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Observational constraints on freezing quintessence in a non-linear $f(R, L_m)$ gravity

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that a non-linear f(R,L_m) gravity model with a hyperbolic scale factor reproduces the observed late-time expansion and yields freezing quintessence behavior.

desk verdict The paper's headline results are built on a scale factor the stated model does not actually produce; fix the parametrization and it's a modest, honest fit-to-data paper. read the letter →

arxiv 2412.10518 v1 pith:BMJV2GHS submitted 2024-12-13 astro-ph.CO

Yerlan Myrzakulov , M. Koussour , A. Caliskan , E. Güdekli , S. Muminov , J. Rayimbaev This is my paper · ORCID
classification astro-ph.CO PACS 98.80.-k95.36.+x
keywords freezingquintessencef(RL_m)gravitylate-timecosmicaccelerationscalefactorparametrizationobservationalconstraintschronometersPantheonsupernovaedarkenergy
topics 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

This paper argues that the non-linear f(R,L_m) gravity model f(R,L_m)=R/2+L_m^$\alpha$, with $\alpha$=1.33, can describe the late-time accelerating universe as freezing quintessence. Choosing the scale factor a(t)=n $\sqrt$($\sinh$ t) and fitting H0 and n to cosmic chronometer, Pantheon supernova, and combined BAO datasets, the model yields H0 about 66 to 67.6 km/s/Mpc and n about 1.18 to 1.405, close to Planck's value. From that single expansion history the paper derives a deceleration parameter that transitions from deceleration to acceleration around redshift 0.3 to 1, an equation-of-state parameter that approaches -1 at late times, freezing trajectories on the omega-omega' plane, and a sound speed between 0 and 1. The authors conclude that the modified gravity model is a credible explanation of the current cosmic acceleration.

What carries the argument

The load-bearing inputs are the functional form f(R,L_m)=R/2+L_m^$\alpha$ and the geometric ansatz a(t)=n $\sqrt$($\sinh$ t). The scale-factor choice closes the otherwise underdetermined Friedmann system: it fixes H(z)=H0/$\sqrt$(2)*$\sqrt$((1+z)^{2n}+1), from which the deceleration parameter, equation-of-state parameter, omega', and sound speed follow algebraically. The parameter $\alpha$ enters the equation-of-state and sound speed but not H(z), and is fixed to 1.33 based on an earlier analysis rather than fitted here.

What would settle it

A precise measurement of H(z) at redshifts above 2, from high-redshift cosmic chronometers or BAO observations, that deviates from H0/$\sqrt$(2) $\sqrt$((1+z)^{2n}+1) with n around 1.4 by more than the quoted uncertainties would rule out the parametrization and thereby the model's conclusion.

Watch

Extended reading notes

Core claim

The central claim is that the non-linear coupling f(R,L_m)=R/2+L_m^$\alpha$, together with the scale-factor parametrization a(t)=n $\sqrt$($\sinh$ t), reproduces the observed late-time expansion history. With H0 and n fitted by MCMC to CC, Pantheon, and combined CC+SN+BAO data, the model predicts a Hubble parameter that decreases with time, a present-day deceleration parameter q0 between -0.30 and -0.41, an equation-of-state parameter in the quintessence range that approaches -1 at late times, and a freezing region in the omega-omega' plane, consistent with an accelerating universe. The model also yields a positive squared sound speed less than 1 throughout cosmic evolution, indicating stability against density perturbations, and its information-criterion differences versus the Lambda CDM model are small for the CC and SN datasets.

Load-bearing premise

The results follow only if the true expansion history is exactly a(t)=n sqrt(sinh t); if the actual scale factor deviates from this hyperbolic form, the fitted parameters and all derived conclusions no longer apply.

