This chapter starts with a description of quantum tunneling as a process taking place in imaginary or even complex time. This physical picture can be extended to include gravity, which leads to a description of Coleman–DeLuccia instantons and the nucleation of bubble universes. The mathematical analysis is complemented by a derivation of negative modes, which puts the tunneling process on a firm theoretical footing. Very similar methods can also describe the decay of spacetime via bubbles of nothing. A semiclassical view of spacetime may also lead to the existence of wormholes, of both the Lorentzian and the Euclidean variety. Their properties as well as associated puzzles are discussed in detail.

Provides a review of how the standard model of cosmology is built up, emphasizing the interplay between theory and observations. The Robertson–Walker line element is derived and used to find the Friedmann equations. Elementary solutions are discussed. In this way the hot big bang model emerges. Its implications are discussed, especially the thermal history of the universe and the existence of the cosmic microwave background radiation. The chapter concludes with a discussion of the main puzzles of the hot big bang model.

How can one describe the appearance of space and time? This chapter reviews the no-boundary proposal, which allows for a concrete calculation of the nucleation of space and time from nothing. After providing heuristic motivations for this idea, concrete examples are presented, and the stability of solutions as well as the numerical methods required to find generic solutions are discussed. A general prescription for characterizing no-boundary instantons is developed, before examining explicit minisuperspace models. A special emphasis is put on the appropriate boundary conditions, both in the path integral formalism and in the Wheeler–DeWitt equation. The robustness of solutions upon the inclusion of expected quantum gravity corrections is discussed, as well as the question of which kinds of complex metrics should be allowed. This leads to a discussion of both postdictions and predictions of the proposal.

A variational principle for gravity, based on the Einstein–Hilbert action, is presented and augmented with a discussion of surface terms and boundary conditions. The ADM or Hamiltonian formalism is introduced, and gravity is rewritten in a (1+3)-dimensional decomposition. The theory is canonically quantized, which leads to the Wheeler–DeWitt equation. The properties of this equation are discussed, as well as those of JWKB semiclassical solutions. In this way it is shown how time is recovered in a semiclassical setting.

A link between horizons, imaginary time, and temperature is developed at the heuristic level first, before being made precise in the following sections with the use of Bogolyubov transformations. This leads to the derivation of the Unruh effect, which shows that an accelerated observer experiences a temperature. Analogous methods allow one to derive the phenomenon of Hawking radiation by which black holes can evaporate, and an explicit calculation of the closely related Hawking–Page transition is provided via path integral methods in which the background spacetime is also quantized. It is further shown that due to the existence of a horizon, one may in the same way associate a temperature with de Sitter spacetime. An explicit discussion of de Sitter mode functions is included, because it relates directly to the quantization of inflationary fluctuations.

After an overview of the observed properties of the cosmic microwave background, we turn to attempts at their explanation. First it is shown that classical statistical fluctuations are not suitable for explaining the primordial perturbations inferred from the temperature fluctuations in the CMB. Then it is shown how to quantize inflationary perturbations, after taking care of diffeomorphism invariance. Exact constant-equation-of-state and approximate slow-roll solutions are derived, both for scalar and tensor fluctuations, and shown to potentially be in accord with observations, if the inflationary model is chosen suitably. A brief discussion of the transition from quantum to effectively classical fluctuations is also included. The chapter concludes with a discussion of the open questions related to inflation.

In the final chapter, the basics of string cosmology are introduced. After a lightning review of string theory, the potential existence of extra dimensions is discussed in some detail. A special emphasis is put on the possible observational signatures of towers of massive Kaluza–Klein modes due to their effects during inflation and in the early universe in general. Then branes are presented as solutions to low-energy approximations to string theory. The difficulties with constructing models of brane inflation are illustrated with a specific example. Finally, a collision of end-of-the-world branes as a model of the big bang is analyzed.

The path integral quantization of gravity is developed, with an emphasis on physical intuition. It is shown how to deal with gauge transformations, and how this results in an ordinary integration over the lapse function. This formalism is applied to minisuperspace models, and transitions in a setting with a cosmological constant are explored for both Dirichlet and Neumann boundary conditions. The relation between the path integral and the Wheeler–DeWitt equation is derived, as well as the rules for composing transition amplitudes.

Inflation is introduced as a possible resolution of the flatness and horizon puzzles. Scalar field dynamics are discussed, and both exact constant-equation-of-state solutions and slow-roll inflationary solutions are presented. This chapter also includes a description of quadratic, Starobinsky, and Higgs inflation, before discussing the end of inflation and reheating. A rather drastically different alternative, ekpyrosis, is also presented. The associated cyclic model of the universe is introduced, and the difficulties in describing cosmic bounces from a contracting to an expanding phase are analyzed.