Equilibrium configurations for a self-gravitating scalar field with self-interaction are constructed. The corresponding Schr\"odinger-Poisson (SP) system is solved using finite differences assuming spherical symmetry. It is shown that equilibrium configurations of the SP system are late-time attractor solutions for initially quite arbitrary density profiles, which relax and virialize through the emission of scalar field bursts; a process dubbed gravitational cooling. Among other potential applications, these results indicate that scalar field dark matter models (in its different flavors) tolerate the introduction of a self-interaction term in the SP equations. This study can be useful in exploring models in which dark matter in galaxies is not point-like.Comment: 10 aastex pages, 12 eps figures. Accepted for publication in Ap

We study a spherically symmetric fluctuation of scalar dark matter in the cosmos and show that it could be the dark matter in galaxies, provided that the scalar field has an exponential potential whose overall sign is negative and whose exponent is constrained observationally by the rotation velocities of galaxies. The local space-time of the fluctuation contains a three-dimensional spacelike hypersurface with a surplus of angle.

In this paper, we develop a Newtonian approach to the collapse of galaxy fluctuations of scalar field dark matter under initial conditions inferred from simple assumptions. The full relativistic system, the so-called Einstein-Klein-Gordon system, is reduced to the Schrödinger-Newton one in the weak-field limit. The scaling symmetries of the SN equations are exploited to track the nonlinear collapse of single scalar matter fluctuations. The results can be applied to both real and complex scalar fields.