# Probing Modified Gravity Theories with Scalar Fields Using Black-Hole Images

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## Abstract

**:**

## 1. Introduction

## 2. Shadow Radius of Compact Objects

#### 2.1. Black Holes

#### 2.2. Wormholes

## 3. The EHT Bounds

## 4. The Einstein-Scalar-GB Theory

#### 4.1. Black Holes

#### 4.1.1. $f\left(\varphi \right)=\alpha \phantom{\rule{0.166667em}{0ex}}\varphi \left(r\right)$

#### 4.1.2. $f\left(\varphi \right)=\frac{\alpha}{2}\phantom{\rule{0.166667em}{0ex}}\varphi {\left(r\right)}^{2}$

#### 4.1.3. $f\left(\varphi \right)=\alpha \phantom{\rule{0.166667em}{0ex}}{\mathrm{e}}^{\gamma \phantom{\rule{0.166667em}{0ex}}\varphi \left(r\right)}$

#### 4.2. Wormholes

## 5. Curvature-Induced Spontaneous Scalarization

#### 5.1. Minimal Model

#### 5.2. Quartic sGB Coupling

## 6. The Einstein–Maxwell-Scalar Theory

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Equations in EsRGB Theory

## Appendix B. Equations in EMS Theory

## Note

1 | Let us note that although we will make use of the bounds on the observed black-hole shadow from Sagittarius A${}^{*}$[104], our analysis will cover also the corresponding bound from the M87${}^{*}$ observation [91,92,93,94,95,96,97,98,120,126] as the latter is less stringent and thus easier to satisfy. |

