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Modified gravity/Dynamical Dark Energy vs $\Lambda$CDM: is the game over?

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The latest Pantheon+ and DESI data exclude the standard cosmological model at about 4σ.

desk verdict A competent data-fitting paper whose '4σ exclusion' headline is not supported by the statistics; the real signal is the large ΔAIC, which deserves a proper non-nested test. read the letter →

arxiv 2412.09409 v1 pith:ENCCR6IW submitted 2024-12-12 gr-qc

Sergei D. Odintsov , Diego Sáez-Chillón Gómez , German S. Sharov This is my paper · ORCID
classification gr-qc PACS 04.50.Kd98.80.-k95.36.+x
keywords F(R)gravityexponentialdarkenergyequationofstateCPLparametrizationPantheon+supernovaeDESIbaryonacousticoscillationsΛCDMmodelAkaikeinformationcriterion
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 claims that the newest cosmic-distance data—the Pantheon+ supernova sample, DESI's first baryon-acoustic-oscillation release, cosmic chronometers, and Planck CMB shift parameters—reverse the previous situation in which ΛCDM fit everything best. Confronting a family of exponential F(R) gravity models and the model-independent CPL dark-energy equation of state with the same data, the authors find that ΛCDM is the worst-fitting model, excluded at about 4σ. If correct, the standard cosmological model would need to be replaced by a modified theory of gravity or by dark energy whose equation of state changes with time, and the Hubble-constant tension would deepen because the preferred H0 drops to about 66 km/s/Mpc. The paper's title question 'is the game over?' is answered in the affirmative.

What carries the argument

The central object is the exponential F(R) gravity family F(R)=R−2Λ(1−$e^{{−β R^α}}$) with R=R/(2Λ), where α=1 recovers standard exponential gravity and ΛCDM is recovered in the limit β→∞ or R→∞. The dynamics is solved as a dynamical system in E(a) and R(a) starting from ΛCDM-like initial conditions at high redshift. The comparison statistic is the minimum of the total χ² over SN Ia, BAO, H(z), and CMB data, judged by the Akaike information criterion AIC=min χ²+2N_p, which penalizes extra parameters; the CPL parametrization w=w0+w1(1−a) serves as a model-independent cross-check.

What would settle it

Simulate the exponential F(R) models and CPL under the null hypothesis that ΛCDM is true, generating mock Pantheon+ and DESI data with the same covariance, and count how often Δχ²≥28.99 arises; if that fraction exceeds the 4σ tail probability of about 6×10⁻⁵, the exclusion claim fails.

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Extended reading notes

Core claim

Using a combined χ² analysis of 1701 Pantheon+ supernovae, DESI DR1 and other BAO data, 32 H(z) cosmic chronometers, and Planck 2018 CMB shift parameters, the paper finds that the standard exponential F(R) model F(R)=R−2Λ(1−$e^{{−βR/(2Λ)}}$) has min χ²=2017.80 versus ΛCDM's 2046.79 (Δχ²=28.99), with an AIC difference ΔAIC=−24.99; the generalized model with α free gives the same min χ² and ΔAIC=−22.99, while CPL gives min χ²=2015.72 and ΔAIC=−27.07. The best-fit β=0.75±0.10 is far from the β→∞ ΛCDM limit, and the best-fit H0≈66 km/s/Mpc with Ωm≈0.314 are mutually excluded with ΛCDM at 1σ to 3σ. The paper concludes that ΛCDM is excluded at 4σ both by exponential F(R) gravity and by the CPL parametrization.

Load-bearing premise

The load-bearing premise is that a drop of about 29 in minimum χ² can be read directly as a 4σ exclusion, even though ΛCDM sits at the edge of the alternative models (β→∞ or w=−1) where standard likelihood-ratio rules do not automatically apply.

Editorial extensions

If this is right

  • If correct, ΛCDM is not merely disfavored but excluded at about 4σ by late-universe data, so the cosmological constant as the sole dark energy is insufficient.
  • The preferred parameter values H0≈66 km/s/Mpc and Ωm≈0.314 imply the Hubble tension with local distance-ladder measurements worsens, while matching Planck's CMB estimate better.
  • Dark energy's equation of state must be dynamical: CPL's best fit (w0≈−0.74, w1≈−0.64) deviates strongly from w=−1, consistent with modified gravity mimicking evolving dark energy.
  • The preference is driven mainly by Pantheon+ supernova data, not by DESI BAO, since removing BAO or using only DESI points leaves ΔAIC nearly unchanged.
  • A viable theory of gravity beyond general relativity must include non-trivial Ricci-scalar terms at cosmological scales.

