserted through the assumed exponential factor in Eq. (27): with H→λ and Hdot→0, w_eff goes to −1 by definition. The same limit is then reported in Section 7 as an asymptotic prediction ('stable late-time cosmological attractor') of the reconstructed model. Since the hybrid law was chosen explicitly to satisfy w_eff≈0 early and w_eff→−1 late, recovering these limits later is a restatement of the ansatz, not a dynamical consequence of the reconstructed f(G).

  2. fitted input called prediction [Section 8.1, Eq. (56); Section 10, Eqs. (83)-(85), Fig. 8]
    "the parameters μ and n may be considered as effective fitting parameters rather than fundamental parameters ... The model parameters are then constrained by minimizing the standard chi-square statistic, χ2 = Σ [H_rec(z_i)−H_obs(z_i)]²/σ_i² ... The parameter values corresponding to the minimum χ2 are adopted for the comparison with the observational Hubble data. ... The fitting yields χ2_min = 14.31 and a reduced chi-square of χ2_ν = 0.55. Hence, it is a satisfactory agreement between the reconstructed model and the observational Hubble data."

    The §10 'agreement' is not an independent prediction: H_rec is built from the power-law f(G)=μG^n whose parameters are explicitly called effective fitting parameters, and they are optimized against the same 31 cosmic-chronometer points via Eq. (85). A small reduced χ² after minimizing on those very points measures in-sample fit quality; it does not test the model against independent data. The conclusion that the model is 'consistent with the observed cosmic expansion history' therefore reduces to restating that the fitted ansatz fits the data used for the fit.

full rationale

The main constructional circularities are (i) the hybrid expansion law Eq. (27) is chosen to force the early matter-dominated and late de Sitter limits, and those same limits are later exhibited as asymptotic predictions of the reconstructed f(G) model; and (ii) the cosmic-chronometer 'support' in §10 is a χ² fit of the reconstruction-inspired f(G)=μG^n to the same 31 data points, with μ,n explicitly labeled effective fitting parameters, so the reported agreement is statistically forced rather than predictive. The Barrow-entropy GSLT and stability plots are demonstrations for selected parameter values, not independent validations. I did not find a load-bearing self-citation chain: the cited [32] and [78] works are external and the Barrow-entropy framework is not used to justify the reconstruction. Note separately (a correctness issue, not counted as circularity) that Eq. (36) has the opposite sign from what Eq. (35) requires if geometric terms are to equal ρ_ghost; solving Eq. (36) as written would give ρ_eff=0, which is an internal-consistency failure rather than a circular reduction. Because the algebraic reconstruction and thermodynamic expressions have independent content but the central observational and asymptotic claims reduce by construction, the circularity score is 6.

Axiom & Free-Parameter Ledger

10 free parameters · 9 axioms · 0 invented entities

The model is a multiparameter construction: ten free parameters (α, β, b², ξ0, ξ1, m, λ, C, μ, n) are chosen by hand or fitted; the only data constraint is the CC χ² fit whose best-fit values are not reported. The expansion history and the functional form of f(G) are assumed, so the 'reconstruction' fits the model to the data rather than predicting it.

free parameters (10)
  • α = 1.0 (representative); varied in CC fit (not reported)
    Coefficient of the linear ghost DE term; fixed by hand in figures; part of the multiparameter CC fit in §10.
  • β = 0.10 (representative); 0.02–0.5 in Fig. 6
    Coefficient of H² ghost DE correction; chosen by hand.
  • b² = 0.03
    Interaction coupling between dark matter and ghost dark energy; chosen as representative; CC fit not reported.
  • ξ0 = 0.01
    Constant bulk viscosity coefficient; chosen by hand.
  • ξ1 = 0.005
    Linear-in-H viscosity coefficient; chosen by hand.
  • m = 2/3 (representative; varied 0.6–0.8)
    Power-law index of hybrid scale factor; motivated by matter-dominated a∝t^{2/3} but free in fits.
  • λ = 0.75 (representative; varied 0.6–0.9)
    Late-time de Sitter rate in hybrid scale factor; chosen by hand.
  • C = 0.5
    Integration constant in ρ_m solution (Eq. 25); chosen in Fig. 1 caption.
  • μ = 0.1 (fitting parameter)
    Overall coupling in power-law f(G)=μG^n; called an 'effective fitting parameter' in §8.1.
  • n = 1.2 (fitting parameter; varied 0.6–1.2)
    Power-law index in f(G)=μG^n; 'effective fitting parameter'.
axioms (9)
  • domain assumption FLRW flat metric with line element (1)
    Assumed spatial flatness and homogeneity/isotropy per standard cosmology.
  • domain assumption f(G) gravity action (2)
    The action with R/2 + f(G) + L_m is assumed; modified Friedmann equations (4)-(5) follow.
  • domain assumption Ghost DE density ρ_ghost = αH + βH² (Eq. 10)
    Standard form from QCD ghost literature ([41]); adopted without re-derivation.
  • domain assumption Bulk viscosity ξ = ξ0 + ξ1H (Eq. 12)
    Phenomenological parametrization from [55]; chosen for simplicity.
  • domain assumption Interaction Q = 3b²H(ρm + ρghost) (Eq. 15)
    Phenomenological coupling widely used in the literature.
  • ad hoc to paper Hybrid expansion law a(t)=a0 t^m e^{λt} (Eq. 27)
    Chosen to interpolate between t^{2/3} and e^{λt}; not derived from the field equations. All later results depend on this ansatz.
  • ad hoc to paper Power-law f(G)=μG^n (Eq. 56)
    'Reconstruction-inspired' ansatz with fitting parameters; not a solution of the reconstruction ODE.
  • domain assumption Barrow entropy form (Eq. 79)
    Borrowed from [75-78]; assumed for horizon thermodynamics.
  • domain assumption Apparent horizon radius R_A=1/H, temperature T=H/(2π)
    Standard apparent-horizon thermodynamics in FLRW cosmology.

