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REVIEW 2 major objections 1 minor 129 references

Testing Exponential $f(R)$ Gravity with CMB, DESI-DR2, and Supernova Data

T0 review · 2 major / 1 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read An exponential f(R) model raises S8 enough to ease the structure tension by up to 1.2 sigma while leaving the Hubble tension untouched.

desk verdict Exponential f(R) raises S8 enough to cut tension by 1.2 sigma with DESI-DR2 but leaves H0 untouched and the Boltzmann implementation unverified. read the letter →

arxiv 2606.02666 v1 pith:T5PXG2X4 submitted 2026-06-01 gr-qc

Saurabh Verma , Archana Dixit , Anirudh Pradhan , M. S. Barak This is my paper · ORCID
classification gr-qc
keywords f(R)gravitycosmologicaltensionsS8tensionH0large-scalestructureDESI-DR2CMBconstraints
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 tests whether a specific exponential modification of gravity can ease the mismatch between early-universe and late-universe observations. When the model is fitted to CMB, DESI-DR2, supernova, and other data, the extra scalar degree of freedom produces slightly lower H0 values than Lambda CDM, so the Hubble tension stays roughly the same. At the same time the model systematically increases the predicted S8, bringing the amplitude of matter fluctuations into better agreement with large-scale structure measurements. The net result is a modest improvement in the description of structure formation without simultaneously fixing both tensions.

What carries the argument

The exponential f(R) function whose extra scalar degree of freedom is evolved inside the Boltzmann solver and jointly constrained by the combined likelihood of PPS, BBN, CC, DESI-DR2 and CMB data.

What would settle it

Repeating the full MCMC analysis with an independent Boltzmann code that evolves the same f(R) scalar degree of freedom and finding that the S8 posterior no longer shifts upward by the amount reported here would falsify the claimed improvement.

Watch

Extended reading notes

Core claim

The exponential f(R) gravity model yields H0 values slightly lower than those obtained in Lambda CDM, indicating no significant relaxation of the H0 tension. In contrast, the model predicts systematically higher values of S8, leading to a moderate alleviation of the S8 tension by up to approximately 1.2 sigma when late-time datasets are included. Overall the results show that the model does not resolve all cosmological tensions simultaneously yet provides a consistent improvement in the description of large-scale structure formation.

Load-bearing premise

The extra scalar degree of freedom is correctly implemented in the Boltzmann solver and the combined datasets carry no unaccounted systematics that would change the reported S8 shift.

Editorial extensions

If this is right

  • The model leaves the Hubble tension at its Lambda CDM level or slightly worse.
  • Inclusion of late-time data produces an S8 value that reduces the tension with weak-lensing surveys by up to 1.2 sigma.
  • The improvement is tied specifically to the higher S8 prediction rather than to any change in the background expansion history.
  • The model remains statistically consistent with all datasets used but does not outperform Lambda CDM on every tension at once.

Reading between the lines

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

  • Future surveys that tighten the S8 measurement from below could quickly rule out or confirm the size of the upward shift reported here.
  • The same f(R) scalar degree of freedom might be tested independently through its effect on the growth rate f sigma8 at intermediate redshifts.
  • If the implementation of the scalar mode in the solver is the dominant source of the S8 shift, repeating the analysis with a different code would be a direct check.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The paper considers an exponential f(R) gravity model as an extension to GR for late-time acceleration. By combining PPS, BBN, CC, DESI-DR2, and CMB datasets, it constrains the free parameter b and standard cosmological parameters. The analysis finds slightly lower H0 values than in ΛCDM, indicating no significant relaxation of the H0 tension, but systematically higher S8 values leading to a moderate alleviation of the S8 tension by up to ~1.2σ with late-time datasets. The model is concluded to provide a consistent improvement in the description of large-scale structure formation without resolving all tensions simultaneously.

Significance. If the scalar perturbation implementation in the Boltzmann solver is accurate, this work demonstrates that exponential f(R) gravity can offer a partial alleviation of the S8 tension through modified growth of structure, while being consistent with the combined datasets. The inclusion of DESI-DR2 data is a positive aspect for the constraints. However, the improvement is obtained by fitting to the same data defining the tension, and the lack of explicit perturbation equations reduces the ability to assess the result independently. The paper does not provide machine-checked proofs or reproducible code.

major comments (2)
  1. [Abstract] Abstract: The abstract states the datasets and the direction of the S8 shift but supplies no equations, error budgets, or description of how the f(R) perturbation equations were solved, so the numerical support for the 1.2-sigma claim cannot be verified.
  2. [Section describing the perturbation implementation (likely §3 or equivalent)] Section describing the perturbation implementation (likely §3 or equivalent): The headline result of higher S8 (~1.2σ tension relief) requires that the extra scalar degree of freedom produces the correct modified growth and slip relations inside the Boltzmann solver. No details are provided on the analytic perturbation equations (effective G, anisotropic stress, or initial conditions for the scalaron), which is load-bearing for the reported S8 posterior.
minor comments (1)
  1. [Abstract] The abstract could be expanded to include the best-fit value and uncertainty on the parameter b for context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight the need for greater transparency on the perturbation implementation, which we address below by committing to specific revisions. We agree that additional details will strengthen the manuscript's reproducibility and verifiability.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The abstract states the datasets and the direction of the S8 shift but supplies no equations, error budgets, or description of how the f(R) perturbation equations were solved, so the numerical support for the 1.2-sigma claim cannot be verified.

