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

We revisit kinetic relaxation and soliton/boson star nucleation in fuzzy scalar dark matter featuring short-ranged self-interactions $H_{\mathrm{int}}=-\lambda |\psi {|}^4/2m^2$, alongside gravitational self-interactions. We map out the full curve of nucleation timescale for both repulsive ($\lambda <0$) and attractive ($\lambda >0$) short-ranged self-interaction strength and in doing so reveal two new points. Firstly, besides the two usual terms, $\propto G^2$ and $\propto {\lambda }^2$, in the total relaxation rate ${\Gamma }_{\text{relax}}$, there is an additional cross term $\propto G\lambda$ arising due to interference between gravitational and short-ranged self-interaction scattering amplitudes. This yields a critical repulsive interaction strength ${\lambda }_{\mathrm{cr}}\simeq -2\pi G{m}^2/v_0^2$, at which the relaxation rate is smallest and serves as the transition point between typical net attractive self-interaction ($\lambda \gtrsim {\lambda }_{\mathrm{cr}}$) and net repulsive self-interaction ($-\lambda \gtrsim -{\lambda }_{\mathrm{cr}}$). Secondly, while in the net attractive regime, nucleation timescale is similar to inverse relaxation timescale ${\tau }_{\mathrm{nuc}}\sim {\Gamma }_{\text{relax}}^{-1}$; in the net repulsive regime, nucleation occurs at a delayed time ${\tau }_{\mathrm{nuc}}\sim (\lambda /{\lambda }_{\mathrm{cr}}){\Gamma }_{\text{relax}}^{-1}$. We confirm our analytical understanding by performing 3D field simulations with varying average mass density $\overline{\rho }$, box size $L$ and grid size $N$.