The main goal of this chapter is the calculation of the noise kernel in de Sitter spacetime, in a de Sitter-invariant vacuum. The geometry of most inflationary models is well approximated by the de Sitter geometry. For this reason, fluctuations around de Sitter and near-de Sitter spacetimes have been extensively studied in the context of inflationary models. Here we study the stress-energy tensor fluctuations of the matter fields described by the noise kernel. We start by reviewing the basic geometric properties of de Sitter spacetime and the invariant bitensors that will be used in this and in later chapters. These tools are employed to write the noise kernel for spacelike separated points in de Sitter-invariant form, and explicit expressions for the case of a free minimally coupled scalar field are derived. Closed results in terms of elementary functions are given for the particular cases of small masses, vanishing mass and large separations. A massless limit discontinuity is found, and is analyzed in some detail. Finally, we discuss the implications of our results for the quantum metric fluctuations around de Sitter spacetime.

In this chapter we present the Schwinger–Keldysh effective action in the so-called ‘in-in’, or ‘closed-time-path’ (CTP) formalism necessary for the derivation of the dynamics of expectation values. The real and causal equation of motion derived therefrom ameliorates the deficiency of the ‘in-out’ effective action which produces an acausal equation of motion for an effective geometry that is complex, thus marring the physical meaning of effects related to backreaction, such as dissipation. We construct the in-in effective action for quantum fields in curved spacetime, show that the regularization required is the same as in the in-out formulation, and show how it can be used to treat problems in nonequilibrium quantum processes by considering initial states described by a density matrix. We then show two applications: (1) the damping of anisotropy in a Bianchi Type I universe from the semiclassical Einstein equation for conformal fields derived from the CTP effective action; and (2) higher-loop calculations, renormalization of the in-in effective action, and the calculation of vacuum expectation values of the stress-energy tensor for a Phi-4 field. The last part describes thermal field theory in its CTP formulation.

We begin with a brief description of the work on (a) the regularization of the stress-energy tensor of quantum fields in Schwarzschild spacetime in the 80s and (b) the black hole end-state and information-loss issues in the 80s, the ‘black hole complementarity principle’ of the 90s and the recent ‘firewall’ conjecture and its controversies. We then treat two classes of problems: (1) the backreaction of Hawking radiation on a black hole in the quasi-stationary regime, which occupies the longest span of a black hole’s life, and (2) the metric fluctuations of the event horizon of an evaporating black hole. In (1) the far field case can be solved analytically via the influence functional, highlighting nonlocal dissipation and colored noise; for the near horizon case we describe a strategy by Sinha et al. for treating the backreaction and fluctuations. In (2) we describe Bardeen’s model and discuss the results of Hu and Roura, who reached the same conclusion as Bekenstein, namely, that even for states regular on the horizon the accumulated fluctuations become significant by the time the black hole mass has changed substantially, well before reaching the Planckian regime. These results have direct implications for the end-state issue.

This chapter presents the familiar Schwinger–DeWitt effective action in the ‘in-out’ formalism, suitable for the computation of S-matrix scattering or transition amplitudes. The effective action method is well suited to the treatment of backreaction problems for quantum processes in dynamical background spacetimes, as it yields equations of motion for both the quantum field and the spacetime in a self-consistent way. In the second part, after a quick refresh of basic field theory and quantum fields in curved spacetime, we construct the ‘in-out’ effective action of an interacting quantum field and apply it to the effects of particle creation and interaction in the Friedmann–Lemaitre–Robertson–Walker universe. We illustrate how dimensional regularization is implemented. The third part treats the case where changes in the background spacetime and fields are gradual enough that one can perform a derivative expansion beyond the constant background, introduce momentum space representation for the propagators and obtain a quasi-local effective action in a closed form. The fourth part discusses dimensional regularization and the derivation of renormalization group equations, using the Phi-4 theory as an example.

As a second application of stochastic gravity, we discuss in this chapter the backreaction problem in cosmology when the gravitational field couples to a quantum conformal matter field, and derive the Einstein–Langevin equations describing the metric fluctuations on the cosmological background. Conformal matter may be a reasonable assumption, because matter fields in the standard model of particle physics are expected to become effectively conformally invariant in the very early universe. We consider a weakly perturbed spatially flat Friedman–Lemaitre–Robertson–Walker spacetime and derive the Einstein–Langevin equation for the metric perturbations off this spacetime, using the CTP functional formalism described in previous chapters. With this calculation we also obtain the probability for particle creation. The CTP effective action is also used to derive the renormalized expectation value of the quantum stress-energy tensor and the corresponding semiclassical Einstein equation.

In this chapter we study the backreaction problem in early universe cosmology, i.e., finding solutions to the semiclassical Einstein equation, which is at the heart of semiclassical gravity theory. Four groups of backreaction problems at the Planck and grand unified theory scales are presented. (1) Effects of the trace anomaly (a) in facilitating possible avoidance of singularity and (b) in engendering inflation, as in the so-called Starobinsky inflation. (2) Effects of particle creation on cosmological singularity and particle horizons, in affecting the equation of state of matter, and in the damping of anisotropy or inhomogeneity. (3) For inflationary cosmology, post-inflationary preheating by the dissipative effects of particle creation and interaction from the nonequilibrium inflaton dynamics, using an O(N) Phi-4 theory as an example. (4) We also mention backreaction problems in stochastic inflation, where the short wavelength modes acting as noise backreact on the long wavelength modes, thereby decohering the latter into classical background modes (5) In terms of quantum cosmology, we consider the validity of minisuperspace approximation by studying the effect of inhomogeneous modes on the homogeneous mode in a Phi-4 model.

