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REVIEW 5 major objections 5 minor 73 references

Physical Parameters for Stable $f(R)$ Models

T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Two stable f(R) gravity models take the universe from deceleration to acceleration without a cosmological constant.

desk verdict Routine re-plot of two known f(R) models; the z=0 agreement is installed as initial conditions and the central ODE looks wrong as written. read the letter →

arxiv 1908.04408 v1 pith:Y5GM3O3K submitted 2019-08-06 gr-qc

Gauranga C Samanta , Nisha Godani This is my paper · ORCID
classification gr-qc PACS 98.80.k98.80.Es
keywords f(R)gravityFRWcosmologymodifiedcosmicaccelerationdecelerationparameterjerkHubbleageoftheuniverse
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 attempts to show that two stable modified-gravity functions, $f(R)=R+R^2+1/R$ and $f(R)=R-\mu R_c R^2/(R^2+R_c^2)$, can produce the present accelerating expansion of a flat FRW universe without a cosmological constant. The authors work with a second-order differential equation for the Hubble parameter $H(z)$, integrate it numerically with $q_0=-0.81$ and $j_0=2.16$, and plot $H(z)$, $q(z)$, $j(z)$, matter density, the effective equation-of-state parameter, and cosmic age against redshift. They report a decelerating-to-accelerating transition, with present values $q_0\approx -0.8$, $j_0=2.16$ or $2.43$, $H_0=0.07$ GYrs$^{-1}$, and ages $12.13$ and $15.11$ GYrs, all stated to be consistent with current observations. If the derivation of the evolution equation is sound, the paper demonstrates that these $f(R)$ models reproduce the observed acceleration without a cosmological constant.

What carries the argument

The load-bearing object is Eq. (15), a second-order nonlinear ordinary differential equation for the Hubble parameter $H(z)$: $$\frac{$d^{2}$H}{$dz^{2}$}=-\frac{1}{H}\left(\frac{dH}{dz}\right)^2+\frac{3}{1+z}\frac{dH}{dz}-\frac{3f'\left[(1+z)\frac{dH}{dz}-$H^{2}$\right]+\left(\frac{f}{2}-\rho_{m0}\right)}{18f''$H^{3}$(1+z)^2},$$ with primes denoting derivatives of $f$ with respect to $R$. The equation is assembled from the Friedmann-type equations (8), the Ricci scalar relation (13), and the derivative relation (14), and it is the single formula from which every plotted quantity follows: $q(z)$ via (16), $j(z)$ via (17), $w_{\mathrm{eff}}$ via (18), and the cosmic age via (19). In the second model the constants are fixed to $\mu=1.6$, $R_c=5.7\times10^{-30}$, satisfying the stated stability bounds.

What would settle it

Re-derive Eq. (15) directly from the metric $f(R)$ field equations (6)-(7) by substituting each of the two $f(R)$ forms. If the resulting ODE contains extra terms involving $R f'$, $\ddot R$, or a different combination than the paper's Eq. (15), then the plotted $q(z)$, $j_0$, $H_0$, and ages are not consequences of the stated models. Alternatively, rerun the numerical integration with $\rho_{m0}$ matched to the quoted $H_0=0.07$ GYrs$^{-1}$ and the stated $q_0$, $j_0$; the quoted age values should be reproduced to the reported precision if the claim is right.

Watch

Extended reading notes

Core claim

The central claim is that the two $f(R)$ functions studied in this paper reproduce the observed expansion history of the universe. Starting from the $f(R)$ field equations in a flat FRW metric, the paper states a second-order nonlinear ODE, Eq. (15), for $H(z)$ and solves it with present-day initial conditions $q_0=-0.81$ and $j_0=2.16$ taken from observational fits. For the first form, and for the second form with $\mu=1.6$ and $R_c=5.7\times10^{-30}$, the integration yields a high-redshift decelerating phase and a low-redshift accelerating phase, with today's parameters $q_0\approx -0.8$, $j_0$ equal to $2.16$ or $2.43$, and $H_0=0.07$ GYrs$^{-1}$. Evaluating the cosmic age integral at large $z$ gives $12.13$ GYrs for the first model and $15.11$ GYrs for the second, which the authors compare with the observational data cited in [73].

Load-bearing premise

The numerical results rest on Eq. (15) being the correct reduction of the $f(R)$ field equations for these two models, together with a valid numerical value for the present matter density $\rho_{m0}$; the paper states the equation and the initial conditions but does not derive the equation or state the value of $\rho_{m0}$ used.

