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Probing Dark Energy Properties in $f Q,C)$ Gravity with FLRW Cosmological Models

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Pith's one-line read This paper claims that the $f(Q,C)$ gravity model $f=a_1Q^{\alpha}+a_2C$, fitted to Hubble, supernova, and BAO data through a transit Hubble parametrization, is observationally viable and reproduces late-time accelerated expansion.

desk verdict The paper is a kinematic fit wearing a modified-gravity costume: alpha, a1, and a2 never enter the likelihood, so the f(Q,C) conclusions do not follow from the data. read the letter →

arxiv 2412.01164 v1 pith:V7MOYIFD submitted 2024-12-02 gr-qc

N. Myrzakulov , Anirudh Pradhan , A. Dixit , S. H. Shekh This is my paper · ORCID
classification gr-qc MSC 83D0583F05 PACS 98.80.-k04.50.Kd
keywords f(QC)gravitydarkenergyFLRWcosmologytransitHubbleparameterconditionsMarkovChainMonteCarlomodifiedequationofstate
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 claims that a modified gravity theory built from the non-metricity scalar $Q$ and its boundary term $C$, with action $f(Q,C)=a_1Q^{\alpha}+a_2C$, can act as a geometric source of dark energy in a flat FLRW universe. By imposing the transit Hubble parametrization $H(z)=H_0(b_1+(1+z)^{b_2})/(1+b_1)$ and fitting to 57 Hubble-parameter measurements, standard-candle distance moduli, and 17 uncorrelated baryon acoustic oscillation points, the authors obtain best-fit parameters $H_0=64.51$, $b_1=1.543$, $b_2=1.141$. With these parameters the equation-of-state parameter stays close to $-1$ throughout and approaches $-1$ in the future, the universe transitions from deceleration to acceleration, and the strong energy condition is violated. The paper reads these results as evidence that $f(Q,C)$ gravity offers a viable alternative to a cosmological constant.

What carries the argument

The load-bearing object is the action $f(Q,C)=a_1Q^{\alpha}+a_2C$, where $Q$ is the non-metricity scalar and $C$ is the boundary term connecting non-metricity to curvature; this choice reduces the FLRW field equations to algebraic expressions for density $\rho$ and pressure $p$ that depend only on $H$ and $\dot H$. The expansion history is then supplied by the assumed transit Hubble parametrization $H(z)=H_0(b_1+(1+z)^{b_2})/(1+b_1)$, which is what produces the smooth deceleration-to-acceleration transition. The MCMC fit of the three free parameters to the combined $H(z)$, standard-candle, and BAO datasets fixes the numbers from which $\omega$, the $\omega$-$\omega'$ plane, sound speed, and energy conditions are evaluated.

What would settle it

Take the same $f(Q,C)$ field equations and fit them to the same Hubble, supernova, and BAO data without assuming the transit form, for example by reconstructing $H(z)$ non-parametrically or with a different parametrization; if the reconstructed history does not show $\omega$ near $-1$ and strong-energy-condition violation, the paper's dark-energy conclusions are artifacts of the chosen ansatz.

Watch

Extended reading notes

Core claim

The central claim is that the field equations of $f(Q,C)$ gravity, once specialized to the linear-in-$C$ power-law model and combined with the transit Hubble ansatz, yield an energy density and isotropic pressure whose late-time behavior matches the observed accelerated expansion. The fitted equation-of-state parameter is nearly $-1$ at all epochs and asymptotically reaches $-1$ in the future, the deceleration-to-acceleration transition occurs at a redshift consistent with current data, and the strong energy condition is violated while the null and dominant energy conditions hold. The authors conclude that the model can describe early deceleration, present acceleration, and a future de Sitter-like phase, and that it passes basic background-level tests against Hubble, supernova, and BAO data.

Load-bearing premise

Every headline result is computed from the hand-picked Hubble form $H(z)=H_0(b_1+(1+z)^{b_2})/(1+b_1)$; if that assumed expansion history is not the real one, the near-$-1$ equation of state, the transition redshift, and the energy-condition behavior are not predictions of $f(Q,C)$ gravity itself.

Editorial extensions

If this is right

  • The fitted $f(Q,C)$ model can reproduce late-time acceleration with $\omega$ near $-1$, offering a geometric alternative to a bare cosmological constant.
  • The best-fit transition redshift is consistent with current observations, so the model can be used to date the onset of cosmic acceleration.
  • Because the strong energy condition is violated while the null and dominant conditions hold, $f(Q,C)$ gravity provides a mechanism for repulsive gravitational behavior without exotic matter.
  • The model passes basic background-level tests against $H(z)$, Type Ia supernova, and BAO data, making it a candidate for further perturbation-level study.
  • The persistently negative squared sound speed implies the model is perturbatively unstable; the paper acknowledges this and links it to non-standard structure formation.

