Can breakdown of perturbation in the $\alpha$-attractor inflation lead to PBH formation?

With the basic $\alpha$-attractor potentials, we investigate an inflationary regime in the high-$k$ limit, where the cosmological perturbation breaks down due to large enhancement in the scalar power spectrum and generation of large negative values of the Bardeen potential. We analyze that, this deep sub-horizon regime creates a situation, which is congenial to the formation of the primordial black holes (PBHs). We work in the spatially flat gauge with $\delta\phi\neq$ 0 and thus explicitly show the roles of perturbations in the inflaton field as well as in the background gravitational field in the mentioned enhancements and thereby in the PBH formation. We calculate the values of $\sigma(M)$, $\beta(M)$ and $f_\mathrm{PBH}(M)$ around the peaks in the density contrast profile and thus estimate the fraction of PBH in the dark matter of the present universe, corresponding to certain mass scales. We observe the formation of PBHs in the $k$ range $0.43\times 10^{13}$ Mpc$^{-1}$ to $9.8\times 10^{13}$ Mpc$^{-1}$ with masses $1.35\times 10^{-13}M_\odot$ to $2.60\times 10^{-16}M_\odot$, evaporation times $7.74\times 10^{33}$ sec to $5.53\times 10^{25}$ sec, Hawking temperatures $3.72\times 10^{-8}$ GeV to $1.93\times 10^{-5}$ GeV and $f_\mathrm{PBH}(M)$ $\sim 6.12\times 10^{-6}$ to $3.63\times 10^{-1}$. The calculated mass range lies in the regions of forecasts by LISA, WD, NS, DECIGO/AI, FL, SIGWs and the $f_\mathrm{PBH} (M)$ results overlap with those of DECIGO/AI, FL, SIGWs.