In this chapter the linearized Riemann tensor correlator on a de Sitter background including one-loop corrections from conformal fields is derived. The Riemann tensor correlation function exhibits interesting features: it is gauge-invariant even when including contributions from loops of matter fields, but excluding graviton loops as it is implemented in the 1/N expansion, it is compatible with de Sitter invariance, and provides a complete characterization of the local geometry. The two-point correlator function of the Riemann tensor is computed by taking suitable derivatives of the metric correlator function found in the previous chapter, and the result is written in a manifestly de Sitter-invariant form. Moreover, given the decomposition of the Riemann tensor in terms of Weyl and Ricci tensors, we write the explicit results for the Weyl and Ricci tensors correlators as well as the Weyl–Ricci tensors correlator and study both their subhorizon and superhorizon behavior. These results are extended to general conformal field theories. We also derive the Riemann tensor correlator in Minkowski spacetime in a manifestly Lorentz-invariant form by carefully taking the flat-space limit of our result in de Sitter.

As a short introduction to this chapter we first briefly summarize the in-in or closed-time-path (CTP) functional formalism and evaluate the CTP effective action for a scalar field in Minkowski spacetime. We then consider N quantum matter fields interacting with the gravitational field assuming an effective field theory approach to quantum gravity and consider the quantization of metric perturbations around a semiclassical background in the CTP formalism. A suitable prescription is given to select an asymptotic initial vacuum state of the interacting theory; this prescription plays an important role in calculations in later chapters. We derive expressions for the two-point metric correlations, which are conveniently written in terms of the CTP effective action that results from integrating out the matter fields by rescaling the gravitational constant and performing a 1/N expansion. These correlations include loop corrections from matter fields but no graviton loops. This is achieved consistently in the 1/N expansion, and is illustrated in a simplified model of matter–gravity interaction.

Structure formation in the early universe is a key problem in modern cosmology. In this chapter we discuss stochastic gravity as an alternative framework for studying the generation of primordial inhomogeneities in inflationary models, which can easily incorporate effects that go beyond the linear perturbations of the inflaton field. We show that the correlation functions that follow from the Einstein–Langevin equation, which emerge in the framework of stochastic gravity, coincide with that obtained with the usual quantization procedures when both the metric perturbations and the inflaton fluctuations are linear. Stochastic gravity, however, can also deal very naturally with the fluctuations of the inflaton field beyond the linear approximation. Here, we illustrate the stochastic approach with one of the simplest chaotic inflationary models in which the background spacetime is a quasi de Sitter universe, and prove the equivalence of the stochastic and quantum correlations to the linear order.

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.