Withholding potentials, absence of ghosts, and relationship between minimal dilatonic gravity andf(R)theories

Abstract

We study the relation between minimal dilatonic gravity (MDG) and $f(R)$ theories of gravity and establish strict conditions for their global equivalence. Such equivalence takes place only for a certain class of cosmological potentials, dubbed here ``withholding potentials,'' since they prevent change of the sign of the dilaton $\ensuremath{\Phi}$. The withholding property ensures the attractive character of gravity, as well as absence of ghosts and a tachyon in the gravi-dilaton sector and yields certain asymptotic of the functions $f(R)$. Large classes of withholding cosmological potentials and functions $f(R)$ are found and described in detail. The particle content of the gravi-dilaton sector is found using perturbation theory around the de Sitter vacuum of MDG. Two phenomena, scalaron waves and induction of gravitational waves by the scalaron field, are discussed using the derived wave equations for MDG scalaron and graviton in the de Sitter background. Seemingly, the MDG and $f(R)$ theories offer a unified description of dark energy and dark matter.