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In this paper, the formation of primordial black holes (PBHs) isreinvestigated using inflationary $\alpha$-attractors. Instead of using theconventional Press-Schechter theory to compute the abundance, the optimizedpeaks theory is used, which was developed in Ref.\cite{Yoo:2018kvb,Yoo:2020dkz}. This method takes into account how curvatureperturbations play a r\^{o}le in modifying the mass of primordial black holes.Analyzing the model proposed in \cite{Mahbub:2019uhl} it is seen that thehorizon mass of the collapsed Hubble patch is larger by $\mathcal{O}(10)$compared to the usual computation. Moreover, PBHs can be formed from curvaturepower spectrum, $\mathcal{P}_{\zeta}(k)$, peaked at lower values usingnumerically favored threshold overdensities. As a result of the generallylarger masses predicted, the peak of the power spectrum can be placed at larger$k$ modes than that is typical with which potential future constraints on theprimordial power spectrum through gravitational waves (GWs) can be evaded.

Primordial black hole formation in $\alpha$-attractor models: an analysis using optimized peaks theory