Pith. sign in

REVIEW 4 major objections 5 minor 91 references

Corrections to inflationary models induced by non-minimal coupling between scalar field and curvature

T0 review · 4 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A power-law non-minimal coupling F=(H/λ)^{2n} deforms the inflationary potential to V∝V_E^{n+1} and rescales the tensor-to-scalar ratio to r=(1−n)r_E while preserving n_T=−r/8 exactly.

desk verdict A careful one-parameter extension of the authors' earlier F=(H/λ)^2 scheme, internally consistent but built on an imposed matching condition that makes the 'corrections' largely a frame re-description. read the letter →

arxiv 2607.10679 v2 pith:L3SRZZIK submitted 2026-07-12 gr-qc

I. V. Fomin , S. V. Chervon , E. S. Dentsel , B. Mishra This is my paper · ORCID
classification gr-qc MSC 83D0583F05 PACS 98.80.Cq04.50.Kd
keywords inflationarycosmologynon-minimalcouplingscalar-tensorgravitytensor-to-scalarratioconsistencyrelationslow-rollapproximationreheatingspectralindex
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 argues that a specific power-law form of non-minimal coupling between the inflaton and curvature—F=(H/λ)^{2n}, with H the Hubble parameter—can be calibrated so that the scalar field follows exactly the same trajectory as in minimal Einstein-gravity inflation. On that trajectory, the potential is deformed to V≃V_E^{n+1}/(1−n), and the tensor-to-scalar ratio is systematically reduced to r=(1−n)r_E, while the standard consistency relation n_T=−r/8 remains exactly intact. This matters observationally because the deformation parameter n becomes a single handle that shifts spectral predictions, allowing first-order slow-roll models to satisfy both Planck and ACT constraints within the standard 50–60 e-folds, whereas second-order attractor-type models require an extended e-fold range 69<ΔN<90. The same parametrization leaves reheating dynamics identical to minimal coupling, since the field equation reduces to the Einstein-frame one. The paper also builds a model-independent classification of inflationary models by the order of the expansion r=r(1−n_S).

What carries the argument

The central object is the power-law parametrization F(φ)=(H/λ)^{2n} of the non-minimal coupling function, together with the matching condition φ̇²=−2Ḣ and H=H_E, φ=φ_E. It converts the modified-gravity background equations into Einstein-frame forms, producing the deformation identities V≃(V_E)^{n+1}/(1−n), F≃(1−n)^{-1}(V_E/V_E(∗))^n, ω≃(V_E/V_E(∗))^n. The companion machinery is the model-independent expansion r=Σ β_k(1−n_S)^k, which lets the paper classify inflationary scenarios by the first-order relation δ=sε, second-order δ=−s√ε, and higher-order relations, and read off how non-minimal coupling changes each class.

What would settle it

Measure the tensor spectral index n_T and tensor-to-scalar ratio r from CMB B-modes at high precision: any significant deviation from n_T=−r/8 would falsify the parametrization's central prediction.

Watch

Extended reading notes

Core claim

Under the ansatz F=(H/λ)^{2n} with −1<n<1, and imposing that the Hubble parameter and scalar field evolve exactly as in the minimally coupled Einstein case (H=H_E, φ=φ_E, φ̇²=−2Ḣ), the generalized scalar-tensor field equation is shown to reduce to the standard Klein-Gordon equation. This equivalence yields explicit slow-roll reconstructions: F≃(1−n)^{-1}(V_E/V_E(∗))^n, V≃(V_E)^{n+1}/(1−n), and ω≃(V_E/V_E(∗))^n, with the scale λ fixed by matching the scalar perturbation amplitude, (λ/H_*)^{2n}=1−n. From these, the perturbation parameters shift to n_S−1=−2(2−n)ε_*+2δ_*, r=16(1−n)ε_*, and n_T=−2(1−n)ε_*, so n_T=−r/8 holds exactly. The paper treats n as a deformation parameter measuring both th

Load-bearing premise

The construction assumes, rather than derives, that the scalar field and Hubble parameter follow exactly the same trajectories as in minimally coupled Einstein gravity (H=H_E, φ=φ_E, φ̇²=−2Ḣ); if real solutions of the non-minimally coupled equations do not stay on that slice, the potential-deformation formulas and the exact n_T=−r/8 result do not follow.

Editorial extensions

If this is right

  • If correct, any potential V_E from Einstein-gravity inflation has a non-minimally coupled counterpart with potential V≃V_E^{n+1}/(1−n), so the whole catalogue of known models can be re-mapped with one parameter.
  • The tensor-to-scalar ratio is reduced by factor (1−n), so for n>0 models with otherwise too-large r can be brought under the current CMB bound r<0.036.
  • The consistency relation n_T=−r/8 is exactly preserved, unlike generic scalar-tensor theories, so this class cannot be distinguished from GR by a broken consistency relation; only via spectral parameter shifts.
  • Reheating after inflation is governed by the same field equation as minimal coupling, so standard reheating constraints and e-fold estimates 50≤ΔN≤60 continue to apply for first-order models.
  • Second-order models (α-attractor type) require 69<ΔN<90 to satisfy ACT data, which can be realized only by modified reheating or additional dark-matter-production scenarios.

