Observational test of ${\cal R}^{2}$ spacetimes with the S2 star in the Milky Way galactic center

A novel class of vacuum metrics expressible in analytical form was recently found for pure $\mathcal R^2$ gravity, based on a groundwork put forth by Buchdahl in 1962. These Buchdahl-inspired solutions offer a practical framework for testing ${\cal R}^2$ gravity through empirical observations. Within a subclass of asymptotically flat Buchdahl-inspired vacuum spacetimes, we identified a parameter $\epsilon$ measuring the deviation from the classic Schwarzschild metric, which corresponds to $\epsilon=0$. In this paper, we employ observational data from the S2 star's orbit around Sgr A* in the Milky Way galactic center and perform Monte Carlo Markov Chain simulations to probe the effects of the new metrics on the orbit of the S2 star. Our analysis presented herein reports a range at 95\% confidence level on the deviation parameter as $\epsilon\in(-0.6690,\ 0.4452)$. While no decisive evidence either in favor or in disfavor of the asymptotically flat Buchdahl-inspired spacetimes has been achieved, the obtained bound is compatible with the tighter results using other data of different nature as recently reported in Eur.\,Phys.\,J.\,C $\bf 84$, 330 (2024). As a meaningful test probing into a strong-field regime, our present study calls for further observations with prolonged period and improved accuracy in order to tighten the bound for $\epsilon$ using the S2 star orbit.