Eternal Inflation and Reheating in the Presence of the Standard Model
  Higgs Field
  We study the details of eternal inflation in the presence of a spectator
Higgs field within the framework of the minimal Standard Model. We have
recently shown that in the presence of scalar field(s) which allow inflation
only within a finite domain of field values, the universe reaches a steady
state where the normalized distribution for the field(s) converges to a steady
state distribution [1]. In this paper, we analyze this eternal inflation
scenario with the renormalized Standard Model Higgs potential, since it also
allows inflation in a finite domain, but turns over at high scales due to the
running of the self-coupling, marking an exit from inflation. We compute the
full steady state distribution for the Higgs using an integral evolution
technique that we formulated in [1] and the fractal dimension of the universe.
We then obtain a bound on the inflationary Hubble scale in order to have a
large observable universe contained within the instability scale of
$H\lesssim\mathcal{O}(10^9)$GeV depending on the top mass. Upon reheating of
the universe, thermal fluctuations in the Higgs field could potentially pose
another problem; however, we compute the rate of thermal bubble production and
find that the probability of tunneling in the post-inflationary era is
negligibly small even for very high reheat temperatures.
