pith. machine review for the scientific record. sign in

arxiv: 2604.04352 · v1 · submitted 2026-04-06 · ✦ hep-th · gr-qc

Recognition: 2 theorem links

· Lean Theorem

Superradiant Suppression of Non-minimally Coupled Scalar fields for a Rotating Charged dS Black Hole in Conformal Weyl Gravity

Owen Gartlan , Jacob March , Leo Rodriguez , Shanshan Rodriguez , Yihan Shen
Authors on Pith no claims yet

Pith reviewed 2026-05-10 20:23 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords superradianceconformal Weyl gravityde Sitter black holesscalar fieldsamplification factorsHeun equationWKB approximationcharged black holes
0
0 comments X

The pith

Superradiant amplification of scalar fields around rotating charged de Sitter black holes is suppressed in conformal Weyl gravity compared to general relativity.

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

The paper compares the superradiant scattering of charged scalar fields in rotating charged de Sitter black hole spacetimes between general relativity and conformal Weyl gravity. For massless fields it uses an exact correspondence between the Heun equation and semiclassical BPZ equations, while for massive fields it applies WKB methods to estimate amplification in different radial regions. The calculations show reduced amplification factors in conformal Weyl gravity across the studied regimes. In the massive sector this reduction becomes strong exponential suppression of order e to the minus two mu over square root of Lambda in the cosmological region. A reader would care because these differences affect black hole energy extraction and stability in modified gravity models of expanding universes.

Core claim

For both the massless and massive sectors, suppression of superradiant amplification in CWG relative to that in GR is observed across the parameter regimes studied. Particularly, in the massive sector, we find strong exponential suppression of superradiant amplification on the order of e^{-2μΛ^{-1/2}} in the cosmological region.

What carries the argument

The Heun-BPZ correspondence for exact solution of massless perturbation equations and WKB approximation for massive fields, used to compute and compare superradiant amplification factors between the two gravity theories.

If this is right

  • Black holes in conformal Weyl gravity are more stable against superradiant instabilities than in general relativity.
  • Energy extraction from rotation via superradiance is less efficient in conformal Weyl gravity spacetimes.
  • The suppression strengthens exponentially with the mass of the scalar field in the region outside the cosmological horizon.
  • The results hold for both massless and massive conformally coupled charged scalars in charged rotating de Sitter backgrounds.

Where Pith is reading between the lines

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

  • If the suppression persists in full nonlinear regimes, conformal Weyl gravity could alter predictions for black hole growth and mergers in cosmological settings.
  • The same methods might be applied to other modified gravity theories to check for similar damping of superradiant effects.
  • Astrophysical signatures such as changes in accretion disk dynamics or gravitational wave signals could distinguish the two theories.

Load-bearing premise

The Heun-BPZ correspondence applies directly to the perturbation equations in these black hole spacetimes and the WKB approximation remains valid in the cosmological region without significant higher-order corrections.

What would settle it

Numerical or observational data showing superradiant amplification factors for charged scalar fields that match general relativity predictions rather than the reduced values found in conformal Weyl gravity.

Figures

Figures reproduced from arXiv: 2604.04352 by Jacob March, Leo Rodriguez, Owen Gartlan, Shanshan Rodriguez, Yihan Shen.

