Abstract Coordinate covariance is a cornerstone of general relativity. At first order in small transformations, it introduces degeneracies in the identification of space-time points in the homogeneous and isotropic description with the corresponding locations in a perturbed universe. These degeneracies are referred to as gauge invariance. In the literature, many gauge choices are employed. We show how the equations can be written in a manifestly gauge-invariant form; how this permits for powerful crosschecks of the computations; and how popular gauge choices can be recovered from the general equations.
Abstract We review the contents of the Einstein equations of general relativity. The ingredients needed for their left-hand side, the Einstein tensor, are explained. The right-hand side, energy-momentum tensor, is specified for typical systems appearing in early universe cosmology (weakly coupled scalar field, thermalized plasma, a coupled system). The concept of a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) universe is introduced, and prototypical ‘background’ solutions of the Einstein equations are displayed. We explain why a ‘spatially flat’ background is generally adopted as a sensible initial condition for the universe’s evolution, even if in the later sections perturbations of the spatial curvature are introduced and turn out to play a key role.
