Topologically protected sheet-like surfaces, called domain walls, form when the potential of a field has a discrete symmetry that is spontaneously broken. Since this condition is commonplace in field theory, it is plausible that many of these walls were produced at some point in the early universe. Moreover, for potentials with a rich enough structure, the walls can join and form a (at large scales) homogeneous and isotropic network that dominates the energy density of the universe for some time before decaying.
In this thesis, we study the faith of large scale perturbations in a cosmology with a short period of domain wall dominance. We start with some background material introducing the unperturbed universe, the physics of inflation and the basics on domain walls in cosmology. We also review the theory of cosmological perturbations in a medium with non-zero anisotropic stress. We then move on to our main result: treating the domain wall network as a relativistic elastic solid at large scales, we show that the perturbations that exited the horizon during inflation get suppressed during the domain wall era, before re-entering the horizon. This power suppression occurs because, unlike a fluid-like universe, a solid-like universe can support sizable anisotropic stress gradients across large scales which effectively act as mass for the scalar and tensor modes. Interestingly, the amplitude of the primordial scalar power spectrum can be closer to one in this cosmology and still give the observed value of 10^(-9) today. As a result, the usual bounds on the energy scale of inflation get relaxed to values closer to the (more natural) Planck scale.
In the last part of this thesis, as an existence proof, we present a hybrid inflation model with N "waterfall" fields that can realize the proposed cosmology. In this model, a domain wall network forms when an approximate O(N) symmetry gets spontaneously broken at the end of inflation, and for N ≥ 5, we show that there is a region in parameter space where the network dominates the energy density for a few e-folds before decaying and reheating the universe
