Scalar Fields in Cosmology

This project is my masters thesis. I started to write it just as the first lockdown hit the UK in 2020 - which was pretty weird. But it all seemed worth it in the end as I was awarded the John Salmon Project Prize for this work. The prize recognizes this project as the top masters project of my year group!

The universe is strange. Everything is getting further away from everything else in every direction and nobody really knows why. The most widely accepted answer is that there is some kind of pressure caused by a mysterious dark energy throughout the universe. But there are problems associated with this theory. This project analyses how the universe would look if dark energy was a scalar field and if that model can help with any of the issues.

Abstract

This report provides an in depth investigation into the dynamics of dark energy using scalar fields with exponential potentials. We begin with an overview of the Friedmann equations and an explanation of the cosmological constant problem, which provides motivation for alternative models of dark energy. This prompts a discussion of quintessence and the use of exponential potentials, for which an exact solution is examined. The dynamics of a scalar field in the presence of a single background fluid are reviewed, then extended to facilitate a second background fluid and the equations of motion are presented. The fixed points of the system are analysed and provided for a general set of fluids $\gamma_1$ and $\gamma_2$. We derive a condition for measuring the time at which the scalar field will cause accelerated expansion, then modify the potential to include a second exponential term, transitioning the system from a scaling regime into an accelerating regime. This allows us to simulate the evolution of the universe, however high levels of fine tuning are found to be required to do so. Constraints are placed on the slopes of the potential and the initial conditions, with which we predict the universe will begin to accelerate at redshift $z_a = 0.61 \pm 0.22$. Further constraints are placed on the free parameters in the potential by using observed values for cosmological parameters, which again confirm the level of fine tuning. Finally we discuss possible applications of the model such as the Hubble tension and investigating domain walls. Using a naive method we find that domains walls must have $\Omega_d < 0.18 \pm 0.02$ at early times for the universe to evolve as expected. The appendix contains a full list of fixed points and eigenvalues for general fluids in each case analysed.