pith. sign in

arxiv: 2605.22348 · v1 · pith:HA24ERU3new · submitted 2026-05-21 · 🌀 gr-qc · astro-ph.CO

Dark Energy in Ghost-free non-local Gravity

S.D. Odintsov , V.K. Oikonomou , G.S. Sharov This is my paper

Pith reviewed 2026-05-22 05:19 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.CO
keywords ghost-free non-local gravitydark energylate-time cosmologyPantheon+ supernovaeDESI observationsequation of stateF(R) gravitycosmic microwave background
0
0 comments X

The pith

Ghost-free non-local gravity fits late-time data from supernovae and DESI but struggles when cosmic microwave background data is added.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tests ghost-free non-local gravity as a model for the universe's late-time accelerated expansion. It compares the model's predictions against data from type Ia supernovae in the Pantheon+ catalogue, the DESI survey, baryon acoustic oscillations, and Hubble parameter measurements. The model performs comparably to the standard Lambda CDM cosmology and to generalized exponential F(R) gravity when these late-time data sets are used. Adding cosmic microwave background radiation data creates clear difficulties for the non-local model, while the F(R) model remains successful because its effective dark energy equation of state transitions from a phantom to a quintessence phase.

Core claim

The ghost-free non-local gravity scenario is successful only when the Pantheon+, DESI and H(z) data are considered. The generalized exponential F(R) model satisfies the viability conditions and in tests with all observational data including CMB surpasses the Lambda CDM model in chi-squared statistics and also with information criteria. This success is related with the dynamical behavior of its effective dark energy equation of state evolving from a phantom to a quintessence phase during the late-time epoch, whereas the ghost-free non-local model demonstrates only a quintessence behavior.

What carries the argument

The ghost-free non-local gravity model, whose modified action removes ghost degrees of freedom while generating late-time acceleration through non-local curvature terms.

If this is right

  • The model accounts for accelerated expansion without a cosmological constant when tested only against late-time observations.
  • Its effective dark energy equation of state remains quintessence-like at all recent epochs.
  • The generalized exponential F(R) model outperforms both Lambda CDM and non-local gravity in comprehensive fits that include CMB data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • High-precision future surveys measuring the dark energy equation of state evolution could separate non-local gravity from F(R) models.
  • If CMB tensions are resolved independently, non-local gravity might regain viability for late-time cosmology.
  • Adding a controlled phantom phase to the non-local action could be tested to improve combined-data fits.

Load-bearing premise

That the primary reason the non-local model fails with combined data is its lack of a phantom-to-quintessence transition in the effective dark energy equation of state.

What would settle it

A precise measurement of the dark energy equation of state at moderate redshifts that shows a clear phantom phase would indicate the non-local model as currently formulated cannot accommodate the full data set.

Figures

Figures reproduced from arXiv: 2605.22348 by G.S. Sharov, S.D. Odintsov, V.K. Oikonomou.

Figure 1
Figure 1. Figure 1: FIG. 1. Evolution of the normalized Hubble parameter [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
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. Figure 3: FIG. 3. Contour plots of [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Contour plots of [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
read the original abstract

