Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations (1809.02127v3)

Abstract: Primordial black holes may have formed in the radiative era of the early Universe from the collapse of large enough amplitude perturbations of the metric. These correspond to non linear energy density perturbations characterized by an amplitude larger than a certain threshold, measured when the perturbations reenter the cosmological horizon. The process of primordial black hole formation is studied here within spherical symmetry, using the gradient expansion approximation in the long wavelength limit, where the pressure gradients are small, and the initial perturbations are functions only of a time-independent curvature profile. In this regime it is possible to understand how the threshold for primordial black hole formation depends on the shape of the initial energy density profile, clarifying the relation between local and averaged measures of the perturbation amplitude. Although there is no universal threshold for primordial black hole formation, the averaged mass excess of the perturbation depends on the amplitude of the energy density peak, and it is possible to formulate a well-defined criterion to establish when a cosmological perturbation is able to form a black hole in terms of one of these two key quantities. This gives understanding of how the abundance of primordial black holes depends on the shape of the the inflationary power spectrum of cosmological perturbations.

• The paper demonstrates that the threshold amplitude for primordial black hole formation is contingent on the precise shape of initial energy density profiles.
• It employs a gradient expansion approach within spherical symmetry, revealing that the local maximum of the compaction function defines a crucial scale for PBH formation.
• The study establishes an invariant criterion linking local and averaged perturbation metrics, underscoring that no universal PBH formation threshold exists.

Threshold for Primordial Black Holes: Dependence on the Shape of Cosmological Perturbations

Primordial black holes (PBHs) potentially formed during the radiation-dominated era of the early Universe from the collapse of large-amplitude perturbations of the metric. This paper primarily investigates the conditions under which PBHs form and how their abundance depends on the shape of initial energy density profiles.

The paper adopts a gradient expansion approximation within spherical symmetry, analyzing cosmological perturbations beyond the linear theory into the nonlinear regime where pressure gradients are small. It provides a detailed examination of how the threshold for PBH formation varies with the curvature profile of perturbations, involving the time-independent curvature perturbation described in real space. The paper elucidates the relation between local and averaged measures of perturbation amplitude, highlighting that there is no universal threshold for PBH formation due to shape dependency.

Key Findings

• Threshold Dependence: The paper identifies that the threshold amplitude, δ_c, for PBH formation is contingent on the shape of the initial energy density profile. An averaged mass excess is tied to the amplitude of the energy density peak, clarifying the varied influence of spatial profiles.
• Compaction Function: A key criterion for PBH formation is based on the local maximum of a compaction function, which measures mass excess relative to areal radius. The location of this peak defines a pertinent scale, r_m, pivotal for analyzing PBH formation potential.
• Shape-independent Criterion: The paper introduces an invariant criterion to measure cosmological perturbation amplitude, showing that evaluating the energy density at r_m is equivalent to assessing mass excess within a spherical region defined by r_m.

Implications

The outlined method for determining the perturbation amplitude has important implications for understanding the cosmological impact of PBHs and their abundance through the inflationary power spectrum. The scalability of this approach to various shapes of perturbation reveals a nuanced dependence that challenges the oversimplified conception of a fixed amplitude threshold.

Additionally, it underscores the necessity of distinguishing between local peak values and averaged perturbation amplitudes when considering PBH formation in cosmological models. This has practical significance for refining predictions about PBHs as viable dark matter candidates and exploring non-linear effects in early Universe dynamics.

Future Developments

Speculation on further AI research, while not explicitly addressed in this paper, might consider using similar computational frameworks to model complex cosmological phenomena, perhaps leveraging machine learning for predictive analyses based on perturbation profiles. Understanding PBH formation dynamically in such models could extend the insights offered here to a broader range of astrophysical and cosmological settings.

Overall, this paper contributes a refined and well-supported perspective on the conditions for primordial black hole formation, establishing groundwork for more detailed cosmic studies involving nonlinear dynamics.