  In this paper, we introduce a weak, transient and periodical derivative
coupling between the inflaton field and gravity, and find that the square of
the sound speed of the curvature perturbations becomes a periodic function,
which results in that the equation of the curvature perturbations can be
transformed into the form of the Mathieu equation in the sub-horizon limit.
Thus, the parametric resonance will amplify the curvature perturbations so as
to generate a formation of abundant primordial black holes (PBHs). We show that
the generated PBHs can make up most of dark matter. Associated with the
generation of PBHs, the large scalar perturbations will give rise to the scalar
induced gravitational waves which may be detected by future gravitational wave
projects.