Reading between the lines

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

  • [editorial inference] The matching condition is a restriction, not a dynamical attractor: the paper's classification samples only trajectories that coincide with Einstein-gravity solutions, so the results do not cover general F(φ)R theories where H and φ evolve differently.
  • [editorial inference] Because n_T=−r/8 is exact, a future high-precision measurement of the tensor tilt that deviates from −r/8 would immediately rule out this entire class; the model is most vulnerable in the tensor sector, not the scalar sector.
  • [editorial inference] The factor (1−n) acts like a dedicated 'knob' that suppresses r without altering n_S predictions at second order, suggesting a testable strategy: compare the n_S–r relation across first- and second-order models to infer n independently of potential choice.
  • [editorial inference] The predicted negative running α_S≈−10^-4 stands in about 1σ tension with the positive ACT hint; a natural extension the authors point to is combining this parametrization with additional corrections that can make α_S positive while keeping n_S within bounds.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 a generalized scalar-tensor action (24) with a nonminimal coupling F(φ)R, and specializes to the power-law parametrization F=(H/λ)^{2n} (Eq. 30). To compare with Einstein gravity, the authors impose the condition that the Hubble parameter and scalar field evolve exactly as in the minimal-coupling case (Eqs. 33–34). Under this ansatz, they derive a deformed Jordan-frame potential V∝(V_E)^{n+1}, a kinetic function ω=(1−n)F, and a normalization λ fixed by matching the scalar amplitude (Eqs. 99–100). They then compute the scalar and tensor perturbation spectra, obtaining a scalar tilt shift Δn_S=2nε*, a tensor ratio r=(1−n)r_E, and an exact preservation of the consistency relation n_T=−r/8. They also argue that the field equation, and hence the reheating dynamics, is identical to the minimal case (Eqs. 36–37). The second half of the paper develops a 'model-independent' classification of inflationary models using the expansion r=r(1−n_S) and the ansatz δ−δ_0=−s(ε−ε_0)^{1/m} (Eq. 114), applying it to first- and second-order examples including hybrid natural inflation and α-attractors, and compares the predictions to Planck and ACT constraints.

Significance. If the calculations are taken at face value, the paper provides a self-consistent and algebraically careful treatment of a particular Jordan-frame parametrization. The derivations in Sections III–V appear internally correct, and the explicit formulas for the spectral parameters, running, non-Gaussianity, and field excursion are useful reference expressions. However, the central physical claim that these are 'corrections induced by non-minimal coupling' is not supported: as shown below, the theory slice explored is conformally equivalent to ordinary single-field inflation with a one-parameter family of Einstein-frame potentials U∝V_E^{1−n}. The paper therefore does not establish new observational signatures beyond selecting that family. The model-independent classification in Section VI is a useful phenomenological exercise, but its scope is narrower than claimed because Eq. (114) is itself an additional ansatz.

major comments (4)
  1. [§V, Eqs. (99)–(100), (111)] The central step is the imposed matching condition H=H_E and φ=φ_E, not derived from the action. All subsequent results—the potential deformation (103), the coupling function (104), the kinetic function (105), and the perturbation shifts (107), (111), (112)—are consequences of this ansatz. A conformal transformation g̃=Fg maps the action (24) to the Einstein frame, where the canonical field χ satisfies dχ/dφ≈√(ω/F)=√(1−n) at leading order and the Einstein-frame potential is U=V/F²∝(1−n)V_E(∗)^n V_E^{1−n}. The slow-roll parameter becomes ε_χ=(1−n)ε_E, so r=(1−n)r_E and n_T=−r/8 are exactly the predictions of a minimal model with potential U∝V_E^{1−n}. Thus the 'non-minimal corrections' are a frame re-description of a one-parameter family of minimal potentials, not generic consequences of F(φ)R gravity. This should be acknowledged and the framing revised.
  2. [§V, Eqs. (99)–(100), (111)] The normalization λ is not a free parameter in any predictive sense: it is fixed by requiring equality of the scalar perturbation amplitude between the minimal and nonminimal cases (Eq. 99), yielding (λ/H_*)^{2n}=1−n (Eq. 100). Consequently, the tensor-to-scalar ratio r=(1−n)r_E (Eq. 111) is inherited from the matching condition rather than independently predicted. Similarly, the tensor tilt n_T=(1−n)n_T(E) follows from the same relation. The paper should state explicitly that n simply re-labels the exponent of the equivalent Einstein-frame potential U∝V_E^{1−n}, and that no new observational discriminant is introduced beyond that choice.
  3. [§IV.C, Eqs. (36)–(37)] The claim that the reheating dynamics is 'completely analogous' to the minimal case is a consequence of the imposed ansatz (33), not a result derived from the full scalar-tensor dynamics. The field equation reduces to the minimal one precisely because H=H_E and φ=φ_E were assumed. This equivalence is therefore a consistency check of the ansatz, not a property of generic scalar-tensor inflation. Moreover, the derivation of (36) uses the slow-roll parametrization through Eqs. (31)–(35) in a regime where slow-roll may not apply, so the extension to reheating is not independently established.
  4. [§VI, Eq. (114)] The phrase 'model-independent analysis' is an overstatement. The expansion r=r(1−n_S) itself is generic, but the concrete classification relies on the additional ansatz δ−δ_0=−s(ε−ε_0)^{1/m} (Eq. 114), which is not shown to cover all inflationary models. The first-order cases δ=sε and δ=sε+b, and the second-order case δ=−s√ε, are specific slices of model space. The paper presents these as examples, but the conclusion and abstract repeatedly call the scheme model-independent. This should be softened to 'parameterized family' or the ansatz should be justified as exhaustive, which it is not.
minor comments (5)
  1. [Throughout] There are several typos: 'non-miminal' in the §III.A heading, 'A important result' in Section IV, and 'Mimimal' in the abstract header of the arXiv text. These should be corrected.
  2. [§IV.A, Eqs. (72)–(73)] The claim that the consistency relation n_T=−r/8 is 'exactly preserved' is stated as if it were an exact result. Equations (72)–(73) are derived under the slow-roll approximation, so the phrase 'exactly' should be replaced with 'at leading order in slow-roll' or similar.
  3. [§V, Eq. (100)] For n=0, the relation (λ/Η_*)^{2n}=1−n is degenerate and does not define λ. The text and Table I leave λ undefined in that limit; a brief comment on this degenerate case would improve clarity.
  4. [§VI.B, Table I] The parameter range table lists λ only for n=0.5 and n=0.9, with '—' for n=0. Since n=0 corresponds to the minimal case, this is fine, but it would be helpful to state in the caption that λ is not defined for n=0.
  5. [§VI, Eq. (114)] The constants ε_0 and δ_0 are introduced as 'small', but their physical meaning in the classification is not specified. For m>1 they are set to zero with a brief justification; this choice should be explained more fully because it affects the allowed parameter ranges in Tables I–III.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's results are explicit algebraic consequences of a clearly stated parametrization and matching condition, not hidden fits or self-citation substitutions.

