### 7.4 Discussion

One important advantage of the Einstein–Langevin approach to the gravitational fluctuations in inflaton
over the approach based on the quantization of the linear perturbations of both the metric and the inflaton
field [218], is that an exact treatment of the inflaton quantum fluctuations is possible. This leads to
corrections to the almost scale invariant spectrum for scalar metric perturbations at large scales, and has
implications for the spectrum of the cosmic microwave background anisotropies. However, in the standard
inflationary models these corrections are subdominant. Furthermore when the full non linear effect of the
quantum field is considered, tensorial metric perturbations are also induced by the inflaton fluctuations. An
estimation of this effect, presumably subdominant over the free tensorial fluctuations, has not been
performed.
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 [51, 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.