  Can a violent process like sudden reheating after supercooling at the onset
of a first-order QCD transition improve the possibility of primordial black
hole formation? Underdensities reheat earlier than overdensities, there is a
short period of huge pressure differences, hence fluid acceleration. Density
perturbations on scales far below the Hubble radius $\lambda\ll R_{\rm H} $ get
an amplification which grows quadratically in wavenumber, the amplifications at
the horizon scale are small. Primordial black hole formation cannot be
sufficiently amplified by the QCD transition unless the initial spectrum is
fine tuned.
