Linearized Gravity in the Starobinsky Model: Perturbative Deviations from General Relativity

Abstract: In this work, we linearize the field equations of $f(R)$ gravity using the Starobinsky model, $R+R^2/(6m^2)$, and examine the modifications to General Relativity. We derive an equation for the trace, $T$, of the energy-momentum tensor, which we then decompose using an auxiliary field. This field satisfies the wave equation with $T$ as its source, while simultaneously acting as an effective source for the classical deviation, $\bar h$, governed by the Klein-Gordon equation. The fields were expressed in terms of Green's functions, whose symmetry properties facilitated the solution of the trace equation. Then $\bar h_{\mu\nu}$ was determined in terms of a modified or effective matter-energy distribution. From this, the effective energy density was obtained as the usual energy density $T_{00}$, plus a perturbative correction proportional to $m^{-2}$, involving the Laplacian of the integral of $T$, weighted by the retarded propagator of the Klein-Gordon equation. Finally, we numerically computed the perturbative term in a binary star system, evaluating it as a function of $m$ and spatial position near the stars. In all cases, the results illustrate how the gravitational influence of the stars diminishes with distance. Additionally, the perturbation decreases as $m$ increases, consistently recovering the relativistic limit. These results highlight the role of modified gravity corrections in the vicinity of compact objects.