Primordial Black Hole Formation in a Scalar Field Dominated Universe: Investigation of the Critical nature of the Collapse

In this paper, we investigate the critical collapse leading to primordial black hole (PBH) formation in a universe dominated by a self-interacting scalar field with a quartic potential, comparing it to the well-known radiation-dominated case. Using fully relativistic nonlinear numerical simulations in spherical symmetry, based on the Misner--Sharp formalism, we analyze the dynamics near the collapse threshold and track the scaling of the black hole mass. Our results confirm that both the scalar field and radiation cases exhibit type II critical behavior with similar -- though not identical -- critical exponents, differing by about 2σ. This suggests that, while a quartic scalar field effectively mimics a radiation fluid even in the nonlinear collapse regime, small differences in the critical exponent persist. Our findings provide direct numerical evidence for the near universality of the critical exponent in PBH formation, with only mild dependence on whether the collapse is driven by a scalar field or a perfect fluid.