Observations have shown that the universe is currently expanding at an accelerating rate due to the presence of a dark energy. A slowly rolling scalar field, known as quintessence, has been proposed as the source of this dark energy. In this thesis I have developed a suite of computer codes that numerically solve the differential equations needed to describe the evolution of this dynamical field. By comparing my theoretically derived results with observations of baryon acoustic oscillations I have discovered the boundaries in solution space which divide constant dark energy solutions from dynamical dark energy solutions. The predictions made by these models can easily be verified, or falsified, by recent and upcoming experiments.