Short Review of the main achievements of the Scalar Field, Fuzzy, Ultralight, Wave, BEC Dark Matter model

Understand

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 the dark matter of the universe is a spinless field $\Phi$ that follows the Klein-Gordon (KG) equation of motion $\Box\Phi-dV/d\Phi=0$, 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 nonrelativistic, weak-field limit of the KG equation where it transforms into the Schr\"odinger equation and the Einstein equations into the Poisson equation, reducing the KG-Einstein system, to the Schr\"odinger-Poisson system.