Spectral line

Quantum mechanics

Theoretical physics

Friedmann–Lemaître–Robertson–Walker metric

Inflation (cosmology)

Cosmology

Mathematical physics

Planck

Scalar field

Inflaton

Classical mechanics

Dark energy

Physics

Friedmann equations

Scalar (mathematics)

Mathematics

Spectral index

WKB approximation

Geometry

Slow roll

Astrophysics and Space Science

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.

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