Primordial black holes and induced gravitational waves in $k$-inflation
  Recent observational constraints indicate that primordial black holes (PBHs)
with the mass scale $\sim 10^{-12}M_{\odot}$ can explain most of dark matter in
the Universe. To produce this kind of PBHs, we need an enhance in the
primordial scalar curvature perturbations to the order of
${\mathcal{O}(10^{-2})}$ at the scale $ k \sim 10^{12}~\rm Mpc^{-1}$. Here, we
investigate the production of PBHs and induced gravitational waves (GWs) in the
framework of \textbf{$k$-inflation}. We solve numerically the Mukhanov-Sasaki
equation to obtain the primordial scalar power spectrum. In addition, we
estimate the PBHs abundance $f_{\text{PBH}}^{\text{peak}}$ as well as the
energy density parameter $\Omega_{\rm GW,0}$ of induced GWs. Interestingly
enough is that for a special set of model parameters, we estimate the mass
scale and the abundance of PBHs as $\sim{\cal O}(10^{-13})M_{\odot}$ and
$f_{\text{PBH}}^{\text{peak}}=0.96$, respectively. This confirms that the
mechanism of PBHs production in our inflationary model can justify most of dark
matter. Furthermore, we evaluate the GWs energy density parameter and conclude
that it behaves like a power-law function $\Omega_{\rm GW}\sim (f/f_c)^n$ where
in the infrared limit $f\ll f_{c}$, the power index reads $n=3-2/\ln(f_c/f)$.