pith-pipeline@v1.3.0-alltime-deepseek · 24889 in / 26255 out tokens · 237230 ms · 2026-08-02T06:25:18.772862+00:00 · methodology

0 comments
read the original abstract

In this work, we investigate an interacting viscous generalized QCD ghost dark energy model in the framework of reconstructed $f(G)$ gravity proposed in Phys.\ Lett.\ B 631, 1--6 (2005). The interaction between dark matter and dark energy together with bulk viscosity is incorporated to describe a more realistic cosmic evolution. A hybrid expansion law is adopted to reconstruct the modified Gauss-Bonnet function, which naturally connects the early matter-dominated epoch with the present accelerated expansion of the universe. Since an exact analytical reconstruction is difficult, the $f(G)$ function is obtained numerically in both the early- and late-time regimes. Motivated by the numerical reconstruction, a reconstruction-inspired power-law form of $f(G)$ is also considered to examine the cosmological implications of the model. The results show that the reconstructed $f(G)$ function evolves smoothly throughout the cosmic history, while the effective equation of state gradually approaches the de Sitter phase at late times. The thermodynamic behavior of the model is further examined using Barrow entropy following Eur.\ Phys.\ J.\ C 81, 644 (2021). The non-negative evolution of the total entropy shows the validity of the generalized second law of thermodynamics. The study finally concludes that the interacting viscous generalized QCD ghost dark energy model in reconstructed $f(G)$ gravity provides a viable and thermodynamically consistent framework for explaining the late-time accelerated expansion of the universe.

Figures

Figures reproduced from arXiv: 2607.13109 by Aziza Altaibayeva, Surajit Chattopadhyay, Ulbossyn Ualikhanova, Zhanar Umurzakhova.

Figure 1
Figure 1. Figure 1: Evolution of the equation of state parameter [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Reconstructed modified Gauss–Bonnet gravitational function [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of the reconstruction-inspired power-law Gauss–Bonnet model for [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the reconstruction-inspired power-law Gauss–Bonnet model for [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Time evolution of the effective equation of state parameter [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Three-dimensional evolution of the total entropy rate [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Three-dimensional evolution of the total entropy rate [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of the reconstructed Hubble parameter with the 31 cosmic [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

77 extracted references · 36 canonical work pages

  1. [1]

    Observational evidence from su- pernovae for an accelerating universe and a cosmological constant,

    A. G. Riess, A. V. Filippenko, P. Challiset al., “Observational evidence from su- pernovae for an accelerating universe and a cosmological constant,” Astron. J.116, 1009–1038 (1998). https://doi.org/10.1086/300499

    doi:10.1086/300499 1998

  2. [2]

    Measurements of Ω and Λ from 42 high-redshift supernovae,

    S. Perlmutter, G. Aldering, G. Goldhaberet al., “Measurements of Ω and Λ from 42 high-redshift supernovae,” Astrophys. J.517, 565–586 (1999). https://doi.org/10.1086/307221

    doi:10.1086/307221 1999

  3. [3]

    Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests,

    K. Bamba, S. Capozziello, S. Nojiri and S. D. Odintsov, “Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests,” As- trophys. Space Sci.342, 155–228 (2012). https://doi.org/10.1007/s10509-012-1181-8

    doi:10.1007/s10509-012-1181-8 2012

  4. [4]

    Dynamics of dark energy,

    E. J. Copeland, M. Sami and S. Tsujikawa, “Dynamics of dark energy,” Int. J. Mod. Phys. D15, 1753–1936 (2006). https://doi.org/10.1142/S021827180600942X

    doi:10.1142/s021827180600942x 1936

  5. [5]

    Modified gravity and cosmol- ogy,

    T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, “Modified gravity and cosmol- ogy,” Phys. Rep.513, 1–189 (2012). https://doi.org/10.1016/j.physrep.2012.01.001

    doi:10.1016/j.physrep.2012.01.001 2012

  6. [6]

    Cosmology intertwined: A re- view of the particle physics, astrophysics, and cosmology associated with the cos- mological tensions and anomalies,

    E. Abdalla, G. F. Abell´ an, A. Aboubrahimet al., “Cosmology intertwined: A re- view of the particle physics, astrophysics, and cosmology associated with the cos- mological tensions and anomalies,” J. High Energy Astrophys.34, 49–211 (2022). https://doi.org/10.1016/j.jheap.2022.04.002

    doi:10.1016/j.jheap.2022.04.002 2022

  7. [7]

    Is dark en- ergy an effective manifestation of non-equilibrium thermodynamics? – insights from DESI,

    S. Bhattacharjee, S. Halder, J. de Haro, S. Pan and E. N. Saridakis, “Is dark en- ergy an effective manifestation of non-equilibrium thermodynamics? – insights from DESI,” Class. Quantum Grav.43, 115005 (2026). https://doi.org/10.1088/1361- 6382/ae6f65

    doi:10.1088/1361- 2026

  8. [8]

    The quintom theory of dark energy after DESI DR2,

    Y. Cai, X. Ren, T. Qiu, M. Li and X. Zhang, “The quintom theory of dark energy after DESI DR2,” Natl. Sci. Rev.13, nwag115 (2026). https://doi.org/10.1093/nsr/nwag115

    doi:10.1093/nsr/nwag115 2026

  9. [9]

    Dynamical dark en- ergy models in light of the latest observations,

    J. de Cruz P´ erez, A. G´ omez-Valent and J. Sol` a Peracaula, “Dynamical dark en- ergy models in light of the latest observations,” Phys. Rev. D113, 083521 (2026). https://doi.org/10.1103/xchk-xlk1 29

    doi:10.1103/xchk-xlk1 2026

  10. [10]

    Quintessence scalar field and cosmological constant: dynamics of a multi-component dark energy model,

    P. Sahoo, N. Roy and H. S. Mondal, “Quintessence scalar field and cosmological constant: dynamics of a multi-component dark energy model,” Gen. Relativ. Gravit. 57, 38 (2025). https://doi.org/10.1007/s10714-025-03372-7

    work page doi:10.1007/s10714-025-03372-7 2025

  11. [11]

    Dy- namics of interacting dark energy and dark matter model in a curved FLR W scalar field cosmology,

    S. Pradhan, G. N. Gadbail, S. Mandal, P. K. Sahoo and K. Bamba, “Dy- namics of interacting dark energy and dark matter model in a curved FLR W scalar field cosmology,” Mon. Not. R. Astron. Soc.543, 1273–1287 (2025). https://doi.org/10.1093/mnras/staf1556

    work page doi:10.1093/mnras/staf1556 2025

  12. [12]

    Late-time acceleration and structure formation in interactingα-attractor dark energy models,

    L. K. Duchaniya, B. Mishra, G. Otalora and M. Gonzalez-Espinoza, “Late-time acceleration and structure formation in interactingα-attractor dark energy models,” JCAP12, 010 (2025). https://doi.org/10.1088/1475-7516/2025/12/010

    work page doi:10.1088/1475-7516/2025/12/010 2025

  13. [13]

    Bouncing Cosmology in Extended Gravity and Its Reconstruction as Dark Energy Model,

    A. S. Agrawal, F. Tello-Ortiz, B. Mishra and S. K. Tripathy, “Bouncing Cosmology in Extended Gravity and Its Reconstruction as Dark Energy Model,” Fortschr. Phys. 70, 2100065 (2022). https://doi.org/10.1002/prop.202100065

    work page doi:10.1002/prop.202100065 2022

  14. [14]

