In the stochastic gravity approach some insights on the exact treatment of the inflaton scalar field
perturbations have been discussed in [316, 353, 354]. The main features that would characterize an exact
treatment of the inflaton perturbations are the following. First, the three types of metric perturbations
(scalar, vectorial and tensorial perturbations) couple to the perturbations of the inflaton field. Second, the
corresponding Einstein–Langevin equation for the linear metric perturbations will explicitly
couple to the scalar and tensorial metric perturbations. Furthermore, although the Fourier
modes (with respect to the spatial coordinates) for the metric perturbations will still decouple in
the Einstein–Langevin equation, any given mode of the noise and dissipation kernels will get
contributions from an infinite number of Fourier modes of the inflaton field perturbations. This fact
will imply, in addition, the need to properly renormalize the ultraviolet divergences arising
in the dissipation kernel, which actually correspond to the divergences associated with the
expectation value of the stress-tensor operator of the quantum matter field evolving on the perturbed
geometry.
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 [66, 316
]. This means that even in the absence of decoherence, the
fluctuations predicted by the Einstein–Langevin equation still give the correct symmetrized
quantum two-point correlation functions. In [66] it was explained how a stochastic description
based on a Langevin-type equation could be introduced to gain information on fully quantum
properties of simple linear open systems. In a forthcoming paper [317] it will be shown that, by
carefully dealing with the gauge freedom and the consequent dynamical constraints, this result
can be extended to the case of
free quantum matter fields interacting with the metric
perturbations around a given background. In particular, the correlation functions for the metric
perturbations obtained using the Einstein–Langevin equation are equivalent to the correlation
functions that would follow from a purely quantum field theory calculation up to the leading order
contribution in the large
limit. This will generalize the results already obtained on a Minkowski
background [203, 204].
These results have important implications on the use of the Einstein–Langevin equation to address
situations in which the background configuration for the scalar field vanishes. This includes not only the
case of a Minkowski background spacetime, but also the remarkably interesting case of the trace
anomaly-induced inflation. That is, inflationary models driven by the vacuum polarization of a large number
of conformal fields [162, 339
, 355
], in which the usual approaches based on the linearization of both the
metric perturbations and the scalar field perturbations and their subsequent quantization can no longer be
applied. More specifically, the semiclassical Einstein equations (8
) for massless quantum fields
conformally coupled to the gravitational field admit an inflationary solution that begins in
an almost de Sitter-like regime and ends up in a matter-dominated-like regime [339, 355]. In
these models the standard approach based on the quantization of the gravitational and the
matter fields to linear order cannot be used because the calculation of the metric perturbations
correspond to having only the last term in the noise kernel in Equation (171
), since there is
no homogeneous field
as the expectation value
and linearization becomes
trivial.
In the trace anomaly induced inflation model Hawking et al. [162] 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 [314],
and in a spatially-flat arbitrary Friedmann–Robertson–Walker model the Einstein–Langevin
equations describing the metric perturbations were first obtained in [73]. The computation
of the corresponding two-point correlation functions for the metric perturbations is now in
progress.
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