  The Standard Model Higgs potential becomes unstable at large field values.
After clarifying the issue of gauge dependence of the effective potential, we
study the cosmological evolution of the Higgs field in presence of this
instability throughout inflation, reheating and the present epoch. We conclude
that anti-de Sitter patches in which the Higgs field lies at its true vacuum
are lethal for our universe. From this result, we derive upper bounds on the
Hubble constant during inflation, which depend on the reheating temperature and
on the Higgs coupling to the scalar curvature or to the inflaton. Finally we
study how a speculative link between Higgs meta-stability and consistence of
quantum gravity leads to a sharp prediction for the Higgs and top masses, which
is consistent with measured values.