    Cosmology and astrophysical constraints of Gauss–Bonnet dark energy,

    T. Koivisto and D. F. Mota, “Cosmology and astrophysical constraints of Gauss–Bonnet dark energy,” Phys. Lett. B644, 104–108 (2007). https://doi.org/10.1016/j.physletb.2006.11.048

    work page doi:10.1016/j.physletb.2006.11.048 2007

  15. [16]

    Modified Gauss–Bonnet theory as grav- itational alternative for dark energy,

    S. Nojiri and S. D. Odintsov, “Modified Gauss–Bonnet theory as grav- itational alternative for dark energy,” Phys. Lett. B631, 1–6 (2005). doi:10.1016/j.physletb.2005.10.010

    doi:10.1016/j.physletb.2005.10.010 2005

  16. [17]

    Dark energy in modified Gauss–Bonnet gravity: Late-time acceleration and the hierarchy problem,

    G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov and S. Zerbini, “Dark energy in modified Gauss–Bonnet gravity: Late-time acceleration and the hierarchy problem,” Phys. Rev. D73, 084007 (2006). doi:10.1103/PhysRevD.73.084007

    work page doi:10.1103/physrevd.73.084007 2006

  17. [18]

    Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution,

    S. Nojiri, S. D. Odintsov and V. K. Oikonomou, “Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution,” Phys. Rept.692, 1–104 (2017). doi:10.1016/j.physrep.2017.06.001

    doi:10.1016/j.physrep.2017.06.001 2017

  18. [19]

    Barrow entropic dark energy: A member of generalized holographic dark energy family,

    S. Nojiri, S. D. Odintsov and T. Paul, “Barrow entropic dark energy: A member of generalized holographic dark energy family,” Phys. Lett. B825, 136844 (2022). doi:10.1016/j.physletb.2021.136844

    arXiv 2022

  19. [20]

    Early and late universe holographic cosmology from a new generalized entropy,

    S. Nojiri, S. D. Odintsov and T. Paul, “Early and late universe holographic cosmology from a new generalized entropy,” Phys. Lett. B831, 137189 (2022). doi:10.1016/j.physletb.2022.137189

    arXiv 2022

  20. [21]

    From nonextensive statistics and black hole entropy to the holographic dark universe,

    S. Nojiri, S. D. Odintsov and V. Faraoni, “From nonextensive statistics and black hole entropy to the holographic dark universe,” Phys. Rev. D105, 044042 (2022). doi:10.1103/PhysRevD.105.044042

    doi:10.1103/physrevd.105.044042 2022

  21. [22]

    Gauss–Bonnet gravitational baryogenesis,

    S. D. Odintsov and V. K. Oikonomou, “Gauss–Bonnet gravitational baryogenesis,” Phys. Lett. B760, 259–262 (2016). https://doi.org/10.1016/j.physletb.2016.06.074 30

    work page doi:10.1016/j.physletb.2016.06.074 2016

  22. [23]

    Modified Gauss–Bonnet gravity with the Lagrange multiplier constraint as mimetic theory,

    A. V. Astashenok, S. D. Odintsov and V. K. Oikonomou, “Modified Gauss–Bonnet gravity with the Lagrange multiplier constraint as mimetic theory,” Class. Quantum Grav.32, 185007 (2015). https://doi.org/10.1088/0264-9381/32/18/185007

    work page doi:10.1088/0264-9381/32/18/185007 2015

  23. [24]

    Viable inflation in scalar–Gauss–Bonnet gravity and reconstruction from observational indices,

    S. D. Odintsov and V. K. Oikonomou, “Viable inflation in scalar–Gauss–Bonnet gravity and reconstruction from observational indices,” Phys. Rev. D98, 044039 (2018). https://doi.org/10.1103/PhysRevD.98.044039

    doi:10.1103/physrevd.98.044039 2018

  24. [25]

    Ghost-free Gauss– Bonnet theories of gravity,

    S. Nojiri, S. D. Odintsov and V. K. Oikonomou, “Ghost-free Gauss– Bonnet theories of gravity,” Phys. Rev. D99, 044050 (2019). https://doi.org/10.1103/PhysRevD.99.044050

    work page doi:10.1103/physrevd.99.044050 2019

  25. [26]

    Inflationary dynamics and swampland criteria for modified Gauss–Bonnet gravity compatible with GW170817,

    S. D. Odintsov, V. K. Oikonomou and F. P. Fronimos, “Inflationary dynamics and swampland criteria for modified Gauss–Bonnet gravity compatible with GW170817,” Phys. Rev. D107, 084007 (2023). https://doi.org/10.1103/PhysRevD.107.084007

    work page doi:10.1103/physrevd.107.084007 2023

  26. [27]

