Basic Principles of 4D Dilatonic Gravity and Some of Their Consequences for Cosmology, Astrophysics and Cosmological Constant Problem

We present a class of simple scalar-tensor models of gravity with one scalar field (dilaton $\Phi$) and only one unknown function (cosmological potential $U(\Phi)$). These models might be considered as a stringy inspired ones with broken SUSY. They have the following basic properties: 1) Positive dilaton mass, $m_\Phi$, and positive cosmological constant $\Lambda$, define two extremely different scales. The models under consideration are consistent with the known experimental facts if $m_\Phi > 10^{-3} eV$ and $\Lambda=\Lambda^{obs}\sim 10^{-56} cm^{-2}$. 2) Einstein week equivalence principle is strictly satisfied and extended to scalar-tensor theories of gravity using a novel form of principle of "constancy of fundamental constants". 3) The dilaton plays simultaneously role of inflation field and quintessence field and yields a sequential hyper-inflation with graceful exit to asymptotic de Sitter space-time which is an attractor, and is approached as $\exp(-\sqrt{3\Lambda^{obs}} ct/2)$. The time duration of inflation is $\Delta t_{infl} \sim m_\Phi^{-1}$. 4) Ultra-high frequency ($\omega_\Phi \sim m_\Phi$) dilatonic oscillations take place in asymptotic regime. 5) No fine tuning. (The Robertson-Walker solutions of general type have the above properties.) 6) A novel adjustment mechanism for cosmological constant problem seems to be possible: the huge value of cosmological constant in the stringy frame is re-scaled to its observed value by dilaton after transition to phenomenological frame.