REVIEW 5 major objections 5 minor 1 cited by
Constraining model parameters in f(Q,C) gravity: Observational analysis and geometric diagnostics
T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read A linear f(Q,C) gravity model with three dark energy equations of state fits Hubble, BAO, and supernova data and supports late-time cosmic acceleration.
desk verdict The f(Q,C) parameters cancel out of the background equations, so the paper constrains only the imposed EoS ansatz, not the gravity model. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the linear Lagrangian $f(Q,C)=\alpha Q+\beta C$, which reduces the $f(Q,C)$ field equations in a flat FRW universe to $\kappa\rho=3\alpha H^2$ and $\kappa p=-3\alpha H^2-2\alpha \dot H$, giving the equation-of-state identity $\omega=-1-\frac{2}{3}\frac{\dot H}{H^2}$. Substituting the three parameterized $\omega(z)$ forms into this identity and integrating yields closed-form Hubble functions $H(z)$ (equations 33, 35, and 37), from which the deceleration parameter, energy density, pressure, energy conditions, statefinder parameters, and sound speed are all derived.
What would settle it
Measure $H(z)$ at several redshifts between $0.4$ and $1.2$ with errors below about 2 percent and compare the deceleration-to-acceleration transition redshift $z_{\rm tr}$: the models predict $z_{\rm tr}\approx0.54$--$0.99$, so observing a transition outside this range, or none at all, would rule out these parameterizations.
Extended reading notes
Core claim
The central claim is that the linear $f(Q,C)=\alpha Q+\beta C$ model, combined with any of three parameterized dark energy equations of state ($\omega=\omega_0+\omega_1 z$, $\omega=\omega_0+\omega_1 z(1+z)/(1+z^2)$, and $\omega=\omega_0+\omega_1 z^2/(1+z^2)$), reproduces the observed expansion history. With best-fit parameters, the deceleration parameter crosses from positive to negative at low redshift, the energy density is positive while the pressure is negative, the NEC, WEC, and DEC hold while the SEC is violated, and the sound speed lies between 0 and 1. The statefinder diagnostics place the first two parameterizations in the quintessence region of the $\{r,s\}$ plane and the third near the $\Lambda$CDM fixed point, supporting $f(Q,C)$ gravity as a viable dark energy framework.
Load-bearing premise
The load-bearing premise is that the universe's total energy content behaves as a single perfect fluid whose equation of state is exactly one of the three adopted redshift parameterizations, with no separate matter or radiation density parameter; if the real universe needs a multi-component energy budget, or if these EoS forms are not the correct effective description of $f(Q,C)$ gravity, the fitted parameters are not predictions of the theory.
Editorial extensions
If this is right
- The model predicts a deceleration-to-acceleration transition at $z_{\rm tr}\approx0.54$--$0.99$ depending on the parameterization and dataset, a directly testable signature.
- The fitted $H_0\approx67.4$--$67.9$ km/s/Mpc sits close to the value from CMB measurements, so the model does not aggravate the Hubble tension.
- Models 1 and 2 describe dark energy as quintessence-like with a time-varying equation of state, while Model 3 behaves nearly like $\Lambda$CDM, so the framework accommodates a spectrum of dark energy behaviors.
- All three parameterizations satisfy the sound-speed stability condition $0<c_s^2<1$, meaning dark energy perturbations do not grow uncontrollably.
- The energy conditions are satisfied except for the SEC, which is violated, matching the standard requirement for accelerated expansion.
Reading between the lines
- The three EoS forms are assumed rather than derived from the action, so the viability claim attaches to the parameterized one-fluid cosmologies; a different EoS history could change the conclusions.
- Adding growth-rate data or explicitly including radiation and baryonic matter would test whether the single-fluid simplification hides tension with CMB-era observations.
- The near-$\Lambda$CDM behavior of Model 3 suggests the framework may mimic a cosmological constant at the background level; checking the growth index would distinguish this from a true constant.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to constrain the parameters of f(Q,C)=αQ+βC gravity using three redshift-dependent dark-energy equation-of-state parameterizations against Hubble, Hubble+BAO, and Hubble+BAO+Pantheon datasets, and to derive deceleration parameters, energy conditions, statefinder diagnostics, and sound speeds from the fits. The central conclusion, stated in the abstract, is that the observational results 'support f(Q,C) gravity as a viable framework for describing diverse dark energy dynamics.' The manuscript derives the background field equations, integrates the Hubble parameter for each EoS ansatz, performs a standard MCMC likelihood analysis, and presents contour plots, error bars, and a battery of cosmological diagnostics.