## References

- Bertotti, B.; Iess, L.; Tortora, P. A test of general relativity using radio links with the Cassini spacecraft. Nature
**2003**, 425, 374–376. [Google Scholar] [CrossRef] [PubMed] - Stairs, I.H. Testing general relativity with pulsar timing. Living Rev. Rel.
**2003**, 6, 5. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Verma, A.; Fienga, A.; Laskar, J.; Manche, H.; Gastineau, M. Use of messenger radioscience data to improve planetary ephemeris and to test general relativity. Astron. Astrophys.
**2014**, 561, A115. [Google Scholar] [CrossRef] - Lambert, S.B.; Le Poncin-Lafitte, C. Improved determination of gamma by VLBI. Astron. Astrophys.
**2011**, 529, A70. [Google Scholar] [CrossRef] [Green Version] - Kramer, M.; Stairs, I.H.; Manchester, R.N.; Wex, N.; Deller, A.T.; Coles, W.A.; Ali, M.; Burgay, M.A.R.T.A.; Camilo, F.; Cognard, I.; et al. Strong-Field Gravity Tests with the Double Pulsar. Phys. Rev. X
**2021**, 11, 041050. [Google Scholar] [CrossRef] - Will, C.M. The Confrontation between General Relativity and Experiment. Living Rev. Rel.
**2014**, 17, 4. [Google Scholar] [CrossRef] [Green Version] - Berti, E.; Barausse, E.; Cardoso, V.; Gualtieri, L.; Pani, P.; Sperhake, U.; Stein, L.C.; Wex, N.; Yagi, K.; Baker, T.; et al. Testing General Relativity with Present and Future Astrophysical Observations. Class. Quant. Grav.
**2015**, 32, 243001. [Google Scholar] [CrossRef] [Green Version] - Ishak, M. Testing General Relativity in Cosmology. Living Rev. Rel.
**2019**, 22, 1. [Google Scholar] [CrossRef] [Green Version] - Capozziello, S.; De Laurentis, M. Extended Theories of Gravity. Phys. Rept.
**2011**, 509, 167–321. [Google Scholar] [CrossRef] [Green Version] - Burgess, C.P. Quantum gravity in everyday life: General relativity as an effective field theory. Living Rev. Rel.
**2004**, 7, 5–56. [Google Scholar] [CrossRef] [Green Version] - Donoghue, J.F. General relativity as an effective field theory: The leading quantum corrections. Phys. Rev. D
**1994**, 50, 3874–3888. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Luckock, H.; Moss, I. Black holes have skyrmion hair. Phys. Lett. B
**1986**, 176, 341–345. [Google Scholar] [CrossRef] - Bizon, P. Colored black holes. Phys. Rev. Lett.
**1990**, 64, 2844–2847. [Google Scholar] [CrossRef] - Campbell, B.A.; Duncan, M.J.; Kaloper, N.; Olive, K.A. Axion hair for Kerr black holes. Phys. Lett. B
**1990**, 251, 34–38. [Google Scholar] [CrossRef] [Green Version] - Campbell, B.A.; Kaloper, N.; Olive, K.A. Classical hair for Kerr-Newman black holes in string gravity. Phys. Lett. B
**1992**, 285, 199–205. [Google Scholar] [CrossRef] [Green Version] - Maeda, K.I.; Tachizawa, T.; Torii, T.; Maki, T. Stability of nonAbelian black holes and catastrophe theory. Phys. Rev. Lett.
**1994**, 72, 450–453. [Google Scholar] [CrossRef] [Green Version] - Kanti, P.; Mavromatos, N.E.; Rizos, J.; Tamvakis, K.; Winstanley, E. Dilatonic black holes in higher curvature string gravity. Phys. Rev. D
**1996**, 54, 5049–5058. [Google Scholar] [CrossRef] [Green Version] - Kanti, P.; Tamvakis, K. Colored black holes in higher curvature string gravity. Phys. Lett. B
**1997**, 392, 30–38. [Google Scholar] [CrossRef] [Green Version] - Kanti, P.; Mavromatos, N.E.; Rizos, J.; Tamvakis, K.; Winstanley, E. Dilatonic black holes in higher curvature string gravity. 2: Linear stability. Phys. Rev. D
**1998**, 57, 6255–6264. [Google Scholar] [CrossRef] [Green Version] - Torii, T.; Yajima, H.; Maeda, K.i. Dilatonic black holes with Gauss-Bonnet term. Phys. Rev. D
**1997**, 55, 739–753. [Google Scholar] [CrossRef] [Green Version] - Barack, L.; Cardoso, V.; Nissanke, S.; Sotiriou, T.P.; Askar, A.; Belczynski, C.; Bertone, G.; Bon, E.; Blas, D.; Brito, R.; et al. Black holes, gravitational waves and fundamental physics: A roadmap. Class. Quant. Grav.
**2019**, 36, 143001. [Google Scholar] [CrossRef] [Green Version] - Tretyakova, D.A.; Latosh, B.N. Scalar-Tensor Black Holes Embedded in an Expanding Universe. Universe
**2018**, 4, 26. [Google Scholar] [CrossRef] [Green Version] - Kerr, R.P. Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett.
**1963**, 11, 237–238. [Google Scholar] [CrossRef] - Israel, W. Event horizons in static vacuum space-times. Phys. Rev.
**1967**, 164, 1776–1779. [Google Scholar] [CrossRef] - Israel, W. Event horizons in static electrovac space-times. Commun. Math. Phys.
**1968**, 8, 245–260. [Google Scholar] [CrossRef] - Carter, B. Global structure of the Kerr family of gravitational fields. Phys. Rev.
**1968**, 174, 1559–1571. [Google Scholar] [CrossRef] [Green Version] - Carter, B. Axisymmetric Black Hole Has Only Two Degrees of Freedom. Phys. Rev. Lett.
**1971**, 26, 331–333. [Google Scholar] [CrossRef] - Hawking, S.W. Black holes in general relativity. Commun. Math. Phys.
**1972**, 25, 152–166. [Google Scholar] [CrossRef] - Price, R.H. Nonspherical perturbations of relativistic gravitational collapse. 1. Scalar and gravitational perturbations. Phys. Rev. D
**1972**, 5, 2419–2438. [Google Scholar] [CrossRef] - Price, R.H. Nonspherical Perturbations of Relativistic Gravitational Collapse. II. Integer-Spin, Zero-Rest-Mass Fields. Phys. Rev. D
**1972**, 5, 2439–2454. [Google Scholar] [CrossRef] - Robinson, D.C. Uniqueness of the Kerr black hole. Phys. Rev. Lett.
**1975**, 34, 905–906. [Google Scholar] [CrossRef] - Teitelboim, C. Nonmeasurability of the quantum numbers of a black hole. Phys. Rev. D
**1972**, 5, 2941–2954. [Google Scholar] [CrossRef] - Bekenstein, J.D. Nonexistence of baryon number for static black holes. Phys. Rev. D
**1972**, 5, 1239–1246. [Google Scholar] [CrossRef] - Bekenstein, J.D. Nonexistence of baryon number for black holes. ii. Phys. Rev. D
**1972**, 5, 2403–2412. [Google Scholar] [CrossRef] - Bekenstein, J.D. Novel ‘‘no-scalar-hair’’ theorem for black holes. Phys. Rev. D
**1995**, 51, R6608. [Google Scholar] [CrossRef] - Guo, Z.K.; Ohta, N.; Torii, T. Black Holes in the Dilatonic Einstein-Gauss-Bonnet Theory in Various Dimensions. I. Asymptotically Flat Black Holes. Prog. Theor. Phys.
