Pith. sign in

REVIEW 6 minor 67 references

Differential Obstructions to Curvature-Dependent Conformal Transformations

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read A curvature-dependent conformal rescaling $\tilde{g}_{\mu\nu}=F(R[g])g_{\mu\nu}$ is a local forward map but not a local change of metric variables: the inverse response carries the nonpolynomial factor $1/\xi^2$, so no finite-jet inverse…

desk verdict The finite-jet obstruction argument is sound, and the paper's parent-theory distinction is the right way to frame off-shell comparisons in metric f(R) gravity; it deserves a serious referee. read the letter →

arxiv 2608.13289 v1 pith:HNCF3QHJ submitted 2026-08-13 gr-qc

David S. Pereira , Francisco S.N. Lobo , José Pedro Mimoso This is my paper · ORCID
classification gr-qc PACS 04.50.Kd
keywords curvature-dependentconformaltransformationsmetricf(R)gravityEinsteinframefinite-jetlocalitynonlocalinversenormaloperatorscalaronoff-shellequivalence
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 asks whether a curvature-dependent conformal rescaling of the metric, $\tilde{g}_{\mu\nu}=F(R[g])g_{\mu\nu}$, counts as a genuine local change of metric variables, the way an ordinary scalar–tensor conformal transformation does. Its answer is no: given the transformed metric, reconstructing the original one is a differential fixed-point problem, because the unknown curvature $R[g]$ appears both as the quantity to be found and inside the operator that determines it. The paper proves that the complete inverse metric response contains a nonpolynomial factor $1/\xi^2$ in its momentum-space symbol, so no formula using only finitely many derivatives at a point can invert the map generically. The practical consequence is a sharp three-way distinction — forward map on known solutions, local transformation on an enlarged parent field space, or off-shell change of variables — and an explicit framework for carrying the differential projection into variational, semiclassical, and quantum calculations in metric $f(R)$ gravity.

What carries the argument

The load-bearing object is the principal symbol of the complete linearized inverse response, $\sigma_0(-N_F^{-1}B_F)=-\frac{F}{3F_X}(\xi^\mu\xi^\nu\gamma_{\mu\nu}/\xi^2-\gamma)$: the nonpolynomial factor $1/\xi^2$ and the momentum-longitudinal projector $\xi^\mu\xi^\nu/\xi^2$ prove that the inverse cannot be a finite-order differential operator, since such an operator would have a polynomial symbol. The same structure reappears in metric $f(R)$ gravity as the normal operator $L_\Phi$ of the projection constraint $\hat{P}_f=0$; its Green inverse $G_P$ constructs the section $s_\star[\tilde{g}]$, and the pulled-back Hessian $\delta^2(S_E\circ\iota)=\delta^2S_E|_{TM_f}+\langle E_s,\delta^2s_\star\rangle$ displays the off-shell embedding term. In the quadratic model $f(R)=R+\alpha R^2$, constrained Gaussian elimination turns this machinery into the explicit nonlocal kernel $R^{(1)}(\tilde{\Box}-m_s^2)^{-1}R^{(1)}$, a normal determinant, and a zero-mode compatibility condition.

What would settle it

Take the quadratic model $F(R)=1+2\alpha R$ on a flat background and write the linearized inverse response from Eq. (33): the pure-trace sector contains $(12f_{RR})^{-1}(\xi^2+m_E^2)^{-1}$, a nonpolynomial factor with a nonzero residue. If a finite-jet inverse existed, this response would have to be a polynomial in $\xi$; a direct attempt to build $g_{\mu\nu}$ from $\tilde{g}_{\mu\nu}$ and finitely many derivatives at the same point — or a check that the response fails polynomial scaling under $\xi\mapsto\lambda\xi$ — would settle the claim.