In this chapter we focus on the stress-energy bitensor and its symmetrized product, with two goals: (1) to present the point-separation regularization scheme, and (2) to use it to calculate the noise kernel that is the correlation function of the stress-energy bitensor and explore its properties. In the first part we introduce the necessary properties and geometric tools for analyzing bitensors, geometric objects that have support at two separate spacetime points. The second part presents the point-separation method for regularizing the ultraviolet divergences of the stress-energy tensor for quantum fields in a general curved spacetime. In the third part we derive a formal expression for the noise kernel in terms of the higher order covariant derivatives of the Green functions taken at two separate points. One simple yet important fact we show is that for a massless conformal field the trace of the noise kernel identically vanishes. In the fourth part we calculate the noise kernel for a conformal field in de Sitter space, both in the conformal Bunch–Davies vacuum and in the static Gibbons–Hawking vacuum. These results are useful for treating the backreaction and fluctuation effects of quantum fields.

In this chapter we describe an important application of stochastic gravity: we derive the Einstein–Langevin equation for the metric perturbations in a Minkowski background. We solve this equation for the linearized Einstein tensor and compute the associated two-point correlation functions, as well as the two-point correlation functions for the metric perturbations. The results of this calculation show that gravitational fluctuations are negligible at length scales larger than the Planck length and predict that the fluctuations are strongly suppressed at small scales. These results also reveal an important connection between stochastic gravity and the 1/N expansion of quantum gravity. In addition, they are used to study the stability of the Minkowski metric as a solution of semiclassical gravity, which constitutes an application of the validity criterion introduced in the previous chapter. This calculation requires a discussion of the problems posed by the so-called runaway solutions and some of the methods of dealing with them.

Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor, stochastic gravity is based on the Einstein–Langevin equation, which in addition has sources due to the noise kernel. The noise kernel is a bitensor which describes the quantum stress-energy tensor fluctuations of the matter fields. In this chapter we describe the fundamentals of this theory using an axiomatic and a functional approach. In the axiomatic approach, the equation is introduced as an extension of semiclassical gravity motivated by the search for self-consistent equations describing the backreaction of the stress-energy fluctuations on the gravitational field. We then discuss the equivalence between the stochastic correlation functions for the metric perturbations and the quantum correlation functions in the 1/N expansion, and illustrate the equivalence with a simple model. Based on the stochastic formulation, a criterion for the validity of semiclassical gravity is proposed. Alternatively, stochastic gravity is formulated using the Feynman–Vernon influence functional based on the open quantum system paradigm, in which the system of interest (the gravitational field) interacts with an environment (the matter fields).

In this epilogue we place the theories of semiclassical and stochastic gravity in perspective, exploring their linkage with quantum gravity, defined as theories for the microscopic structures of spacetime, not necessarily and most likely not from quantizing general relativity. We distinguish two categorical approaches, ‘top-down’ (Planck energy) and ‘bottom-up’. The tasks of the ‘top-down’ approach, which include string theory and other proposed theories for the microstructures of spacetime, lie in explaining how the micro-constituents give rise to macroscopic structure. They are thus more appropriately called emergent gravity. warnings are issued not to blindly follow the dogma that quantizing general relativity naturally yields a microscopic structure of spacetime, or to accept, without checking the emergent mechanisms, the dictum that some micro-constituent is the theory that gives us everything. Stochastic gravity takes the more conservative ‘bottom-up’ approach. For the linkage with quantum gravity we mention (a) the kinetic theory approach, relying on the structure of a correlation hierarchy and the role played by noise and fluctuations, and (b) the effective theory approach, using large N techniques. The ingredients of both approaches have been developed in earlier chapters systematically. We end with a description of the advantages and limitations of stochastic gravity.

This chapter provides a pedagogical guide to research works on the infrared behavior of interacting quantum fields in de Sitter space which began in the 80s but has seen vibrant activities in the last decade. It aims to help orient readers who wish to enter into research into this area but are bewildered by the vast and diverse literature on the subject. We describe the three main veins of activities – the Euclidean zero-mode dominance, the Lorentzian interacting quantum field theory and the classical stochastic field theory approaches – in some detail, explaining the underlying physics and the technicalities of each. This includes the identification of zero mode in Euclidean quantum field theory, the use of 2PI effective action, the concept of effective infrared dimension, dimensional reduction, dynamical finite size effect, the late time behavior described by Langevin and Fokker–Planck equations, functional resummation techniques and nonperturbative renormalization group methods. We show how these approaches are interconnected, and highlight recent papers that hold promise for future developments.

This chapter is an overview, placing the body of work described in this book in perspective and describing its overarching structure, namely, how the three levels of structure are related: quantum field theory in curved spacetime established in the 1970s, semiclassical gravity developed in the 80s and stochastic gravity introduced in the 90s, a manifestation of the almost ubiquitous existence of a semiclassical and a stochastic regime in relation to quantum and classical in the description of physical systems. We describe the main physical issues in semiclassical and stochastic gravity, namely, backreaction and fluctuations, the mathematical tools used, and their applications to physical problems in early universe cosmology and black hole physics. In terms of connection to related disciplines, it is pointed out that the popular Newton–Schrödinger equation cherished in alternative quantum theories does not belong to semiclassical gravity, as it is not derivable from quantum field theory and general relativity. However, stochastic gravity is needed for quantum information issues involving gravity. These theories enter even in the low-energy, weak-gravity realm where laboratory experiments are carried out. We finish with a summary of the contents of each chapter and a guide to readers.