Analysis of the cosmological evolution parameters, energy conditions, and linear matter perturbations of an exponential-type model in $f(Q)$ gravity

A. Oliveros; Ivan R. Vasquez

REVIEW 2 major objections 5 minor 3 cited by

Analysis of the cosmological evolution parameters, energy conditions, and linear matter perturbations of an exponential-type model in $f(Q)$ gravity

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A two-parameter exponential modification of nonmetric gravity can mimic dark energy and change how cosmic structure grows, while staying close to ΛCDM.

desk verdict Solid background extension of an exponential f(Q) model, but the structure-growth predictions rest on a perturbation equation whose validity for this model is not demonstrated. read the letter →

arxiv 2501.12585 v2 pith:4U3CBUOB submitted 2025-01-22 gr-qc astro-ph.COhep-th

Ivan R. Vasquez , A. Oliveros This is my paper · ORCID

classification gr-qcastro-ph.COhep-th MSC 83D0583F05 PACS 04.50.Kd95.36.+x98.80.-k

keywords f(Q)gravitynonmetricitydarkenergyquintessencephantomgrowthindexfσ8conditions

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

The paper studies a modified theory of gravity in which the usual action is altered by an exponential term involving the non-metricity scalar $Q$. The model $f(Q)=Q+2\Lambda e^{-(b\Lambda/Q)^n}$ is a smooth small perturbation of $\Lambda$CDM, and the paper derives analytic expressions for the Hubble parameter at late times for $n=1$ and $n=2$. Using these, it calculates the effective dark-energy equation of state, energy conditions, deceleration and statefinder parameters, and the growth of linear matter perturbations. The central conclusion is that the model is a workable dark-energy surrogate: it produces late-time accelerated expansion, behaves like quintessence for $b>0$ and like phantom energy for $b<0$, and its growth predictions, including $f\sigma_8$, are consistent with observational data within the quoted errors. A sympathetic reader should care because the model offers a geometric origin for dark-energy phenomenology without new fields, and it makes a concrete prediction that growth observables deviate from $\Lambda$CDM near the present.

What carries the argument

The central object is the exponential nonmetricity function $f(Q)=Q+2\Lambda e^{-(b\Lambda/Q)^n}$, a smooth perturbative deformation of the $\Lambda$CDM action. Expanding in the small parameter $b$ turns the Friedmann equation into an algebraic equation for $u=H^2/H_0^2$, which is solved order by order to give analytic $H(z)$ for $n=1$ and $n=2$. The same expansion controls the effective dark-energy density and pressure, and its derivative $f_Q$ enters the perturbation equation through $G_{\rm eff}=G/f_Q$, which is what carries the model's influence on structure growth.

What would settle it

Compute the full second-order action for $f(Q)=Q+2\Lambda e^{-(b\Lambda/Q)^n}$ in the parameter range used here and check for ghosts or strong coupling; if an instability appears, the sub-horizon growth equation is not trustworthy. Observationally, a measurement of $f\sigma_8(z)$ at several redshifts with errors smaller than the model's percent-level deviations from $\Lambda$CDM would distinguish the $b>0$ and $b<0$ branches.

Watch

Extended reading notes

Core claim

The paper argues that the exponential nonmetricity model $f(Q)=Q+2\Lambda e^{-(b\Lambda/Q)^n}$, which reduces to $\Lambda$CDM when $b=0$, is a viable late-time cosmology. With $|b|$ small it yields analytic Hubble parameters $H(z)$ for $n=1$ and $n=2$ that stay within about $0.008\%$ of the numerical solution. The effective dark energy arising from the geometry then behaves as quintessence for $b>0$ ($w_{DE}>-1$) and as phantom-like for $b<0$ ($w_{DE}<-1$), while the effective gravitational constant $G_{\rm eff}=G/f_Q$ changes near the present, altering the linear growth of matter perturbations, the growth index $\gamma$, and the observable $f\sigma_8$. The model's $f\sigma_8$ curves fall within the error bars of 23 published growth measurements, and the authors take this as showing that the model reproduces the main effects attributed to dark energy.

Load-bearing premise

The growth-of-structure part assumes the standard sub-horizon perturbation equation of $f(Q)$ gravity, with the gravitational constant replaced by $G/f_Q$, remains valid for this model; if the strong-coupling or ghost problems noted in the paper invalidate that equation, the growth predictions would not follow.