Watch

Extended reading notes

Core claim

For any smooth nondegenerate branch with $F>0$ and $F_R\neq 0$, the transformation $\tilde{g}_{\mu\nu}=F(R[g])g_{\mu\nu}$ is a well-defined forward assignment on known metrics, yet its inverse is a differential fixed-point problem: the preimage curvature must satisfy $R_J=T_F[R_J]$, with the d'Alembertian of $R_J$ appearing on the right-hand side. Linearizing about a projected background, the complete metric response has principal symbol containing the factor $1/\xi^2$ and the longitudinal projector $\xi^\mu\xi^\nu/\xi^2$; since a finite-order differential operator would have polynomial momentum dependence, no differentiable finite-jet metric-only inverse exists on an open set of unrestricted metric configurations. Branchwise functional inverses do exist once a functional domain, boundary or Cauchy data, and a Green prescription are specified. The paper's constructive counterpart is the parent construction: introducing an independent auxiliary scalar makes the forward map algebraic, and the original metric theory is recovered by a differential constraint whose normal operator is $L_\Phi=3\Phi(\tilde{\Box}-\tilde{\nabla}^\mu s\tilde{\nabla}_\mu)+X(\Phi)-\Phi X_\Phi(\Phi)$ in metric $f(R)$ gravity. The pulled-back metric-only Hessian then acquires an embedding correction $\langle E_s,\delta^2 s_\star\rangle$ that vanishes on the common classical shell, which is why exact parent and metric solutions remain equivalent while off-shell variational and fluctuation problems differ.

Load-bearing premise

The proof assumes that the conformal factor's carrier is eliminated through a genuinely second-order differential constraint whose principal symbol is $-3F_X\xi^2$; if the carrier were fixed algebraically instead — as in Palatini $f(R)$ — or if the branch had $F_R=0$, the $1/\xi^2$ obstruction would not follow, and the claim also presupposes that the inverse, if it existed, would be $C^1$ on an open set of metrics.

Editorial extensions

If this is right

  • In metric $f(R)$ gravity, the Einstein-frame scalar–tensor action is a local parent theory; a metric-only Einstein-side description exists only after solving the differential projection $\hat{P}_f=0$, so the scalaron is reconstructed through a Green operator plus homogeneous data rather than being a freely adjustable field.
  • Off shell, the metric-theory Hessian and the unrestricted parent Hessian differ by the embedding correction, so one-loop effective actions computed in the two settings have different fluctuation domains and quadratic kernels; the difference vanishes on the common classical shell.
  • The same scalaron denominator that obstructs local inversion governs observable response: the static Yukawa scalar charge of compact bodies, the quasistatic effective Newton coupling and slip, and the cosmological scalaron response all inherit the pole of the normal operator.
  • Pure Palatini $f(R)$ gravity escapes the obstruction because its scalar carrier is fixed algebraically by the trace equation before the connection is eliminated; hybrid and metric-affine theories require auditing with coupled normal-operator matrices rather than a single scalar block.
  • Reconstructing an $f(R)$ model from a prescribed Einstein-frame expansion history requires solving the nonlinear section equation with branch and Cauchy data, so Einstein-side reconstruction is more constrained than generic scalar–tensor reconstruction.

