

7.1 The model
In this chaotic inflationary model [199] the inflaton field
of mass
is described by the following
Lagrangian density:
The conditions for the existence of an inflationary period, which
is characterized by an accelerated cosmological expansion, is that
the value of the field over a region with the typical size of the
Hubble radius is higher than the Planck mass
. This is because in order to solve the cosmological
horizon and flatness problem more than 60 e-folds of expansion are
needed; to achieve this the scalar field should begin with a value
higher than
. The inflaton mass is small: As we will
see, the large scale anisotropies measured in the cosmic background
radiation [265
] restrict the
inflaton mass to be of the order of
. We will not
discuss the naturalness of this inflationary model and we will
simply assume that if one such region is found (inside a much
larger universe) it will inflate to become our observable universe.
We want to study the metric perturbations
produced by the stress-energy tensor fluctuations of the inflaton
field on the homogeneous background of a flat
Friedmann-Robertson-Walker model, described by the cosmological
scale factor
, where
is the conformal
time, which is driven by the homogeneous inflaton field
. Thus we write the inflaton field in
the following form:
where
corresponds to a free massive quantum
scalar field with zero expectation value on the homogeneous
background metric,
. We will restrict ourselves
to scalar-type metric perturbations, because these are the ones
that couple to the inflaton fluctuations in the linear theory. We
note that this is not so if we were to consider inflaton
fluctuations beyond the linear approximation; then tensorial and
vectorial metric perturbations would also be driven. The perturbed
metric
can be written in the
longitudinal gauge as
where the scalar metric perturbations
and
correspond to Bardeen’s gauge invariant
variables [12].

