# 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. |

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**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