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Exploring Hubble Tension Alleviation through Neutrino-Coupled Perturbed $f(R)$ Gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Coupling neutrinos to perturbed Hu–Sawicki $f(R)$ gravity moves the fitted Hubble constant to about $70$ km/s/Mpc and lowers $S_8$, leaving residual tensions near one sigma.

desk verdict Reuses the author's prior neutrino-coupled f(R) model and adds S8 constraints, but the free-Γ fit contradicts the derived-Γ theory and no code connects the equations to the quoted likelihood results. read the letter →

arxiv 2502.03190 v2 pith:4VI4KDW4 submitted 2025-02-05 gr-qc astro-ph.CO

Muhammad Yarahmadi This is my paper · ORCID
classification gr-qcastro-ph.CO PACS 04.50.Kd98.80.-k95.36.+x14.60.Pq
keywords Hubbletensionf(R)gravityneutrinocouplingHu-SawickimodelS8modifiedcosmologicalparameterestimation
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 tries to establish that adding a neutrino–gravity interaction to perturbed $f(R)$ gravity—in the Hu–Sawicki form—raises the inferred Hubble constant from the CMB-preferred value to about $70$ km/s/Mpc, easing the roughly $4\sigma$ mismatch with local distance-ladder measurements, while also lowering the clustering amplitude $S_8$ toward the values preferred by weak-lensing surveys. It compares the same model with and without a single neutrino interaction parameter $\Gamma$ on five dataset combinations built from CMB, lensing, BAO, cosmic chronometers, and supernova data. In the coupled model, the residual tension with the CMB-based $H_0$ can drop to $0.74\sigma$ and the tension with the local value to $0.95\sigma$, and the $S_8$ tension with weak-lensing priors falls below $1\sigma$ for several combinations. The paper's conclusion is not that the tensions disappear, but that the neutrino-coupled model aligns the datasets better than $f(R)$ gravity alone.

What carries the argument

The machinery is the Hu–Sawicki $f(R)$ function, $f(R) = -m^2 c_1 (R/m^2)^n / (c_2 (R/m^2)^n + 1)$ with $n=4$, together with the neutrino interaction $\Gamma = u^\mu\nabla_\mu f_R$ and $Q_\nu=-\Gamma\rho_\nu$. The paper rewrites the perturbed field equations as a first-order autonomous system in the eight phase-space variables $\eta_1,\dots,\eta_8$ (normalized perturbations of the metric potentials, curvature, matter density, and neutrino density), evolving in $N=\ln a$ according to equations (25)–(34). That system is what the numerical likelihood calculation integrates; the interaction parameter $\Gamma$ and the total neutrino mass are the extra degrees of freedom that shift $H_0$ upward and $S_8$ downward relative to the uncoupled model.

What would settle it

Fix $\Gamma=0$ and re-run the same likelihood pipeline on the same datasets; the recovered $H_0$ and $S_8$ should match the paper's uncoupled $f(R)$ tables, and if they instead reproduce the coupled-model values, the attribution of tension relief to neutrino coupling is an implementation artifact.

Watch

Extended reading notes

Core claim

The central claim, on the paper's own terms, is that a neutrino interaction term in perturbed $f(R)$ gravity acts as a lever on both the expansion rate and the growth of structure. The paper defines the coupling as $Q_\nu=-\Gamma\rho_\nu$ with $\Gamma=u^\mu\nabla_\mu f_R$, treats $\Gamma$ as a free constant alongside the neutrino mass sum, and finds that including it shifts $H_0$ upward and $S_8$ downward relative to the uncoupled model. Across all dataset combinations the coupled model returns $H_0$ between $69.82$ and $70.57$ km/s/Mpc, and for the CMB+lensing combination the residual tensions are $0.74\sigma$ with the early-universe reference and $0.95\sigma$ with the local distance ladder. For the full dataset the model gives $S_8=0.785\pm0.046$, with tensions of $1.13\sigma$ against the CMB reference and $0.70\sigma$ and $0.23\sigma$ against the two weak-lensing priors. The paper reads this as evidence that modified gravity combined with neutrino physics can adjust both the background expansion and matter clustering in a way the standard model cannot.

Load-bearing premise

The load-bearing premise is that the numerical code used in the likelihood analysis actually solves the paper's phase-space equations for the Hu–Sawicki model with the neutrino interaction term, since the paper supplies the equations but not the code or a detailed mapping from those equations to the chi-squared likelihoods.

Editorial extensions

If this is right

  • If the model is right, a single mechanism can move $H_0$ toward late-time measurements without wrecking the CMB fit, so the Hubble tension no longer has to be blamed entirely on systematics in one of the two probes.
  • The same coupling suppresses $S_8$, addressing the growth-structure tension simultaneously; with the full dataset the residual tension with the stronger weak-lensing prior is $0.23\sigma$.
  • The full dataset fixes the interaction parameter at $\Gamma = 0.64 \pm 0.14$, meaning the data prefer a nonzero energy exchange between neutrinos and the modified gravity sector.
  • The full dataset also tightens the neutrino mass sum to $\sum m_\nu < 0.119$ eV at 95% confidence, consistent with the oscillation lower bound and more restrictive than the CMB-only bound.
  • Since the uncoupled $f(R)$ model leaves tensions in the $1.1\sigma$–$1.5\sigma$ range, the comparison isolates the neutrino coupling as the ingredient doing most of the work.

