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REVIEW 4 major objections 5 minor 83 references

Parameterized Deceleration in $f(Q,C)$ Gravity: A Logarithmic Approach

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that a logarithmic deceleration parameter in $f(Q,C)$ gravity can reproduce the observed cosmic expansion with a single geometric dark-energy fluid, transitioning from deceleration to acceleration near $z\approx0.8$.

desk verdict Internal inconsistency in the reported q0 and transition redshift undermines an otherwise standard kinematic fit; the f(Q,C) label adds nothing because the linear C term drops out. read the letter →

arxiv 2412.19852 v3 pith:UBCJD6DO submitted 2024-12-25 gr-qc

S.R. Bhoyar , Yash B. Ingole This is my paper · ORCID
classification gr-qc
keywords f(QC)gravitydarkenergydecelerationparameterlogarithmicparameterizationnonmetricitytransitionredshiftobservationalHubbledataPantheon+SH0ES
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 tries to establish that one chosen logarithmic formula for the deceleration parameter, $q(z)=q_0+q_1[\ln(\alpha+z)/(1+z)-\beta]$, can serve as the expansion history of a universe governed by $f(Q,C)$ gravity with action $f(Q,C)=\gamma_1Q^n+\gamma_2C$. The model is fitted to 31 cosmic-chronometer Hubble points and 1701 Pantheon+SH0ES supernovae, yielding $H_0\approx70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ and a smooth deceleration-to-acceleration crossover at $z_t\approx0.98$ (OHD) and $z_t\approx0.76$ (Pantheon+SH0ES). If the claim holds, late-time acceleration arises from geometry alone: the nonmetricity sector acts as an effective dark-energy fluid whose equation of state remains in the quintessence window, so no cosmological constant or scalar field is needed. The paper openly notes that the logarithmic choice is somewhat arbitrary and is used to close the system, so the gravitational theory is tested only through this assumed history.

What carries the argument

The load-bearing object is the logarithmic deceleration parameterization in Eq. (27), $q(z)=q_0+q_1[\ln(\alpha+z)/(1+z)-\beta]$, which is integrated through the kinematic relation $q(z)=-1+(1+z)H'(z)/H(z)$ to give the Hubble solution in Eq. (30). That Hubble solution is then put into the modified Friedmann equations of $f(Q,C)$ gravity, whose effective dark-energy density and pressure are given by Eqs. (23)--(24). The action $f(Q,C)=\gamma_1Q^n+\gamma_2C$ with integer $n>1$ supplies the geometric reinterpretation of the fluid, while the logarithmic factor is what makes the transition smooth and keeps $q(z)$ finite at high redshift.

What would settle it

Apply a model-independent reconstruction of $H(z)$ from the same OHD and Pantheon+SH0ES data and read off the deceleration parameter $q(z)=-1+(1+z)H'(z)/H(z)$. If the reconstructed curve does not cross zero between $z\approx0.7$ and $z\approx1.0$, or if it deviates from Eq. (27) by more than the reported uncertainties at $z>2$, the assumed logarithmic parameterization is falsified independently of the $f(Q,C)$ action.

Watch

Extended reading notes

Core claim

The paper's central claim is that nonmetricity-modified gravity can reproduce the observed transition from a decelerating to an accelerating universe without exotic matter. Starting from the action $f(Q,C)=\gamma_1Q^n+\gamma_2C$ with $n=2$ and $\gamma_1=0.235$, the Friedmann-like equations produce an effective dark-energy density and pressure; inserting the logarithmic ansatz gives a Hubble rate that tracks the data. The best fits place the current deceleration parameter at $q_0\approx-0.28$ (OHD) and $q_0\approx-0.26$ (Pantheon+SH0ES), with equation-of-state parameters $\omega_0\approx-0.55$ and $\omega_0\approx-0.70$, all in the quintessence regime. The transition redshifts are $z_t\approx0.98$ and $z_t\approx0.76$ respectively, and the statefinder and $Om(z)$ diagnostics behave consistently with a quintessence-like dark energy that approaches the $\Lambda$CDM point.