Figure 1
Figure 1. Figure 1: Parameter space (a, Λ) for the KN dSCG (Panel (a)) and KN dS (Panel (b)) spacetimes. The region bounded by the blue and red curves in the lower left corner denotes the black hole region with at least one event horizon. The red curve indicates rE = rE,c where the cosmological ergosphere and the black hole ergosphere coincide. To its right side, the black hole no longer possesses an ergosphere. For both case… view at source ↗
Figure 2
Figure 2. Figure 2: Parameter space (Q, Λ) for the KN dSCG (Panel (a)) and KN dS (Panel (b)) spacetimes. The region bounded by the blue and red curves in the lower left corner denotes the black hole region with at least one event horizon. The red curve indicates rE = rE,c where the cosmological ergosphere and the black hole ergosphere coincide. To its right side, the black hole no longer possesses an ergosphere. For both case… view at source ↗
Figure 3
Figure 3. Figure 3: Outer event horizon, r+, as a function of Q/M (Panel (a)) and a/M (Panel (b)). In both graphs, the solid lines represent the KN dSCG spacetime and the dashed lines represent the KN dS spacetime [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: rISCO as a function of Q/M (Panel (a)) and a/M (Panel (b)). In both graphs, the solid lines represent the KN dSCG spacetime and the dashed lines represent the KN dS spacetime. Finally, we consider the ergoregion, which plays a central role in black hole energy extraction mechanisms and is defined by the surface gtt = 0. As shown in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Meridional cross section of the ergoregion for varying [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Amplification factor for the l = 1, m = 1 mode Z11 as a function of scalar field frequency Mω for varying black hole charge Q (Panel (a)) and spin a (Panel (b)). In both graphs, the solid lines represent the KN dSCG spacetime and the dashed lines represent the KN dS spacetime. Figs. 6 and 7 show that both superradiant amplification factors Z11 and Z22 sourced by our conformal Weyl gravity metric (4) are co… view at source ↗
Figure 7
Figure 7. Figure 7: Amplification factor for the l = 2, m = 2 mode Z22 as a function of scalar field frequency Mω for varying black hole charge Q (Panel (a)) and spin a (Panel (b)). In both graphs, the solid lines represent the KN dSCG spacetime and the dashed lines represent the KN dS spacetime [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Amplification factors Z00 (Panel (a)) and Z11 (Panel(b)) for varying scalar field charge q. In both graphs, the solid lines represent the KN dSCG spacetime and the dashed lines represent the KN dS spacetime. regular singular points remain irreducible. We therefore analyze this case using the WKB approximation. For the remainder of this section, we focus on the KN dSCG background, and therefore treat the in… view at source ↗
Figure 9
Figure 9. Figure 9: Effective potential V (r) as a function of r over r ∈ [30, 2000] for varying black hole charge Q (Panel (a)) and scalar field mass µ (Panel (b)). V (rtp) plotted in dashed line as a reference [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Effective potential V (r) as a function of r over r ∈ [2000, 500000] for varying blak hole charge Q (Panel (a)) and scalar field µ (Panel (b)). 6.2. Amplification Factor Approximation at Cosmological Region in Terms of Far Region Amplification Since the lower bound of the superradiance condition (48) contains ω, the allowed superradiant frequency range remains implicit. In this section, we conduct the sup… view at source ↗
Figure 11
Figure 11. Figure 11: Panel (a) shows the WKB parameters ϵ(r) as a function of r. Panel (b) shows the WKB parameter S as a function of scalar field µ for varying black hole charge Q and cosmological constant Λ [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
read the original abstract

In this study, we present an analytical investigation of the superradiant scattering of a massive charged conformally coupled scalar field in rotating charged $de~Sitter$ black hole spacetimes within two gravitational theories: General Relativity (GR) and fourth-order Conformal (Weyl-squared) Gravity (CWG). For the massless charged conformally coupled scalar, we exploit a recently discovered correspondence between the Heun equation and the semiclassical limit of Belavin-Polyakov-Zamolodchikov (BPZ) equations in two-dimensional conformal field theory to solve for the superradiant amplification factors as controlled expansions in a small parameter scaling. For the massive charged conformally coupled scalar, we use WKB methods to derive an order of magnitude approximation for the amplification factors in the cosmological region in terms of those in the region $r_+\ll r \ll r_c$ where $r_+$ and $r_c$ are the outer and cosmological event horizons, respectively. For both the massless and massive sectors, suppression of superradiant amplification in CWG relative to that in GR is observed across the parameter regimes studied. Particularly, in the massive sector, we find strong exponential suppression of superradiant amplification on the order of $e^{-2\mu\Lambda^{-1/2}}$ in the cosmological region.

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 / 1 minor

Summary. The manuscript presents an analytical study of superradiant scattering for massive and massless charged conformally coupled scalar fields in rotating charged de Sitter black hole backgrounds, comparing General Relativity to Conformal Weyl Gravity (CWG). It utilizes the Heun-BPZ correspondence for the massless case to derive amplification factors as expansions in a small parameter, and WKB approximations for the massive case to estimate suppression in the cosmological region, claiming overall suppression in CWG relative to GR, with a specific exponential form e^{-2μΛ^{-1/2}} for the massive sector.