Editorial extensions

If this is right

  • If the model is correct, the late-time accelerating phase is driven by freezing quintessence rather than a cosmological constant, with the equation of state approaching -1 only asymptotically.
  • The fitted Hubble constants, spanning 66.0 to 67.6 km/s/Mpc, sit close to the Planck value and below local distance-ladder measurements, so the model offers a possible route to address the Hubble tension.
  • The predicted present deceleration q0 in the range -0.41 to -0.30 and transition redshift around 0.3 to 1 can be tested against independent geometric measurements such as those from BAO and gravitational lensing.
  • The stability condition 0<nu_s^2<1 implies that density perturbations grow without exponential instability, so structure formation proceeds normally under this model.
  • Information-criterion comparisons show the hyperbolic parametrization is statistically comparable to Lambda CDM for the CC and SN datasets, indicating it is a viable alternative, though it becomes mildly disfavored when BAO data are included.

Reading between the lines

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

  • The qualitative conclusions, including the deceleration-to-acceleration transition and freezing behavior, are inherited from the assumed a(t)=n sqrt(sinh t) and would hold for any gravity theory that adopts that expansion history; they do not by themselves test the f(R,L_m) action.
  • If alpha were treated as a free parameter in the MCMC instead of being fixed to 1.33, the constraints on n and H0 could shift and the equation-of-state and sound-speed predictions could change, so a direct fit would strengthen the model claim.
  • The model's H0 predictions are systematically lower than local distance-ladder values, so combining the model with a high-redshift early-universe prior could sharpen whether it genuinely resolves the Hubble tension.
  • A falsifiable extension would be to use the same scale-factor ansatz in general relativity and in f(R,L_m) gravity; any difference in the implied H(z) would isolate the effect of the matter-geometry coupling.
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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

3 major / 6 minor

Summary. The paper studies late-time cosmic acceleration in a non-linear f(R,L_m) gravity model with f(R,L_m)=R/2+L_m^alpha. It adopts the scale-factor parametrization a(t)=n sqrt(sinh t), derives the corresponding Hubble parameter, fits the parameters H0 and n to cosmic chronometer, Pantheon, and combined CC+SN+BAO datasets via MCMC, and then analyzes the deceleration parameter, matter-energy density, EoS parameter, the omega-omega' plane, and the sound speed. The authors report H0 around 66-67.6 km/s/Mpc, n around 1.18-1.405, and conclude that the model shows freezing quintessence behavior, late-time acceleration, stability against density perturbations, and consistency with LCDM, claiming that the f(R,L_m) gravity is a credible approach to cosmic acceleration.

Significance. If the scale-factor parametrization is corrected, the paper would provide a straightforward reconstruction of late-time cosmology in a non-linear f(R,L_m) model, with transparent MCMC fits to standard CC, Pantheon, and BAO data and a comparison to LCDM via AIC/BIC. The algebra from the f(R,L_m) action to Eqs. (12)-(13) is correct, and the subsequent formulas (17), (25), (26), (28), (30), (32) are mutually consistent once Eq. (17) is taken as the input expansion history. The paper does not provide code or data products, so the MCMC results are not fully reproducible from the manuscript alone. More importantly, the central claimed prediction of freezing quintessence is an algebraic consequence of the assumed H(z), not a result of the modified-gravity dynamics, and the current version of the paper contains a direct contradiction between the stated scale factor and the fitted H(z). The paper therefore cannot, as written, support its conclusion that the f(R,L_m) gravity is a credible explanation of cosmic acceleration.