Load-bearing premise

The load-bearing premise is that the true expansion history really is the assumed logarithmic curve $q(z)=q_0+q_1[\ln(\alpha+z)/(1+z)-\beta]$; the gravity theory only reinterprets that imposed history, so if the curve is wrong the fitted transition redshifts and equation of state carry no predictive weight.

Editorial extensions

If this is right

  • If the central claim is right, cosmic acceleration can be obtained from nonmetricity geometry alone, removing the need for a cosmological constant or scalar-field dark energy.
  • The model predicts that acceleration began recently, with a transition redshift between about 0.76 and 0.98 depending on the dataset.
  • The equation of state stays in the quintessence interval $-1<\omega<-1/3$ and does not cross the phantom divide, so the future evolution approaches a de Sitter-like regime.
  • The statefinder trajectory ends at the $\Lambda$CDM point and $Om(z)$ has a negative slope, giving two diagnostics that distinguish this geometric dark-energy fluid from phantom models.

Reading between the lines

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

  • Because $\gamma_2$ multiplies the boundary term $C$ and cancels from the explicit density and pressure in Eqs. (37)--(38), the fits presented here effectively test the $f(Q)=\gamma_1Q^2$ sector; the advertised $f(Q,C)$ framework is not yet distinguished from plain $f(Q)$ by these data.
  • The same logarithmic ansatz could be fitted in general relativity with a purely phenomenological dark-energy fluid; if the goodness of fit is statistically indistinguishable, the data do not select nonmetricity gravity over a direct parameterization of dark energy.
  • A sharper test would add BAO and CMB distance priors or growth data, since the model's $H(z)$ rises steeply at high redshift and the two datasets already disagree on the transition redshift by roughly 0.2.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a logarithmic ansatz for the deceleration parameter, q(z)=q0+q1[ln(alpha+z)/(1+z)-beta], and uses it to close the FLRW equations of f(Q,C)=gamma1 Q^n + gamma2 C gravity. It fits the parameters (H0, q0, q1, alpha, beta) to 31 OHD points and 1701 Pantheon+SH0ES points via MCMC, reports best fits in Table I, and then derives H(z), the deceleration parameter, effective dark-energy density and pressure, the equation of state, statefinder diagnostics, and the Om diagnostic. The headline results are a deceleration-to-acceleration transition at zt about 0.98 (OHD) and 0.76 (Pantheon+SH0ES), with current deceleration q0 about -0.28 and -0.25 and current EoS about -0.55 and -0.70 for the two datasets.

Significance. If the results were correct, the paper would demonstrate that a nonmetricity-based modified gravity can reproduce late-time cosmic acceleration without scalar fields or an explicit cosmological constant. The paper includes an MCMC analysis, compares with LambdaCDM, and uses standard diagnostics such as statefinder and Om, which are appropriate tools for this kind of study. However, the central quantitative claim is not reproducible from the stated equations and fitted parameters, and the chosen action makes the boundary term C dynamically inert. As it stands, the paper does not support its conclusions.

