The radial field in the axion kinetic misalignment mechanisms

The axion kinetic misalignment mechanism introduces an initial velocity for the phase of a complex scalar field thereby creating axion dark matter with a lower decay constant. The introduction of a velocity in angular direction however gives rise to the question of what would happen if we also introduced a velocity in radial direction and allowed the radial field to change its value away from the PQ symmetry breaking scale fa. In this work we study the behaviour of the radial field by solving the coupled equations of motion for the radial and angular fields at different times. We allow the radial field to decay into Standard Model particles, which damps the velocity in radial direction.