Editorial extensions

If this is right

  • The two models each show a decelerating phase at high redshift and an accelerating phase at low redshift, with $q_0\approx-0.8$ today.
  • The jerk parameter today comes out at $2.16$ for the first model and $2.43$ for the second, close to the observed value cited in [72].
  • The Hubble parameter at present is $0.07$ GYrs$^{-1}$, and the age integral converges to $12.13$ GYrs (first model) and $15.11$ GYrs (second model), both compared with the observational data cited in [73].
  • The effective equation-of-state parameter ranges between $-0.9$ and $0.3$ (first model) and $-0.3$ and $0.3$ (second model), within current observational bounds.
  • Should Eq. (15) be correct, these two stable $f(R)$ forms give a background cosmology that matches observed acceleration with no cosmological constant.

Reading between the lines

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

  • My inference: the same $H(z)$ solutions could be used to compute the Type Ia supernova distance modulus and compare directly with supernova samples; the paper stops at the background parameters, so this is a test it does not run.
  • My inference: the 2.98 GYr difference in age between the two models implies measurably different deceleration-to-acceleration transition redshifts; a precise measurement of the transition redshift would discriminate between the two functions.
  • My inference: because $\rho_{m0}$ never appears numerically, the results are not yet reproducible; fixing $\rho_{m0}$ from independent cosmological data is the minimal next step needed to make the reported values checkable.
  • My inference: the same Eq. (15) machinery could scan the second model's parameter space in $\mu$ and $R_c$, rather than the single point used here, to see which combinations survive the observational bounds.
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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 / 5 minor

Summary. The paper studies flat FRW cosmology in two f(R) gravity models, f(R)=R+R^2+1/R and f(R)=R-μR_c R^2/(R^2+R_c^2), and aims to compute the Hubble, deceleration, jerk, matter-density, effective equation-of-state, and age of the universe as functions of redshift. The authors introduce a second-order ODE for H(z) as Eq. (15), integrate it numerically for the two models, and report that the present values q0≈-0.8, j0≈2.16 or 2.43, H0≈0.07 GYrs^-1, and ages 12.13 and 15.11 GYrs are consistent with current observations. The central quantitative claims rest entirely on Eq. (15).

Significance. If Eq. (15) were a correct reduction of the f(R) field equations and the numerical inputs were fully specified, the paper would provide two worked examples of cosmological evolution in viable f(R) models; this would be a modest but useful check. However, the advertised agreement with current data is not obtained as a prediction: q0 and j0 are imposed as initial conditions, and the only dynamical equation is presented without derivation and appears inconsistent with the field equations written in Section 2. As it stands, the paper does not establish that the plotted parameters follow from the stated f(R) models, so its significance is limited to a numerical exercise with no demonstrated connection to the models.

major comments (5)
  1. [Section 3, Eq. (15)] Eq. (15) is stated without derivation, and I am unable to reproduce it from Eqs. (8), (13), and (14). Substituting R=6H(2H-(1+z)H') and \dot R=6(1+z)H[(1+z)H''-H'] into Eq. (8) gives an equation of the form 18(1+z)^2 H^2 f'' H'' = ρm0(1+z)^3 - f/2 + 3H^2 f' - 3(1+z)HH'f' + 18(1+z)H^2 H'f'', which differs from Eq. (15) in the matter term (ρm0(1+z)^3 versus ρm0), in the f' combination, and in the presence of the H'^2 and H'/(1+z) terms. Since Eq. (15) is the only dynamical input used for all figures, the reported H(z), q(z), j(z), ρm(z), weff(z), and ages do not follow from the stated f(R) field equations.
  2. [Section 3 and Section 4] The authors state in Section 3 that the present values of the deceleration and jerk parameters are taken to be q0=-0.81 and j0=2.16 from Ref. [72], and then in Section 4 report that q(0) is nearly -0.8 and j(0) is 2.16 (or 2.43) as close to observational values. At z=0, Eq. (16) gives q(0)=-1+H'(0)/H(0), and Eq. (17) gives j(0) in terms of the initial H, H', and H''; hence the z=0 agreement is an identity imposed by the chosen initial conditions, not a prediction of the model.
  3. [Section 3, after Eq. (15)] The quantity ρm0 appears in Eq. (15) but its numerical value, units, and relation to the matter density parameter Ωm0 are never given. Without this input the integration of Eq. (15) cannot be reproduced or checked. This is not a minor omission because the matter term is central to the evolution of H(z) and to the age integral in Eq. (19).
  4. [Section 2, Eqs. (13)-(14)] Eq. (14), as printed, is not the direct derivative of Eq. (13). Differentiating R=6H(2H-(1+z)H') with respect to cosmic time gives \dot R=6(1+z)H[(1+z)H''-H'], whereas Eq. (14) contains additional terms proportional to H H'' and (H')^2. This further indicates that the input used to obtain Eq. (15) is not the standard reduction of the f(R) cosmological equations.
  5. [Section 3 and Figures 1, 7] The text says that the Hubble parameter is scaled by H0 so that its present value is unity, yet Figures 1 and 7 show H(0)≈0.07 and the text states that the present value is 0.07 GYrs^-1. This inconsistency makes it unclear whether the initial condition for Eq. (15) is H(0)=1 or H(0)=0.07, which changes the numerical solution and the derived age.
minor comments (5)
  1. [Throughout] There are several typographical errors: 'Odintosov' in Refs. [1] and [38], 'Palataini' in the Introduction, 'Eintein-Λ' in the Introduction, and 'verses' in the captions of Figures 1-12; these should be corrected.
  2. [Section 3, Case II] The parameter constraints 'µ≥ 8√3/9 and Rc≤ 5.7735×10^-30' are stated without a derivation or reference, and the adopted values µ=1.6 and Rc=5.7×10^-30 are not justified in the text.
  3. [Figures 1-12] The figures lack units and error bars, and the claimed observational values are not marked on the plots, which would make the claimed consistency easier to assess.
  4. [Section 4] The quoted ages 12.13 GYrs and 15.11 GYrs are presented as derived results, but the description 't tends to 12.13 as z tends to infinity' in the text is confusing because the age is an integral from z=0 to infinity, not a limiting value of t(z) at large z; the notation should be clarified.
  5. [Introduction] The sentence 'The smallest estimates for its value are of order 55 [59,60]' is unclear; the intended meaning (a discrepancy of order 10^55 relative to naive estimates) should be stated explicitly.

