Pith. sign in

REVIEW 1 major objections 5 minor 87 references

The paper claims that the stability of the exact de Sitter solution in a generic sixth-order gravity is controlled by the sign of 3γ1+γ2, even though these two couplings do not affect the solution's expansion rate.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 11:50 UTC pith:L46TETM5

load-bearing objection The instability bound 3γ1+γ2<0 is clean and likely right, but it sits on unverified EL field equations, and the paper has a concrete algebra error in Section II.C plus an unwarranted ACT claim. the 1 major comments →

arxiv 2607.20575 v1 pith:L46TETM5 submitted 2026-07-22 gr-qc

Unstable de Sitter inflationary solution in sixth-order gravity

Tuan Q. Do This is my paper
classification gr-qc PACS 04.50.Kd98.80.-k
keywords sixth-order gravityde Sitter solutioninflationstability analysishigher-derivative gravityFLRW cosmologydynamical systems
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper studies the most general gravity action built from Ricci and Riemann curvature terms with up to six metric derivatives, in a spatially flat FLRW universe. It derives the reduced field equations, finds an exact de Sitter solution β(t)=ζt, and then asks whether that solution is stable. The central result is that the coefficients γ1 and γ2 of R□R and Rμν□Rμν do not change the solution's expansion rate ζ, but their combination 3γ1+γ2 decides its fate: whenever it is negative, the de Sitter fixed point is unstable, so trajectories are repelled from it. That makes the model a candidate for inflation, since an unstable de Sitter phase can end, while the constrained second-order limit gives a stable de Sitter attractor suitable for late-time acceleration. The fourth-order limit, by contrast, can be unstable or stable depending on the coefficients.

Core claim

For the action (2.1), an exact de Sitter solution exists whenever 144γ3+36γ4+9γ5+24γ7+4γ8>0, and it is given by β(t)=ζt with ζ^4=α/(144γ3+36γ4+9γ5+24γ7+4γ8). The R□R, Rμν□Rμν, R², and Rμν² terms drop out of this algebraic equation, so they do not fix the Hubble rate. But in the stability polynomial for the de Sitter fixed point, the leading coefficient a4=4(3γ1+γ2) appears; since a0>0 follows from the reality of ζ, a4<0 forces at least one positive eigenvalue λ, making the fixed point a repeller. A direct perturbation of Eq. (2.16) cross-checks the same conclusion through c0c4<0 when 3γ1+γ2<0. The paper therefore establishes a clean separation: the value of the de Sitter solution is governed

What carries the argument

The load-bearing object is the sixth-order gravity action (2.1) reduced to the lapse-N, scale-factor-β variables, with field equations (2.16)-(2.17) obtained by Euler-Lagrange variation. The reparametrization B=1/β̇², Q=β̈/β̇², Q2=β(3)/β̇³, etc., converts these into a five-dimensional autonomous dynamical system; the de Sitter solution is the fixed point (B0,0,0,0,0) with B0²=(144γ3+36γ4+9γ5+24γ7+4γ8)/α. The decisive identity is the stability polynomial (3.49): the eigenvalues are λ=0 plus roots of a4λ⁴+a3λ³+a2λ²+a1λ+a0=0 with a4=4(3γ1+γ2). Because a0>0, the sign of a4 alone forces a positive root when 3γ1+γ2<0. This single coefficient carries the entire stability claim, and the same structu

Load-bearing premise

The whole stability criterion rests on the correctness of the reduced FLRW field equations (2.16)-(2.17)—in particular the coefficient -4(3γ1+γ2) multiplying β(6) in Eq. (2.17); the paper cites a software package for these reductions but does not include an independent derivation, and a missing boundary term or mis-assembled six-derivative invariant that changed this coefficient would move the stability boundary.

