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

TL;DR

This work analyzes linearized gravity in the Starobinsky $f(R)$ model, $f(R)=R+\frac{1}{6m^2}R^2$, in the weak-field limit to quantify deviations from GR. It introduces an auxiliary field and uses Green's functions to solve the trace equation as a two-stage Klein-Gordon-like system, yielding a trace $\bar{h}$ with a massive propagation kernel $G_2$ and a massless kernel $G_1$. The tensor perturbation $\bar{h}_{\mu\nu}$ is then obtained from an effective stress-energy $\mathcal{T}_{\mu\nu}$, leading to a perturbative correction to the energy density and a modified quadrupole moment. The authors perform a numerical study in a binary star system, finding that the modification decays with $m$ and distance, recovering GR in the $m\to\infty$ limit, and highlighting finite-range effects near compact objects.

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 $\bar h_{μν}$ 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.

Figures (4)

• Figure 1: Numerical evaluation of $\int T\left(z^{\sigma}\right) G_2\left(x^{\sigma}-z^{\sigma}\right)d^4z$, in $L^{-2}$, as a function of $m$, in $L^{-1}$, for different values of $x^1$ and $x^0=x^2=x^3=0$. The curves, from top to bottom, correspond to $x^1=3R, 4R, 5R$, and $6R$. As $x^1$ increases, the amplitude of the integral decreases, and the curves decay to zero more rapidly (i.e., at lower values of $m$).
• Figure 2: Correction term, Eq. (\ref{['correction']}), in units of $L^{-2}$ as a function of $m$ for different values of $x^1$ (bottom to top: $2R$ [continuous line], $3R$, $4R$, $5R$, $6R$), obtained using a global adaptive integration method, and with $x^0=x^2=x^3=0$. Each curve starts at a negative value and increases smoothly toward zero, exhibiting a concave-down shape. As $x^1$ increases, the amplitude of the correction term (i.e., its value at $m\to0$) decreases, tending to zero.
• Figure 3: Spatial dependence of the correction term as a function of $x^1$, in units of $L$, for $m=0$ and $m=50$, and for $x^0=x^2=x^3=0$. The curves reflect the suppression of modified gravity effects at larger distances. The units of the vertical axis are $L^{-2}$.
• Figure 4: Perturbation ($L^{-2}$) as a function of $x^1$ ($L$) for different values of $x^3$ (shown in the inset box, corresponding to each curve from bottom to top), plotted in the limit $m\to 0$, and for $x^0=x^2=0$. All curves tend to zero, consistent with the expected suppression of perturbations far from the source.