Quintessence dark energy model in non-linear $f(Q)$ theory with bulk-viscosity

Dinesh Chandra Maurya

REVIEW 2 major objections 5 minor 138 references

Quintessence dark energy model in non-linear $f(Q)$ theory with bulk-viscosity

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A quadratic $f(Q)$-gravity model with bulk viscosity, fitted to CC, BAO, and Pantheon data, yields an ever-accelerating quintessence universe with $H_0 \approx 68$ km/s/Mpc, $q_0 \approx -0.32$, and an age near 13.8 Gyr.

desk verdict The MCMC constraints in this paper do not apply to the model's field equations: the key reduction drops nonzero terms and the fitted H(z) contradicts the solved H(t). read the letter →

arxiv 2506.02543 v2 pith:WHTZZQYG submitted 2025-06-03 gr-qc

Dinesh Chandra Maurya This is my paper

classification gr-qc PACS 98.80-k98.80.Jk04.50.Kd

keywords f(Q)gravityLRSBianchitype-IbulkviscosityquintessencedarkenergyHubbleparameterOmdiagnosticstatefinderobservationalconstraints

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 tries to establish that a quadratic version of $f(Q)$ gravity — gravity sourced by non-metricity rather than curvature — combined with a bulk-viscous fluid can reproduce late-time cosmic acceleration without adding a cosmological constant. The author solves the modified Einstein equations in an LRS Bianchi type-I spacetime (a homogeneous but anisotropic geometry with two equal spatial directions) under the viscosity ansatz $\xi = \xi_1 \dot{H} - \xi_0$, obtaining a hyperbolic Hubble law, and constrains the free parameters with Markov-chain Monte Carlo fits to cosmic-chronometer, BAO, and Pantheon supernova data. With the fitted parameters, the model gives $H_0 \approx 68$ km/s/Mpc, $q_0 \approx -0.32$, an effective equation of state $\omega_{\rm eff} \approx -0.55$, and a current age near 13.8 Gyr, and its Om diagnostic has a negative slope, which classifies the dark energy as quintessence. If the derivation holds, this is a no-$\Lambda$ route to an ever-accelerating universe that converges toward the $\Lambda$CDM behavior at late times.

What carries the argument

The load-bearing object is the hyperbolic solution $H(t) = k_0 \coth(k_1 t + c_0)$, obtained from Eq. (35), $\dot{H} + \frac{36\alpha(2m+1)}{36\alpha(2m+1)+\xi_1(m+2)^2} H^2 - \frac{\xi_0(m+2)^2}{36\alpha(2m+1)+\xi_1(m+2)^2} = 0$. That equation comes from substituting the anisotropy ansatz $A = B^m$ (equivalently $A = a^{3m/(m+2)}$, $B = a^{3/(m+2)}$), the quadratic $f(Q) = -\alpha Q^2$, the non-metricity scalar $Q = -\frac{18(2m+1)}{(m+2)^2} H^2$, and the viscosity law $\xi = \xi_1\dot{H} - \xi_0$ into the difference of the pressure field equations. This one equation fixes the Hubble law, and through it every derived quantity: the deceleration parameter, effective equation of state, cosmographic coefficients, age integral, Om function, and statefinder pair.

What would settle it

Plug the claimed solution $H(t) = k_0\coth(k_1t + c_0)$ back into the original field equations (17)-(19) with $f(Q) = -\alpha Q^2$, and evaluate the difference equation (31) numerically at the best-fit parameters. If the residual is not identically zero (or numerically far below the fitted $H^2$ terms), the central claim fails at the level of the field equations; a symbolic computer-algebra check of the step from Eq. (31) to Eq. (35) gives the same verdict.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a closed-form viscous solution in non-linear $f(Q)$ theory. Choosing $f(Q) = -\alpha Q^2$ and $\xi = \xi_1\dot{H} - \xi_0$ in an LRS Bianchi type-I metric with the anisotropy relation $A = B^m$ reduces the difference of the pressure field equations to a first-order equation for the average Hubble parameter, whose solution is $H(t) = k_0 \coth(k_1 t + c_0)$ and whose scale factor is $a(t) = c_1[\sinh(k_1 t + c_0)]^n$. With the constrained parameters, the deceleration parameter stays between $-1$ and $0$ at all redshifts, the effective equation of state stays inside the quintessence window, the Om diagnostic has a negative slope, and the statefinder and cosmographic curves move toward the $\Lambda$CDM point at late times. The paper reads these as evidence of a quintessence, ever-accelerating, mildly anisotropic dark-energy model produced by non-linear non-metricity gravity plus bulk viscosity.