Reading between the lines

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

  • The paper does not report an information-criterion comparison; adding $\Gamma$ and loosening the neutrino mass adds parameters, so the improved agreement might not survive a penalty for model complexity.
  • Treating $\Gamma$ as a constant is the simplest parametrization; a redshift-dependent interaction would trace a different $H_0$–$S_8$ trajectory that future BAO and weak-lensing data could distinguish.
  • Because $\Gamma$ and the neutrino mass sum are fitted simultaneously, a laboratory measurement that pins down the mass sum would sharpen the allowed range of $\Gamma$.
  • The same phase-space system could be used to predict growth observables such as $f\sigma_8$; comparing those predictions to redshift-space distortion measurements at low redshift would test the model independently of the $H_0$ fit.
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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

4 major / 5 minor

Summary. The paper studies two extensions of ΛCDM, perturbed f(R) gravity and perturbed f(R) gravity coupled to neutrinos, fitting them to CMB, BAO, cosmic chronometer, Pantheon, and lensing data with MontePython/CLASS. The central claim is that the coupled model shifts H0 toward the local R22 value, reducing the tension with Planck 2018 to 0.74σ and with R22 to 0.95σ, and also reduces the S8 tension relative to Planck, KiDS-1000, and DES-Y3. Parameter tables for H0, S8, Γ, and neutrino mass limits are reported for several dataset combinations.

Significance. If the numerical results were reliable, the conclusion that a neutrino coupling in perturbed f(R) gravity can simultaneously ease the Hubble and S8 tensions would be of interest to the modified-gravity and cosmological-tensions communities. The paper assembles a broad set of public datasets and presents parameter tables in a transparent format. However, the central claim is currently undermined by an internal inconsistency in the treatment of Γ, by the absence of any demonstration that the claimed equations are what the likelihood code actually solves, and by inconsistencies between the quoted tension values and the tabulated errors. No machine-checked proofs, modified Boltzmann code, chain files, or parameter files are provided, so the numerical constraints cannot be independently verified. For these reasons the significance of the result, as presented, is not established.

major comments (4)
  1. [§IV and §VII, Eqs. (23), (34)] The parameter Γ is defined in Eq. (20) as Γ = u^μ∇_μ f_R and in Eq. (23) is stated to be explicitly written in terms of the Hu-Sawicki model as Γ = d f_R/dN, i.e., a derived function of the Ricci scalar and background dynamics. In contrast, Section VII treats Γ as an independent free constant and reports constraints such as Γ = 0.64 ± 0.14. A derived quantity cannot be varied independently of the Hu-Sawicki parameters (c1, c2, n) and the background H(N); either the reported fits are not fits of the stated model, or Eq. (23) is not part of the fitted model and the connection to Hu-Sawicki f(R) is broken. This ambiguity directly affects the headline claim that neutrino physics in f(R) gravity alleviates the Hubble tension.
  2. [§VI and §V, Eqs. (24)–(39)] The manuscript states that the likelihood analysis is performed with MontePython and CLASS (refs. [111–113]), but it never shows that the phase-space system (24)–(39), the Hu-Sawicki form (22), or the neutrino interaction (20)–(21) are implemented in the Boltzmann solver. Standard CLASS does not contain this autonomous system, and no modified code, parameter files, or mapping between the η_i variables and CLASS perturbation variables is given. Without such a mapping, the quoted H0 and S8 constraints cannot be checked against the claimed theory, so the numerical results do not currently constrain the model presented in Sections II–IV.
  3. [Table VII and abstract] The tension values quoted in Table VII are not consistent with the quoted errors. For the CMB+All row, H0 = 70.46 ± 2.01 km/s/Mpc gives a tension with Planck 2018 (67.4 ± 0.5) of (70.46 − 67.4)/sqrt(2.01^2 + 0.5^2) ≈ 1.48σ, not 1.42σ, and with R22 (73.5 ± 1.04) of (73.5 − 70.46)/sqrt(2.01^2 + 1.04^2) ≈ 1.34σ, not 1.12σ. Similar discrepancies appear in the CMB+Lensing row (0.77σ versus 0.74σ for Planck; 1.12σ versus 0.95σ for R22). Since the abstract's claim of reduction to 0.74σ and 0.95σ is based on these numbers, the headline quantitative result is not supported by the tabulated data.
  4. [§VII, Eqs. (53)–(55)] The neutrino-mass section introduces η9 in Eq. (53) but the autonomous system (24)–(39) defines only η1 through η8, and no evolution equation for η9 is provided. In addition, Eq. (55) states that substituting ρν into 'Equation (1)' yields the expression for η9, but the relevant definition is Eq. (53), not the action Eq. (1). This makes the connection between the reported Σm_ν constraints and the coupled f(R)+neutrino system unclear and should be corrected before the neutrino-mass bounds can be assessed.
minor comments (5)
  1. [Table numbering and cross-references] The text refers to 'Table VII' and 'Table IX' in places that do not match the printed table captions; for example, the Hubble-tension table for the coupled model is Table VII, while the S8 tables are Tables VIII and IX. The cross-references in Sections VIII and IX need to be harmonized with the actual table numbers.
  2. [Section III, Eq. (17)] The action (17) includes L_m and L_int, and the text mentions L_ν in the same sentence; it should be clarified whether L_ν is part of L_m or a separate term, since the subsequent field equation (18) contains only T^(m) and T^(int).
  3. [Section II, Eqs. (3)–(5)] The background equations contain notation that is not defined at first use, such as the meaning of the prime in Eq. (3) and the role of c_s^2 before the dust-dominated assumption is introduced; defining these quantities in place would improve readability.
  4. [Section V, Table I] Several BAO entries in Table I are listed without uncertainties (e.g., the rows at z = 0.38 and z = 0.51), which is inconsistent with their use in a likelihood; the source of each value and error should be stated.
  5. [Section VII, Fig. 1] Figure 1 is described as showing the comparison of Σm_ν and H0 (top) and Γ and H0 (low), but the axes and contours are not legible in the text version; a higher-resolution figure with axis labels and caption definitions would be needed.

Circularity Check

2 steps flagged · score 7.0 of 10

Neutrino-coupled f(R) 'alleviation' is partly built from its own inputs: KiDS/DES S8 values are used as Gaussian priors and then reported as low residual tensions, while Γ is defined as a derived Hu-Sawicki quantity (Eq. 23) yet fitted as an independent constant.

