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Spherical and nonspherical models of primordial black hole formation: exact solutions

We construct spacetimes which provide spherical and nonspherical models of black hole formation in the flat Friedmann--Lemaitre--Robertson--Walker (FLRW) universe with the Lemaitre--Tolman--Bondi solution and the Szekeres quasispherical solution, respectively. These dust solutions may contain both shell-crossing and shell-focusing naked singularities. These singularities can be physically regarded as the breakdown of dust description, where strong pressure gradient force plays a role. We adopt the simultaneous big bang condition to extract a growing mode of adiabatic perturbation in the flat FLRW universe. If the density perturbation has a sufficiently homogeneous central region and a sufficiently sharp transition to the background FLRW universe, its central shell-focusing singularity is globally covered. If the density concentration is sufficiently large, no shell-crossing singularity appears and a black hole is formed. If the density concentration is not sufficiently large, a shell-crossing singularity appears. In this case, a large dipole moment significantly advances shell-crossing singularities and they tend to appear before the black hole formation. In contrast, a shell-crossing singularity unavoidably appears in the spherical and nonspherical evolution of cosmological voids. The present analysis is general and applicable to cosmological nonlinear structure formation described by these dust solutions.