Constraining $f(R)$ gravity and evolving dark energy via large-scale structure and phase-space trajectories

Alvaro de la Cruz-Dombriz; Savvas Nesseris; Tshepo Mathibela

REVIEW

Constraining $f(R)$ gravity and evolving dark energy via large-scale structure and phase-space trajectories

Tshepo Mathibela, Alvaro de la Cruz-Dombriz, Savvas Nesseris

General Physics · Submitted 9 Sep 2026 · 58 references

Pith's verdict: Solid MCMC analysis of f(R) models, but the missing wavenumber specification for growth predictions is a load-bearing flaw that undermines the claimed constraints.

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CitationAlvaro de la Cruz-Dombriz, Savvas Nesseris, Tshepo Mathibela. "Constraining $f(R)$ gravity and evolving dark energy via large-scale structure and phase-space trajectories." arXiv:2609.10651 (2026). https://pith.science/paper/4EK6TQFV

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Editor's note

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This paper does two things well. First, it runs a clean MCMC pipeline on two popular f(R) models using DESI DR2 BAO, Pantheon+, CMB distance priors, and an updated growth compilation. The methodology is standard and clearly described. Second, it builds phase-space diagrams (μ–γ and μ–Σ) as a diagnostic null test for f(R) models, extending an idea previously applied to Horndeski theories. That is a useful visual tool, provided the caveats are respected. The authors honestly report that the BIC does not confirm the mild AIC preference over ΛCDM, so they are not over-claiming.

The soft spot you flagged is real and bigger than the reader's report suggests. The stress-test note is correct: the growth likelihood (Eq. 25–27, plus μ from Eq. 26) depends strongly on wavenumber k in f(R) gravity. The paper never states what k value was used to compute the theoretical fσ8, f, and σ8 predictions when fitting the 30+ data points in Table IV. Figures 3 and 4 use k=300 H0 for illustration, but that choice is not carried into the MCMC section. For bHS ~ −6, the difference between k=30 H0 and k=300 H0 in fσ8 is a few percent, which could shift best-fit parameters by an amount comparable to the quoted error bars. This is not a minor omission; it is a missing specification that makes the constraints ambiguous. The reader's rationalisation about the sub-horizon approximation does not cover this.

Other soft spots: the phase-space exclusion regions are drawn from best-fit parameters, not from a full posterior propagation, so they are indicative rather than rigorous. The n=1 assumption for both models is noted but not tested. The Starobinsky parameter bS is essentially unconstrained (0.8±61.0×10^{-4}), so the Starobinsky constraints add little. Chameleon screening is invoked but not modeled; the analysis assumes the fifth force operates linearly on all scales used, which is a standard but potentially optimistic assumption.

Who this paper is for: readers who want updated MCMC bounds on two canonical f(R) models, and who are interested in the phase-space diagnostic as a complement to parameter constraints. The paper is worth engaging with, but only after the wavenumber issue is resolved.

Recommendation: The paper deserves a serious referee, but the review should condition acceptance on the authors specifying the wavenumber(s) used in the growth likelihood, justifying that choice with reference to the effective scales of the RSD data, and ideally testing sensitivity to k. If they provide that, the paper is publishable. If they cannot, the constraints are not reliable.

Referee report

Major objections (3)

