Primordial black hole formation during the QCD phase transition: threshold, mass distribution and abundance

Primordial black hole (PBH) formation during cosmic phase transitions and annihilation periods, such as the QCD transition or the $e^+e^-$-annihilation, is thought to be particularly efficient due to a softening of the equation of state. We present a detailed numerical study of PBH formation during the QCD epoch in order to derive an accurate PBH mass function. We also briefly consider PBH formation during the $e^+e^-$-annihilation epoch. Our investigation confirms that, for nearly scale-invariant spectra, PBH abundances on the QCD scale are enhanced by a factor $\sim 10^3$ compared to a purely radiation dominated Universe. For a power spectrum producing an (almost) scale-invariant PBH mass function outside of the transition, we find a peak mass of $M_{\rm pbh}\approx 1.9 M_{\odot}$ with a fraction $f\approx 1.5\times 10^{-2}$ of the PBHs having a mass of $M_{\rm pbh} > 10 M_{\odot}$, possibly contributing to the LIGO-Virgo black hole merger detections. We point out that the physics of PBH formation during the $e^+e^-$-annihilation epoch is more complex as it is very close to the epoch of neutrino decoupling. We argue that neutrinos free-streaming out of overdense regions may actually hinder PBH formation.