Revisiting primordial black holes formation from preheating instabilities: the case of Starobinsky inflation

The primordial black holes (PBHs) formation in the early universe inflationary cosmology has received a lot of attention in recent years. One of the ways PBHs formation can be a possibility is the preheating stage after inflation and this particular scenario does not require any ad-hoc fine tuning of the scalar field potential. In this paper, we focus on the growth of primordial density perturbation and the consequent possibility of PBHs formation in the preheating stage of the Starobinsky model for inflation. The typical mechanism for PBH formation during preheating is based on the collapse of primordial fluctuations that become super-horizon during inflation (type I) and re-enter the particle horizon in the different phases of cosmic expansion. In this work, we show that there exists a certain range of modes that remain in the sub-horizon (not exited) during inflation (type II modes). Those can, in the later phase of evolution, lead to large density perturbation above the threshold and can potentially also contribute to the PBH formation. We obtain in detail the conditions that determine the possible collapse of type I and/or type II modes. Since the preheating stage is an 'inflaton' (approximately) matter-dominated phase with the equation of state $w\ll 1$, we follow the framework of the critical collapse of fluctuations and compute the mass fraction using the well-known Press-Schechter and the Khlopov-Polnarev formalisms, and compare the two. Finally, we comment on the implications of our study for the investigations concerned with primordial accretion and consequent PBH contribution to the dark matter.