  1. fitted input called prediction [Introduction (S8 priors paragraph) and Section VIII, 'S8 Tension with Planck 2018, KiDS, and DES for Perturbed f(R) Gravity Coupled with Neutrinos']
    "In our analysis, we also consider Gaussian priors on S8 based on measurements from KiDS-1000x{2dFLenS+BOSS} (S8 = 0.766+0.02−0.014) [22] and DES-Y3 (S8 = 0.776 ± 0.017) [21]. ... The KiDS-1000x{2dFLenS+BOSS} result of S8 = 0.766+0.02−0.014 [21] shows a tension of approximately 0.70σ with our findings, while the DES-Y3 value of S8 = 0.776 ± 0.017 [22] indicates a tension of around 0.23σ."

    The 'tensions' reported against KiDS-1000 and DES-Y3 are computed against the same S8 values that the paper says it used as Gaussian priors in its analysis. A posterior that has been pulled toward S8 = 0.766–0.776 by construction will show small residual offsets from those inputs; reporting those offsets as evidence that the model 'reduces' the S8 tension is comparing the fit to its own input. The central S8 improvement claim therefore reduces to the prior, not to a parameter-free prediction of perturbed f(R)+neutrino gravity.

  2. self definitional [Section IV, Eq. (23); Section IV autonomous system, Eq. (34); Section VII, 'Constraint on the Total Mass of Neutrinos']
    "The parameter Γ is explicitly written in terms of the Hu-Sawicki model [101]): Γ = d/dN (fR) = ... (23) ... dη8/dN = η8(3ων − 1) − η3η8 + Γη2η8 ... where Γ represents a coupling term between neutrinos and the gravitational sector. ... Our analysis shows that the most precise determination of Γ is achieved when utilizing the full dataset combination, yielding Γ = 0.64 ± 0.14."

    Eq. (23) defines Γ as the derivative of Hu-Sawicki f_R with respect to N, so Γ is determined once c1, c2, n, H(N), and R are specified. Section VII nevertheless treats Γ as an independent 'interaction parameter' and reports a fitted value Γ = 0.64 ± 0.14. If Γ is derived, the quoted constraint is not an independent measurement and cannot be used to show that neutrino coupling 'plays a significant role'; if Γ is free, then Eqs. (20)–(23) do not define the model actually sampled, and the H0/S8 constraints do not test the stated Γ = d f_R/dN relation. Either reading removes the numerical support for the headline alleviation claim.

full rationale

The paper is not entirely circular: the f(R)-only fits use standard external datasets, and the R22 Hubble comparison is a genuine out-of-fit reference. The neutrino-coupled H0 constraints also involve external late-time data. However, two load-bearing reductions make the headline claims weaker than presented. First, the claimed low S8 tensions with KiDS-1000 and DES-Y3 use the same values that the introduction says were adopted as Gaussian priors, so those residual tensions are partly forced by construction. Second, Γ is defined in Eq. (23) as a derived function of Hu-Sawicki parameters but is then fitted as an independent constant in Section VII; the reported Γ = 0.64 ± 0.14 is either redundant or tests a different model, and in either case cannot support the statement that neutrino interactions 'play a significant role.' Additionally, the numerical link between the η_i phase-space equations and the CLASS/MontePython likelihoods is not provided, which is a reproducibility gap rather than a circularity. The self-citations [98–101] are frequent and include the source of the model [101], but the model equations are reproduced in the text and the data are external, so I do not treat the self-citations as the primary circularity. Overall: multiple predictions reduce to fitted inputs, so score 7.

Assumptions & free parameters 6 free parameters · 4 assumptions · 1 invented entities

The central claim rests on a newly fitted interaction Γ and on the Hu-Sawicki parameters c1, c2, n, which are all tuned to the data being explained. The numerical implementation is not described, so the constraints cannot be traced to the stated equations. One invented entity (the neutrino interaction) is introduced with no independent falsifiable handle.

free parameters (6)
  • Γ (neutrino interaction strength) = 0.64 ± 0.14 for CMB+All; range 0.62-0.64 across datasets
    Introduced as d f_R/dN in Eq (23), but treated as a free constant in the MontePython fit (Section VII). Its fitted value is used to claim neutrinos shift H0.
  • c1 (Hu-Sawicki parameter) = 1.15e-3 for CMB+All
    Best-fit values from Section IV; sets the amplitude of the f(R) modification.
  • c2 (Hu-Sawicki parameter) = 6.55e-5 for CMB+All
    Best-fit value from Section IV; shape parameter of the Hu-Sawicki function.
  • n (Hu-Sawicki exponent) = 4
    Chosen by hand (often chosen to be n=4 for phenomenological reasons, Section IV).
  • Σ m_ν (sum of neutrino masses) = <0.119 eV at 95% for CMB+All
    Fitted in the MCMC; the H0 constraints depend on it according to Section VII.
  • ΛCDM parameters (Ω_b h^2, Ω_c h^2, 100θ_MC, τ, ln(10^10 A_s), n_s) = e.g., 0.02219±0.00018, 0.1187±0.0025, 1.0408±0.0005, 0.054, 3.042, 0.969 for CMB+All (Table IV)
    Standard cosmological parameters fitted to the same data; the H0 and S8 constraints depend on them.
assumptions (4)
  • domain assumption The Hu-Sawicki form f(R) = -m^2 c1 (R/m^2)^n / (c2 (R/m^2)^n + 1) (Eq 22) describes the modified gravity sector.
    A specific phenomenological model assumed without derivation; the paper relies on it for all perturbation and background dynamics.
  • ad hoc to paper The neutrino interaction is Q_ν = -Γρ_ν with Γ = u^μ ∇_μ f_R (Eqs 19-20), and its back-reaction is fully captured by the modified continuity equation (Eq 21).
    Postulated coupling term; the paper provides no microphysical derivation, and in the numerical analysis Γ is reduced to a fitted constant, inconsistent with the definition.
  • ad hoc to paper The perturbed field equations (10)-(14) and the phase-space system (24)-(39) are equivalent to the dynamics implemented in the likelihood code.
    No derivation or code link is shown; the chi-squared likelihoods in Section VI are standard ΛCDM-like likelihood functions and no modified-gravity transfer function is described.
  • domain assumption The datasets and likelihoods are correctly implemented in MontePython/CLASS (Section VI).
    The paper gives no chain properties, no convergence diagnostics, and no modified code; this is a standard but unverified assumption.
invented entities (1)
  • Neutrino-modified-gravity interaction term Γ (Q_ν = -Γρ_ν)
    purpose: Energy exchange between neutrinos and the extra scalar degree of freedom of f(R) gravity, introduced to shift late-time expansion and H0.
    The only handle is the fitted value Γ≈0.64 from the same datasets; the paper presents no independent prediction (e.g., a signature in neutrino observables or redshift-dependent interaction) that could falsify it outside the fit.