Reading between the lines

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

  • A direct test not performed in the paper would be to fit $f(Q,C)$ gravity without imposing the transit Hubble ansatz; the fact that all conclusions are computed from that ansatz means the model's apparent success is not yet separated from the choice of background.
  • The best-fit $H_0=64.51$ sits below local distance-ladder measurements; if this pattern persists in a model-independent reconstruction, the $f(Q,C)$ framework could have something to say about the Hubble tension, though the paper does not explore that.
  • The negative squared sound speed suggests that a full linear-perturbation analysis is needed before claiming the model is viable on structure-formation scales; the paper's conclusions are background-level only.
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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. This manuscript studies f(Q,C) modified gravity with the specific model f(Q,C) = a1 Q^α + a2 C in a flat FLRW universe. A transit Hubble parametrization H(z) = b0 (b1 + (1+z)^b2) is proposed, and the parameters H0, b1, b2 are fitted to a combination of H(z), Type Ia supernova, and uncorrelated BAO data using MCMC, yielding H0 = 64.51, b1 = 1.543, b2 = 1.141. The fitted H(z) is then substituted into the field equations to compute the energy density, pressure, equation of state, (ω−ω′) plane, squared sound speed, and energy conditions. On this basis the paper concludes that the f(Q,C) model can replicate late-time accelerated expansion and dark-energy-like behavior, including ω near −1 and SEC violation.

Significance. If the central claim were valid, the paper would provide an observational constraint on a specific f(Q,C) gravity model, which would be of interest to the modified-gravity community. However, the analysis does not actually test the f(Q,C) action: the MCMC likelihood in Eq. (20) contains no dependence on the gravity parameters α, a1, or a2, so it constrains only the kinematic Hubble ansatz. The derived physical quantities in Figs. 2–7 are algebraic consequences of substituting that same H(z) into the model equations, and the key conclusion (ω near −1, SEC violation, transition redshift) is an artifact of the assumed H(z) form rather than a prediction of the gravitational theory. The manuscript does have some strengths: it uses standard datasets, reports best-fit values with uncertainties, and clearly explains the geometric motivations for f(Q,C) gravity. Nevertheless, the central claim in the abstract and conclusion is not supported by the analysis as presented.

major comments (4)
  1. [§III.D–III.E, Eq. (20)] The MCMC likelihood in Eq. (20) is constructed from χ²_H(z), χ²_SN, and χ²_unCorBAO, all of which depend only on H(z) through Eq. (14) and the standard distance formulas (16)–(19). The f(Q,C) model parameters α, a1, and a2 from Eq. (7) do not appear in any of these terms. Therefore the best-fit values reported in §III.E constrain only a kinematic Hubble law, not the gravitational theory. The subsequent physical outputs, including the transition redshift, ω close to −1, and SEC violation, are obtained by inserting the fitted H(z) into Eqs. (8)–(13); they are consequences of the chosen transit ansatz and would be the same for any theory with identical kinematics. This undermines the paper's central claim that the f(Q,C) model is observationally viable.
  2. [§II, Eqs. (8)–(9)] The passage from the general field equations (5)–(6) to the model-specific expressions (8)–(9) is not derived. For f = a1 Q^α + a2 C, equation (5) contains contributions from f_C = a2 and the boundary term C, yet these disappear without justification in Eq. (8). Moreover, the resulting ρ and p still depend on α and a1, but the paper never states the numerical values of α and a1 used for the figures. Consequently, Figures 2–7 have no specified parameter values or units and cannot be reproduced from the information given.
  3. [§IV, Eq. (10) and Figs. 2–7] The equation of state in Eq. (10) is independent of a1 but depends explicitly on α. Since α is not reported anywhere in the manuscript, the quantitative claims about ω(z) remaining near −1, the magnitudes of ρ and p, and the energy-condition violations are not checkable. The figures must have been generated with some implicit choice of α (and a1), but without those values the curves are not reproducible. This is a load-bearing problem because the conclusion that the model 'successfully describes' cosmic evolution rests on these plots.
  4. [Data Availability, p. 9] The data availability statement says 'No data was used for the research described in the article,' but Section III explicitly employs 57 H(z) points, the Pantheon Type Ia supernova sample, and 17 uncorrelated BAO points. This is a direct contradiction that must be corrected to list the actual datasets and their sources.
minor comments (5)
  1. [§II, Eqs. (5)–(6)] The definitions of Q and C for the FLRW metric are not provided, so a reader cannot verify the substitution that leads to Eqs. (8)–(9); please give explicit expressions for Q and C in this geometry.
  2. [Figure captions] Figures 5 and 6 are mislabeled. Figure 5 is described in the text as the (ω−ω′) plane, but its caption reads 'The behavior of squared velocity of sound'; Figure 6, which actually shows the squared sound speed, has a caption that also references the same quantity. The captions should be swapped.
  3. [§III.D–III.E] The MCMC implementation is not described: no priors, chain lengths, burn-in, or convergence diagnostics are reported, so the 1σ and 2σ contours in Figure 1 cannot be independently verified.
  4. [Throughout] There are many typographical and grammatical errors, including repeated phrases in figure descriptions, inconsistent punctuation, and an incorrect year for reference [7] (2023 for what is a 2013 release); careful proofreading is needed.
  5. [§III.B, Eqs. (16)–(18)] The distance modulus in Eq. (16) uses µ0, and the luminosity distance in Eq. (17) uses c/H0, but the treatment of the absolute magnitude nuisance parameter and the covariance matrix in χ²_SN is not explained; please clarify whether the standard Pantheon likelihood with marginalization over the absolute magnitude is used.