Reading between the lines

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

  • The paper's framework gives a structural explanation of the known off-shell mismatch between metric and scalar–tensor one-loop results: the mismatch is not a regularization artifact but a difference of fluctuation domains, and a full gauge-fixed projected one-loop computation — which the paper leaves to future work — would quantify it.
  • The derived nonlocal kernel $R^{(1)}(\tilde{\Box}-m_s^2)^{-1}R^{(1)}$ connects metric $f(R)$ gravity to nonlocal-gravity phenomenology in reverse: here the inverse d'Alembertian is derived from a local parent theory by constrained elimination, so the nonlocality carries a definite Green prescription (retarded for causal response, symmetric for variational kernels).
  • A practical test: in the Starobinsky model on a Euclidean four-sphere, the graph breaks at the spectral locus $m_E^2=-\ell(\ell+3)/a^2$; computing the projected one-loop determinant there would reveal whether the zero-mode collective-coordinate treatment changes physical predictions relative to the parent theory.
  • Numerical and cosmological codes that evolve the Einstein-frame parent without enforcing $\hat{P}_f=0$ are solving a different off-shell theory; enforcing the projection is equivalent to solving the differential inverse identified here, and the principal-symbol analysis predicts where iterative metric-reconstruction schemes lose convergence.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper studies conformal transformations of the form g~_{μν}=F(R[g])g_{μν} with F>0 and F_R≠0, asking whether they can be viewed as local changes of metric variables. The main claim is that although the forward map is a local assignment on known metrics, the inverse is not a finite-jet (finite-derivative) local functional on any open set of unrestricted metrics. The proof introduces an independent auxiliary scalar X to localize the forward map, linearizes the constraint δP=B[γ]+N_F χ, and computes the principal symbols σ2(N_F)=−3F_X ξ² and σ2(B_F)γ=F(−ξ^μξ^νγ_{μν}+ξ²γ). The resulting linearized inverse response contains the nonpolynomial factor 1/ξ² and the projector ξ^μξ^ν/ξ², which cannot be the symbol of a finite-order differential operator. The paper then applies this framework to metric f(R) gravity, showing that the Einstein-frame scalar–tensor representation is a local parent theory while a metric-only Einstein-side description requires a differential projection whose inverse is a Green operator with boundary or Cauchy data. Further sections derive the pulled-back Hessian with an off-shell embedding correction, the nonlocal scalaron kernel in R+αR², the normal determinant and zero-mode compatibility condition, the weak-field Yukawa response, and the cosmological projection equation. The authors are careful to state that exact parent and metric solutions remain equivalent; the obstruction concerns locality and off-shell variational/fluctuation domains.

Significance. The result is significant for the modified-gravity community because it cleanly separates three operations that are often conflated: using a curvature-dependent conformal rule as a forward map, as a local transformation on an enlarged parent field space, and as a genuine local change of variables on the off-shell metric configuration space. The central no-finite-jet theorem is supported by a self-contained principal-symbol computation that involves no parameter fitting and does not rely on controversial assumptions. The paper also provides concrete technical payoffs: the projected Hessian with its embedding correction (Eq. 96), the nonlocal quadratic form factor 1/(□~−m_s²) in Eq. (211), and a constrained Gaussian integral that simultaneously yields the nonlocal kernel and the normal determinant. The cross-checks with the standard Yukawa response and quasistatic f(R) observables reinforce the physical relevance of the obstruction. If correct, the paper clarifies why one-loop equivalence between the metric and scalar–tensor forms of f(R) gravity is a subtle question involving functional domains, measures, and boundary data, rather than a trivial field redefinition.

minor comments (6)
  1. [Section XII] In the Discussion and Conclusions, the phrase "configuration pace" should read "configuration space".
  2. [Section II B] The symbol "≃" is used in Eq. (12) to denote a linearized equality; defining this notation or replacing it with "=" at first use would improve clarity.
  3. [Section II C] The relation F − X F_X = 0 in Eq. (26) is called a "homogeneous algebraic diagnostic"; the authors might state more explicitly that this condition only diagnoses constant-mode degeneracy and does not by itself establish a global obstruction, a point that is later emphasized in Sec. IX.
  4. [Section IV B] In Eq. (87), the trace γ^ of the metric perturbation is used without prior definition; writing γ^ = g~^{μν}γ_{μν} in the surrounding text would avoid confusion.
  5. [Section X] The determinant in Eq. (213) is presented with an absolute value, and the phase convention for Lorentzian signature is mentioned only briefly; an additional sentence explaining the convention for the real delta functional would be helpful for readers.
  6. [References] Reference [23] is cited as a 2026 preprint; if a journal version exists, it would be helpful to update the citation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the finite-jet obstruction is derived from the paper's own linearized projection and symbol computation, with no fitted parameter renamed as a prediction.