full rationale

The construction is transparent: the paper posits F=(H/λ)^{2n} (Eq. 30) and, to compare theories, imposes the matching conditions H=H_E and φ=φ_E (Eqs. 33–34). The potential deformation V∝V_E^{n+1}/(1−n) (Eqs. 44, 103), the tensor ratio r=(1−n)r_E (Eqs. 73, 111), and the identity of the reheating field equation (Eqs. 36–37) are derived consequences of these assumptions rather than empirical predictions obtained by fitting. The normalization λ is fixed by requiring equal scalar amplitudes (Eqs. 99–100); this does not determine r, n_S, or n_T, which follow from the perturbation formulas. The r=r(1−n_S) expansion (Eq. 113) is a Taylor expansion, and the slow-roll relation δ−δ0=−s(ϵ−ϵ0)^{1/m} (Eq. 114) is explicitly introduced as an ansatz, not imported as a proved theorem. The self-citations [50–53] and [54] motivate the parametrization and the classification scheme, but the derivation in this paper is self-contained once its stated assumptions are granted. The restricted scope—a single trajectory slice of scalar-tensor gravity—is a limitation on generality, but it is not circular reasoning.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The framework rests on the chosen power-law ansatz, the identical-dynamics matching condition, and the slow-roll approximation. The main free parameter n is not fitted but it is not predicted either; it varies freely within −1<n<1. The expansion coefficients s,b play a fitting role in the model-independent analysis.

free parameters (4)
  • n = not fitted; −1<n<1
    Power-law index in F=(H/λ)^{2n}; controls the potential deformation and the r rescaling. It is a free theory parameter varied in Tables I–III, not derived from data or first principles.
  • λ = fixed by amplitude matching: (λ/H_*)^{2n}=1−n
    Normalization scale λ is fixed by requiring A_S equality between minimal and non-minimal cases (Eq. 99–100), i.e., calibrated to AS=2.1e−9, not predicted.
  • s (model-independent expansion slope) = ranges in Tables I–III
    Slope in δ=sϵ or δ=−s√ϵ; fitted to ranges by requiring experimental constraints, not derived.
  • b (first-order offset) = b in [−0.0171,−0.0001] in Table I
    Offset in δ=sϵ+b; effectively fit to the spectral index n_S.
assumptions (5)
  • ad hoc to paper H=H_E, ϕ=ϕ_E, ˙ϕ²=−2 ˙H (identical dynamics between minimal and non-minimal cases)
    Introduced in §III.A, Eqs. (33)–(34), explicitly 'to compare inflationary models with and without non-minimal coupling'. This is the premise that maps the theories, not a derived property; the paper even states it is 'ensured by the specific choice' of F, ω, V.
  • ad hoc to paper F=(H/λ)^{2n} with constant n, λ as an ansatz
    Eq. (30). It is a chosen parametrization rather than derived from an action, and the paper explicitly calls it 'the proposed parametrization'.
  • domain assumption Slow-roll approximation with ϵ≪1, |δ|≪1, and cold canonical scalar field
    Used throughout §III–IV to derive (43)–(45), (60), (62), and the perturbation observables. Standard in inflationary theory, but it limits validity and is invoked without dedicated error control.
  • ad hoc to paper The model-independent ansatz δ−δ_0=−s(ϵ−ϵ_0)^{1/m} in Eq. (114) covers the physically relevant inflationary models
    This is the basis for all Section VI classifications. It is justified only by examples (power-law, HNI, α-attractors). The paper admits higher orders are analytically intractable.
  • domain assumption Standard linear perturbation theory for scalar-tensor gravity (Eqs. 53–59) remains valid
    Imported from refs [21,22,50–53]; standard within the field but unproven in the paper.
invented entities (1)
  • Power-law coupling F=(H/λ)^{2n}
    purpose: To model non-minimal scalar-curvature coupling as a deformation of Einstein-frame inflation
    Possibly constrained indirectly via r=(1−n)r_E and n_S shifts, but the paper does not derive n or λ from a deeper principle; λ is fixed by the CMB amplitude, so it has no independent falsifiable handle outside the paper. The parametrization itself is invented to organize the analysis.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Corrections to inflationary models induced by non-minimal coupling between scalar field and curvature." pith.science (2026). https://pith.science/paper/L3SRZZIK

