Modified hoop conjecture in expanding spacetimes and primordial black hole production in FRW universe

TL;DR

This paper generalizes the hoop conjecture to expanding spacetimes, deriving a formula that predicts black hole formation from particle collisions in an FRW universe, and analyzes primordial black hole production rates.

Contribution

It introduces a new formula for black hole formation in expanding universes, extending the hoop conjecture to include effects of cosmic expansion.

Findings

The critical distance for black hole formation depends on the expansion rate.

In rapid expansion, the critical distance equals the particle horizon, ensuring causal contact.

Black hole production is significant at high temperatures but does not dominate the radiation background.

Abstract

According to a variant of the hoop conjecture, if we localize two particles within the Schwarzschild radius corresponding to their center of mass energy, then a black hole will form. Despite a large body of work on the formation of primordial black holes, so far this conjecture has not been generalized to expanding spacetimes. We derive a formula which gives the distance within which two particles must be localized to give a black hole, and which crucially depends on the expansion rate of the background space. In the limit of a very slow expansion, we recover the flat spacetime case. In the opposite limit of the large expansion rate when the inverse Hubble radius is smaller than the Schwarzschild radius of a "would be" black hole, the new critical distance between two particles that can make a black hole becomes equal to the particle horizon, which is just a requirement that the particles are in a causal contact. This behavior also nicely illustrates why the Big Bang singularity is not a black hole. We then use our formula to calculate the number density, energy density and production rate of black holes produced in collisions of particles. We find that though black holes might be numerous at high temperatures, they never dominate over the background radiation below the Planck temperature.