**2008**, 120, 581–607. [Google Scholar] [CrossRef] [Green Version] - Pani, P.; Cardoso, V. Are black holes in alternative theories serious astrophysical candidates? The Case for Einstein-Dilaton-Gauss-Bonnet black holes. Phys. Rev. D
**2009**, 79, 084031. [Google Scholar] [CrossRef] [Green Version] - Pani, P.; Macedo, C.F.B.; Crispino, L.C.B.; Cardoso, V. Slowly rotating black holes in alternative theories of gravity. Phys. Rev. D
**2011**, 84, 087501. [Google Scholar] [CrossRef] [Green Version] - Kleihaus, B.; Kunz, J.; Radu, E. Rotating Black Holes in Dilatonic Einstein-Gauss-Bonnet Theory. Phys. Rev. Lett.
**2011**, 106, 151104. [Google Scholar] [CrossRef] [Green Version] - Yagi, K.; Yunes, N.; Tanaka, T. Slowly Rotating Black Holes in Dynamical Chern-Simons Gravity: Deformation Quadratic in the Spin. Phys. Rev. D
**2012**, 86, 044037. [Google Scholar] [CrossRef] [Green Version] - Sotiriou, T.P.; Zhou, S.Y. Black hole hair in generalized scalar-tensor gravity. Phys. Rev. Lett.
**2014**, 112, 251102. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sotiriou, T.P.; Zhou, S.Y. Black hole hair in generalized scalar-tensor gravity: An explicit example. Phys. Rev. D
**2014**, 90, 124063. [Google Scholar] [CrossRef] [Green Version] - Kleihaus, B.; Kunz, J.; Mojica, S.; Radu, E. Spinning black holes in Einstein–Gauss-Bonnet–dilaton theory: Nonperturbative solutions. Phys. Rev. D
**2016**, 93, 044047. [Google Scholar] [CrossRef] [Green Version] - Blázquez-Salcedo, J.L.; Macedo, C.F.B.; Cardoso, V.; Ferrari, V.; Gualtieri, L.; Khoo, F.S.; Kunz, J.; Pani, P. Perturbed black holes in Einstein-dilaton-Gauss-Bonnet gravity: Stability, ringdown, and gravitational-wave emission. Phys. Rev. D
**2016**, 94, 104024. [Google Scholar] [CrossRef] [Green Version] - Antoniou, G.; Bakopoulos, A.; Kanti, P. Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-Bonnet Theories. Phys. Rev. Lett.
**2018**, 120, 131102. [Google Scholar] [CrossRef] [Green Version] - Antoniou, G.; Bakopoulos, A.; Kanti, P. Black-Hole Solutions with Scalar Hair in Einstein-Scalar-Gauss-Bonnet Theories. Phys. Rev. D
**2018**, 97, 084037. [Google Scholar] [CrossRef] [Green Version] - Doneva, D.D.; Yazadjiev, S.S. New Gauss-Bonnet Black Holes with Curvature-Induced Scalarization in Extended Scalar-Tensor Theories. Phys. Rev. Lett.
**2018**, 120, 131103. [Google Scholar] [CrossRef] [Green Version] - Silva, H.O.; Sakstein, J.; Gualtieri, L.; Sotiriou, T.P.; Berti, E. Spontaneous scalarization of black holes and compact stars from a Gauss-Bonnet coupling. Phys. Rev. Lett.
**2018**, 120, 131104. [Google Scholar] [CrossRef] [Green Version] - Brihaye, Y.; Hartmann, B.; Urrestilla, J. Solitons and black hole in shift symmetric scalar-tensor gravity with cosmological constant. JHEP
**2018**, 6, 074. [Google Scholar] [CrossRef] [Green Version] - Doneva, D.D.; Kiorpelidi, S.; Nedkova, P.G.; Papantonopoulos, E.; Yazadjiev, S.S. Charged Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories. Phys. Rev. D
**2018**, 98, 104056. [Google Scholar] [CrossRef] [Green Version] - Bakopoulos, A.; Antoniou, G.; Kanti, P. Novel Black-Hole Solutions in Einstein-Scalar-Gauss-Bonnet Theories with a Cosmological Constant. Phys. Rev. D
**2019**, 99, 064003. [Google Scholar] [CrossRef] [Green Version] - Witek, H.; Gualtieri, L.; Pani, P.; Sotiriou, T.P. Black holes and binary mergers in scalar Gauss-Bonnet gravity: Scalar field dynamics. Phys. Rev. D
**2019**, 99, 064035. [Google Scholar] [CrossRef] [Green Version] - Minamitsuji, M.; Ikeda, T. Scalarized black holes in the presence of the coupling to Gauss-Bonnet gravity. Phys. Rev. D
**2019**, 99, 044017. [Google Scholar] [CrossRef] [Green Version] - Macedo, C.F.B.; Sakstein, J.; Berti, E.; Gualtieri, L.; Silva, H.O.; Sotiriou, T.P. Self-interactions and Spontaneous Black Hole Scalarization. Phys. Rev. D
**2019**, 99, 104041. [Google Scholar] [CrossRef] [Green Version] - Doneva, D.D.; Staykov, K.V.; Yazadjiev, S.S. Gauss-Bonnet black holes with a massive scalar field. Phys. Rev. D
**2019**, 99, 104045. [Google Scholar] [CrossRef] [Green Version] - Zou, D.C.; Myung, Y.S. Scalarized charged black holes with scalar mass term. Phys. Rev. D
**2019**, 100, 124055. [Google Scholar] [CrossRef] [Green Version] - Cunha, P.V.P.; Herdeiro, C.A.R.; Radu, E. Spontaneously Scalarized Kerr Black Holes in Extended Scalar-Tensor–Gauss-Bonnet Gravity. Phys. Rev. Lett.
**2019**, 123, 011101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Brihaye, Y.; Hartmann, B.; Aprile, N.P.; Urrestilla, J. Scalarization of asymptotically anti–de Sitter black holes with applications to holographic phase transitions. Phys. Rev. D
**2020**, 101, 124016. [Google Scholar] [CrossRef] - Ellis, H.G. Ether flow through a drainhole: A particle model in general relativity. J. Math. Phys.
**1973**, 14, 104. [Google Scholar] [CrossRef] - Bronnikov, K.A. Scalar-tensor theory and scalar charge. Acta Phys. Polon. B
**1973**, 4, 251–266. [Google Scholar] - Visser, M.; Kar, S.; Dadhich, N. Traversable wormholes with arbitrarily small energy condition violations. Phys. Rev. Lett.
**2003**, 90, 201102. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bronnikov, K.A. Extra dimensions, nonminimal couplings, horizons and wormholes. Grav. Cosmol.
**1996**, 2, 221–226. [Google Scholar] - Bronnikov, K.A.; Fabris, J.C. Weyl space-times and wormholes in D-dimensional Einstein and dilaton gravity. Class. Quant. Grav.
**1997**, 14, 831–842. [Google Scholar] [CrossRef] [Green Version] - Armendariz-Picon, C. On a class of stable, traversable Lorentzian wormholes in classical general relativity. Phys. Rev. D
**2002**, 65, 104010. [Google Scholar] [CrossRef] [Green Version] - Bronnikov, K.A.; Grinyok, S.V. Conformal continuations and wormhole instability in scalar-tensor gravity. Grav. Cosmol.
**2004**, 10, 237. [Google Scholar] - Lobo, F.S.N. Phantom energy traversable wormholes. Phys. Rev. D
**2005**, 71, 084011. [Google Scholar] [CrossRef] [Green Version] - Lobo, F.S.N. Chaplygin traversable wormholes. Phys. Rev. D
**2006**, 73, 064028. [Google Scholar] [CrossRef] [Green Version] - Lobo, F.S.N.; Oliveira, M.A. Wormhole geometries in f(R) modified theories of gravity. Phys. Rev. D
**2009**, 80, 104012. [Google Scholar] [CrossRef] [Green Version] - Bronnikov, K.A.; Skvortsova, M.V.; Starobinsky, A.A. Notes on wormhole existence in scalar-tensor and F(R) gravity. Grav. Cosmol.
