Primordial black hole formation caused by the QCD transition ?

We consider the evolution of cosmological perturbations at the QCD transition, in particular the sudden reheating from a supercooled universe to the transition temperature. Sudden reheating happens at a specific temperature, hence density, and singles out one specific hypersurface. Underdensities reach the reheating earlier than overdensities, there is a short period of huge pressure differences which leads to a jump in the fluid velocity. Density perturbations of scales far below the Hubble radius $\lambda\ll R_H$ get an amplification which grows quadratically in wavenumber. Primordial black hole formation will not be sufficiently amplified by the QCD transition unless the initial spectrum is fine tuned.