The abundance of primordial black holes depends on the shape of the inflationary power spectrum

Abstract: In this letter, combining peak theory and the numerical analysis of gravitational collapse in the radiation dominated era, we show that the abundance of primordial blacks holes, generated by an enhancement in the inflationary power spectrum, is extremely dependent on the shape of the peak. Given the amplitude of the power spectrum, we show that the density of primordial black holes generated from a narrow peak, is exponentially smaller than in the case of a broad peak. Specifically, for a top-hat profile of the power spectrum in Fourier space, we find that for having primordial black holes comprising all of the dark matter, one would only need a power spectrum amplitude an order of magnitude smaller than suggested previously whereas in the case of a narrow peak, one would instead need a much larger power spectrum amplitude, which in many cases would invalidate the perturbative analysis of cosmological perturbations. Finally, we show that, although critical collapse gives a broad mass spectrum, the density of primordial black holes formed is dominated by masses roughly equal to the cosmological horizon mass measured at horizon crossing.

• The paper demonstrates that the abundance of primordial black holes critically depends on the shape of the inflationary power spectrum, with broad peaks enhancing formation.
• It employs peak theory and numerical simulations to show that a broad peak reduces the required amplitude for PBH formation by an order of magnitude compared to a narrow peak.
• The results challenge previous threshold estimates and refine the predicted PBH mass spectrum centered near the cosmological horizon mass at crossing.

The Influence of Inflationary Power Spectrum Shape on Primordial Black Hole Abundance

The paper by Cristiano Germani and Ilia Musco presents a detailed examination of the abundance of primordial black holes (PBHs) and their dependency on the shape of the inflationary power spectrum. This study employs peak theory in conjunction with numerical simulations of gravitational collapse during the radiation-dominated era to assess how variations in the inflationary power spectrum can significantly impact the formation of PBHs.

Key Findings and Numerical Results

1. Dependence on Power Spectrum Shape: The abundance of PBHs is found to be highly sensitive to the power spectrum's configuration. Specifically, the study reveals that a narrow peak in the power spectrum results in exponentially fewer PBHs compared to a broad peak. Numerical simulations indicate that for broad peaks, the amplitude required for PBH formation can be an order of magnitude smaller than previously assumed.
2. Threshold Values and Critical Collapse: The paper shows the pivotal role played by the threshold of the energy density peak in PBH formation. The simulations unveil that for varying power spectrum shapes, the critical density profile remains consistent, leading to stable threshold values for PBH formation. The findings challenge earlier estimates of these thresholds, emphasizing the need for revised computations.
3. Mass Spectrum: PBH masses, influenced by the shape of the power spectrum, are primarily on the order of the cosmological horizon mass at horizon crossing. The critical collapse mechanism provides a broad mass spectrum, but the peak remains centered around this mass.

Implications and Future Research Directions

The implications of this research are twofold. Practically, the findings suggest that models aiming to explain dark matter entirely through PBHs should consider the broad spectrum configuration due to its efficiency in PBH formation with lesser amplitude. Theoretically, the study advocates for enhanced scrutiny of inflationary models, especially regarding the perturbative analysis.

Future directions involve comprehensively exploring the contributions of non-Gaussianities to PBH formation, as these perturbations might affect the abundance and mass spectrum significantly. Additionally, extending simulations to consider non-spherical deformations in primordial perturbations could yield further insights into the formation pathways of PBHs.

This paper invites reconsiderations in both cosmological perturbation theory and observational strategies aimed at identifying PBH-driven dark matter, underscoring the necessity of integrating detailed spectrum shape analyses into theoretical models and simulations.