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The possibility that dark matter may be in the form of a Bose-EinsteinCondensate (BEC) has been extensively explored at galactic scale. Inparticular, good fits for the galactic rotations curves have been obtained, andupper limits for the dark matter particle mass and scattering length have beenestimated. In the present paper we extend the investigation of the propertiesof the BEC dark matter to the galactic cluster scale, involving dark matterdominated astrophysical systems formed of thousands of galaxies each. Byconsidering that one of the major components of a galactic cluster, theintra-cluster hot gas, is described by King's $\beta$-model, and that bothintra-cluster gas and dark matter are in hydrostatic equilibrium, bound by thesame total mass profile, we derive the mass and density profiles of the BECdark matter. In our analysis we consider several theoretical models,corresponding to isothermal hot gas and zero temperature BEC dark matter,non-isothermal gas and zero temperature dark matter, and isothermal gas andfinite temperature BEC, respectively. The properties of the finite temperatureBEC dark matter cluster are investigated in detail numerically. We compare ourtheoretical results with the observational data of 106 galactic clusters. Usinga least-squares fitting, as well as the observational results for the darkmatter self-interaction cross section, we obtain some upper bounds for the massand scattering length of the dark matter particle. Our results suggest that themass of the dark matter particle is of the order of $\mu $eV, while thescattering length has values in the range of $10^{-7}$ fm.

Testing the Bose-Einstein Condensate dark matter model at galactic cluster scale