Load-bearing premise

Everything hinges on the algebraic reduction from Eq. (31) to Eq. (35): after substituting $A = a^{3m/(m+2)}$, $B = a^{3/(m+2)}$, $f(Q) = -\alpha Q^2$, and $\xi = \xi_1\dot H - \xi_0$, the leftover terms proportional to $(m-1)H^2\dot H$ and $(m-1)H^4$ must cancel; if they do not, the $\coth$ Hubble law and every number derived from it do not describe the proposed model.

Editorial extensions

If this is right

The three fits (CC, CC+BAO, CC+Pantheon) return $H_0 = 68.2 \pm 1.3$, $68.11 \pm 0.52$, and $68.4 \pm 1.6$ km/s/Mpc, so if the model is right, the expansion rate today sits near the lower side of the measured range.
$q(z)$ is negative over the whole fitted range with $q_0 \approx -0.32$, and the effective equation of state obeys $-1 \le \omega_{\rm eff} < -1/3$, so the model universe is ever-accelerating rather than a transient-transition model.
The negative Om slope and the late-time approach of the statefinder pair toward $(r, s) = (1, 0)$ classify the dark energy as quintessence converging to $\Lambda$CDM.
The age integral gives $t_0$ between 13.81 and 13.89 Gyr across the three datasets, close to the conventional age of the universe.
The anisotropy parameter is small ($\Delta \approx 0.00002$ to $0.0007$), so the anisotropic background remains observationally close to a flat, isotropic universe.

Reading between the lines

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

Editorial inference: the whole parameter set inherits the algebraic step from Eq. (31) to Eq. (35); a user of these numbers should first verify that the substitution leaves no uncancelled $H^2\dot{H}$ or $H^4$ terms, since that check is more direct than re-running the fits.
Editorial inference: because the fitted $H_0$ lands near 68 km/s/Mpc rather than the higher local-distance-ladder value, this class of viscous $f(Q)$ models is a natural ingredient in Hubble-tension discussions, although the paper itself does not quantify that comparison.
Editorial inference: fitting the same model to independent high-redshift expansion data would test whether the $\coth$ Hubble law remains viable beyond the redshifts covered by the current $H(z)$ and Pantheon samples.

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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

Quick take: the MCMC results and all diagnostics in arXiv:2506.02543 are not constraints on the model defined by the field equations. The reduction from Eq. (31) to Eq. (35) drops nonzero terms, and the H(z) used in the likelihood, Eq. (38), is not the redshift form of the solved H(t), Eq. (36).

To its credit, the paper is a readable example of a standard program: LRS Bianchi I, f(Q) = αQ², bulk viscosity ξ = ξ₁Ḣ − ξ₀, a hyperbolic solution, and MCMC with CC, BAO, and Pantheon, followed by Om, statefinder, and cosmographic analyses. The data handling looks standard, including CC covariance, and the presentation is mostly clear. That is about the extent of the novelty; the same author's prior works [66–68] already present anisotropic bulk-viscous f(Q) quintessence models with the same qualitative conclusions.

The load-bearing step is wrong. Substituting A = a^{3m/(m+2)}, B = a^{3/(m+2)}, f(Q) = −αQ², and ξ = ξ₁Ḣ − ξ₀ into Eq. (31), and using Q = −18(2m+1)H²/(m+2)², gives an equation with terms proportional to HḢ and H³—specifically 3cHḢ + 3cH³ + 3ξ₁Ḣ − 3ξ₀ = 0 with c = 36α(2m+1)/(m+2)²—not the linear ODE in Eq. (35). The hyperbolic solution H = k₀coth(k₁t + c₀) solves (35) by construction; it does not solve (31). So the fitted parameters sit on a function that is not a solution of the stated field equations.

Separately, Eq. (38) is not the redshift transform of Eq. (36). The scale factor (37) gives H(t) = k₀coth(k₁t + c₀), which grows as a power of (1+z) at high z, whereas Eq. (38) decays to zero. The H(z) used in the likelihood is a different function, so the chains and all derived quantities (q₀, ω_v, Om slope, age) do not speak to the solved model.

