Beyond isolated curvature peaks: collective collapse and multiple Primordial Black Hole formation

Abstract

Primordial black hole (PBH) calculations usually treat rare curvature peaks as isolated collapsing regions. Using fully nonlinear $3+1$ numerical-relativity simulations in a radiation-dominated Universe, we demonstrate that neighbouring primordial curvature perturbations need not map one-to-one onto PBHs: they may ultimately disperse, collapse collectively into a single PBH, or undergo distinct local collapses and form more than one PBH. In the family of two-component profiles studied here, the latter outcome is a pair of PBHs, identified by the coexistence of two disconnected apparent horizons on at least one time slice. We introduce a nonspherical quasi-local compaction diagnostic based on the Hawking mass and referenced to a round flat-FLRW sphere of equal area. It retains the angular structure of the curvature field and reduces to the standard Misner-Sharp compaction function in spherical symmetry. The global maximum of this diagnostic, $\mathcal K_{\rm form}$, provides an empirical indicator of whether at least one PBH forms, with a transition near $\mathcal K_{\rm form,c}\approx0.56$ and modest profile-dependent scatter. For forming bimodal profiles, we supplement the nonlinear compactness with a signed linear surface strength evaluated on the same probing spheres. The viability of the weaker local branch and the competition between the local and common-enclosing branches improve the empirical discrimination between single- and double-PBH outcomes. Our results therefore show that the maximum curvature amplitude alone does not determine the collapse outcome, which also depends on the spatial extent, characteristic scales, and geometry of the surrounding curvature environment.