full rationale

The central claim is self-contained and does not reduce to its inputs by construction. The paper defines the forward map (Eq. 3), localizes it with an auxiliary scalar (Eqs. 17-19), linearizes the projection condition (Eq. 28), computes the principal symbols (Eqs. 29-30), and obtains the nonpolynomial inverse response (Eqs. 31-33) directly from those definitions. No parameter in that chain is fitted from the result it is meant to predict; alpha is an input coupling, m_s^2 = 1/(6 alpha) follows from the trace equation, and the weak-field Yukawa and quasistatic expressions are used as independent cross-checks of the same denominator. The only coauthor citations are [8], used as background for conformal-frame equivalence discussions, and [64], used to define the hybrid metric-Palatini example; neither citation supplies a load-bearing step, and neither is invoked as an external uniqueness theorem or as a justification for the finite-jet obstruction. The paper also explicitly scopes out Palatini algebraic reductions and the characteristic set xi^2 = 0, acknowledging where a Green prescription is needed. I found no equation that is equivalent to its own input by definition or to a fitted parameter, so the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted: the alpha in R+alpha R^2 is an input model parameter, and m_s^2=1/(6 alpha) follows from the field equations. The derivation imports standard conformal identities, the f(R) scalar-tensor equivalence, and standard microlocal facts about symbols and Green operators; none of these are authored by the present authors in a way that forces the result. No new physical entities are introduced; terms such as 'normal operator', 'graph section', and 'embedding correction' are descriptive terminology for derived quantities.

assumptions (4)
  • standard math Conformal transformation of the Ricci scalar under g-tilde=F(R[g])g: R = F R-tilde + 3 F_R box-tilde R + 3(F_RR - 3/2 F_R^2/F)(grad-tilde R)^2 (Eq. (6)).
    Standard differential-geometric identity; it is the fixed-point equation (5) on which the entire inverse-response computation rests.
  • domain assumption On a regular Legendre branch (f_XX != 0, Phi = f_X > 0), metric f(R) gravity is locally equivalent to the scalar-tensor action (47) on shell (Sec. III A).
    Defines the parent-field-space construction used to separate local parent transformations from the differential projection to the metric sector.
  • standard math A C^1 finite-jet functional of a metric has a linearization that is a finite-order differential operator with polynomial principal symbol (Sec. II C).
    The no-go proof compares this polynomial symbol with the nonpolynomial 1/xi^2 response to derive the contradiction.
  • domain assumption For the chosen function spaces and boundary or Cauchy data, the normal operator L_Phi admits a right inverse or generalized inverse (Secs. IV and V).
    The inverse section s_star[g-tilde], the pulled-back Hessian, and the nonlocal kernel are all defined through this Green prescription; kernel and cokernel obstructions are treated separately in Sec. IX.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Differential Obstructions to Curvature-Dependent Conformal Transformations." pith.science (2026). https://pith.science/paper/HNCF3QHJ

@misc{pith2026260813289,
  author       = {Pith},
  title        = {Pith review of: Differential Obstructions to Curvature-Dependent Conformal Transformations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HNCF3QHJ}},
  note         = {Machine review of arXiv:2608.13289}
}
abstract

Curvature-dependent conformal rules are local forward assignments on known metrics, but they are not generically local changes of metric variables. We study the nondegenerate class $\widetilde g_{\mu\nu}=F(R[g])g_{\mu\nu}$, with $F>0$ and $F_R\neq0$. Introducing an independent auxiliary scalar localizes the forward map, while recovering the original metric requires a differential constraint. The complete inverse metric tangent map contains the nonpolynomial projector $\xi^\mu\xi^\nu/\xi^2$; hence no differentiable finite-jet inverse, i.e., a formula involving only finitely many derivatives at the same point, exists on an open set of unrestricted metrics. Branchwise functional inverses may nevertheless exist after boundary or Cauchy data are specified. Metric $f(R)$ gravity gives an explicit realization: its local Einstein-frame scalar--tensor representation is a parent theory, whereas a metric-only Einstein-side description requires a differential section governed by a normal operator. We derive the pulled-back classical Hessian and its off-shell embedding term, and in the quadratic model show how constrained Gaussian elimination produces the scalaron nonlocal kernel, the corresponding normal determinant for the displayed measure, and the zero-mode compatibility condition. Exact parent and metric solutions remain equivalent; the obstruction concerns locality and the off-shell variational and fluctuation domains. These results provide a precise framework for assessing curvature-dependent frame transformations in modified gravity and clarify their implications for effective actions, semiclassical analyses, and quantum frame equivalence.