Also worth noting: those diagnostics are computed after fitting H₀, α, m, ξ₁, and c₁, so they are restatements of the fitted curve rather than independent predictions. The 'ever-accelerating quintessence' conclusion is baked into the hyperbolic ansatz.

Bottom line: I would not send this to a referee. The central derivation fails, the likelihood uses an inconsistent H(z), and the qualitative claims repeat earlier work. Desk reject. If the algebra is fixed and the fit rerun with a consistent H(z), the model might become a routine example in the genre, but as it stands the constraints are about an equation that is not the model.

Referee Report

2 major / 5 minor

Summary. The paper constructs an LRS Bianchi type-I cosmological model in quadratic f(Q) gravity with a bulk-viscous fluid whose viscosity coefficient is taken as ξ(t)=ξ1 Ḣ−ξ0. The author claims to derive a hyperbolic Hubble solution H(t)=k0 coth(k1t+c0) from the difference of the modified field equations, then uses MCMC on CC, BAO, and Pantheon datasets to constrain the parameters H0, ξ1, m, α. From the fitted function the paper reports an ever-accelerating quintessence behavior (q0≈−0.32, ωeff≈−0.55, ωv≈−0.46), a present age ≈13.8 Gyr, and negative-slope Om diagnostic, along with statefinder and cosmographic analyses.

Significance. If the derivations were correct, the paper would offer a new anisotropic viscous dark-energy model in non-linear f(Q) gravity and demonstrate its consistency with late-time cosmological observations. The manuscript includes the standard ingredients of such an analysis: exact field equations, explicit ansätze, MCMC implementation, and comparison with CC, BAO, and Pantheon data. However, the central step from Eq. (31) to Eq. (35) is algebraically wrong, and the subsequent H(z) formula (38) is inconsistent with the solved scale factor. These are load-bearing errors, not presentation issues: they invalidate the fitted model and all derived cosmological diagnostics. Because the core claim—that this model fits the data and behaves as quintessence—rests entirely on these steps, the paper cannot be accepted in its current form. The negative slope of the Om diagnostic and the quintessence classification are direct artifacts of the erroneous H(z).

major comments (2)

[Section 3, Eqs. (31)–(35)] The reduction leading to Eq. (35) is algebraically incorrect. Substituting A=a^{3m/(m+2)}, B=a^{3/(m+2)}, f(Q)=−αQ^2, Q=−18(2m+1)/(m+2)^2 H^2, and ξ=ξ1 Ḣ−ξ0 into Eq. (31) gives, after dividing by the non-zero factor 3(m−1)/(m+2)H, the equation 108α(2m+1)/(m+2)^2 H Ḣ + 108α(2m+1)(m^2+m−1)/((m−1)(m+2)^3) H^3 + 3ξ1 Ḣ − 3ξ0 = 0. This contains terms proportional to H Ḣ and H^3 that do not appear in Eq. (35), and these terms do not vanish for the best-fit values (e.g., m≈1.05, α≈1.09 in the CC case). Consequently the hyperbolic solution (36) solves Eq. (35) but does not solve Eq. (31), so the solution (36)–(37) is not a solution of the modified field equations actually defined by the paper.
[Section 3, Eq. (38)] The Hubble function H(z) in Eq. (38) is not the redshift-space form of the scale factor (37). From a(t)=c1 sinh^n(k1t+c0) one obtains H(t)=k0 coth(k1t+c0), which behaves as H ∝ a^{−1/n} ∝ (1+z)^{1/n} at high redshift (early times). In contrast, Eq. (38) gives H(z) ∝ (1+z)^{−1/n}, decaying to zero as z→∞. Thus Eq. (38) represents a different model, and the MCMC fits in Section 4, which use Eq. (38) in the χ² definitions (43)–(44) and (50)–(51), constrain parameters of that spurious function rather than the model derived in Section 3. All parameter constraints in Table 2 and the subsequent cosmological conclusions are therefore invalid.

minor comments (5)