Significance. If the analysis were a genuine test of f(Q,C) gravity, the paper would provide useful observational constraints on a modified-gravity model and a comparison of three dark-energy parameterizations. The paper has some strengths: the H(z) integrals in Eqs. (33), (35), and (37) are standard and, as algebraic exercises, are carried out correctly; the MCMC pipeline is conventional; and the use of three data combinations allows a check of parameter stability. However, the central inference is not supported because, as the paper's own equations show, the free parameters α and β of f(Q,C) cancel from the background dynamics, so the fits constrain only the hand-chosen EoS parameters and H0. Consequently, the paper does not test f(Q,C) gravity; it tests phenomenological EoS forms in a background that is indistinguishable from GR with a perfect fluid. This is a load-bearing problem that cannot be repaired by a local correction.
major comments (5)
- [Section 3, Eqs. (19)-(21)] The background field equations for f(Q,C)=αQ+βC reduce to κρ=3αH² and κp=-3αH²-2αḢ. The resulting EoS, ω=p/ρ=-1-(2/3)Ḣ/H², is independent of both α and β, and α merely rescales the effective gravitational constant. Therefore the Hubble parameter in Eqs. (33), (35), and (37) is determined solely by the assumed ω(z) and H0; the MCMC fits constrain only H0, ω0, and ω1. The choice α=0.5 made in Section 6.2 is arbitrary and cancels from all diagnostics. Because of this degeneracy, the abstract's statement that the findings 'support f(Q,C) gravity as a viable framework' does not follow from the analysis: any theory with the same background fluid would produce identical fits, transition redshifts, statefinders, and sound speeds. The paper's observational analysis is thus not a test of f(Q,C) gravity.
- [Section 5.3, Eq. (40)] The deceleration parameter for Model 5.3 is inconsistent with the standard relation q=0.5+1.5ω(z). Inserting ω=ω0+ω1z²/(1+z²) gives q(z)=0.5+1.5ω0+1.5ω1z²/(1+z²), whereas Eq. (40) contains the terms 3ω1z(1+z)/(4(1+z²)) and -3ω1/(4(1+z²)), which do not reduce to the required quadratic-in-z² form. Differentiating ln H from Eq. (37) yields a constant prefactor of 3(2+2ω0+ω1)/2, not 3(2+2ω0+ω1)/4 as written, and a last term proportional to (1+z)/(1+z²). This error propagates into the q0 and ztr values reported in Section 6.1 and Figure 7(c).
- [Section 6.3 and Tables 4-6] The present-day EoS values quoted in the text do not match the MCMC best-fit values in the tables. For Model 5.1, Table 4 lists ω0≈-0.588 to -0.590, while Section 6.3 reports ω0=-0.635, -0.657, and -0.675. For Model 5.2, Table 5 lists ω0≈-0.650 to -0.654, while the text reports -0.9630, -0.9016, and -0.8401. For Model 5.3, Table 6 lists ω0≈-0.569 to -0.571, while the text reports -0.6214, -0.5396, and -0.52345. Because ω0 and ω1 are the only physically meaningful free parameters in the fits, these inconsistencies affect every subsequently derived diagnostic, including the deceleration parameter, statefinders, and sound speed.
- [Section 6.4, Eqs. (47)-(55)] The SEC inequalities are written as ρ+3p≥0 for Models 5.1, 5.2, and 5.3, but the surrounding text and Figures 11-13 state that the SEC is negative at all redshifts. For an accelerating universe the SEC should be violated, i.e., ρ+3p<0, and the figures indeed plot negative values. The equations as written contradict the interpretation and the plotted results, so the energy-condition analysis is internally inconsistent.
- [Section 6.6, Figure 16] The text claims all models satisfy 0<c_s²<1, but Figure 16(b) for Model 5.2 appears to show sound speed values orders of magnitude larger than 1; the axis labels are corrupted, and the plotted curves are not in the claimed range. As written, the figure contradicts the causality/stability conclusion, and the typesetting artifacts make the quantitative claim impossible to verify.
minor comments (5)
- [Abstract and Section 4.4] The abstract states that the Pantheon sample contains 1408 data points, whereas Section 4.4 and Table 3 state 1048; the discrepancy should be reconciled.
