How does gravity save or kill Q-balls?

We explore stability of gravitating Q-balls with potential $V_4(ϕ)={m^2\over2}ϕ^2-λϕ^4+\frac{ϕ^6}{M^2}$ via catastrophe theory, as an extension of our previous work on Q-balls with potential $V_3(ϕ)={m^2\over2}ϕ^2-μϕ^3+λϕ^4$. In flat spacetime Q-balls with $V_4$ in the thick-wall limit are unstable and there is a minimum charge $Q_{\rm min}$, where Q-balls with $Q<Q_{\rm min}$ are nonexistent. If we take self-gravity into account, on the other hand, there exist stable Q-balls with arbitrarily small charge, no matter how weak gravity is. That is, gravity saves Q-balls with small charge. We also show how stability of Q-balls changes as gravity becomes strong.

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