@misc{pith2026260710679,
  author       = {Pith},
  title        = {Pith review of: Corrections to inflationary models induced by non-minimal coupling between scalar field and curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L3SRZZIK}},
  note         = {Machine review of arXiv:2607.10679}
}
read the original abstract

In this paper, we consider possible corrections to the characteristics of inflationary models based on a specific parametrization of the non-minimal coupling between the scalar field and curvature. At the inflationary stage, these corrections lead to a deformation of the scalar field potential and a corresponding deviation in the determination of the cosmological perturbation parameters. At the same time, it is shown that the proposed parametrization yields a description of the reheating stage dynamics completely analogous to the case of Einstein gravity with minimal coupling between the scalar field and curvature. For a model-independent analysis of inflationary corrections induced by a non-minimal coupling, a classification of inflationary scenarios based on the expansion in series of the dependence of the tensor-to-scalar ratio on the spectral index of scalar perturbations is considered. It is also shown that this approach allows for the inclusion of well-known inflationary models as special cases.

Figures

Figures reproduced from arXiv: 2607.10679 by the authors.

Figure 1
Figure 1. FIG. 1: Dependencies [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 1
Figure 1. FIG. 1: Dependencies [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Dependencies [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figures from the paper (1 more)
Figure 2
Figure 2. Figure 2: FIG. 2: Dependencies [PITH_FULL_IMAGE:figures/full_fig_p015_2.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

91 extracted references · 73 linked inside Pith

  1. [1]

    Generalized Hybrid Natural Inflation As the example of inflationary models with relation (140) between the slow-roll parameters we consider the power-law generalized Hybrid Natural Inflation. A no- table departure from standard single-field realisations is obtained by combining the inflationary trajectory of Nat- ural Inflation [66–70] with the exit mecha...

  2. [2]

    ,(55) w1 =F,(56) w2 = 2HF+ ˙F ,(57) w3 =−9F H2 −9H ˙F+ 3 2 (H ˙F−2F ˙H− ¨F),(58) w4 =F.(59) The velocity of the scalar perturbations in this case c2 S = 1 in the chosen system of units [50–53], which can be verified by direct substitution (56)–(59) into (55). Also, in the slow-roll regimeϵ≪1,|δ| ≪1, from (30), (54) and (56)–(59) we obtain QS = H λ 2n ϵ (n...

  3. [3]

    UlGPU named after I.N. Ulyanov

    Modifiedα-attractor models As an example of the second-order inflationary model, we considerα-attractors models. Cosmologicalα- attractors models are a class of inflationary theories that arise naturally in the context of supergravity theories. Their distinctive feature is the presence of universal be- havior in their predictions, independent of the detai...

    2026

  4. [4]

    Baumann and L

    D. Baumann and L. McAllister,Inflation and String The- ory, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2015) arXiv:1404.2601 [hep-th]

    arXiv 2015

  5. [5]

    Chervon, I

    S. Chervon, I. Fomin, V. Yurov, and A. Yurov,Scalar Field Cosmology, Series on the Foundations of Natural Science and Technology, Vol. 13 (WSP, Singapur, 2019)

    2019

  6. [6]

    Martin, C

    J. Martin, C. Ringeval, and V. Vennin, Encyclopædia Inflationaris: Opiparous Edition, Phys. Dark Univ.5-6, 75 (2014), arXiv:1303.3787 [astro-ph.CO]

    arXiv 2014

  7. [7]

    Aghanimet al.(Planck), Planck 2018 results

    N. Aghanimet al.(Planck), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph]

    arXiv 2018

  8. [8]

    P. A. R. Adeet al.(BICEP, Keck), Improved Con- straints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season, Phys. Rev. Lett.127, 151301 (2021), arXiv:2110.00483 [astro-ph.CO]

    arXiv 2018

Show all 91 references
  1. [9]

    Louiset al.(Atacama Cosmology Telescope), The Atacama Cosmology Telescope: DR6 power spectra, likelihoods and ΛCDM parameters, JCAP11, 062, arXiv:2503.14452 [astro-ph.CO]

    T. Louiset al.(Atacama Cosmology Telescope), The Atacama Cosmology Telescope: DR6 power spectra, likelihoods and ΛCDM parameters, JCAP11, 062, arXiv:2503.14452 [astro-ph.CO]

    arXiv

  2. [10]

    Calabreseet al.(Atacama Cosmology Telescope), The Atacama Cosmology Telescope: DR6 constraints on extended cosmological models, JCAP11, 063, arXiv:2503.14454 [astro-ph.CO]