**2010**, 16, 216–222. [Google Scholar] [CrossRef] [Green Version] - Garcia, N.M.; Lobo, F.S.N. Wormhole geometries supported by a nonminimal curvature-matter coupling. Phys. Rev. D
**2010**, 82, 104018. [Google Scholar] [CrossRef] [Green Version] - Kanti, P.; Kleihaus, B.; Kunz, J. Wormholes in Dilatonic Einstein-Gauss-Bonnet Theory. Phys. Rev. Lett.
**2011**, 107, 271101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kanti, P.; Kleihaus, B.; Kunz, J. Stable Lorentzian Wormholes in Dilatonic Einstein-Gauss-Bonnet Theory. Phys. Rev. D
**2012**, 85, 044007. [Google Scholar] [CrossRef] [Green Version] - Bolokhov, S.V.; Bronnikov, K.A.; Skvortsova, M.V. Magnetic black universes and wormholes with a phantom scalar. Class. Quant. Grav.
**2012**, 29, 245006. [Google Scholar] [CrossRef] [Green Version] - Bronnikov, K.A.; Galiakhmetov, A.M. Wormholes without exotic matter in Einstein–Cartan theory. Grav. Cosmol.
**2015**, 21, 283–288. [Google Scholar] [CrossRef] [Green Version] - Shaikh, R. Lorentzian wormholes in Eddington-inspired Born-Infeld gravity. Phys. Rev. D
**2015**, 92, 024015. [Google Scholar] [CrossRef] [Green Version] - Mehdizadeh, M.R.; Kord Zangeneh, M.; Lobo, F.S.N. Einstein-Gauss-Bonnet traversable wormholes satisfying the weak energy condition. Phys. Rev. D
**2015**, 91, 084004. [Google Scholar] [CrossRef] [Green Version] - Kuhfittig, P.K.F. Wormholes in f(R) gravity with a noncommutative-geometry background. Indian J. Phys.
**2018**, 92, 1207–1212. [Google Scholar] [CrossRef] [Green Version] - Ibadov, R.; Kleihaus, B.; Kunz, J.; Murodov, S. Wormhole solutions with NUT charge in higher curvature theories. Arabian J. Math.
**2021**, 11, 31–41. [Google Scholar] [CrossRef] - Karakasis, T.; Papantonopoulos, E.; Vlachos, C. f(R) gravity wormholes sourced by a phantom scalar field. Phys. Rev. D
**2022**, 105, 024006. [Google Scholar] [CrossRef] - Ghosh, B.; Mitra, S. Wormhole solutions in f(R) gravity theory for Chaplygin gas scenario. Int. J. Mod. Phys. A
**2021**, 36, 18. [Google Scholar] [CrossRef] - Fisher, I.Z. Scalar mesostatic field with regard for gravitational effects. Zh. Eksp. Teor. Fiz.
**1948**, 18, 636–640. [Google Scholar] - Janis, A.I.; Newman, E.T.; Winicour, J. Reality of the Schwarzschild Singularity. Phys. Rev. Lett.
**1968**, 20, 878–880. [Google Scholar] [CrossRef] - Wyman, M. Static Spherically Symmetric Scalar Fields in General Relativity. Phys. Rev. D
**1981**, 24, 839–841. [Google Scholar] [CrossRef] - Agnese, A.G.; La Camera, M. GRAVITATION WITHOUT BLACK HOLES. Phys. Rev. D
**1985**, 31, 1280–1286. [Google Scholar] [CrossRef] [PubMed] - Roberts, M.D. Scalar Field Counterexamples to the Cosmic Censorship Hypothesis. Gen. Rel. Grav.
**1989**, 21, 907–939. [Google Scholar] [CrossRef] - Kleihaus, B.; Kunz, J.; Kanti, P. Particle-like ultracompact objects in Einstein-scalar-Gauss-Bonnet theories. Phys. Lett. B
**2020**, 804, 135401. [Google Scholar] [CrossRef] - Kleihaus, B.; Kunz, J.; Kanti, P. Properties of ultracompact particlelike solutions in Einstein-scalar-Gauss-Bonnet theories. Phys. Rev. D
**2020**, 102, 024070. [Google Scholar] [CrossRef] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abernathy, M.R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett.
**2016**, 116, 061102. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral. Phys. Rev. Lett.
**2017**, 119, 161101. [Google Scholar] [CrossRef] [Green Version] - Abbott, R.; Abbott, T.D.; Abraham, S.; Acernese, F.; Ackley, K.; Adams, A.; Adams, C.; Adhikari, R.X.; Adya, V.B.; Affeldt, C.; et al. GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run. Phys. Rev. X
**2020**, 11, 021053. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. Astrophys. J. Lett.
**2019**, 875, L1. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First M87 Event Horizon Telescope Results. II. Array and Instrumentation. Astrophys. J. Lett.
**2019**, 875, L2. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First M87 Event Horizon Telescope Results. III. Data Processing and Calibration. Astrophys. J. Lett.
**2019**, 875, L3. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole. Astrophys. J. Lett.
**2019**, 875, L4. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring. Astrophys. J. Lett.
**2019**, 875, L5. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole. Astrophys. J. Lett.
**2019**, 875, L6. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First M87 Event Horizon Telescope Results. VII. Polarization of the Ring. Astrophys. J. Lett.
**2021**, 910, L12. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near The Event Horizon. Astrophys. J. Lett.
**2021**, 910, L13. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way. Astrophys. J. Lett.
**2022**, 930, L12. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First Sagittarius A* Event Horizon Telescope Results. II. EHT and Multiwavelength Observations, Data Processing, and Calibration. Astrophys. J. Lett.
**2022**, 930, L13. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First Sagittarius A* Event Horizon Telescope Results. III. Imaging of the Galactic Center Supermassive Black Hole. Astrophys. J. Lett.
**2022**, 930, L14. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First Sagittarius A* Event Horizon Telescope Results. IV. Variability, Morphology, and Black Hole Mass. Astrophys. J. Lett.
**2022**, 930, L15. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First Sagittarius A* Event Horizon Telescope Results. V. Testing Astrophysical Models of the Galactic Center Black Hole. Astrophys. J. Lett.
**2022**, 930, L16. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First Sagittarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric. Astrophys. J. Lett.
**2022**, 930, L17. [Google Scholar] [CrossRef] - Virbhadra, K.S.; Ellis, G.F.R. Schwarzschild black hole lensing. Phys. Rev. D
**2000**, 62, 084003. [Google Scholar] [CrossRef] [Green Version] - Claudel, C.M.; Virbhadra, K.S.; Ellis, G.F.R. The Geometry of photon surfaces. J. Math. Phys.
**2001**, 42, 818–838. [Google Scholar] [CrossRef] [Green Version] - Virbhadra, K.S.; Ellis, G.F.R. Gravitational lensing by naked singularities. Phys. Rev. D
**2002**, 65, 103004. [Google Scholar] [CrossRef] - Zakharov, A.F.; De Paolis, F.; Ingrosso, G.; Nucita, A.A. Direct measurements of black hole charge with future astrometrical missions. Astron. Astrophys.