major comments (4)
  1. [Section III, Eq. (27); Table I vs. Table II and Fig. 5] The reported fits are internally inconsistent. Substituting the Table I best-fit values (q0=3.53, q1=-1.151, alpha=1.863, beta=1.58) into Eq. (27) at z=0 gives q(0)=3.53+(-1.151)(ln 1.863 - 1.58)=+4.63, a strongly decelerating present epoch. Even the paper's own rewritten expression, Eq. (36), evaluated at z=0 with the same numbers gives approximately +2.7. Both are positive, contradicting Table II, which reports q0 about -0.28, and Fig. 5, which shows negative q at low redshift. Consequently, the claimed transition redshifts zt about 0.98 and 0.76, the negative current EoS, and all the derived conclusions in Section V are not consequences of the stated parameterization with the stated fitted parameters.
  2. [Section III, Eqs. (25)-(30) and Eq. (36)] The derivation connecting the ansatz Eq. (27) to the Hubble solution is not correct as printed. Integrating H'/H=(1+q)/(1+z) with q given by Eq. (27) yields a factor ((alpha+z)/(1+z))^{-q1/(alpha-1)} in H(z), whereas Eq. (28) and Eq. (30) have the opposite sign in the exponent, ((alpha+z)/(1+z))^{q1/(alpha-1)}. The subsequent expression for q(z) in Eq. (36) does not reduce to Eq. (27) when the relation q=-1+(1+z)H'/H is used, and it gives a positive q(0) numerically, as noted above. The stated limiting condition after Eq. (27), q(0)=q0+q1 log(alpha-beta), is also wrong; the correct limit is q0+q1(ln alpha - beta). These are load-bearing algebraic errors, not presentation issues.
  3. [Section II, Eq. (31); Eqs. (23)-(24) and Eqs. (37)-(38)] With the chosen action f(Q,C)=gamma1 Q^n + gamma2 C, one has f_C=gamma2, a constant, so all terms involving derivatives of f_C vanish, and the gamma2 C contribution cancels identically in the effective dark-energy density and pressure. This is visible in Eqs. (37)-(38), where gamma2 does not appear at all. The analysis is therefore dynamically an f(Q) model, not an f(Q,C) model; the title, abstract, and conclusions overstate the role of the boundary term C in producing the reported cosmic dynamics.
  4. [Section III and Section VII] The q(z) ansatz is imposed ad hoc rather than derived from the f(Q,C) field equations, and Section VII concedes that the choice is 'somewhat arbitrary.' As a result, the transition redshift, the current deceleration, and the EoS are re-expressions of the parameters fitted to the same Hubble and supernova data, not independent predictions of the gravity theory. The data agreement validates the ansatz, but it does not provide evidence for f(Q,C) gravity unless the theory is shown to single out this expansion history or to produce it dynamically.
minor comments (5)
  1. [Table II] The quantity labeled q0 in Table II is the derived current value of the deceleration parameter, not the fitted parameter q0 of Eq. (27). Rename it q(0) or q_cur to avoid the direct contradiction with Table I.
  2. [Section V.D] The text says the r-s trajectory converges to the LambdaCDM point (0,1), while the same section earlier correctly identifies LambdaCDM as (r=1, s=0). One of these statements is a typo and should be corrected.
  3. [Section IV] The MCMC description is incomplete: the priors, burn-in length, and convergence diagnostics are not stated, and the text interchangeably says chi-square minimization and Bayesian sampling. Please report the effective chi-square or a comparable goodness-of-fit statistic for each dataset.
  4. [Section IV.A] The treatment of the cosmic chronometer data should specify whether the 31 H(z) points are treated as independent or with a covariance matrix; the distinction can affect the derived uncertainties in Table I.
  5. [References and figures] References [38] and [39] are identical, and the caption of Fig. 3 contains the typo 'Pantheon + SHE0ES.' These should be cleaned up in a revision.

Circularity Check

3 steps flagged · score 7.0 of 10

The reported transition redshift and present acceleration are re-expressions of the assumed logarithmic q(z) fit, not independent predictions of f(Q,C) gravity.

  1. self definitional [Section III, Eq. (27); Section VII]
    "our study adopts the parameterization of a specific form of the deceleration parameter: q(z) = q0 + q1 [ ln[α+z]/(1+z) − β ], where q0, q1, α, and β are arbitrary model parameters. ... To summarize, the choice of q(z) (Eqn. 27) with a logarithmic term is somewhat arbitrary and is adopted here to explore the impact of the logarithmic term on the resulting cosmological model."

    The H(z) used in the fits is obtained by integrating this assumed q(z), and the f(Q,C) field equations are never used to determine q(z). Therefore the late-time acceleration and the deceleration-to-acceleration transition are properties inserted through the ansatz, not derived from the gravity theory. Reporting the resulting transition as a finding of f(Q,C) gravity equates the output with the input definition of q(z).

