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Dynamics of late time universe in $f(Q)$ gravity

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Pith's one-line read Three common f(Q) gravity models, each assuming a constant jerk parameter, all reproduce the observed late-time accelerating expansion and predict a quintessence-like effective fluid with a violated strong energy condition.

desk verdict A standard reconstruction paper where the constant-jerk ansatz fixes H(z), the f(Q) parameters are hand-picked, and the central equations contain algebra errors; not publishable as is. read the letter →

arxiv 2504.15680 v1 pith:P7ROKI3A submitted 2025-04-22 gr-qc hep-th

Bikash Chandra Roy , Aroonkumar Beesham , Bikash Chandra Paul This is my paper · ORCID
classification gr-qchep-th MSC 83D0583F05 PACS 04.50.Kd98.80.-k
keywords f(Q)gravitynonmetricitylate-timeaccelerationjerkparameterequationofstateenergyconditionsPantheondatasetcosmicchronometers
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 show that modified gravity built from the nonmetricity scalar $Q$ — with $f(Q)$ replacing the Einstein–Hilbert Lagrangian — can account for the late-time accelerated expansion of the universe without introducing a cosmological constant. It fixes the background expansion by assuming the cosmological jerk parameter $j$ is constant, calibrates the free parameters $H_0$, $j_0$, and $q_0$ against 31 cosmic chronometer measurements and the 1048-point Pantheon supernova sample, and then feeds that $H(z)$ into three popular $f(Q)$ forms: power-law, log-square-root, and exponential. The central conclusions are that the effective equation-of-state parameter sits in the quintessence band ($-1<\omega_{\rm eff}<-1/3$) at the present epoch for all three forms, that the universe transitions from deceleration to acceleration at $z_t\simeq 0.61$–$0.62$, and that the effective fluid violates the strong energy condition. If these conclusions hold, $f(Q)$ gravity is a viable route to dark energy that avoids the fine-tuning issues of a bare cosmological constant.

What carries the argument

The load-bearing object is the constant jerk assumption $j(z)=j_0$, where the jerk is defined by $j=(1/aH^3)\,d^3a/dt^3$. With jerk fixed, the standard relation between jerk and deceleration becomes a differential equation for $q(z)$, and the relation $dH/dz=(1+q)H/(1+z)$ then fixes $H(z)$ by integration. That single $H(z)$ is fed into the $f(Q)$ Friedmann equations, in which the nonmetricity scalar is $Q=6H^2$, to generate the dark-energy density, pressure, equation-of-state parameters, and energy conditions for each of the three $F(Q)$ forms. The mechanism is a reconstruction: the assumed jerk drives the kinematics, while the $f(Q)$ ansatz controls how that kinematics is split between ordinary matter and geometric dark energy.

What would settle it

A direct test is to fit the same cosmic chronometer and Pantheon data with the jerk left free to vary with redshift (for example, a constant-jerk model compared with a model where $j(z)$ has a linear or power-law drift) and check whether $j$ is statistically consistent with a constant at the reported $j_0\approx0.93$ or $1.208$. If a significantly varying jerk is preferred, the kinematic prior collapses. A complementary test is to compute $j(z)$ directly from each $f(Q)$ field equation with the best-fit matter densities and see whether the equations themselves force a constant jerk.

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Extended reading notes

Core claim

The paper's central claim is that a constant-jerk cosmological background, when interpreted through the field equations of $f(Q)=Q+F(Q)$ gravity, yields viable late-time cosmologies for three different forms of $F(Q)$. For the power-law model $F(Q)=\alpha(Q/Q_0)^n$, the log-square-root model $F(Q)=nQ_0\sqrt{Q/(\lambda Q_0)}\ln(\lambda Q_0/Q)$, and the exponential model $F(Q)=Q e^{\beta Q_0/Q}-Q$, the derived effective energy density stays positive while the effective pressure becomes sufficiently negative to drive acceleration. The paper reports best-fit values $H_0=68.13\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$, $j_0=0.93$, $q_0=-0.45$ from cosmic chronometers alone and $H_0=69.418$, $j_0=1.208$, $q_0=-0.604$ from the joint CC+Pantheon analysis. From these it finds present-day $\omega_{\rm eff}$ values of $-0.89$ (power-law, CC), $-0.94$ (power-law, joint), $-0.6$ (log-square-root), and $-0.79$ or $-0.76$ (exponential), all in the quintessence band, and finds $\rho_{\rm eff}+3p_{\rm eff}<0$ at late times, signalling SEC violation. The paper concludes that the matter content favours a quintessence-type fluid in all the $f(Q)$ models considered.

Load-bearing premise

Everything follows from assuming that the jerk—the third Taylor coefficient of the scale factor—is strictly constant, $j(z)=j_0$; if the true jerk varies with redshift, the derived Hubble parameter, equation-of-state curves, and energy conditions all change, because they are solved from that kinematic prior rather than from the $f(Q)$ dynamics alone.