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Pith. "Pith review of Exploring Hubble Tension Alleviation through Neutrino-Coupled Perturbed $f(R)$ Gravity." pith.science (2026). https://pith.science/paper/4VI4KDW4

@misc{pith2026250203190,
  author       = {Pith},
  title        = {Pith review of: Exploring Hubble Tension Alleviation through Neutrino-Coupled Perturbed $f(R)$ Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4VI4KDW4}},
  note         = {Machine review of arXiv:2502.03190}
}
abstract

This work examines the Hubble constant (\(H_0\)) tension within the frameworks of perturbed \(f(R)\) gravity and perturbed \(f(R)\) gravity coupled with neutrinos, using lastest observational data. The datasets incorporate the Cosmic Microwave Background (CMB), Baryon Acoustic Oscillations (BAO), Cosmic Chronometers (CC), lensing, and Pantheon supernovae. We compare the ability of these models to bridge the discrepancy between Planck 2018 (\(H_0 = 67.4 \pm 0.5 \ \text{km/s/Mpc}\)) and the local R22 measurement (\(H_0 = 73.5 \pm 1.04 \ \text{km/s/Mpc}\)). In perturbed \(f(R)\) gravity, the derived \(H_0\) values align closely with Planck, leaving a substantial tension with R22. The inclusion of neutrino interactions introduces additional parameters that shift \(H_0\) toward higher values, reducing the tension with local measurements. Notably, the coupled model achieves a smaller residual tension compared to the standalone perturbed \(f(R)\) model, indicating that neutrino physics plays a significant role in modifying the late-time expansion dynamics. While both models provide insights into addressing the Hubble tension, the coupled \(f(R)\) gravity with neutrinos offers a more consistent alignment across the datasets.

Figures

Figures reproduced from arXiv: 2502.03190 by the authors.

Figure 1
Figure 1. Top panel: The comparison of Pmν, and H0. Low panel: The comparison of Γ, and H0 from the variety of dataset combination in perturbed f(R) + mν model values against the established Planck 2018 measurement (HPlanck 2018 0 = 67.4 ± 0.5 km/s/Mpc) [3] and the R22 result (HR22 0 = 73.5 ± 1.04 km/s/Mpc), highlighting the corresponding statistical tensions. For the Cosmic Microwave Background (CMB) and lensing combination,… view at source ↗
Figure 2
Figure 2. The comparison of Ωm vs H0 measurement for different combination of data sets(CMB+Other)for perturbed f(R) gravity model. Dataset Combination H0 (km/s/Mpc) Tension with Planck 2018 Tension with R22 CMB + Lensing 69.82 ± 3.12 0.74σ 0.95σ CMB + BAO + Lensing 70.37 ± 2.32 1.14σ 1.03σ CMB + Lensing + CC 70.26 ± 2.38 1.13σ 0.98σ CMB + Lensing + Pantheon 70.57 ± 2.54 1.17σ 0.88σ CMB + All Data (BAO, CC, Pantheon, Lensing)… view at source ↗
Figure 3
Figure 3. The comparison of cosmological parameters for different combination of data sets(CMB+Other) for perturbed [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The comparison of Ωm vs H0 measurement for different combination of data sets(CMB+Other) perturbed f(R) gravity model coupled with neutrinos. S8 Tension with Planck 2018, KiDS, and DES for Perturbed f(R) Gravity Coupled with Neutrinos In our analysis within the framewo…
Figure 5
Figure 5. Figure 5: The comparison of cosmological parameters for different combination of data sets(CMB+Other) for perturbed [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: The comparison of S8 for different combination of data sets(CMB+Other) for perturbed f(R) gravity model. neutrinos, highlighting the impact of neutrino coupling on the clustering amplitude. In contrast, [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: The comparison of S8 for different combination of data sets(CMB+Other) for perturbed f(R) gravity model coupled with neutrinos. late-Universe measurements persisted, particularly when combining CMB data with other late-time probes. In contrast, the perturbed f(R) gravi…

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Reference graph

Works this paper leans on

112 extracted references · 19 canonical work pages

  1. [98]

    2024, MNRAS,Pages 3055–3067, https://doi.org/10.1093/mnras/stae2257

    M Yarahmadi, A Salehi. 2024, MNRAS,Pages 3055–3067, https://doi.org/10.1093/mnras/stae2257

    doi:10.1093/mnras/stae2257 2024

  2. [101]

    Yarahmadi et al

    M. Yarahmadi et al. ,2025,physics of dark universe, https://doi.org/10.1016/j.dark.2025.101824

    work page arXiv 2025

  3. [1]

    Planck Collaboration, P. A. R. Ade et al. Astron. Astrophys. 571 (2014) A16, arXiv:1303.5076

    arXiv 2014

  4. [2]

