Do metric fluctuations affect the Higgs dynamics during inflation?
  We show that the dynamics of the Higgs field during inflation is not affected
by metric fluctuations if the Higgs is an energetically subdominant light
spectator. For Standard Model parameters we find that couplings between Higgs
and metric fluctuations are suppressed by $\mathcal{O}(10^{-7})$. They are
negligible compared to both pure Higgs terms in the effective potential and the
unavoidable non-minimal Higgs coupling to background scalar curvature. The
question of the electroweak vacuum instability during high energy scale
inflation can therefore be studied consistently using the Jordan frame action
in a Friedmann--Lema\^itre--Robertson--Walker metric, where the Higgs-curvature
coupling enters as an effective mass contribution. Similar results apply for
other light spectator scalar fields during inflation.
