Emergent vacua and stability constraints on black hole solutions in higher-dimensional $f(R)$ gravity

Abstract: We investigate static spherically symmetric vacuum solutions in higher-dimensional $f(R)$ gravity, beginning with the five-dimensional Starobinsky model governed by the action $f(R) = R + αR² - 2Λ$. By enforcing the ghost-free stability criterion $f&#39;(R) &gt; 0$ on constant scalar curvature spacetimes, we show that a stable effective cosmological constant cannot be dynamically generated from pure $R<sup>2$ geometric corrections in five dimensions; its existence is inextricably tied to a bare cosmological constant. Generalizing this analysis to arbitrary dimensions $D$ and single-term curvature corrections $f(R) = R + αR<sup>n$ with a vanishing bare cosmological constant, we derive a universal stability bound, $n &gt; D/2$, required for the existence of stable emergent vacua. Finally, we demonstrate that expanding the gravitational action to a multi-term polynomial hierarchy circumvents this strict limitation. By including curvature corrections up to $\mathcal{O}(R<sup>3)$, the extended geometric degrees of freedom simultaneously satisfy the trace constraint and the stability criterion. Furthermore, we establish the exact parameter space boundaries that ensure not only asymptotic vacuum stability but strict global stability ($f&#39;(R) &gt; 0$ for all $R$), allowing for the dynamical generation of exact, globally ghost-free vacuum spacetimes in $D \ge 5$ purely from higher-order geometric terms. The scalaron mass requirement, $m_s<sup>2&gt;0$ is not imposed in full in our analysis. A detailed investigation of its implications on the models considered in this work are left for future study.