The Probability for Primordial Black Holes

TL;DR

This work analyzes the likelihood of primordial black hole formation during an inflationary epoch using the no-boundary proposal. By evaluating complex Euclidean saddlepoints in minisuperspace for two topologies, $S^3$ (no holes) and $S^1 x S^2$ (holes), it shows that black holes are highly suppressed unless the initial cosmological constant is near the Planck scale. Extending to a dynamical effective cosmological constant from a massive scalar, it finds that black hole horizons can grow during inflation, but the overall probability remains tiny unless the initial scalar value is extremely large, implying that only Planck-scale primordial black holes have non-negligible formation probability. The results highlight the stringent constraints NB boundary conditions place on PBH formation in simple inflationary settings and set the stage for further exploration of the global spacetime structure.

Abstract

We consider two quantum cosmological models with a massive scalar field: an ordinary Friedmann universe and a universe containing primordial black holes. For both models we discuss the complex solutions to the Euclidean Einstein equations. Using the probability measure obtained from the Hartle-Hawking no-boundary proposal, we find that the only unsuppressed black holes start at the Planck size but can grow with the horizon scale during the roll down of the scalar field to the minimum.