- [Figure 16] The axis labels in Figure 16 contain obvious typesetting artifacts (e.g., '3 4 3', '7 4 3', 'G F H'), rendering the figure unintelligible; it should be regenerated with proper mathematical notation.
- [Section 4.2] Eq. (25) defines χ²_BAO using D_obs and D_th, but the text calls D_th a theoretical distance modulus, while Table 2 lists H(z) values; the notation needs to be clarified so the reader can identify which quantity is actually compared.
- [Section 6.1, Figure 7(b)] In the caption of Figure 7(b), the value '-303 842' appears to be a typo for '-0.3842'.
- [Section 6.3] The sentence 'The values from Model 5.2 show a strong negative trend, suggesting a more pronounced dark energy component that could hint at phantom behavior' is not supported by the reported ω0 values, which are all greater than -1; this should be reworded.
Circularity Check
The f(Q,C) parameters α and β cancel from the background equations, so the MCMC fits only the hand-chosen ω(z) ansatz; the reported H(z), q(z), statefinders and sound speeds are algebraic functions of the fitted ω0, ω1, not predictions of f(Q,C) gravity.
-
self definitional
[Section 3, Eqs. (19)–(21)]
"κρ = 3αH 2 , (19) κp = −3αH 2 − 2α ˙H. (20) ... The EoS parameter’s expression is derived by using the formula ( ω = p ρ ) in the following manner: ω = −1 − 2 3 ˙H H 2 . (21)"
Substituting f(Q,C)=αQ+βC into Eqs. (16)–(17) makes the β terms cancel identically and leaves ρ and p proportional to αH² and αHḢ. Since α appears in both ρ and p, the equation of state ω=p/ρ reduces to the standard single-fluid identity ω = −1 − (2/3)Ḣ/H². Thus the model-specific parameters α and β carry no background information, and every quantity later compared with data is fixed by the assumed ω(z), not by f(Q,C).
-
fitted input called prediction
[Section 5.1, Eqs. (32)–(33); analogous constructions in Eqs. (34)–(35) and (36)–(37)]
"By inserting the above linear form into equation (21) and using the fo rmula dH dt = −H(z)(1 + z) dH dz , we obtain the explicit form of H(z) as H(z) = H0(1 + z) 3 2 (1+ω 0−ω 1)exp [ 3ω 1z 2 ] , (33)"
The 'model' H(z) is obtained by inverting Eq. (21) for the adopted ω(z) ansatz. The MCMC then reports best-fit H0, ω0, ω1 and the paper presents the resulting H(z) as the prediction of f(Q,C) gravity, but α and β never appear in the likelihood. Any single-fluid cosmological model with the same ω(z) would yield identical H(z), identical χ² values, and identical constraints; the observational fit therefore tests only the EoS parameterization, not f(Q,C).
2 more flagged steps
-
fitted input called prediction
[Section 6.1, Eq. (38) (also Eqs. (39)–(40)) and discussion of ztr and q0]
"For our H(z) models, we derive the deceleration parameter expressions by utilising the equations (33), (35) and (37) as follows : • Model 5.1: q(z) = −1 + 3(1 + ω 0 − ω 1) 2 + 3ω 1 2 (1 + z), (38)"
Using Eq. (21), this q(z) is exactly q(z) = 1/2 + (3/2)ω(z). Consequently q0 is the fitted ω0 renamed, q0 = 1/2 + 3ω0/2, and the reported transition redshift solves ω(ztr) = −1/3. The claimed deceleration-to-acceleration transition is therefore imposed by the assumed parameterization crossing the acceleration threshold; it is not a new result derived from f(Q,C) gravity.
-
fitted input called prediction
[Section 6.5–6.6, Eqs. (56)–(60) and Figures 14–16]
"The statefinder diagnostics, {r, s }, introduced by [63, 64] ... r = ... a aH 3 = 2q2 + q − ˙q H , (56) s = (r − 1) 3(q − 1 2 ) . (57) ... The formula for finding c2 s is: dp dρ ."