Reading between the lines

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

  • The paper does not run a formal hypothesis test: the 4σ significance is read off from Δχ² or ΔAIC, and because ΛCDM sits at the boundary of each alternative (β→∞ or w=−1), a likelihood-ratio calibration with boundary corrections could shrink the claimed significance.
  • If the Pantheon+ covariance or its zero-point calibration shifts, part of the Δχ²≈29 could be systematic; reanalyzing with an alternative supernova covariance matrix would test this directly.
  • A concrete forecast: if DESI DR2 data strengthen the CPL w1 preference, the dynamical-dark-energy interpretation gains support; if w1 moves back toward 0, the case for modified gravity weakens.
  • The same fitting machinery could be applied to an F(R) model with an explicit R² inflation term to check whether the late-time preference survives when early-time cosmological consistency is imposed.
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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 / 5 minor

Summary. The paper fits the generalized exponential F(R) gravity model (7), its α=1 limit, wCDM, and CPL parametrizations to a combined dataset of Pantheon+ supernovae, DESI DR1 BAO, cosmic chronometers, and Planck CMB shift parameters, and compares them with ΛCDM. The central claim, repeated in the abstract and Section VI, is that ΛCDM is excluded at 4σ by both the exponential F(R) model and the CPL parametrization, with reported Δχ² values of about 29 and 31 relative to ΛCDM and ΔAIC values of about -25 and -27. However, no formal hypothesis test or null distribution is presented; the '4σ' language is not justified.

Significance. If the 4σ exclusion were statistically valid, the paper would report a decisive tension between ΛCDM and both modified gravity and dynamical dark energy using current public data, which would be a major result. The paper does provide a transparent tabulation of best-fit parameters, χ² values, and AIC differences for the models considered, and it correctly computes reduced chi-square values. Yet because the central significance claim is derived from Δχ² and ΔAIC without a calibrated test, the conclusion is not established; the reported improvements are plausibly model ranking rather than exclusion.

major comments (3)
  1. [Abstract, Section VI, Table II] The central claim that ΛCDM 'is excluded at 4σ' is not supported by any stated statistical test. The paper quotes Δχ² = 28.99 for exp(-βR) and Δχ² = 31.07 for CPL relative to ΛCDM in Table II, but a difference in minimum chi-square is not a significance level; one must state a test statistic and its distribution under the null hypothesis. For the exponential F(R) model, ΛCDM is recovered in the limit β → ∞, so the null is at the boundary of the parameter space and the parameters are unidentifiable under the null; Wilks' theorem does not apply. For CPL, ΛCDM is nested at (w0, w1) = (-1, 0), but no p-value is computed. The '4σ' language therefore has no probabilistic content in the manuscript.
  2. [Section IV, Table II] All models have reduced chi-square appreciably larger than unity: ΛCDM has min χ²/d.o.f. = 2046.79/1769 ≈ 1.157, exp(-βR) has 2017.80/1767 ≈ 1.142, and CPL has 2015.72/1767 ≈ 1.141. This indicates either that the covariance model underestimates the errors or that the models are misspecified, and it can inflate the observed Δχ² between models. The paper neither discusses this nor corrects for it, so the reported Δχ² values cannot be taken at face value as evidence against ΛCDM.
  3. [Section IV, Eq. (28); Section VI] The Akaike information criterion is used as if its differences were hypothesis-test significance: ΔAIC values of -24.99 for exp(-βR) and -27.07 for CPL are presented together with the claim of 'excluded at 4σ'. AIC is a model-selection criterion with relative weights; it does not provide a p-value or Gaussian sigma. Without a simulation or a properly calibrated evidence measure for the non-nested F(R) versus ΛCDM comparison, the AIC differences do not license the exclusion language.
minor comments (5)
  1. [Abstract] There is a typo in the abstract: 'tehir power' should be 'their power'.
  2. [Section II, around Eq. (7)] The text says 'an additional term Finf might be considered in the action (7)' and later writes 'is considered tp become negligible'; 'tp' should be 'to'.
  3. [Section IV, Eq. (27)] The likelihood expression L(θ_j) uses mabs, the absolute minimum of χ², but mabs is not defined before its use in Eq. (27); define it explicitly.
  4. [Section VI] There are several grammatical slips in the conclusions, for example 'scenarioa' should be 'scenarios', 'the the large difference' should be 'the large difference', and 'are not connected not with DESI BAO data' should be 'are not connected with DESI BAO data'.
  5. [Table III] In Table III, the column header for dataset (b) lacks the degrees-of-freedom values that are given for dataset (a); adding the d.o.f. would make the reduced chi-squares directly comparable.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's ΛCDM-exclusion claim rests on chi-square fits to public datasets; the 4σ wording is a statistical-calibration issue, not a self-referential derivation.

