We numerically study the primordial black hole (PBH) formation by an isocurvature perturbation of a massless scalar field on super Hubble scales in the radiation-dominated Universe. As a first step we perform simulations of spherically symmetric configurations. For the initial condition, we employ the spatial gradient expansion and provide the general form of the growing mode solutions valid up through the second order in this expansion. The initial scalar field profile is assumed to be Gaussian with a characteristic comoving wave number k; ∼exp(-k2R2), where R is the radial coordinate. We find that a PBH is formed for a sufficiently large amplitude of the scalar field profile. Nevertheless, we find that the late time behavior of the gravitational collapse is dominated by the dynamics of the fluid but not by the scalar field, which is analogous to the PBH formation from an adiabatic perturbation in the radiation-dominated Universe.
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