Applications of Cosmological Perturbation Theory

TL;DR

This paper explores advanced cosmological perturbation theory, revealing how second-order effects generate vorticity and magnetic fields, with implications for cosmic microwave background measurements and early universe magnetogenesis.

Contribution

It develops second and third order perturbation formalisms, including new equations for metric and gauge transformations, and uncovers the sourcing of vorticity at second order.

Findings

Vorticity is sourced by quadratic energy density and entropy perturbations at second order.

Vorticity decays at linear order but persists and grows at higher orders.

Derived the metric tensor and gauge transformation rules for third order perturbations.

Abstract

Cosmological perturbation theory is crucial for our understanding of the universe. The linear theory has been well understood for some time, however developing and applying the theory beyond linear order is currently at the forefront of research in theoretical cosmology. This thesis studies the applications of perturbation theory to cosmology and, specifically, to the early universe. Starting with some background material introducing the well-tested 'standard model' of cosmology, we move on to develop the formalism for perturbation theory up to second order giving evolution equations for all types of scalar, vector and tensor perturbations, both in gauge dependent and gauge invariant form. We then move on to the main result of the thesis, showing that, at second order in perturbation theory, vorticity is sourced by a coupling term quadratic in energy density and entropy perturbations. This source term implies a qualitative difference to linear order. Thus, while at linear order vorticity decays with the expansion of the universe, the same is not true at higher orders. This will have important implications on future measurements of the polarisation of the Cosmic Microwave Background, and could give rise to the generation of a primordial seed magnetic field. Having derived this qualitative result, we then estimate the scale dependence and magnitude of the vorticity power spectrum, finding, for simple power law inputs a small, blue spectrum. The final part of this thesis concerns higher order perturbation theory, deriving, for the first time, the metric tensor, gauge transformation rules and governing equations for fully general third order perturbations. We close with a discussion of natural extensions to this work and other possible ideas for off-shooting projects in this continually growing field.