Ellipsoidal collapse and primordial black hole formation

Abstract

We reinvestigate gravitational ellipsoidal collapse with special focus on its impact on primordial black-hole formation. For a generic model we demonstrate that the abundance and energy density of the produced primordial black holes will be significantly decreased when the non-sphericity of the overdensities is taken into account.The process of gravitational collapse is of utmost importance to our understanding of the Universe. From the formation of galaxies [1,2], clusters of galaxies [3,4], haloes [5][6][7][8][9] (for a recent review see [10]) or even to the possible formation of primordial black holes [11][12][13], the nature of the collapse is crucial in determining characteristics like abundance, mass or shape.In many cases, both for its calculational simplicity as well as being a reasonable first approximation, spherical symmetry has been an integral assumption to investigate gravitational collapse processes [14][15][16]. Although in most cases initial non-sphericity is either small or eventually leads to (approximately) spherical objects, its effect might nevertheless be consequential. For instance, estimates of the abundance of small galactic haloes receive considerable corrections [17], the formation of space-time singularities might be very different [18], or, the precise geometric way in which collapse proceeds may lead to major intermediate deformations [19,20].The space of all possible shapes a collapsing overdensitiy might have is enormously large. Hence, one needs to focus on the most relevant structures. One of the simplest and most studied, deviating from spherical symmetry, is an ellipsoidal one, also because it gives a fairly good approximation to objects of many shapes. This has been and still is the focus of a vast amount of literature (cf. [17, 21-27]), including the seminal work of Sheth, Mo and Tormen [17] who obtained a fitting formula for the mentioned collapse threshold which they found to be supported by numerical evidence. More recent evidence for the improvement of fits with an ellipsoidal collapse model can be found for instance in [28,29].While most of the quoted references on ellipsoidal collapse deal with the formation of dark-matter haloes, the investigation of how the shape distribution of initial overdensities may affect the formation of primordial black holes is relatively modest. In [30] the authors studied tri-axial collapse of black holes and critical collapse in a way which is relevant also for primordial black-hole formation, and in [31] a non-spherical critical collapse was considered. The authors of [32][33][34] discussed effects of non-spherical geometry in the formation of primordial black holes if it occurred during an intermediate phase of matter domination due to a superheavy unified field theory particle in the very early universe. However, to the best of our knowledge, there has not been a thorough investigation of the effect of the abundance, or, the energy density of primordial back holes when lifting the spherical assumption on the overdensities in a genera...

Spins of primordial black holes formed in the matter-dominated phase of the Universe

“…Based on this estimate of the production rate, Carr et al [31] discussed the inflaton and spectator field perturbations. It should be noted that the nonspherical effect in primordial black hole formation was also discussed by Kühnel and Sandstad [32] in a very different way.…”

Cosmological Signature of the Standard Model Higgs Vacuum Instability: Primordial Black Holes as Dark Matter

“…In our findings, we have not included the fact that the mass of the PBH is not precisely the mass contained in the corresponding horizon volume, but in fact, it obeys a scaling relation with initial perturbations [16] or the fact that the threshold is shape-dependent [18]. Furthermore, we have not accounted for the fact that the threshold amplitude and the final black hole mass depend on the initial density profile of the perturbation [19].…”

Uncertainties in primordial black-hole constraints on the primordial power spectrum

“…If the shapes of the overdensities are non-spherical, which is to be expected in a realistic distribution, this may also lead to large effects, as has been pointed out recently in Ref. [30]. This non-sphericity effect is likely to strongly reduce the overall production of PBHs, yielding significantly weaker constraints on the primordial power spectrum from PBH non-observations.…”

“…Although corresponding detailed numerical studies are still lacking, utilizing the estimate for ellipsoidal collapse of Ref. [30], we are essentially led to changes in the threshold of the density contrast δ c , which increases as…”

“…Refs. [29,30]). There are several important effects which need to be accounted for in order for the transformation between the primordial power spectrum and the PBH mass distribution to yield reliable answers.…”