    Authors: We agree that the abstract is too concise and omits key methodological information. In the revised manuscript we will expand the abstract to include a brief statement on the solution of the f(R) perturbation equations within the Boltzmann solver and the resulting impact on the S8 posterior. revision: yes

  2. Referee: [Section describing the perturbation implementation (likely §3 or equivalent)] Section describing the perturbation implementation (likely §3 or equivalent): The headline result of higher S8 (~1.2σ tension relief) requires that the extra scalar degree of freedom produces the correct modified growth and slip relations inside the Boltzmann solver. No details are provided on the analytic perturbation equations (effective G, anisotropic stress, or initial conditions for the scalaron), which is load-bearing for the reported S8 posterior.

    Authors: The referee correctly identifies that explicit analytic expressions for the perturbation equations are absent from the current text. We will add a new subsection (or expand the existing section on the model) that derives and presents the effective gravitational constant, the anisotropic stress, the slip relation, and the initial conditions for the scalaron, together with their numerical implementation. This will allow independent verification of the modified growth responsible for the reported S8 shift. revision: yes

Circularity Check

1 steps flagged · score 4.0 of 10

S8 'prediction' and tension alleviation are direct outputs of parameter fit to the same combined datasets

  1. fitted input called prediction [Abstract]
    "the model predicts systematically higher values of S8, leading to a moderate alleviation of the S8 tension by up to ∼1.2σ when late-time datasets are included."

    S8 is obtained by fitting the free parameter b together with standard cosmological parameters to the combined PPS+BBN+CC+DESI-DR2+CMB likelihood; the 'prediction' and tension relief are therefore the best-fit posterior values to the tension-defining data rather than an a-priori model output.

full rationale

The paper performs standard MCMC parameter estimation on the exponential f(R) model using the joint likelihood from PPS+BBN+CC+DESI-DR2+CMB. The reported higher S8 values and ~1.2σ alleviation are the posterior means from that fit; no independent forward prediction or external benchmark is shown. This matches the 'fitted input called prediction' pattern but remains a normal statistical outcome rather than a definitional reduction or self-citation chain. No load-bearing self-citation or ansatz smuggling is evident from the provided text. The central claim retains independent content in the form of the specific model implementation and dataset combination.

Assumptions & free parameters 1 free parameters · 0 assumptions · 0 invented entities

Only the abstract is available, so the ledger is limited to the single free parameter named in the text.

free parameters (1)
  • b
    Single extra parameter of the exponential f(R) model that is fitted to the combined cosmological datasets.

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Cite this review

Pith. "Pith review of Testing Exponential $f(R)$ Gravity with CMB, DESI-DR2, and Supernova Data." pith.science (2026). https://pith.science/paper/T5PXG2X4

@misc{pith2026260602666,
  author       = {Pith},
  title        = {Pith review of: Testing Exponential $f(R)$ Gravity with CMB, DESI-DR2, and Supernova Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T5PXG2X4}},
  note         = {Machine review of arXiv:2606.02666}
}
abstract

One of the most popular competitors to the CDM paradigm as an explanation for the late-time acceleration of the universe is the modification of general relativity (GR), with models such as $f(R)$ gravity among the main motivations. In this study, we consider an exponential $f(R)$ gravity model as a possible extensions of the GR. The extra scalar degrees of freedom and their effects on the cosmic expansion and structure formation are continuously considered in this scenario. By combining the PPS, BBN, CC, DESI-DR2, and CMB datasets, we imposed limitations on this model. We performed a detailed statistical analysis of the free model parameter $b$ together with the standard cosmological parameters. Our analysis yields values of $H_0$ that are slightly lower than those obtained in $\Lambda$CDM, indicating no significant relaxation of the $H_0$ tension. In contrast, the model predicts systematically higher values of $S_8$, leading to a moderate alleviation of the $S_8$ tension by up to $\sim 1.2\sigma$ when late-time datasets are included. Overall, these results demonstrate that although the considered $f(R)$ gravity model does not resolve all cosmological tensions simultaneously, it provides a consistent improvement in the description of large-scale structure formation.

Figures

Figures reproduced from arXiv: 2606.02666 by the authors.

Figure 1
Figure 1. FIG. 1. The Exponential [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Marginalized 68% and 95% confidence contours in the (Ω [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Two-dimensional marginalized 68% and 95% confidence contours in the ( [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Marginalized 68% and 95% confidence contours in the [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. One-D posterior distributions and Two-D marginalized confidence regions (68% CL and 95% CL) for all considered [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

129 extracted references · 2 canonical work pages

  1. [1]

    Importantly, local constraints restrict the behavior of the model only in the high-curvature regime R≫RE, while cosmological evolution probes the regime R∼H2

  2. [2]

    Therefore, local bounds do not force the model to become observationally indistinguishable from ΛCDM at cosmological scales

    Since RE = Λb∼0.1H2 0, the ratio R/RE at late times is of order unity, and the exponential term is not negligibly small. Therefore, local bounds do not force the model to become observationally indistinguishable from ΛCDM at cosmological scales. This behavior differs from the Hu-Sawicki model, where the suppression typically follows a power-law dependence...

