Quintessential inflation for exponential type potentials: scaling and tracker behavior

We will show that for exponential type potentials of the form V(phi)similar to e-gamma phi n/Mpln, which are used to depict quintessential inflation, the solutions whose initial conditions take place during the slow roll phase in order to describe correctly the inflationary period, do not belong for large values of the parameter n to the basin of attraction of the scaling solution - a solution of the scalar field equation whose energy density scale as the one of the fluid component of the universe during radiation or the matter domination period -, meaning that a late time mechanism to exit this behavior and depict correctly the current cosmic acceleration is not needed. However, in such cases, namely n large enough, these potentials cannot correctly depict the current cosmic acceleration. This is the reason why the potential must be improved introducing another parameter -as the one in the well-known Peebles-Vilenkin quintessential inflation model, which depends on two parameters, one to describe inflation and the other one to correctly depict the present accelerated evolution - able to deal with the late time acceleration of our universe.

Automated summary

For exponential type potentials used to depict quintessential inflation, solutions with large n values do not belong to the basin of attraction of the scaling solution. A late time mechanism to exit this behavior and depict correctly the current cosmic acceleration is not needed, but an improvement introducing another parameter is required to deal with the late time acceleration of our universe.