$f(R)$-gravity after the detection of the orbital precession of the S2 star around the Galactic centre massive black hole

The GRAVITY Collaboration achieved the remarkable detection of the orbital precession of the S2 star around the Galactic centre supermassive black hole, providing yet another proof of the validity of the General Relativity. The departure from the Schwarzschild precession is encoded in the parameter $f_{\rm SP}$ which multiplies the predicted general relativistic precession. Such a parameter results to be $f_{\rm SP}=1.10\pm0.19$, which is consistent with General Relativity ($f_{\rm SP}=1$) at 1$\sigma$ level. Nevertheless, this parameter may also hide an effect of modified theories of gravity. Thus, we consider the orbital precession due to the Yukawa-like gravitational potential arising in the weak field limit of $f(R)$-gravity, and we use the current bound on the $f_{SP}$ to constrain the strength and the scale length of the Yukawa-like potential. No deviation from GR are revealed at scale of $\lambda<6592$ AU with the strength of the Yukawa potential restricted to $\delta = -0.02^{+0.26}_{-0.13}$. Our constraint rules out $f(R)$ models whose second derivatives is substantially different from zero. Nevertheless, models having higher order derivatives different from zero are still possible and deserve further studies.