  It is generally accepted that the Moon accreted from the disk formed by an
impact between the proto-Earth and impactor, but its details are highly
debated. Some models suggest that a Mars-sized impactor formed a silicate
melt-rich (vapor-poor) disk around Earth, whereas other models suggest that a
highly energetic impact produced a silicate vapor-rich disk. Such a vapor-rich
disk, however, may not be suitable for the Moon formation, because moonlets,
building blocks of the Moon, of 100 m-100 km may experience strong gas drag and
fall onto Earth on a short timescale, failing to grow further. This problem may
be avoided if large moonlets ($\gg 100$ km) form very quickly by streaming
instability, which is a process to concentrate particles enough to cause
gravitational collapse and rapid formation of planetesimals or moonlets. Here,
we investigate the effect of the streaming instability in the Moon-forming disk
for the first time and find that this instability can quickly form $\sim 100$
km-sized moonlets. However, these moonlets are not large enough to avoid strong
drag and they still fall onto Earth quickly. This suggests that the vapor-rich
disks may not form the large Moon, and therefore the models that produce
vapor-poor disks are supported. This result is applicable to general
impact-induced moon-forming disks, supporting the previous suggestion that
small planets ($<1.6 R_\oplus$) are good candidates to host large moons because
their impact-induced disks would be likely vapor-poor. We find a limited role
of streaming instability in a satellite formation in an impact-induced disk,
whereas it plays a key role during planet formation.
