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.

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 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.