2 From Semiclassical to Stochastic Gravity

There are three main steps that lead to the recent development of stochastic gravity. The first step begins with quantum field theory in curved spacetime [75, 25, 100, 285, 113], which describes the behavior of quantum matter fields propagating in a specified (not dynamically determined by the quantum matter field as source) background gravitational field. In this theory the gravitational field is given by the classical spacetime metric determined from classical sources by the classical Einstein equations, and the quantum fields propagate as test fields in such a spacetime. An important process described by quantum field theory in curved spacetime is indeed particle creation from the vacuum, and effects of vacuum fluctuations and polarizations, in the early universe [234, 260, 300, 301, 146, 21, 22, 23, 75, 96, 65], and Hawking radiation in black holes [130, 131, 174, 235, 281].

The second step in the description of the interaction of gravity with quantum fields is backreaction, i.e., the effect of the quantum fields on the spacetime geometry. The source here is the expectation value of the stress-energy operator for the matter fields in some quantum state in the spacetime, a classical observable. However, since this object is quadratic in the field operators, which are only well defined as distributions on the spacetime, it involves ill defined quantities. It contains ultraviolet divergences, the removal of which requires a renormalization procedure [75, 67, 68]. The final expectation value of the stress-energy operator using a reasonable regularization technique is essentially unique, modulo some terms which depend on the spacetime curvature and which are independent of the quantum state. This uniqueness was proved by Wald [282, 283] who investigated the criteria that a physically meaningful expectation value of the stress-energy tensor ought to satisfy.

The theory obtained from a self-consistent solution of the geometry of the spacetime and the quantum field is known as semiclassical gravity. Incorporating the backreaction of the quantum matter field on the spacetime is thus the central task in semiclassical gravity. One assumes a general class of spacetime where the quantum fields live in and act on, and seek a solution which satisfies simultaneously the Einstein equation for the spacetime and the field equations for the quantum fields. The Einstein equation which has the expectation value of the stress-energy operator of the quantum matter field as the source is known as the semiclassical Einstein equation. Semiclassical gravity was first investigated in cosmological backreaction problems [203, 115, 158, 159, 124, 3, 4, 123, 90, 129]; an example is the damping of anisotropy in Bianchi universes by the backreaction of vacuum particle creation. Using the effect of quantum field processes such as particle creation to explain why the universe is so isotropic at the present was investigated in the context of chaotic cosmology [214, 19, 20] in the late 1970s prior to the inflationary cosmology proposal of the 1980s [117, 2, 197, 198], which assumes the vacuum expectation value of an inflaton field as the source, another, perhaps more well-known, example of semiclassical gravity.