  3. [3]

    The scalaron tracks the minimum ofV eff at all times, withm 2 ϕ/H2 0≈37≫1 forb∼0.1 [Eq. (B11)]

  4. [4]

    The effective equation of state satisfies weff(0)≈−0.977, close to−1 and fully consistent with the near-ΛCDM constraints in Table II [Eq. (32)]

  5. [5]

    The analytical estimate correctly predicts the sign and sub-percent magnitude of theH0 shift: ∆ H0 < 0, consistent with the sub-0.2σshift in Table II [Eq. (33)]

  6. [6]

    The enhanced effective gravitational coupling Geff > Ganalytically predicts ∆ S8≈+0.005, in quantitative agreement with the numerically obtained ∆S 8 = +0.0053 from Table II. These results demonstrate that the exponentialf(R) model is not only numerically well-constrained but also analytically understood, with all major observational signatures traceable ...

    2018

  7. [7]

    Within DESI-DR2+BBN+CMB, the best-fit value of the f(R) model is S8 = 0.8129±0.0086, which is larger than the corresponding ΛCDM value S8 = 0.8076±0.0083

    The opposite trend was observed for the growth parameter S8. Within DESI-DR2+BBN+CMB, the best-fit value of the f(R) model is S8 = 0.8129±0.0086, which is larger than the corresponding ΛCDM value S8 = 0.8076±0.0083. This trend is stable when PPS and CC datasets are further added; for the f(R) model, S8 = 0.8114±0.0087 and 0.8101±0.0087, while in ΛCDM the ...

  8. [8]

    For the Hubble constant, the modified gravity scenario yields values ofH0 that are systematically slightly lower than those obtained in the ΛCDM model for all dataset combinations

    Using the constraints reported in Table II, we quantified the impact of the exponential f(R) model on the existing cosmological tensions in a dataset–dependent manner. For the Hubble constant, the modified gravity scenario yields values ofH0 that are systematically slightly lower than those obtained in the ΛCDM model for all dataset combinations. As a res...

    2018

Show all 129 references
  1. [9]

    is m2 ϕ ⏐⏐⏐ cosmo = 1 3fRR ⏐⏐⏐⏐ R∼H2 0 = Λb 2 6 eH2 0/(Λb).(B9) The ratio to the present Hubble rate is therefore m2 ϕ H2 0 = Λb 2 6H2 0 eH2 0/(Λb)∼b2 6 e1/b,(B10) where we used Λ∼H2

  2. [10]

    The scalaron thereforeadiabatically tracks ϕmin(ρ(t)) throughout the cosmological evolution, justifying the quasi-static approximation used in the background equations (4) and (5)

    For the cosmologically favoured valueb∼0.1, this gives m2 ϕ H2 0 ⏐⏐⏐⏐ b=0.1 ≈(0.1)2 6 e10≈37≫1.(B11) Since mϕ≫H0, the scalar degree of freedom rapidly relaxes to the minimum of the effective potential on timescales much shorter than the Hubble time. The scalaron thereforeadiab...

  3. [11]

    A. G. Riess et al., Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, Astron. J., 116, 1009 (1998)

    1998

  4. [12]

    Perlmutter et al., Measurements of Omega and Lambda from 42 High-Redshift Supernovae, Astrophys

    S. Perlmutter et al., Measurements of Omega and Lambda from 42 High-Redshift Supernovae, Astrophys. J., 517, 565 (1999)

    1999

  5. [13]

    P. J. E. Peebles and B. Ratra, The Cosmological Constant and Dark Energy, Rev. Mod. Phys., 75, 559 (2003)

    2003

  6. [14]

    E. J. Copeland, M. Sami, and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D, 15, 1753 (2006)

    2006

  7. [15]

    Sahni and A

    V. Sahni and A. Starobinsky, Reconstructing Dark Energy, Int. J. Mod. Phys. D, 15, 2105 (2006)

    2006

  8. [16]

    Bamba, et al.,Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests, Astrophys

    K. Bamba, et al.,Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests, Astrophys. Space Sci., 342, 155 (2012)

    2012

  9. [17]

    Li, et al., Dark Energy: a Brief Review , Front

    M. Li, et al., Dark Energy: a Brief Review , Front. Phys., 8, 828 (2013)

    2013

  10. [18]

    Motta, et al.,Taxonomy of Dark Energy Models, Universe, 7, 163 (2021)

    V. Motta, et al.,Taxonomy of Dark Energy Models, Universe, 7, 163 (2021)

    2021

  11. [19]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Introduction to Modified Gravity and Gravitational Alternative for Dark Energy, Int.J.Geom.Meth.Mod.Phys., 4, 115 (2007)

    2007

  12. [20]

    T. P. Sotiriou and V. Faraoni,f(R) Theories Of Gravity, Rev. Mod. Phys., 82, 451 (2010)

    2010

  13. [21]

    De Felice and S

    A. De Felice and S. Tsujikawa,f(R) theories ,Living Rev. Rel., 13, 3 (2010)

    2010

  14. [22]

    Clifton, et al.,Modified Gravity and Cosmology, Phys

    T. Clifton, et al.,Modified Gravity and Cosmology, Phys. Rept., 513, 1 (2012)

    2012

  15. [23]

    Capozziello and M

    S. Capozziello and M. De Laurentis, Extended Theories of Gravity, Phys. Rept., 509, 167 (2011)

    2011

  16. [24]

