A Hydrodynamic Model of Galactic Halos

Abstract

The question about the existence of dark matter in the Universe is nowadays an open problem in cosmology. In this work we will present how we can build a hydrodynamic model in order to study dark matter halos of galaxies. The theoretical general idea is to start with the Einstein–Hilbert Lagrangian in which we have added a complex scalar field minimally coupled to the geometry. Then, by making variations of the corresponding action we come up with the Einstein field equations for the geometry and a Klein–Gordon equation for the scalar field. This set of coupled partial differential equations is non-linear. If we assume that dark matter halos can be described in the Newtonian limit we obtain a set of equations known as Schrödinger–Poisson equations. This set of equations can be written in the form of Euler equations for a fluid by making a Madelung transformation, where the self-interaction potential of the fluid is present. Also, there appears a quantum-like potential which depends non-linearly on the density of the fluid. We present results on the Jeans’ instability of the fluid model for dark matter and show how the physical parameters of the model can be determined, in particular, we compute the mass of the scalar field.