Figures

Figures reproduced from arXiv: 2608.13289 by the authors.

Figure 1
Figure 1. FIG. 1. Projection to the geometric metric [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

67 extracted references · 19 canonical work pages

  1. [1]

    J. S. R. Chisholm, Nucl. Phys.26, 469 (1961)

    work page 1961

  2. [2]

    Kamefuchi, L

    S. Kamefuchi, L. O’Raifeartaigh, and A. Salam, Nucl. Phys.28, 529 (1961)

    1961

  3. [3]

    R. H. Dicke, Phys. Rev.125, 2163 (1962)

    1962

  4. [4]

    Faraoni, E

    V. Faraoni, E. Gunzig, and P. Nardone, Fund. Cosmic Phys.20, 121 (1999), arXiv:gr-qc/9811047

    arXiv 1999

  5. [5]

    Fujii and K.-i

    Y. Fujii and K.-i. Maeda,The Scalar-Tensor Theory of Gravitation(Cambridge University Press, Cambridge, 2003)

    work page 2003

  6. [6]

    E. E. Flanagan, Class. Quant. Grav.21, 3817 (2004), arXiv:gr-qc/0403063

    arXiv 2004

  7. [7]

    Faraoni and S

    V. Faraoni and S. Nadeau, Phys. Rev. D75, 023501 (2007), arXiv:gr-qc/0612075

    arXiv 2007

  8. [8]

    The cosmological frame principle and cosmic acceleration

    S. Cotsakis, J. P. Mimoso, and J. Miritzis, Eur. Phys. J. C83, 735 (2023), arXiv:2304.12733 [gr-qc]

    work page Pith review arXiv 2023

Show all 67 references
  1. [9]

    Babichev, K

    E. Babichev, K. Izumi, N. Tanahashi, and M. Yamaguchi, Adv. Theor. Math. Phys.25, 309 (2021), arXiv:1907.12333 [hep-th]

    2021 arXiv

  2. [10]

    Babichev, K

    E. Babichev, K. Izumi, N. Tanahashi, and M. Yamaguchi, PTEP2022, 013A01 (2022), arXiv:2109.00912 [hep-th]

    2022 arXiv

  3. [11]

    Whitt, Phys

    B. Whitt, Phys. Lett. B145, 176 (1984)

    1984

  4. [12]

    Maeda, Phys

    K.-i. Maeda, Phys. Rev. D39, 3159 (1989)

    1989

  5. [13]

    Teyssandier and P

    P. Teyssandier and P. Tourrenc, J. Math. Phys.24, 2793 (1983)

    1983

  6. [14]

    Wands, Class

    D. Wands, Class. Quant. Grav.11, 269 (1994), arXiv:gr- qc/9307034

    1994

  7. [15]

    Magnano and L

    G. Magnano and L. M. Sokolowski, Phys. Rev. D50, 5039 (1994), arXiv:gr-qc/9312008

    1994 arXiv

  8. [16]

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

    2010 arXiv

  9. [17]

    De Felice and S

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

    2010 arXiv

  10. [18]

    A. Y. Kamenshchik and C. F. Steinwachs, Phys. Rev. D 91, 084033 (2015), arXiv:1408.5769 [gr-qc]

    2015 arXiv

  11. [19]

    M. S. Ruf and C. F. Steinwachs, Phys. Rev. D97, 044049 (2018), arXiv:1711.04785 [gr-qc]

    2018 arXiv

  12. [20]

    M. S. Ruf and C. F. Steinwachs, Phys. Rev. D97, 044050 (2018), arXiv:1711.07486 [gr-qc]

    2018 arXiv

  13. [21]

    Ohta, PTEP2018, 033B02 (2018), arXiv:1712.05175 [hep-th]

