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The QCD axion is among the best motivated candidates for Dark Matter. In a scenario in which the Peccei-Quinn symmetry is restored after inflation, the axion field acquires random initial values in causally disconnected patches of our universe. When the axion potential develops around the QCD phase transition, fluctuations in the axion field are transferred into order 1 differences in the density contrast on comoving scales of roughly 0.02 pc. Besides, the decay of cosmic strings and domain walls, which are present as remnant of the phase transition, might add further inhomogeneities to the axion density. The regions of high overdensity collapse already around matter radiation equality, forming so called axion miniclusters.
The existence of axion miniclusters is crucial to the outcome of axion dark matter direct detection experiments but also of possible indirect signatures. In order to accurately predict the properties of miniclusters detailed knowledge of the density contrast previous to gravitational collapse is crucial.
In this talk I explain the production of axions from misalignment, string and wall decay and the difficulties in modeling these processes numerically. I continue by showing recent results of our numerical simulations, which follow the evolution of the axion field around the time of the QCD phase transition and determine the resulting density contrast, for the first time including all three relevant production processes. Our simulations indicate that the inclusion of strings and domain walls puts a lot of fluctuation power in scales which are smaller than the horizon at the time of the QCD phase transition and we expect a large hierarchy of masses extending down to those smaller scales.