Scalar Field Wave Dark Matter and Galactic Halos

The question of ``What is Dark Matter?" has been a focus of cosmological research since the turn of the 20th century. Though the composition of Dark Matter is unknown, the existence of Dark Matter is crucial to the modern theory of cosmology. We focus on a theory of Dark Matter referred to as \textit{Scalar Field Wave Dark Matter} (SF$\psi$DM), which has received an increasing amount of interest from the research community since the late 2000s. SF$\psi$DM is a peculiar theory in which Dark Matter is composed of ultralight bosonic particles. As a result, SF$\psi$DM has an astronomically large deBroglie wavelength, generating complicated wave dynamics on the largest cosmological scales.

This thesis focuses on describing the status of SF$\psi$DM theory, SF$\psi$DM halos, and how SF$\psi$DM halos are affected by the wave-like features of the scalar field. In particular, we offer an analysis of galactic rotation curves and how they relate to SF$\psi$DM excited states. This analysis yields a novel model for an observed galactic trend referred to as the Baryonic Tully-Fisher Relation. Furthering this model, we formulate an eigenfunction decomposition which can be used to describe superpositions of excited states.