**2005**, 442, 795–799. [Google Scholar] [CrossRef] [Green Version] - Virbhadra, K.S.; Keeton, C.R. Time delay and magnification centroid due to gravitational lensing by black holes and naked singularities. Phys. Rev. D
**2008**, 77, 124014. [Google Scholar] [CrossRef] [Green Version] - Virbhadra, K.S. Relativistic images of Schwarzschild black hole lensing. Phys. Rev. D
**2009**, 79, 083004. [Google Scholar] [CrossRef] [Green Version] - Johannsen, T.; Psaltis, D. A Metric for Rapidly Spinning Black Holes Suitable for Strong-Field Tests of the No-Hair Theorem. Phys. Rev. D
**2011**, 83, 124015. [Google Scholar] [CrossRef] [Green Version] - Zakharov, A.F. Constraints on a charge in the Reissner-Nordström metric for the black hole at the Galactic Center. Phys. Rev. D
**2014**, 90, 062007. [Google Scholar] [CrossRef] [Green Version] - Psaltis, D.; Wex, N.; Kramer, M. A Quantitative Test of the No-Hair Theorem with Sgr A* using stars, pulsars, and the Event Horizon Telescope. Astrophys. J.
**2016**, 818, 121. [Google Scholar] [CrossRef] [Green Version] - Ezquiaga, J.M.; Zumalacárregui, M. Dark Energy After GW170817: Dead Ends and the Road Ahead. Phys. Rev. Lett.
**2017**, 119, 251304. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Baker, T.; Bellini, E.; Ferreira, P.G.; Lagos, M.; Noller, J.; Sawicki, I. Strong constraints on cosmological gravity from GW170817 and GRB 170817A. Phys. Rev. Lett.
**2017**, 119, 251301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Creminelli, P.; Vernizzi, F. Dark Energy after GW170817 and GRB170817A. Phys. Rev. Lett.
**2017**, 119, 251302. [Google Scholar] [CrossRef] [Green Version] - Sakstein, J.; Jain, B. Implications of the Neutron Star Merger GW170817 for Cosmological Scalar-Tensor Theories. Phys. Rev. Lett.
**2017**, 119, 251303. [Google Scholar] [CrossRef] [Green Version] - Cunha, P.V.P.; Herdeiro, C.A.R. Shadows and strong gravitational lensing: A brief review. Gen. Rel. Grav.
**2018**, 50, 42. [Google Scholar] [CrossRef] [Green Version] - Khodadi, M.; Allahyari, A.; Vagnozzi, S.; Mota, D.F. Black holes with scalar hair in light of the Event Horizon Telescope. JCAP
**2020**, 9, 026. [Google Scholar] [CrossRef] - Psaltis, D.; Medeiros, L.; Christian, P.; Özel, F.; Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Ball, D.; et al. Gravitational Test Beyond the First Post-Newtonian Order with the Shadow of the M87 Black Hole. Phys. Rev. Lett.
**2020**, 125, 141104. [Google Scholar] [CrossRef] - Psaltis, D.; Talbot, C.; Payne, E.; Mandel, I. Probing the Black Hole Metric. I. Black Hole Shadows and Binary Black-Hole Inspirals. Phys. Rev. D
**2021**, 103, 104036. [Google Scholar] [CrossRef] - Oikonomou, V.K.; Fronimos, F.P. Reviving non-minimal Horndeski-like theories after GW170817: Kinetic coupling corrected Einstein–Gauss–Bonnet inflation. Class. Quant. Grav.
**2021**, 38, 035013. [Google Scholar] [CrossRef] - Odintsov, S.D.; Oikonomou, V.K.; Fronimos, F.P. Rectifying Einstein-Gauss-Bonnet Inflation in View of GW170817. Nucl. Phys. B
**2020**, 958, 115135. [Google Scholar] [CrossRef] - Odintsov, S.D.; Oikonomou, V.K. Swampland implications of GW170817-compatible Einstein-Gauss-Bonnet gravity. Phys. Lett. B
**2020**, 805, 135437. [Google Scholar] [CrossRef] - Oikonomou, V.K. A refined Einstein–Gauss–Bonnet inflationary theoretical framework. Class. Quant. Grav.
**2021**, 38, 195025. [Google Scholar] [CrossRef] - Kocherlakota, P.; Rezzolla, L.; Falcke, H.; Fromm, C.M.; Kramer, M.; Mizuno, Y.; Nathanail, A.; Olivares, H.; Younsi, Z.; Akiyama, K.; et al. Constraints on black-hole charges with the 2017 EHT observations of M87*. Phys. Rev. D
**2021**, 103, 104047. [Google Scholar] [CrossRef] - Völkel, S.H.; Barausse, E.; Franchini, N.; Broderick, A.E. EHT tests of the strong-field regime of general relativity. Class. Quant. Grav.
**2021**, 38, 21LT01. [Google Scholar] [CrossRef] - Vagnozzi, S.; Roy, R.; Tsai, Y.D.; Visinelli, L. Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A
^{*}. arXiv**2022**, arXiv:2205.07787. [Google Scholar] - Falcke, H.; Melia, F.; Agol, E. Viewing the shadow of the black hole at the galactic center. Astrophys. J. Lett.
**2000**, 528, L13. [Google Scholar] [CrossRef] [Green Version] - Psaltis, D. Testing General Metric Theories of Gravity with Bursting Neutron Stars. Phys. Rev. D
**2008**, 77, 064006. [Google Scholar] [CrossRef] [Green Version] - Antoniou, G.; Bakopoulos, A.; Kanti, P.; Kleihaus, B.; Kunz, J. Novel Einstein–scalar-Gauss-Bonnet wormholes without exotic matter. Phys. Rev. D
**2020**, 101, 024033. [Google Scholar] [CrossRef] [Green Version] - Perlick, V.; Tsupko, O.Y. Calculating black hole shadows: Review of analytical studies. Phys. Rept.
**2022**, 947, 1–39. [Google Scholar] [CrossRef] - Abuter, R.; Amorim, A.; Anugu, N.; Bauböck, M.; Benisty, M.; Berger, J.P.; Blind, N.; Bonnet, H.; Brandner, W.; Buron, A.; et al. Detection of the gravitational redshift in the orbit of the star S2 near the Galactic centre massive black hole. Astron. Astrophys.
**2018**, 615, L15. [Google Scholar] [CrossRef] [Green Version] - Amorim, A.; Bauböck, M.; Berger, J.P.; Brandner, W.; Clénet, Y.; Du Foresto, V.C.; de Zeeuw, P.T.; Dexter, J.; Duvert, G.; Ebert, M.; et al. Test of the Einstein Equivalence Principle near the Galactic Center Supermassive Black Hole. Phys. Rev. Lett.
**2019**, 122, 101102. [Google Scholar] [CrossRef] [Green Version] - Abuter, R.; Amorim, A.; Bauböck, M.; Berger, J.P.; Bonnet, H.; Brandner, W.; Cardoso, V.; Clénet, Y.; De Zeeuw, P.T.; Dexter, J.; et al. Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole. Astron. Astrophys.
**2020**, 636, L5. [Google Scholar] [CrossRef] [Green Version] - Do, T.; Hees, A.; Ghez, A.; Martinez, G.D.; Chu, D.S.; Jia, S.; Sakai, S.; Lu, J.R.; Gautam, A.K.; O’neil, K.K.; et al. Relativistic redshift of the star S0-2 orbiting the Galactic center supermassive black hole. Science
**2019**, 365, 664–668. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hees, A.; Dehghanfar, A.; Do, T.; Ghez, A.M.; Martinez, G.D.; Campbell, R.; Lu, J.R. An adaptive scheduling tool to optimize measurements to reach a scientific objective: Methodology and application to the measurements of stellar orbits in the Galactic Center. Astrophys. J.