What would settle it

Recompute Eqs. (2.16)-(2.17) by direct metric variation of the action (2.1) or with independent computer algebra and verify that the coefficient of β(6) in Eq. (2.17) is exactly -4(3γ1+γ2). Then, with parameters satisfying 3γ1+γ2<0 and 144γ3+36γ4+9γ5+24γ7+4γ8>0, integrate the full field equations from initial data infinitesimally close to the de Sitter solution; if β(t) converges back to ζt instead of moving away, the central claim is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Under 3γ1+γ2<0, no stable de Sitter attractor exists in the full sixth-order theory: generic FLRW trajectories leave the de Sitter point and flow to a non-de Sitter fixed point, so inflation can end.
  • The expansion rate of the exact de Sitter solution is fixed entirely by α and the cubic coupling combination; tuning γ1, γ2, R², or Rμν² leaves ζ unchanged.
  • In the fourth-order limit (3γ1+γ2=0), the same de Sitter solution is unstable whenever the coefficient ā2 in Eq. (3.79) is positive, so this regime can also serve as an inflationary repeller.
  • In the second-order limit, with the constraints (2.19), (2.28), (2.29), and (2.30) satisfied, the de Sitter fixed point is a stable attractor with perturbations decaying as e^{-3τ}, compatible with late-time cosmic acceleration.
  • Since δ(β(6)/β̇⁶) can be defined even when β1=β2=0, the quadratic curvature terms are not needed for a complete perturbation system, unlike the generalized Einsteinian cubic gravity case.

Where Pith is reading between the lines

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

  • Because the stability boundary is the single combination 3γ1+γ2, any future measurement of inflationary observables in such a theory—such as the scalar spectral index n_s—would constrain this sum, not the two coefficients separately.
  • The same mechanism may extend to even higher-derivative theories: for eighth-order gravity built on R□²R, the de Sitter stability should likewise be set by the coefficient of the highest-derivative term, since lower-derivative terms do not determine ζ.
  • A direct tensorial variation of the action (2.1), rather than the effective lapse-function reduction, would be a natural independent check; if the coefficient multiplying β(6) in the ii-equation differs from -4(3γ1+γ2), the inequality would have to be shifted.
  • The repeller interpretation suggests that the non-de Sitter fixed point acts as the actual late-time attractor for early-universe trajectories; computing the curvature invariants at that point would show what kind of geometry inflation relaxes into.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper studies the sixth-order gravity (2.1) in a flat FLRW background, derives the field equations (2.16)-(2.17) by an effective Euler-Lagrange method, and finds the exact de Sitter solution β(t)=ζt with ζ^4=α/(144γ3+36γ4+9γ5+24γ7+4γ8). The central claim is that, although γ1 and γ2 do not affect the de Sitter radius, the solution is unstable whenever 3γ1+γ2<0, because the characteristic polynomial (3.49) then has a4a0<0 and hence at least one positive root. The paper also analyzes the second-order and fourth-order limits, and provides numerical trajectories illustrating that the de Sitter point is a repeller. An Appendix A cross-check repeats the sign-of-root argument using a direct perturbation of Eq. (2.16).

Significance. If the field equations (2.16)-(2.17) are correct, the paper gives a clean, parameter-free condition — 3γ1+γ2<0 — for an unstable de Sitter solution, with the coefficients γ1,γ2 absent from the solution value but controlling its stability. The sign-of-root argument itself is rigorous and is independently cross-checked in Appendix A, and the numerical evolution in Section III.F supports the repeller interpretation. The distinction between the value of the de Sitter solution and its stability is a genuinely interesting feature. However, the entire result rests on lengthy coefficient algebra that is not independently verified: the paper cites the xCoba package but ships no notebook or explicit derivation of the β(6) coefficient in Eq. (2.17). A concrete algebraic error in the related constraint-solving of Section II.C reduces confidence in the unverified long equations.

major comments (1)
  1. [II.C, Eqs. (2.21), (2.26)] The claim that constraints (2.21) and (2.26) force γ1=γ2=0 is false. A counterexample is γ1=1, γ2=-3, γ3=-2/3, γ4=5/2, γ5=γ7=γ8=0, which satisfies all constraints (2.19)-(2.27). This error is not directly used in the main de Sitter stability analysis, but it demonstrates that the long coefficient algebra leading to (2.16)-(2.17) is error-prone. The central criterion 3γ1+γ2<0 depends on the coefficient of β(6) in Eq. (2.17) being exactly -4(3γ1+γ2) and on the constant term a0 in Eq. (3.54), and the Appendix A cross-check uses the same unverified Eq. (2.16). The paper cites xCoba in footnote [82] but provides no notebook or independent derivation. This is load-bearing: without an independent check of these coefficients, the main instability theorem is conditional.
minor comments (5)
  1. [III.E, Eq. (3.49)] The characteristic polynomial is written as a4λ4 + a3λ3 + a3λ3 + a2λ2 + a1λ + a0, duplicating the a3λ3 term. It should be a4λ4 + a3λ3 + a2λ2 + a1λ + a0.
  2. [III.E, after Eq. (3.55)] The statement that a polynomial with all positive coefficients has only non-positive roots is not generally true. For example, λ3+λ2+λ+10 has all positive coefficients but two roots with positive real part. The conclusion that a stable de Sitter point is 'possible' when all coefficients are positive is therefore not established by the stated reasoning, although this does not affect the main instability criterion.
  3. [III.E, Eq. (3.48)] The matrix in Eq. (3.48) is typeset ambiguously; the rows are not clearly separated. Please rewrite with explicit matrix entries.
  4. [Abstract] Typo: 'give raise' should be 'give rise'.
  5. [II.C, Eqs. (2.31)-(2.34)] The reduction to Eqs. (2.32) and (2.34) is said to use 'suitable linear combinations' of the constraints, but the combinations are not displayed. Please state them explicitly, especially because the preceding constraints were shown to be satisfied by a non-trivial γ1,γ2 solution, making the logical status of the reduction unclear.