Significance. If the central mappings and approximations are rigorously verified, the results would demonstrate that higher-derivative terms in CWG suppress superradiant amplification relative to GR for these scalar fields. This could be relevant for understanding modified gravity effects on black hole instabilities. The analytical approach using the Heun-BPZ correspondence for controlled expansions and WKB for order-of-magnitude estimates is a methodological strength, providing explicit parameter dependence rather than purely numerical outputs.

major comments (2)
  1. [Perturbation equations and Heun-BPZ application] The application of the Heun-BPZ correspondence for the massless sector assumes that the radial perturbation equation in the CWG background reduces to the required Heun form. However, the fourth-order Weyl gravity metric functions differ from GR Kerr-dS, and the effective potential for the conformally coupled scalar may acquire additional derivative terms from the Weyl-squared action. The manuscript must explicitly derive the radial equation in CWG (likely in the perturbation equations section) and demonstrate the singularity structure and reduction steps to confirm the correspondence holds without alteration. This step is load-bearing for the reported suppression factors.
  2. [WKB approximation for massive scalars] For the massive sector, the WKB approximation is invoked to obtain the exponential suppression e^{-2μΛ^{-1/2}} in the cosmological region. No error estimates, validity conditions, or assessment of higher-order corrections are provided, particularly in the r_+ ≪ r ≪ r_c and cosmological regions where the potential may differ due to CWG. The manuscript should include a justification of the WKB regime and leading-order accuracy to support the quantitative claim.
minor comments (1)
  1. [Abstract] The abstract mentions 'controlled expansions in a small parameter scaling' without identifying the small parameter; this should be stated explicitly for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which will help strengthen the presentation of our results. We address each major comment below and outline the revisions we will implement.

read point-by-point responses

  1. Referee: [Perturbation equations and Heun-BPZ application] The application of the Heun-BPZ correspondence for the massless sector assumes that the radial perturbation equation in the CWG background reduces to the required Heun form. However, the fourth-order Weyl gravity metric functions differ from GR Kerr-dS, and the effective potential for the conformally coupled scalar may acquire additional derivative terms from the Weyl-squared action. The manuscript must explicitly derive the radial equation in CWG (likely in the perturbation equations section) and demonstrate the singularity structure and reduction steps to confirm the correspondence holds without alteration. This step is load-bearing for the reported suppression factors.

    Authors: We agree that an explicit derivation of the radial perturbation equation is required to rigorously justify the Heun-BPZ correspondence in the CWG background. In the revised manuscript we will add a dedicated subsection deriving the radial equation for the conformally coupled scalar from the Weyl-squared action, explicitly displaying the metric functions, the resulting effective potential, and the singularity structure. We will then detail the coordinate transformations and reduction steps that map the equation onto the standard Heun form, confirming that no additional derivative terms alter the correspondence for the conformally coupled case. This addition will directly support the reported suppression factors. revision: yes

  2. Referee: [WKB approximation for massive scalars] For the massive sector, the WKB approximation is invoked to obtain the exponential suppression e^{-2μΛ^{-1/2}} in the cosmological region. No error estimates, validity conditions, or assessment of higher-order corrections are provided, particularly in the r_+ ≪ r ≪ r_c and cosmological regions where the potential may differ due to CWG. The manuscript should include a justification of the WKB regime and leading-order accuracy to support the quantitative claim.

    Authors: We acknowledge that the WKB treatment requires additional justification to substantiate the quantitative exponential suppression. In the revised manuscript we will expand the relevant section to include: (i) the precise validity conditions for the WKB regime in both the r_+ ≪ r ≪ r_c and cosmological regions, (ii) an estimate of the leading-order error and the size of higher-order corrections, and (iii) a brief comparison of the CWG effective potential with its GR counterpart to show why the leading exponential form e^{-2μΛ^{-1/2}} remains robust. These additions will clarify the regime of applicability of our order-of-magnitude estimate. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external correspondences to distinct metrics

full rationale

The paper derives superradiant amplification factors by applying the Heun-BPZ correspondence (for massless sector) and WKB approximation (for massive sector) to the radial perturbation equations of the given rotating charged dS metrics in both GR and CWG. These methods are invoked as external mathematical tools, and the reported suppression (including the exponential factor in the cosmological region) emerges from explicit comparison of the resulting expressions across the two theories. No step reduces a claimed prediction to a parameter fitted from the same data, a self-defined quantity, or a load-bearing self-citation chain; the central results remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard mathematical tools from differential equations and approximation methods in black hole physics, with no new free parameters or invented entities introduced beyond the standard black hole metrics and field couplings.

axioms (2)
  • standard math The Heun equation arises from the radial perturbation equation and admits a correspondence to the semiclassical BPZ equations in 2D CFT
    Invoked to obtain controlled expansions for the massless amplification factors.
  • domain assumption WKB approximation is applicable in the cosmological region for the massive scalar field
    Used to derive the order-of-magnitude estimate relating different radial regions.

