Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories

We study the relation between Minimal Dilatonic Gravity (MDG) and f(R) theories of gravity and establish strict conditions for their {\em global} equivalence. Such equivalence takes place only for a certain class of cosmological potentials, dubbed here {\em withholding potentials}, since they prevent change of the sign of dilaton $\Phi$. 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 admissible functions $f(R)$. Large classes of withholding cosmological potentials and functions $f(R)$ are found and described in detail. It is shown that the popular choices of $f(R)$ functions are not withholding ones. The particle content of the gravi-dilaton sector is found using perturbation theory around de Sitter vacuum of MDG. The graviton remains massless, since it obeys conformal invariant field equation in the de Sitter space-time. The $R/6$ term in the conformal invariant wave operator introduces a very small mass scale $m_{R} \approx 1.5 \times 10^{-38} m_e$, $m_e$ being the mass of the electron. The mass of the dilaton is much larger: $m\gtrapprox 10^{29} m_{R}$. Two new phenomena: scalaron waves and induction of gravitational waves by the scalaron field are discussed using the derived wave equations for scalaron and graviton. The MDG and f(R) theories are shown to predict physical deviations from GR. Seemingly, the MDG and f(R) theories, when globally equivalent, offer a unified description of dark energy and dark matter.