[Section 5.5, Eqs. (65)–(67)] The reported values σ/θ ≈ −0.010582 and σ/H ≈ −0.031748 for the CC+BAO dataset are negative, whereas σ, θ, and H are non-negative quantities; the expression should use the absolute value |m−1|/(m+2).
[Section 5.2, Eq. (60)] The conversion factor '978' in Eq. (60) is not defined or justified; moreover this age formula inherits the incorrect H(z) from Eq. (38), so the quoted age t0≈13.8 Gyr is not a property of the solved model.
[Section 4, Table 2 and Figs. 1–2] The parameter c1 is fixed to 1.5 rather than marginalized or fitted, although c1 appears explicitly in H(z) in Eq. (38) and controls the redshift dependence of the model; its arbitrary choice influences the fitted values of H0, ξ1, m, and α.
[Throughout] Several typos and grammatical issues remain, e.g., 'Binachi' for 'Bianchi' and 'Tise condition' for 'This condition' in Section 3; these should be corrected in any revision.
[Section 5.4, Om diagnostic] The negative slope of Om(z) is a direct mathematical consequence of the incorrect H(z) in Eq. (38), which decreases with z; this diagnostic should not be reported as a prediction of the model until the H(z) error is fixed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported diagnostics are functions of the fitted model, not inputs to the fit, and the self-citations are not load-bearing.

full rationale

The derivation chain is: field equations in f(Q) gravity, the ansatz f(Q) = -alpha Q^2, the anisotropy relation A = B^m, the bulk-viscosity assumption xi = xi1 Hdot - xi0, the resulting first-order ODE for H(t), the hyperbolic solution, the H(z) expression, and then MCMC fits to CC, BAO, and Pantheon data. The quantities highlighted as 'results' (q0, omega_eff, Omega_m slope, age, statefinder pairs) are all evaluated from the best-fit H(z) after the parameters H0, xi1, m, alpha and M have been fitted. That is a standard model-interpretation step, not a circular one: those diagnostics are not used to define the model or to determine the fitted parameters, and no fitted parameter is renamed as an independent prediction. The model does impose a fixed functional form before fitting, so the diagnostics are constrained by that form, but that is a limitation of the model-building approach rather than circular reasoning. The self-citations to the author's previous f(Q) and bulk-viscosity papers are background references and are not used as load-bearing uniqueness theorems or as justification for the core equations; the core assumptions (quadratic f(Q) and linear-in-Hdot bulk viscosity) are stated explicitly in the paper itself. Note that the algebraic step from Eq. (31) to Eq. (35) appears not to close: substituting the anisotropic relations and the quadratic f(Q) into Eq. (31) yields terms proportional to H Hdot and H^3 that are absent from Eq. (35). This is a correctness or consistency flaw, not a circularity, and therefore does not raise the circularity score. The paper is self-contained against its external H(z) and Pantheon data in the sense that its claimed numerical outputs are not assumed in advance of the fit.

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

The model introduces no new particles or forces; it relies on known bulk viscosity and a chosen f(Q) Lagrangian. The five fitted or hand-assigned parameters (H0, xi1, m, alpha, c1, plus xi0 implicitly and M for Pantheon) drive all derived diagnostics, so the physical content beyond the ansatz is limited.

free parameters (7)

H0 = 68.2 +/- 1.3 (CC), 68.11 +/- 0.52 (CC+BAO), 68.4 +/- 1.6 (CC+Pantheon)
Present Hubble constant, fitted by MCMC in Eq. (38). Central normalization of all derived quantities.
xi1 = 0.166, 0.0047, 0.183
Bulk viscosity coefficient in xi=xi1 Hdot-xi0; fitted by MCMC.
m = 1.05, 0.946, 1.01
Anisotropy exponent in A=B^m; fitted by MCMC.
alpha = 1.09, 1.26, 0.96
Coefficient of the quadratic f(Q)=-alpha Q^2; fitted by MCMC.
c1 = 1.5 (fixed by hand)
Integration constant in the scale factor; fixed to 1.5, not marginalized. Affects all derived parameter values.
xi0 = not directly reported
Constant part of the bulk viscosity coefficient; degenerate with H0, alpha, m and c1 through k0. Not independently constrained.
M = 23.8477 +/- 0.0051
Nuisance absolute magnitude parameter fitted for the Pantheon distance modulus.

