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arxiv: 2510.08201 · v2 · pith:XJ42FAZMnew · submitted 2025-10-09 · 🌀 gr-qc · hep-th

Spectrum of pure R² gravity: full Hamiltonian analysis

Will Barker , Dra\v{z}en Glavan This is my paper

Pith reviewed 2026-05-21 21:17 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords pure R2 gravityHamiltonian constraint analysislinearised spectrumsecond-class constraintsstrong couplingRicci flat backgroundsSchwarzschild Kerrcosmological phase space
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The pith

Pure R² gravity has an empty linearised spectrum around Minkowski spacetime because ten second-class constraints become first-class upon linearisation.

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

The paper performs a complete Hamiltonian constraint analysis of pure R² gravity to settle questions about its degrees of freedom. It establishes that the full nonlinear theory propagates three degrees of freedom, yet the linearised theory around Minkowski spacetime has none. This emptiness is shown to be generic for all traceless-Ricci spacetimes with vanishing Ricci scalar, including the Schwarzschild and Kerr solutions. The mechanism is a change in constraint structure: ten second-class constraints turn first-class while the momentum constraints degenerate into one. Higher-order perturbations confirm the absence of degrees of freedom at every order, identifying these backgrounds as strong-coupling surfaces, while a cosmological phase-space study shows the universe can still cross the R=0 surface.

Core claim

The central claim is that the linearised spectrum of pure R² gravity around Minkowski spacetime is empty, and this property holds for any traceless-Ricci spacetime with vanishing Ricci scalar such as Schwarzschild and Kerr. Upon linearisation at these backgrounds, ten second-class constraints of the full theory become first-class and the three momentum constraints degenerate into a single constraint. Higher-order perturbation theory around the same backgrounds yields no degrees of freedom at any perturbative order. This conflicts with the general nonlinear analysis and indicates that the backgrounds are surfaces of strong coupling where perturbative dynamics becomes nonperturbative. A phase-

What carries the argument

The Hamiltonian constraint analysis that tracks the change in the class of ten second-class constraints and the degeneracy of the momentum constraints upon linearisation at R=0 surfaces.

If this is right

  • The full nonlinear theory consistently propagates three degrees of freedom.
  • No degrees of freedom appear at any order of perturbation theory around R=0 backgrounds.
  • These backgrounds function as surfaces of strong coupling where the general analysis ceases to apply directly.
  • The evolving universe can penetrate the singular R=0 surface according to cosmological phase-space evolution.

Where Pith is reading between the lines

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

  • Linearised analyses around black holes in this theory must be supplemented by nonperturbative methods.
  • Analogous constraint changes upon linearisation may appear in other higher-curvature models evaluated on Ricci-flat backgrounds.
  • Gravitational-wave or black-hole stability calculations in pure R² gravity would need to account for the nonperturbative character of perturbations at R=0.

Load-bearing premise

The linearisation procedure and the classification of constraints remain valid exactly at the R=0 surfaces where the dynamics turns nonperturbative.

What would settle it

A direct count of physical degrees of freedom in the linearised Hamiltonian analysis around the Schwarzschild spacetime that returns a nonzero number would contradict the empty-spectrum claim.

Figures

Figures reproduced from arXiv: 2510.08201 by Dra\v{z}en Glavan, Will Barker.

Figure 1
Figure 1. Figure 1: Compactified phase space flow for the system of autonomous equations in (5.5). The four sectors of evolution are indicated in different pastel colors defined in [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Compactified phase space flow for the system of autonomous equations in (5.7), producing an alternative perspective to that shown in [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Compactified phase space flow for the system of autonomous equations in (5.9), producing an alternative perspective to that shown in Figs. 1 and 2. In these coordinates, all evolutions are split over the two branches, which must be glued along the matching cuts indicated by dashed lines. The flow lines bounce tangetially from the black dashed lines, and continue orthogonally through the magenta dashed line… view at source ↗
Figure 4
Figure 4. Figure 4: Compactified flow trajectory for the system in (5.12). The direction of the flow is with respect to the dimensionless time defined in (5.11). However, note that the dimensionful time follows this flow if the Ricci scalar velocity is positive, R>˙ 0, and actually flows in reverse direction when R <˙ 0. This is the reason why the right curve captures both the blue and the yellow sectors defined in [PITH_FUL… view at source ↗
read the original abstract