Editorial extensions

If this is right

If the model is correct, late-time cosmic acceleration can be produced by the geometry of nonmetric gravity alone, with no additional scalar field or separate dark-energy fluid.
The sign of the deviation parameter $b$ controls the effective dark-energy sector: $b>0$ gives quintessence-like behavior with $w_{DE}>-1$, while $b<0$ gives phantom-like behavior with $w_{DE}<-1$.
Because $G_{\rm eff}=G/f_Q$ varies with redshift, structure growth is modified near the present; the model predicts different clustering compared with $\Lambda$CDM for the two signs of $b$.
The $n=2$ variant differs from $\Lambda$CDM only at order $b^2$ in the background, so its expansion history is nearly indistinguishable from $\Lambda$CDM and growth data become the main way to detect it.

Reading between the lines

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

The authors assign $b$ by hand in the growth plots; a full statistical fit to the $f\sigma_8$ compilation could tighten the model and reveal whether the preferred branch is the quintessence or phantom side.
The perturbation-sector caveat the authors cite implies a decisive check: computing the full action-level kinetic structure for this $f(Q)$ could determine whether ghosts or strong coupling appear at the parameter values used, in which case the background and energy-condition results would survive but the growth results would need revision.
Because the model's effective gravitational constant changes with redshift, the same exponential ansatz could be tested with cosmic-shear or CMB-lensing statistics, where a time-varying gravitational strength leaves distinctive scale-dependent signatures.

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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: this is a careful, workmanlike extension of the authors' own exponential f(Q) model. The genuinely new piece is the n=2 analytic H(z) solution, and they check both analytic solutions against the full numerical ODE with a maximum relative error of about 0.008%. That check is real evidence. The background and energy-condition phenomenology are handled cleanly and the ΛCDM limit is correctly recovered. The paper deserves a referee, but not a clean acceptance.

Strengths: the small-b expansion is done honestly, the cubic and quartic roots are treated without hand-waving, and the statefinder and Om(z) diagnostics are standard but competently executed. The quintessence-like (b>0) versus phantom-like (b<0) classification is a direct consequence of the model's form, and the authors do not oversell it as a discovery. They also explicitly flag that a full statistical constraint on b is future work, which is the right level of modesty.

Soft spots, in order of seriousness. First and most important: Section 7's growth results all rest on Eq. (92), the sub-horizon perturbation equation with G_eff = G/f_Q from [48]. The paper never addresses whether that equation is valid for this model. Since f_QQ ≠ 0 for every b ≠ 0, the scalar perturbations around FLRW in f(Q) can be strongly coupled or ghostly (Gomes et al., PRL 132, 141401, which is their [61]). The conclusions mention that ghosts 'may be propagating' and cite a ghost-free class [62], but they never show this exponential model belongs to that class. So the fσ8 curves and growth index are conditional at best. If Eq. (92) is not the correct effective description, those numbers are not trustworthy predictions. This does not sink the background analysis, but it does mean the paper's most observationally flavored claim is not established.

Minor issues: the H0 inconsistency (71.99 km/s/Mpc in the numerical check, 67.36 elsewhere) needs a comment. The fσ8 comparison with 23 data points is visual only; the authors admit a statistical constraint is future work, but the text's 'aligns nicely' is overstatement without a chi-square. Also, the model is engineered as a small perturbation around ΛCDM, so small deviations are built in, not discovered.

Bottom line: conditional accept. The background and analytic solution work is publishable after minor revision. The perturbation section should either demonstrate that the model evades the strong-coupling/ghost issue or be explicitly labeled as illustrative pending that question. I would send this to a serious referee, not desk reject.

Referee Report

2 major / 5 minor

Summary. The manuscript analyzes the exponential f(Q) model f(Q) = Q + 2Λ exp[-(bΛ/Q)^n] in a flat FLRW universe. It derives approximate analytic Hubble-parameter solutions for n = 1 and n = 2 using a small-b expansion around ΛCDM, checks these against a numerical integration of the Friedmann ODE, and then computes the effective dark-energy equation of state, deceleration and statefinder parameters, Om(z), energy conditions, and linear matter perturbations. The perturbation analysis uses a sub-horizon equation with G_eff = G/f_Q to obtain δ_m, the growth factor f_g, the growth index γ, and fσ8, which is compared visually with 23 observational data points. The authors conclude that b > 0 gives quintessence-like behavior and b < 0 gives phantom-like behavior, and that the model produces small late-time deviations from ΛCDM that affect structure growth.

