"Invisible" QCD axion rolling through the QCD phase transition

Visible matter in the current Universe is a consequence of the phase transition of the strong force, quantum chromodynamics (QCD). This phase transition has occurred at the Universe temperature around $T_c\simeq 165\,$MeV while it was expanding. Strongly interacting matter particles are quarks above $T_c$, while they are pions, protons and neutrons below $T_c$. The spin degrees of freedom 37 in the quark and gluon phase just above $T_c$ are converted to 3 (pions) after the phase transition. This phase transition might have been achieved mostly at supercooled temperatures. The supercooling was provided by the expansion of the Universe. We obtain the effective bubble formation rate $\alpha(T)\approx 10^{4-5}\,$MeV and the completion temperature of the phase change (to the hadronic phase), $T_f\simeq 126\,$MeV. During the phase transition, the scale factor $R$ has increased by a factor of 2.4. This provides a key knowledge on the energy density of "invisible" QCD axion at the full hadronic-phase commencement temperature $T_f$, and allows for us to estimate the current energy density of cold dark matter composed of "invisible" QCD axions.