Editorial extensions

If this is right

  • If the constant-jerk reconstruction is correct, $f(Q)$ gravity with any of the three forms reproduces the observed late-time acceleration with a positive effective energy density, offering a dark-energy alternative without a cosmological constant.
  • In the power-law model the dark-energy equation of state crosses $\omega_{\rm DE}=-1$ for negative $n$ (phantom) and stays above it for positive $n$ (quintessence), so the same functional form can accommodate either side of the phantom divide.
  • The log-square-root model keeps $\omega_{\rm DE}$ in the quintessence band under CC data but dips below $-1$ under CC+Pantheon, so future measurements of the dark-energy equation of state can discriminate between those behaviours.
  • All three models satisfy the weak, null, and dominant energy conditions but violate the strong energy condition at late times, matching the standard signature of accelerated expansion.
  • The deceleration parameter flips sign at $z_t\simeq0.62$ (CC) and $z_t\simeq0.61$ (CC+Pantheon), marking the transition from deceleration to acceleration within each model.

Reading between the lines

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

  • Because $H(z)$ is fixed entirely by the constant-jerk prior, the three $f(Q)$ models are not independent predictions: they are three different mappings from one assumed kinematics to fluid variables. A measurement of a non-constant jerk would change all three sets of equation-of-state and energy-condition curves at once.
  • The $f(Q)$ parameters $n$, $\lambda$, and $\beta$ are largely set by hand rather than marginalized in the statistical fit; a comparison that varies those parameters would be needed to decide which of the three forms is actually preferred by the data.
  • The same reconstruction could screen other $F(Q)$ ansatze: any proposed form can be plugged into the constant-jerk $H(z)$ and checked for positive energy density and a viable effective EoS, so the paper's method is a general filter for $f(Q)$ models.
  • The paper's 'quintessence in all $f(Q)$' statement applies to the effective total fluid; the dark-energy component itself is phantom for the exponential model, so the summary claim does not mean each model has a quintessence dark-energy sector.
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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

5 major / 4 minor

Summary. The paper proposes late-time cosmological models in f(Q) gravity using three functional forms (power-law, log-square-root, and exponential). A constant jerk parameter j is assumed to derive the Hubble parameter H(z) in Sec. 4; the parameters H0, j0, and q0 are then fitted to 31 cosmic-chronometer Hubble data points and to the Pantheon supernova sample in Sec. 5. The fitted H(z) is subsequently inserted into the three f(Q) models to compute dark-energy and effective equation-of-state parameters, and to test energy conditions in Secs. 6 and 7. The paper concludes that the effective fluid favours quintessence in all three models and that the strong energy condition is violated in the late universe.

Significance. If the central claim were established, the work would provide a simple cosmographic reconstruction of late-time acceleration in f(Q) gravity and a comparison of three common f(Q) forms against CC and Pantheon data. The use of standard chi-square minimization and the explicit presentation of the reconstructed cosmological quantities are strengths, and the paper is clearly written in its broad structure. However, the main results are not dynamical predictions of the f(Q) field equations: the same kinematic H(z) is imposed on all three models, the model-specific parameters are chosen by hand rather than constrained by the data, and the Model-II algebra contains internal inconsistencies. The significance of the paper is therefore contingent on resolving these methodological and technical issues.