[§IV.D, §V, Eq. (25)–(26)] The growth likelihood (§IV.D) is constructed using fσ₈, f, and σ₈ data, but the paper never specifies which wavenumber k is used to evaluate the effective gravitational coupling μ(k, z) in Eq. (26) when solving Eq. (25). This is a critical omission because μ—and thus the predicted fσ₈—is strongly scale-dependent; for log₁₀ b_HS ≈ -6 at z=0, the enhancement over ΛCDM at k=300 H₀ is ~10%, while at k=30 H₀ it is negligible. The constraints on b_HS and b_S derive entirely from the growth data, and without stating the effective k (or the range of scales assumed to be representative of the RSD measurements), the analysis is ambiguous and the reported error bars cannot be taken at face value. The authors must state the exact k value(s) used, justify that choice relative to the effective scales of the surveys in Table IV, and ideally test the sensitivity of the constraints to varying k within the linear regime.
[§VI, Figs. 9–10] The phase-space diagrams are constructed using the best-fit parameters from Table II, which themselves are derived from the same datasets that the null test is intended to test. This makes the test a consistency check rather than an independent prediction. The paper should clarify that these regions are forecasted allowed regions conditional on the current best-fit model, and that a truly null test would require model-independent measurements of μ, γ, Σ that do not assume the f(R) form. The statement that 'should future surveys provide data points with μ-1<0 and γ-1>1 or Σ-1<0, then the models could be directly ruled out' is valid as a sufficiency condition, but the paper should note that the boundaries of the allowed zones depend on the best-fit values and their uncertainties.
[§III, §V, Table II] In Table II, the best-fit σ₈,₀ for the Hu & Sawicki model with the full dataset is reported as 0.7764^{+0.2115}_{-0.0216}. The asymmetric errors are unusually large and strongly skewed; the upper error is an order of magnitude larger than the lower error. This suggests that the posterior for σ₈,₀ is highly non-Gaussian or that the MCMC chain has not converged properly for that parameter. The authors should comment on this and provide a convergence diagnostic (e.g., Gelman-Rubin statistic) for all chains.

Minor objections (7)

[§IV.B, Eq. (49)] The χ² expression for Pantheon+ includes a term (1^T C^{-1} 1) / (2π) inside a logarithm. This term arises from marginalisation over M_B, but the standard expression in the literature (e.g., Scolnic et al. 2018, Eq. (27)) does not include the constant log(1^T C^{-1} 1 / 2π) because it cancels in χ² differences. While this does not affect parameter constraints, it should be justified or removed for clarity.
[Table III] The column labelled 'χ²-reduced' is given but the number of degrees of freedom for each dataset combination is not defined. The authors should either state the number of data points used in each combination or show how χ²-reduced is computed.
[Table IV] Many entries list data at the same redshift from multiple references (e.g., z=0.02 has three different fσ₈ values). The text should explain whether these are independent measurements or alternative estimates from the same survey, and why all are included without, e.g., using a covariance that accounts for overlap.
[Fig. 1 caption] The potentials are said to be 'normalized' but the normalization factor is not defined. Please specify whether these are Φ/Φ_in and Ψ/Ψ_in where Φ_in, Ψ_in are the initial values at a particular redshift.
[§II, Eq. (9)] In the dynamical system, the variable Q ≡ f_R / (R f_RR). The authors should note that this quantity becomes ill-defined if f_RR = 0, which occurs in the GR limit. Since the numerical solutions approach GR, the code must handle this limit carefully; this should be mentioned.
[§III, after Eq. (23)] The text states that 'in the standard ΛCDM model ... the metric potentials turn out to be identical.' This is true only if anisotropic stress from e.g. neutrinos is neglected. A qualifying phrase such as 'at late times and ignoring neutrino free-streaming' would be more precise.
[Abstract] The phrase 'γ-1>1' in the abstract should read 'γ-1 > 0' (or keep as γ-1 > 1? Actually the paper says 'γ-1>1' which is inconsistent with Figs. 9–10 where γ-1 is negative. Please correct to 'γ-1 > 0' or clarify the intended threshold.

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The paper

The paper claims

This paper uses large-scale structure data to constrain two popular f(R) gravity models and constructs a null-test based on phase-space trajectories of gravitational parameters that could rule them out with future observations.