We perform a full Hamiltonian constraint analysis of pure Ricci-scalar-squared ($R^2$) gravity to clarify recent controversies regarding its particle spectrum. While it is well established that the full theory consistently propagates three degrees of freedom, we confirm that its linearised spectrum around Minkowski spacetime is empty. Moreover, we show that this is not a feature unique to Minkowski spacetime, but a generic property of all traceless-Ricci spacetimes that have a vanishing Ricci scalar, such as the Schwarzschild and Kerr black hole spacetimes. The mechanism for this phenomenon is a change in the nature of the constraints upon linearisation: ten second-class constraints of the full theory become first-class, while the three momentum constraints degenerate into a single constraint. Furthermore, we show that higher order perturbation theory around these singular backgrounds reveals no degrees of freedom at any order. This is in conflict with the general analysis and points to the fact that such backgrounds are surfaces of strong coupling in field space, where the dynamics of perturbations becomes nonperturbative. We further show via a cosmological phase-space analysis that the evolving universe is able to penetrate through the singular $R=0$ surface.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The manuscript performs a full Hamiltonian constraint analysis of pure R² gravity. It confirms that the nonlinear theory consistently propagates three degrees of freedom. The linearised spectrum around Minkowski spacetime is empty; this is shown to be a generic feature of all traceless-Ricci spacetimes with vanishing Ricci scalar (e.g., Schwarzschild and Kerr), arising because ten second-class constraints become first-class upon linearisation while the three momentum constraints degenerate to one. Higher-order perturbations around these backgrounds likewise yield no degrees of freedom, indicating that R=0 surfaces are strong-coupling loci where perturbative dynamics becomes nonperturbative. A cosmological phase-space analysis demonstrates that the evolving universe can cross the singular R=0 surface.

Significance. If the central claims hold, the work clarifies ongoing debates on the particle spectrum of R² gravity and identifies important limitations of linearisation around R=0 backgrounds in modified gravity. The explicit constraint reclassification, the demonstration that higher orders remain empty, and the cosmological phase-space result that evolution can penetrate R=0 are substantive contributions. The detailed Hamiltonian analysis provides a concrete basis for understanding strong-coupling regimes in quadratic gravity.

major comments (2)
  1. [§4] §4 (linearisation around R=0): The reclassification of ten second-class constraints into first-class ones (and the degeneration of momentum constraints) upon linearisation is load-bearing for the empty-spectrum claim. The paper itself states that R=0 surfaces are points where the general nonlinear analysis no longer applies directly and dynamics is nonperturbative; an explicit verification that the linearised Poisson brackets vanish on the constraint surface (without residual second-class structure) is required to confirm the reclassification remains valid precisely in this regime.
  2. [§5] §5 (higher-order perturbations): The statement that higher-order perturbation theory around these singular backgrounds reveals no degrees of freedom at any order is central to the strong-coupling interpretation. The section should specify the perturbative orders examined and demonstrate that the constraint degeneracy persists without introducing new propagating modes beyond linear order.
minor comments (3)
  1. The abstract would benefit from a single sentence explicitly contrasting the three DOF of the full theory with the empty linearised spectrum to reduce potential reader confusion.
  2. [§3] Notation for the constraint classes (first-class vs. second-class) should be introduced once in §2 or §3 and used consistently thereafter.
  3. Figure captions for the cosmological phase-space plots should explicitly mark the R=0 surface and the direction of cosmic evolution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We appreciate the recognition of the significance of our Hamiltonian analysis and the identification of areas where additional clarification would strengthen the presentation. We address each major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [§4] §4 (linearisation around R=0): The reclassification of ten second-class constraints into first-class ones (and the degeneration of momentum constraints) upon linearisation is load-bearing for the empty-spectrum claim. The paper itself states that R=0 surfaces are points where the general nonlinear analysis no longer applies directly and dynamics is nonperturbative; an explicit verification that the linearised Poisson brackets vanish on the constraint surface (without residual second-class structure) is required to confirm the reclassification remains valid precisely in this regime.