pith-pipeline@v0.9.0 · 5561 in / 1505 out tokens · 33019 ms · 2026-05-10T20:23:24.703707+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

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

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

Reference graph

Works this paper leans on

84 extracted references · 68 canonical work pages · 6 internal anchors

  1. [1]

    Abbott B Pet al.(LIGO Scientific, Virgo) 2016Phys. Rev. Lett.116061102 (Preprint 1602.03837)

    work page internal anchor Pith review arXiv

  2. [2]

    Abbott B Pet al.(LIGO Scientific, Virgo) 2019Phys. Rev. D100104036 (Preprint1903.04467)

    work page internal anchor Pith review arXiv

  3. [3]

    Akiyama Ket al.(Event Horizon Telescope) 2019Astrophys. J. Lett.875L4 (Preprint 1906.11241)

    work page Pith review arXiv 1906

  4. [4]

    Akiyama Ket al.(Event Horizon Telescope) 2019Astrophys. J. Lett.875L1 (Preprint 1906.11238)

    work page internal anchor Pith review arXiv 1906

  5. [5]

    Akiyama Ket al.(Event Horizon Telescope) 2022Astrophys. J. Lett.930L12 (Preprint 2311.08680)

    work page internal anchor Pith review arXiv

  6. [6]

    Akiyama Ket al.(Event Horizon Telescope) 2022Astrophys. J. Lett.930L14 (Preprint 2311.09479)

    work page arXiv

  7. [7]

    Nuovo Cim.1252–276

    Penrose R 1969Riv. Nuovo Cim.1252–276

  8. [8]

    Misner C W 1972Phys. Rev. Lett.28994–997

  9. [10]

    Penrose R and Floyd R M 1971Nature Physical Science229177–179 ISSN 0300-8746, 2058-1106

    2058

  10. [11]

    Penrose R and Floyd R M 1971Nature229177–179

  11. [12]

    Damour T, Deruelle N and Ruffini R 1976Lettere al Nuovo Cimento (1971-1985)15257–262 ISSN 1827-613X

    1971

  12. [13]

    Di Menza L and Nicolas J P 2015Class. Quant. Grav.32145013 (Preprint1411.3988)

    work page arXiv

  13. [14]

    Benone C L, de Oliveira E S, Dolan S R and Crispino L C B 2014Phys. Rev. D89104053 (Preprint1404.0687)

    work page arXiv

  14. [15]

    Benone C L and Crispino L C B 2016Phys. Rev. D93024028 (Preprint1511.02634)

    work page arXiv

  15. [16]

    Balakumar V, Winstanley E, Bernar R P and Crispino L C B 2020Phys. Lett. B811135904 (Preprint2010.01630)

    work page arXiv

  16. [17]

    Glampedakis K and Andersson N 2001Class. Quant. Grav.181939–1966 (Preprint gr-qc/0102100)

    work page Pith review arXiv 1966

  17. [18]

    Dolan S R 2008Class. Quant. Grav.25235002 (Preprint0801.3805)

    work page Pith review arXiv

  18. [19]

    Benone C L and Crispino L C B 2019Phys. Rev. D99044009 (Preprint1901.05592)

    work page arXiv

  19. [21]

    Wang C H, Pang C Q and Wei S W 2022Physical Review D106124050 ISSN 2470-0010, 2470-0029 (Preprint2209.08837)

    work page arXiv

  20. [22]

    Yuan C, Jiang Y and Huang Q G 2022Physical Review D106ISSN 2470-0010, 2470-0029 (Preprint2204.03482)

    work page arXiv

  21. [23]

    Berti E, Brito R, Macedo C F B, Raposo G and Rosa J L 2019Phys. Rev. D99104039 (Preprint1904.03131)

    work page arXiv

  22. [24]

    Press W H and Teukolsky S A 1972Nature238211–212

  23. [25]

    Cardoso V, Dias O J C, Lemos J P S and Yoshida S 2004Phys. Rev. D70044039 [Erratum: Phys.Rev.D 70, 049903 (2004)] (Preprinthep-th/0404096)

    work page arXiv 2004

  24. [26]

    Cardoso V 2013Gen. Rel. Grav.452079–2097 (Preprint1307.0038)

    work page arXiv 2097

  25. [27]

    Herdeiro C A R, Degollado J C and R´ unarsson H F 2013Phys. Rev. D88063003 (Preprint 1305.5513)

    work page arXiv

  26. [28]

