Applications of Cosmological Perturbation Theory in the Late Universe

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.