Ghost-free non-local gravity is investigated with regards to its late-time dynamics. Viable solutions in this model are confronted with the observational data including the Pantheon+ catalogue of Type Ia supernovae, the Dark Energy Spectroscopic Instrument, the measurements of baryon acoustic oscillations and the Hubble parameter estimations $H(z)$. The ghost-free non-local gravity is found to be successful in these tests in comparison to the $\Lambda$CDM model and can be also comparable with the generalized exponential $F(R)$ gravity scenario. However the model encounters difficulties when the data from the above observations and the cosmic microwave background radiation data are combined. In tests with the whole set of Pantheon+, DESI, $H(z)$ and CMB data, the generalized exponential $F(R)$ model is essentially more successful. This success is related with the dynamical behavior of its effective dark energy equation of state evolving from a phantom to a quintessence phase during the late-time epoch, whereas the ghost-free non-local model demonstrates only a quintessence behavior. Hence the ghost-free non-local gravity scenario is successful only when the Pantheon+, DESI and $H(z)$ data are considered. The generalized exponential $F(R)$ model satisfies the viability conditions and in tests with all observational data including CMB surpasses the $\Lambda$CDM model in $\chi^2$ statistics and also with information criteria.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the late-time dynamics of ghost-free non-local gravity and tests its viability against cosmological observations such as the Pantheon+ Type Ia supernovae catalogue, DESI baryon acoustic oscillations, Hubble parameter H(z) measurements, and cosmic microwave background (CMB) data. It concludes that the model is successful in fits to Pantheon+, DESI, and H(z) data, performing comparably to the ΛCDM model and generalized exponential F(R) gravity, but faces difficulties when CMB data is added to the combination. The paper attributes the differing success to the effective dark energy equation of state: quintessence-only behavior in the non-local model versus a phantom-to-quintessence transition in F(R) gravity. The F(R) model is reported to outperform ΛCDM in χ² statistics and information criteria with the full dataset.

Significance. If the central claims regarding the model's performance and the role of the dark energy equation of state hold after addressing the methodological details, this work would provide valuable insights into the observational viability of non-local gravity theories as alternatives to dark energy. It highlights potential challenges in reconciling such models with CMB observations and compares them to other modified gravity scenarios, contributing to the broader effort of testing gravity modifications at cosmological scales. The comparative analysis with F(R) gravity and ΛCDM adds to the understanding of how different dynamical behaviors affect fit quality.

major comments (2)
  1. [Abstract and Results] The abstract and results section report comparative fits to Pantheon+, DESI, H(z), and CMB data but provide no details on derivation steps, error analysis, or exact fitting procedures. This absence limits the verifiability of the claims that the non-local model is successful in the Pantheon+ + DESI + H(z) tests and encounters specific difficulties when CMB data are included.
  2. [Discussion] The manuscript attributes the failure when CMB data are added to the non-local model's purely quintessence w_DE behavior (contrasted with the phantom-to-quintessence transition in F(R) gravity), but does not isolate this factor from other model differences such as perturbation spectra, sound-horizon calibration, or growth-rate constraints. A dedicated comparison or sensitivity test demonstrating that w_DE dominates the likelihood difference is needed to support the central explanatory claim.
minor comments (2)
  1. [Model Setup] Clarify the exact definition and computation of the effective dark energy equation of state w_DE in the non-local model, including any assumptions about the background expansion history.
  2. [Abstract] The abstract mentions χ² statistics and information criteria for the F(R) model outperforming ΛCDM; providing the numerical values or a dedicated table would improve clarity for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help improve the clarity and verifiability of our work. We address each major comment below and indicate the revisions planned for the manuscript.

read point-by-point responses

  1. Referee: [Abstract and Results] The abstract and results section report comparative fits to Pantheon+, DESI, H(z), and CMB data but provide no details on derivation steps, error analysis, or exact fitting procedures. This absence limits the verifiability of the claims that the non-local model is successful in the Pantheon+ + DESI + H(z) tests and encounters specific difficulties when CMB data are included.

    Authors: We agree that the current presentation lacks sufficient methodological detail for full reproducibility. In the revised manuscript we will add a new subsection to the Results section that outlines the derivation of the background field equations from the ghost-free non-local action, the numerical integration scheme employed for the late-time cosmology, the precise chi-squared construction for each dataset (including covariance matrices for Pantheon+ and DESI), and the error estimation procedure. These additions will directly address the verifiability concern without altering the reported conclusions. revision: yes

  2. Referee: [Discussion] The manuscript attributes the failure when CMB data are added to the non-local model's purely quintessence w_DE behavior (contrasted with the phantom-to-quintessence transition in F(R) gravity), but does not isolate this factor from other model differences such as perturbation spectra, sound-horizon calibration, or growth-rate constraints. A dedicated comparison or sensitivity test demonstrating that w_DE dominates the likelihood difference is needed to support the central explanatory claim.