**2019**, 880, 87. [Google Scholar] [CrossRef] - Della Monica, R.; de Martino, I.; de Laurentis, M. Orbital precession of the S2 star in Scalar–Tensor–Vector Gravity. Mon. Not. Roy. Astron. Soc.
**2022**, 510, 4757–4766. [Google Scholar] [CrossRef] - de Martino, I.; della Monica, R.; de Laurentis, M. f(R) gravity after the detection of the orbital precession of the S2 star around the Galactic Center massive black hole. Phys. Rev. D
**2021**, 104, L101502. [Google Scholar] [CrossRef] - Fragione, G.; Loeb, A. Implication of Spin Constraints by the Event Horizon Telescope on Stellar Orbits in the Galactic Center. Astrophys. J. Lett.
**2022**, 932, L17. [Google Scholar] [CrossRef] - Lee, B.H.; Lee, W.; Ro, D. Expanded evasion of the black hole no-hair theorem in dilatonic Einstein-Gauss-Bonnet theory. Phys. Rev. D
**2019**, 99, 024002. [Google Scholar] [CrossRef] [Green Version] - Papageorgiou, A.; Park, C.; Park, M. Rectifying no-hair theorems in Gauss-Bonnet theory. Phys. Rev. D
**2022**, 106, 084024. [Google Scholar] [CrossRef] - Arnowitt, R.L.; Deser, S.; Misner, C.W. The Dynamics of general relativity. Gen. Rel. Grav.
**2008**, 40, 1997–2027. [Google Scholar] [CrossRef] [Green Version] - Popova, A.D.; Petrov, A.N. Exact Dynamic Theories on a Given Background in Gravitation. Int. J. Mod. Phys. A
**1988**, 3, 2651–2679. [Google Scholar] [CrossRef] - Petrov, A.N.; Lompay, R.R. Covariantized Noether identities and conservation laws for perturbations in metric theories of gravity. Gen. Rel. Grav.
**2013**, 45, 545–579. [Google Scholar] [CrossRef] [Green Version] - Alexeyev, S.O.; Petrov, A.N.; Latosh, B.N. Maeda-Dadhich Solutions as Real Black Holes. Phys. Rev. D
**2015**, 92, 104046. [Google Scholar] [CrossRef] [Green Version] - Perkins, S.E.; Nair, R.; Silva, H.O.; Yunes, N. Improved gravitational-wave constraints on higher-order curvature theories of gravity. Phys. Rev. D
**2021**, 104, 024060. [Google Scholar] [CrossRef] - Gross, D.J.; Sloan, J.H. The Quartic Effective Action for the Heterotic String. Nucl. Phys. B
**1987**, 291, 41–89. [Google Scholar] [CrossRef] - Blázquez-Salcedo, J.L.; Doneva, D.D.; Kunz, J.; Yazadjiev, S.S. Radial perturbations of the scalarized Einstein-Gauss-Bonnet black holes. Phys. Rev. D
**2018**, 98, 084011. [Google Scholar] [CrossRef] [Green Version] - Antoniou, G.; Macedo, C.F.B.; McManus, R.; Sotiriou, T.P. Stable spontaneously-scalarized black holes in generalized scalar-tensor theories. Phys. Rev. D
**2022**, 106, 024029. [Google Scholar] [CrossRef] - Andreou, N.; Franchini, N.; Ventagli, G.; Sotiriou, T.P. Spontaneous scalarization in generalised scalar-tensor theory. Phys. Rev. D
**2019**, 99, 124022. [Google Scholar] [CrossRef] [Green Version] - Silva, H.O.; Macedo, C.F.B.; Sotiriou, T.P.; Gualtieri, L.; Sakstein, J.; Berti, E. Stability of scalarized black hole solutions in scalar-Gauss-Bonnet gravity. Phys. Rev. D
**2019**, 99, 064011. [Google Scholar] [CrossRef] [Green Version] - Ventagli, G.; Lehébel, A.; Sotiriou, T.P. Onset of spontaneous scalarization in generalized scalar-tensor theories. Phys. Rev. D
**2020**, 102, 024050. [Google Scholar] [CrossRef] - Antoniou, G.; Bordin, L.; Sotiriou, T.P. Compact object scalarization with general relativity as a cosmic attractor. Phys. Rev. D
**2021**, 103, 024012. [Google Scholar] [CrossRef] - Ventagli, G.; Antoniou, G.; Lehébel, A.; Sotiriou, T.P. Neutron star scalarization with Gauss-Bonnet and Ricci scalar couplings. Phys. Rev. D
**2021**, 104, 124078. [Google Scholar] [CrossRef] - Antoniou, G.; Lehébel, A.; Ventagli, G.; Sotiriou, T.P. Black hole scalarization with Gauss-Bonnet and Ricci scalar couplings. Phys. Rev. D
**2021**, 104, 044002. [Google Scholar] [CrossRef] - Herdeiro, C.A.R.; Radu, E.; Sanchis-Gual, N.; Font, J.A. Spontaneous Scalarization of Charged Black Holes. Phys. Rev. Lett.
**2018**, 121, 101102. [Google Scholar] [CrossRef] [Green Version] - Fernandes, P.G.S.; Herdeiro, C.A.R.; Pombo, A.M.; Radu, E.; Sanchis-Gual, N. Spontaneous Scalarisation of Charged Black Holes: Coupling Dependence and Dynamical Features. Class. Quant. Grav.
**2019**, 36, 134002. [Google Scholar] [CrossRef] [Green Version] - Blázquez-Salcedo, J.L.; Herdeiro, C.A.R.; Kunz, J.; Pombo, A.M.; Radu, E. Einstein-Maxwell-scalar black holes: The hot, the cold and the bald. Phys. Lett. B
**2020**, 806, 135493. [Google Scholar] [CrossRef] - Zajaček, M.; Tursunov, A.; Eckart, A.; Britzen, S. On the charge of the Galactic centre black hole. Mon. Not. Roy. Astron. Soc.
**2018**, 480, 4408–4423. [Google Scholar] [CrossRef] [Green Version] - Zajaček, M.; Tursunov, A.; Eckart, A.; Britzen, S.; Hackmann, E.; Karas, V.; Stuchlík, Z.; Czerny, B.; Zensus, J.A. Constraining the charge of the Galactic centre black hole. J. Phys. Conf. Ser.
**2019**, 1258, 012031. [Google Scholar] [CrossRef] - Blackburn, L.; Doeleman, S.; Dexter, J.; Gómez, J.L.; Johnson, M.D.; Palumbo, D.C.; Weintroub, J.; Bouman, K.L.; Chael, A.A.; Zhao, G.; et al. Studying Black Holes on Horizon Scales with VLBI Ground Arrays. arXiv
**2019**, arXiv:1909.01411. [Google Scholar] - Doeleman, S. Black Hole Imaging: First Results and Future Vision. In Proceedings of the American Astronomical Society Meeting Abstracts, virtual. 7–9 June 2021; Volume 53, p. 221.01. [Google Scholar]
- Misner, C.W.; Sharp, D.H. Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse. Phys. Rev.
**1964**, 136, B571–B576. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Qualitative representation of a light ray reaching an observer at an angle $\alpha $, located at distance ${r}_{\mathrm{obs}}$ from the point singularity. The blue line traces a light ray escaping from a closed orbit around the black hole to infinity. The red line aligns with the inferred angle of approach for the light ray to an asymptotic observer. The point of closest approach for the light ray with respect to the black hole is located at $r={r}_{0}$. If ${r}_{0}={r}_{\mathrm{sh}}$ the light ray escapes the photon sphere. The shaded, circular area denotes the interior of the black-hole horizon, while the dashed, circular line corresponds to the location of the photon sphere. (

