  Even if the Higgs field does not affect the evolution of the background
geometry, its massive inhomogeneities induce large-scale gauge fields whose
energy density depends on the slow-roll parameters, on the effective scalar
mass and, last but not least, on the dimensionless coupling to the space-time
curvature. Since the non-Abelian gauge modes are screened, the non-minimal
coupling to gravity predominantly affects the evolution of the hypercharge and
electromagnetic fields. While in the case of minimal coupling the obtained
constraints are immaterial, as soon as the coupling increases beyond one fourth
the produced fields become overcritical. We chart the whole parameter space of
this qualitatively new set of bounds. Whenever the limits on the curvature
coupling are enforced, the magnetic field may still be partially relevant for
large-scale magnetogenesis and exceed $10^{-20}$ G for the benchmark scale of
the protogalactic collapse.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology;High Energy Physics - Theory