Unified $f(R)$ gravity at local scales

We explore the shifted $f(R) (\propto R^{1+\delta})$ model with ${\delta}$ as a distinguishing physical parameter for the study of constraints at local scales. The corresponding dynamics confronted with different geodesics (null and non-null) along with its conformal analogue is investigated. For null geodesics, we discuss the light deflection angle, whereas for non-null geodesics under the weak field limit, we investigate the perihelion advance of the Mercury orbit in $f(R)$ Schwarzschild background, respectively. The extent of an additional force, appearing for non-null geodesics, depends on $\delta$. Such phenomenological investigations allow us to strictly constrain $\delta$ to be approximately $\mathcal{O}(10^{-6})$ with a difference of unity in orders at galactic and planetary scales and seems to provide a unique $f(R)$ at local scales. Further, at late cosmic time, we analyse the constraint on $\delta$ via the bare scalar self-interaction Einstein frame potential to provide a null test of dark energy. We constrain the deviation parameter, $\mid\delta\mid$ to $(\approx 0.6)$ which is in a close agreement with the results obtained through various observations in the Jordan frame by several authors. Our results suggest that the present form of model is suitable for the alternate explanation of dark matter-like effects at local scales, whereas at large scales the deviations grow higher and must be addressed in terms of the accelerated background.