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7.1 The model

In this chaotic inflationary model [245] the inflaton field φ of mass m is described by the following Lagrangian density,
ℒ(φ ) = 1gab∇a φ∇b φ + 1m2 φ2. (164 ) 2 2
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 mP. 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 3mP. The inflaton mass is small; as we will see, the large scale anisotropies measured in the cosmic background radiation [334Jump To The Next Citation Point] restrict the inflaton mass to be on the order of 10− 6m P. 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 a(η), where η is the conformal time, which is driven by the homogeneous inflaton field φ(η) = ⟨φˆ⟩. Thus we write the inflaton field in the following form

ˆ φ = φ(η) + ˆϕ(x ), (165 )
where ˆϕ(x) corresponds to a free massive quantum scalar field with zero expectation value on the homogeneous background metric: ⟨ϕˆ⟩ = 0. 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 &tidle;gab = gab + hab can be written in the longitudinal gauge as,
ds2 = a2(η)[− (1 + 2Φ (x))dη2 + (1 − 2Ψ (x))δijdxidxj ], (166 )
where the scalar metric perturbations Φ (x ) and Ψ (x) correspond to Bardeen’s gauge-invariant variables [17].
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