    Planck Collaboration, P. A. R. Ade et al. Astron. Astrophys. 594 (2016) A13, arXiv:1502.01589

    arXiv 2016

  5. [3]

    Aghanim et al

    Planck Collaboration, N. Aghanim et al. Planck 2018 results. VI. Cosmological parameters. arXiv:1807.06209

    arXiv 2018

  6. [4]

    A. G. Riess, S. Casertano, W. Yuan et al., Astrophys J. 861 (2018) 126, arXiv: 1804.10655

    arXiv 2018

  7. [5]

    A. G. Riess, S. Casertano, W. Yuan, L. M. Macri and D. Scolnic, Astrophys J. 876 (2019) 85, arXiv: 1903.07603

    arXiv 2019

  8. [6]

    Verde, T

    L. Verde, T. Treu and A. G. Riess, Nature Astronomy 3 (2019) 891, arXiv:1907.10625

    arXiv 2019

Show all 112 references
  1. [7]

    W. L. Freedman, B. F. Madore, D. Hatt et al., Astrophys J. 882 (2019) 34, arXiv:1907.05922

    2019 arXiv

  2. [8]

    W. L. Freedman B. F. Madore, T. Hoyt et al., arXiv:2002.01550

    2002 arXiv

  3. [9]

    K. C. Wong, S. H. Suyu, G. C.-F. Chen et al., 2019, Mon. Not. Roy. Astron. Soc., doi:10.1093/mnras/stz3094, arXiv:1907.04869

    2019 arXiv

  4. [10]

    Wei and F

    J.-J. Wei and F. Melia, Astrophys.J. 897 (2020) 2, 127, arXiv:2005.10422

    2020 arXiv

  5. [11]

    Denzel, J

    P. Denzel, J. P. Coles, P. Saha and L. L. R. Williams, The Hubble constant from eight time-delay galaxy lenses, arXiv:2007.14398

    2007 arXiv

  6. [12]

    E. J. Baxter and B. D. Sherwin, arXiv:2007.04007

    2007 arXiv

  7. [13]

    C. D. Huang, A. G. Riess, W. Yuan et al., 2019, Astrophys. J. 889 (2020) 1, 5, arXiv:1908.10883

    2020 arXiv

  8. [14]

    Comments on the Scalar-Tensor Theory

    Bergmann, Peter G. 1968. “Comments on the Scalar-Tensor Theory.” International Journal of Theoretical Physics 1(1):25–36. doi: 10.1007/BF00668828

    1968 doi

  9. [15]

    F(R) Theories

    De Felice, Antonio, and Shinji Tsujikawa. 2010. “F(R) Theories.”

    2010

  10. [16]

    F(R) Gravities, Killing Spinor Equations, ‘BPS’ Domain Walls and Cosmology

    Liu, Haishan, H. L¨ u, and Zhao-Long Wang. 2012. “F(R) Gravities, Killing Spinor Equations, ‘BPS’ Domain Walls and Cosmology.” Journal of High Energy Physics 2012(2):83. doi: 10.1007/JHEP02(2012)083

    2012 doi

  11. [17]

    Non-Linear Lagrangians and Cosmological Theory

    Buchdahl, H. A. 1970. “Non-Linear Lagrangians and Cosmological Theory.” Monthly Notices of the Royal Astronomical Society 150(1):1–8. doi: 10.1093/mnras/150.1.1

    1970 doi

  12. [18]

    On Continued Gravitational Contraction

    Oppenheimer, J. R., and H. Snyder. 1939. “On Continued Gravitational Contraction.” Physical Review 56(5):455–59. doi: 10.1103/PhysRev.56.455

    1939 doi

  13. [19]

    Einstein Static Universe in Rastall Theory of Gravity

    Darabi, F., K. Atazadeh, and Y. Heydarzade. 2017. “Einstein Static Universe in Rastall Theory of Gravity.”

    2017

  14. [20]

    Note on the Evolution of the Gravitational Potential in Rastall Scalar Field Theories

    Fabris, J. C., M. Hamani Daouda, and O. F. Piattella. 2012. “Note on the Evolution of the Gravitational Potential in Rastall Scalar Field Theories.” Physics Letters B 711(3–4):232–37. doi: 10.1016/j.physletb.2012.04.020

    2012 doi

  15. [21]

    Joint analysis of DES Year 3 data and CMB lensing from SPT and Planck III: Combined cosmological constraints,

    T. M. C. Abbott et al. (DES, SPT), “Joint analysis of DES Year 3 data and CMB lensing from SPT and Planck III: Combined cosmological constraints,” (2022), arXiv:2206.10824 [astro-ph.CO]

    2022 arXiv

  16. [22]

    KiDS-1000 Cosmology: Multi-probe weak gravitational lensing and spectroscopic galaxy clustering constraints,

    Catherine Heymans et al., “KiDS-1000 Cosmology: Multi-probe weak gravitational lensing and spectroscopic galaxy clustering constraints,” Astron. Astrophys. 646, A140 (2021), arXiv:2007.15632 [astro-ph.CO]

    2021 arXiv

  17. [23]

    Rastall, Phys

    P. Rastall, Phys. Rev. D6, 3357 (1972)

    1972

  18. [24]

    Heydarzade and F

    Y. Heydarzade and F. Darabi, Phys. Lett. B771, 365 (2017), arXiv:1702.07766

    2017 arXiv

  19. [25]

    J. P. Morais Graca and I. P. Lobo, Eur. Phys. J. C78, 101 (2018), arXiv:1711.08714

    2018 arXiv

  20. [26]

    K. A. Bronnikov, J. C. Fabris, O. F. Piattella, and E. C. Santos, Gen. Rel. Grav. 48, 162 (2016), arXiv:1606.06242

    2016 arXiv

  21. [27]

    Heydarzade, H

    Y. Heydarzade, H. Moradpour, and F. Darabi, Can. J. Phys. 95, 1253 (2017), arXiv:1610.03881

    2017 arXiv

  22. [28]

