In this work, we explore the production of primordial black holes (PBHs) within the context of $\alpha$-attractor inflationary models, focusing on the preheating phase following inflation. During this phase, self-resonance instabilities arise due to deviations of the inflationary potential from a quadratic form. PBH formation is analyzed using three criteria: (i) the perturbation must lie within the instability band, (ii) its characteristic length must exceed the Jeans length, and (iii) it must have sufficient time to collapse based on the estimations of massive scalar field spherical collapse in an Einstein–de Sitter universe. Based on these criteria, we calculate the PBH mass fraction using the altered Press-Schechter (PS) that is generally implemented in preheating scenarios and Khlopov-Polnarev (KP) formalism that considers nonspherical effects. Our results show that the altered PS formalism tends to overestimate PBH abundance during preheating in contrast to the KP formalism. We provide a detailed comparison between these two frameworks with observational constraints from evaporating PBHs. Notably, the altered PS formalism is excluded by these constraints, which are based on Hawking radiation, while the KP formalism remains viable. These findings underscore the importance of accounting for nonspherical effects and accurate collapse dynamics in studies of PBH formation during preheating.

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• Inflation
• Primordial black holes