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

TL;DR

This study uses EHT-shadow observations of M87* and Sgr A* to constrain a Yukawa-type modification of gravity, parameterized by κ and λ in Verlinde’s emergent gravity framework. By computing the black hole shadow via the critical curve and the second photon ring, and by modeling the accretion flow with a MAD-like synchrotron emission, the authors translate angular shadow measurements into bounds on κ(λ). Across the two sources, no significant deviation from general relativity is found, with tighter constraints at larger λ (e.g., κ≈-0.04 to -0.08 for λ>1 AU in Sgr A*). Looking ahead, next-generation VLBI (ngEHT, BHEX) and direct second-ring imaging could substantially tighten these limits, reducing allowed Yukawa couplings and providing a more stringent test of strong-field gravity.

Abstract

Recently, the \textit{EHT} collaboration unveiled the shadow images of the supermassive black hole (SMBH) M87* and Sgr A*, with angular radii of $42\pm3$\,$μ$as and $48.7\pm7.0$\,$μ$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 $λ$ and $κ$. 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 $κ=-0.04^{+0.09}_{-0.10}$ for $λ>1$\,AU with the Keck prior, while $κ=-0.08^{+0.09}_{-0.06}$ with the VLTI prior. As $λ$ decreases, the constraints weaken, reaching $-0.37<κ<0.17$ (Keck prior) and $-0.47<κ<0.04$ (VLTI prior) at $λ=0.1$\,AU. For M87*, with a mass significantly larger than Sgr A*, this system can only put constraints on $κ$ at larger $λ$. For $λ>1.5\times10^4$\,AU, the \textit{EHT} observation of M87* yields $κ=-0.01^{+0.17}_{-0.17}$. 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.

Figures (8)

• Figure 1: The gravitational potential probed by different tests against the mass of the central body that generates gravity in these tests. The black hole shadow observations explore a new region in this parameter space. The figure is inspired by 2004AIPC..714...29P and Hees2017PRL.
• Figure 2: The total number of orbits, $n\equiv\phi/2\pi$, as a function of the impact parameter $b$ for varying Yukawa parameters $\lambda$ and $\kappa$. The gray dashed line represents the critical value $n=3/4$ and $n=5/4$. Those photons with the orbits numbers $3/4<n<5/4$ constitute the second photon ring.
• Figure 3: The intensity profiles of the radio emission resulting from the Yukawa metric with different parameters $\lambda$ and $\kappa$. The solid part of each curve represents the intensity from the second photon ring which is composed of photons who satisfy $3/4<n<5/4$, while the filled dots represent the peaks of the second photon rings. Left and right columns show the cases with $\kappa=-0.5$ and $0.5$, respectively. Top to bottom panels show the cases with $\lambda=1$, $2$, $10$, and $100$, respectively.
• Figure 4: Constraints on $\kappa$ for varying $\lambda$ from EHT Sgr A* results of Schwarzschild shadow deviation $\delta$. Based on Keck and VLTI priors, the EHT gives two different measurements for $\delta$: $\delta = -0.04^{+0.09}_{-0.10}$ for Keck prior and $\delta = -0.08 ^{+0.09}_{-0.09}$ for VLTI prior. The blue lines correspond to $\delta = -0.04$ for the Keck prior and $\delta=-0.08$ for the VLTI prior, while the red lines show the upper and lower bounds for $\kappa$. For comparison, the constraints obtained from the motion of S-stars are shown by the green dashed lines 2024PhRvD.109d4047T.
• Figure 5: Legends are similar to that for Figure \ref{['constrain_SgrA*']}, except for M87* with $\delta=-0.01\pm 0.17$.