major comments (3)
  1. [Sec. III, Eqs. (14)-(17)] The scale-factor parametrization is internally inconsistent. From Eq. (14), a(t)=n sqrt(sinh t), direct differentiation gives H(t)=a'/a=0.5 coth t, not coth(t)/n as stated in Eq. (15). Inverting a=1/(1+z) gives sinh t=1/[n^2(1+z)^2], not Eq. (16), and the resulting redshift evolution is H(z)=0.5 sqrt(n^4(1+z)^4+1), with H0=0.5 sqrt(n^4+1). This is very different from Eq. (17), H/H0=sqrt((1+z)^{2n}+1)/sqrt(2), which is the expression actually fitted in Sec. IV. Consequently, all derived quantities in Secs. V-VII (q, omega, omega', nu_s^2) describe a different expansion history than the model stated in Eq. (14). For example, for n=1.405, the stated scale factor gives q0=(n^4-1)/(n^4+1)=+0.59, a decelerating universe today, whereas Eq. (26) gives q0=-1+n/2=-0.30. The authors should either correct Eq. (14) to a(t)=(sinh t)^{1/n}, which does yield Eq. (17), or re-derive the full redshift dependence for a(t)=n sqrt(sinh t) and repeat the MCMC analysis.
  2. [Sec. IV and Sec. V] The parameter alpha is not constrained in this work. The abstract calls alpha a free parameter, but the MCMC analysis fits only H0 and n, and the end of Sec. IV fixes alpha=1.33 by importing the result of Ref. [53]. The uncertainty in alpha is not propagated into omega(z), omega'(z), nu_s^2(z), or the conclusions about quintessence behavior and stability. Because the sign and magnitude of the EoS corrections in Eqs. (28) and (30) depend on alpha (quantitatively, alpha>1/2 is required for the claimed freezing property), the paper should either fit alpha jointly with H0 and n for each dataset or explicitly state that the derived cosmological parameters are conditional on the external value alpha=1.33 and should not be presented as constraints from the datasets used here.
  3. [Sec. VI, Eq. (30)] The freezing behavior is a built-in property of the assumed H(z), not a prediction of the f(R,L_m) action. Equation (30) gives omega'(z) < 0 for all z whenever alpha > 1/2 and n > 0, so the trajectory in the omega-omega' plane cannot enter the thawing region. More generally, q, rho, omega, and nu_s^2 are all obtained from the same fitted H(z) through Eqs. (17), (25), (28), and (32); their agreement with observational trends therefore tests the adopted parametrization of the scale factor, not the modified-gravity action. The concluding claim in Sec. VIII that 'the modified f(R,L_m) gravity is a credible approach' is stronger than the analysis supports and should be qualified accordingly.
minor comments (6)
  1. [Abstract and Sec. IV] The abstract and conclusion refer to the 'Pantheon+ (SN)' dataset, but Sec. IV uses the 1048-point Pantheon sample (Refs. [74,75]); please clarify which supernova compilation was actually used and cite Pantheon+ accordingly if it was used.
  2. [Sec. IV, Fig. 1 and Table I] The text reports BAO-only constraints H0=70.0^{+10}_{-9} km/s/Mpc and n=1.417^{+0.026}_{-0.025}, but Fig. 1 and Table I do not show the BAO-only contours; please include them or remove the BAO-only statement.
  3. [Sec. V, Eq. (32)] The derivation of the sound speed in Eq. (32) is not shown; starting from Eq. (31), one also needs the time derivatives of rho(z) from Eq. (12) and of omega(z) from Eq. (28), so a brief derivation or at least a statement of the intermediate steps would improve reproducibility.
  4. [References] There are several reference errors: [68] should be 'Stern' rather than 'Stren'; [69] lists '2010' as the year but the journal issue is 2012; and [46] is a loop-quantum-cosmology preprint that does not appear to support the statement about solar system constraints on f(R,L_m) gravity. Please correct these citations.
  5. [Sec. VIII] The abstract says the results are 'in excellent agreement with observational data,' but Table I shows Delta AIC = 5.3 for the combined CC+SN+BAO dataset, which the authors themselves describe as 'mild tension' according to Jeffreys' scale; consider using more measured language.
  6. [Sec. III] After correcting Eqs. (14)-(17), the normalization of a0=1 and the role of n in the redshift inversion should be stated explicitly, since the multiplicative constant n in Eq. (14) changes the relation between t and z.

Circularity Check

3 steps flagged · score 7.0 of 10

Freezing quintessence and q0 results are algebraic consequences of the fitted H(z) ansatz, while alpha=1.33 is imported from same-group prior work.

  1. self definitional [Sec. VI, Eq. (30), Fig. 7]
    "Also, we have identified only the freezing region in that plane because ω′ < 0 for ω < 0. Therefore, there is no thawing region available in our model."