major comments (5)
  1. [Sec. 4, Eqs. (44)-(47)] The constant-jerk ansatz is a kinematic prior imposed before any f(Q) dynamics are introduced, and the fitted q0 is already negative. The same H(z) is then substituted into all three models, while the likelihood analysis in Table 2 constrains only H0, j0, and q0; the f(Q) parameters n and beta are fixed by hand in Sec. 6.3 to keep energy densities positive. Consequently, the deceleration-acceleration transition, the EoS evolution, and the SEC violation shown in Figs. 3-14 are outputs of the assumed expansion history, not tests of the f(Q) gravity models. The abstract's claim that the models predict the observed late-time behaviour is therefore not supported by the analysis.
  2. [Sec. 3, Eqs. (28)-(43)] The modified Friedmann constraint (15), 3H^2 = rho_m + rho_r + rho_DE, is never enforced after the kinematic H(z) is substituted. The effective densities in Eqs. (28), (35), and (42) are defined as matter/radiation densities plus the model's rho_DE, but with the constant-jerk H(z) the right-hand side does not equal 3H^2(z) away from z=0; the closure relations (30) and (38) impose equality only at z=0. As a result, the plotted rho_eff, p_eff, omega_eff, and energy conditions do not describe solutions of the f(Q) field equations but rather algebraic combinations constructed from the assumed H(z).
  3. [Sec. 4, Eqs. (44) and (47)] Equation (44) is not the standard kinematic relation between j and q: for a constant q it gives j = 3q, whereas the standard identity is j = q(1+2q) + (1+z)dq/dz (or an equivalent form). In addition, Eq. (47) contains sqrt(-1-8j0), which is imaginary for the best-fit values j0 = 0.93 and 1.208 in Table 2, and no branch or real-part prescription is given. The H(z) used in all subsequent plots is therefore not a well-defined real function for the fitted parameters, which undermines the numerical results throughout the paper.
  4. [Sec. 3.2, Eqs. (32)-(38)] Model-II contains algebraic inconsistencies. Equation (32) gives rho_DE = (6n / lambda^{3/2} H0) H^3(z), whereas Eq. (35) and the closure relation (38) correspond to rho_DE = (6n / sqrt(lambda)) H0 H(z); these differ by powers of H/H0 and lambda. Similarly, Eq. (34) does not follow from Eqs. (32)-(33): substituting (32) into p_DE/rho_DE leaves factors of lambda and H0 in the second term, not the claimed -1 + (1+z)H'(z)/(3H(z)). The Model-II EoS curves and energy conditions are therefore computed from mutually inconsistent definitions.
  5. [Abstract and Sec. 8] The abstract states that the effective EoS favours quintessence in all f(Q) models, but the paper's own results show phantom DE for the power-law model with n = -1 (Sec. 6.3.1) and for the exponential model with beta = 0.37 (Sec. 6.3.3). Section 8 correctly lists these cases, so the abstract overclaims a universal quintessence behaviour that the body of the paper does not support.
minor comments (4)
  1. [Sec. 5.1, Eq. (48)] The text says 31 cosmic-chronometer data points are used, but the chi-square sum in Eq. (48) runs over 57 terms; please align the dataset size with the index range.
  2. [Sec. 4, Eq. (45)] The parameter d appearing in Eqs. (45) and (47) is never defined; if it denotes q0, this should be stated explicitly before use.
  3. [Table 2] The table reports best-fit values of H0, j0, and q0 without uncertainties, so the statistical significance of the constraints cannot be assessed from the paper as written.
  4. [Throughout] There are numerous typographical errors and garbled figure labels, including 'Cosmoligical' in the keywords, 'evoluation' in the captions of Figs. 8 and 9, and 'Flrw' in Eq. (10); a careful proofreading and regeneration of the figures with clean axis labels are needed.

Circularity Check

2 steps flagged · score 7.0 of 10

The headline behavior is kinematic reconstruction, not f(Q) prediction: the same constant-jerk H(z) is imposed on all three models, so neither quintessence nor SEC violation tests the f(Q) gravity.

  1. fitted input called prediction [Sec. 4 (Eqs. 44-47), Sec. 5 (Table 2), Sec. 6.3]
    "For a constant j(z) = j0 the deceleration parameter is given by [Eq. 45]. ... On integration we determine the Hubble parameter which is [Eq. 47]."

    The H(z) used to construct ρ_DE, p_DE, ω_DE, and ω_eff in all three f(Q) models is obtained from the kinematic ansatz j(z)=j0, with H0, j0, q0 fitted to the same CC and Pantheon data (Table 2 gives q0=-0.45 and -0.604). The late-time acceleration, transition redshift, and quintessence-like effective EoS are therefore inputs inherited from the fitted q0, not outputs of the f(Q) field equations. The same H(z) is imposed on all three F(Q) forms, so the exercise cannot discriminate the models or test f(Q).

  2. self definitional [Sec. 7 (Eqs. 61-64) vs. Sec. 4 (Eq. 46) and Sec. 5 (Table 2)]
    "3H 2 = ρm + ρr + ρDE , (15) ... 2 ˙H + 3H 2 = − 1/3 pr − pDE , (16) ... Strong energy condition (SEC): ρ ef f + 3pef f ≥ 0. (64)"

    Using the paper's effective Friedmann equations, ρ_eff=3H^2 and p_eff=-3H^2-2\dot H, so with q=-1-\dot H/H^2 one obtains ρ_eff+3p_eff=6qH^2. The reported SEC violation at late times is therefore exactly the statement that the fitted deceleration parameter is negative (q0<0 from Table 2). The SEC plots restate Eq. (45) rather than providing an independent f(Q) prediction; if H(z) is not a solution of the f(Q) field equations at all redshifts, the SEC curves are not valid dynamical outputs at all.