    Cosmology of modified Gauss–Bonnet gravity,

    B. Li, J. D. Barrow and D. F. Mota, “Cosmology of modified Gauss–Bonnet gravity,” Phys. Rev. D76, 044027 (2007). https://doi.org/10.1103/PhysRevD.76.044027

    work page doi:10.1103/physrevd.76.044027 2007

  27. [28]

    Construction of cosmologically viablef(G) gravity models,

    A. De Felice and S. Tsujikawa, “Construction of cosmologically viablef(G) gravity models,” Phys. Lett. B675, 1–8 (2009). https://doi.org/10.1016/j.physletb.2009.03.060

    work page doi:10.1016/j.physletb.2009.03.060 2009

  28. [29]

    Exploring cylindrical solutions in modifiedf(G) gravity,

    M. J. S. Houndjo, M. E. Rodrigues, D. Momeni and R. Myrzakulov, “Exploring cylindrical solutions in modifiedf(G) gravity,” Can. J. Phys.92, 1528–1540 (2014). https://doi.org/10.1139/cjp-2014-0070

    work page doi:10.1139/cjp-2014-0070 2014

  29. [30]

    Sta- bility of anisotropy pressure in self-gravitational systems inf(G) gravity,

    Z. Yousaf, M. Z. Bhatti, S. Khan, A. Malik, H. I. Alrebdi and M. Z. Ahmad, “Sta- bility of anisotropy pressure in self-gravitational systems inf(G) gravity,” Axioms 12, 740 (2023). https://doi.org/10.3390/axioms12030257

    work page doi:10.3390/axioms12030257 2023

  30. [31]

    f(G) modified gravity and the energy conditions,

    N. M. Garc ´ ıa, F. S. N. Lobo, J. P. Mimoso and T. Harko, “f(G) modified gravity and the energy conditions,” J. Phys. Conf. Ser.314, 012056 (2011). https://doi.org/10.1088/1742-6596/314/1/012056

    work page doi:10.1088/1742-6596/314/1/012056 2011

  31. [32]

    Ghost dark energy model inf(G) gravity,

    M. Sharif and S. Saba, “Ghost dark energy model inf(G) gravity,” Chin. J. Phys. 58, 202–211 (2019). https://doi.org/10.1016/j.cjph.2018.12.023

    work page doi:10.1016/j.cjph.2018.12.023 2019

  32. [33]

    S. R. Green and R. M. Wald, How well is our universe described by an FLR W model?, Class. Quant. Grav.31, 234003 (2014)

    2014

  33. [34]

    Holman, How problematic is the near-Euclidean spatial geometry of the large- scale universe?, Found

    M. Holman, How problematic is the near-Euclidean spatial geometry of the large- scale universe?, Found. Phys.48, 1617–1647 (2018)

    2018

  34. [35]

    Koussour, S

    M. Koussour, S. H. Shekh, A. Hanin, Z. Sakhi, S. R. Bhoyer and M. Bennai, Flat FLR W universe in logarithmic symmetric teleparallel gravity with observational con- straints, Class. Quant. Grav.39, 195021 (2022)

    2022

  35. [36]

    Calcagni, B

    G. Calcagni, B. de Carlos and A. De Felice, Ghost conditions for Gauss–Bonnet cosmologies, Nucl. Phys. B752, 404–438 (2006). https://doi.org/10.1016/j.nuclphysb.2006.06.020 31

    work page doi:10.1016/j.nuclphysb.2006.06.020 2006

  36. [37]

    The 4D Einstein– Gauss–Bonnet theory of gravity: a review,

    P. G. S. Fernandes, P. Carrilho, T. Clifton and D. J. Mulryne, “The 4D Einstein– Gauss–Bonnet theory of gravity: a review,” Class. Quantum Grav.39, 063001 (2022). https://doi.org/10.1088/1361-6382/ac500a

    doi:10.1088/1361-6382/ac500a 2022

  37. [38]

    Mandal and S

    S. Mandal and S. Shankaranarayanan, Modified theories of gravity at different cur- vature scales, arXiv:2502.07437 (2025)

    Pith/arXiv arXiv 2025

  38. [40]

    The cosmological constant from the QCD Veneziano ghost,

    F. R. Urban and A. R. Zhitnitsky, “The cosmological constant from the QCD Veneziano ghost,” Phys. Lett. B688, 9–12 (2010). https://doi.org/10.1016/j.physletb.2010.03.080

    work page doi:10.1016/j.physletb.2010.03.080 2010

  39. [41]

