Limiting the Yukawa Gravity through the Black Hole Shadows of Sgr A* and M87*

Abstract: Recently, the \textit{EHT} collaboration unveiled the shadow images of the supermassive black hole (SMBH) M87* and Sgr A*, with angular radii of $42\pm3$\,$\mu$as and $48.7\pm7.0$\,$\mu$as, respectively. These observations are consistent with the shadow of a Kerr black hole in general relativity (GR). Observations of the shadow of SMBHs can be used to test modified gravity theories, including Yukawa gravity, in extremely strong fields. In this paper, we illustrate the shadows of Yukawa black holes, showing that their sizes are significantly influenced by the Yukawa parameters $\lambda$ and $\kappa$. Using the EHT observations of M87* and Sgr A*, we obtain constraints on the Yukawa parameters. For Sgr A*, Keck and VLTI provide different priors on its gravitational radius. The Sgr A* shadow yields $\kappa=-0.04<sup>{+0.09}_{-0.10}$ for $\lambda&gt;1$\,AU with the Keck prior, while $\kappa=-0.08<sup>{+0.09}_{-0.06}$ with the VLTI prior. As $\lambda$ decreases, the constraints weaken, reaching $-0.37&lt;\kappa &lt;0.17$ (Keck prior) and $-0.47&lt;\kappa&lt;0.04$ (VLTI prior) at $\lambda=0.1$\,AU. For M87*, with a mass significantly larger than Sgr A*, this system can only put constraints on $\kappa$ at larger $\lambda$. For $\lambda&gt;1.5\times10<sup>4$\,AU, the \textit{EHT} observation of M87* yields $\kappa=-0.01<sup>{+0.17}_{-0.17}$. No significant deviation from GR is detected in our analysis. Additionally, we explore potential constraints using the next-generation VLBI, like \textit{ngEHT} and the Black Hole Explorer (BHEX), which promise the detection of the second ring of photons. The improved angular resolution and the measurements of the second ring could substantially refine constraints on the Yukawa parameters, enhancing our ability to test deviations from GR in the strong-field regime.