    Koyama, Cosmological Tests of Modified Gravity, Rept

    K. Koyama, Cosmological Tests of Modified Gravity, Rept. Prog. Phys., 79, 046902 (2016)

    2016

  17. [25]

    Cai,et al.,f(T) teleparallel gravity and cosmology, Rept

    Y.-F. Cai,et al.,f(T) teleparallel gravity and cosmology, Rept. Prog. Phys., 79, 106901 (2016)

    2016

  18. [26]

    Nojiri, et al., Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution, Phys

    S. Nojiri, et al., Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution, Phys. Rept., 692, 1 (2017)

    2017

  19. [27]

    P. G. Ferreira, Cosmological Tests of Gravity, Ann. Rev. Astron. Astrophys., 57, 335 (2019)

    2019

  20. [28]

    Bahamonde, et al., Teleparallel Gravity: From Theory to Cosmology, Rep

    S. Bahamonde, et al., Teleparallel Gravity: From Theory to Cosmology, Rep. Prog. Phys., 86 026901 (2023)

    2023

  21. [29]

    A. G. Riess, et al., Cluster Cepheids with High Precision Gaia Parallaxes, Low Zeropoint Uncertainties, and Hubble Space Telescope Photometry, ApJ, 938, 36 2022

    2022

  22. [30]

    A. G. Riess et al., A Comprehensive Measurement of the Local Value of the Hubble Constant with 1km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team, Astrophys. J. Lett., 934, L7 (2022)

    2022

  23. [31]

    Aghanim et al., Planck 2018 results

    N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys., 641, A6 (2020)

    2018

  24. [32]

    Di Valentino, et al., Reconciling Planck with the local value of H0 in extended parameter space, Phys

    E. Di Valentino, et al., Reconciling Planck with the local value of H0 in extended parameter space, Phys. Lett. B, 761, 242 (2016)

    2016

  25. [33]

    Yang, et al., Observational constraints on one-parameter dynamical dark-energy parametrizations and the H0 tension, Phys

    W. Yang, et al., Observational constraints on one-parameter dynamical dark-energy parametrizations and the H0 tension, Phys. Rev. D, 99, 043543 (2019)

    2019

  26. [34]

    Vagnozzi, New physics in light of theH 0 tension: an alternative view, Phys

    S. Vagnozzi, New physics in light of theH 0 tension: an alternative view, Phys. Rev. D, 102, 023518 (2020)

    2020

  27. [35]

    Visinelli, et al., Revisiting a negative cosmological constant from low-redshift data , Symmetry, 11, 1035 (2019)

    L. Visinelli, et al., Revisiting a negative cosmological constant from low-redshift data , Symmetry, 11, 1035 (2019)

    2019

  28. [36]

    Alestas, et al., H0 Tension, Phantom Dark Energy and Cosmological Parameter Degeneracies, Phys

    G. Alestas, et al., H0 Tension, Phantom Dark Energy and Cosmological Parameter Degeneracies, Phys. Rev. D, 101, 123516 (2020). 18

    2020

  29. [37]

    Di Valentino, et al., Dark Energy with Phantom Crossing and theH 0 tension, Entropy, 23, 404 (2021)

    E. Di Valentino, et al., Dark Energy with Phantom Crossing and theH 0 tension, Entropy, 23, 404 (2021)

    2021

  30. [38]

    Yang, et al., Dynamical dark energy after Planck CMB final release and H0 tension, Mon

    W. Yang, et al., Dynamical dark energy after Planck CMB final release and H0 tension, Mon. Not. Roy. Astron. Soc., 501, 5845 (2021)

    2021

  31. [39]

    Yang, et al., 2021−H0 Odyssey: Closed, Phantom and Interacting Dark Energy Cosmologies, JCAP, 10, 008 (2021)

    W. Yang, et al., 2021−H0 Odyssey: Closed, Phantom and Interacting Dark Energy Cosmologies, JCAP, 10, 008 (2021)

    2021

  32. [40]

    Ballardini, et aol., Phantom scalar-tensor models and cosmological tensions, JCAP, 04, 029 (2023)

    M. Ballardini, et aol., Phantom scalar-tensor models and cosmological tensions, JCAP, 04, 029 (2023)

    2023

  33. [41]

    Di Valentino, et al., In the Realm of the Hubble tension -a Review of Solutions, Class

    E. Di Valentino, et al., In the Realm of the Hubble tension -a Review of Solutions, Class. Quant. Grav., 38, 153001 (2021)

    2021

  34. [43]

    Sch¨ oneberg, et al., TheH0 Olympics: A fair ranking of proposed models, Phys

    N. Sch¨ oneberg, et al., TheH0 Olympics: A fair ranking of proposed models, Phys. Rept., 984, 1 (2022)

    2022

  35. [44]

    Kamionkowski and A

    M. Kamionkowski and A. G. Riess, The Hubble Tension and Early Dark Energy, Ann.Rev.Nucl.Part.Sci., 73, 153 (2023)

    2023

  36. [45]

    E. Abdalla, et al., Cosmology intertwined: A review of the particle physics, astrophysics, and cos- mology associated with the cosmological tensions and anomalies, JHEAp, 34, 49 (2022)

    2022

  37. [46]

    E. D. Valentino, et al., In the realm of the Hubble ten- sion—a review of solutions, Class. Quantum Grav., 38, 153001 (2021)

    2021

  38. [47]

