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We study the process of primordial black hole (PBH) formation at the beginning of the radiation era for the cosmological scenario in which the inflaton is a pseudo-Nambu-Goldstone boson (axion) and there is a coupling of the inflaton with some gauge field. In this model inflation is accompanied by the gauge quanta production, and a strong rise of the curvature power spectrum amplitude at small scales (along with non-Gaussianity) is predicted. We show that data on PBH searches can be used for a derivation of essential constraints on the model parameters in such an axion inflation scenario. We compare our numerical results with the similar results published earlier, in the work A. Linde, S. Mooij, and E. Pajer, Phys. Rev. D 87, 103506 (2013).
Bugaev et al. (Wed,) studied this question.