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 $n=3-2/\ln(f_c/f)$.