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, while in flat spacetime there is no upper bound on $Q$ 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 $Q_{\rm max}$. 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 $Q\approx Q_{\rm max}$ 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.

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