Solving the standard model problems in softened gravity

The Higgs naturalness problem is solved if the growth of Einstein's gravitational interaction is softened at an energy. less than or similar to 10(11) GeV (softened gravity). We work here within an explicit realization where the Einstein-Hilbert Lagrangian is extended to include terms quadratic in the curvature and a nonminimal coupling with the Higgs. We show that this solution is preserved by adding three right-handed neutrinos with masses below the electroweak scale, accounting for neutrino oscillations, dark matter and the baryon asymmetry. The smallness of the right-handed neutrino masses (compared to the Planck scale) and the QCD theta-term are also shown to be natural. We prove that a possible gravitational source of CP violation cannot spoil the model, thanks to the presence of right-handed neutrinos. Inflation is approximately described by the Starobinsky model in this context and can occur even if we live in a metastable vacuum.