The possibility of the “ invisible ” axion being cold dark matter relies on the acceptable estimates of the current axion energy density .
The estimate depends on the nature of QCD phase transition at a few hundred MeV and the evolution of the misalignment angle \bar { \theta } .
The onset of \bar { \theta } oscillation undergoes a bottleneck period which occurred during the QCD phase transition .
In addition , the anharmonic coupling of order a ^ { 4 } affects the \bar { \theta } evolution .
From the time that the anharmonic effect is negligible , it is rather simple to calculate the ratio of \bar { \theta } ’ s between early and late times .
For multi GHz oscillations , the current age of the Universe needs at least 10 ^ { 27 } oscillations which limits an exact calculation of \bar { \theta } .
We establish a stepwise approximation for numerical solutions of the differential equation and obtain \bar { \theta } _ { now } / \bar { \theta } _ { f } \approx 3 \times 10 ^ { -17 } for m _ { a } \simeq 10 ^ { -4 } \textrm { eV } , where t _ { f } is the first time that the full hadronic phase ( after the QCD phase transition ) was established .