    Authors: We acknowledge that a complete isolation of w_DE from all other differences would strengthen the explanatory claim. Our present analysis emphasizes the background expansion history, which is directly governed by w_DE and controls the distance-redshift relations probed by the late-time datasets; the CMB contribution to the combined likelihood is dominated by the angular-diameter distance to recombination, again set by the integrated expansion. Both models are evolved with comparable perturbation treatments, so the principal distinction remains the w_DE trajectory. We will expand the Discussion to make this reasoning explicit and add a brief qualitative sensitivity exercise in which the effective equation-of-state parameters are varied while holding perturbation and sound-horizon settings fixed. A fully quantitative, multi-parameter isolation test lies beyond the scope of the present study but can be noted as a worthwhile direction for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; observational viability tests are independent of model inputs

full rationale

The paper sets up the ghost-free non-local gravity action, derives the background equations for late-time dynamics, obtains numerical solutions for the effective dark energy equation of state, and then performs separate statistical comparisons (chi-squared, information criteria) against external datasets (Pantheon+, DESI, H(z), CMB). No derivation step reduces by construction to a fitted parameter or to a self-citation that itself assumes the target result. The reported difference in w_DE behavior (quintessence-only versus phantom-to-quintessence) is extracted from the solved dynamics of each model and offered as a post-hoc interpretation of why the combined-data likelihoods differ; it is not used to define the model or to force the fit outcome. All viability conclusions rest on external observational benchmarks rather than on internal re-labeling of inputs. This is a standard, non-circular model-selection exercise.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based on abstract only; specific free parameters, axioms, or invented entities cannot be identified. The non-local gravity action likely contains unspecified parameters fitted to data, but details are unavailable.

pith-pipeline@v0.9.0 · 5779 in / 1337 out tokens · 63769 ms · 2026-05-22T05:19:27.748282+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

80 extracted references · 80 canonical work pages · 33 internal anchors

  1. [1]

    preliminary

    and the Hubble parameter estimations H(z). The calculated solutions H(z) are tested with these 0 1 2 50 100 150 200 z H(z) km/s Mpc F(R,φ) ΛCDM Exp F(R) CC H(z) −8 −4 0 0 5 10 log a log E F(R,φ) ΛCDM Exp F(R) −8 −4 0 0 10 20 log a log(R/2Λ) F(R,φ) ΛCDM Exp F(R) −8 −4 0 0 10 20 log a log(− Φ), logΨ − Φ Ψ 0 5 10 −1 0 z ωDE F(R,φ) ΛCDM Exp F(R) FIG. 1. Evolu...

    work page 2025

  2. [2]

    Note that the Lagrang ian ( 3.1) of the generalized exponential F (R) model in the limit β → ∞ tends to the ΛCDM Lagrangian F (R) = R − 2Λ

    The best fits for β with the CMB data are slightly enhanced, but the error boxes and 1 σ , 2 σ CL domains become noticeably more narrow. Note that the Lagrang ian ( 3.1) of the generalized exponential F (R) model in the limit β → ∞ tends to the ΛCDM Lagrangian F (R) = R − 2Λ. The best fitted values β = 0 . 807+0. 104 − 0. 082 are low, hence the considered e...

    work page 2020

  3. [3]

    A. G. Adame et al. [DESI], JCAP 02 (2025), 021, [arXiv:2404.03002 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  4. [4]

    DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints

    M. Abdul Karim et al. [DESI], Phys. Rev. D 112 (2025) no.8, 083515, [arXiv:2503.14738 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  5. [5]

    The Pantheon+ Analysis: The Full Dataset and Light-Curve Release

    D. Scolnic et al. , Astrophys. J. 938 (2022) 113, arXiv:2112.03863

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  6. [6]

    Union Through UNITY: Cosmology with 2,000 SNe Using a Unified Bayesian Framework

    D. Rubin et al. , Astrophys. J. 986 (2025) no.2, 231, arXiv:2311.12098 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  7. [7]

