In this paper, 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 $r_m$, where $r_m$ 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 the numerical simulations, that this limit is ${\delta }_c=0.40$, 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 normalized compaction function curvature at $r_m$.

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.

Primordial black holes, which could have been formed in the very early Universe due to the collapse of large curvature fluctuations, are currently one of the most attractive and fascinating research areas in cosmology for their possible theoretical and observational implications. This review article presents the current results and developments on the conditions for primordial black hole formation from the collapse of curvature fluctuations in spherical symmetry on a Friedman–Lemaître–Robertson–Walker background and its numerical simulation. We review the appropriate formalism for the conditions of primordial black hole formation, and we detail a numerical implementation. We then focus on different results regarding the threshold and the black hole mass using different sets of curvature fluctuations. Finally, we present the current state of analytical estimations for the primordial black hole formation threshold, contrasted with numerical simulations.

Naive dimensional analysis seems to suggest possible unitarity violations in the framework of the Higgs and new Higgs inflationary scenarios. These violations seem to happen around the value in which the potential energy, per a given Higgs boson’s vacuum expectation value, crosses the perturbative cutoff scale calculated around the electroweak vacuum. Conversely to these expectations, we show that, by using an exact analysis of the background dependent cutoff scale, and by including the contribution of the phase-space volume in the perturbative scattering amplitudes of scalars, no violation of (perturbative) unitarity might ever happen during the whole Universe evolution.