    Eq. (30) gives ω′(z) = −4(2α−1)n²(1+z)^{2n} / [3α((1+z)^{2n}+1)²]. With the adopted α=1.33>1/2 and best-fit n>0, this quantity is negative for every z. The freezing-region placement is therefore an algebraic identity of the H(z) ansatz inserted in Sec. III, not an output of the f(R,L_m) field equations or of the MCMC fits. The data can move the trajectory along a curve that is everywhere in the freezing region; it cannot test thawing versus freezing. The claimed qualitative result is thus equivalent to the input parametrization by construction.

  2. self citation load bearing [Sec. IV (after Table I) and Sec. V]
    "Since the model parameter α does not explicitly appear in the expression for the Hubble parameter, we fixed its value to study the evolution of matter-energy density and the EoS parameter. We used the value α = 1.33, as constrained by observational datasets in Ref. [53]."

    All EoS, freezing-region, and sound-speed results (Eqs. (28), (30), (32)) depend on α, but α is neither fitted to the CC/SN/BAO data used here nor derived from the f(R,L_m) action. It is taken from Ref. [53], a previous paper by the same research group (Myrzakulov et al.). The freezing conclusion specifically requires α>1/2, so the central qualitative claim is carried by a self-citation rather than by an independent parameter-free derivation or by the present datasets.

1 more flagged steps
  1. fitted input called prediction [Sec. V B, Eq. (26), Fig. 4; Table I]
    "q(z) = −1 − ˙H/H² = n − 1 − n/((1 + z)^{2n} + 1) ... the present values of q are determined to be q0 = −0.38+0.06−0.06, q0 = −0.41+0.10−0.11, and q0 = −0.30+0.01−0.01 for the CC, SN, and combined datasets, respectively."

    Eq. (26) is obtained solely by substituting the fitted H(z) of Eq. (17) into q = −1 − ˙H/H²; at z=0 it reduces to q0 = n/2 − 1, a one-to-one algebraic function of the fitted parameter n. The reported q0 values and the claimed 'transition from deceleration to acceleration' are therefore restatements of the best-fit n posterior, not independent predictions of the f(R,L_m) model. The same applies to ω(z), ω′(z), and ν_s²(z), which are all computed from the same fitted H(z) and the imported α.

full rationale

The paper is not vacuous: the MCMC fits to CC, Pantheon+, and BAO data are genuine external comparisons, and the reported H0 constraints carry independent information. However, the distinctive physical conclusions—freezing behavior in the ω−ω′ plane, the present-day deceleration parameter, and the late-time acceleration transition—are not independent outputs of the f(R,L_m) action. The H(z) ansatz of Eq. (17) is inserted to close the underdetermined field equations, and q, ω, ω′, and ν_s² are then computed from that same fitted H(z) with α imported from Ref. [53]. Eq. (30) makes ω′<0 for all z for α>1/2, so the freezing-region claim is an identity; q0 = n/2−1 is a restatement of the fit; and the α dependence is supplied by a same-group prior paper. Separately, there is an algebraic inconsistency between Eq. (14) as written and Eqs. (15)–(17): differentiating a(t)=n√sinh t gives H=0.5 coth t, not coth(t)/n, and the resulting q0=(n⁴−1)/(n⁴+1) is positive for n=1.405, whereas the fitted q0<0 comes from Eq. (17). That inconsistency is a correctness problem rather than circularity itself, but it reinforces that the published conclusions belong to the inserted H(z), not to the stated model. Overall, the central qualitative claims are partially forced by construction and by self-citation, while the quantitative fits retain external content; score 7.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim depends on three fitted or imported numbers: H0 and n are fitted here; alpha is imported from Ref. [53]. The analysis also assumes the FLRW geometry, the identification L_m=rho, the ad hoc hyperbolic scale factor, and the sound-speed positivity criterion as a stability proxy. No new particles, forces, or dimensions are introduced; the non-conservation of the energy-momentum tensor is inherited from the f(R,L_m) framework. These inputs are the real cost of the model: the expansion history is not derived from the gravity action, and the equation-of-state results are not independent of the fitted curve.

