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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 Private nonprofit colleges are increasingly using tuition resets, or a decrease in sticker price by at least 5%, to attract new students and counter declining demand. While discounting tuition with institutional aid is a common practice to get accepted students to matriculate and to increase affordability, a tuition reset is a more transparent approach that moves colleges away from a high aid/high tuition model. The authors find minimal evidence that these policies increase student enrollment in the long run, but that there may be short-term impacts. As expected, institutional aid decreases and varies directly with the size of the sticker price reduction. The average net price students pay decreases, but this effect may be driven by changes in the estimated non-tuition elements of the total cost of attendance. Finally, net tuition revenue appears unrelated to tuition resets. These findings call into question the efficacy of this practice.