    Dolan S R, Ponglertsakul S and Winstanley E 2015Phys. Rev. D92124047 (Preprint 1507.02156) 29

    work page arXiv

  27. [29]

    Dias O J C and Masachs R 2018Class. Quant. Grav.35184001 (Preprint1801.10176)

    work page arXiv

  28. [30]

    Brito R, Cardoso V and Pani P 2020Superradiance – the 2020 Editionvol 971 (Preprint 1501.06570)

    work page arXiv 2020

  29. [31]

    Suzuki H, Takasugi E and Umetsu H 1998Progress of Theoretical Physics100491–505 ISSN 0033-068X, 1347-4081 (Preprintgr-qc/9805064)

    work page arXiv

  30. [32]

    Mascher G, Destounis K and Kokkotas K D 2022Physical Review D105084052 ISSN 2470-0010, 2470-0029 (Preprint2204.05335)

    work page arXiv

  31. [33]

    Green S R, Hollands S, Ishibashi A and Wald R M 2016Class. Quant. Grav.33125022 (Preprint1512.02644)

    work page arXiv

  32. [34]

    Li R, Zhao Y, Zi T and Chen X 2019Phys. Rev. D99084045

  33. [35]

    Yang H and Miao Y G 2023Chin. Phys. C47075101 (Preprint2211.15130)

    work page arXiv

  34. [36]

    Chen W X and Zheng Y G 2021 (Preprint2103.04239)

    work page arXiv 2021

  35. [37]

    Cuadros-Melgar B, Fontana R D B and de Oliveira J 2021Phys. Rev. D104104039 (Preprint 2108.04864)

    work page arXiv

  36. [38]

    Khodadi M and Pourkhodabakhshi R 2021Phys. Lett. B823136775 (Preprint2111.03316)

    work page arXiv

  37. [39]

    Richarte M G, Martins ´E L and Fabris J C 2022Phys. Rev. D105064043 (Preprint 2111.01595)

    work page arXiv

  38. [40]

    Ishii T, Kaku Y and Murata K 2022JHEP10024 (Preprint2207.03123)

    work page arXiv

  39. [41]

    Zhu Z, Zhang S J, Pellicer C E, Wang B and Abdalla E 2014Phys. Rev. D90044042 [Addendum: Phys.Rev.D 90, 049904 (2014)] (Preprint1405.4931)

    work page arXiv 2014

  40. [42]

    Destounis K 2019Phys. Rev. D100044054 (Preprint1908.06117)

    work page arXiv

  41. [43]

    Zouros T J M and Eardley D M 1979Annals Phys.118139–155

  42. [44]

    Berti E, Cardoso V and Starinets A O 2009Class. Quant. Grav.26163001 (Preprint0905.2975)

    work page internal anchor Pith review arXiv

  43. [45]

    Konoplya R A and Zhidenko A 2011Rev. Mod. Phys.83793–836 (Preprint1102.4014)

    work page internal anchor Pith review arXiv

  44. [46]

    Xia Z W, Yang H and Miao Y G 2024Class. Quant. Grav.41165010 (Preprint2310.00253)

    work page arXiv

  45. [47]

    Iyer S and Will C M 1987Phys. Rev. D353621

  46. [48]

    Brensinger S and Rodgers V G J 2019Int. J. Mod. Phys. A331850223 (Preprint1712.05394)

    work page arXiv

  47. [49]

    Brensinger S, Heitritter K, Rodgers V G J and Stiffler K 2021Phys. Rev. D103044060 (Preprint2009.06730)

    work page arXiv

  48. [50]

    Brensinger S, Heitritter K, Rodgers V G J, Stiffler K and Whiting C A 2020Class. Quant. Grav. 37055003 (Preprint1907.05334)

    work page arXiv

  49. [51]

    Mannheim P D 2011Gen. Rel. Grav.43703–750 (Preprint0909.0212)

    work page arXiv

  50. [52]

    Edery A and Paranjape M B 1998Phys. Rev. D58024011 (Preprintastro-ph/9708233)

    work page arXiv

  51. [53]

    Edery A and Paranjape M B 1999Gen. Rel. Grav.311031–1047 (Preprintastro-ph/9808345)

    work page arXiv

  52. [54]

    Edery A, Methot A A and Paranjape M B 2001Gen. Rel. Grav.332075–2079 (Preprint astro-ph/0006173)

    work page arXiv 2079

  53. [55]

    Zou D C and Myung Y S 2020Phys. Rev. D101084021 (Preprint2001.01351)

    work page arXiv

  54. [56]