free parameters (3)
  • H0 = 66.0 to 67.6 km/s/Mpc (CC, SN, CC+SN+BAO)
    Present Hubble rate, enters Eq. (17); fitted with MCMC and determines the normalization of H(z).
  • n = 1.18 to 1.405 (CC, SN, CC+SN+BAO)
    Shape parameter in a(t)=n sqrt(sinh t); sets the redshift dependence of H(z) and all derived cosmological parameters.
  • alpha = 1.33 (fixed from Ref. [53])
    Power in L_m^alpha. Not fitted in this paper; controls the EoS, omega-omega' plane, and sound speed. Its uncertainty is not propagated.
assumptions (6)
  • standard math The f(R,L_m) field equations (3) and their flat FLRW reduction (9)-(10) from Refs. [37,52] are correct.
    The paper uses these as the starting point for all later equations and does not re-derive or benchmark them.
  • domain assumption The universe is described by a flat FLRW metric and a single perfect fluid with energy density rho and pressure p (Sec. II, Eqs. 6-8).
    This is a standard but nontrivial restriction: anisotropic or inhomogeneous effects, radiation, and separate dark-matter components are ignored.
  • domain assumption The matter Lagrangian density equals the energy density, L_m = rho (Sec. III).
    This identification is conventional; L_m = -rho or L_m = p would give different Friedmann equations. It is adopted from Ref. [44] without a physical justification for the cosmic fluid.
  • ad hoc to paper The scale factor is parametrized as a(t) = n sqrt(sinh t), with n > 0 (Sec. III, Eq. 14).
    This ansatz closes the system and fixes H(z) in Eq. (17). It is not derived from the f(R,L_m) action or field equations, so the expansion history is an input rather than an output.
  • ad hoc to paper The value alpha = 1.33, taken from Ref. [53], is a valid constraint for the datasets used in this paper.
    The EoS, omega-omega' trajectory, and sound speed all depend on alpha, but alpha is not fitted here and its uncertainty is not propagated.
  • domain assumption Stability against density perturbations is judged by the sign of the adiabatic squared sound speed (Sec. VII, Eq. 31).
    Full linear perturbation equations for f(R,L_m) gravity are not derived; positive nu_s^2 is a necessary but not sufficient condition for stability.

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Pith. "Pith review of Observational constraints on freezing quintessence in a non-linear $f(R, L_m)$ gravity." pith.science (2026). https://pith.science/paper/BMJV2GHS

@misc{pith2026241210518,
  author       = {Pith},
  title        = {Pith review of: Observational constraints on freezing quintessence in a non-linear $f(R, L_m)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BMJV2GHS}},
  note         = {Machine review of arXiv:2412.10518}
}
abstract

In this paper, we investigate the freezing quintessence scenario in late-time cosmic expansion using a non-linear $f(R, L_m)$ gravity model, $f(R,L_m)=\frac{R}{2}+L_m^\alpha$, where $\alpha$ is a free parameter. We consider a solution for this model using an appropriate parametrization of the scale factor, and then the model is constrained by observational datasets, including CC, Pantheon+ (SN), and CC+SN+BAO. Our analysis yields results aligning closely with observational data. The Hubble parameter, deceleration parameter, matter-energy density, and EoS parameter of our model exhibit expected trends over cosmic time, supporting its physical validity. Furthermore, the model demonstrates consistency with the $\Lambda$CDM model in late times, displaying freezing behavior in the $\omega - \omega'$ plane and stability against density perturbations. Our findings suggest that the modified $f(R, L_m)$ gravity model is a credible approach to describing the universe's accelerating phase.

Figures

Figures reproduced from arXiv: 2412.10518 by the authors.

Figure 1
Figure 1. FIG. 1: The model parameters, namely [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The plot displays the Hubble parameter [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The plot displays the distance modulus [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Plot of deceleration parameter [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Plot of matter-energy density [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Plot of EoS parameter [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 8
Figure 8. Figure 8: shows that the square of the sound speed re￾mains positive and 0 < ν 2 s < 1 throughout the cosmic evolution, indicating the stability of our model. This be￾havior is crucial for ensuring that density perturbations do not lead to instabilities in the system. CC SN CC+S…
Figure 7
Figure 7. Figure 7: FIG. 7: Plot of [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Observationally Constrained Cosmological model in $f(Q,\mathcal{L}_{m})$ Gravity with $H(z)$ parameterization

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