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, , and examine the modifications to general relativity (GR). 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, , 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 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 GR, 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.