Significance. If the background and perturbation results were fully established, the paper would provide a convenient closed-form approximate treatment of a two-parameter f(Q) model, including analytic H(z) expressions with a documented numerical check and a concrete fσ8 comparison. The paper is transparent in stating that statistical constraints and the ghost-field issue are left for future work. However, the perturbation sector, which carries the main new claim about structure growth, rests on an equation whose validity for f_QQ ≠ 0 is not demonstrated, and the numerical validation of the background solutions contains an apparent algebraic error. These issues make the current support for the central claims conditional rather than conclusive.

major comments (2)

[Section 7, Eqs. (92) and (100)] The linear growth equation (92) is taken from Ref. [48] in the quasistatic sub-horizon limit with G_eff = G/f_Q. For every b ≠ 0 in this model, f_Q ≠ 1 and f_QQ ≠ 0 (Eqs. 94–95), and Ref. [61] shows that generic f(Q) modifications develop strong coupling or ghosts in the scalar sector. The manuscript never checks whether the exponential model belongs to the ghost-free class of Ref. [62] over the parameter range used here (b ∈ [−0.2, 0.2], n = 1, 2). Since Eqs. (100) and (101) and all of the δ_m, f_g, γ, and fσ8 results in Section 7 are built on Eq. (92), the structure-growth part of the central claim is not established. Section 8 explicitly acknowledges the possible ghost propagation but does not resolve it; a demonstration of the validity of the quasistatic approximation for this model is needed before the perturbation results can be accepted.
[Section 4.3, Eq. (58)] The numerical ODE displayed in Eq. (58) does not follow from Eq. (25) for n = 1. With g' = bΛ/(6H^2)^2, the coefficient of H_dot in Eq. (25) is 1 + Λ e^g [24H^2(g'' + g'^2) + 2g'], which reduces to 1 + e^{-H0^2 ΩΛ b/(2H^2)} [−(3/2) b H0^4 ΩΛ^2/H^4 + (1/2) b^2 H0^6 ΩΛ^3/H^6]. Equation (58) instead contains the combination b (2/3) H0^4 ΩΛ^2/H^4 − b H0^4 ΩΛ^2/(2H^4), which has the opposite sign and a different magnitude for the linear-in-b term. Because this equation is the stated basis for the reported 0.008% agreement between the analytic and numerical H(z), the numerical validation needs to be corrected and re-run before the analytic solutions are relied on.

minor comments (5)

[Equation (33)] The bracket in Eq. (33) appears to have an algebraic typo: 12H^2 n (bΛ)^n (6H^2)^{-n-1} reduces to 2n [bΛ/(6H^2)]^n, which becomes 2n [H0^2 ΩΛ b/(2H^2)]^n after substituting Λ = 3H0^2 ΩΛ; the printed expression with H0^2 factors and (2H^2)^{-n-1} is off by a factor of 3. The subsequent n = 1 and n = 2 equations are consistent with the corrected form, so this should be fixed for consistency.
[Equation (42)] The first-order coefficient δu1 in Eq. (42) should be −3 ΩΛ^2/(2ξ), not −7 ΩΛ^2/(2ξ). The final solution (44) is consistent with the corrected coefficient, so this is a typographical error that should be corrected.
[Section 4.3] The numerical check uses H0(−0.1) = 71.99 km/s/Mpc, while the rest of the paper uses H0 = 67.36 km/s/Mpc (e.g., Figs. 2–17). Please clarify the fiducial H0 convention and how H0(b) in Eq. (45) is related to the Planck value, since this affects the interpretation of the numerical comparison.
[Section 7, Fig. 23] The statement that the model 'fits nicely' the 23 fσ8 data points is based on a visual comparison; since no χ² or likelihood is quoted, please either quantify the agreement or explicitly label the plot as illustrative. The text already says that a statistical constraint is future work, so this is a presentation issue rather than a fatal one.
[Figures and captions] Some figure captions are not in English (e.g., 'Evolución de δm...' in Fig. 19) and the Fig. 1 caption contains 'Der' in parentheses; please unify the language and correct the caption typos.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: predictions are derived from the cited f(Q) ansatz with b treated as a free parameter, and the only self-citation (prior MCMC work [24]) is contextual, not load-bearing.

