  The measured Standard Model parameters lie in a range such that the Higgs
potential, once extrapolated up to high scales, develops a minimum of negative
energy density. This has important cosmological implications. In particular,
during inflation, quantum fluctuations could have pushed the Higgs field beyond
its potential barrier, triggering the formation of anti-de Sitter regions, with
fatal consequences for our universe. By requiring that this did not happen, one
can in principle connect (and constrain) Standard Model parameters with the
energy scale of inflation. In this context, we highlight the sensitivity of the
fate of our vacuum to seemingly irrelevant physics. In particular, the
departure of inflation from an exact de Sitter phase, as well as
Planck-suppressed derivative operators, can, already and surprisingly, play a
decisive role in (de)stabilizing the Higgs during inflation. Furthermore, in
the stochastic dynamics, we quantify the impact of the amplitude of the noise
differing from the one of a massless field, as well as of going beyond the
slow-roll approximation by using a phase-space approach. On a general ground,
our analysis shows that relating the period of inflation to precision particle
physics requires a knowledge of these "irrelevant" effects.
