Abstract
The $(3+1)$-dimensional evolution of an inhomogeneous axion field configuration around the QCD epoch is studied numerically, including important non-linear effects due to the attractive self-interaction. It is found that axion perturbations on scales corresponding to causally disconnected regions at $T \sim 1 \, {\rm GeV}$ can lead to very dense pseudo-soliton configurations we call axitons. These configurations evolve to axion miniclusters with present density $\rho_a \ga 10^{-8}\,{\rm g \, cm^{-3}}$. This is high enough for the collisional $2a \rightarrow 2a$ process to lead to Bose--Einstein relaxation in the gravitationally bound clumps of axions, forming Bose stars.
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