    Perivolaropoulos and F

    L. Perivolaropoulos and F. Skara, Challenges for ΛCDM: An update, New Astron. Rev., 95, 101659 (2022)

    2022

  39. [48]

    Verma, et al., Observational Constraints onf(R) Gravity Model with DESI BAO Data., G&C, 31, 553 (2025)

    S. Verma, et al., Observational Constraints onf(R) Gravity Model with DESI BAO Data., G&C, 31, 553 (2025)

    2025

  40. [49]

    Verma, et al., Testing f(T) Gravity with Cosmological Observations: Confronting the Hubble Tension and Implications for the Late-Time Universe, JHEAP, 49, 100440 (2025)

    S. Verma, et al., Testing f(T) Gravity with Cosmological Observations: Confronting the Hubble Tension and Implications for the Late-Time Universe, JHEAP, 49, 100440 (2025)

    2025

  41. [50]

    Dixit, et al., Observational constraints and geometric diagnostics of Barboza–Alcaniz and logarithmic dark energy parametrizations, AOP, 488, 170390 (2026)

    A. Dixit, et al., Observational constraints and geometric diagnostics of Barboza–Alcaniz and logarithmic dark energy parametrizations, AOP, 488, 170390 (2026)

    2026

  42. [51]

    Verma et al., Alleviating the hubble tension with logarithmic dark energy: constraints on the wlogCDM model, Eur

    S. Verma et al., Alleviating the hubble tension with logarithmic dark energy: constraints on the wlogCDM model, Eur. Phys. J. Plus, 141, 195 (2026)

    2026

  43. [52]

    Verma, et al., Late-Time Cosmic Acceleration in tanh f(R) Gravity: Constraints from DESI DR2, Supernovae, and Big-Bang Nucleosynthesis, JHEAP, 52, 100581 (2026)

    S. Verma, et al., Late-Time Cosmic Acceleration in tanh f(R) Gravity: Constraints from DESI DR2, Supernovae, and Big-Bang Nucleosynthesis, JHEAP, 52, 100581 (2026)

    2026

  44. [53]

    Dixit, et al., Easing the Hubble Tension in f(R,Lm) Gravity: A Bayesian MCMC Analysis with CC and Pantheon Plus & SH0ES Datasets, Universe, 12, 66 (2026)

    A. Dixit, et al., Easing the Hubble Tension in f(R,Lm) Gravity: A Bayesian MCMC Analysis with CC and Pantheon Plus & SH0ES Datasets, Universe, 12, 66 (2026)

    2026

  45. [54]

    Krishnan, et al., Running Hubble Tension and aH 0 Diagnostic, Phy

    C. Krishnan, et al., Running Hubble Tension and aH 0 Diagnostic, Phy. Rev. D, 103, 103509 (2021)

    2021

  46. [55]

    Efstathiou, ToH 0 or not toH 0 ?, MNRAS, 505, 3866 (2021)

    G. Efstathiou, ToH 0 or not toH 0 ?, MNRAS, 505, 3866 (2021)

    2021

  47. [56]

    Camarena and V

    D. Camarena and V. Marra, On the use of the local prior on the absolute magnitude of Type Ia supernovae in cosmological inference, MNRAS, 504, 5164 (2021)

    2021

  48. [57]

    Camarena1 and V

    D. Camarena1 and V. Marra, A new method to build the (inverse) distance ladder, MNRAS, 495, 2630 (2020)

    2020

  49. [58]

    R. C. Nunes and E. D. Valentino, Dark sector inter- action and the supernova absolute magnitude tension, Phy. Rev. D, 104, 063529 (2021)

    2021

  50. [59]

    Alestas, D

    G. Alestas, D. Camarena et al., Late-transition vs smooth H(z) deformation models for the resolution of the Hubble crisis, Phy. Rev. D, 105, 063538 (2022)

    2022

  51. [60]

    Abdalla et al., Cosmology Intertwined: A Review of the Particle Physics, Astrophysics, and Cosmology Associated with the Cosmological Tensions and Anomalies, JHEAp, 34, 49 (2022)

    E. Abdalla et al., Cosmology Intertwined: A Review of the Particle Physics, Astrophysics, and Cosmology Associated with the Cosmological Tensions and Anomalies, JHEAp, 34, 49 (2022)

    2022

  52. [61]

    Bessa, et al., Observational constraints on Starobinsky f(R) cosmology from cosmic expansion and structure growth data, Eur

    P. Bessa, et al., Observational constraints on Starobinsky f(R) cosmology from cosmic expansion and structure growth data, Eur. Phys. J. C, 82, 506 (2022)

    2022

  53. [62]

    Leizerovich, et al., Testingf(R) gravity models with quasar X-ray and UV fluxes , Phys

    M. Leizerovich, et al., Testingf(R) gravity models with quasar X-ray and UV fluxes , Phys. Rev. D, 105, 103526 (2022)

    2022

  54. [63]

    ] C. R. Farrugia, et al., Spatial Curvature inf(R) Gravity, Phys. Rev. D, 104, 123503 (2021)

    2021

  55. [64]

    Pan, et al., Testing f(R) gravity with the simulated data of gravitational waves from the Einstein Telescope, Astrophys

    Y. Pan, et al., Testing f(R) gravity with the simulated data of gravitational waves from the Einstein Telescope, Astrophys. J., 911, 135 (2021)

    2021

  56. [65]

    Negrelli, ET AL., Solar System tests and chameleon effect inf(R) gravity, Phys

    C. Negrelli, ET AL., Solar System tests and chameleon effect inf(R) gravity, Phys. Rev. D, 101, 064005 (2020)

    2020

  57. [66]

    Capozziello, et al., Kinematic model-independent reconstruction of Palatini f(R) cosmology, Gen

    S. Capozziello, et al., Kinematic model-independent reconstruction of Palatini f(R) cosmology, Gen. Rel. Grav., 51, 2 (2019).]

