Axion fragmentation on the lattice

We analyze the phenomenon of axion fragmentation when an axion field rolls over many oscillations of a periodic potential. This is particularly relevant for the case of relaxion, in which fragmentation provides the necessary energy dissipation to stop the field evolution. We compare the results of a linear analysis with the ones obtained from a classical lattice simulation, finding an agreement in the stopping time of the zero mode between the two within an O(1) difference. We finally speculate on the generation of bubbles with different VEVs of the axion field, and discuss their cosmological consequences.