Inflaton Potential Reconstruction and Generalized Equations of State

We extend a previous analysis concerning cosmological fluids with generalized equations of state in order to study inflationary scenarios. In the framework of the slow-roll approximation we find the expressions for the perturbation parameters $ϵ$, $\eta$ and the density perturbation spectra in terms of the adiabatic index $\text{ga}$ as a function of the universe scale factor. This connection allows to find straightforwardly $\text{ga}$ corresponding, for example, to the simplest {\it chaotic} model and to the Harrison-Zeldovich potential and shows its capability to be applied to more complicate situations. Finally, we use this description to develop a new approach to the early universe dynamics, based on a $1\mathrm{/}N$ expansion, where $N$ is the e-fold number. To this aim, we introduce a set of suitable dimensionless variables and show that at the zero-th order in $1\mathrm{/}N$, an improved slow-roll approximation is obtained.