    Spallucci and A

    E. Spallucci and A. Smailagic, Int. J. Mod. Phys. D27, 1850003 (2017), arXiv:1709.05795

    2017 arXiv

  23. [29]

    Kumar and S

    R. Kumar and S. G. Ghosh, Eur. Phys. J. C78, 750 (2018), arXiv:1711.08256

    2018 arXiv

  24. [30]

    Z. Xu, X. Hou, X. Gong, and J. Wang, Eur. Phys. J. C78, 513 (2018), arXiv:1711.04542

    2018 arXiv

  25. [31]

    Sadeghi, (2018), arXiv:1809.08698

    M. Sadeghi, (2018), arXiv:1809.08698

    2018 arXiv

  26. [32]

    I. P. Lobo, H. Moradpour, J. P. Morais Gracurlca, and I. G. Salako, Int. J. Mod. Phys. D27, 1850069 (2018), arXiv:1710.04612

    2018 arXiv

  27. [33]

    K. Lin, Y. Liu, and W.-L. Qian, Gen. Rel. Grav. 51, 62 (2019), arXiv:1809.10075

    2019 arXiv

  28. [34]

    Lin and W.-L

    K. Lin and W.-L. Qian, Chin. Phys. C43, 083106 (2019), arXiv:1812.10100

    2019 arXiv

  29. [35]

    A. S. Al-Rawaf and M. O. Taha, Phys. Lett. B366, 69 (1996)

    1996

  30. [36]

    A. S. Al-Rawaf and M. O. Taha, Gen. Rel. Grav. 28, 935 (1996)

    1996

  31. [37]

    C. E. M. Batista, M. H. Daouda, J. C. Fabris, O. F. Piattella, and D. C. Rodrigues, Phys. Rev. D85, 084008 (2012), 22 arXiv:1112.4141

    2012 arXiv

  32. [38]

    J. C. Fabris, O. F. Piattella, D. C. Rodrigues, C. E. M. Batista, and M. H. Daouda, Int. J. Mod. Phys. Conf. Ser. 18, 67 (2012), arXiv:1205.1198

    2012 arXiv

  33. [39]

    K. A. Bronnikov, J. C. Fabris, O. F. Piattella, D. C. Rodrigues, and E. C. Santos, Eur. Phys. J. C77, 409 (2017), arXiv:1701.06662

    2017 arXiv

  34. [40]

    Darabi, K

    F. Darabi, K. Atazadeh, and Y. Heydarzade, Eur. Phys. J. Plus 133, 249 (2018), arXiv:1710.10429

    2018 arXiv

  35. [41]

    Yuan and P

    F.-F. Yuan and P. Huang, Class. Quant. Grav. 34, 077001 (2017), arXiv:1607.04383

    2017 arXiv

  36. [42]

    J. C. Fabris, M. H. Daouda, and O. F. Piattella, Phys. Lett. B711, 232 (2012), arXiv:1109.2096

    2012 arXiv

  37. [43]

    C. E. M. Batista, J. C. Fabris, O. F. Piattella, and A. M. Velasquez-Toribio, Eur. Phys. J. C73, 2425 (2013), arXiv:1208.6327

    2013 arXiv

  38. [44]

    Moradpour, Y

    H. Moradpour, Y. Heydarzade, F. Darabi, and I. G. Salako, Eur. Phys. J. C77, 259 (2017), arXiv:1704.02458

    2017 arXiv

  39. [45]

    Weinberg, Rev

    S. Weinberg, Rev. Mod. Phys. 61, 1 (1989), [,569(1988)]

    1989

  40. [46]

    Padmanabhan, Phys

    T. Padmanabhan, Phys. Rept. 380, 235 (2003), arXiv:hep-th/0212290

    2003 arXiv

  41. [47]

    E. J. Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod. Phys. D15, 1753 (2006), arXiv:hep-th/0603057

    2006 arXiv

  42. [48]

    Bamba, S

    K. Bamba, S. Capozziello, S. Nojiri, and S. D. Odintsov, Astrophys. Space Sci. 342, 155 (2012), arXiv:1205.3421

    2012 arXiv

  43. [49]

    Li, X.-D

    M. Li, X.-D. Li, S. Wang, and Y. Wang, Commun. Theor. Phys. 56, 525 (2011), arXiv:1103.5870

    2011 arXiv

  44. [50]

    R. R. Caldwell, R. Dave, and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998), arXiv:astro-ph/9708069

    1998 arXiv

  45. [51]

    Sen, JHEP 07, 065 (2002), arXiv:hep-th/0203265

    A. Sen, JHEP 07, 065 (2002), arXiv:hep-th/0203265

    2002 arXiv

  46. [52]

    Armendariz-Picon, V

    C. Armendariz-Picon, V. F. Mukhanov, and P. J. Steinhardt, Phys. Rev. Lett. 85, 4438 (2000), arXiv:astro-ph/0004134

    2000 arXiv

  47. [53]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Phys. Lett. B562, 147 (2003), arXiv:hep-th/0303117

    2003 arXiv

  48. [54]

    A. Yu. Kamenshchik, U. Moschella, and V. Pasquier, Phys. Lett. B511, 265 (2001), arXiv:gr-qc/0103004

    2001 arXiv

  49. [55]

    A. G. Cohen, D. B. Kaplan, and A. E. Nelson, Phys. Rev. Lett. 82, 4971 (1999), arXiv:hep-th/9803132

    1999 arXiv

  50. [56]

    Li, Phys

    M. Li, Phys. Lett. B603, 1 (2004), arXiv:hep-th/0403127

    2004 arXiv

  51. [57]

    Wang, Y.-g

    B. Wang, Y.-g. Gong, and E. Abdalla, Phys. Lett. B624, 141 (2005), arXiv:hep-th/0506069

    2005 arXiv

  52. [58]

