Beyond dimensional analysis: Higgs and new Higgs inflations {\it do not}
  violate unitarity
  Na\"ive dimensional analysis seems to suggest possible unitarity violations
in the framework of the Higgs and new Higgs inflationary scenarios. These
violations seem to happen around the value in which the potential energy, per
given Higgs boson's vacuum expectation value, crosses the perturbative cut-off
scale calculated around the electroweak vacuum. Conversely to these
expectations, in this paper we show that, by using an exact analysis of the
background dependent cut-off scale, and by including the contribution of the
phase-space volume in the perturbative scattering amplitudes, no violation of
(perturbative) unitarity might ever happen during the whole Universe evolution.