    2019

  58. [67]

    Akarsu, et al., Scalar perturbations in cosmological f(R) models: the cosmic screening approach, Eur

    O. Akarsu, et al., Scalar perturbations in cosmological f(R) models: the cosmic screening approach, Eur. Phys. J. C, 78, 609 (2018)

    2018

  59. [68]

    Lazkoz, et al.,f(R) gravity modifications: from the action to the data, Eur

    R. Lazkoz, et al.,f(R) gravity modifications: from the action to the data, Eur. Phys. J. C, 78, 213 (2018)

    2018

  60. [69]

    R. C. Nunes, et al., New observational constraints on f(R) gravity from cosmic chronometers, JCAP, 01, 005 (2017)

    2017

  61. [70]

    D’Agostino and R

    R. D’Agostino and R. C. Nunes, Measurements of H0 in modified gravity theories: The role of lensed quasars in the late-time Universe, Phys. Rev. D, 101, 103505 (2020)

    2020

  62. [71]

    Wang, Canf(R) gravity relieveH0 andσ 8 tensions?, Eur

    D. Wang, Canf(R) gravity relieveH0 andσ 8 tensions?, Eur. Phys. J. C, 81, 482 (2021)

    2021

  63. [72]

    S. D. Odintsov, et al., Analyzing the H0 tension inF(R)gravity models, Nucl. Phys. B, 966, 115377 (2021)

    2021

  64. [73]

    Gaztanaga, et al., Bounds on the possible evolution of the Gravitational Constant from Cosmological Type-Ia Supernovae, Phys

    E. Gaztanaga, et al., Bounds on the possible evolution of the Gravitational Constant from Cosmological Type-Ia Supernovae, Phys. Rev. D, 65, 023506 (2002)

    2002

  65. [74]

    B. S. Wright and B. Li, Type Ia supernovae, standardisable candles, and gravity, Phys. Rev. D, 97, 083505 (2018)

    2018

  66. [75]

    Perivolaropoulos and F

    L. Perivolaropoulos and F. Skara, Hubble tension or a transition of the Cepheid SnIa calibrator parameters?, Phys. Rev. D, 104, 123511 (2021)

    2021

  67. [76]

    Alestas, et al., Late-transition vs smooth H(z) deformation models for the resolution of the Hubble crisis, Phys

    G. Alestas, et al., Late-transition vs smooth H(z) deformation models for the resolution of the Hubble crisis, Phys. Rev. D, 105, 063538 (2022)

    2022

  68. [77]

    Sapone, et al., Is there any measurable redshift dependence on the SN Ia absolute magnitude?, Phys

    D. Sapone, et al., Is there any measurable redshift dependence on the SN Ia absolute magnitude?, Phys. Dark Univ., 32, 100814 (2021)

    2021

  69. [78]

    Kumar, et al., New cosmological constraints on f(T) gravity in light of full Planck-CMB and type Ia supernovae data, Phys

    S. Kumar, et al., New cosmological constraints on f(T) gravity in light of full Planck-CMB and type Ia supernovae data, Phys. Rev. D, 107, 063529 (2023)

    2023

  70. [79]

    Ballardini and F

    M. Ballardini and F. Finelli, Type Ia supernovae data with scalar-tensor gravity, Phys. Rev. D, 106, 063531 (2022)

    2022

  71. [80]

    R. C. Nunes, et al., New observational constraints onf(R) gravity from cosmic chronometers, JCAP, 1701, 005 (2017)

    2017

  72. [81]

    Sultana, et al., Constraining f(R) models with cosmic chronometers and the HII galaxy Hub- ble diagram, Mon

    J. Sultana, et al., Constraining f(R) models with cosmic chronometers and the HII galaxy Hub- ble diagram, Mon. Not. R. Astron. Soc., 514, 5827 (2022)

    2022

  73. [82]

    Cognola, et al., A Class of viable modified f(R) gravities describing inflation and the onset of accelerated expansion, 19 Phys

    G. Cognola, et al., A Class of viable modified f(R) gravities describing inflation and the onset of accelerated expansion, 19 Phys. Rev. D, 77, 046009 (2008)

    2008

  74. [83]

    Elizalde, et al., Non-singular exponential gravity: a simple theory for early and late-time accelerated expansion, Phys

    E. Elizalde, et al., Non-singular exponential gravity: a simple theory for early and late-time accelerated expansion, Phys. Rev. D, 83, 086006 (2011)

    2011

  75. [84]

    Zhang, Testingf(R) gravity against the large scale structure of the universe, Phys

    P. Zhang, Testingf(R) gravity against the large scale structure of the universe, Phys. Rev. D 73, 123504 (2006)

    2006

  76. [85]

    Brax, et al., Chameleon cosmology, Phys

    P. Brax, et al., Chameleon cosmology, Phys. Rev. D, 70, 123518 (2004)

    2004

  77. [86]

