A review of basic results on the Bose–Einstein condensate dark matter model

We review basic results on the Bose–Einstein condensate dark matter (BECDM) model. Self-gravitating BECs experience a collisionless process of gravitational cooling and violent relaxation, leading to BECDM halos with a “core-envelope” structure. The quantum core (soliton), which is the ground state of the Gross–Pitaevskii–Poisson (GPP) equations, may solve the core–cusp problem of the cold dark matter (CDM) model. The approximately isothermal envelope, resulting from the quantum interferences of the excited states, is similar to the Navarro–Frenk–White (NFW) profile of CDM halos and accounts for the flat rotation curves of the galaxies. We derive the core mass–radius relation, the halo mass–radius relation, and the core mass–halo mass relation of BECDM halos. We show that the core mass increases with the halo mass and we discuss the possibility that it collapses above a maximum mass arising from general relativity or from the attractive self-interaction of the bosons. We discuss the secular evolution of BECDM halos induced by the formation of granules (or quasiparticles) in the envelope, and we mention the analogy with the evolution of globular clusters. We also discuss the basic elements of BECDM cosmology. Throughout this review, we emphasize the importance of the maximum mass of dilute axion stars with an attractive self-interaction [P.H. Chavanis, Phys. Rev. D 84, 043531 (2011)] and its consequences.