    N. Ohta, PTEP2018, 033B02 (2018), arXiv:1712.05175 [hep-th]

    2018 arXiv

  14. [22]

    Falls and M

    K. Falls and M. Herrero-Valea, Eur. Phys. J. C79, 595 (2019), arXiv:1812.08187 [hep-th]

    2019 arXiv

  15. [23]

    Kuntz and S

    I. Kuntz and S. Liberati, (2026), arXiv:2607.12644 [hep- th]

    2026 arXiv

  16. [24]

    T. S. Koivisto, Phys. Rev.D78, 123505 (2008), arXiv:0807.3778 [gr-qc]

    2008 arXiv

  17. [25]

    Deser and R

    S. Deser and R. P. Woodard, JCAP1311, 036, arXiv:1307.6639 [astro-ph.CO]

    arXiv

  18. [26]

    de la Cruz-Dombriz, A

    A. de la Cruz-Dombriz, A. Dobado, and A. L. Maroto, Phys. Rev.D77, 123515 (2008), arXiv:0802.2999 [astro- ph]

    2008 arXiv

  19. [27]

    Deffayet and R

    C. Deffayet and R. P. Woodard, JCAP0908, 023, arXiv:0904.0961 [gr-qc]

    arXiv

  20. [28]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, M. Sasaki, and Y.-l. Zhang, Phys. Lett.B696, 278 (2011), arXiv:1010.5375 [gr-qc]

    2011 arXiv

  21. [29]

    I. L. Shapiro, Class. Quant. Grav.25, 103001 (2008), arXiv:0801.0216 [gr-qc]

    2008 arXiv

  22. [30]

    Deser and R

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

    2007 arXiv

  23. [31]

    Koivisto and D

    T. Koivisto and D. F. Mota, JCAP0808, 021, arXiv:0805.4229 [astro-ph]

    arXiv

  24. [32]

    Bamba, S

    K. Bamba, S. Nojiri, and S. D. Odintsov, JCAP10, 045, arXiv:0807.2575 [hep-th]

    arXiv

  25. [33]

    Barnaby and N

    N. Barnaby and N. Kamran, JHEP02, 008, arXiv:0709.3968 [hep-th]

    arXiv

  26. [34]

    S. B. Giddings, Mod. Phys. Lett.A22, 2949 (2007), arXiv:0705.2197 [hep-th]

    2007 arXiv

  27. [35]

    Wetterich, Gen

    C. Wetterich, Gen. Rel. Grav.30, 159 (1998), arXiv:gr- qc/9704052 [gr-qc]

    1998

  28. [36]

    A. D. Dolgov and M. Kawasaki, Phys. Lett. B573, 1 (2003), arXiv:astro-ph/0307285

    2003 arXiv

  29. [37]

    Bär, Commun

    C. Bär, Commun. Math. Phys.333, 1585 (2015), arXiv:1310.0738 [math-ph]

    2015 arXiv

  30. [38]

    Salgado, Class

    M. Salgado, Class. Quant. Grav.23, 4719 (2006), arXiv:gr- qc/0509001

    2006

  31. [39]

    Bandara, M

    L. Bandara, M. Goffeng, and H. Saratchandran, Adv. Math.420, 108968 (2023), arXiv:2104.01919 [math.DG]

    2023 arXiv

  32. [40]

    Dyer and K

    E. Dyer and K. Hinterbichler, Phys. Rev. D79, 024028 (2009), arXiv:0809.4033 [gr-qc]

    2009 arXiv

  33. [41]

    Guarnizo, L

    A. Guarnizo, L. Castaneda, and J. M. Tejeiro, Gen. Rel. Grav.42, 2713 (2010), arXiv:1002.0617 [gr-qc]

    2010 arXiv

  34. [42]

    G. A. Vilkovisky, Nucl. Phys. B234, 125 (1984)

    1984

  35. [43]

    B. S. DeWitt,The Global Approach to Quantum Field Theory, International Series of Monographs on Physics, Vol. 114 (Oxford University Press, Oxford, 2003)

    2003

  36. [44]