    Momennia M and Hendi S H 2019Phys. Rev. D99124025 (Preprint1905.12290)

    work page arXiv

  55. [57]

    Momennia M and Hendi S H 2020Eur. Phys. J. C80505 (Preprint1910.00428)

    work page arXiv

  56. [58]

    Momennia M, Hossein Hendi S and Soltani Bidgoli F 2021Phys. Lett. B813136028 (Preprint 1807.01792)

    work page arXiv

  57. [59]

    Xu H and Yung M H 2019Phys. Lett. B79397–103 (Preprint1811.07309)

    work page arXiv

  58. [60]

    Bambi C, Modesto L, Porey S and Rachwa l L 2018Eur. Phys. J. C78116 (Preprint 1711.04575)

    work page arXiv

  59. [61]

    Biedke E, Quaid S and Rodgers V 2026 (Preprint2603.06995)

    work page arXiv 2026

  60. [62]

    Aggarwal D, Chang D, Helmers Q D, Sivrioglu N, Ram-Mohan L R, Rodriguez L, Rodriguez S and Suleiman R 2023IJMPD322350017

  61. [63]

    Stelle K S 1977Phys. Rev. D16953–969

  62. [64]

    Edery A, Fabbri L and Paranjape M B 2006Class. Quant. Grav.236409–6423 (Preprint hep-th/0603131)

    work page arXiv

  63. [65]

    Making the Case for Conformal Gravity,

    Mannheim P D 2012Found. Phys.42388–420 (Preprint1101.2186) 30

    work page arXiv

  64. [66]

    Kiefer C and Nikolic B 2017Phys. Rev. D95084018 (Preprint1702.04973)

    work page arXiv

  65. [67]

    J.342635–638

    Mannheim P D and Kazanas D 1989Astrophys. J.342635–638

  66. [68]

    Mannheim P D 2006Prog. Part. Nucl. Phys.56340–445 (Preprintastro-ph/0505266)

    work page arXiv

  67. [69]

    Mannheim P D and Kazanas D 1991Phys. Rev. D44(2) 417–423 URL https://link.aps.org/doi/10.1103/PhysRevD.44.417

    work page doi:10.1103/physrevd.44.417

  68. [70]

    Maldacena J 2011 (Preprint1105.5632)

    work page arXiv 2011

  69. [71]

    Anastasiou G and Olea R 2016Phys. Rev. D94086008 (Preprint1608.07826)

    work page arXiv

  70. [72]

    Anastasiou G, Araya I J and Olea R 2021JHEP01134 (Preprint2010.15146)

    work page arXiv

  71. [73]

    Mannheim P D and Kazanas D 1991Physical Review D44417–423 ISSN 0556-2821

  72. [74]

    Liu H S and Lu H 2013Journal of High Energy Physics2013139 ISSN 1029-8479 (Preprint 1212.6264)

    work page arXiv

  73. [75]

    Batic D and Schmid H 2007Journal of Mathematical Physics48042502 ISSN 0022-2488, 1089-7658 (Preprintgr-qc/0701064)

    work page arXiv

  74. [76]

    Hortacsu M 2018Advances in High Energy Physics20181–14 ISSN 1687-7357, 1687-7365 (Preprint1101.0471)

    work page arXiv

  75. [77]

    Lisovyy O and Naidiuk A 2022Journal of Physics A: Mathematical and Theoretical55434005 ISSN 1751-8113, 1751-8121 (Preprint2208.01604)

    work page arXiv

  76. [78]

    Bonelli G, Iossa C, Lichtig D P and Tanzini A 2023Communications in Mathematical Physics 397635–727 ISSN 0010-3616, 1432-0916

  77. [79]

    Aminov G, Arnaudo P, Bonelli G, Grassi A and Tanzini A 2023Journal of High Energy Physics 202359 ISSN 1029-8479 (Preprint2307.10141)

    work page arXiv

  78. [80]

    Faraoni V Conformally coupled inflation (Preprint1309.4900) URL http://arxiv.org/abs/1309.4900

    work page arXiv

  79. [81]

    Santos J S, Cruz M B and Brito F A811082 ISSN 1434-6044, 1434-6052 (Preprint2103.11212) URLhttp://arxiv.org/abs/2103.11212

    work page arXiv

  80. [82]

    Calz` a M, Rinaldi M and Vagnozzi S112104055 ISSN 2470-0010, 2470-0029 (Preprint 2510.12257) URLhttp://arxiv.org/abs/2510.12257

    work page arXiv

Showing first 80 references.