Probing hybrid modified gravity by stellar motion around Galactic Centre

We consider possible signatures for the so called {\it hybrid gravity} within the Galactic Central Parsec. This modified theory of gravity consists of a superposition of the metric Einstein-Hilbert Lagrangian with an $f(R)$ term constructed {\it \`{a} la Palatini} and can be easily reduced to an equivalent scalar-tensor theory. Such an approach is introduced in order to cure the shortcomings related to $f(R)$ gravity, in general formulated either in metric or in metric-affine frameworks. Hybrid gravity allows to disentangle the further gravitational degrees of freedom with respect to those of standard General Relativity. The present analysis is based on the S2 star orbital precession around the massive compact dark object at the Galactic Centre where the simulated orbits in hybrid modified gravity are compared with astronomical observations. These simulations result with constraints on the range of hybrid gravity interaction parameter $\phi_0$, showing that in the case of S2 star it is between -0.0009 and -0.0002. At the same time, we are also able to obtain the constraints on the effective mass parameter $m_{\phi}$, and found that it is between -0.0034 and -0.0025 AU$^{-1}$ for S2 star. Furthermore, the hybrid gravity potential induces precession of S2 star orbit in the same direction as General Relativity. In previous papers, we considered other types of extended gravities, like metric power law $f(R)\propto R^n$ gravity, inducing Yukawa and Sanders-like gravitational potentials, but it seems that hybrid gravity is the best among these models to explain different gravitational phenomena at different astronomical scales.