  According to the best-fit parameters of the Standard Model, the Higgs field's
potential reaches a maximum at a field value $h \sim 10^{10-11}$ GeV and then
turns over to negative values. During reheating after inflation, resonance
between the inflaton and the Higgs can cause the Higgs to fluctuate past this
maximum and run down the dangerous side of the potential if these fields couple
too strongly. In this paper, we place constraints on the inflaton-Higgs
couplings such that the probability of the Higgs entering the unstable regime
during reheating is small. To do so, the equations of motion are approximately
solved semi-analytically, then solved fully numerically. Next the growth in
variance is used to determine the parameter space for $\kappa$ and $\alpha$,
the coupling coefficients for inflaton-Higgs cubic and quartic interactions,
respectively. We find the upper bounds of $\kappa < 1.6 \times 10^{-5} m_\phi
\sim 2.2 \times 10^8$ GeV and $\alpha < 10^{-8}$ to allow the Higgs to remain
stable in most Hubble patches during reheating, and we also find the full two
parameter joint constraints. We find a corresponding bound on the reheat
temperature of $T_\text{reh} \lesssim 9.2 \times 10^9$ GeV. Additionally, de
Sitter temperature fluctuations during inflation put a lower bound on
inflaton-Higgs coupling by providing an effective mass for the Higgs, pushing
back its hilltop during inflation. These additional constraints provide a lower
bound on $\alpha$, while $\kappa$ must also be non-zero for the inflaton to
decay efficiently.