    Brax, et al.,f(R) gravity and chameleon theories, Phys

    P. Brax, et al.,f(R) gravity and chameleon theories, Phys. Rev. D, 78, 104021 (2008)

    2008

  78. [87]

    Brax, et al., Tuning the mass of chameleon fields in Casimir force experiments, Phys

    P. Brax, et al., Tuning the mass of chameleon fields in Casimir force experiments, Phys. Rev. Lett., 104, 241101 (2010)

    2010

  79. [88]

    Brax, Screening mechanisms in modified gravity, Class

    P. Brax, Screening mechanisms in modified gravity, Class. Quantum Grav., 30, 214005 (2013)

    2013

  80. [89]

    Brax and S

    P. Brax and S. Fichet, Quantum Chameleons, Phys. Rev. D, 99, 104049 (2019)

    2019

  81. [90]

    Brax, et al., Axio-Chameleons: A Novel String-Friendly Multi-field Screening Mechanism, JCAP, 03, 015 (2024)

    P. Brax, et al., Axio-Chameleons: A Novel String-Friendly Multi-field Screening Mechanism, JCAP, 03, 015 (2024)

    2024

  82. [91]

    Martin Pernot-Borr` as, et al., A general study of chameleon fifth force in gravity space experiments, Phys. Rev. D, 100, 084006 (2019)

    2019

  83. [92]

    Brax, et al., Detecting Dark Energy in Orbit—the Cosmological Chameleon, Phys

    P. Brax, et al., Detecting Dark Energy in Orbit—the Cosmological Chameleon, Phys. Rev. D, 70, 123518 (2004)

    2004

  84. [93]

    Anastassopoulos, Improved search for solar chameleons with a GridPix detector at CAST, JCAP, 01, 032 (2019)

    V. Anastassopoulos, Improved search for solar chameleons with a GridPix detector at CAST, JCAP, 01, 032 (2019)

    2019

  85. [94]

    Brax and P

    P. Brax and P. Valageas, Cosmological perturbations in modified gravity models with screening, Phys. Rev. D, 90, 023507 (2014)

    2014

  86. [95]

    Sutiono, et al., Dynamical System Analysis Of Chameleon Mechanism in Brans-Dicke Scalar-Tensor Model, Int

    A. Sutiono, et al., Dynamical System Analysis Of Chameleon Mechanism in Brans-Dicke Scalar-Tensor Model, Int. J. Theor. Phys., 63, 187 (2024)

    2024

  87. [96]

    Singha and A

    R. Singha and A. Singh, Non-minimally coupled scalar field model: Late-time constraints and the H0 tension, Phys. Dark Uni., 50, 102131 (2025)

    2025

  88. [97]

    Singh, A

    A. Singh, A. Pradhan and A. Beesham, Cosmological aspects of anisotropic chameleonic Brans-Dicke gravity, New Astronomy, 100, 101995 (2023)

    2023

  89. [98]

    Singh and A

    A. Singh and A. Pradhan, Cosmic acceleration and ekpyrotic bounce with Chameleon field, Int. J. Geom. Methods Mod. Phys., 19, 2250216 (2022)

    2022

  90. [99]

    Chaubey, A

    R. Chaubey, A. Singh and R. Raushan, Chameleon scalar field in LRS Bianchi type I cosmological model, Gravit. Cosmol., 22, 54 (2016)

    2016

  91. [100]

    Basilakos, et al., Observational constraints on viable f(R) parametrizations with geometrical and dynamical probes, Phys

    S. Basilakos, et al., Observational constraints on viable f(R) parametrizations with geometrical and dynamical probes, Phys. Rev. D, 87, 123529 (2013)

    2013

  92. [101]

    Oliveros, A viable f(R) gravity model without oscillations in the effective dark energy, Int

    A. Oliveros, A viable f(R) gravity model without oscillations in the effective dark energy, Int. J. Mod. Phys. D, 32, 2350086 (2023)

    2023

  93. [102]

    Khoury and A

    J. Khoury and A. Weltman, Chameleon fields: Awaiting surprises for tests of gravity in space, Phys. Rev. Lett., 93, 171104 (2004)

    2004

  94. [103]

    P. S. Corasaniti and E. J. Copeland, Constraining the quintessence equation of state with cosmic microwave background acoustic peaks, Phys. Rev. D, 67, 063521 (2003)

    2003

  95. [104]

    Aghanim, et al., Planck collaboration, Planck 2018 results -V

    N. Aghanim, et al., Planck collaboration, Planck 2018 results -V. CMB power spectra and likelihoods, A&A, 641, A5 (2020)

    2018

  96. [105]

    Aghanim, et al., Planck collaboration, Planck 2018 results -VIII

    N. Aghanim, et al., Planck collaboration, Planck 2018 results -VIII. Gravitational lensing, A&A 641, A8 (2020)

    2018

  97. [106]

    Jimenez, A

    R. Jimenez, A. Loeb, et al., Constraining cosmological parameters based on relative galaxy ages, Astrophys. J., 573, 37 (2002)

    2002

  98. [107]

    Moresco, L

    M. Moresco, L. Pozzetti, et al., A 6% measurement of the Hubble parameter at z∼0.45 : direct evidence of the epoch of cosmic re-acceleration, J. Cosmol. Astropart. Phys. 2016, 014 (2016)