    R. G. Cai, Z. L. Tuo, Y. B. Wu and Y. Y. Zhao, More on QCD ghost dark energy, Phys. Rev. D86, 023511 (2012). doi:10.1103/PhysRevD.86.023511

    work page doi:10.1103/physrevd.86.023511 2012

  40. [42]

    Sharif and M

    M. Sharif and M. Ajmal, Cosmography of non-interacting ghost and generalized ghost dark energy models inf(Q) gravity, Commun. Theor. Phys.77, 045402 (2025). https://doi.org/10.1088/1572-9494/ad8db8 arXiv:2507.02280 [gr-qc]

    Pith/arXiv arXiv 2025

  41. [43]

    Khurshudyan, A

    M. Khurshudyan, A. Khurshudyan and R. Myrzakulov, Interacting varying ghost dark energy models in general relativity, Astrophys. Space Sci.357, 113 (2015). doi:10.1007/s10509-015-2341-4

    work page doi:10.1007/s10509-015-2341-4 2015

  42. [44]

    Gauss-Bonnet dark energy,

    S. Nojiri, S. D. Odintsov and M. Sasaki, “Gauss-Bonnet dark energy,” Phys. Rev. D 71, 123509 (2005). https://doi.org/10.1103/PhysRevD.71.123509

    doi:10.1103/physrevd.71.123509 2005

  43. [45]

    Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models,

    S. Nojiri and S. D. Odintsov, “Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models,” Phys. Rept.505, 59-144 (2011). https://doi.org/10.1016/j.physrep.2011.04.001

    doi:10.1016/j.physrep.2011.04.001 2011

  44. [46]

    Extended Theories of Gravity,

    S. Capozziello and M. De Laurentis, “Extended Theories of Gravity,” Phys. Rept. 509, 167-321 (2011). https://doi.org/10.1016/j.physrep.2011.09.003

    doi:10.1016/j.physrep.2011.09.003 2011

  45. [47]

    JCAP06, 010 (2008) https://iopscience.iop.org/article/10.1088/1475-7516/2008/06/010/meta

    He, J.H., Wang, B.: Effects of the interaction between dark energy and dark matter on cosmological parameters. JCAP06, 010 (2008) https://iopscience.iop.org/article/10.1088/1475-7516/2008/06/010/meta

    work page doi:10.1088/1475-7516/2008/06/010/meta 2008

  46. [48]

    Wang, B., Abdalla, E., Atrio-Barandela, F., Pavon, D.: Dark mat- ter and dark energy interactions: theoretical challenges, cosmological impli- cations and observational signatures. Rep. Prog. Phys.79, 096901 (2016) https://iopscience.iop.org/article/10.1088/0034-4885/79/9/096901/meta

    work page doi:10.1088/0034-4885/79/9/096901/meta 2016

  47. [49]

    Wang, J.S., Wang, F.Y.: Cosmological model of the interaction be- tween dark matter and dark energy. Astron. Astrophys.564, A137 (2014) https://doi.org/10.1051/0004-6361/201322606

    work page doi:10.1051/0004-6361/201322606 2014

  48. [50]

    Hu, B., Ling, Y.: Interacting dark energy, holographic princi- ple, and coincidence problem. Phys. Rev. D73, 123510 (2006) https://doi.org/10.1103/PhysRevD.73.123510 32

    work page doi:10.1103/physrevd.73.123510 2006

  49. [51]

    Sadjadi, H.M., Alimohammadi, M.: Cosmological coincidence problem in interacting dark energy models. Phys. Rev. D74, 103007 (2006) https://doi.org/10.1103/PhysRevD.74.103007

    doi:10.1103/physrevd.74.103007 2006

  50. [52]

    Caldera-Cabral, G., Maartens, R., Urena-Lopez, L.A.: Dynam- ics of interacting dark energy. Phys. Rev. D79, 063518 (2009) https://doi.org/10.1103/PhysRevD.79.063518

    doi:10.1103/physrevd.79.063518 2009

  51. [53]

    Berger, M.S., Shojaei, H.: Interacting dark energy and the cosmic coincidence prob- lem. Phys. Rev. D74, 043530 (2006) https://doi.org/10.1103/PhysRevD.73.083528

    work page doi:10.1103/physrevd.73.083528 2006

  52. [54]

    Avelino, P.P., Gomes, A.R., Tamayo, D.A.: Note on bulk viscosity as an alternative to dark energy. Phys. Rev. D111, 083537 (2025) https://doi.org/10.1103/2kf6-413m

    work page doi:10.1103/2kf6-413m 2025

  53. [55]