    T. M. C. Abbott et al. (DES), Astrophys. J. Lett. 973, no.1 L14 (2024), arXiv:2401.02929 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  8. [8]

    P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003 ), arXiv:astro-ph/0207347

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  9. [9]

    Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests

    K. Bamba, S. Capozziello, S. Nojiri and S. D. Odintsov, As trophys. Space Sci. 342 (2012), 155-228, [arXiv:1205.3421 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  10. [10]

    Capozziello, M

    S. Capozziello, M. De Laurentis, Phys. Rept. 509, 167 (2011); V. Faraoni and S. Capozziello, Fundam. Theor. Phys. 170 (2010)

    work page 2011

  11. [11]

    Introduction to Modified Gravity and Gravitational Alternative for Dark Energy

    S. Nojiri, S.D. Odintsov, [Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007)], [arXiv:hep-th/0601213]; S. Nojiri, S.D. Odintsov, Phys. Rept. 505, 59 (2011), [arXiv:1011.0544 [gr-qc]]; S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Phys. Rept. 692 (2017) 1 [arXiv:1705.11098 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  12. [12]

    Planck 2018 results. VI. Cosmological parameters

    Planck collaboration: N. Aghanim et al. , Astron. Astrophys. 641 (2020), A6 [arXiv:1807.06209 [astro- ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  13. [13]

    A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team

    A.G. Riess, W. Yuan, L.M. Macri and D. Scolnic, Astrophy s. J. Lett. 908 (2021), L6, arXiv:2112.04510 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  14. [14]

    Di Valentino, A

    E. Di Valentino, A. Mukherjee and A. A. Sen, Entropy 23 (2021) no.4, 404 doi:10.3390/e23040404 [arXiv:2005.12587 [astro-ph.CO]]

    work page doi:10.3390/e23040404 2021

  15. [15]

    Di Valentino, S

    E. Di Valentino, S. Gariazzo, O. Mena and S. Vagnozzi, JC AP 07 (2020) no.07, 045. [arXiv:2005.02062[astro- ph.CO]]

    work page arXiv 2020

  16. [16]

    Krishnan, E

    C. Krishnan, E. ´O. Colg´ ain, Ruchika, A. A. Sen, M. M. Sheikh-Jabbari and T. Y ang, Phys. Rev. D 102 (2020) no.10, 103525 [arXiv:2002.06044 [astro-ph.CO]]

    work page arXiv 2020

  17. [17]

    Krishnan, R

    C. Krishnan, R. Mohayaee, E. ´O. Colg´ ain, M. M. Sheikh-Jabbari and L. Yin, Class. Quant. G rav. 38 (2021) no.18, 184001 doi:10.1088/1361-6382/ac1a81 [arXiv:2105 .09790 [astro-ph.CO]]

    work page doi:10.1088/1361-6382/ac1a81 2021

  18. [18]

    S. D. Odintsov, D. S´ aez-Chill´ on G´ omez and G. S. Sharo v, Nucl. Phys. B. 966, (2021), 115377, arXiv:2011.03957

    work page arXiv 2021

  19. [19]

    Vagnozzi, L

    S. Vagnozzi, L. Visinelli, P. Brax, A. C. Davis and J. Sak stein, Phys. Rev. D 104 (2021) no.6, 063023, [arXiv:2103.15834 [hep-ph]]

    work page arXiv 2021

  20. [20]

    Vagnozzi, F

    S. Vagnozzi, F. Pacucci and A. Loeb, JHEAp 36 (2022), 27-35, [arXiv:2105.10421 astro-ph.CO]]

    work page arXiv 2022

  21. [21]

    G. Ye, J. Zhang and Y. S. Piao, [arXiv:2107.13391 [astro -ph.CO]]

    work page arXiv

  22. [22]

    Ferlito, S

    F. Ferlito, S. Vagnozzi, D. F. Mota and M. Baldi, Mon. Not . Roy. Astron. Soc. 512 (2022) no.2, 1885-1905, [arXiv:2201.04528 [astro-ph.CO]]

    work page arXiv 2022

  23. [23]

