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Characteristic Size and Mass of Galaxies in the Bose-Einstein Condensate Dark Matter Model

We study the characteristic length scale of galactic halos in the Bose-Einstein condensate (or scalar field) dark matter model. Considering the evolution of the density perturbation we show that the average background matter density determines the quantum Jeans mass and hence the spatial size of galaxies at a given epoch. In this model the minimum size of galaxies increases while the minimum mass of the galaxies decreases as the universe expands. The observed values of the mass and the size of the dwarf galaxies are successfully reproduced with the dark matter particle mass $m\simeq 5\times 10^{-22}eV$. The minimum size is about $6\times 10^{-3}\sqrt{m/H}λ_c$ and the typical rotation velocity of the dwarf galaxies is $O(\sqrt{H/m}$) c, where $H$ is the Hubble parameter and $λ_c$ is the Compton wave length of the particle. We also suggest that ultra compact dwarf galaxies are the remnants of the dwarf galaxies formed in the early universe.