On Nonlinear Schrödinger Equation as a Model for Dark Matter

Abstract

In this chapter, we present an overview of the problem of dark matter and the scalar field dark matter model, which assumes the existence of a cosmological matter wave describing a condensate of ultralight axions. The mathematical description is in terms of a nonlinear Schrodinger-Poisson system of equations. We introduce the framework in a pedagogical way, for readers interested in nonlinear science assuming no prior knowledge of cosmology. We describe a split-step pseudospectral numerical method which is useful to compute the evolution in time of dark matter distributions. We then discuss two aspects of the model: an explanation of the so-called offsets between dark matter and stars in galactic clusters and the laws relating supermassive black holes and dark matter distributions. Finally, we emphasize the formal connections to particular situations of other physical systems, including cold atom Bose-Einstein condensates and laser beam propagation in thermo-optical media, which may lead to tabletop laboratory analogues of cosmological phenomena.