  We study the amplification of the curvature perturbations due to a small
sound speed and find that its origin is different completely from that due to
the ultraslow-roll inflation. This is because when the sound speed is very
small the enhancement of the power spectrum comes from the fact that the
curvature perturbations at the scales smaller than the cosmic microwave
background (CMB) scale becomes scale-variant, rather than growing that leads to
the amplification of the curvature perturbations during the ultraslow-roll
inflation. At large scales the power spectrum of the curvature perturbations
remains to be scale invariant, which is consistent with the CMB observations,
and then it will have a transient $k^2$ growth and finally approach a $k^4$
growth as the scale becomes smaller and smaller. Thus the power spectrum can be
enhanced to generate a sizable amount of primordial black holes. Furthermore,
when the high order correction in the dispersion relation of the curvature
perturbations is considered the growth of the power spectrum of the curvature
perturbations has the same origin as that in the case without this correction.