    Cai, Phys

    R.-G. Cai, Phys. Lett. B657, 228 (2007), arXiv:0707.4049

    2007 arXiv

  53. [59]

    Wei and R.-G

    H. Wei and R.-G. Cai, Phys. Lett. B660, 113 (2008), arXiv:0708.0884

    2008 arXiv

  54. [61]

    F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne’eman, Phys. Rept. 258, 1 (1995), arXiv:gr-qc/9402012

    1995 arXiv

  55. [62]

    Harko, F

    T. Harko, F. S. N. Lobo, S. Nojiri, and S. D. Odintsov, Phys. Rev. D84, 024020 (2011), arXiv:1104.2669

    2011 arXiv

  56. [63]

    Brans and R

    C. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961), [,142(1961)]

    1961

  57. [64]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Phys. Lett. B631, 1 (2005), arXiv:hep-th/0508049

    2005 arXiv

  58. [65]

    Lovelock, J

    D. Lovelock, J. Math. Phys. 12, 498 (1971)

    1971

  59. [66]

    Horava, Phys

    P. Horava, Phys. Rev. D79, 084008 (2009), arXiv:0901.3775

    2009 arXiv

  60. [67]

    Horava, Phys

    P. Horava, Phys. Rev. Lett. 102, 161301 (2009), arXiv:0902.3657

    2009 arXiv

  61. [68]

    K. Lin, A. Wang, Q. Wu, and T. Zhu, Phys. Rev. D84, 044051 (2011), arXiv:1106.1486

    2011 arXiv

  62. [69]

    K. Lin, S. Mukohyama, A. Wang, and T. Zhu, Phys. Rev. D89, 084022 (2014), arXiv:1310.6666

    2014 arXiv

  63. [70]

    and De Laurentis M.,(2011), Phys

    Capozziello S. and De Laurentis M.,(2011), Phys. Rept. 509 , 167-321 [arXiv:1108.6266 [gr-qc]]

    2011 arXiv

  64. [71]

    Capozziello S.,(2002), Int. J. Mod. Phys. D 11 , 483-492 [arXiv:gr-qc/0201033 [gr-qc]]

    2002 arXiv

  65. [72]

    and Francaviglia M.,(2008) , Gen

    Capozziello S. and Francaviglia M.,(2008) , Gen. Rel. Grav. 40 357-420, arXiv:0706.1146 [astro-ph]. 3, 16, 39, 50

    2008 arXiv

  66. [73]

    and Tsujikawa S.,(2010), Living Rev

    De Felice A. and Tsujikawa S.,(2010), Living Rev. Rel. 13 , 3 [arXiv:1002.4928 [gr-qc]]

    2010 arXiv

  67. [74]

    and Odintsov S

    Nojiri S. and Odintsov S. D.,(2006), eConf C0602061 , 06 [arXiv:hep-th/0601213 [hep-th]]

    2006 arXiv

  68. [75]

    D.,(2003), Phys

    Nojir S.i and Odintsov S. D.,(2003), Phys. Rev. D 68 , 123512 [arXiv:hep-th/0307288 [hep-th]]

    2003 arXiv

  69. [76]

    Sotiriou T. P. and Faraoni V.,(2010), Rev. Mod. Phys. 82 , 451-497 [arXiv:0805.1726 [gr-qc]]

    2010 arXiv

  70. [77]

    M., 1993, ApJ, 413, L105

    Phillips M. M., 1993, ApJ, 413, L105

    1993

  71. [78]

    Scolnic, D. et al. ,(2022), ApJ, Volume 938, Number 2

    2022

  72. [79]

    Shulei Cao, Bharat Ratra, Phys. Rev. D 107, 103521 (2023)

    2023

  73. [81]

    et al., arXiv:1907.12875 , Planck Collaboration

    Aghanim, N. et al., arXiv:1907.12875 , Planck Collaboration

    1907 arXiv

  74. [82]

    Planck 2018 results-VIII

    Aghanim, N.; Akrami, Y.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.; Bartolo, N.; Basak, S. Planck 2018 results-VIII. Gravitational lensing. Astron. Astrophys. 2020, 641, A8

    2018

  75. [83]

    and Rubin, D

    Gelman, A. and Rubin, D. B.,1992, Statist. Sci. 7, 457–472

    1992

  76. [84]

    and Bridle, S.,2002, Phys

    Lewis, A. and Bridle, S.,2002, Phys. Rev. D 66, 103511

    2002

  77. [85]

    Lewis, A.,2013, Phys. Rev. D 87, 103529

    2013

  78. [86]

    Kai Lin, Wei-Liang Qian,Eur. Phys. J. C (2020) 80: 561 https://doi.org/10.1140/epjc/s10052-020-8116-2

    2020 doi

  79. [87]

    Jeans Type Instability of a Complex Self-Interacting Scalar Field in General Relativity

    Su´ arez, Abril, and Pierre-Henri Chavanis. 2018. “Jeans Type Instability of a Complex Self-Interacting Scalar Field in General Relativity.” Physical Review D 98(8):083529. doi: 10.1103/PhysRevD.98.083529

    2018 doi

  80. [88]

    Myrzakulov, R

    Wali Hossain, Md. Myrzakulov, R. Sami, M. Saridakis, E. N.,(2014),Phys. Rev. D 90, 023512 (2014)

    2014

  81. [89]

    The European Physical Journal C 83 (910 (2023))

    M Yarahmadi, A Salehi. The European Physical Journal C 83 (910 (2023))

    2023

  82. [90]

    Cardona, W., & Sabogal, M.A., 2023, JCAP, 2023, 045

    2023

  83. [91]

    , 2021; 23(4):404

    Di Valentino E, Mukherjee A, Sen AA. , 2021; 23(4):404. https://doi.org/10.3390/e23040404

    2021 doi

  84. [92]