Circularity Check

3 steps flagged · score 7.0 of 10

The headline results (transition redshift, ω(z)≈−1, and SEC violation) are obtained by inserting the fitted transit Hubble ansatz Eq. (14) into formulas that depend only on H and Ḣ; the MCMC likelihood constrains only H0, b1, b2, so the f(Q,C) action is never tested and the 'model validation' reduces to the kinematic fit.

  1. self definitional [Section II, Eq. (14), and Section III E]
    "This parametric form, which corresponds to the so-called transit scale factor, is particularly useful in describing a smooth transition between different cosmic epochs, accommodating deceleration at early times and acceleration in the present era."

    The deceleration-to-acceleration transition is written into Eq. (14) before any data analysis. The results section then reads this same input back out: 'The parameter constraints indicate a transition redshift that aligns well with current observational data.' Because Eq. (14) was adopted as a transit form, the later detection of a transition is an algebraic property of the ansatz, not an emergent prediction of f(Q,C) gravity.

  2. fitted input called prediction [Section IV.3, after Eq. (10), Fig. 4]
    "The equation of state parameter is mathematically expressed as given in Equation (10). To visualize its graphical behavior, we substitute Equation (14) into Equation (10). ... As the universe transitions towards the present epoch (z ≈ 0), ω(z) stabilizes near −1, aligning closely with observational data."

    Equation (10) contains only H, Ḣ, and α; substituting the fitted Eq. (14) fixes ω(z) as a deterministic function of the kinematic fit. Algebraically, Eq. (10) reduces to ω = −1 − (2α/3)(Ḣ/H²), independent of a1 and a2. The near-−1 behavior, and the associated SEC violation in Eq. (13), therefore follows from the chosen transit H(z), not from the f(Q,C) action. The claim that f(Q,C) 'retains the essential features of the ΛCDM model' is a restatement of the input ansatz.

1 more flagged steps
  1. fitted input called prediction [Section III D–E, Eq. (20)]
    "Using observational datasets, the model parameters H0, b1, and b2 are constrained using the Markov Chain Monte Carlo (MCMC) method. ... The MCMC method is used to constrain the model parameters by minimizing the total chi-square function: χ² = χ²H(z) + χ²SN + χ²unCorBAO."

    The likelihood contains only the kinematic Hubble expression Eq. (14) (and the associated distance integrals); the f(Q,C) parameters α, a1, and a2 never enter χ², and a2 in fact cancels from Eqs. (8)–(9). Thus the observational constraints validate the Hubble parametrization, not the modified-gravity model. Reporting this fit as evidence that the f(Q,C) framework 'effectively models the universe's evolution' equates the model's viability with the fitted ansatz by construction.

