  In this thesis, we discuss some of the applications of cosmological
perturbation theory in the late universe. We begin by reviewing the tools used
to understand the standard model of cosmology theoretically and to compute its
observational consequences, including a detailed exposition of cosmological
perturbation theory. We then describe the results in this thesis; we present
novel analytical solutions for linear-order gravitational waves or tensor
perturbations in a flat Friedmann-Robertson-Walker universe containing two
perfect fluids -- radiation and pressureless dust -- and allowing for neutrino
anisotropic stress. One of the results applies to any sub-horizon gravitational
wave in such a universe. Another result applies to gravitational waves of
primordial origin (for example, produced during inflation) and works both
before and after they cross the horizon. These results improve on analytical
approximations previously set out in the literature. Comparison with numerical
solutions shows that both these approximations are accurate to within 1% or
better, for a wide range of wave-numbers relevant for cosmology. We present a
new and independent approach to computing the relativistic galaxy number counts
to second order in cosmological perturbation theory. We also derive analytical
expressions for the full second-order relativistic observed redshift, for the
angular diameter distance and the volume spanned by a survey. We then compare
our result with previous works which compute the general distance-redshift
relation, finding that our result is in agreement at linear and leading
nonlinear order. Lastly, we briefly study a class of almost scale-invariant
Gauss-Bonnet modified gravity theory and derive the Einstein-like field
equations to first order in cosmological perturbation theory in longitudinal
gauge.
