  We clarify the conditions of the cosmic quantum chromodynamics (QCD)
first-order phase transition in the early universe by carefully distinguishing
the chiral and deconfinement phase transitions. While the chiral one with light
quarks at zero chemical potential is unlikely to be first order based on the
recent lattice QCD calculations, the latter one can be naturally extended with
one extra rolling scalar to be first order. The argument is also valid for the
dark QCD theory with arbitrary $N_c$ with a wide range of phase transition
temperatures, which can be from hundreds of MeV up to beyond TeV. Notably, here
we derive the general formula for the deconfinement phase transition potential
of SU($N_c$) gauge theory characterized by the Polyakov loop. With the
effective potential in hand, the gravitational wave spectrum is then determined
via the sound shell model, which then enables us to give for the first time the
quantitative analysis of the gravitational wave signals coming from the QCD
deconfinement phase transition and awaits the check from future space
interferometers.
