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Our goal is to reproduce inflation through the coupling between the non-minimal first derivative of the scalar field and the Einstein tensor in which we introduced a potential. We analyse the inflation by examining the equation of state, the expansion parameter and the scale factor. We have shown that when the potential is proportional to the field *φ* and proportional to the square of the field, inflation does not appear; but when the potential is an exponential function of the scalar field, this model brings up inflation. Inflation does not occur when the time t is near minus infinity but it is noticed a few units of Planck time.

Based on observations [1-3], we can say that the universe today is almost flat. If this is the case, i.e., it has remained flat since the beginning of time? To this problem of flatness, we can add the horizon problem. To solve these problems Englert and Guth proposed in [

nicknamed John in Fab Four. The second part consists of a minimal coupling with a multiplicative potential [

After building the action, we deduced the different equations by considering a flat space. We then presented some cosmological models and which followed by a discussion of those models.

Let be the reduced Planck mass.

Let,

and

Which is equivalent to

where ε and γ are constants dimensionless coupling; κ is 1/L^{2}, is 1/L. This action can be written

ε = 0, we find the models studied by [

Consider a spatially-flat cosmological model with the metric

With the scale factor and the Euclidian metric. If one assume, then the cosmological equations which derive from the action (4) can be written

where

The Equations (6) and (7) are the equation of movement; 8 is the Klein-Gordon equation. If one poses

where and are respectively the pressure and the matter density of the field. We can use 6 and 7 to determine equation of state (EoS)

Given the complexity of the equations, we did not obtain solutions analytics.

Before starting the numerical solution, we will work Equations (7) and (8) to separate and

We perform numerical integration by using matlab ode 45 precisely embedded Runge-Kutta method.

When, the evolution of the state equation shows that with the presence of this potential, the field acts like the dark matter for ε negative or positive. The acceleration parameter goes to zero. The field ϕ vs the scale factor is a constant. This model does not accommodate inflation. These plots (Figures 1-4) have been obtained by numerical integration for an initial condition of φ = 10 in natural units in the case γ = 1.

Here also, the field starts with an EoS but this field likes the dark matter with state equation equal 0; the acceleration parameter is negative; the Universe only decelerates as can been shown the four following plots (Figures 5-8)

We find that we can have cosmological models for only λ negative. We have shown the curves for. compared with those obtained for which is studied in the model [