Primordial black hole abundances and scalar induced gravitational waves from finite-width power spectra in a stiff thermal history

We study the formation of primordial black holes (PBHs) from large primordial perturbations that re-enter the horizon during an epoch with equation of state ${\rm w}\geq1/3$. We consider a log-normal curvature power spectrum of finite width $Δ$ and determine the collapse amplitude by numerical-relativity simulations of a self-gravitating perfect fluid. Threshold scans are performed for five values of $\rm w$ and five spectral widths, and the resulting numerical thresholds are used in the PBH abundance and scalar-induced gravitational-wave (SIGW) calculations. For comparison, we also evaluate the semi-analytical $q$-function prescription. It reproduces the numerical trend close to radiation domination and for nearly monochromatic profiles, but it is not a reliable threshold estimator for generic finite-width profiles in stiffer backgrounds. We show how collapse thresholds increase with both $Δ$ and $\rm w$, changing the curvature amplitude required for PBHs to constitute all of the dark matter and, consequently, the normalization of the accompanying SIGW signal.