What happens to Q-balls if $Q$ is so large?

In the system of a gravitating Q-ball, there is a maximum charge $Q_{{\rm max}}$inevitably,whileinflatspacetimethereisnoupperboundon$Q$ in typical models such as the Affleck-Dine model. Theoretically the charge $Q$ is a free parameter, and phenomenologically it could increase by charge accumulation. We address a question of what happens to Q-balls if $Q$ is close to $Q_{\mathrm{m}\mathrm{a}\mathrm{x}}$. First, without specifying a model, we show analytically that inflation cannot take place in the core of a Q-ball, contrary to the claim of previous work. Next, for the Affleck-Dine model, we analyze perturbation of equilibrium solutions with $Q\approx Q_{\mathrm{m}\mathrm{a}\mathrm{x}}$ by numerical analysis of dynamical field equations. We find that the extremal solution with $Q=Q_{{\rm max}}$ and unstable solutions around it are “critical solutions”, which means the threshold of black-hole formation.