    E. Calabreseet al.(Atacama Cosmology Telescope), The Atacama Cosmology Telescope: DR6 constraints on extended cosmological models, JCAP11, 063, arXiv:2503.14454 [astro-ph.CO]

    arXiv

  3. [11]

    A. G. Adameet al.(DESI), DESI 2024 VI: cosmologi- cal constraints from the measurements of baryon acous- tic oscillations, JCAP02, 021, arXiv:2404.03002 [astro- ph.CO]

    2024 arXiv

  4. [12]

    S. V. Chervon, Chiral Cosmological Models: Dark Sector Fields Description, Quant. Matt.2, 71 (2013), arXiv:1403.7452 [gr-qc]

    2013 arXiv

  5. [13]

    Paliathanasis, G

    A. Paliathanasis, G. Leon, and S. Pan, Exact Solutions in Chiral Cosmology, Gen. Rel. Grav.51, 106 (2019), arXiv:1811.10038 [gr-qc]

    2019 arXiv

  6. [14]

    S. V. Chervon, I. V. Fomin, E. O. Pozdeeva, M. Sami, and S. Y. Vernov, Superpotential method for chiral cos- mological models connected with modified gravity, Phys. Rev. D100, 063522 (2019), arXiv:1904.11264 [gr-qc]

    2019 arXiv

  7. [15]

    I. V. Fomin and S. V. Chervon, New method of exponen- tial potentials reconstruction based on given scale fac- tor in phantonical two-field models, JCAP04(04), 025, arXiv:2112.09359 [gr-qc]

    arXiv

  8. [16]

    Gomez-Valent and J

    A. Gomez-Valent and J. Sol` a Peracaula, Phantom Mat- ter: A Challenging Solution to the Cosmological Ten- sions, Astrophys. J.975, 64 (2024), arXiv:2404.18845 [astro-ph.CO]

    2024 arXiv

  9. [17]

    Barbosa-Cendejas, J

    N. Barbosa-Cendejas, J. De-Santiago, G. German, J. C. Hidalgo, and R. R. Mora-Luna, Theoretical and observa- tional constraints on Tachyon Inflation, JCAP03, 015, arXiv:1711.06693 [astro-ph.CO]

    arXiv

  10. [18]

    Nautiyal, Reheating constraints on Tachyon Inflation, Phys

    A. Nautiyal, Reheating constraints on Tachyon Inflation, Phys. Rev. D98, 103531 (2018), arXiv:1806.03081 [astro- ph.CO]

    2018 arXiv

  11. [19]

    A. L. Ferreira Junior, N. Pinto-Neto, and J. Zanelli, In- flation and late-time accelerated expansion driven by k- essence degenerate dynamics, Phys. Rev. D109, 023515 (2024), arXiv:2311.01456 [hep-th]

    2024 arXiv

  12. [20]

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

    2010 arXiv

  13. [21]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models, Phys. Rept.505, 59 (2011), arXiv:1011.0544 [gr-qc]

    2011 arXiv

  14. [22]

    De Felice and S

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

    2010 arXiv

  15. [23]

    De Felice and S

    A. De Felice and S. Tsujikawa, Conditions for the cosmo- logical viability of the most general scalar-tensor theories and their applications to extended Galileon dark energy models, JCAP02, 007, arXiv:1110.3878 [gr-qc]

    arXiv

  16. [24]

    De Felice and S

    A. De Felice and S. Tsujikawa, Primordial non- Gaussianities in general modified gravitational models of inflation, JCAP04, 029, arXiv:1103.1172 [astro-ph.CO]

    arXiv

  17. [25]

    De Felice, S

    A. De Felice, S. Tsujikawa, J. Elliston, and R. Tavakol, Chaotic inflation in modified gravitational theories, JCAP08, 021, arXiv:1105.4685 [astro-ph.CO]

    arXiv

  18. [26]

    Clifton, P

    T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Modified Gravity and Cosmology, Phys. Rept.513, 1 (2012), arXiv:1106.2476 [astro-ph.CO]

    2012 arXiv

  19. [27]

    Capozziello and M

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

    2011 arXiv

  20. [28]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, Modi- fied Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution, Phys. Rept.692, 1 (2017), arXiv:1705.11098 [gr-qc]

    2017 arXiv

  21. [29]

    Ishak, Testing General Relativity in Cosmology, Liv- ing Rev

    M. Ishak, Testing General Relativity in Cosmology, Liv- ing Rev. Rel.22, 1 (2019), arXiv:1806.10122 [astro- ph.CO]

    2019 arXiv

  22. [30]

    Han, Loop quantum cosmological dynamics of scalar- tensor theory in the Jordan frame, Phys

    Y. Han, Loop quantum cosmological dynamics of scalar- tensor theory in the Jordan frame, Phys. Rev. D100, 123541 (2019), arXiv:1911.01128 [gr-qc]

    2019 arXiv

  23. [31]

    Akramiet al.(CANTATA),Modified Gravity and Cos- mology

    Y. Akramiet al.(CANTATA),Modified Gravity and Cos- mology. An Update by the CANTATA Network, edited by E. N. Saridakis, R. Lazkoz, V. Salzano, P. Vargas Moniz, S. Capozziello, J. Beltr´ an Jim´ enez, M. De Laurentis, and G. J. Olmo (Springer, 2021) arXiv:2105.12582 [gr-qc]

