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

TL;DR

This paper linearizes the Starobinsky $f(R)$ gravity model to analyze deviations from General Relativity, deriving equations for perturbations and numerically evaluating their effects near binary stars, showing how modifications diminish with distance and increasing $m$.

Contribution

It provides a detailed perturbative analysis of the Starobinsky model, deriving explicit equations and numerical results for gravitational deviations near compact objects.

Findings

Perturbative corrections to gravity decrease with distance from stars.

The deviation diminishes as the parameter $m$ increases.

Results recover General Relativity in the limit of large $m$.

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.