  The cosmological Higgs vacuum stability has been an attractive research
subject and it is crucial to accurately follow the development of the Higgs
fluctuations. In this work, we thoroughly investigate how the vacuum
fluctuations of the Higgs field affect the stability of the electroweak vacuum
in Friedmann-Lemaitre-Robertson-Walker (FLRW) background. Adopting adiabatic
(WKB) approximation or adiabatic regularization methods, we clearly show that
vacuum fluctuations of the Higgs field in the FLRW background depend on the
curvature and also masses of the Higgs or other scalar fields. The Higgs
fluctuations can generate true vacuum bubbles and trigger off a collapse of the
electroweak vacuum. Furthermore we clearly show that the effective Higgs
potential in the FLRW background is modified by the Higgs vacuum fluctuations.
The vacuum fluctuations of the standard model fields can stabilize or
destabilize the effective Higgs potential through backreaction effects.
Considering the improved effective Higgs potential with the Higgs vacuum
fluctuations $\left< { \delta \phi }^{ 2 } \right>$ in various backgrounds, we
provide new cosmological constraints on the mass of the Higgs-coupled scalar
fields and a quantitative description of the Higgs stability in the FLRW
background.
