### 7.1 The model

In this chaotic inflationary model [245] the inflaton field of mass is described by the following
Lagrangian density,
The condition 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 [334] restrict the inflaton mass to be on the order of
. We will not discuss the naturalness of this inflationary model but 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 arise. The perturbed metric can be written in
the longitudinal gauge as,
where the scalar metric perturbations and correspond to Bardeen’s gauge-invariant
variables [17].