In practice, to make the explicit computation of the Hadamard function, we will assume that the field
state is in the Euclidean vacuum and the background spacetime is de Sitter. Furthermore, we will compute
the Hadamard function for a massless field, and will make a perturbative expansion in terms of the
dimensionless parameter . Thus we consider
with
and where
are the positive frequency -modes for a massless minimally-coupled scalar field on a de Sitter
background, which define the Euclidean vacuum state
[34
].
The assumption of a massless field for the computation of the Hadamard function is made because
massless modes in de Sitter are much simpler to deal with than massive modes. We can see that this is
nonetheless a reasonable approximation as follows: For a given mode, the approximation is
reasonable when its wavelength
is shorter than the Compton wavelength,
. In our case we
have a very small mass
and the horizon size
, where
is the Hubble constant
(here
with
the physical time
), satisfies
. Thus, for modes inside the
horizon,
and
are a good approximation. Outside the horizon, massive modes decay in
amplitude as
whereas massless modes remain constant. Thus, when modes leave the
horizon, the approximation will eventually break down. However, we only need to ensure that the
approximation is still valid after
e-folds, i.e.,
(
being the time between horizon exit
and the end of inflation). But this is the case provided
, since the decay factor
will not be too different from unity for those modes that left the horizon during
the last sixty e-folds of inflation. This condition is indeed satisfied given that
in most
slow-roll inflationary models [233, 284] and in particular for the model considered here, in which
.
We note that the background geometry is not exactly that of de Sitter spacetime, for which
with
. One can expand in terms of the “slow-roll” parameters and assume
that, to first order,
, where
is the physical time. The correlation function for the
metric perturbation (175
) can then be easily computed; see [315, 316
] for details. The final result, however,
is very weakly dependent on the initial conditions as one may understand from the fact that the accelerated
expansion of the quasi-de Sitter spacetime during inflation erases the information about the initial
conditions. Thus, one may take the initial time to be
and obtain to lowest order in
the
expression
We now comment on some differences with [53, 60, 262, 263], which used a self-interacting scalar field or
a scalar field interacting nonlinearly with other fields. In these works an important relaxation of the ratio
was found. The long wavelength modes of the inflaton field were regarded as an open
system in an environment made out of the shorter wavelength modes. Then, Langevin type
equations were used to compute the correlations of the long wavelength modes driven by the
fluctuations of the shorter wavelength modes. In order to get a significant relaxation on the above
ratio, however, one had to assume that the correlations of the free long-wavelength modes,
which correspond to the dispersion of the system’s initial state, were very small. Otherwise they
dominate by several orders of magnitude those fluctuations that come from the noise of the
environment. This would require a great amount of fine-tuning for the initial quantum state of each
mode [316
].
We should remark that in the linear model discussed here there is no environment for the inflaton fluctuations. When one linearizes with respect to both the scalar metric perturbations and the inflaton perturbations, the system cannot be regarded as a true open quantum system. The reason is that Fourier modes decouple and the dynamical constraints due to diffeomorphism invariance link the metric perturbations of scalar type with the perturbations of the inflaton field so that only one true dynamical degree of freedom is left for each Fourier mode. Nevertheless, the inflaton fluctuations are responsible for the noise that induces the metric perturbations.
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