  2. fitted input called prediction [Section V.A, Table II; Eq. (27)]
    "The transition redshifts are obtained as zt ≈ 0.98 and, zt ≈ 0.76 for OHD and Pantheon + SH0ES datasets, respectively. ... The current values of the deceleration parameter for the OHD and Pantheon + SH0ES samples are observed to be q0 ≈ −0.30 and q0 ≈ −0.25."

    Table I fits the parameters (q0,q1,α,β) of Eq. (27) to the very same OHD and Pantheon+SH0ES data. Table II's q0, zt, and ω0 are then computed by evaluating that fitted q(z) (or the equivalent Eq. (36)) and the derived ρDE, pDE. These are functions of the best-fit parameters, so the 'revealed' current acceleration and transition redshift are forced by the fit, not independent predictions.

1 more flagged steps
  1. fitted input called prediction [Section V.B]
    "To preserve a positive energy density and the accelerating features of the EoS parameter, we then set the values of our model parameters γ1 and γ2, appropriately. ... Therefore, we use γ1 = 0.235 and n = 2, to keep the Hubble and deceleration parameters within the ranges suggested by cosmological discoveries."

    The gravitational parameters are not fitted or derived from the field equations; they are chosen after the fact so that ρDE is positive and the EoS has the accelerating (quintessence) behavior. The subsequent report of ω0 ≈ −0.55 (OHD) and −0.70 (Pantheon+SH0ES) as a successful prediction is thus a restatement of the parameter choice made to produce that behavior.

full rationale

The paper's central results—current deceleration q0 ≈ −0.30/−0.25, transition redshifts zt ≈ 0.98/0.76, and quintessence EoS ω0 ≈ −0.55/−0.70—are not derived from the f(Q,C) field equations but are consequences of the assumed logarithmic parameterization Eq. (27) together with parameter values chosen or fitted to the same OHD and Pantheon+SH0ES datasets. The expansion history H(z) is obtained by integrating the assumed q(z), so the acceleration is an input of the model, not an output of the gravitational theory. Additionally, the gravitational parameters γ1 and n are explicitly set after the fact to preserve positive energy density and the accelerating features of the EoS, making the reported EoS behavior partially self-imposed. Separately, the paper contains an internal inconsistency: using Table I values in Eq. (27) gives q(0) ≈ +4.6, and even the rewritten Eq. (36) gives q(0) ≈ +2.7, contradicting Table II's q0 ≈ −0.28 and Fig. 5; this is a correctness defect independent of the circularity assessment. There is no load-bearing self-citation chain or imported uniqueness theorem; the circularity is instead that the headline 'predictions' reduce by construction to the fitted ansatz.

Assumptions & free parameters 7 free parameters · 5 assumptions · 1 invented entities

The model rests on the assumed q(z) parameterization, the hand-chosen f(Q,C) form with n=2 and gamma1=0.235, and the standard FLRW/f(Q,C) background. The boundary term C is present in the action but carries no dynamics in this ansatz, so it is not an independent input. The fitted parameters q0, q1, alpha, beta, H0 are all free numbers tuned to the data; the derived 'predictions' (zt, q0_current, omega0) are deterministic functions of those fits.

