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

In this work, we linearize the field equations of $f(R)$ gravity using the Starobinsky model, R+R²/(6m²), 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, h, governed by the Klein-Gordon (KG) equation. The fields were expressed in terms of Green's functions, whose symmetry properties facilitated the solution of the trace equation. Then hμν 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 T₀₀, plus a perturbative correction proportional to m⁻², involving the Laplacian of the integral of T, weighted by the retarded propagator of the KG equation. As an illustrative example, 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, and how the perturbation decreases as m increases, consistently recovering the relativistic limit. Finally we computed the quadrupole components I₁₁, I₂₂, and I₃₃ for $m=1$ in the modified gravity model. The results show the same sinusoidal-squared structure as in General Relativity, with I₁₁ and I₂₂ having equal but larger amplitudes, and I₃₃ being negligible. We also numerically demonstrated that increasing m reduces the support of the Bessel-type function J₁ modulated by a Heaviside factor, which governs the propagation close to the light cone, a physically expected effect. These results highlight the role of modified gravity corrections in the vicinity of compact objects.