  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. 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 $N \geq 5$, 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.
