Axion Misalignment Abundance When the Bounce Caps the History Before Oscillation

The required initial angle is 13.80 at a decay constant of 10^10 GeV, 0.94 at 10^12 GeV, and 0.25 at 10^13 GeV. An angle of unity at 10^13 GeV would overproduce dark matter by the factor shown in the deposited table, with Omega h^2 reaching 1.99e+00. The turnaround itself occurs at a thermal scale of 2.82e+18 eV for the bound density of 9.72e+54 g per cubic centimetre, far above the oscillation epoch, so the field has a long, ordinary radiation-era interval in which a homogeneous displacement can persist unchanged. A required value above pi, the largest physical displacement of the field, means that at the lowest decay constant shown, misalignment with any allowed angle falls short of the full dark-matter density, and an additional production channel would be needed there. This study is computed from stated assumptions and published comparison values; it is not a measurement.