Circularity Check

0 steps flagged

No significant circularity: instability condition 3γ1+γ2<0 follows algebraically from the stated field equations and is not fitted or imported.

full rationale

The derivation chain is self-contained. The action (2.1) is reduced to FLRW form and varied via the explicit Euler-Lagrange equations (2.14)-(2.15) to produce the displayed field equations (2.16)-(2.17); the de Sitter value ζ^4=α/(144γ3+36γ4+9γ5+24γ7+4γ8) follows by direct substitution, and the stability criterion is obtained by linearizing the explicit dynamical system (3.7)-(3.11) around that fixed point. No parameter is fitted to data, and no target claim is used as an input: the sign of a4=4(3γ1+γ2) in the characteristic polynomial (3.49) is the condition cited for instability, with a0>0 following from the independently derived existence condition (3.5) and the sign of a4. The self-citations (Refs. [53,70-75]) are methodological references for the effective Lagrangian and dynamical-system techniques; the equations are written out, so those citations are not load-bearing for the central result. Appendix A provides an independent direct-perturbation cross-check. The erroneous constraint claim in Section II.C (that Eqs. (2.21) and (2.26) force γ1=γ2=0) is an algebraic mistake in a special limit; it does not make the derivation circular, because it is not the source of the stability condition. A missing notebook for the xCoba verification is a reproducibility concern, not a circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 0 invented entities

The paper is a conditional analysis: given the sixth-order action and the FLRW assumption, the derived inequality controls stability. It introduces no fitted constants and no new degrees of freedom, but it imports the completeness of the six-derivative action and the correctness of the EL field equations from prior work and from xCoba, which is not shipped.

free parameters (2)
  • 3γ1+γ2
    The instability result is conditional on this combination being negative; its value is not fitted to data.
  • C=144γ3+36γ4+9γ5+24γ7+4γ8
    Positivity of C is required for a real de Sitter radius; no numerical value is fitted.
axioms (6)
  • domain assumption Action (2.1), attributed to Refs. [56,57], is the most general six-derivative extension of the Einstein-Hilbert action in four dimensions.
    The completeness of the invariant basis is imported from prior work; if other independent six-derivative invariants contribute in FLRW, the field equations and stability criterion could change.
  • domain assumption Spatially flat FLRW metric (2.3) with lapse N is sufficient; the cosmic time slicing is non-singular.
    Stability is assessed only for homogeneous isotropic backgrounds; anisotropic or non-flat perturbations are not covered.
  • domain assumption Euler-Lagrange equations (2.14)-(2.15) derived from the reduced action give the correct field equations; total-derivative and boundary terms can be discarded.
    This is the effective method used in the author's earlier papers; no explicit tensorial field equations are written.
  • domain assumption For 3γ1+γ2≠0, Eq. (2.17) can be solved algebraically for β(6)/βdot^6, defining the dynamical system.
    The fourth-order limit is treated separately; the main stability claim is restricted to the sixth-order regime.
  • domain assumption α>0 and the real-root condition (3.5) hold.
    α>0 matches GR normalization; positivity of the cubic combination is needed for a real positive ζ.
  • standard math A real polynomial with negative leading coefficient and positive constant term has a positive real root.
    Used to infer instability from a4<0, a0>0 without computing eigenvalues.

pith-pipeline@v1.3.0-alltime-deepseek · 25783 in / 18896 out tokens · 180253 ms · 2026-08-01T11:50:01.334248+00:00 · methodology

