Saturation Equations of State in Critical Gravitational Collapse: The Primordial Black Hole Threshold

Abstract

The threshold and scaling laws of gravitational critical collapse depend sensitively on the matter equation of state. We investigate how these quantities are modified by a generic feature of dense matter that is absent from the radiation fluid commonly assumed in primordial black hole (PBH) studies: pressure stiffening as a maximum density is approached. As an analytically tractable proxy, we adopt the closed-form equation of state of a single-occupancy lattice gas, \(p=-T\ln(1-ρ)\), which exhibits a density-dependent sound speed and a saturation density. Using general-relativistic simulations of spherically symmetric collapse, we show that this nonlinear pressure feedback increases the PBH formation threshold by \(0.50\pm0.02\%\) relative to the radiation equation of state within the causal regime of the model. At the same time, the critical mass-scaling exponent remains \(γ=0.357\pm0.001\), consistent with the radiation-fluid value to within our numerical precision. This agreement reflects the fact that the lattice equation of state approaches the radiation fluid at low density and remains only a mild perturbation over the near-critical regime, rather than indicating a universal critical exponent. Our results provide a proof of principle that saturation-induced stiffening can stabilize gravitational collapse and shift the PBH threshold, while introducing a linear-response framework for assessing the impact of more realistic equations of state on primordial black hole formation.