Why the Universal Threshold for Primordial Black Hole Formation is Universal

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Why the Universal Threshold for Primordial Black Hole Formation is Universal

We show why the threshold for primordial black hole formation is universal (independent from the shape of the perturbation) when expressed in terms of the volume averaged compaction function. The proof is rooted in the self-similarity of the gravitational collapse phenomenon at criticality.

One. INTRODUCTION

The topic of Primordial Black Holes has become much debated in the last years since they might provide an explanation for some of the signals from binary black hole mergers measured by gravitational wave detectors and an important component of the dark matter in the universe. Furthermore, the next generation of gravitational wave experiments will provide an armoury of smoking guns to distinguish Primordial Black Holes from their astrophysical counterparts at high redshifts.

The most common scenario for the formation of the Primordial Black Holes is the one where they are originated from the collapse of large fluctuations formed during inflation upon horizon re-entry. The formation probability, and therefore their current abundance, is extremely sensitive to the critical threshold of collapse, that is on the minimum value Cc that the amplitude of the compaction function (or the energy overdensity) must have in order for a fluctuation to form a black hole.

Numerical simulations have shown that the critical threshold depends sensitively on the curvature at the peak of the compaction function, that is on the shape of the perturbation. The threshold of the compaction function varies from two-fifths for broad profiles to two-thirds for very peaked profiles. This is certainly not a pleasant feature since the theory delivers only stochastic quantities and one can only calculate the averaged profile of a perturbation. Furthermore, in realistic cases the critical threshold for formation is determined by the broadest possible compaction function, and not by the mean profile.

This ambiguity can be eliminated by the interesting observation made in Ref. Based on a numerical approach, it has been shown that the threshold to form Primordial Black Holes from an initial spherically symmetric perturbation is universal when the compaction function is averaged over a sphere of radius equal to the location of the maximum of the compaction function. Its critical value in radiation turns out to be two-fifths independently from the shape of the perturbation and within the numerical errors.

The goal of this short paper is to provide a simple proof of why such universal threshold for the formation of Primordial Black Holes is indeed universal and independent of the shape of the collapsing compaction function. As we shall see, our proof is rooted in the fundamental property of the phenomenon of gravitational collapse: the existence of attractors within the black hole threshold, that is, attractors of codimension one in phase space, and which are self-similar. More specifically, the persistence of self-similarity at criticality will play a key role.

Two. SOME BASIC CONCEPTS

Three. THE SELF-SIMILARITY AT CRITICALITY

Four. Why the Universal Threshold for Primordial Black Hole Formation is Universal

Overview

The paper explores how the threshold for primordial black hole formation remains consistent regardless of perturbation shapes, relying on the concept of self-similarity in gravity collapse at critical points. It aims to clarify the universal nature of this threshold through theoretical proof.

Key Points

1The threshold for primordial black hole formation is universal and independent of initial perturbation shapes
2The principle of self-similarity plays a crucial role in the gravitational collapse phenomenon
3The volume-averaged compaction function provides a reliable measure for determining the threshold
4Numerical simulations have confirmed that the critical threshold value remains constant across different scenarios
5Understanding this threshold has implications for explaining observations from gravitational wave detectors.

Details

Authors: Alex Kehagias, Davide Perrone, Antonio Riotto
Category: Physical Sciences

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