  Primordial black holes (PBHs) are a useful tool in cosmology to probe
primordial inhomogeneities on small scales that reenter the Hubble radius
during the radiation dominated epoch. In this paper, a phenomenon we call the
double formation of PBHs, described below, is explored. Suppose there exists a
highly perturbed region which will collapse to form a PBH after the horizon
crossing of this region, and farther that this region is superposed on much
larger region, which also collapses upon reentry. One then expects the collapse
of the central smaller region at the time of the crossing of this region,
followed by another collapse of the larger perturbation at the time of its
respective crossing. The smaller PBH, formed earlier, should be swallowed in
the second collapse leading to a single larger PBH as the final state. This
paper reports the first direct numerical confirmation of such double PBH
formation. Related to this, we also discuss the effects of high-frequency modes
on the formation of PBHs.