0 comments
read the original abstract

A sixth-order gravity, which involves not only two leading terms, $\gamma_1 R\Box R$ and $\gamma_2 R_{\mu\nu} \Box R^{\mu\nu}$, but also quadratic curvature terms along with cubic curvature ones, will be investigated in this paper to see if it admits an exact stable de Sitter solution. First, we will derive sixth-order differential field equations of this gravity under the homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker background spacetime, using the effective method based on the Euler-Lagrange equations. Then, we will analytically solve these field equations to figure out an exact de Sitter solution, which turns out to be equivalent to a fixed point of the corresponding dynamical system of the studied gravity. Interestingly, two coefficients, $\gamma_1$ and $\gamma_2$, do not contribute to the value of the obtained de Sitter solution. However, they affect on the stability of the de Sitter solution. In particular, if these two coefficients obey the following inequality, $3\gamma_1+\gamma_2 <0$, then the de Sitter solution will always be unstable. Furthermore, numerical calculations will be performed to verify that the de Sitter fixed point is indeed a repeller of the dynamical system once this inequality is satisfied. All these results indicate that the sixth-order gravity is more suitable for an inflationary phase of early universe. To be complete, two special limits of the studied gravity model, in which field equations are reduced to second-order and fourth-order, respectively, will be investigated. As expected, only the second-order limit can always give raise a stable de Sitter solution, compatible with an accelerated expansion of late-time universe.

Figures

Figures reproduced from arXiv: 2607.20575 by Tuan Q. Do.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

discussion (0)

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

Reference graph

Works this paper leans on

87 extracted references · 1 canonical work pages

  1. [1]

    A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B91, 99 (1980). 19

    1980

  2. [2]

    A. H. Guth, The inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D23, 347 (1981)

    1981

  3. [3]

    A. D. Linde, A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett.108B, 389 (1982)

    1982

  4. [4]

    A. D. Linde, Chaotic inflation, Phys. Lett.129B, 177 (1983)

    1983

  5. [5]

    Baumann, Cosmology, Cambrigde University Press (2022), chapters 4 and 8

    D. Baumann, Cosmology, Cambrigde University Press (2022), chapters 4 and 8

    2022

  6. [6]

    Akramiet al.[Planck], Planck 2018 results

    Y. Akramiet al.[Planck], Planck 2018 results. X. Constraints on inflation, Astron. Astrophys.641, A10 (2020) [arXiv:1807.06211]

    Pith/arXiv arXiv 2018

  7. [7]

    Calabreseet al.[Atacama Cosmology Telescope], The Atacama Cosmology Telescope: DR6 constraints on extended cosmological models, J

    E. Calabreseet al.[Atacama Cosmology Telescope], The Atacama Cosmology Telescope: DR6 constraints on extended cosmological models, J. Cosmol. Astropart. Phys.11, 063 (2025) [arXiv:2503.14454]

    Pith/arXiv arXiv 2025

  8. [8]

    Addazi, Y

    A. Addazi, Y. Aldabergenov, and S. V. Ketov, Curvature corrections to Starobinsky inflation can explain the ACT results, Phys. Lett. B869, 139883 (2025) [arXiv:2505.10305]

    arXiv 2025

  9. [9]

    Bianchi and M

    E. Bianchi and M. Gamonal, Precision predictions of Starobinsky inflation with self-consistent Weyl-squared corrections, Phys. Rev. D112, 124006 (2025) [arXiv:2506.10081]

    arXiv 2025

  10. [10]

    S. V. Ketov, E. O. Pozdeeva, and S. Y. Vernov, Inflation in F(R) gravity models revisited after ACT, J. Cosmol. Astropart. Phys.12, 040 (2025) [arXiv:2508.08927]

    arXiv 2025

  11. [11]

    Ellis, M

    J. Ellis, M. A. G. Garc ´ ıa, N. Nagata, D. V. Nanopoulos, and K. A. Olive, Deformations of Starobinsky inflation in no-scale SU(5) and SO(10) GUTs, J. Cosmol. Astropart. Phys.12, 038 (2025) [arXiv:2508.13279]

    Pith/arXiv arXiv 2025

  12. [12]

    M. G. Park, D. Y. Cheong, and S. C. Park, Beyond the Starobinsky model after ACT, Phys. Lett. B873, 140217 (2026) [arXiv:2509.04105]

    arXiv 2026

  13. [13]