assumptions (5)

domain assumption Shear is proportional to expansion scalar, giving A=B^m with m != 1.
Used to close the system of field equations in Section 3. Cited to Collins et al., but it is a modeling assumption about this spacetime.
ad hoc to paper Bulk viscosity takes the form xi=xi1 Hdot-xi0.
Chosen in Eq. (34) so that Eq. (35) becomes integrable; no thermodynamic derivation is provided.
ad hoc to paper f(Q) is exactly quadratic, f(Q)=-alpha Q^2.
Selected in Eq. (30) to make the field equations tractable; not derived from a deeper principle.
domain assumption LRS Bianchi type-I metric Eq. (13) describes the spacetime.
Assumes a homogeneous but anisotropic geometry; the paper does not derive this from observations.
standard math The connection is torsion-free and matter has no hypermomentum.
Adopts the symmetric teleparallel f(Q) setup of [30], meaning the connection field equations reduce to Eq. (11).

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Cite this review

Pith. "Pith review of Quintessence dark energy model in non-linear $f(Q)$ theory with bulk-viscosity." pith.science (2026). https://pith.science/paper/WHTZZQYG

@misc{pith2026250602543,
  author       = {Pith},
  title        = {Pith review of: Quintessence dark energy model in non-linear $f(Q)$ theory with bulk-viscosity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WHTZZQYG}},
  note         = {Machine review of arXiv:2506.02543}
}

abstract

In this study, we investigate a locally rotationally symmetric (LRS) Bianchi type-I cosmological model in non-linear form of $f(Q)$ gravity with observational constraints. We solved the modified Einstein's field equations with a viscous fluid source and got a hyperbolic solution. First, we apply MCMC analysis to the cosmic chronometer (CC), Baryon Acoustic Oscillation (BAO) and Pantheon datasets to place observational constraints on the model parameters. Using constrained values of model parameters, we study the behavior of cosmological parameters, such as the Hubble parameter $H$, the deceleration parameter $q$, and the equation of state (EoS) parameter $\omega_{v}$ with the skewness parameter $\delta_{v}$ for the viscous fluid. In addition, we perform the Om diagnostics and statefinder analysis to categorize dark energy models. Also, we studied cosmographic series coefficients to explore the whole evolution of the derived universe model. We estimated the current age of the universe as $t_{0}\approx13.8$ Gyrs. We obtained a quintessential and ever-accelerating model with bulk viscosity fluid.

Figures

Figures reproduced from arXiv: 2506.02543 by the authors.

**Figure 1.** The contour plots of H0, ξ1, m, α at σ1, σ2 confidence levels for CC dataset and CC+BAO datasets, respectively. Figure 1a and 1b show the contour plots of H0, ξ1, m, α at a fixed value of arbitrary constant c1 = 1.5 at σ1, σ2 confidence levels for CC and CC+BAO datasets, respectively. We have estimated the constrained values of model parameters by applying wide range of priors which mentioned in [PITH_FULL_IMAGE:fi… view at source ↗

**Figure 2.** The contour plots of H0, ξ1, m, α and M at σ1, σ2 confidence levels for CC+Pantheon datasets [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** The plots of deceleration parameter q(z) and effective EoS parameter ωef f versus z, respectively. The dimensionless parameter q characterizes the phase of the expanding universe; a positive value indicates a decelerating phase, whereas a negative value signifies an accelerating phase of expansion. The deceleration parameter q(z) as a function of z is presented in Equation (39). Figure 3a illustrates the variation o… view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: The plots of EoS parameter ωv and skewness parameter δv versus z, respectively. 5.1 Cosmographic Analysis The cosmological principle specifies a scale factor as the only degree of freedom that rules the universe. By expanding the current Taylor series of a(t) around pr…

**Figure 5.** Figure 5: The plots of jerk parameter j(z) and snap parameter s(z) over z, respectively. Using these variables, researchers investigate the dynamics of the universe in its later stages. In order to ascertain the physical properties of the coefficients, the form of the Hubble exp…

**Figure 6.** Figure 6: The plots of lerk parameter l(z) and max-out parameter m(z) over z, respectively. 5.2 Age of the present universe We define the age of universe as follows: t0 − t = − Z t t0 dt = Z z 0 dz′ (1 + z ′)H(z ′) (59) Using (38) in (59) and integrating, we get t0 − t = n q 1 +…

**Figure 7.** Figure 7: The evolution of cosmic age of the universe versus [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: The variations of statefinder parameters [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: The variations of s(z) versus r(z), and r(z) versus q(z), respectively. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: The variation of Om(z) versus z [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

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