Hamilton's approach in cosmological inflation with an exponential potential and its observational constraints

TL;DR

This paper explores Hamilton's approach to cosmological inflation with an exponential potential, deriving exact solutions without slow-roll approximation and constraining the model using Planck 2018 observational data.

Contribution

It introduces a Hamiltonian-based method to analyze inflation with exponential potentials, providing exact solutions and observational constraints without slow-roll assumptions.

Findings

Exact solutions to Einstein-Klein-Gordon equations obtained.

Inflationary parameters constrained by Planck 2018 data.

Method offers a simple alternative to traditional slow-roll analysis.

Abstract

The Friedmann-Robertson-Walker (FRW) cosmology is analyzed with a general potential $\rm V(\phi)$ in the scalar field inflation scenario. The Bohmian approach (a WKB-like formalism) was employed in order to constraint a generic form of potential to the most suited to drive inflation, from here a family of potentials emerges; in particular we select an exponential potential as the first non trivial case and remains the object of interest of this work. The solution to the Wheeler-DeWitt (WDW) equation is also obtained for the selected potential in this scheme. Using Hamilton's approach and equations of motion for a scalar field $\rm \phi$ with standard kinetic energy, we find the exact solutions to the complete set of Einstein-Klein-Gordon (EKG) equations without the need of the slow-roll approximation (SR). In order to contrast this model with observational data (Planck 2018 results), the inflationary observables: the tensor-to-scalar ratio and the scalar spectral index are derived in our proper time, and then evaluated under the proper condition such as the number of e-folding corresponds exactly at 50-60 before inflation ends. The employed method exhibits a remarkable simplicity with rather interesting applications in the near future.