    C. M. G. R. Morais, G. Rodrigues-da-Silva, and L. G. Medeiros, Higher-order gravity models: corrections up to cubic curvature invariants and inflation, Eur. Phys. J. C85, 1439 (2025) [arXiv:2509.09167]

    arXiv 2025

  14. [14]

    Bezerra-Sobrinho and L

    J. Bezerra-Sobrinho and L. G. Medeiros, Starobinsky inflation and the latest CMB data: a subtle tension?, Eur. Phys. J. C86, 416 (2026) [arXiv:2511.06640]

    Pith/arXiv arXiv 2026

  15. [15]

    Whitt, Fourth order gravity as general relativity plus matter, Phys

    B. Whitt, Fourth order gravity as general relativity plus matter, Phys. Lett. B145, 176 (1984)

    1984

  16. [16]

    K. i. Maeda, Inflation as a transient attractor inR 2 cosmology, Phys. Rev. D37, 858 (1988)

    1988

  17. [17]

    J. D. Barrow and S. Cotsakis, Inflation and the conformal structure of higher order gravity theories, Phys. Lett. B214, 515 (1988)

    1988

  18. [18]

    Faraoni, De Sitter attractors in generalized gravity, Phys

    V. Faraoni, De Sitter attractors in generalized gravity, Phys. Rev. D70, 044037 (2004) [gr-qc/0407021]

    Pith/arXiv arXiv 2004

  19. [19]

    Faraoni, The stability of modified gravity models, Phys

    V. Faraoni, The stability of modified gravity models, Phys. Rev. D72, 124005 (2005) [gr-qc/0511094]

    Pith/arXiv arXiv 2005

  20. [20]

    Faraoni, de Sitter space and the equivalence between f(R) and scalar-tensor gravity, Phys

    V. Faraoni, de Sitter space and the equivalence between f(R) and scalar-tensor gravity, Phys. Rev. D75, 067302 (2007) [gr-qc/0703044]

    Pith/arXiv arXiv 2007

  21. [21]

    J. D. Barrow and S. Hervik, On the evolution of universes in quadratic theories of gravity, Phys. Rev. D74, 124017 (2006) [gr-qc/0610013]

    Pith/arXiv arXiv 2006

  22. [22]

    A. V. Toporensky and P. V. Tretyakov, De Sitter stability in quadratic gravity, Int. J. Mod. Phys. D16, 1075 (2007) [gr-qc/0611068]

    Pith/arXiv arXiv 2007

  23. [23]

    Q. G. Huang, A polynomial f(R) inflation model, J. Cosmol. Astropart. Phys.02, 035 (2014) [arXiv:1309.3514]

    Pith/arXiv arXiv 2014

  24. [24]

    Elizalde, S

    E. Elizalde, S. D. Odintsov, E. O. Pozdeeva, and S. Y. Vernov, Renormalization-group improved inflationary scalar elec- trodynamics and SU(5) scenarios confronted with Planck 2013 and BICEP2 results, Phys. Rev. D90, 084001 (2014) [arXiv:1408.1285]

    Pith/arXiv arXiv 2013

  25. [25]

    E. O. Pozdeeva, M. Sami, A. V. Toporensky, and S. Y. Vernov, Stability analysis of de Sitter solutions in models with the Gauss-Bonnet term, Phys. Rev. D100, 083527 (2019) [arXiv:1905.05085]

    Pith/arXiv arXiv 2019

  26. [26]

    Vernov and E

    S. Vernov and E. Pozdeeva, De Sitter Solutions in Einstein–Gauss–Bonnet Gravity, Universe7, 149 (2021) [arXiv:2104.11111]

    Pith/arXiv arXiv 2021

  27. [27]

    Myrzakulov, S

    R. Myrzakulov, S. Odintsov, and L. Sebastiani, Inflationary universe from higher-derivative quantum gravity, Phys. Rev. D91, 083529 (2015) [arXiv:1412.1073]

    Pith/arXiv arXiv 2015

  28. [28]

    Aoki and S

    K. Aoki and S. Mukohyama, Ghostfree quadratic curvature theories with massive spin-2 and spin-0 particles, Phys. Rev. D100, 064061 (2019) [arXiv:1907.09690]

    Pith/arXiv arXiv 2019

  29. [29]

    D. Y. Cheong, H. M. Lee, and S. C. Park, Beyond the Starobinsky model for inflation, Phys. Lett. B805, 135453 (2020) [arXiv:2002.07981]

