Mikko Laine

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.

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.

We describe how the left and right-hand sides of the Einstein equations can be perturbed to first order around the background solution. The perturbations are then decomposed into scalar, vector and tensor components under helicity , or two-dimensional rotations. The resulting Einstein equations for the different types of perturbations are displayed. We demonstrate how taking linear combinations of the Einstein equations, or their integrals (if an equation relates two total derivatives), or their derivatives (like when considering energy-momentum conservation), can lead to simpler equations. We also illustrate how anisotropic stress can be included within the framework of viscous hydrodynamics, and how the steps mentioned can be implemented with computer algebra.

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.

Combining Einstein equations and a scalar field evolution equation, obtained in Chap. 3 , we derive a Mukhanov-Sasaki equation for a gauge-invariant curvature perturbation, defined in Chap. 4 . Proceeding to its solution, we review the case in which the inflationary period corresponds to an almost de Sitter space-time. We show, analytically, how the amplitude and the phase of the perturbations can be fixed in the distant-past vacuum, and how the perturbations evolve as a function of conformal time, until they cross outside of the Hubble horizon. We demonstrate that this produces an almost scale-invariant spectrum of curvature perturbations, which can subsequently be compared with observational data, as they were reviewed in Chap. 2 .

The equations for scalar perturbations from Chap. 7 have a fairly complicated structure. We explain how their solution gets simplified once the wavelength redshifts enough that the mode ‘exits the Hubble horizon’, whereby it effectively decouples from local dynamics. Subsequently, the nature of the background solution may change, from vacuum-energy dominated expansion, via a possible intermediate period, ultimately into a radiation-dominated universe. A key feature of the inflationary paradigm is that the cosmological perturbations visible in the CMB are outside of the Hubble horizon throughout these processes, and are almost unaffected by them, or ‘frozen’. We also explain how the frozen solution may have a richer structure in multi-field inflationary models, leading to the appearance of so-called isocurvature perturbations.

Our current universe is homogeneous and isotropic on scales that are larger than about 1% of its visible size, and spatially flat to a good precision. Its structure at smaller scales may have formed via gravitational collapse of early inhomogeneities. Reviewing experimental evidences for the universe until the emission of the Cosmic Microwave Background (CMB) radiation and the onset of structure formation, we collect the target quantities that the theoretical framework presented in the book aims to describe. The relations between present-day frequencies and wavelengths and those that play a role in the early universe are summarized. Finally, we derive basic formulae for the statistical description of small perturbations on a homogeneous and isotropic background, and compare the procedures adopted in different contexts.

Motivated by axion-like inflation and its warm embedding within the Standard Model, we study the early stages of the energy transfer between an axion condensate and an SU(2) gauge ensemble, by employing non-linear classical real-time lattice simulations. The discretized equations of motion are worked out, elaborating on Gauss constraints. A numerical solution is implemented on the CosmoLattice platform. Adopting a quadratic potential, and omitting universe expansion, we establish initial exponential growth of the low-momentum gauge modes; damping of axion oscillations after some delay; and subsequent energy equipartition between axion and gauge ensembles. A clear difference between the SU(2) and U(1) dynamics is observed, likely associated with non-Abelian self-interactions. We elaborate on what this implies for the possible thermalization of the SU(2) ensemble.

A bstract: Axions coupled to thermal non-Abelian gauge fields may have cosmological significance. As the heat bath defines a frame, its influence depends separately on energy and momentum. A light-like momentum ( k ≈ ω ) is relevant for the axion contribution to the effective number of light neutrinos, ∆ N _eff , whereas a vanishing momentum ( k = 0) plays a role for warm natural inflation or ultralight dark matter, and has been employed in lattice estimates (both classical and quantum-statistical) of the strong sphaleron rate. Focussing on soft energies ( α _s T ≪ ω ≪ πT ), we carry out an HTL computation to show how the domains k = 0 and k ≈ ω interpolate to each other. We then compare with lattice data at k = 0, and connect our analysis to NLO computations at k ≈ ω ≥ πT . Assembling the current best input, we re-investigate light QCD axion decoupling dynamics at T ≥ 200 MeV, showing that efficient interactions in the ultrasoft domain increase ∆ N _eff from ∼ 0.03 to ∼ 0.04 at f _a = 4 × 10 ⁸ GeV.

Working in an arbitrary gauge, we derive the source term for scalar-induced gravitational waves (SIGW) valid during a general reheating epoch. The dominant energy component is allowed to transition smoothly from an inflaton field to a set of fluids, possibly via a period of matter domination. Gauge invariance is verified up to second order.