    2021 arXiv

  24. [32]

    Bahamonde, K

    S. Bahamonde, K. F. Dialektopoulos, C. Escamilla- Rivera, G. Farrugia, V. Gakis, M. Hendry, M. Hohmann, J. Levi Said, J. Mifsud, and E. Di Valentino, Teleparallel gravity: from theory to cosmology, Rept. Prog. Phys.86, 026901 (2023), arXiv:2106.13793 [gr-qc]

    2023 arXiv

  25. [33]

    S. D. Odintsov, V. K. Oikonomou, I. Giannakoudi, F. P. Fronimos, and E. C. Lymperiadou, Recent Advances in Inflation, Symmetry15, 1701 (2023), arXiv:2307.16308 [gr-qc]

    2023 arXiv

  26. [34]

    Barroso Varela, O

    M. Barroso Varela, O. Bertolami, and A. Mantziris, Infla- tionary dynamics of non-minimally coupled f(R) matter- curvature theories, JCAP01, 061, arXiv:2509.01532 [gr- qc]. 20

  27. [35]

    R. Roy, G. Otalora, J. Saavedra, and S. Capozziello, Non- minimally coupled loop quantum inflation with inverse- volume corrections, Nucl. Phys. B1029, 117530 (2026), arXiv:2603.04182 [gr-qc]

    2026 arXiv

  28. [36]

    D’Eramo, N

    F. D’Eramo, N. Fernandez, and S. Profumo, When the Universe Expands Too Fast: Relentless Dark Matter, JCAP05, 012, arXiv:1703.04793 [hep-ph]

    arXiv

  29. [37]

    Maharana and I

    A. Maharana and I. Zavala, Postinflationary scalar tensor cosmology and inflationary parameters, Phys. Rev. D97, 123518 (2018), arXiv:1712.07071 [hep-ph]

    2018 arXiv

  30. [38]

    M. R. Haque, S. Pal, and D. Paul, Improved predictions on Higgs-Starobinsky inflation and reheating with ACT DR6 and primordial gravitational waves, Phys. Lett. B 869, 139852 (2025), arXiv:2505.04615 [astro-ph.CO]

    2025 arXiv

  31. [39]

    D. G. Figueroa and E. H. Tanin, Ability of LIGO and LISA to probe the equation of state of the early Universe, JCAP08, 011, arXiv:1905.11960 [astro-ph.CO]

    1905 arXiv

  32. [40]

    S. D. Odintsov and T. Paul, ACT inflation and its influ- ence on reheating era in Einstein-Gauss-Bonnet gravity, Phys. Lett. B870, 139930 (2025), arXiv:2508.11377 [gr- qc]

    2025

  33. [41]

    B. P. Abbottet al., Multi-messenger Observations of a Binary Neutron Star Merger, Astrophys. J. Lett.848, L12 (2017), arXiv:1710.05833 [astro-ph.HE]

    2017 arXiv

  34. [42]

    Fujii and K

    Y. Fujii and K. Maeda,The scalar-tensor theory of gravi- tation, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2007)

    2007

  35. [43]

    Faraoni,Cosmology in scalar tensor gravity, Funda- mental Theories of Physics (Springer Dordrecht, 2004)

    V. Faraoni,Cosmology in scalar tensor gravity, Funda- mental Theories of Physics (Springer Dordrecht, 2004)

    2004

  36. [44]

    Garcia-Bellido and D

    J. Garcia-Bellido and D. Wands, General relativity as an attractor in scalar - tensor stochastic inflation, Phys. Rev. D52, 5636 (1995), arXiv:gr-qc/9503049

    1995 arXiv

  37. [45]

    E. O. Pozdeeva, M. A. Skugoreva, A. V. Toporensky, and S. Y. Vernov, Possible evolution of a bouncing universe in cosmological models with non-minimally coupled scalar fields, JCAP12, 006, arXiv:1608.08214 [gr-qc]

    arXiv

  38. [46]

    J. A. Belinch´ on, T. Harko, and M. K. Mak, Exact scalar–tensor cosmological models, Int. J. Mod. Phys. D 26, 1750073 (2017), arXiv:1612.05446 [gr-qc]

    2017 arXiv

  39. [47]

    Bhattacharya, K

    S. Bhattacharya, K. Das, and K. Dutta, Attractor models in scalar–tensor theories of inflation, Int. J. Mod. Phys. D27, 1850079 (2018), arXiv:1706.07934 [gr-qc]

    2018 arXiv

  40. [48]

    Fomin and S

    I. Fomin and S. Chervon, Inflation with explicit parametric connection between general relativity and scalar–tensor gravity, Mod. Phys. Lett. A33, 1850161 (2018), arXiv:1802.10462 [gr-qc]

    2018 arXiv

  41. [49]

    I. V. Fomin and S. V. Chervon, Reconstruction of general relativistic cosmological solutions in modified gravity the- ories, Phys. Rev. D100, 023511 (2019), arXiv:1903.03974 [gr-qc]

    2019 arXiv

  42. [50]

    Motohashi and A

    H. Motohashi and A. A. Starobinsky, Constant-roll inflation in scalar-tensor gravity, JCAP11, 025, arXiv:1909.10883 [gr-qc]

