We revisit the formation of primordial black holes (PBHs) in the radiationdominated era for both linear and nonlinear regimes, elaborating on the concept of an apparent horizon. Contrary to the expectation from vacuum models, we argue that in a cosmological setting a density fluctuation with a high density does not always collapse to a black hole. To this end, we first elaborate on the perturbation theory for spherically symmetric space times in the linear regime. Thereby, we introduce two gauges. This allows to introduce a well defined gaugeinvariant quantity for the expansion of null geodesics. Using this quantity, we argue that PBHs do not form in the linear regime irrespective of the density of the background. Finally, we consider the formation of PBHs in nonlinear regimes, adopting the spherical collapse picture. In this picture, overdensities are modeled by closed FRW models in the radiationdominated era. The difference of our approach is that we start by finding an exact solution for a closed radiationdominated universe. This yields exact results for turnaround time and radius. It is important that we take the initial conditions from the linear perturbation theory. Additionally, instead of using uniform Hubble gauge condition, both density and velocity perturbations are admitted in this approach. Thereby, the matching condition will impose an important constraint on the initial velocity perturbations Î´h0 = âÎ´0/2. This can be extended to higher orders. Using this constraint, we find that the apparent horizon of a PBH forms when Î´ > 3 at turnaround time. The corrections also appear from the third order. Moreover, a PBH forms when its apparent horizon is outside the sound horizon at the reentry time. Applying this condition, we infer that the threshold value of the density perturbations at horizon reentry should be larger than Î´th > 0.7.
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