We should remark that although the gravitational
fluctuations are here assumed to be classical, the correlation
functions obtained correspond to the expectation values of the
symmetrized quantum metric perturbations [49, 254]. This means that
even in the absence of decoherence the fluctuations predicted by
the Einstein-Langevin equation, whose solutions do not describe the
actual dynamics of the gravitational field any longer, still give
the correct symmetrized quantum two-point functions.
Another important advantage of the stochastic
gravity approach is that one may also compute the gravitational
fluctuations in inflationary models which are not driven by an
inflaton field, such as Starobinsky inflation which is driven by
the trace anomaly due to conformally coupled quantum fields. In
fact, Einstein’s semiclassical equation (7) for a massless
quantum field which is conformally coupled to the gravitational
field admits an inflationary solution which is almost de Sitter
initially and ends up in a matter-dominated-like regime [269, 280]. In these models the
standard approach based on the quantization of the gravitational
and the matter fields to linear order cannot be used. This is
because the calculation of the metric perturbations correspond to
having only the last term in the noise kernel in Equation (147
), since there is no
homogeneous field
as the expectation value
, and linearization becomes trivial.
In the trace anomaly induced inflation framework,
Hawking et al. [132] were able
to compute the two-point quantum correlation function for scalar
and tensorial metric perturbations in a spatially closed de Sitter
universe, making use of the anti-de Sitter conformal field theory
correspondence. They find that short scale metric perturbations are
strongly suppressed by the conformal matter fields. This is similar
to what we obtained in Section 6 for the induced metric
fluctuations in Minkowski spacetime. In the stochastic gravity
context, the noise kernel in a spatially closed de Sitter
background was derived in [252]. However, in a
spatially flat arbitrary Friedmann-Robertson-Walker model the
Einstein-Langevin equation describing the metric perturbations was
first obtained in [58] (see
also [169
]). The two-point
correlation functions for the metric perturbations can be derived
from its solutions, but this is work still in progress.