Unscreening of f(R) gravity near the galactic center black hole: Testability through pericenter shift below S0-2's orbit

General Relativity (GR) has been tested extensively in the solar system and is being tested in the new environment of the Galactic Centre (GC) black hole where the dimensionless gravitational potential ($GM/c^2r$) is 100 times stronger than the one encountered in solar system. Therefore, the neighbourhood of the GC black hole is a naive opportunity to test modified theories of gravity. In this work, effect of $f(R)$ gravity near the black hole is studied. The difference of pericentre shift between GR and $f(R)$ gravity is studied for compact orbits having semi-major axis equal to and below $a=1000$ au (S0-2 like orbits). In a model dependent approach, we choose $f(R) \propto R^2$ (power law gravity) model which is cosmologically motivated and study the deviation in orbital pericentre shift for both zero spin and non-zero spin of the black hole. It is found that effect of $f(R)$ gravity becomes prominent for compact orbits. In model independent approach to $f(R)$ gravity with the generic scalaron fields ($ψ=f’(R)$), we extract the parameters of $f(R)$ gravity from the current bounds on Parametrised Post Newtonian (PPN) parameters ($γ, β$) near the GC black hole. The screening of $f(R)$ gravity is also investigated for these bounds on PPN parameters. It has been found that sufficiently massive scalarons ($10^{-16}$ eV) are completely screened but light and intermediate mass scalarons ($10^{-22}$ eV and $10^{-19}$ eV) are unscreened towards S0-2 like orbits as well as in the orbit of the newly discovered short period star S4716 ($a=407$ au). The possibility of detection of the $f(R)$ gravity effects due to these unscreened scalarons is forecasted with existing and upcoming astrometric capabilities of Extremely Large Telescopes (ELTs).

💡 Research Summary

The paper investigates the viability of testing f(R) modified gravity in the strong‑field environment of the Galactic Centre (GC) super‑massive black hole (Sgr A*), focusing on the pericentre (peri‑apsis) shift of short‑period stars whose semi‑major axes are ≤ 1000 au (the regime of the well‑studied S0‑2 star and the newly discovered S4716). Two complementary strategies are employed.

Model‑dependent analysis (R² gravity).
The authors adopt the Starobinsky‑type quadratic correction f(R)=αR² (n = 2) which, in the weak‑field limit, modifies the Newtonian potential by a term proportional to (r/r_c)^δ with δ≈0.666… and a characteristic scale r_c (chosen as 100 au and 1000 au). By subtracting the Newtonian part they obtain a perturbing potential V(r) and compute the resulting pericentre shift using the integral formalism of Adkins & McDonnell (2007). The analytic result involves a hyper‑geometric function (Eq. 9). The shift for a Schwarzschild black hole (Eq. 13) and for a Kerr black hole (Eqs. 15‑16) is then compared to the f(R) prediction, yielding the differences shown in Figures 1 and 2. The key finding is that the deviation grows with semi‑major axis but only becomes appreciable (≥10⁻⁴ rad) for very compact orbits (a of a few tens of au). For wider orbits the f(R) correction is suppressed and essentially indistinguishable from General Relativity (GR).

Model‑independent analysis (scalaron/Yukawa description).
In a generic f(R) theory the extra scalar degree of freedom ψ = df/dR gives rise to a Yukawa‑type correction to the Newtonian potential: −(GM/r)