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 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.

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.