The reading

The authors confront two viable f(R) modified gravity models—the Hu & Sawicki and Starobinsky models—with combined background and growth-of-structure data (BAO, supernovae, CMB distance priors, and measurements of fσ₈, f, and σ₈). Using MCMC sampling, they obtain tight constraints on the deviation parameters: log₁₀ b_HS ≈ −6.3 and b_S ≈ 8×10⁻⁵, both consistent with General Relativity. Model comparison via the Akaike Information Criterion shows a mild preference for the f(R) models over flat ΛCDM (|ΔAIC| ≥ 4), but the Bayesian Information Criterion does not confirm this preference (|ΔBIC| ≤ 2). The paper also introduces a new diagnostic using two-dimensional phase-space diagrams in the (μ, γ) and (μ, Σ) planes, where ΛCDM corresponds to the fixed point (1,1). If future weak-lensing and galaxy surveys find points with μ−1 < 0 and γ−1 > 1 or Σ−1 < 0, the Hu & Sawicki and Starobinsky models could be ruled out.

What carries the argument

The dynamical system approach for the background evolution (variables x, y, χ, Ω, h) and the linear perturbation equations in the sub-horizon approximation, which produce the effective gravitational coupling μ, the slip parameter γ, and the light-deflection parameter Σ for f(R) gravity. The chi-squared likelihood construction for multiple datasets and the MCMC sampling (using wide flat priors) are the statistical machinery that delivers parameter constraints. The phase-space null-test then uses the predicted trajectories of (μ,γ) and (μ,Σ) across redshifts and scales to define the region where these models live.

Extended reading notes

Core claim

The central claim is that the Hu & Sawicki and Starobinsky f(R) models, with their deviation parameters set to best-fit values, are not excluded by current cosmological probes, but the data prefer ΛCDM when the Bayesian penalty for extra parameters is accounted for. The paper further claims that the allowed region in the (μ−1, γ−1) and (μ−1, Σ−1) phase spaces occupies the second and first quadrants respectively, providing a crisp null-test: any future data point falling in the opposite quadrants would falsify these models.

Load-bearing premise

The analysis assumes that the Hu & Sawicki and Starobinsky models with n=1 are the correct description of dark energy, that the fifth force is unscreened on cosmological scales (i.e., the chameleon mechanism does not suppress it at the scales probed), and that the sub-horizon approximation (k ≫ aH) holds for the perturbation equations down to k ≈ 30 H₀.

Editorial extensions

If this is right

If the AIC-based mild preference for f(R) models holds, future data with smaller errors could either strengthen the preference or rule it out, pushing the deviation parameters further toward the GR limit.
The phase-space null-test provides a model-specific but observationally accessible way to discard the Hu & Sawicki and Starobinsky models using upcoming lensing and clustering surveys (e.g., Euclid, Rubin) without requiring explicit parameter fitting.
The constraints on the DESI dark-energy equation of state parameters w₀ and wₐ are also reported, allowing comparison with w₀wₐCDM models that show a stronger AIC preference over ΛCDM.
The fact that the BIC does not support f(R) models over ΛCDM suggests that the additional degree of freedom is not warranted by the current data, reinforcing the standard cosmological model.
The analytical expressions for μ, γ, and Σ in the sub-horizon limit can be used in future full-shape analyses of galaxy clustering and weak lensing to perform a joint test of f(R) gravity.

Reading between the lines

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

The phase-space diagnostic introduced here could be extended to other f(R) models beyond Hu & Sawicki and Starobinsky; if those models also occupy the same quadrants, the test becomes a broader null-test for a class of modified gravity theories.
Because the sub-horizon approximation may break down at very large scales (k ~ 30 H₀), future surveys with wide angular coverage might see deviations from the predicted trajectories that are not due to the f(R) models but to the approximation itself; a full calculation without the sub-horizon assumption would strengthen the diagnostic.
The double-parameter comparison (AIC vs. BIC) highlights the well-known tension between model selection criteria; a reader interested in Bayesian model comparison might want to see the Bayes factor explicitly, which the paper does not compute.
The chameleon screening mechanism is invoked to pass solar-system tests but is not modelled in the growth equations; the derived constraints on b_HS and b_S assume the fifth force is active on cosmological scales, so a failure of screening on those scales would require reinterpreting the bounds.

