Primordial black hole abundance calculation in concert with optimized peaks theory-- what changes compared to traditional Press-Schechter?

The abundance of primordial black hole (PBH) formation in the early Universe has been long studied using the Press-Schechter formalism. Although very easy to apply, the theory is built on a rather flimsy theoretical framework (despite making robust predictions). Peaks theory is an alternative to the study of matter condensation in the Universe (PBHs by extention) which was developed under stronger theoretical foundations while avoiding certain pitfalls of Press-Schechter like the window function ambiguity. Here, I will talk about PBH abundance calculated using the optimized peaks theory. This particular variant introduces a more accurate expression for PBH mass, giving rise to an O(10) increase compared to the traditional calculation. This has very interesting consequences in model building, viz a viz newer limits to where the peak in the curvature power spectra can be placed in order to evade evaporation constraints. Furthermore, peaks theory can help model builders evade certain future, small scale constraints on P_ζ arising from LISA. Applying these on a PBH forming α-attractor model, it can be shown that PBHs of mass M~10¹⁷ g can be formed from the collapse of smaller scale modes k~8x10¹⁴ Mpc^-1 which would have previously predicted M<10¹⁶ g. I also comment on the collapse threshold and show that the numerically favored δ_th≈0.5 can be achieved using smaller peaks in P_ζ.