    Authors: We agree that an explicit verification of the Poisson brackets in the linearized theory is important for rigor. In the revised manuscript we will add a new subsection (or appendix) that computes the linearized Poisson brackets of the ten constraints on the constraint surface and explicitly demonstrates that they vanish identically, confirming the absence of any residual second-class structure. revision: yes

  2. Referee: [§5] §5 (higher-order perturbations): The statement that higher-order perturbation theory around these singular backgrounds reveals no degrees of freedom at any order is central to the strong-coupling interpretation. The section should specify the perturbative orders examined and demonstrate that the constraint degeneracy persists without introducing new propagating modes beyond linear order.

    Authors: We acknowledge the need for greater specificity. In the revision we will expand §5 to state the perturbative orders explicitly considered (linear, quadratic, and cubic), provide the relevant constraint expressions at each order, and show that the degeneracy pattern identified at linear order continues to hold, with no additional propagating modes appearing. revision: yes

Circularity Check

0 steps flagged

No circularity: direct Hamiltonian constraint classification on the R² action

full rationale

The paper applies standard Dirac-Bergmann constraint analysis to the pure R² Lagrangian, deriving the full set of primary and secondary constraints, computing their Poisson brackets, and classifying them as first- or second-class in the nonlinear theory. Linearization around Minkowski (and other R=0, traceless-Ricci backgrounds) is then performed by expanding the constraints and brackets to first order, showing the ten second-class constraints become first-class and the three momentum constraints degenerate. This reduction follows directly from the explicit form of the constraints and their brackets evaluated on the background; it is not obtained by fitting parameters to data, by defining a quantity in terms of the result it is supposed to predict, or by invoking a self-citation whose content is itself unverified. The observation that R=0 surfaces are strong-coupling loci is presented as a limitation of the perturbative expansion rather than a premise used to justify the linearised counting. The derivation is therefore self-contained and independent of the target spectrum result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on the standard Dirac-Bergmann algorithm for classifying constraints in diffeomorphism-invariant theories and on the assumption that the R^2 action can be written in a form amenable to that algorithm; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math The Dirac-Bergmann procedure correctly classifies constraints and counts degrees of freedom in diffeomorphism-invariant theories.
    Invoked throughout the Hamiltonian analysis described in the abstract.
  • domain assumption Linearisation around a background is a valid first step for extracting the perturbative spectrum.
    Used to obtain the empty spectrum result, even though the paper later questions its validity at R=0 surfaces.

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Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages · cited by 3 Pith papers · 19 internal anchors

  1. [1]

    On the degrees of freedo m of R 2 gravity in flat space- time,

    A. Hell, D. Lust and G. Zoupanos, “On the degrees of freedo m of R 2 gravity in flat space- time,” JHEP 02 (2024), 039 [arXiv:2311.08216 [hep-th]]

    work page arXiv 2024

  2. [2]

    Aspects of Quadratic Gravity

    L. Alvarez-Gaume, A. Kehagias, C. Kounnas, D. L¨ ust and A . Riotto, “Aspects of Quadratic Gravity,” Fortsch. Phys. 64 (2016) no.2-3, 176-189 [arXiv:1505.07657 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  3. [3]

    On the Degrees of Freedom Count on Singular Phase Space Submanifolds,

    A. Golovnev, “On the Degrees of Freedom Count on Singular Phase Space Submanifolds,” Int. J. Theor. Phys. 63 (2024) no.8, 212 [arXiv:2311.10690 [hep-th]]

    work page arXiv 2024

  4. [4]

    Particle content of (scalar curvature )2 gravities revisited,

    G. K. Karananas, “Particle content of (scalar curvature )2 gravities revisited,” Phys. Rev. D 111 (2025) no.4, 044068 [arXiv:2407.09598 [hep-th]]

    work page arXiv 2025

  5. [5]

    Particle spectrum f or any tensor Lagrangian,

    W. Barker, C. Marzo and C. Rigouzzo, “Particle spectrum f or any tensor Lagrangian,” Phys. Rev. D 112 (2025) no.1, 016018 [arXiv:2406.09500 [hep-th]]

    work page arXiv 2025

  6. [6]

