In this letter, we argue and show numerically that the threshold to form primordial black holes from an initial spherically symmetric perturbation is, to an excellent approximation, universal, whenever given in terms of the compaction function averaged over a sphere of radius rm, where rm is the scale on which the compaction function is maximum. This can be understood as the requirement that, for a black hole to form, each shell of the averaged compaction function should have an amplitude exceeding the so-called Harada-Yoo-Kohri limit. For a radiation dominated universe we argued, supported by numerical evidence, that this limit is δc = 0.40, which is slightly below the one quoted in the literature. Additionally, we show that the profile dependence of the threshold for the compaction function is only sensitive to its curvature at the maximum. We use these results to provide an analytic formula for the threshold amplitude of the compaction function at its maximum in terms of the normalised compaction function curvature at rm.

Primordial black holes (PBHs) may result from high peaks in a random field of cosmological perturbations. In single field inflationary models, such perturbations can be seeded as the inflaton overshoots a small barrier on its way down the potential. PBHs are then produced through two distinct mechanisms, during the radiation era. The first one is the familiar collapse of large adiabatic overdensities. The second one is the collapse induced by relic bubbles where the inflaton field is trapped in a false vacuum. The latter are due to rare backward fluctuations of the inflaton which prevented it from overshooting the barrier in horizon sized regions. We consider (numerically and analytically) the effect of non-Gaussianities on the threshold for overdensities to collapse into a PBH. Since typical high peaks have some dispersion in their shape or profile, we also consider the effect of such dispersion on the corresponding threshold for collapse. With these results we estimate the most likely channel for PBH production as a function of the non-Gaussianity parameter f NL . We also compare the threshold for collapse coming from the perturbative versus the non perturbative template for the non-Gaussianity arising in this model. We show that i) for f NL 3.5, the population of PBH coming from false vacuum regions dominates over that which comes from the collapse of large adiabatic overdensities, ii) the non-perturbative template of the non-Gaussianities is important to get accurate results. iii) the effect of the dispersion is small in determining the threshold for the compaction function, although it can be appreciable in determining the threshold amplitude for the curvature perturbation at low f NL . We also confirm that the volume averaged compaction function provides a very accurate universal estimator for the threshold.3 These are sometimes refered to as black holes with a baby universe inside. Note, however, that the baby universe is not in the trapped region, or "interior" of the black hole. Rather, the trapped region separates two normal regions, one in the parent ambient universe and the other in the baby universe, which were once causally connected but are not anymore, after the trapped region forms.4 In a nutshell, the problems is that ζ diverges when the amplitude of ζg reaches a critical value µ * , and it is not even defined for larger amplitude of ζg, for which there is a finite probability.5 Refs. [44,[51][52][53] consider the non-Gaussianity in the density perturbation δ due to the non-linear relation between δ and ζ. Note that such discussion would be redundant in our approach, where the initial conditions for numerical evolution, as well as the threshold estimators for gravitational collapse, are expressed directly in terms of ζ.

We consider black holes which form from an initially spherically symmetric super-Hubble perturbation of a cosmological background filled by a perfect fluid p = wρ with w ∈ (0, 1]. Previous work has shown that when w = 1/3 (radiation), there is a critical threshold for black hole formation (δ c ), which, to a very good approximation, only depends upon the curvature of the compaction function around its peak value. We find that this generalizes to all w 1/3; for smaller ws the knowledge of the full shape of the compaction function is necessary. We provide analytic approximations for δ c which are accurate for w ∈ [1/3, 1].

In this work we have used for the first time pseudo-spectral methods to perform numerical simulations of spherically symmetric black hole formations on a Friedman-Robertson-Walker universe. With these methods, the differential equations describing the gravitational collapse are partially solved algebraically. With our publicly available code we then independently check, and confirm, previous numerical estimations of the thresholds to form primordial black holes. By using an excision technique and analytical estimations of accretion rates, we were also able to estimate the black holes mass even in the case of large deviations from the threshold. There, we confirm, with an explicit example, that the estimation of the black hole mass via the self-similar scaling law is only accurate up to O(15%), for the largest allowed mass.

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