    Pith/arXiv arXiv 2020

  30. [30]

    P. A. Cano, K. Fransen, and T. Hertog, Novel higher-curvature variations ofR 2 inflation, Phys. Rev. D103, 103531 (2021) [arXiv:2011.13933]

    Pith/arXiv arXiv 2021

  31. [31]

    Rodrigues-da-Silva, J

    G. Rodrigues-da-Silva, J. Bezerra-Sobrinho, and L. G. Medeiros, Higher-order extension of Starobinsky inflation: Initial conditions, slow-roll regime, and reheating phase, Phys. Rev. D105, 063504 (2022) [arXiv:2110.15502]

    Pith/arXiv arXiv 2022

  32. [32]

    V. R. Ivanov, S. V. Ketov, E. O. Pozdeeva, and S. Y. Vernov, Analytic extensions of Starobinsky model of inflation, J. Cosmol. Astropart. Phys.03, 058 (2022) [arXiv:2111.09058]

    Pith/arXiv arXiv 2022

  33. [33]

    S. V. Ketov, Starobinsky–Bel–Robinson Gravity, Universe8, 351 (2022) [arXiv:2205.13172]

    Pith/arXiv arXiv 2022

  34. [34]

    Shtanov, V

    Y. Shtanov, V. Sahni, and S. S. Mishra, Tabletop potentials for inflation from f(R) gravity, J. Cosmol. Astropart. Phys. 03, 023 (2023) [arXiv:2210.01828]

    Pith/arXiv arXiv 2023

  35. [35]

    Modak, L

    T. Modak, L. R¨ over, B. M. Sch¨ afer, B. Schosser, and T. Plehn, Cornering extended Starobinsky inflation with CMB and SKA, SciPost Phys.15, 047 (2023) [arXiv:2210.05698]

    Pith/arXiv arXiv 2023

  36. [36]

    S. V. Ketov, E. O. Pozdeeva, and S. Y. Vernov, On the superstring-inspired quantum correction to the Starobinsky model of inflation, J. Cosmol. Astropart. Phys.12, 032 (2022) [arXiv:2211.01546]. 20

    Pith/arXiv arXiv 2022

  37. [37]

    Asorey, F

    M. Asorey, F. Ezquerro, and M. Pardina, Stability of cosmological singularity-free solutions in quadratic gravity, Phys. Rev. D111, 064020 (2025) [arXiv:2412.10111]

    Pith/arXiv arXiv 2025

  38. [38]

    Lambiase, S

    G. Lambiase, S. Mukohyama, T. K. Poddar, and A. C. Rescigno, Exorcising ghosts with gravitational waves: cases of ghostful and ghost-free fourth-order gravity, arXiv:2510.17789

    arXiv

  39. [39]

    H. J. Schmidt, Fourth order gravity: Equations, history, and applications to cosmology, eConfC0602061, 12 (2006) [gr-qc/0602017]

    Pith/arXiv arXiv 2006

  40. [40]

    Alvarez-Gaume, A

    L. Alvarez-Gaume, A. Kehagias, C. Kounnas, D. L¨ ust, and A. Riotto, Aspects of quadratic gravity, Fortsch. Phys.64, 176 (2016) [arXiv:1505.07657]

    Pith/arXiv arXiv 2016

  41. [41]

    Salvio, Quadratic gravity, Front

    A. Salvio, Quadratic gravity, Front. in Phys.6, 77 (2018) [arXiv:1804.09944]

    Pith/arXiv arXiv 2018

  42. [42]

    E. J. Copeland, M. Sami, and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D15, 1753 (2006) [hep- th/0603057]

    arXiv 2006

  43. [43]

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

    Pith/arXiv arXiv 2010

  44. [44]

    De Felice and S

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

    Pith/arXiv arXiv 2010

  45. [45]

    S. M. Carroll, A. De Felice, V. Duvvuri, D. A. Easson, M. Trodden, and M. S. Turner, The cosmology of generalized modified gravity models, Phys. Rev. D71, 063513 (2005) [astro-ph/0410031]

    Pith/arXiv arXiv 2005

  46. [46]

    Amendola, R

    L. Amendola, R. Gannouji, D. Polarski, and S. Tsujikawa, Conditions for the cosmological viability of f(R) dark energy models, Phys. Rev. D75, 083504 (2007) [gr-qc/0612180]

    Pith/arXiv arXiv 2007

  47. [47]