All higher-order diagnostics are computed from q(z) and from ρ, p, which are proportional to αH² times functions of the fitted ω0, ω1. The arbitrary α = 0.5 chosen by hand in §6.2 cancels in c²s, and β cancels from the start. Thus the sound-speed stability and the quintessence/ΛCDM classification merely restate properties of the assumed ω(z) ansatz, giving no independent evidence about f(Q,C).
full rationale
The paper's central observable chain is: assume ω(z), integrate Eq. (21) to obtain H(z), fit H0, ω0, ω1 to Hubble/BAO/Pantheon data, and then present q(z), energy conditions, statefinders and sound speed as consequences of f(Q,C) gravity. But the linear model f(Q,C)=αQ+βC has β cancelling identically from Eqs. (19)–(20) and α cancelling from ω, so the gravitational action contributes nothing to the background equations beyond the standard relation ω = −1 − (2/3)Ḣ/H². The paper's own text states that the EoS parameterizations are assumed in §5, and Eq. (21) is then inverted to produce H(z) in Eqs. (33), (35) and (37). Every derived diagnostic, including q0, ztr, {r0, s0} and c²s, is an algebraic function of the fitted ω0 and ω1; no f(Q,C) parameter is constrained by the MCMC. The choice α=0.5 is made by hand in §6.2, and β never appears in any likelihood or diagnostic. The abstract's conclusion that the results 'support f(Q,C) gravity as a viable framework' therefore does not follow from the data analysis: the fit supports the assumed dark-energy EoS forms, not the specific gravity theory. Self-citations in the reference list are not load-bearing; the circularity is internal, via Eq. (21) plus the EoS ansatz. This is a case where the 'predictions' reduce by construction to the fitted inputs, so a score of 8 is appropriate rather than a lower self-citation-only score.
Assumptions & free parameters
free parameters (5)
- ω0 =
-0.57 to -0.65 depending on model/dataset
- ω1 =
0.27 to 0.58 depending on model/dataset
- H0 =
67.4 to 67.9 km/s/Mpc
- α =
0.5 (chosen by hand)
- β =
unconstrained
assumptions (5)
- domain assumption The universe is described by a spatially flat FRW metric
- domain assumption The affine connection is flat (Γ=0)
- ad hoc to paper The matter content is a single perfect fluid whose total EoS is one of the three parameterizations
- ad hoc to paper The linear form f(Q,C)=αQ+βC captures the relevant f(Q,C) dynamics
- ad hoc to paper α>0 so that ρ≥0
Cite this review
Pith. "Pith review of Constraining model parameters in f(Q,C) gravity: Observational analysis and geometric diagnostics." pith.science (2026). https://pith.science/paper/G6EGCIFP
@misc{pith2026241117754,
author = {Pith},
title = {Pith review of: Constraining model parameters in f(Q,C) gravity: Observational analysis and geometric diagnostics},
year = {2026},
howpublished = {\url{https://pith.science/paper/G6EGCIFP}},
note = {Machine review of arXiv:2411.17754}
}
abstract
We investigate the cosmological implications of $f(Q,C)$ gravity with $f(Q,C)=\alpha Q+\beta C$, where $Q$ is the non-metricity scalar and $C$ encapsulates cosmological expansion terms. Three parameterizations of the EoS for dark energy, $\omega=\omega_{0}+\omega_{1}z$, $\omega=\omega_{0}+\frac{\omega_{1}z(1+z)}{1+z^{2}}$ and $\omega=\omega_{0}+\frac{\omega_{1}z^{2}}{1+z^{2}}$ are tested using the Hubble, Hubble plus BAO, and Hubble plus BAO plus Pantheon datasets to constrain model parameters. The resulting Hubble and deceleration parameters reveal a transition from deceleration to acceleration, supporting current cosmic acceleration observations. Analysis of the energy density and pressure confirms positive energy density and a negative pressure for dark energy, potentially driving the late-time acceleration. We examine energy conditions, showing compliance with NEC, WEC and DEC, while SEC remains negative, supporting an accelerated expansion. Statefinder diagnostics suggest that two of the EoS parameterizations lead to Quintessence-like behavior with a time-varying dark energy component, while the third closely approaches $\Lambda$CDM showing slight deviations consistent with recent observations. Sound speed analysis demonstrates the physical stability of all parameterizations.
Figures
Figures from the paper (13 more)
Forward citations
Cited by 1 Pith paper
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Parameterized Deceleration in $f(Q,C)$ Gravity: A Logarithmic Approach
A logarithmic q(z) ansatz fitted to OHD and Pantheon+SH0ES in f(Q,C)=gamma1 Q^2 + gamma2 C claims zt about 0.98 and 0.76, but C is dynamically irrelevant and Table I's q0 contradicts the reported q(z=0).
Reference graph
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