full rationale

The paper's central claim is that exponential F(R) gravity and the CPL parametrization fit recent cosmological data substantially better than ΛCDM, with ΛCDM 'excluded at 4σ'. This is an empirical model-comparison result: the authors solve the field equations (12)–(13), compute χ² from public Pantheon+, DESI DR1 BAO, cosmic-chronometer, and Planck data, and report best-fit parameters and ΔAIC in Tables II and III. The F(R) model in Eq. (7) is openly defined so that ΛCDM is recovered in the limits β→∞ and/or R→∞, and the paper explicitly uses this limit to set asymptotic initial conditions. This creates a genuine statistical subtlety — ΛCDM sits at a boundary or limit of the alternative model space, so converting Δχ² or ΔAIC into '4σ' requires a null distribution that the paper never states — but that is a soundness/calibration problem, not circularity. No fitted parameter is renamed as a prediction, and no result is defined in terms of itself. The paper's heavy self-citation (Refs. [10,15,17,21,22] among others) supplies the model ansatz, numerical integration techniques, and prior best-fit values; these are inputs and methods, not load-bearing evidence for the new exclusion claim. The new conclusion is falsifiable against external data and does not reduce to the self-citations. Therefore no circular step is exhibited, and the appropriate circularity score is 0.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a standard cosmological likelihood pipeline plus several assumptions: a flat FLRW background, F(R) field equations, a Planck-fixed radiation ratio, asymptotic ΛCDM initial conditions, compressed CMB priors, and an unproved conversion from Δχ²/ΔAIC to sigma. The excluded 4σ significance is the fragile item.

free parameters (8)
  • alpha (generalized F(R) exponent) = 1.002 +0.184 -0.173
    This exponent generalizes the standard exponential model; the best fit is consistent with α=1, reducing it to the standard case.
  • beta (F(R) exponential amplitude) = 0.733 +0.377 -0.273 (generalized); 0.750 +0.099 -0.079 (α=1)
    This parameter controls the size of the exponential correction; β→∞ recovers ΛCDM, and the small best-fit value drives the claimed preference over ΛCDM.
  • Omega0m = 0.3138 +0.0054 -0.0052 (F(R)); 0.2914 +0.0012 -0.0011 (ΛCDM); 0.3153 +0.0053 -0.0053 (CPL)
    This is the present matter density parameter; best fits differ by more than 3σ between F(R)/CPL and ΛCDM.
  • Omega_Lambda = 0.571 +0.058 -0.057 (generalized); 0.570 +0.010 -0.007 (α=1)
    This is the effective cosmological-constant parameter in the exponential F(R) model, fitted to the data.
  • H0 = 66.06 +1.61 -1.59 km/s/Mpc (F(R)); 68.51 +1.56 -1.53 km/s/Mpc (ΛCDM)
    This is the present Hubble constant, treated as a free parameter and also as a SN nuisance parameter.
  • w (wCDM) = -0.926 +0.018 -0.018
    This is the constant dark energy equation-of-state parameter; its value differs from -1 by several sigma in a simple Gaussian reading.
  • w0 (CPL) = -0.741 +0.054 -0.053
    This is the dark energy equation-of-state at present in the CPL parametrization.
  • w1 (CPL) = -0.635 +0.175 -0.183
    This is the time-dependence slope in the CPL parametrization; a nonzero value signals evolving dark energy.
assumptions (6)
  • domain assumption Flat FLRW metric (2) is assumed throughout.
    Standard cosmological background; no curvature or anisotropy degrees of freedom are tested.
  • standard math F(R) field equations (3) and dynamical system (12)-(13) correctly describe the model.
    Derived from action (1) by standard variation; we did not find an algebraic error, but the equations are not machine-checked.
  • domain assumption Radiation-matter ratio Xr is fixed to Planck value 2.9656e-4 (Eq. 9).
    External input from Planck reduces free parameters but can influence the fitted H0 and Ωm.
  • domain assumption Initial conditions are set by the asymptotic ΛCDM solution (10) with ε=1e-9 (Eq. 14).
    Assumes the exponential correction is negligible at the integration start; for the preferred β≈0.75, the start redshift is only z≈4, where ΛCDM asymptotics are marginal.
  • domain assumption Compressed CMB priors x_Pl=(R, ℓ_A, ωb) with z* from Planck (Eqs. 21-23) are valid for modified gravity models.
    The compressed likelihood and z* are calibrated under ΛCDM; using them for F(R) models may bias constraints.
  • ad hoc to paper Δχ²/ΔAIC can be converted into Gaussian significance '4σ' for non-nested or boundary models.
    No likelihood-ratio test, simulation, or boundary correction is given; this is the load-bearing statistical premise. Appears in the Abstract and Section VI.