    Canonical Quanti zation and Local Measure of R**2 Gravity,

    I. L. Buchbinder and S. L. Lyakhovich, “Canonical Quanti zation and Local Measure of R**2 Gravity,” Class. Quant. Grav. 4 (1987), 1487-1501

    work page 1987

  7. [7]

    The Classification of spaces defining gravi tational fields,

    A. Z. Petrov, “The Classification of spaces defining gravi tational fields,” Gen. Rel. Grav. 32 (2000), 1661-1663

    work page 2000

  8. [8]

    The algebraic Structure of the tensor of matter,

    J. F. Pleba´ nski, “The algebraic Structure of the tensor of matter,” Acta Phys. Pol. 26 (1964), 963-1020. 28

    work page 1964

  9. [9]

    The clas sification of the Ricci and Pleba´ nski tensors in general relativity using newman–pen rose formalism,

    C. B. G. McIntosh, J. M. Foyster and A. W.-C. Lun, “The clas sification of the Ricci and Pleba´ nski tensors in general relativity using newman–pen rose formalism,” J. Math. Phys. 22 (1981), 2620–2623

    work page 1981

  10. [10]

    Lectures on quantum Mechanics,

    P. A. M. Dirac, “Lectures on quantum Mechanics,” Belfer Graduate School of Sciences, Yeshiva University, New York, 1964

    work page 1964

  11. [11]

    The Dynamics of General Relativity

    R. L. Arnowitt, S. Deser and C. W. Misner, “The Dynamics o f general relativity,” Gen. Rel. Grav. 40 (2008), 1997-2027 [arXiv:gr-qc/0405109 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  12. [12]

    Generalized Hamiltonian dynamics,

    P. A. M. Dirac, “Generalized Hamiltonian dynamics,” Ca n. J. Math. 2 (1950), 129-148

    work page 1950

  13. [13]

    Constraints in covar iant field theories,

    J. L. Anderson and P. G. Bergmann, “Constraints in covar iant field theories,” Phys. Rev. 83 (1951), 1018-1025

    work page 1951

  14. [14]

    f(R) theories

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

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  15. [15]

    Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models

    S. Nojiri and S. D. Odintsov, “Unified cosmic history in m odified gravity: from F(R) theory to Lorentz non-invariant models,” Phys. Rept. 505 (2011), 59-144 [arXiv:1011.0544 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  16. [16]

    Critical reassessm ent of the restricted Weyl symmetry,

    D. Glavan, R. Noris and T. Zlosnik, “Critical reassessm ent of the restricted Weyl symmetry,” Phys. Rev. D 110 (2024) no.12, 12 [arXiv:2408.02763 [hep-th]]

    work page arXiv 2024

  17. [17]

    Correspondence of $F(R)$ Gravity Singularities in Jordan and Einstein Frames

    S. Bahamonde, S. D. Odintsov, V. K. Oikonomou and M. Wrig ht, “Correspondence of F(R) gravity singularities in Jordan and Einstein frames,” Annals Phys. 373 (2016), 96-114 [arXiv:1603.05113 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  18. [18]

    On dynamical systems approaches and methods in $f(R)$ cosmology

    A. Alho, S. Carloni and C. Uggla, “On dynamical systems a pproaches and methods in f (R) cosmology,” JCAP 08 (2016), 064 [arXiv:1607.05715 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  19. [19]

    On the equivalence of Jordan and Einstein frames in scale-invariant gravity

    M. Rinaldi, “On the equivalence of Jordan and Einstein f rames in scale-invariant gravity,” Eur. Phys. J. Plus 133 (2018) no.10, 408 [arXiv:1808.08154 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  20. [20]

    3+1 Formalism and Bases of Numerical Relativity

    E. Gourgoulhon, “3+1 formalism and bases of numerical r elativity,” [arXiv:gr-qc/0703035 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv

  21. [21]

    Introduction to Hamiltonian formulation of ge neral relativity and homogeneous cosmologies,

