3+1 Approach to Cosmological Perturbations : Deriving the First Order Scalar Perturbations of the Einstein Field Equations

Experimental data suggest that the universe is homogeneous and isotropic on sufficiently large scales. An exact solution of the Einstein field equations exists for a homogeneous and isotropic universe, also known as a Friedmann-Lemaître-Robertson-Walker (FLRW) universe. However, this model is only a first approximation since we know that, locally, the universe has anisotropic and inhomogeneous structures such as galaxies and clusters of galaxies. In order to successfully introduce inhomogeneities and anisotropies to the model one uses perturbative methods. In cosmological perturbations the FLRW universe is considered the zeroth order term in a perturbation expansion and perturbation theory is used to derive higher order terms which one tries to match with observations. In this thesis I present a review of the main concepts of general relativity, discuss the 3+1 formalism which gives us the Einstein field equations in a useful form for the perturbative analysis, and lastly, I derive the first order scalar perturbations of the Einstein field equations.