The Higgs not only induces the masses of all SM particles, the Higgs, given its special mass value, is the natural candidate for the inflaton and in fact is ruling the evolution of the early universe, by providing the necessary dark energy which remains the dominant energy density. SM running couplings not only allow us to extrapolate SM physics up to the Planck scale, but equally important they are triggering the Higgs mechanism. This is possible by the fact that the bare mass term in the Higgs potential changes sign at about mu_0 = 1.4x10^16 GeV and in the symmetric phase is enhanced by quadratic terms in the Planck mass. Such a huge Higgs mass term is able to play a key role in triggering inflation in the early universe. In this article we extend our previous investigation by working out the details of a Higgs inflation scenario. We show how different terms contributing to the Higgs Lagrangian are affecting inflation. Given the SM and its extrapolation to scales mu>mu_0 we find a calculable cosmological constant V(0) which is weakly scale dependent and actually remains large during inflation. This is different to the Higgs fluctuation field dependent Delta V(phi), which decays exponentially during inflation, and actually would not provide a sufficient amount of inflation. The fluctuation field has a different effective mass which shifts the bare Higgs transition point to a lower value mu'_0 = 7.7x10^14 GeV. The vacuum energy V(0) being proportional to M_Pl^4 has a coefficient which vanishes near the Higgs transition point, such that the bare and the renormalized cosmological constant match at this point. The role of the Higgs in reheating and baryogenesis is emphasized.

39 pages, 25 figures, 1 table. Replacement: typos corrected, Eq (3) corrected, notation adjusted