Emerging Newtonian potential in pure $R^2$ gravity on a de Sitter background

Abstract

In Fortsch. Phys. $\textbf{64}$, 176 (2016), Alvarez-Gaume et al established that pure $R^2$ theory propagates $\textit{massless}$ spin-2 graviton on a de Sitter (dS) background but $\textit{not}$ on a locally flat background. We build on this insight to derive a Newtonian limit for the theory. Unlike most previous works that linearized the metric around a locally flat background, we explicitly employ the dS background to start with. We directly solve the field equation of the action $(2\kappa)^{-1}\int d^{4}x\sqrt{-g}\,R^2$ coupled with the stress-energy tensor of normal matter in the form $T_{\mu\nu}=Mc^2\,\delta(\vec{r})\,\delta_\mu^0\,\delta_\nu^0$. We obtain the following Schwarzschild-de Sitter metric $ds^{2}=-\Bigl(1-\frac{\Lambda}{3}r^{2}-\frac{\kappa c^{2}}{48\pi\Lambda}\frac{M}{r}\Bigr)c^2dt^2+\Bigl(1-\frac{\Lambda}{3}r^2-\frac{\kappa c^2}{48\pi\Lambda}\frac{M}{r}\Bigr)^{-1}dr^2+r^2d\Omega^2$ which features a potential $V(r)=-\frac{\kappa c^4}{96\pi\Lambda}\frac{M}{r}$ with the correct Newtonian tail. The parameter $\Lambda$ plays a dual role: (i) it sets the scalar curvature for the background dS metric, and (ii) it partakes in the Newtonian potential $V(r)$. We reach two key findings. Firstly, the Newtonian limit only emerges owing to the de Sitter background. Most existing studies of the Newtonian limit in modified gravity chose to linearize the metric around a locally flat background. However, this is a $\textit{false}$ vacuum to start with for pure $R^2$ gravity. These studies unknowingly omitted the information about $\Lambda$ of the de Sitter background, hence incapable of attaining a Newtonian behavior in pure $R^2$ gravity. Secondly, as $\Lambda$ appears in $V(r)$ in a $\textit{singular}$ manner, viz. $V(r)\propto\Lambda^{-1}$, the Newtonian limit for pure $R^2$ gravity cannot be obtained by any perturbative approach treating $\Lambda$ as a small parameter.