Bosonic ultra-light dark matter (ULDM) would form cored density distributions at the center of galaxies. These cores, seen in numerical simulations, admit analytic description as the lowest energy bound state solution (soliton) of the Schroedinger-Poisson equations. Numerical simulations of ULDM galactic halos found empirical scaling relations between the mass of the large-scale host halo and the mass of the central soliton. We discuss how the simulation results of different groups can be understood in terms of the basic properties of the soliton. Importantly, simulations imply that the energy per unit mass in the soliton and in the virialised host halo should be approximately equal. This relation lends itself to observational tests, because it predicts that the peak circular velocity, measured for the host halo in the outskirts of the galaxy, should approximately repeat itself in the central region. Contrasting this prediction to the measured rotation curves of well-resolved near-by galaxies, we show that ULDM in the mass range $msim (10^{-22}div 10^{-21})$ eV, which has been invoked as a possible solution to the small-scale puzzles of $Lambda$CDM, is in tension with the data. We suggest that a dedicated analysis of the Milky Way inner gravitational potential could probe ULDM up to $mlesssim 10^{-19}$ eV.
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