    1909 arXiv

  43. [51]

    L´ opez, G

    M. L´ opez, G. Otalora, and N. Videla, Chaotic inflation and reheating in generalized scalar-tensor gravity, JCAP 10, 021, arXiv:2107.07679 [gr-qc]

    arXiv

  44. [52]

    Amaek, A

    W. Amaek, A. Payaka, and P. Channuie, Warm inflation in general scalar-tensor theory of gravity, Phys. Rev. D 105, 083501 (2022), arXiv:2111.07141 [gr-qc]

    2022 arXiv

  45. [53]

    I. V. Fomin and S. V. Chervon, Non-minimal cou- pling influence on the deviation from de Sitter cos- mological expansion, Eur. Phys. J. C78, 918 (2018), arXiv:1711.06870 [gr-qc]

    2018 arXiv

  46. [54]

    I. V. Fomin, S. V. Chervon, and A. V. Tsyganov, Gener- alized scalar-tensor theory of gravity reconstruction from physical potentials of a scalar field, Eur. Phys. J. C80, 350 (2020), arXiv:2004.08544 [gr-qc]

    2020 arXiv

  47. [55]

    Fomin and S

    I. Fomin and S. Chervon, Exact and Slow-Roll Solu- tions for Exponential Power-Law Inflation Connected with Modified Gravity and Observational Constraints, Universe6, 199 (2020), arXiv:2006.16074 [gr-qc]

    2020 arXiv

  48. [56]

    I. V. Fomin, S. V. Chervon, A. N. Morozov, and I. S. Golyak, Relic gravitational waves in verified inflationary models based on the generalized scalar–tensor gravity, Eur. Phys. J. C82, 642 (2022)

    2022

  49. [57]

    I. V. Fomin, S. V. Chervon, L. K. Duchaniya, and B. Mishra, The scalar-torsion gravity corrections in the first-order inflationary models, Phys. Dark Univ.48, 101895 (2025), arXiv:2407.14542 [gr-qc]

    2025 arXiv

  50. [58]

    Lin and M

    W. Lin and M. Ishak, Testing gravity theories using tensor perturbations, Phys. Rev. D94, 123011 (2016), arXiv:1605.03504 [astro-ph.CO]

    2016 arXiv

  51. [59]

    Scalisi and I

    M. Scalisi and I. Valenzuela, Swampland distance con- jecture, inflation andα-attractors, JHEP08, 160, arXiv:1812.07558 [hep-th]

    arXiv

  52. [60]

    M. S. Turner, Coherent Scalar Field Oscillations in an Expanding Universe, Phys. Rev. D28, 1243 (1983)

    1983

  53. [61]

    Martin and C

    J. Martin and C. Ringeval, First CMB Constraints on the Inflationary Reheating Temperature, Phys. Rev. D 82, 023511 (2010), arXiv:1004.5525 [astro-ph.CO]

    2010 arXiv

  54. [62]

    Martin, C

    J. Martin, C. Ringeval, and V. Vennin, Observing Infla- tionary Reheating, Phys. Rev. Lett.114, 081303 (2015), arXiv:1410.7958 [astro-ph.CO]

    2015 arXiv

  55. [63]

    M. A. G. Garcia, K. Kaneta, Y. Mambrini, and K. A. Olive, Inflaton Oscillations and Post-Inflationary Reheat- ing, JCAP04, 012, arXiv:2012.10756 [hep-ph]

    2012 arXiv

  56. [64]

    Martin, C

    J. Martin, C. Ringeval, R. Trotta, and V. Vennin, The Best Inflationary Models After Planck, JCAP03, 039, arXiv:1312.3529 [astro-ph.CO]

    arXiv

  57. [65]

    Grøn, Predictions of Spectral Parameters by Several Inflationary Universe Models in Light of the Planck Re- sults, Universe4, 15 (2018)

    Ø. Grøn, Predictions of Spectral Parameters by Several Inflationary Universe Models in Light of the Planck Re- sults, Universe4, 15 (2018)

    2018

  58. [66]

    S. S. Mishra, V. Sahni, and A. V. Toporensky, Initial conditions for Inflation in an FR W Universe, Phys. Rev. D98, 083538 (2018), arXiv:1801.04948 [gr-qc]

    2018 arXiv

  59. [67]

    N. A. Avdeev and A. V. Toporensky, Ruling Out Infla- tion Driven by a Power Law Potential: Kinetic Cou- pling Does Not Help, Grav. Cosmol.28, 416 (2022), arXiv:2203.14599 [gr-qc]

    2022 arXiv

  60. [68]

    S. D. Odintsov, S. D’Onofrio, and T. Paul, Entropic In- flation in Presence of Scalar Field, Universe10, 4 (2024), arXiv:2312.13587 [gr-qc]

    2024 arXiv

  61. [69]

    Freese, J

    K. Freese, J. A. Frieman, and A. V. Olinto, Natural Infla- tion with Pseudo - Nambu-Goldstone Bosons, Phys. Rev. Lett.65, 3233 (1990)

    1990

  62. [70]