    B. H. Lee, W. Lee, E. ´O. Colg´ ain, M. M. Sheikh-Jabbari and S. Thakur, JCAP 04 (2022) no.04, 004, [arXiv:2202.03906 [astro-ph.CO]]

    work page arXiv 2022

  24. [24]

    S. A. Adil, U. Mukhopadhyay, A. A. Sen and S. Vagnozzi, JC AP 10 (2023), 072, [arXiv:2307.12763 [astro- ph.CO]]

    work page arXiv 2023

  25. [25]

    H¨ og ˚ as and E

    M. H¨ og ˚ as and E. M¨ ortsell, Phys. Rev. D 108 (2023) no.12, 124050 doi:10.1103/PhysRevD.108.124050 [arXiv:2309.01744 [astro-ph.CO]]

    work page doi:10.1103/physrevd.108.124050 2023

  26. [26]

    Menci, S

    N. Menci, S. A. Adil, U. Mukhopadhyay, A. A. Sen and S. Vag nozzi, JCAP 07 (2024), 072, [arXiv:2401.12659 [astro-ph.CO]]

    work page arXiv 2024

  27. [27]

    The CosmoVerse White Paper: Addressing observational tensions in cosmology with systematics and fundamental physics

    E. Di Valentino, J. Levi Said, A. Riess, A. Pollo, V. Poul in, A. G´ omez-Valent, A. Weltman, A. Palmese, C. D. Huang and C. van de Bruck, et al. [arXiv:2504.01669 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv

  28. [28]

    Y. Cai, X. Ren, T. Qiu, M. Li and X. Zhang, [arXiv:2505.24 732 [astro-ph.CO]]

    work page

  29. [29]

    G. Ye, M. Martinelli, B. Hu and A. Silvestri, Phys. Rev. L ett. 134 (2025) no.18, 181002, [arXiv:2407.15832 [astro-ph.CO]]

    work page arXiv 2025

  30. [30]

    Chaudhary, S

    H. Chaudhary, S. Capozziello, V. K. Sharma, I. G´ omez-V argas and G. Mustafa, [arXiv:2508.10514 [astro- ph.CO]]

    work page arXiv

  31. [31]

    Chaudhary, S

    H. Chaudhary, S. Capozziello, S. Praharaj, S. K. J. Paci f and G. Mustafa, JHEAp 50 (2026), 100507, [arXiv:2509.17124[gr-qc]]. 13

    work page arXiv 2026

  32. [32]

    Giar` e, M

    W. Giar` e, M. A. Sabogal, R. C. Nunes and E. Di Valentino, Phys. Rev. Lett. 133 (2024) no.25, 251003, [arXiv:2404.15232 [astro-ph.CO]]

    work page arXiv 2024

  33. [33]

    S. Pan, S. Paul, E. N. Saridakis and W. Yang, Phys. Rev. D 113 (2026) no.2, 023515, [arXiv:2504.00994[astro- ph.CO]]

    work page arXiv 2026

  34. [34]

    Y. Yang, Q. Wang, X. Ren, E. N. Saridakis and Y. F. Cai, Ast rophys. J. 988 (2025) no.1, 123 doi:10.3847/1538-4357/ade43f [arXiv:2504.06784 [astro -ph.CO]]

    work page doi:10.3847/1538-4357/ade43f 2025

  35. [35]

    Zhang, Y

    X. Zhang, Y. H. Xu and Y. Sang, Commun. Theor. Phys. 78 (2026) no.3, 035404 doi:10.1088/1572- 9494/ae1a5b [arXiv:2511.02220 [astro-ph.CO]]

    work page doi:10.1088/1572- 2026

  36. [36]

    D. D. Y. Ong, D. Yallup and W. Handley, [arXiv:2511.1063 1 [astro-ph.CO]]

    work page arXiv

  37. [37]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and V. K. Oikonomou, [arXiv:25 12.06279 [gr-qc]]

    work page

  38. [38]

