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In recent years, the formation of primordial black holes (PBH) in the early universe inflationary cosmology has garnered significant attention. One plausible scenario for primordial black hole (PBH) formation arises during the preheating stage following inflation. Notably, this scenario does not necessitate any ad-hoc fine-tuning of the scalar field potential. This paper focuses on the growth of primordial density perturbation and the consequent possibility of PBH 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) but evolve identically to type I modes if they fall into the instability band, leading to large density perturbation above the threshold and can potentially also contribute to the PBH formation. We outline the conditions that govern the potential collapse of typeI and type II modes with wavelengths exceeding the Jeans length,which we derive based on the effective sound speed of scalar fieldfluctuations. Since the preheating stage is an `inflaton' (approximately) matter-dominated phase, 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. 

Abstract We investigate the effect of dark stars (DSs) on the reionization history of the universe, and the interplay between them and feedback due to Lyman–Werner (LW) radiation in reducing the cosmic microwave background (CMB) optical depth to a value within the τ = 0.054 ± 0.007 range measured by Planck. We use a semianalytic approach to evaluate reionization histories and CMB optical depths, which includes Population II stars in atomic cooling halos and Population III stars in minihalos with LW feedback, preceded by a DS phase. We show that while LW feedback by itself can reduce the integrated optical depth to the last scattering surface to ∼0.05 only if the Population III star formation efficiency is less than ∼0.2%, the inclusion of a population of DSs can naturally lead to the measured CMB optical depth for much larger Population III star formation efficiencies ≳1%.