Short review of the main achievements of the scalar field, fuzzy, ultralight, wave, BEC dark matter model

The Scalar Field Dark Matter model has been known in various ways throughout its history; Fuzzy, BEC, Wave, Ultralight, Axion-like Dark Matter, etc. All of them consist in proposing that dark matter of the universe is a spinless field Φ that follows the Klein-Gordon (KG) equation of motion □Φ − dV/dΦ = 0, for a given scalar field potential V. The difference between different models is sometimes the choice of the scalar field potential V. In the literature we find that people usually work in the non-relativistic, weak-field limit of the Klein-Gordon equation, where it transforms into the Schrödinger equation and the Einstein equations into the Poisson equation, reducing the KG-Einstein system, to the Schrödinger-Poisson system. In this paper, we review some of the most interesting achievements of this model from the historical point of view and its comparison with observations, showing that this model could be the last answer to the question about the nature of dark matter in the universe.