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.
Abstract While the motivation for inflationary cosmology comes from scalar perturbations (the source of anisotropies in the CMB and of structure formation, cf. Sect. 2.2 ), a future probe of this period might be offered by tensor perturbations, manifesting themselves as gravitational waves (cf. Eq. ( 3.106 )). A key property of gravitational waves is that they propagate almost freely until present time, i.e. that their transfer function is simple (cf. Eq. ( 2.18 )). Apart from the inflationary epoch, we discuss how gravitational waves could originate at or after reheating, from hydrodynamic fluctuations, second-order scalar perturbations, or elementary particle decays and scatterings. Many other sources have been proposed in the literature, such as preheating, topological defects, or phase transitions, but these are strongly model-dependent, and not discussed here. As gravitational-wave science will grow in importance in the next decades, we end the book with a summary of the various frequency domains that can hopefully be empirically investigated one day.
Abstract As time goes by, other matter components than the inflaton field play an increasingly important role. The expectation is that some of them should be Standard Model particles, interacting fairly strongly with each other, and eventually thermalizing, setting up the required environment for big-bang nucleosynthesis. The equilibrated system is called a primordial plasma, while the equilibration process, culminating in a radiation-dominated universe, is known as ‘reheating’. Introducing generic couplings between the inflaton and the plasma, we write down the corresponding background and perturbed equations. We show how a ‘seed’ temperature may emerge as a fixed point of the background solution already during the slow-roll stage of inflation. We demonstrate how reheating influences inflationary predictions, through the overall redshift between horizon crossing and the late universe. We indicate how interactions damp initial quantum fluctuations, but also generate new thermal fluctuations, via thermal noise.
Abstract Moving on from the idealized quasi de Sitter space-time to a more general situation in which the Hubble rate is a function of time, we show how the equation for the curvature perturbation can be set up in a manageable form. The concept of an overdamped ‘slow-roll’ regime is elaborated upon, and we show how observable quantities can be derived as a power series in small parameters. We demonstrate how the basic equations can be rephrased through the so-called stochastic formalism, introducing the notion of quantum noise. We also explain how the stochastic formalism can be simplified under a number of further assumptions, leading to a framework that has been used in the literature for simulating non-linear aspects of inflationary dynamics.
