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.

Zeta-function regularization is arguably the most elegant of the four major regularization methods used for quantum fields in curved spacetime, linked to the heat kernel and spectral theorems in mathematics. The only drawback is that it can only be applied to Riemannian spaces (also called Euclidean spaces), whose metrics have a ++++ signature, where the invariant operator is of the elliptic type, as opposed to the hyperbolic type in pseudo-Riemannian spaces (also called Lorentzian spaces) with a −+++ signature. Besides, the space needs to have sufficiently large symmetry that the spectrum of the invariant operator can be calculated explicitly in analytic form. In the first part we define the zeta function, showing how to calculate it in several representative spacetimes and how the zeta-function regularization scheme works. We relate it to the heat kernel and derive the effective Lagrangian from it via the Schwinger proper time formalism. In the second part we show how to obtain the correlation function of the stress-energy bitensor, also known as the noise kernel, from the second metric variation of the effective action. Noise kernel plays a central role in stochastic gravity as much as the expectation values of stress-energy tensor do for semiclassical gravity.

In this chapter we derive the full two-point quantum metric perturbations on a de Sitter background including one-loop corrections from conformal fields. We do the calculation using the CTP effective action with the 1/N expansion, and select an asymptotic initial state by a suitable prescription that defines the vacuum of the interacting theory. The decomposition of the metric perturbations into scalar, vector and tensor perturbations is reviewed, and the effective action is given in terms of that decomposition. We first compute the two-point function of the tensor perturbations, which are dynamical degrees of freedom. The relation with the intrinsic and induced fluctuations of stochastic gravity is discussed. We then compute the two-point metric perturbations for the scalar and vector modes, which are constrained degrees of freedom. The result for the full two-point metric perturbations is invariant under spatial rotations and translations as well as under a simultaneous rescaling of the spatial and conformal time coordinates. Finally, our results are extended to general conformal field theories, even strongly interacting ones, by deriving the effective action for a general conformal field theory.

In this chapter we construct the closed-time-path (CTP) two-particle-irreducible (2PI) effective action to two-loop order. The CTP formalism introduced in Chapter 3 is needed to track the dynamics of expectation values and to produce real and causal equations of motion. The composite particle or 2PI formalism introduced in Chapter 6 is needed to treat critical phenomena, because the correlation function and the mean field act as independent variables, instead of the former being a derivative of the latter, as in the 1PI formulation. The large N expansion has the advantage of yielding nonperturbative evolution equations in the regime of strong mean field and a covariantly conserved stress-energy tensor. To leading order in large N, the quantum effective action for the matter fields can be interpreted as a leading-order term in the expansion of the full matter plus gravity quantum effective action, which produces equations of motion for semiclassical gravity and, at the next-to-leading order in large N, stochastic gravity. Two types of quantum fields are treated: (a) O(N) self-interacting Phi-4 fields, and (b) Yukawa coupling between scalar and spinor fields, as an example of dealing with fermions in curved spacetime.

Distant points of light – quasars – show us the intervening universe in silhouette. The result is a map of the gas in and between galaxies, expanding with space.

Distant points of light – quasars – show us the intervening universe in silhouette. The result is a map of the gas in and between galaxies, expanding with space.

The universe is smooth on the largest scales, with roughly the same number of galaxies in every large cosmic neighbourhood. But the standard history of the universe won't allow any process to smooth out an initially smooth universe. An addition to the standard model, called cosmic inflation, aims to fill this void.

At night, the sky is dark. This familiar fact has some remarkable implications for the history and structure of the universe.