Nontrivial vacua in higher-derivative gravitation

A discussion of an extended class of higher-derivative classical theories of gravity is presented. A procedure is given for exhibiting the new propagating degrees of freedom, at the full nonlinear level, by transforming the higher-derivative action to a canonical second-order form. For general fourth-order theories, described by actions which are general functions of the scalar curvature, the Ricci tensor and the full Riemann tensor, it is shown that the higher-derivative theories may have multiple stable vacua. The vacua are shown to be, in general, nontrivial, corresponding to de Sitter or anti-de Sitter solutions of the original theory. It is also shown that around any vacuum the elementary excitations remain the massless graviton, a massive scalar field, and a massive ghostlike spin-two field. The analysis is extended to actions which are arbitrary functions of terms of the form ${\nabla }^{2k}R$, and it is shown that such theories also have a nontrivial vacuum structure.