    S. D. Odintsov, D. S´ aez-Chill´ on G´ omez and G. S. Sharo v, Eur. Phys. J. C 85 (2025) no.3, 298, [arXiv:2412.09409 [gr-qc]]

    work page arXiv 2025

  39. [39]

    S. D. Odintsov, V. K. Oikonomou and G. S. Sharov, JHEAp 50 (2026), 100471, [arXiv:2506.02245[gr-qc]]

    work page arXiv 2026

  40. [40]

    S. D. Odintsov, V. K. Oikonomou and G. S. Sharov, JHEAp 52 (2026), 100579, [arXiv:2601.06949[gr-qc]]

    work page arXiv 2026

  41. [41]

    Modified gravity with negative and positive powers of the curvature: unification of the inflation and of the cosmic acceleration

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

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  42. [42]

    Reconciling dark energy models with f(R) theories

    S. Capozziello, V. F. Cardone and A. Troisi, Phys. Rev. D 71 (2005), 043503, [arXiv:astro-ph/0501426[astro- ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  43. [43]

    J. c. Hwang and H. Noh, Phys. Lett. B 506 (2001), 13-19 doi:10.1016/S0370-2693(01)00404-X [arXiv :astro- ph/0102423 [astro-ph]]

    work page doi:10.1016/s0370-2693(01)00404-x 2001

  44. [44]

    Y. S. Song, W. Hu and I. Sawicki, Phys. Rev. D 75 (2007), 044004 doi:10.1103/PhysRevD.75.044004 [arXiv:astro-ph/0610532 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.75.044004 2007

  45. [45]

    Faulkner, M

    T. Faulkner, M. Tegmark, E. F. Bunn and Y. Mao, Phys. Rev. D 76 (2007), 063505 doi:10.1103/PhysRevD.76.063505 [arXiv:astro-ph/06125 69 [astro-ph]]

    work page doi:10.1103/physrevd.76.063505 2007

  46. [46]

    G. J. Olmo, Phys. Rev. D 75 (2007), 023511 doi:10.1103/PhysRevD.75.023511 [arXiv:g r-qc/0612047 [gr-qc]]

    work page doi:10.1103/physrevd.75.023511 2007

  47. [47]

    Sawicki and W

    I. Sawicki and W. Hu, Phys. Rev. D 75 (2007), 127502 doi:10.1103/PhysRevD.75.127502 [arXiv:a stro- ph/0702278 [astro-ph]]

    work page doi:10.1103/physrevd.75.127502 2007

  48. [48]

    Faraoni, Phys

    V. Faraoni, Phys. Rev. D 75 (2007), 067302 doi:10.1103/PhysRevD.75.067302 [arXiv:g r-qc/0703044 [gr-qc]]

    work page doi:10.1103/physrevd.75.067302 2007

  49. [49]

    The evolution of density perturbations in f(R) gravity

    S. Carloni, P. K. S. Dunsby and A. Troisi, Phys. Rev. D 77 (2008), 024024 doi:10.1103/PhysRevD.77.024024 [arXiv:0707.0106 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.77.024024 2008

  50. [50]

    Unifying inflation with LambdaCDM epoch in modified f(R) gravity consistent with Solar System tests

    S. Nojiri and S. D. Odintsov, Phys. Lett. B 657 (2007), 238-245 doi:10.1016/j.physletb.2007.10.027 [arXiv:0707.1941 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2007.10.027 2007

  51. [51]

    Junction Conditions in f(R) Theories of Gravity

    N. Deruelle, M. Sasaki and Y. Sendouda, Prog. Theor. Phy s. 119 (2008), 237-251 doi:10.1143/PTP.119.237 [arXiv:0711.1150 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1143/ptp.119.237 2008

  52. [52]

    S. A. Appleby and R. A. Battye, JCAP 05 (2008), 019 doi:10.1088/1475-7516/2008/05/019 [arXiv:0 803.1081 [astro-ph]]

    work page doi:10.1088/1475-7516/2008/05/019 2008

  53. [53]