Circularity Check

3 steps flagged · score 6.0 of 10

The quoted present q and j values are the initial conditions the paper says it 'took', so the Section 4 consistency checks are identities; the age and H(z) outputs remain partly model-dependent.

  1. fitted input called prediction [Section 3 (numerical setup), echoed in Section 4]
    "To find the numerical plots of these physical parameters, the present values of deceleration parameter and jerk parameter are taken to be equal to -0.81 and 2.16 respectively [72]."

    Section 4 then reports: 'At z = 0, it is nearly equal to -0.8, in each case, which is also very much closed to recent experimental value [72].' With H(0)=H0 and H'(0) chosen so that Eq. (16) gives q(0)=-0.81, the plotted q(0) ≈ -0.8 is the imposed initial condition, not a prediction of the f(R) models. The consistency claim is identity by construction.

  2. fitted input called prediction [Section 4 (jerk results), with input fixed in Section 3]
    "In the first case, its present values are obtained as 2.16 and in the second case, it is found as 2.43 which is very much closed to the current measured value [72]."

    The same Section 3 sentence fixes the present jerk to 2.16 before integration, and Eq. (17) evaluated at z=0 is determined by the z=0 initial data on H (H(0), H'(0), H''(0), or, for the stated second-order ODE, by the chosen ρm0). Reporting j(0)=2.16 for Case I as 'obtained' and then comparing 2.43 against the same input is a restatement of the input, not an independent model prediction.

1 more flagged steps
  1. fitted input called prediction [Section 3 (normalization) and Section 4 (Hubble result)]
    "Hubble parameter is scaled as H/H0 so that its value at present is unity. ... its present value is found to be equal to 0.07 GYrs−1 which is consistent with the current observational data [73]."

    The normalization enforces H(0)=H0 by definition; the dimensional value 0.07 GYrs^{-1} is the adopted observational scale, not a value produced by integrating the f(R) equations. Presenting it as 'found to be equal' to the WMAP value labels an input as an output, though this is a secondary reporting issue beside the q and j identities.

full rationale

The paper's models and external data (Nojiri-Odintsov and Hu-Sawicki forms, Rapetti et al. values, WMAP H0) are genuine external inputs, and no load-bearing self-citation chain is present. The circularity is localized to the 'present value' consistency claims: q0=-0.81 and j0=2.16 are explicitly stated as taken values, then Section 4 presents them as being close to observational values, so those two checks reduce to the inputs by construction. H0=0.07 is similarly the normalization scale rather than a derived result. The age values (12.13 and 15.11 GYrs), the H(z) curves, and the deceleration-to-acceleration transition retain model-dependent content, assuming Eq. (15) is a correct reduction of Eqs. (8), (13), and (14). The derivation of Eq. (15) is not shown and appears to contain term and matter-scaling discrepancies, and ρm0 is never specified; those are correctness and reproducibility concerns rather than circularity. Overall the central claim is only partially forced, so the appropriate score is 6.

