Primordial black holes formation in the inflationary model with
  field-dependent kinetic term for quartic and natural potentials
  Within the framework of inflationary model with field-dependent kinetic term
for quartic and natural potentials, we investigate generation of the primordial
black holes (PBHs) and induced gravitational waves (GWs). In this setup, we
consider a kinetic function as $G(\phi)=g_I(\phi)\big(1+g_{II}(\phi)\big)$ and
show that in the presence of first term $g_I(\phi)$ both quartic and natural
potentials, in contrast to the standard model of inflation, can be consistent,
with the 68\% CL of Planck observations. Besides, the second term
$g_{II}(\phi)$ can cause a significant enhancement in the primordial curvature
perturbations at the small scales which results the PBHs formation. For the
both potentials, we obtain an enhancement in the scalar power spectrum at the
scales $k\sim10^{12}~\rm Mpc^{-1}$, $10^{8}~\rm Mpc^{-1}$, and $10^{5}~\rm
Mpc^{-1}$, which causes PBHs production in mass scales around
$10^{-13}M_{\odot}$, $10^{-5}M_{\odot}$, and $10 M_{\odot}$, respectively.
Observational constraints confirm that PBHs with a mass scale of
$10^{-13}M_{\odot}$ can constitute the total of dark matter in the universe.
Furthermore, we estimate the energy density parameter of induced GWs which can
be examined by the observation. Also we conclude that it can be parametrized as
a power-law function $\Omega_{\rm GW}\sim (f/f_c)^n$, where the power index
equals $n=3-2/\ln(f_c/f)$ in the infrared limit $f\ll f_{c}$.