    Viscous cosmol- ogy for early- and late-time universe,

    I. Brevik, Ø. Grøn, J. de Haro, S.D. Odintsov and E.N. Saridakis, “Viscous cosmol- ogy for early- and late-time universe,” Int. J. Mod. Phys. D26, 1730024 (2017). https://doi.org/10.1142/S0218271817300245

    doi:10.1142/s0218271817300245 2017

  54. [56]

    Viscous cosmology inf(T) gravity,

    J. Yang, R.H. Lin and X.H. Zhai, “Viscous cosmology inf(T) gravity,” Eur. Phys. J. C82, 45 (2022). https://doi.org/10.1140/epjc/s10052-022-11008-2

    work page doi:10.1140/epjc/s10052-022-11008-2 2022

  55. [57]

    Viscous cosmology in new holographic dark en- ergy model and the cosmic acceleration,

    C.P. Singh and M. Srivastava, “Viscous cosmology in new holographic dark en- ergy model and the cosmic acceleration,” Eur. Phys. J. C78, 741 (2018). https://doi.org/10.1140/epjc/s10052-018-5683-6

    work page doi:10.1140/epjc/s10052-018-5683-6 2018

  56. [58]

    Holographic representation of the unified early and late universe via a viscous dark fluid,

    I. Brevik and A.V. Timoshkin, “Holographic representation of the unified early and late universe via a viscous dark fluid,” Int. J. Geom. Methods Mod. Phys.19, 2250186 (2022). https://doi.org/10.1142/S0219887822501134

    work page doi:10.1142/s0219887822501134 2022

  57. [59]

    Extracting the scaling exponents of a self-affine, non-Gaussian process from a finite-length time series,

    K. Kiyani, S. C. Chapman and B. Hnat, “Extracting the scaling exponents of a self-affine, non-Gaussian process from a finite-length time series,” Phys. Rev. E74, 051122 (2006). https://doi.org/10.1103/PhysRevE.74.051122

    work page doi:10.1103/physreve.74.051122 2006

  58. [60]

    Cosmology Due to Thermodynamics of Apparent Horizon,

    S. I. Kruglov, “Cosmology Due to Thermodynamics of Apparent Horizon,” Ann. Phys. (Berlin)537, 2500204 (2025). https://doi.org/10.1002/andp.202500204

    doi:10.1002/andp.202500204 2025

  59. [61]

    Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FR W universe,

    M. Akbar and R.-G. Cai, “Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FR W universe,” Phys. Rev. D75, 084003 (2007). https://doi.org/10.1103/PhysRevD.75.084003

    doi:10.1103/physrevd.75.084003 2007

  60. [62]

    Hawking radiation of an appar- ent horizon in a FR W universe,

    R.-G. Cai, L.-M. Cao and Y.-P. Hu, “Hawking radiation of an appar- ent horizon in a FR W universe,” Class. Quantum Grav.26, 155018 (2009). https://doi.org/10.1088/0264-9381/26/15/155018

    work page doi:10.1088/0264-9381/26/15/155018 2009

  61. [63]

    Thermodynamics of the FLR W apparent horizon,

    L. M. S´ anchez and H. Quevedo, “Thermodynamics of the FLR W apparent horizon,” Phys. Lett. B843, 137778 (2023). https://doi.org/10.1016/j.physletb.2023.137778

    arXiv 2023

  62. [64]

    Unified first law and the thermodynamics of the apparent horizon in the FR W universe,

    R.-G. Cai and L.-M. Cao, “Unified first law and the thermodynamics of the apparent horizon in the FR W universe,” Phys. Rev. D75, 064008 (2007). https://doi.org/10.1103/PhysRevD.75.064008 33

    work page doi:10.1103/physrevd.75.064008 2007

  63. [65]

    Thermodynamical properties of appar- ent horizon in warped DGP braneworld,

    A. Sheykhi, B. Wang and R.-G. Cai, “Thermodynamical properties of appar- ent horizon in warped DGP braneworld,” Nucl. Phys. B779, 1–12 (2007). https://doi.org/10.1016/j.nuclphysb.2007.04.028

    work page doi:10.1016/j.nuclphysb.2007.04.028 2007

  64. [66]

    Bekenstein–Hawking entropy,

    J. D. Bekenstein, “Bekenstein–Hawking entropy,” Scholarpedia3(10), 7375 (2008). https://doi.org/10.4249/scholarpedia.7375

    doi:10.4249/scholarpedia.7375 2008

  65. [67]