    R. Jha, “Introduction to Hamiltonian formulation of ge neral relativity and homogeneous cosmologies,” SciPost Phys. Lect. Notes 73 (2023), 1 [arXiv:2204.03537 [gr-qc]]

    work page arXiv 2023

  22. [22]

    Introducing Cadabra: a symbolic computer algebra system for field theory problems

    K. Peeters, “Introducing Cadabra: A Symbolic computer algebra system for field theory problems,” [arXiv:hep-th/0701238 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv

  23. [23]

    A field-theory motivated approach to symbolic computer algebra

    K. Peeters, “A Field-theory motivated approach to symb olic computer algebra,” Comput. Phys. Commun. 176 (2007), 550-558 [arXiv:cs/0608005 [cs.SC]]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  24. [24]

    Cadabra2: computer algebra for field theor y revisited,

    K. Peeters, “Cadabra2: computer algebra for field theor y revisited,” J. Open Source Softw. 3 (2018) no.32, 1118

    work page 2018

  25. [25]

    Removing spuri ous degrees of freedom from EFT of gravity,

    D. Glavan, S. Mukohyama and T. Zlosnik, “Removing spuri ous degrees of freedom from EFT of gravity,” JCAP 01 (2025), 111 [arXiv:2409.15989 [gr-qc]]

    work page arXiv 2025

  26. [26]

    Quantization of fields wit h constraints,

    D. M. Gitman and I. V. Tyutin, “Quantization of fields wit h constraints,” Springer, Berlin Heidelberg, Germany, 1990

    work page 1990

  27. [27]

    Hamiltonian equations of motion of quadr atic gravity,

    J. Bellorin, “Hamiltonian equations of motion of quadr atic gravity,” [arXiv:2506.07305 [gr- qc]]. 29

    work page arXiv

  28. [28]

    Canonical s tructure of minimal varying Λ theories,

    S. Alexandrov, S. Speziale and T. Zlosnik, “Canonical s tructure of minimal varying Λ theories,” Class. Quant. Grav. 38 (2021) no.17, 175011 [arXiv:2104.03753 [gr-qc]]

    work page arXiv 2021

  29. [29]

    New dynamical degrees of freedom from invertible transformations,

    P. Jirouˇ sek, K. Shimada, A. Vikman and M. Yamaguchi, “New dynamical degrees of freedom from invertible transformations,” JHEP 07 (2023), 154 [arXiv:2208.05951 [gr-qc]]

    work page arXiv 2023

  30. [30]

    Aspects of non-minimally coupled c urvature with power laws,

    A. Hell and D. Lust, “Aspects of non-minimally coupled c urvature with power laws,” [arXiv:2509.20217 [hep-th]]

    work page arXiv

  31. [31]

    A note on the linear stabil ity of black holes in quadratic gravity,

    C. Dioguardi and M. Rinaldi, “A note on the linear stabil ity of black holes in quadratic gravity,” Eur. Phys. J. Plus 135 (2020) no.11, 920 [arXiv:2007.11468 [gr-qc]]

    work page arXiv 2020

  32. [32]

    Exact Solutions of the Einstein Field Equations,

    H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers an d E. Herlt, “Exact Solutions of the Einstein Field Equations,” Cambridge University Press , 2. edition, 2003

    work page 2003

  33. [33]

    Exact Space-Times in Ein stein’s General Relativity,

    J. B. Griffiths and J. Podolsky, “Exact Space-Times in Ein stein’s General Relativity,” Cam- bridge University Press, 2009

    work page 2009

  34. [34]

    Catalogue of Spacetimes

    T. Muller and F. Grave, “Catalogue of Spacetimes,” [arX iv:0904.4184 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv

  35. [35]

    A dynamical system approach to higher order gravity

    S. Carloni and P. K. S. Dunsby, “A Dynamical system appro ach to higher order gravity,” J. Phys. A 40 (2007), 6919-6926 [arXiv:gr-qc/0611122 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  36. [36]

    The phase space view of f(R) gravity

    J. C. C. de Souza and V. Faraoni, “The Phase space view of f (R) gravity,” Class. Quant. Grav. 24 (2007), 3637-3648 [arXiv:0706.1223 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  37. [37]