Original abstract

We present a joint observational analysis confronting viable $f(R)$ modified gravity theories, specifically the Hu \& Sawicki and Starobinsky models, with background and large-scale structure (LSS) data. Utilizing Monte-Carlo Markov chain (MCMC) sampling across datasets including baryon acoustic oscillations (BAO), type Ia supernovae (SNeIa), cosmic microwave background (CMB) distance priors, and linear growth measurements ($f\sigma_8$, $f$, $\sigma_8$), we place tight constraints on the model parameters governing deviations from General Relativity. For the full dataset combination, we obtain $\log_{10} b_\mathrm{HS} = -6.325_{-1.138}^{+1.216}$ for the Hu \& Sawicki model and $b_\mathrm{S} = (0.8\pm61.0)\times10^{-4}$ for the Starobinsky model. Model comparison based on the Akaike Information Criterion indicates that these $f(R)$ extensions are statistically favored over flat $\Lambda\text{CDM}$ ($|\Delta\text{AIC}| \ge 3.99$) for the combined data. However, when considering the Bayesian Information Criterion, the evidence for support is significantly reduced. Furthermore, we construct two-dimensional phase-space diagrams in the $(\mu, \gamma)$ and $(\mu, \Sigma)$ planes across several redshifts, establishing a novel diagnostic null-test allowing us to probe for deviations from $\Lambda\text{CDM}$, corresponding to the fixed point $(1,1)$ in both planes, using LSS observables. Should future weak-lensing and galaxy surveys provide data points with $\mu-1<0$ and $\gamma-1>1$ or $\Sigma-1 < 0 $, then the aforementioned models could be directly ruled out.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,140 input tokens.

What the abstract itself states, as probabilities: claims first 0.15 · states limitation 0.85 · human participants 0.01 · controlled comparison 0.72 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present a joint observational analysis confronting viable $f(R)$ modified gravity theories, specifically the Hu \& Sawicki and Starobinsky models, with background and large-scale structure (LSS) data. Utilizing Monte-Carlo Markov chain (MCMC) sampling across datasets including baryon acoustic oscillations (BAO), type Ia supernovae (SNeIa), cosmic microwave background (CMB) distance priors, and linear growth measurements ($f\sigma_8$, $f$, $\sigma_8$), we place tight constraints on the model parameters governing deviations from General Relativity. For the full dataset combination, we obtain $\log_{10} b_\mathrm{HS} = -6.325_{-1.138}^{+1.216}$ for the Hu \& Sawicki model and $b_\mathrm{S} = (0.8\pm61.0)\times10^{-4}$ for the Starobinsky model. Model comparison based on the Akaike Information Criterion indicates that these $f(R)$ extensions are statistically favored over flat $\Lambda\text{CDM}$ ($|\Delta\text{AIC}| \ge 3.99$) for the combined data. However, when considering the Bayesian Information Criterion, the evidence for support is significantly reduced. Furthermore, we construct two-dimensional phase-space diagrams in the $(\mu, \gamma)$ and $(\mu, \Sigma)$ planes across several redshifts, establishing a novel diagnostic null-test allowing us to probe for deviations from $\Lambda\text{CDM}$, corresponding to the fixed point $(1,1)$ in both planes, using LSS observables. Should future weak-lensing and galaxy surveys provide data points with $\mu-1<0$ and $\gamma-1>1$ or $\Sigma-1 < 0 $, then the aforementioned models could be directly ruled out.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a joint observational analysis confronting viable $f(R)$ modified gravity theories, specifically the Hu \& Sawicki and Starobinsky models, with background and large-scale structure (LSS) data.	contribution 1.00	0.93	0.05	asserted 0.98
s1	Utilizing Monte-Carlo Markov chain (MCMC) sampling across datasets including baryon acoustic oscillations (BAO), type Ia supernovae (SNeIa), cosmic microwave background (CMB) distance priors, and linear growth measurements ($f\sigma_8$, $f$, $\sigma_8$), we place tight constraints on the model parameters governing deviations from General Relativity.	method 1.00	0.95	0.07	asserted 1.00
s2	For the full dataset combination, we obtain $\log_{10} b_\mathrm{HS} = -6.325_{-1.138}^{+1.216}$ for the Hu \& Sawicki model and $b_\mathrm{S} = (0.8\pm61.0)\times10^{-4}$ for the Starobinsky model.	result 1.00	0.96	0.12	asserted 1.00
s3	Model comparison based on the Akaike Information Criterion indicates that these $f(R)$ extensions are statistically favored over flat $\Lambda\text{CDM}$ ($\|\Delta\text{AIC}\| \ge 3.99$) for the combined data.	result 0.91	0.94	0.33	hedged possibility 0.51
s4	However, when considering the Bayesian Information Criterion, the evidence for support is significantly reduced.	result 0.53	0.84	0.09	asserted 0.98
s5	Furthermore, we construct two-dimensional phase-space diagrams in the $(\mu, \gamma)$ and $(\mu, \Sigma)$ planes across several redshifts, establishing a novel diagnostic null-test allowing us to probe for deviations from $\Lambda\text{CDM}$, corresponding to the fixed point $(1,1)$ in both planes, using LSS observables.	contribution 0.88	0.96	0.12	asserted 1.00
s6	Should future weak-lensing and galaxy surveys provide data points with $\mu-1<0$ and $\gamma-1>1$ or $\Sigma-1 < 0 $, then the aforementioned models could be directly ruled out.	future work 0.59	0.19	0.09	hedged possibility 1.00