    2016

  99. [108]

    Moresco, et al., Raising the bar: new constraints on the Hubble parameter with cosmic chronometers at z∼2, Mon

    M. Moresco, et al., Raising the bar: new constraints on the Hubble parameter with cosmic chronometers at z∼2, Mon. Not. R. Astron. Soc. Lett., 450, L16 (2015)

    2015

  100. [109]

    Simon, L

    J. Simon, L. Verde, R. Jimenez, et al., Constraints on the redshift dependence of the dark energy potential, Phys. Rev. D, 71, 123001 (2005)

    2005

  101. [110]

    Stern, R

    D. Stern, R. Jimenez, et al., Cosmic chronometers: constraining the equation of state of dark energy. I: H(z) measurements, J. Cosmol. Astropart. Phys. 2010, 008 (2010)

    2010

  102. [111]

    Zhang, H

    C. Zhang, H. Zhang, et al., Four new observational H(z) data from luminous red galaxies in the Sloan Digital Sky Survey data release seven., Res. Astron. Astrophys, 14, 1221 (2014)

    2014

  103. [112]

    A. G. Riess, L. M. Macri, et al., A 2 .4% determination of the local value of the Hubble constant, Astrophys. J., 826, 56 (2016)

    2016

  104. [113]

    Abdul-Karim, et al., DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints, Phys

    M. Abdul-Karim, et al., DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints, Phys. Rev. D, 112, 083515 (2025)

    2025

  105. [114]

    Abdul-Karim, et al., DESI DR2 Results I: Baryon Acoustic Oscillations from the Lyman Alpha Forest, arXiv:2503.14739

    M. Abdul-Karim, et al., DESI DR2 Results I: Baryon Acoustic Oscillations from the Lyman Alpha Forest, arXiv:2503.14739

    arXiv

  106. [115]

    E. Aver, K. A. Olive, and E. D. Skillman, The effects of He I λ10830 on helium abundance determinations, JCAP, 07, 011 (2015)

    2015

  107. [116]

    R. J. Cooke, M. Pettini, and C. C. Steidel,One percent determination of the primordial deuterium abundance, Astrophys. J., 855, 102 (2018)

    2018

  108. [117]

    Aver et al., The effects of He I 10830 on helium abundance determinations, JCAP, 07, 011 (2015)

    E. Aver et al., The effects of He I 10830 on helium abundance determinations, JCAP, 07, 011 (2015)

    2015

  109. [118]

    R. J. Cooke et al., One percent determination of the primordial deuterium abundance, Astrophys. J., 855, 102 (2018)

    2018

  110. [119]

    D. Blas, J. Lesgourgues and T. Tram, The Cosmic Lin- ear Anisotropy Solving System (CLASS) II: Approxima- tion schemes, JCAP, 07, 034 (2011)

    2011

  111. [120]

    Lesgourgues, The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview, arXiv:1104.2932

    J. Lesgourgues, The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview, arXiv:1104.2932

    arXiv

  112. [121]

    Brinckmann and J

    T. Brinckmann and J. Lesgourgues, MontePython 3: boosted MCMC sampler and other features, Phys. Dark Univ., 24, 100260 (2019)

    2019

  113. [122]

    A. G. Riess, et al., Large Magellanic Cloud Cepheid standards provide a 1% foundation for the determination of the Hubble constant, Astrophys. J., 876, 85 (2019)

    2019

  114. [123]

    A. G. Riess, et al., A comprehensive measurement of the local value of the Hubble constant with 1 km s −1 Mpc−1 uncertainty from the SH0ES team, Astrophys. J. Lett., 934, L7 (2022). 20

    2022

  115. [124]

    Aghanim, et al., Planck 2018 results

    N. Aghanim, et al., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys., 641, A6 (2020)

    2018

  116. [125]

    Hildebrandt, et al., KiDS-1000 cosmology: constraints from weak gravitational lensing, Astron

    H. Hildebrandt, et al., KiDS-1000 cosmology: constraints from weak gravitational lensing, Astron. Astrophys., 633, A69 (2020)

    2020

  117. [126]

    T. M. C. Abbott, et al., Dark Energy Survey year 1 results: cosmological constraints from galaxy clustering and weak lensing, Phys. Rev. D, 98, 043526 (2018)

    2018

  118. [127]

    Hamana, et al., Cosmological constraints from cosmic shear power spectra with Subaru Hyper Suprime-Cam first-year data, Publ

    S. Hamana, et al., Cosmological constraints from cosmic shear power spectra with Subaru Hyper Suprime-Cam first-year data, Publ. Astron. Soc. Jpn., 72, 16 (2020)

    2020

  119. [128]

    Verde, T

    L. Verde, T. Treu, and A. G. Riess, Tensions between the early and late Universe, Nat. Astron., 3, 891 (2019)

    2019

  120. [129]

    Di Valentino, et al., Cosmology intertwined II: the Hubble constant tension, Class

    E. Di Valentino, et al., Cosmology intertwined II: the Hubble constant tension, Class. Quantum Grav., 38, 153001 (2021)

    2021

  121. [130]

    Di Valentino, et al., In the realm of the Hubble tension: a review of solutions, Astropart

    E. Di Valentino, et al., In the realm of the Hubble tension: a review of solutions, Astropart. Phys., 131, 102605 (2021)

    2021

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