**b**) Same but for a wormhole geometry. Here we show the embedding diagram depicting a finite radius throat along the vertical axis. The blue line traces a light ray escaping from the photon sphere to infinity, while the red straight line corresponds to the inferred line of approach to an asymptotic observer.

**Figure 2.**Shadow radius for EsGB theory with linear coupling. The blue line scans the full range of values of $\beta $ defined in (28) with the left and right endpoints corresponding to $\beta =-1$ and $\beta =1$, respectively. The horizontal solid and dashed lines denote the EHT 1-$\sigma $ and 2-$\sigma $ allowed ranges, respectively; the blue lines correspond to the mG-ring bound and the red lines to the eht-imaging bound.

**Figure 3.**Shadow radius for EsGB theory with a quadratic coupling. Each colorful line scans the full range of parameter $\beta $ for a different fixed value of ${\varphi}_{h}$. The endpoint of the lines in the negative and positive regime of the horizontal axis correspond to $\beta =-1$ and $\beta =1$, respectively. The red dots denote the point in the parameter space at which condition (30) is satisfied. The horizontal solid and dashed lines denote the EHT bounds as before.

**Figure 4.**Shadow radius for EsGB theory with a dilatonic coupling with $\gamma =1$. The colored lines have the same meaning as in Figure 3, while the horizontal solid and dashed lines denote the EHT bounds as before.