    Generalized second law of thermodynamics in black-hole physics,

    J. D. Bekenstein, “Generalized second law of thermodynamics in black-hole physics,” Phys. Rev. D9, 3292–3300 (1974). https://doi.org/10.1103/PhysRevD.9.3292

    doi:10.1103/physrevd.9.3292 1974

  66. [68]

    Dark energy and the generalized second law,

    G. Izquierdo and D. Pav´ on, “Dark energy and the generalized second law,” Phys. Lett. B633, 420–426 (2006). https://doi.org/10.1016/j.physletb.2005.12.040

    doi:10.1016/j.physletb.2005.12.040 2006

  67. [69]

    The generalized second law of thermodynamics in generalized gravity theories,

    S. F. Wu, B. Wang, G. H. Yang and P. M. Zhang, “The generalized second law of thermodynamics in generalized gravity theories,” Class. Quantum Grav.25, 235018 (2008). https://doi.org/10.1088/0264-9381/25/23/235018

    work page doi:10.1088/0264-9381/25/23/235018 2008

  68. [70]

    Ten proofs of the generalized second law,

    A. C. Wall, “Ten proofs of the generalized second law,” JHEP06, 021 (2009). https://doi.org/10.1088/1126-6708/2009/06/021

    work page doi:10.1088/1126-6708/2009/06/021 2009

  69. [71]

    Thermodynamics in closed universe with entropy corrections,

    M. Sharif and A. Jawad, “Thermodynamics in closed universe with entropy corrections,” Int. J. Mod. Phys. D22, 1350057 (2013). https://doi.org/10.1142/S0218271813500144

    work page doi:10.1142/s0218271813500144 2013

  70. [72]

    Apparent horizon and gravitational thermody- namics of the universe: Solutions to the temperature and entropy confu- sions and extensions to modified gravity,

    D. W. Tian and I. Booth, “Apparent horizon and gravitational thermody- namics of the universe: Solutions to the temperature and entropy confu- sions and extensions to modified gravity,” Phys. Rev. D92, 024001 (2015). https://doi.org/10.1103/PhysRevD.92.024001

    doi:10.1103/physrevd.92.024001 2015

  71. [73]

    Effects of the interaction between dark en- ergy and dark matter on cosmological parameters,

    J. H. He and B. Wang, “Effects of the interaction between dark en- ergy and dark matter on cosmological parameters,” JCAP06, 010 (2008). https://doi.org/10.1088/1475-7516/2008/06/010

    doi:10.1088/1475-7516/2008/06/010 2008

  72. [74]

    Cosmological implications of an interacting model of dark matter & dark energy,

    K. R. Mishra, S. K. J. Pacif, R. Kumar and K. Bamba, “Cosmological implications of an interacting model of dark matter & dark energy,” Phys. Dark Univ.42, 101211 (2023). https://doi.org/10.1016/j.dark.2023.101211

    arXiv 2023

  73. [75]

    Big Bang Nucleosyn- thesis constraints on Barrow entropy,

    J. D. Barrow, S. Basilakos, and E. N. Saridakis, “Big Bang Nucleosyn- thesis constraints on Barrow entropy,” Phys. Lett. B815, 136134 (2021). doi:10.1016/j.physletb.2021.136134

    arXiv 2021

  74. [76]

    Modified cosmology through spacetime thermodynamics and Bar- row horizon entropy,

    E. N. Saridakis, “Modified cosmology through spacetime thermodynamics and Bar- row horizon entropy,” J. Cosmol. Astropart. Phys.07, 031 (2020). doi:10.1088/1475- 7516/2020/07/031

    doi:10.1088/1475- 2020

  75. [77]

    Barrow holographic dark energy,

    E. N. Saridakis, “Barrow holographic dark energy,” Phys. Rev. D102, 123525 (2020). doi:10.1103/PhysRevD.102.123525

    doi:10.1103/physrevd.102.123525 2020

  76. [78]

    The generalized second law of thermodynamics with Barrow entropy,

    E. N. Saridakis and S. Basilakos, “The generalized second law of thermodynamics with Barrow entropy,” Eur. Phys. J. C81, 644 (2021). doi:10.1140/epjc/s10052-021- 09431-y. 34

    doi:10.1140/epjc/s10052-021- 2021

  77. [79]

    H. Yu, B. Ratra, and F.-Y. Wang, “Hubble parameter and baryon acoustic oscillation measurement constraints on the Hubble constant, the deviation from the spatially flat ΛCDM model, the deceleration–acceleration transition redshift, and spatial cur- vature,” Astrophys. J.856(1), 3 (2018). doi:10.3847/1538-4357/aab0a2. 35

    doi:10.3847/1538-4357/aab0a2 2018