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 $\mathcal{H}_{\rm int} = -\lambda|\psi|^4/2m^2$, 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 $\Gamma_{\rm relax}$, 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 $\lambda_{\rm cr} \simeq - 2\pi Gm^2/v_{0}^2$, at which the relaxation rate is smallest and serves as the transition point between typical net attractive self-interaction ($\lambda \gtrsim \lambda_{\rm cr}$), and net repulsive self-interaction ($-\lambda \gtrsim -\lambda_{\rm cr}$). Secondly, while in the net attractive regime, nucleation time scale is similar to 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 our analytical understanding by performing 3D field simulations with varying average mass density $\bar{\rho}$, box size $L$ and grid size $N$.