    De Felice, A., & Tsujikawa, S. (2010). f(R) Theories. Living Reviews in Relativity, 13(3). https://doi.org/10.12942/ lrr-2010-3

    2010

  85. [93]

    W., & Turner, M

    Kolb, E. W., & Turner, M. S. (1990). The Early Universe. Frontiers in Physics. Addison-Wesley Publishing Company. ISBN: 0201626748 23

    1990

  86. [94]

    P., & Faraoni, V

    Sotiriou, T. P., & Faraoni, V. (2010). f(R) Theories of Gravity. Reviews of Modern Physics, 82(1), 451. https://doi. org/10.1103/RevModPhys.82.451

    2010 doi

  87. [95]

    Nojiri, S., & Odintsov, S. D. (2011). Unified cosmic history in modified gravity: from f (R) theory to Lorentz non-invariant models. Physics Reports, 505(2-4), 59-144. https://doi.org/10.1016/j.physrep.2011.04.001

    2011 doi

  88. [96]

    Lesgourgues, J., & Pastor, S. (2006). Massive neutrinos and cosmology. Physics Reports, 429(6), 307-379. https://doi. org/10.1016/j.physrep.2006.04.001

    2006 doi

  89. [99]

    M Yarahmadi.2024, Physics of dark universe.Volume 47,101733,https://doi.org/10.1016/j.dark.2024.101733

    2024

  90. [100]

    M Yarahmadi.2024, https://doi.org/10.1140/epjc/s10052-024-12805-7

    2024 doi

  91. [102]

    Salehi et al

    A. Salehi et al. ,2020,EPJC, Volume 80, article number 753, (2020)

    2020

  92. [103]

    Low redshift baryon acoustic oscillation measurement from the reconstructed 6-degree field galaxy survey,

    P. Carter, F. Beutler, W. J. Percival, C. Blake, J. Koda, and A. J. Ross, “Low redshift baryon acoustic oscillation measurement from the reconstructed 6-degree field galaxy survey,” Mon. Not. R. Astron. Soc., vol. 481, p. 2371, 2018, arXiv:1803.01746 [astro-ph.CO]

    2018 arXiv

  93. [104]

    Gil-Mar ´ ın, J

    H. Gil-Mar ´ ın, J. E. Bautista, R. Paviot, M. Vargas-Maga˜ na, S. de la Torre, S. Fromenteau, S. Alam, S.´Avila, E. Burtin, C.-H. Chuang, et al., “The Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: measurement of the BAO and growth rate of structure of th...

    2020 arXiv

  94. [105]

    J. E. Bautista, R. Paviot, M. Vargas Maga˜ na, S. de la Torre, S. Fromenteau, H. Gil-Mar ´ ın, A. J. Ross, E. Burtin, K. S. Dawson, J. Hou, et al., “The completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: measurement of the BAO and growth rate of structure of t...

    2021 arXiv

  95. [106]

    Dark Energy Survey Year 3 results: A 2.7% measurement of baryon acoustic oscillation distance scale at redshift 0.835,

    DES Collaboration, “Dark Energy Survey Year 3 results: A 2.7% measurement of baryon acoustic oscillation distance scale at redshift 0.835,” Phys. Rev. D, vol. 105, p. 043512, 2022, arXiv:2107.04646 [astro-ph.CO]

    2022 arXiv

  96. [107]

    Neveux, E

    R. Neveux, E. Burtin, A. de Mattia, A. Smith, A. J. Ross, J. Hou, J. Bautista, J. Brinkmann, C.-H. Chuang, K. S. Dawson, et al., “The completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: BAO and RSD measurements from the anisotropic power spectrum of the quasar ...

    2020

  97. [108]

    J. Hou, A. G. S´ anchez, A. J. Ross, A. Smith, R. Neveux, J. Bautista, E. Burtin, C. Zhao, R. Scoccimarro, K. S. Dawson, et al., “The completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: BAO and RSD measurements from anisotropic clustering analysis of the quasar...

    2021 arXiv

  98. [109]

    du Mas des Bourboux, J

    H. du Mas des Bourboux, J. Rich, A. Font-Ribera, V. de Sainte Agathe, J. Farr, T. Etourneau, J.-M. Le Goff, A. Cuceu, C. Balland, J. E. Bautista, et al., ’The Completed SDSS-IV Extended Baryon Oscillation Spectroscopic Survey: Baryon Acoustic Oscillations with Ly α Forests,’ A...

    2020 arXiv

  99. [110]

    Jaime, Leonardo Patino, Marcelo Salgado,2012, https://doi.org/10.48550/arXiv.1206.1642

    Luisa G. Jaime, Leonardo Patino, Marcelo Salgado,2012, https://doi.org/10.48550/arXiv.1206.1642

    2012 doi arXiv

  100. [111]

    Conservative Constraints on Early Cosmology with Mon- tePython,

    B. Audren, J. Lesgourgues, K. Benabed, and S. Prunet, “Conservative Constraints on Early Cosmology with Mon- tePython,” JCAP, vol. 2013, no. 02, p. 001, 2013,

    2013

  101. [112]

    MontePython 3: Boosted MCMC sampler and other features,

    T. Brinckmann and J. Lesgourgues, “MontePython 3: Boosted MCMC sampler and other features,” Phys. Dark Univ., vol. 24, p. 100260, 2019,

    2019

  102. [113]

    The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes,

    D. Blas, J. Lesgourgues, and T. Tram, “The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes,” JCAP, vol. 2011, no. 07, p. 034, 2011,

    2011

  103. [114]

    Planck 2018 results. VI. Cosmological parameters,

    Planck Collaboration, “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys., vol. 641, p. A6, 2020,

    2018

  104. [115]

    The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample,

    D. M. Scolnic et al., “The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample,” Astrophys. J., vol. 859, no. 2, p. 101, 2018, 24

    2018

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