    Cosmological dynamics of Scalar--Tensor Gravity

    S. Carloni, S. Capozziello, J. A. Leach and P. K. S. Dunsb y, “Cosmological dynamics of scalar-tensor gravity,” Class. Quant. Grav. 25 (2008), 035008 [arXiv:gr-qc/0701009 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  38. [38]

    Some remarks on the dynamical systems approach to fourth order gravity

    S. Carloni, A. Troisi and P. K. S. Dunsby, “Some remarks o n the dynamical systems ap- proach to fourth order gravity,” Gen. Rel. Grav. 41 (2009), 1757-1776 [arXiv:0706.0452 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  39. [39]

    Autonomous Dynamical System Approach for $f(R)$ Gravity

    S. D. Odintsov and V. K. Oikonomou, “Autonomous dynamic al system approach for f (R) gravity,” Phys. Rev. D 96 (2017) no.10, 104049 [arXiv:1711.02230 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  40. [40]

    A not e on the dynamical system formulations in f(R) gravity,

    S. Chakraborty, P. K. S. Dunsby and K. Macdevette, “A not e on the dynamical system formulations in f(R) gravity,” Int. J. Geom. Meth. Mod. Phys . 19 (2022) no.08, 2230003 [arXiv:2112.13094 [gr-qc]]

    work page arXiv 2022

  41. [41]

    Cosmological dynamics of R^n gravity

    S. Carloni, P. K. S. Dunsby, S. Capozziello and A. Troisi , “Cosmological dynamics of R**n gravity,” Class. Quant. Grav. 22 (2005), 4839-4868 [arXiv:gr-qc/0410046 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  42. [42]

    The Pha se space view of inflation. 2: Fourth order models,

    S. Capozziello, F. Occhionero and L. Amendola, “The Pha se space view of inflation. 2: Fourth order models,” Int. J. Mod. Phys. D 1 (1993), 615-639

    work page 1993

  43. [43]

    Dynamical Systems Perspective of Cosmological Finite-time Singularities in $f(R)$ Gravity and Interacting Multifluid Cosmology

    S. D. Odintsov and V. K. Oikonomou, “Dynamical Systems P erspective of Cosmological Finite-time Singularities in f (R) Gravity and Interacting Multifluid Cosmology,” Phys. Rev. D 98 (2018) no.2, 024013 [arXiv:1806.07295 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  44. [44]

    The particle content of (scalar curva ture)2 metric-affine gravity,

    G. K. Karananas, “The particle content of (scalar curva ture)2 metric-affine gravity,” [arXiv:2408.16818 [hep-th]]

    work page arXiv

  45. [45]

    Hamiltonian analysis of metric-affine-R 2 theory,

    D. Glavan, T. Zlosnik and C. Lin, “Hamiltonian analysis of metric-affine-R 2 theory,” JCAP 04 (2024), 072 [arXiv:2311.17459 [gr-qc]]

    work page arXiv 2024

  46. [46]

    Conformal and pure scale-invarian t gravities in d dimensions,

    A. Hell and D. Lust, “Conformal and pure scale-invarian t gravities in d dimensions,” [arXiv:2506.18775 [hep-th]]. 30

    work page arXiv

  47. [47]

    Hamiltonian analysis of the cuscuton

    H. Gomes and D. C. Guariento, Phys. Rev. D 95 (2017) no.10, 104049 [arXiv:1703.08226 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  48. [48]

    On the stro ng coupling of Einsteinian Cubic Gravity and its generalisations,

    J. Beltr´ an Jim´ enez and A. Jim´ enez-Cano, “On the stro ng coupling of Einsteinian Cubic Gravity and its generalisations,” JCAP 01 (2021), 069 [arXiv:2009.08197 [gr-qc]]

    work page arXiv 2021

  49. [49]

    Propagating degrees of freedom on maximally symmetric backgrounds in f(R) theories of grav ity,

    A. Casado-Turri´ on, ´A. de la Cruz-Dombriz and A. Dobado, “Propagating degrees of freedom on maximally symmetric backgrounds in f(R) theories of grav ity,” Phys. Rev. D 111 (2025) no.4, 044030 [arXiv:2412.09366 [gr-qc]]. 31

    work page arXiv 2025