  We investigate whether Higgs inflation can occur in the Standard Model
starting from natural initial conditions or not. The Higgs has a non-minimal
coupling to the Ricci scalar. We confine our attention to the regime where
quantum Einstein gravity effects are small in order to have results that are
independent of the ultraviolet completion of gravity. At the classical level we
find no tuning is required to have a successful Higgs inflation, provided the
initial homogeneity condition is satisfied. On the other hand, at the quantum
level we obtain that the renormalization for large non-minimal coupling
requires an additional degree of freedom that transforms Higgs inflation into
Starobinsky $R^2$ inflation, unless a tuning of the initial values of the
running parameters is made.