    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]

    Pith/arXiv arXiv 2011

  48. [48]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution, Phys. Rept.692, 1 (2017) [arXiv:1705.11098]

    Pith/arXiv arXiv 2017

  49. [49]

    Capozziello and M

    S. Capozziello and M. De Laurentis, Extended Theories of Gravity, Phys. Rept.509, 167-321 (2011) [arXiv:1108.6266]

    Pith/arXiv arXiv 2011

  50. [50]

    K. S. Stelle, Renormalization of higher derivative quantum gravity, Phys. Rev. D16, 953 (1977)

    1977

  51. [51]

    Bellorin, C

    J. Bellorin, C. B´ orquez, and B. Droguett, Batalin-Fradkin-Vilkovisky quantization of quadratic gravity, Phys. Rev. D113, 106006 (2026) [arXiv:2511.15474]

    Pith/arXiv arXiv 2026

  52. [52]

    R. P. Woodard, Ostrogradsky’s theorem on Hamiltonian instability, Scholarpedia10, 32243 (2015) [arXiv:1506.02210]

    Pith/arXiv arXiv 2015

  53. [53]

    10 20 30 40 50 60 τ 2.×10 -6 4.×10 -6 6.×10 -6 8.×10 -6 0.00001 B[τ] FIG

    that stable de Sitter solutions tend to emerge from the second-order gravity theories, whereas unstable ones seem to exist in higher-order cases such as fourth-order gravities. 10 20 30 40 50 60 τ 2.×10 -6 4.×10 -6 6.×10 -6 8.×10 -6 0.00001 B[τ] FIG. 2.The de Sitter fixed point of the second-order limit acts as an attractor. The parameters have been chose...

    2023

  54. [54]

    T. Q. Do, Effect ofR 2 on the stability of de Sitter solution of the generalized Einsteinian cubic gravity, Eur. Phys. J. C 86, 569 (2026) [arXiv:2605.26173]

    Pith/arXiv arXiv 2026

  55. [55]

    De Felice and S

    A. De Felice and S. Tsujikawa, Excluding static and spherically symmetric black holes in Einsteinian cubic gravity with unsuppressed higher-order curvature terms, Phys. Lett. B843, 138047 (2023) [arXiv:2305.07217]

    Pith/arXiv arXiv 2023

  56. [56]

    S. V. Ketov, On Legacy of Starobinsky Inflation, arXiv:2501.06451

    Pith/arXiv arXiv

  57. [57]

    B. L. Giacchini and I. Kol´ aˇ r, Black holes and other exact solutions in six-derivative gravity, Phys. Rev. D112, 024051 (2025) [arXiv:2503.17318 [gr-qc]]

    Pith/arXiv arXiv 2025

  58. [58]

    B. L. Giacchini and I. Kol´ aˇ r, Toward regular black holes in sixth-derivative gravity, Phys. Rev. D110, 104056 (2024) [arXiv:2406.00997 [gr-qc]]

    Pith/arXiv arXiv 2024

  59. [59]

    D. Y. Xu, Two important invariant identities, Phys. Rev. D35, 769 (1987)

    1987

  60. [60]

    Gottlober, H

    S. Gottlober, H. J. Schmidt, and A. A. Starobinsky, Sixth order gravity and conformal transformations, Class. Quant. Grav.7, 893 (1990)

    1990

  61. [61]

    H. J. Schmidt, Variational derivatives of arbitrarily high order and multiinflation cosmological models, Class. Quant. Grav. 7, 1023 (1990)

    1990

  62. [62]

    A. L. Berkin and K. i. Maeda, Effects ofR 3 andR□Rterms onR 2 inflation, Phys. Lett. B245, 348 (1990)

    1990

  63. [63]

    Quandt and H

    I. Quandt and H. J. Schmidt, The Newtonian limit of fourth and higher order gravity, Astron. Nachr.312, 97 (1991) [gr-qc/0109005]

    Pith/arXiv arXiv 1991

  64. [64]

    S. A. Fulling, R. C. King, B. G. Wybourne, and C. J. Cummins, Normal forms for tensor polynomials. 1: The Riemann tensor, Class. Quant. Grav.9, 1151 (1992)

    1992

  65. [65]

    Amendola, A

    L. Amendola, A. Battaglia Mayer, S. Capozziello, F. Occhionero, S. Gottlober, V. Muller, and H. J. Schmidt, Generalized sixth order gravity and inflation, Class. Quant. Grav.10, L43 (1993)

