Semianalytic Analysis of Primordial Black Hole Formation During a First-order QCD Phase Transition

It has recently been suggested that cosmologically significant numbers of black holes could form during a first-order QCD phase transition. Further, it has been asserted that these black holes would have masses corresponding naturally to the inferred mass ($\sim 1 M_{\odot}$) of the MACHOs responsible for the observed gravitational microlensing events. In this model, the underlying spectrum of primordial density perturbations provides the fluctuations that give rise to black holes at the epoch of the QCD transition. We employ a simplified model to estimate the reduction in the critical overdensity of a horizon-sized primordial perturbation required for collapse to a black hole. We find that a first-order QCD transition does indeed produce a sharp peak in the black hole mass spectrum, but that this peak corresponds to the horizon mass at an epoch somewhat earlier than the cosmological transition itself. Assuming a COBE normalized primordial density perturbation spectrum with constant spectral index, for the black holes so produced to be cosmologically significant would require an extremely finely tuned ``blue'' primordial density perturbation spectrum. Specifically, in the context of our simplified model, a spectral index in the range $n=1.37-1.42$ corresponds to the range $\Omega \sim 10^{-5}-10^3$ of the black hole contribution to the present-day density parameter.