What kind of paper the classifier reports this is: empirical study 0.68 · evidence from analysis of existing data 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The evolution of the normalized potentials

**Figure 2.** The redshift evolution of the growth of structure

**Figure 3.** The evolution of the light deflection parameter

**Figure 4.** The evolution of the gravitational light deflection parameter

**Figure 5.** The constraints for the parameters {Ωm,0, h, σ8} for the standard ΛCDM model. The three contour levels correspond to the 1σ, 2σ, and 3σ interval levels. {27, 3}. For a given set of cosmological parameters θ, the residual vectors for the quantities ∈ {j} are given by, ∆X = fσ8(zi , θ) − (fσ8)obs, (59) ∆Y = f(zi , θ) − (f)obs, (60) ∆Q = σ8(zi , θ) − (σ8)obs, (61) where zi are the i th observed redshifts and (fσ8)obs,(f)obs,(σ8)obs are the observed data for the quantities in {j} as shown in Table IV. V. MCMC RESULTS In this section we provide the cosmological fits for Hu & Sawicki, and…

**Figure 6.** Constraints for the parameters {Ωm,0, h, σ8, log10bHS} for the standard Hu & Sawicki model. The three contour levels correspond to the 1σ, 2σ, and 3σ interval levels. the parameters {w0, wa} are set to {−1, 0}. Lastly, the initial values for {log10bHS, bS} = {−3, 0.01} for the Hu & Sawicki and Starobinsky models respectively. For all combinations of datasets, we adopt the uniform wide flat priors on the parameters as presented in Table I. Parameter Prior Ωm,0 U[0.1, 0.6] h U[0.4, 1] σ8,0 U[0.2, 0.9] w0 U[−2, 1] wa U[−3, 2] log10bHS U[−8, 0] bS U[−0.5, 0.5] Table I. Priors imposed on…

**Figure 8.** The constraints for the parameters {Ωm,0, h, σ8} for the w0waCDM model. The three contour levels correspond to the 1σ, 2σ, and 3σ interval levels

**Figure 9.** The scale and redshift evolution of the LSS functions

**Figure 10.** The scale and redshift evolution of the LSS functions

Showing 9 of the paper's 10 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2609.10651 by the authors.

References

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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Again, m and n are positive constants with n∈N +. Some algebraic manipulations entail that Eq. (17) can be written as [18], fHS(R) =R− 2ΛHS 1 + (bHSΛHS R )n.(18) withΛ HS =m2c1/2c2 andbHS = 2c(1− 1 n )/c1. It is easy to map Eq. (18) into Eq. (16) and realize thaty(R,b HS) =[ 1 + (bHSΛHS R )n]−1 . Moreover, we see that whenbHS→ 0 the functiony(R,b HS)→ 1an...

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