Consistent higher derivative gravitational theories with stable de Sitter and anti–de Sitter backgrounds

In this paper we provide the criteria for any generally covariant, parity preserving, and torsion-free theory of gravity to possess a stable de Sitter (dS) or anti–de Sitter (AdS) background. By stability we mean the absence of tachyonic or ghostlike states in the perturbative spectrum that can lead to classical instabilities and violation of quantum unitarity. While we find that the usual suspects, the $F(R)$ and $F(G)$ theories, can indeed possess consistent (A)dS backgrounds, $G$ being the Gauss-Bonnet term, another interesting class of theories, string-inspired infinite derivative gravitational theories, can also be consistent around such curved vacuum solutions. Our study should not only be relevant for quantum gravity and early universe cosmology involving ultraviolet physics, but also for modifications of gravity in the infrared sector vying to replace dark energy.