free parameters (7)
  • q0 = 3.53 +/- 0.11 (Table I)
    Free parameter in Eq. (27) controlling q(z); fitted to OHD/Pantheon, but inconsistent with Table II's current q about -0.28.
  • q1 = -1.151 +/- 0.094
    Slope of the logarithmic term in Eq. (27); fitted to data.
  • alpha = 1.863 +/- 0.087
    Parameter inside the logarithm in Eq. (27); fitted to data.
  • beta = 1.58 +/- 0.11
    Additive shift in the parameterization of Eq. (27); fitted to data.
  • H0 = 70.01 km/s/Mpc
    Present Hubble constant; fitted with tight errors, not predicted by the model.
  • gamma1 = 0.235
    Coefficient of Q^n in f(Q,C); chosen by hand (Section V.B) to keep rho_DE positive, not fit to data.
  • n = 2
    Exponent in f(Q,C); chosen by hand as an integer greater than 1, not fit to data.
assumptions (5)
  • domain assumption FLRW flat metric and Friedmann-like equations (21)-(22) from f(Q,C) theory
    Assumed homogeneous-isotropic background; standard in cosmology, cited in Section II.
  • ad hoc to paper The ad hoc logarithmic deceleration parameterization Eq. (27) describes the true expansion history
    Imposed without derivation; paper itself calls the choice 'somewhat arbitrary' (Section VII).
  • ad hoc to paper f(Q,C)=gamma1 Q^n + gamma2 C with n=2 and linear C
    Not derived; with linear C, f_C is constant and C contributes only a total derivative, so the boundary term has no dynamical effect (no gamma2 in Eqs. 37-38).
  • domain assumption Vanishing affine connection in the symmetric teleparallel gauge
    Standard coincidence gauge in f(Q,C) background cosmology, cited to [51].
  • domain assumption Matter sector is negligible in the late-time kinematic fit
    The likelihood uses only H(z) and distance modulus; matter density is not explicitly modeled in the fit.
invented entities (1)
  • Effective geometric dark-energy fluid (rho_DE, p_DE)
    purpose: Absorbs modified-gravity terms into a fluid that drives acceleration
    Defined after H(z) is imposed via Eqs. (37)-(38); has no independent observational handle and is not an external prediction.

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Pith. "Pith review of Parameterized Deceleration in $f(Q,C)$ Gravity: A Logarithmic Approach." pith.science (2026). https://pith.science/paper/UBCJD6DO

@misc{pith2026241219852,
  author       = {Pith},
  title        = {Pith review of: Parameterized Deceleration in $f(Q,C)$ Gravity: A Logarithmic Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UBCJD6DO}},
  note         = {Machine review of arXiv:2412.19852}
}
abstract

This study explores a distinctive logarithmic parameterization of the deceleration parameter within the $f(Q, C)$ gravity framework, incorporating a nonlinear functional form $f(Q, C) = \gamma_1 Q^n + \gamma_2 C$, where $Q$ and $C$ denote the nonmetricity scalar and boundary term, respectively, and $n \geq 1$. This approach provides a unique perspective on the universe's accelerated expansion without resorting to exotic fields. Using observational data from Hubble measurements (OHD) and the Pantheon+SH0ES Type Ia supernovae dataset, the model parameters were constrained through a $\chi^2$ minimization technique. The analysis reveals a transition from deceleration to acceleration in the expansion history of the universe, with the transition redshifts $z_t \approx 0.98$ (OHD) and $z_t \approx 0.76$ (Pantheon+SH0ES). The model demonstrates consistency with observations, offering insights into the dynamics of dark energy and alternative gravity theories, while effectively modeling cosmic evolution across epochs.

Figures

Figures reproduced from arXiv: 2412.19852 by the authors.

Figure 1
Figure 1. FIG. 1: Error bar plots for 31 data points from the Hubble datasets, together with best-fit plots. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Marginalized constraints on the coefficients in the expression of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Error bar plots for 1701 data points from the [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The MCMC confidence contours derived from constraining the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: depicts the evolution of the universe, highlighting its transition from early-time deceleration to late-time acceleration, as determined by the constrained values of the model parameters derived from the observational data used in this article. The transition redshifts…
Figure 6
Figure 6. Figure 6: presents the evolution of the DE density ρDE as a function of redshift z. At low redshifts (z ≈ 0), the results from both datasets demonstrate similar trends in energy density, which approach near zero. This behavior aligns with the current epoch of accelerated cosmic …
Figure 7
Figure 7. Figure 7: shows the variation in DE pressure pDE as a function of redshift z for two datasets: OHD and Pantheon + SH0ES. The plot illustrates the evolution of pressure across different cosmic epochs. In the past (z > 0), corresponding to the early universe, the pressure was high…
Figure 8
Figure 8. Figure 8: FIG. 8: Plot of the EoS parameter [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Plot of [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: shows that Om(z) has a negative slope, which denotes the quintessence-like behavior of DE in a slowly evolving EoS for both datasets. VI. DISCUSSION It is important to mention that model-independent techniques reconstruct cosmic evolution using observational data with…

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