full rationale

The formulaic derivation is internally consistent: Eqs. (8)–(9) are reductions of the f(Q,C) field equations for the chosen action, and the MCMC fit to H(z) is a standard kinematic exercise. The circularity lies in what is presented as a prediction. The transition redshift, ω(z)≈−1, and SEC violation are all evaluated by inserting the fitted transit ansatz Eq. (14) into formulas that, after cancellation of a1 and a2, depend only on H, Ḣ, and the unconstrained α. The likelihood in Eq. (20) constrains only H0, b1, and b2, so the f(Q,C) action is never tested; any theory producing the same Hubble law would generate identical physical curves. This is partial circularity: the reported results reduce to the assumed Hubble parametrization, while the model-specific content is incidental and unconstrained. There is no load-bearing self-citation chain or imported uniqueness theorem; the issue is the fitted-input-called-prediction structure of the central claim.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim depends on two unreported parameters (alpha, a1), a fitted kinematic Hubble ansatz, and an unproved cancellation of the boundary term. The model therefore imports most of its content from the parametrization rather than from independent physics.

free parameters (6)
  • H0 = 64.51+0.21/-0.20
    Present-day Hubble constant, fitted via MCMC from H(z), SC, and BAO data; b0 is derived from H0/(1+b1) and is not independent.
  • b1 = 1.543+0.013/-0.012
    Shape parameter in the assumed Hubble ansatz, fitted to the combined likelihood.
  • b2 = 1.141+0.042/-0.038
    Shape parameter controlling the deceleration-acceleration transition, fitted to the combined likelihood.
  • alpha = not reported
    Exponent in f=a1Q^alpha+a2C; appears in equations (8)-(10) and determines the deviation of omega from -1, but no best-fit, prior, or assumed value is given.
  • a1 = not reported
    Coefficient multiplying Q^alpha; scales energy density and pressure in equations (8)-(9), but is never constrained or reported.
  • a2 = not reported
    Coefficient of the boundary term C in the action; is absent from the working equations (8)-(9) with no derivation showing why its contributions cancel.
assumptions (6)
  • domain assumption The universe is described by the flat FLRW metric (Eq. 3).
    The paper restricts to a homogeneous, isotropic, spatially flat background; all subsequent equations assume this geometry.
  • domain assumption Matter is a perfect fluid with energy-momentum tensor (Eq. 4).
    The definitions of rho and p and the energy conditions are built on this fluid description.
  • domain assumption The f(Q,C) field equations (5)-(6) are the correct variation of action (1).
    The equations are stated without derivation and inherit all conventions of the f(Q,C) framework from prior literature.
  • ad hoc to paper The action is restricted to f = a1 Q^alpha + a2 C (Eq. 7).
    This power-law model is selected for mathematical convenience; it is not derived from the theory or from data.
  • ad hoc to paper The Hubble parameter has the transit form H(z) = b0 (b1 + (1+z)^b2) (Eq. 14).
    This is the load-bearing input: the transition from deceleration to acceleration is embedded in this ansatz before any data are used.
  • ad hoc to paper The a2 (boundary term) contributions cancel or vanish in Eqs. (8)-(9).
    No a2 term appears in the final density and pressure expressions, but the paper never shows the cancellation; if these terms survive, the results change.

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Pith. "Pith review of Probing Dark Energy Properties in $f Q,C)$ Gravity with FLRW Cosmological Models." pith.science (2026). https://pith.science/paper/V7MOYIFD

@misc{pith2026241201164,
  author       = {Pith},
  title        = {Pith review of: Probing Dark Energy Properties in $f Q,C)$ Gravity with FLRW Cosmological Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V7MOYIFD}},
  note         = {Machine review of arXiv:2412.01164}
}
abstract

This study delves into the cosmological implications of the $f(Q,C)$ modified gravity framework within the context of the FLRW spacetime which offers a dynamic alternative to the standard $\Lambda$CDM cosmology. Here, we define the transit form of Hubble's parameter to explain several geometrical and physical aspects. The chosen parametric form of the Hubble parameter represents a smooth transition from the decelerating early universe to the accelerating present and late-time evolution. Employing observational datasets such as the Hubble parameter, Type Ia supernovae, Baryon Acoustic Oscillations (BAO), and Standard Candles (SC), we constrain the model parameters using the Markov Chain Monte Carlo (MCMC) method. The isotropic pressure, energy density, equation of state parameter, and energy conditions were analyzed to explore the physical viability of the $f(Q,C)$ framework. The results highlight the model's ability to replicate key cosmological behaviors, including the accelerated expansion driven by dark energy.

Figures

Figures reproduced from arXiv: 2412.01164 by the authors.

Figure 1
Figure 1. FIG. 1. This figure corresponds to 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The behavior of energy density for the transit form of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The behavior of isotropic pressure for the transit form [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. The behavior of squared velocity of sound for the tran [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The behavior of energy conditions for the transit form [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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