**Figure 5.**Shadow radius for EsGB theory with a dilatonic coupling with $\gamma =2$. The colored lines have the same meaning as in Figure 3, while the horizontal solid and dashed lines denote the EHT bounds as before.

**Figure 6.**Wormhole solutions in EsGB theory with coupling function $f\left(\varphi \right)=\alpha {e}^{-\varphi}$, for ${f}_{0}=\{1,1.25,1.5,2,3\}$.

**Figure 7.**(

**a**) Shadow radius of the fundamental mode $(n=0)$ for spontaneously scalarized black holes in the EsRGB theory with quadratic couplings between the scalar field and curvature. The values of $\varphi -R$ coupling for the lines plotted are $\beta =0,5,10,50,100$. At the same time, the $\varphi -\mathcal{G}$ coupling spans all the allowed values for which spontaneously scalarized solutions are retrieved. (

**b**) Same as left panel but for the first overtone $n=1$. The $\beta =100$ case is not presented here for illustrative purposes as it extends to values of $\widehat{M}$ that are much smaller than the rest.

**Figure 8.**Shadow radius of the fundamental modes $(n=0)$ for spontaneously scalarized black holes in EsGB theory with a quartic $\varphi -GB$ coupling, for different ratios $\alpha /\zeta =\{0,-1,-2,-10\}$.

**Figure 9.**(

**a**) Onset of scalarization for different overtone numbers. The threshold does not depend on the coupling function. (

**b**) Shadow radius for the fundamental mode for spontaneously scalarized EMS black holes with an exponential coupling function $f\left(\varphi \right)=-{e}^{-\alpha {\varphi}^{2}}$, for an s-EM coupling with values $\alpha =\{-5,-10,-20\}$. The solid line corresponds to the GR limit (RN). (

**c**) Same as top right but for a quadratic coupling function $f\left(\varphi \right)=\alpha {\varphi}^{2}-1$. (

**d**) Same as top right but for a hyperbolic coupling function of the form $f\left(\varphi \right)=-cosh\left(\sqrt{-2\alpha}\varphi \right)$.

**Table 1.**Sagittarius A* bounds on the deviation parameter $\delta $. The colored bounds are the ones we use in the plots in the main part.

Sgr ${\mathrm{A}}^{*}$ Estimates | ||||
---|---|---|---|---|

Deviation $\mathbf{\delta}$ | 1-$\mathbf{\sigma}$ Bounds | 2-$\mathbf{\sigma}$ Bounds | ||

eht-img | VLTI | $-0.{08}_{-0.09}^{+0.09}$ | $4.31\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.25$ | $3.85\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.72$ |

Keck | $-0.{04}_{-0.10}^{+0.09}$ | $4.47\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.46$ | $3.95\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.92$ | |

${\mathrm{Avg}}$ | $-0.{06}_{-0.067}^{+0.064}$ | ${4.54}{\le}{\displaystyle {\frac{{{r}}_{{\mathrm{sh}}}}{{M}}}}{\le}{5.22}$ | ${4.19}{\le}{\displaystyle {\frac{{{r}}_{{\mathrm{sh}}}}{{M}}}}{\le}{5.55}$ | |

SMILI | VLTI | $-0.{10}_{-0.10}^{+0.12}$ | $4.16\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.30$ | $3.64\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.92$ |

Keck | $-0.{06}_{-0.10}^{+0.13}$ | $4.36\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.56$ | $3.85\le \frac{{r}_{\mathrm{sh}}}{M}\le 6.24$ | |

DIFMAP | VLTI | $-0.{12}_{-0.08}^{+0.10}$ | $4.16\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.09$ | $3.74\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.61$ |

Keck | $-0.{08}_{-0.09}^{+0.09}$ | $4.31\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.25$ | $3.85\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.72$ | |

mG-ring | VLTI | $-0.{17}_{-0.10}^{+0.11}$ | $3.79\le \frac{{r}_{\mathrm{sh}}}{M}\le 4.88$ | $3.27\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.46$ |

Keck | $-0.{13}_{-0.11}^{+0.11}$ | $3.95\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.09$ | $3.38\le \frac{{r}_{\mathrm{sh}}}{M}\le 5.66$ | |

${\mathrm{Avg}}$ | $-0.{15}_{-0.074}^{+0.078}$ | ${4.03}{\le}{\displaystyle {\frac{{{r}}_{{\mathrm{sh}}}}{{M}}}}{\le}{4.82}$ | ${3.64}{\le}{\displaystyle {\frac{{{r}}_{{\mathrm{sh}}}}{{M}}}}{\le}{5.23}$ |

M${87}^{*}$ Estimates | |||
---|---|---|---|

Deviation $\mathbf{\delta}$ | 1-$\mathbf{\sigma}$ Bounds | 2-$\sigma $ Bounds | |

EHT | $-0.{01}_{-0.17}^{+0.17}$ | $4.26\le \frac{{r}_{\mathrm{sh}}}{M}\le 6.03$ | $3.38\le \frac{{r}_{\mathrm{sh}}}{M}\le 6.91$ |

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**MDPI and ACS Style**

Antoniou, G.; Papageorgiou, A.; Kanti, P.
Probing Modified Gravity Theories with Scalar Fields Using Black-Hole Images. *Universe* **2023**, *9*, 147.
https://doi.org/10.3390/universe9030147

**AMA Style**

Antoniou G, Papageorgiou A, Kanti P.
Probing Modified Gravity Theories with Scalar Fields Using Black-Hole Images. *Universe*. 2023; 9(3):147.
https://doi.org/10.3390/universe9030147

**Chicago/Turabian Style**

Antoniou, Georgios, Alexandros Papageorgiou, and Panagiota Kanti.
2023. "Probing Modified Gravity Theories with Scalar Fields Using Black-Hole Images" *Universe* 9, no. 3: 147.
https://doi.org/10.3390/universe9030147