The effects of asymptotically flat $R^2$ spacetime on black hole image of Sagittarius A*

Abstract: A new class of analytically expressible vacuum solutions has recently been discovered for pure ${R}^2$ gravity, building upon Buchdahl's seminal work from 1962. These solutions, inspired by Buchdahl's framework, offer a promising avenue for testing ${R}^2$ gravity against astrophysical observations. Within a subset of asymptotically flat Buchdahl-inspired vacuum spacetimes, we introduce a free parameter $\epsilon$ to characterize deviations from the Schwarzschild metric, which is recovered in the limit $\epsilon = 0$. In this study, we employ the publicly available code \textit{ipole} to simulate black hole images under the Buchdahl-inspired metric, with a focus on the black hole at the center of the Milky Way, Sagittarius A* (Sgr A*). Our simulations show that both the shadow size and photon ring diameter decrease monotonically with increasing $\epsilon$. By exploring a range of observational inclination angles, we find that the photon ring diameter being a direct observable is only weakly sensitive to the inclination angle. We further constrain the parameter $\epsilon$ by comparing our simulation results with the Event Horizon Telescope (EHT) observations of Sgr A*. The obtained bounds are consistent with those previously derived from the orbital motion of the S2 star, but provide tighter constraints. In addition, we analyze the influence of the Buchdahl-inspired spacetime on the polarization patterns near the black hole and find its impact to be minimal. In contrast, the observational inclination angle has a substantial effect on the observed polarization structure, highlighting the dominant role of viewing geometry in shaping polarization features.