    F. C. Adams, J. R. Bond, K. Freese, J. A. Frieman, and A. V. Olinto, Natural inflation: Particle physics models, power law spectra for large scale structure, and constraints from COBE, Phys. Rev. D47, 426 (1993), arXiv:hep-ph/9207245

    1993 arXiv

  63. [71]

    Freese and W

    K. Freese and W. H. Kinney, On: Natural inflation, Phys. Rev. D70, 083512 (2004), arXiv:hep-ph/0404012

    2004 arXiv

  64. [72]

    J. E. Kim, H. P. Nilles, and M. Peloso, Completing nat- ural inflation, JCAP01, 005, arXiv:hep-ph/0409138

    arXiv

  65. [73]

    de la Fuente, P

    A. de la Fuente, P. Saraswat, and R. Sundrum, Natural Inflation and Quantum Gravity, Phys. Rev. Lett.114, 151303 (2015), arXiv:1412.3457 [hep-th]

    2015 arXiv

  66. [74]

    G. G. Ross and G. German, Hybrid natural inflation from 21 non Abelian discrete symmetry, Phys. Lett. B684, 199 (2010), arXiv:0902.4676 [hep-ph]

    2010 arXiv

  67. [75]

    E. D. Stewart and J. D. Cohn, Inflationary models with a flat potential enforced by nonAbelian discrete gauge symmetries, Phys. Rev. D63, 083519 (2001), arXiv:hep- ph/0002214

    2001

  68. [76]

    J. D. Cohn and E. D. Stewart, NonAbelian discrete gauge symmetries and inflation, Phys. Lett. B475, 231 (2000), arXiv:hep-ph/0001333

    2000 arXiv

  69. [77]

    Antusch, S

    S. Antusch, S. F. King, M. Malinsky, L. Velasco-Sevilla, and I. Zavala, Flavon Inflation, Phys. Lett. B666, 176 (2008), arXiv:0805.0325 [hep-ph]

    2008 arXiv

  70. [78]

    G. G. Ross and G. German, Hybrid Natural Low Scale Inflation, Phys. Lett. B691, 117 (2010), arXiv:1002.0029 [hep-ph]

    2010 arXiv

  71. [79]

    J. A. V´ azquez, M. Carrillo-Gonz´ alez, G. Germ´ an, A. Herrera-Aguilar, and J. C. Hidalgo, Constrain- ing Hybrid Natural Inflation with recent CMB data, JCAP02, 039, [Addendum: JCAP 10, A01 (2015)], arXiv:1411.6616 [astro-ph.CO]

    2015 arXiv

  72. [80]

    J. B. Munoz and M. Kamionkowski, Equation-of-State Parameter for Reheating, Phys. Rev. D91, 043521 (2015), arXiv:1412.0656 [astro-ph.CO]

    2015 arXiv

  73. [81]

    Kitabayashi, Generalized hybrid natural inflation, Phys

    T. Kitabayashi, Generalized hybrid natural inflation, Phys. Rev. D108, 043514 (2023), arXiv:2305.03905 [hep- ph]

    2023 arXiv

  74. [82]

    Kallosh, A

    R. Kallosh, A. Linde, and D. Roest, Superconformal In- flationaryα-Attractors, JHEP11, 198, arXiv:1311.0472 [hep-th]

    arXiv

  75. [83]

    Kallosh and A

    R. Kallosh and A. Linde, Universality Class in Conformal Inflation, JCAP07, 002, arXiv:1306.5220 [hep-th]

    arXiv

  76. [84]

    Kallosh, A

    R. Kallosh, A. Linde, and D. Roest, Large field inflation and doubleα-attractors, JHEP08, 052, arXiv:1405.3646 [hep-th]

    arXiv

  77. [85]

    Galante, R

    M. Galante, R. Kallosh, A. Linde, and D. Roest, Unity of Cosmological Inflation Attractors, Phys. Rev. Lett.114, 141302 (2015), arXiv:1412.3797 [hep-th]

    2015 arXiv

  78. [86]

    J. J. M. Carrasco, R. Kallosh, and A. Linde, Cosmological Attractors and Initial Conditions for Inflation, Phys. Rev. D92, 063519 (2015), arXiv:1506.00936 [hep-th]

    2015 arXiv

  79. [87]

    German, On theα-attractor T-models, JCAP09, 017, arXiv:2105.05426 [astro-ph.CO]

    G. German, On theα-attractor T-models, JCAP09, 017, arXiv:2105.05426 [astro-ph.CO]

    arXiv

  80. [88]

    S. Aoki, H. Otsuka, and R. Yanagita, Heavy field effects on inflationary models in light of ACT data, JCAP11, 088, arXiv:2509.06739 [hep-ph]

  81. [89]

    Frolovsky and S

    D. Frolovsky and S. V. Ketov, Are single-field models of inflation and PBHs production ruled out by ACT observations?, Mod. Phys. Lett. A40, 2550182 (2025), arXiv:2505.17514 [astro-ph.CO]

    2025

  82. [90]

    S. V. Ketov, E. O. Pozdeeva, and S. Y. Vernov, Inflation in F(R) gravity models revisited after ACT, JCAP12, 040, arXiv:2508.08927 [gr-qc]

  83. [91]

    W. J. Wolf, Inflationary attractors and radiative corrections in light of ACT data, JCAP02, 088, arXiv:2506.12436 [astro-ph.CO]

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.