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Cite this review

Pith. "Pith review of Modified gravity/Dynamical Dark Energy vs $\Lambda$CDM: is the game over?." pith.science (2026). https://pith.science/paper/ENCCR6IW

@misc{pith2026241209409,
  author       = {Pith},
  title        = {Pith review of: Modified gravity/Dynamical Dark Energy vs $\Lambda$CDM: is the game over?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ENCCR6IW}},
  note         = {Machine review of arXiv:2412.09409}
}
abstract

Over the last decades, tests on the standard model of cosmology, the so-called $\Lambda$CDM model, have been widely analysed and compared with many different models for describing dark energy. Modified gravities have played an important role in this sense as an alternative to $\Lambda$CDM model. Previous observational data has been always favouring $\Lambda$CDM model in comparison to any other model. While statistically speaking, alternative models have shown tehir power, fitting in some cases the observational data slightly better than $\Lambda$CDM, the significance and goodness of the fits were not significantly relevant to exclude the standard model of cosmology. In this paper, a generalisation of exponential $F(R)$ gravity is considered and compared with $\Lambda$CDM model by using the latest observational data. Also some well-known model independent parameterisations for the equation of state (EoS) of dark energy are explored. These scenarios are confronted with the renewed observational data involving the Pantheon plus datasets of supernovae type Ia, the Hubble parameter estimations, data from the cosmic microwave background and baryon acoustic oscillations, where the latter includes the data provided by Dark Energy Spectroscopic Instrument collaboration. Results of this analysis suggest that standard exponential $F(R)$ models provide much better fits than $\Lambda$CDM model, which is excluded at 4$\sigma$. Moreover, the parameterisations of the equation of state suggest a non-constant EoS parameter for dark energy, where $\Lambda$CDM model is also excluded at 4$\sigma$.

Figures

Figures reproduced from arXiv: 2412.09409 by the authors.

Figure 1
Figure 1. FIG. 1. Contour plots of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Contour plots of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison of 1 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

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Forward citations

Cited by 8 Pith papers

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

  1. BAO miscalibration cannot rescue late-time solutions to the Hubble tension

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    Even after rescaling BAO data to prefer H0≈73 km/s/Mpc, none of six tested late-time dark-energy models can resolve the Hubble tension once unanchored SNeIa and CMB geometry are included.

  2. Signatures of Modified Gravity on Linear Scales in a Dynamical Dark Energy Background

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    Using Planck, DESI, supernovae, and redshift-space distortions, the authors infer that modified gravity, if present, must act only below ~30 Mpc and cannot modify the large scales previously considered.

  3. Exponential $f(R)$ cosmology with massive neutrinos as a dynamical dark energy framework

    gr-qc 2025-11 conditional novelty 5.0 of 10

    Exponential f(R) gravity with massive neutrinos fits current expansion data about as well as ΛCDM and yields slightly different H0 and Σmν constraints, but does not eliminate the Hubble tension.

  4. Revisiting cosmic acceleration with DESI BAO

    astro-ph.CO 2025-07 conditional novelty 5.0 of 10

    Constraints from DESI BAO, Planck CMB, and three SN catalogs yield a negative jerk parameter at z=0 in the CPL model, suggesting cosmic acceleration is slowing.

  5. Robustness of dark energy phenomenology across different parameterizations

    astro-ph.CO 2025-02 conditional novelty 5.0 of 10

    The viability of minimally and non-minimally coupled quintessence models is robust across CPL, JBP, BA, and EXP parameterizations, with all four reproducing the models' predicted observables accurately.

  6. Is Phantom Divide Crossing in General Relativity Completely Impossible? Shortcomings in canonical and minimally coupled scalar field and Possible Solutions in $k$-essence Models

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    Phantom-to-quintessence crossing is impossible for a canonical scalar field, but can be engineered in finely-tuned k-essence theories where ghost instabilities are suppressed by a chosen kinetic coefficient.

  7. Evidence of dynamical dark energy found via the DESI DR2 Lyman$\alpha$ forest

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    Fitting seven dark-energy parameterizations to DESI DR2 Lyman-alpha BAO plus CMB, galaxy BAO, and three supernova samples yields 2–2.5σ hints of evolving dark energy (w0>−1, wa<0) that weaken below 2σ when supernovae ...

  8. An overview of what current data can (and cannot yet) say about evolving dark energy

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    The apparent preference for evolving dark energy depends strongly on which supernova catalog and which BAO survey are used, and is not robust across all independent data combinations.

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