We study an overpopulated gas of gravitationally interacting bosons surrounding a droplet of Bose-Einstein condensate—the Bose star. We argue that kinetic evolution of this gas approaches with time a self-similar attractor solution to the kinetic equation. If the scale symmetry of the equation is broken by external conditions, the attractor solution exists, remains approximately self-similar, but has a slowly drifting scaling dimension. The latter new regime of adiabatic self-similarity can determine growth of dark matter Bose stars in cosmological models.

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• Dark matter
• Kinetic theory
• Turbulence
• Axion-like particles
• Bose-Einstein condensates
• Ultracold gases
• Numerical techniques