content_image/392320.jpg content_image/392321.jpg content_image/392319.jpg content_image/392322.jpg content_image/392323.jpg content_image/392328.jpg content_image/392329.jpg content_image/392330.jpg content_image/392331.jpg content_image/392314.jpg content_image/392334.jpg content_image/392336.jpg content_image/392337.jpg content_image/392338.jpg content_image/392342.jpg content_image/392343.jpg
<img src='content_image/392320.jpg'>

<img src='content_image/392321.jpg'>

n$_{S}$ is the slope of the scalar primordial power spectrum, n$_{t}$ is the corresponding tensor one, r is the scalar to tensor ratio and Δ 2 R is the amplitude of the initial perturbations. All these numbers are constrained by current cosmological observations [10, 11, 12]. We will use their constraints to explore the naturalness of our inflationary trajectories. In- flation takes place when the slow-roll parameters are much smaller than 1. 

We will use the amplitude of initial perturbations and the number of efoldings to fit some of the paramenters of the toy model in the previous section. Recall that the potential in the range of interest is: 

<img src='content_image/392319.jpg'>

which appears in figure 1. We can compute ϵ, η while rolling in the β direction: 

<img src='content_image/392322.jpg'>

<img src='content_image/392323.jpg'>

since β << 1, ϵ is naturally small. We can make η small by a slight fine tuning of the difference A$_{1}$ − B$_{1}$ . We will write η later as a ratio of the inflaton and gravitino masses. Once the slow roll conditions are satisfied, we can compute the number of efoldings (see for instance [16, 17]): 

<img src='content_image/392328.jpg'>

From (19) we get: 

<img src='content_image/392329.jpg'>

In most models of supersymmetry breaking, the gravitino mass is given by: 

<img src='content_image/392330.jpg'>

hence, we can rewrite the parameters and masses in (9) as: 

<img src='content_image/392331.jpg'>

thus: 

<img src='content_image/392314.jpg'>

The number of efoldings is considered normally to be be- tween 50 − 100. Finally we will use the amplitude of initial perturbations to get one extra condition in the parameters of our potential. Using [11] (16) can be written as: 

<img src='content_image/392334.jpg'>

where β is taken at N -efoldings before the end of inflation. Summarizing, the two cosmological constraints we get on the parameters of our potential can be written as: 

<img src='content_image/392336.jpg'>

<img src='content_image/392337.jpg'>

and the η parameter can be written as: 

<img src='content_image/392338.jpg'>

We take β$_{i}$ above the supersymmetry breaking scale √ f/M = µ/M , and β$_{f}$ close to m$_{soft}$/M , therefore we can easily get values for N between 50 − 100 for moderate values of η , which is expressed here as the square of the ratio of the inflaton to the gravitino mass. It is interesting to notice that from (27), we can write the supersymmetry breaking scale µ in terms of the η -parameter: 

<img src='content_image/392342.jpg'>

Hence for a value of η ∼ . 1 we can get µ ∼ 10 13 GeV. Lower values of the supersymmetry breaking scale can be obtained by reducing η . However, since the inflaton mass is 

<img src='content_image/392343.jpg'>

we may end up with an inflaton whose mass is substantially lighter than the gravitino. For these values of η, µ , we have that β$_{i}$ ∼ 10 $^{13}$/M, β$_{f}$ ∼ 10 $^{3}$/M , and the number of efoldings is ∼ 110. 

