More about Q-ball with elliptical orbit

Q-balls formed from the Affleck-Dine field have rich cosmological implications and have been extensively studied from both theoretical and simulational approaches. From the theoretical point of view, the exact solution of the Q-ball was obtained and it shows a circular orbit in the complex plane of the field value. In practice, however, it is reported that the Q-ball that appears after the Affleck-Dine mechanism has an $elliptical$ orbit, which carries larger energy per unit $U(1)$ charge than the well-known solution with a circular orbit. We call them "elliptical" Q-balls. In this paper, we report the first detailed investigation of the properties of the elliptical Q-balls by $3$D lattice simulation. The simulation results indicate that the elliptical Q-ball has an almost spherical spatial profile with no nodes, and we observed a highly elliptic orbit that cannot be described through small perturbations around the ground state Q-ball. Higher ellipticity leads to more excitation of the energy, whose relation is also derived as a dispersion relation. Finally, we derive two types of approximate solutions by extending the Gaussian approximation and considering the time-averaged equation of motion and we also show the consistency with the simulation results.