    Capozziello, A

    S. Capozziello, A. Stabile, and A. Troisi, Phys. Rev. D 76, 104019 (2007), arXiv:0708.0723 [gr-qc]

    2007 arXiv

  37. [45]

    Capozziello, A

    S. Capozziello, A. Stabile, and A. Troisi, Mod. Phys. Lett. A24, 659 (2009), arXiv:0901.0448 [gr-qc]

    2009 arXiv

  38. [46]

    Stabile, Phys

    A. Stabile, Phys. Rev. D82, 064021 (2010), arXiv:1004.1973 [gr-qc]

    2010 arXiv

  39. [47]

    C. P. L. Berry and J. R. Gair, Phys. Rev. D83, 104022 (2011), [Erratum: Phys.Rev.D 85, 089906 (2012)], arXiv:1104.0819 [gr-qc]

    2011 arXiv

  40. [48]

    A. A. Starobinsky, Phys. Lett. B91, 99 (1980)

    1980

  41. [49]

    Khoury and A

    J. Khoury and A. Weltman, Phys. Rev. Lett.93, 171104 (2004), arXiv:astro-ph/0309300

    2004 arXiv

  42. [50]

    Khoury and A

    J. Khoury and A. Weltman, Phys. Rev. D69, 044026 (2004), arXiv:astro-ph/0309411

    2004 arXiv

  43. [51]

    Hu and I

    W. Hu and I. Sawicki, Phys. Rev. D76, 064004 (2007), arXiv:0705.1158 [astro-ph]

    2007 arXiv

  44. [52]

    Faraoni, Phys

    V. Faraoni, Phys. Rev. D74, 104017 (2006), arXiv:astro- 25 ph/0610734

    2006

  45. [53]

    Y.-S. Song, W. Hu, and I. Sawicki, Phys. Rev. D75, 044004 (2007), arXiv:astro-ph/0610532

    2007 arXiv

  46. [54]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Phys. Rev. D74, 086005 (2006), arXiv:hep-th/0608008

    2006 arXiv

  47. [55]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, and D. Saez-Gomez, Phys. Lett. B681, 74 (2009), arXiv:0908.1269 [hep-th]

    2009 arXiv

  48. [56]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Phys. Rept.505, 59 (2011), arXiv:1011.0544 [gr-qc]

    2011 arXiv

  49. [57]

    Faraoni, Phys

    V. Faraoni, Phys. Rev. D75, 067302 (2007), arXiv:gr- qc/0703044

    2007

  50. [58]

    A. O. Barvinsky, Mod. Phys. Lett. A30, 1540003 (2015), arXiv:1408.6112 [hep-th]

    2015 arXiv

  51. [59]

    J. Z. Simon, Phys. Rev. D41, 3720 (1990)

    1990

  52. [60]

    C. P. Burgess, Living Rev. Rel.7, 5 (2004), arXiv:gr- qc/0311082

    2004

  53. [61]

    A. R. Solomon and M. Trodden, JCAP02, 031, arXiv:1709.09695 [hep-th]

    arXiv

  54. [62]

    G. J. Olmo, Int. J. Mod. Phys. D20, 413 (2011), arXiv:1101.3864 [gr-qc]

    2011 arXiv

  55. [63]

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

    1995 arXiv

  56. [64]

    Harko, T

    T. Harko, T. S. Koivisto, F. S. N. Lobo, and G. J. Olmo, Phys. Rev. D85, 084016 (2012), arXiv:1110.1049 [gr-qc]

    2012 arXiv

  57. [65]

    K. S. Stelle, Phys. Rev. D16, 953 (1977)

    1977

  58. [66]

    Hindawi, B

    A. Hindawi, B. A. Ovrut, and D. Waldram, Phys. Rev. D53, 5583 (1996), arXiv:hep-th/9509142

    1996 arXiv

  59. [67]

    D. C. Rodrigues, F. de O. Salles, I. L. Shapiro, and A. A. Starobinsky, Phys. Rev. D83, 084028 (2011), arXiv:1101.5028 [gr-qc]

    2011 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.