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.

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 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.