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,
[25].
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, however, a reasonable approximation as
follows. For a given mode the approximation is
reasonable when its wavelength
is shorter that 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 that
. Thus,
for modes inside the horizon,
and
is a reasonable 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.,
, but this is
the case since
given that
, and
as in most inflationary
models [190, 229].
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 (150
) can then be easily
computed; see [253, 254
] 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 de 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
From this result two main conclusions are
derived. First, the prediction of an almost Harrison-Zel’dovich
scale-invariant spectrum for large scales, i.e., small values of
. Second, since the correlation function is of order
of
, a severe bound to the mass
is imposed by the gravitational fluctuations derived from the small
values of the Cosmic Microwave Background (CMB) anisotropies
detected by COBE. This bound is of the order of
[265, 218
].
We should now comment on some differences with
those works in [45, 213, 212, 51] which used a
self-interacting scalar field or a scalar field interacting
nonlinearly with other fields. In those 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 initial state, had to be 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 [254
]. We should remark
that in the model discussed here there is no environment for the
inflaton fluctuations. The inflaton fluctuations, however, are
responsible for the noise that induces the metric perturbations.