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

The central plots rest on the chosen initial values q0=-0.81 and j0=2.16, the scale H0=0.07 GYrs^-1, and an unstated matter density ρm0. The model parameters μ and Rc are selected from viability constraints. The paper introduces no new entity, but relies on the unproved correctness of Eq. (15).

free parameters (6)
  • q0 (present deceleration parameter) = -0.81
    Taken from ref. [72] as an input initial condition, then reported as recovered at z=0.
  • j0 (present jerk parameter) = 2.16
    Taken from ref. [72] as an input initial condition; Case I yields exactly 2.16. The paper presents this as a confirmation.
  • H0 (present Hubble parameter) = 0.07 GYrs^-1
    Used to normalize H/H0 at z=0; value matches the observed Hubble constant. It is an input scale, not derived.
  • μ (Hu-Sawicki parameter) = 1.6
    Chosen in Case-II from the allowed stability region μ≥1.539; not fitted in this paper.
  • Rc (Hu-Sawicki curvature scale) = 5.7e-30
    Chosen near the upper bound 5.7735e-30 from the model's stability constraints.
  • ρm0 (present matter density) = not stated
    Appears explicitly in Eq. (15) and controls the solution, but the paper never gives its value or Ωm0. This hidden input is required to reproduce the plots.
assumptions (6)
  • domain assumption The universe is described by a flat FRW metric
    Eq. (5); the entire analysis uses a homogeneous, isotropic, spatially flat background.
  • domain assumption Matter content is pressureless dust with standard conservation
    Pressure p=0 and Eq. (11); the matter density redshifts as (1+z)^3.
  • domain assumption The f(R) field equations as written in Eqs. (6) and (7) are the correct equations of motion
    These are the standard metric f(R) equations, though Eq. (2) is mistyped.
  • domain assumption The two f(R) models are stable and pass local gravity and solar system constraints
    Attributed to refs. [1,2] in Section 3; the paper does not re-derive these conditions but relies on them.
  • ad hoc to paper Eq. (15) correctly reduces Eqs. (8), (13), and (14) to an ODE for H(z)
    Eq. (15) is stated without derivation; it is the load-bearing equation for all plots and appears algebraically incomplete.
  • domain assumption The observed q0 and j0 from Rapetti et al. [72] are accurate present values
    Used as initial conditions; if these are not the true values, the claimed consistency fails.

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Cite this review

Pith. "Pith review of Physical Parameters for Stable $f(R)$ Models." pith.science (2026). https://pith.science/paper/Y5GM3O3K

@misc{pith2026190804408,
  author       = {Pith},
  title        = {Pith review of: Physical Parameters for Stable $f(R)$ Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y5GM3O3K}},
  note         = {Machine review of arXiv:1908.04408}
}
abstract

Nojiri \& Odintsov \cite{noj1} and Hu \& Sawicki \cite{hu} have studied non-linear functions in modified gravity that explain the cosmic acceleration without cosmological constant, fulfil the conditions of local gravity \& stability and pass the solar system tests. In this paper, FRW model, a best fitted and fruitful mathematical model of the physical universe \cite{rob, hub, alph, pen} is studied in the context of these non-linear functions. The cosmological implications such as Hubble parameter, deceleration parameter, jerk parameter, matter density and the effective equation of state parameter of the universe are plotted with respect to redshift. Subsequently, the age of the universe is predicted in $f(R)$ gravity. All are found to represent the features of present phase of the universe.

Figures

Figures reproduced from arXiv: 1908.04408 by the authors.

Figure 1
Figure 1. Hubble parameter H verses redshift z 0 1 2 3 4 -0.5 0.0 0.5 1.0 z q [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. The deceleration parameter q versus redshift z In this case, the function f(R) is taken as f(R) = R + R2 + 1 R which is free from parameters. The term 1 R dominates at low curvature and produces the cosmic acceleration. This model is stable and satisfy local gravity constraints [1]. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The jerk parameter j versus redshift z 0 1 2 3 4 5 0 50 000 100 000 150 000 200 000 250 000 300 000 350 000 z ρm [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Matter density ρm versus redshift z 8 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: The equation of state parameter wef f versus redshift z 0 5 10 15 20 25 0 2 4 6 8 10 12 z t [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Age t versus redshift z 9 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Hubble parameter H verses redshift z 10 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: The deceleration parameter q versus redshift z 0 1 2 3 4 0 500 1000 1500 2000 2500 z j [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: The jerk parameter j versus redshift z 4 Results & Discussion Various parameters describing the evolution of the universe are plotted with respect to red￾shift in the context of two f(R) gravity models in the previous section (Figs. (1-12)). These 11 [PITH_FULL_IMAGE:…
Figure 10
Figure 10. Figure 10: Matter density ρm versus redshift z 0 1 2 3 4 -0.8 -0.6 -0.4 -0.2 0.0 0.2 z weff [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: The equation of state parameter wef f versus redshift z include Hubble parameter, deceleration parameter, jerk parameter, matter density, the effec￾tive equation of state parameter and age of the universe. The results obtained are discussed below: In Figures (1) and (…
Figure 12
Figure 12. Figure 12: Age t versus redshift z cases, it tends to infinity as z → ∞ and its present value is found to be equal to 0.07 GYrs−1 which is consistent with the current observational data [73]. In Figures (2) and (8), deceleration parameter is plotted with respect to redshift z fo…

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