full rationale

The paper is a kinematic reconstruction rather than a first-principles f(Q) derivation. In Sec. 4 the Hubble function is not obtained by solving the f(Q) Friedmann equations; it is obtained by integrating the constant-jerk ansatz j(z)=j0 (Eqs. 44-47), and H0, j0, q0 are then fitted to the CC and joint CC+Pantheon data (Table 2). This single H(z) is substituted into all three F(Q) models, so the transition redshift, the effective EoS, and the energy-condition plots are outputs of the assumed kinematics with the fitted q0<0. The SEC violation is the clearest example: with the paper's effective Friedmann equations, ρ_eff+3p_eff=6qH^2, so the 'violation' is literally q(z)<0. Model parameters are not fixed by the likelihoods: n=-1 and β=0.37 are chosen 'in such a way so as to ensure that the energy density remains positive' (Sec. 6.3), while α and λ are set by present-epoch closure relations (30) and (38). The abstract's universal-quintessence statement is also contradicted by the paper's own figures (phantom ω_DE for power-law n<0 and for the exponential model), so it is an overstatement independent of the circularity issue. The self-citation [76] for the constant-jerk ansatz is not load-bearing because the integration is shown in the paper. Overall, the central 'predictions' reduce by construction to the fitted constant-jerk expansion history, giving partial but substantial circularity (7/10).

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a kinematic prior (constant jerk) and hand-picked model parameters; the only quantities actually fitted to data are H0, j0, q0. The matter density parameters needed to fix the f(Q) amplitudes are never specified, so the plotted densities and EoS curves are not fully determined by the paper. No new entities are introduced.

free parameters (7)
  • H0 = 68.13 (CC), 69.418 (CC+Pantheon) km/s/Mpc
    Fitted by chi-square minimization of the constant-jerk H(z) against cosmic chronometer and Pantheon data (Table 2).
  • j0 = 0.93 (CC), 1.208 (CC+Pantheon)
    Constant jerk parameter fitted to the same data; it fixes q(z) through Eq. (45).
  • q0 = -0.45 (CC), -0.604 (CC+Pantheon)
    Present deceleration parameter fitted with j0 and H0; the negative value forces present acceleration by construction.
  • n (power-law model) = 0.33, -0.5, -1 (hand-chosen)
    Chosen ad hoc to keep energy densities positive (Sec. 6.3.1); not constrained by the data.
  • n (log-square-root model) = not specified
    Required by Eq. (38) to set lambda, but its value is never given in the text or figure captions.
  • beta (exponential model) = 0.25, 0.37 (hand-chosen)
    Chosen by hand for the plots (Sec. 6.3.3); not constrained by the data.
  • Omega_m0, Omega_r0 = not stated
    Needed to fix alpha (Eq. 30) and lambda (Eq. 38), but never provided; all density and EoS plots depend on them.
assumptions (6)
  • domain assumption FLRW flat metric with coincident gauge, Q=6H^2
    Used throughout Sec. 2 to derive the Friedmann equations (10)-(12).
  • standard math f(Q) field equations from Jimenez et al. [54] as given in Eq. (8)
    The modified Einstein equations are taken from the cited literature without re-derivation.
  • ad hoc to paper Constant jerk parameter j(z)=j0
    Central kinematic prior in Sec. 4; not derived from f(Q) dynamics and not tested against a varying-j model.
  • domain assumption Matter content is pressureless dust plus radiation, with independent conservation (Eq. 23)
    Assumed in Secs. 2 and 6; radiation is subdominant at late times.
  • ad hoc to paper Three F(Q) functional forms (power-law, log-square-root, exponential) are taken from literature [54, 63, 66]
    No preference principle is given; they are selected because they are common in the f(Q) literature.
  • ad hoc to paper Model parameters n and beta chosen so that rho_DE > 0
    Stated in Sec. 6.3; this selects parameter values after seeing the model outputs.

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Pith. "Pith review of Dynamics of late time universe in $f(Q)$ gravity." pith.science (2026). https://pith.science/paper/P7ROKI3A

@misc{pith2026250415680,
  author       = {Pith},
  title        = {Pith review of: Dynamics of late time universe in $f(Q)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P7ROKI3A}},
  note         = {Machine review of arXiv:2504.15680}
}
abstract

We construct cosmological model in nonmetricity scalar functional gravitational Lagrangian $f(Q)$ which describes the dynamical evolution of the late accelerating universe. Cosmological models are constructed considering different functional of $f(Q)$ gravity where $Q$ in the gravitational action. We obtain cosmological model probing late universe with a constant jerk parameter. The observational constraints that are imposed on the model parameters for a realistic scenario estimated using the observational Hubble data and the Pantheon dataset. The evolution of the deceleration parameter, energy density and the equation of state (EoS) parameter are also explored. The transition of the universe from a deceleration to an accelerating phase is investigated in different framework of $f(Q)$ theories. We also analyzed the variation of the effective EoS parameter and found that the matter content in the universe favours quintessence type fluid in all the $f(Q)$-gravity. The energy conditions for a realistic scenario are examined and noted that the effective fluid violates the strong energy condition.

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Forward citations

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