full rationale

The derivation chain is self-contained: the model f(Q) = Q + 2Λ exp[-(bΛ/Q)^n] is explicitly introduced as an ansatz from the authors' prior work [24], not as a first-principles result, and the paper clearly states that it is a perturbative expansion around ΛCDM. The analytical Hubble solutions (Eqs. 44 and 52) are obtained by expanding in b, and subsequent quantities (w_DE, q, statefinders, Om, energy conditions) are direct algebraic and numerical consequences of those solutions. The parameter b is not fitted in this paper to the predicted observables; the paper explicitly says 'we are assigning values for the b parameter, to provide a better fit, a statistical constraint must be performed using this dataset' when discussing fσ8. The growth-sector results use Eq. (92) from the external reference [48] with G_eff = G/f_Q; this is an imported literature equation, not a self-citation or a redefinition of the model output. The manuscript also acknowledges in Sec. 8 the possibility of ghost fields ([61]) and the existence of ghost-free constructions ([62]), which is a correctness caveat rather than a circular step. The generic outcome of small late-time deviations from ΛCDM is guaranteed by the perturbative ansatz, but the sign-dependent quintessence/phantom behavior and the growth-index deviations follow from the derived expressions rather than from fitting those very behaviors. There is one minor self-citation to [24] for the model and for the MCMC-preferred negative values of b, but it is used as context and parameter motivation, not as the logical basis for the central derivations, so the paper has no significant circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The model introduces no new particles, fields, or forces; the dark energy is a purely geometric effective contribution. The only free inputs are the model parameters b and n, and standard cosmological parameters taken from Planck/Riess.

free parameters (2)

b = best-fit values from [24] (negative preferred, e.g., b = -0.1); used here as input at b = ±0.05, ±0.1, ±0.2
Deviation-from-ΛCDM parameter. The central claims about quintessence vs phantom and growth-rate changes depend directly on its sign and magnitude. Constraints come from the authors' earlier MCMC analysis [24], not from this paper.
n = 1 or 2 (selected)
Exponent in the exponential model. Only n=1 and n=2 are treated; the paper does not justify why these are preferred beyond analytic tractability.

assumptions (5)

domain assumption The flat FLRW field equations (13) and (14) with Q = 6H² are the correct cosmological reduction of f(Q) gravity.
Standard f(Q) literature result [13-15]; all background calculations depend on it.
domain assumption The perturbation equation δ''_m + H δ'_m - (4π ρ̄ G a²/f_Q) δ_m = 0 (Eq. 92) describes linear matter growth in f(Q), with G_eff = G/f_Q.
Taken from [48]. The paper does not address strong-coupling/ghost concerns [61] that may invalidate this sub-horizon equation for the model.
ad hoc to paper The exponential expansion e^{-x} ≈ 1 - x + x²/2 with x = (bΛ/6H²)^n is valid, i.e., bΛ/6H² << 1 for all relevant z.
Enabling approximation for the analytic H(z) solutions. Only validated numerically for b = ±0.1 (n=1) and b = 0.1 (n=2); the range b ∈ (-0.2, 0.2) used in figures is not error-checked.
domain assumption Radiation is neglected: Ωr,0 = 0.
Late-time approximation, standard in the literature; could affect results at z > 10, but the paper focuses on late times.
domain assumption The geometric dark energy can be described as a perfect fluid with ρDE and pDE from (26)-(27), satisfying standard conservation.
Modeling choice used to define energy conditions and wDE; not independently justified from the f(Q) action.

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Pith. "Pith review of Analysis of the cosmological evolution parameters, energy conditions, and linear matter perturbations of an exponential-type model in $f(Q)$ gravity." pith.science (2026). https://pith.science/paper/4U3CBUOB

@misc{pith2026250112585,
  author       = {Pith},
  title        = {Pith review of: Analysis of the cosmological evolution parameters, energy conditions, and linear matter perturbations of an exponential-type model in $f(Q)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4U3CBUOB}},
  note         = {Machine review of arXiv:2501.12585}
}

abstract

We study cosmological evolution in a flat FLRW spacetime in the context of modified STEGR gravity or $f(Q)$, using an exponential two-parameter model which represents a smooth perturbative expansion around the $\Lambda$CDM model. The cosmological analysis is carried out by calculating the Hubble parameter as a function of redshift, for selected values of the parameters. The Hubble parameter is obtained analytically by means of several approximations good enough to deviate slightly from the $\Lambda$CDM case. Several late-time cosmological parameters are computed, such as: dark energy state parameter, deceleration parameter, statefinder parameters. Additionally, we analyzed the behavior of the classical energy conditions WEC, SEC, NEC, and DEC for both the combination of matter and geometrical contribution and the geometrical contribution alone. Beyond the background level, linear matter perturbations are studied by calculating parameters relevant to structure growth and formation. The overall results indicate that the model may exhibit quintessence-like and phantom-like behavior and it impacts the growth of structures in the universe by means late-time deviations from the $\Lambda$CDM model.

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Forward citations

Cited by 3 Pith papers

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Two new logarithmic f(Q) gravity models fit current cosmological data and predict contrasting, testable deviations in the effective gravitational coupling and gravitational-wave damping.
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