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We revisit kinetic relaxation and soliton/Boson star nucleation in fuzzyscalar dark matter featuring short-ranged self-interactions $\mathcal{H}_{\rmint} = -\lambda|\psi|^4/2m^2$, alongside gravitational self-interactions. Wemap out the full curve of nucleation timescale for both repulsive ($\lambda <0$) and attractive ($\lambda > 0$) short-ranged self-interaction strength, andin doing so reveal two new points. Firstly, besides the two usual terms,$\propto G^2$ and $\propto \lambda^2$, in the total relaxation rate$\Gamma_{\rm relax}$, there is an additional cross term $\propto G\lambda$arising due to interference between gravitational and short-rangedself-interaction scattering amplitudes. This yields a critical repulsiveinteraction strength $\lambda_{\rm cr} \simeq - 2\pi Gm^2/v_{0}^2$, at whichthe relaxation rate is smallest and serves as the transition point betweentypical net attractive self-interaction ($\lambda \gtrsim \lambda_{\rm cr}$),and net repulsive self-interaction ($-\lambda \gtrsim -\lambda_{\rm cr}$).Secondly, while in the net attractive regime, nucleation time scale is similarto inverse relaxation time scale $\tau_{\rm nuc} \sim \Gamma^{-1}_{\rm relax}$,in the net repulsive regime nucleation occurs at a delayed time $\tau_{\rm nuc}\sim (\lambda/\lambda_{\rm cr})\Gamma^{-1}_{\rm relax}$. We confirm ouranalytical understanding by performing 3D field simulations with varyingaverage mass density $\bar{\rho}$, box size $L$ and grid size $N$.

Kinetic relaxation and nucleation of Bose stars in self-interacting wave dark matter