    1993

  66. [66]

    Harvey, Identities of the scalars of the four-dimensional Riemannian manifold, J

    A. Harvey, Identities of the scalars of the four-dimensional Riemannian manifold, J. Math. Phys.36, 356 (1995)

    1995

  67. [67]

    Decanini and A

    Y. Decanini and A. Folacci, Irreducible forms for the metric variations of the action terms of sixth-order gravity and approximated stress-energy tensor, Class. Quant. Grav.24, 4777 (2007) [arXiv:0706.0691]

    Pith/arXiv arXiv 2007

  68. [68]

    Skugoreva, A

    M. Skugoreva, A. Toporensky, and P. Tretyakov, Cosmological dynamics in six-order gravity, Grav. Cosmol.17, 110 (2011) [arXiv:1007.3365]

    Pith/arXiv arXiv 2011

  69. [69]

    Iihoshi, Mutated hybrid inflation inf(R,□R)-gravity, J

    M. Iihoshi, Mutated hybrid inflation inf(R,□R)-gravity, J. Cosmol. Astropart. Phys.02, 022 (2011) [arXiv:1011.3927]

    Pith/arXiv arXiv 2011

  70. [70]

    Khodabakhshi, M

    H. Khodabakhshi, M. Farhang, F. Shojai, and H. L¨ u, Cosmology with higher-derivative gravities, Phys. Rev. D110, L061504 (2024) [arXiv:2405.02879]

    arXiv 2024

  71. [71]

    T. Q. Do, No-go theorem for inflation in Ricci-inverse gravity, Eur. Phys. J. C81, 431 (2021) [arXiv:2009.06306]

    Pith/arXiv arXiv 2021

  72. [72]

    T. Q. Do, No-go theorem for inflation in an extended Ricci-inverse gravity model, Eur. Phys. J. C82, 15 (2022) [arXiv:2101.08538]

    Pith/arXiv arXiv 2022

  73. [73]

    T. Q. Do, D. H. Nguyen, and T. M. Pham, Stability investigations of isotropic and anisotropic exponential inflation in the Starobinsky–Bel–Robinson gravity, Int. J. Mod. Phys. D32, 2350087 (2023) [arXiv:2303.17283]

    Pith/arXiv arXiv 2023

  74. [74]

    T. M. Pham, D. H. Nguyen, T. Q. Do and W. F. Kao, Stability investigations of de Sitter inflationary solutions in power-law extensions of the Starobinsky model, Eur. Phys. J. C84, 729 (2024) [arXiv:2403.02623]

    Pith/arXiv arXiv 2024

  75. [75]

    T. Q. Do, Stability analysis of de Sitter solution in the Einstein-Grisaru-Zanon gravity using the dynamical system method, Int. J. Geom. Meth. Mod. Phys. (2026) 2650168 [DOI: 10.1142/S0219887826501689] [arXiv:2606.15119]. 21

    arXiv 2026

  76. [76]

    T. Q. Do, Stability analysis of de Sitter solution in the Starobinsky-Grisaru-Zanon gravity using the dynamical system method, Int. J. Geom. Meth. Mod. Phys. (2026) 2650267 [DOI: 10.1142/S0219887826502671]

    work page doi:10.1142/s0219887826502671 2026

  77. [77]

    Bueno and P

    P. Bueno and P. A. Cano, Einsteinian cubic gravity, Phys. Rev. D94, 104005 (2016) [arXiv:1607.06463]

    Pith/arXiv arXiv 2016

  78. [78]

    R. A. Hennigar, D. Kubizˇ n´ ak, and R. B. Mann, Generalized quasitopological gravity, Phys. Rev. D95, 104042 (2017) [arXiv:1703.01631]

    Pith/arXiv arXiv 2017

  79. [79]

    Arciniega, J

    G. Arciniega, J. D. Edelstein, and L. G. Jaime, Towards geometric inflation: the cubic case, Phys. Lett. B802, 135272 (2020) [arXiv:1810.08166]

    Pith/arXiv arXiv 2020

  80. [80]

    Arciniega, P

    G. Arciniega, P. Bueno, P. A. Cano, J. D. Edelstein, R. A. Hennigar, and L. G. Jaime, Geometric Inflation, Phys. Lett. B 802, 135242 (2020) [arXiv:1812.11187]

    Pith/arXiv arXiv 2020

Showing first 80 references.