    E. V. Linder, Phys. Rev. D 80 (2009) 123528, arXiv:0905. 2962

    work page 2009

  54. [54]

    P. K. S. Dunsby, E. Elizalde, R. Goswami, S. Odintsov and D. S. Gomez, Phys. Rev. D 82 (2010), 023519 doi:10.1103/PhysRevD.82.023519 [arXiv:1005.2205 [gr-q c]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.82.023519 2010

  55. [55]

    Hu and I

    W. Hu and I. Sawicki, Phys. Rev. D 76 (2007), 064004 doi:10.1103/PhysRevD.76.064004 [arXiv:0 705.1158 [astro-ph]]

    work page doi:10.1103/physrevd.76.064004 2007

  56. [56]

    Cosmic history of viable exponential gravity: Equation of state oscillations and growth index from inflation to dark energy era

    K. Bamba, A. Lopez-Revelles, R. Myrzakulov, S. D. Odint sov and L. Sebastiani, Class. Quant. Grav. 30 (2013), 015008 doi:10.1088/0264-9381/30/1/015008 [arXi v:1207.1009 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/30/1/015008 2013

  57. [57]

    S. D. Odintsov, V. K. Oikonomou, I. Giannakoudi, F. P. Fr onimos and E. C. Lymperiadou, Symmetry 15 (2023) no.9, 1701, [arXiv:2307.16308 [gr-qc]]

    work page arXiv 2023

  58. [58]

    D’Onofrio, S

    S. D’Onofrio, S. Odintsov and T. Schiavone, [arXiv:251 1.06924 [gr-qc]]

    work page

  59. [59]

    S. D. Odintsov, D. Saez-Chillon Gomez, G. S. Sharov. Eur . Phys. J. C 77 (2017) 862, arXiv:1709.06800

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  60. [60]

    S. D. Odintsov, D. Saez-Chillon Gomez and G. S. Sharov, P hys. Rev. D. 99 (2019) 024003, arXiv:1807.02163

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  61. [61]

    S. D. Odintsov, D. S´ aez-Chill´ on G´ omez and G. S. Sharo v, Phys. Dark Univ. 42 (2023) 101369, [arXiv:2310.20302 [gr-qc]]

    work page arXiv 2023

  62. [62]

    Nonlocal gravity. Conceptual aspects and cosmological predictions

    L. Modesto and L. Rachwal, Int. J. Mod. Phys. D 26 (2017) n o.11, 1730020; E. Belgacem, Y. Dirian, S. Foffa and M. Maggiore, JCAP 1803 (2018) 002 doi:10.1088/1475-751 6/2018/03/002 [arXiv:1712.07066 [hep-th]]; A. S. Koshelev, L. Modesto, L. Rachwal and A. A. Starobinsky, JHEP 1611 (2016) 067 doi:10.1007/JHEP11(2016)067 [arXiv:1604.03127 [hep-th ]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475-751 2017

  63. [63]

    Nonlocal Cosmology

    S. Deser and R. P. Woodard, Phys. Rev. Lett. 99 (2007), 111301 doi:10.1103/PhysRevLett.99.111301 [arXiv:0706.2151 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.99.111301 2007

  64. [64]

    Deffayet and R

    C. Deffayet and R. P. Woodard, JCAP 08 (2009), 023 doi:10.1088/1475-7516/2009/08/023 [arXiv:0 904.0961 [gr-qc]]

    work page doi:10.1088/1475-7516/2009/08/023 2009

  65. [65]

    Deser and R

    S. Deser and R. P. Woodard, JCAP 11 (2013), 036 doi:10.1088/1475-7516/2013/11/036 [arXiv:1 307.6639 [astro-ph.CO]]

    work page doi:10.1088/1475-7516/2013/11/036 2013

  66. [66]

    Deser and R

    S. Deser and R. P. Woodard, JCAP 06 (2019), 034 doi:10.1088/1475-7516/2019/06/034 [arXiv:1 902.08075 [gr-qc]]

    work page doi:10.1088/1475-7516/2019/06/034 2019

  67. [67]

