Electroweak Vacuum Instability and Renormalized Higgs Field Vacuum
  Fluctuations in the Inflationary Universe
  In this work, we investigated the electroweak vacuum instability during or
after inflation. In the inflationary Universe, i.e., de Sitter space, the
vacuum field fluctuations $\left< {\delta \phi }^{ 2 } \right>$ enlarge in
proportion to the Hubble scale $H^{2}$. Therefore, the large inflationary
vacuum fluctuations of the Higgs field $\left< {\delta \phi }^{ 2 } \right>$
are potentially catastrophic to trigger the vacuum transition to the
negative-energy Planck-scale vacuum state and cause an immediate collapse of
the Universe. However, the vacuum field fluctuations $\left< {\delta \phi }^{ 2
} \right>$, i.e., the vacuum expectation values have an ultraviolet divergence,
and therefore a renormalization is necessary to estimate the physical effects
of the vacuum transition. Thus, in this paper, we revisit the electroweak
vacuum instability from the perspective of quantum field theory (QFT) in curved
space-time, and discuss the dynamical behavior of the homogeneous Higgs field
$\phi$ determined by the effective potential ${ V }_{\rm eff}\left( \phi
\right)$ in curved space-time and the renormalized vacuum fluctuations $\left<
{\delta \phi }^{ 2 } \right>_{\rm ren}$ via adiabatic regularization and
point-splitting regularization. We simply suppose that the Higgs field only
couples the gravity via the non-minimal Higgs-gravity coupling $\xi(\mu)$. In
this scenario, the electroweak vacuum stability is inevitably threatened by the
dynamical behavior of the homogeneous Higgs field $\phi$, or the formations of
AdS domains or bubbles unless the Hubble scale is small enough $H< \Lambda_{I}
$.