    Modified non-local-F(R) gravity as the key for the inflation and dark energy

    S. Nojiri and S. D. Odintsov, Phys. Lett. B 659 (2008) 821 doi:10.1016/j.physletb.2007.12.001 [arXiv:0708.0924 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2007.12.001 2008

  68. [68]

    Screening of cosmological constant in non-local gravity

    S. Nojiri, S. D. Odintsov, M. Sasaki and Y. l. Zhang, Phys . Lett. B 696 (2011) 278 doi:10.1016/j.physletb.2010.12.035 [arXiv:1010.5375 [ gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2010.12.035 2011

  69. [69]

    Dynamics in Nonlocal Cosmological Models Derived from String Field Theory

    L. Joukovskaya, Phys. Rev. D 76 (2007) 105007 doi:10.11 03/PhysRevD.76.105007 [arXiv:0707.1545 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  70. [70]

    Localization of nonlocal theories

    G. Calcagni, M. Montobbio and G. Nardelli, Phys. Lett. B 662 (2008) 285 doi:10.1016/j.physletb.2008.03.024 [arXiv:0712.2237 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2008.03.024 2008

  71. [71]

    Phantom and non-phantom dark energy: The cosmological relevance of non-locally corrected gravity

    S. Jhingan, S. Nojiri, S. D. Odintsov, M. Sami, I. Thongk ool and S. Zerbini, Phys. Lett. B 663 (2008) 424 14 doi:10.1016/j.physletb.2008.04.054 [arXiv:0803.2613 [ hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2008.04.054 2008

  72. [72]

    Accelerating cosmologies from non-local higher-derivative gravity

    S. Capozziello, E. Elizalde, S. Nojiri and S. D. Odintso v, Phys. Lett. B 671 (2009) 193 doi:10.1016/j.physletb.2008.11.060 [arXiv:0809.1535 [ hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2008.11.060 2009

  73. [73]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Phys. Dark Univ. 28 (2020), 100541 doi:10.1016/j.dark.2020.100541 [arXiv:1911.07329 [gr- qc]]

    work page doi:10.1016/j.dark.2020.100541 2020

  74. [74]

    Nojiri, S

    S. Nojiri, S. Odintsov and V. K. Oikonomou, Phys. Lett. B 874 (2026), 140290 doi:10.1016/j.physletb.2026.140290 [arXiv:2601.07879 [gr-qc]]

    work page doi:10.1016/j.physletb.2026.140290 2026

  75. [75]

    The Atacama Cosmology Telescope: DR6 Constraints on Extended Cosmological Models

    E. Calabrese et al. [Atacama Cosmology Telescope], JCAP 11 (2025), 063 doi:10.1088/1475- 7516/2025/11/063 [arXiv:2503.14454 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1475- 2025

  76. [76]

    Planck 2018 results. X. Constraints on inflation

    Y. Akrami et al. [Planck], Astron. Astrophys. 641 (2020), A10 [arXiv:1807.06211 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  77. [77]

    P. A. R. Ade et al. [BICEP and Keck], Phys. Rev. Lett. 127 ( 2021) no.15, 151301, [arXiv:2110.00483 [astro- ph.CO]]

    work page arXiv 2021

  78. [78]

    S. D. Odintsov and V. K. Oikonomou, Phys. Lett. B 797, 134874 (2019), [arXiv:1908.07555 [gr-qc]]. arXiv:1908.07555

    work page arXiv 2019

  79. [79]

    A. R. Liddle, Mon. Not. Roy. Astron. Soc. 377 (2007), L74-L78, [arXiv:astro-ph/0701113 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  80. [80]

    Distance Priors from Planck Final Release

    L. Chen, Q.-G. Huang and K. Wang, J. Cosmol. Astropart. P hys. 1902 (2019) 028, arXiv:1808.05724

    work page internal anchor Pith review Pith/arXiv arXiv 1902