Separatrices in the Hamilton-Jacobi Formalism of Inflaton Models

TL;DR

This paper investigates the existence and characteristics of separatrix solutions in inflaton models within the Hamilton-Jacobi formalism, providing insights into inflationary dynamics and criteria for potential growth.

Contribution

It introduces a broad class of inflaton models with separatrices and analyzes their properties, including asymptotic behaviors and growth conditions.

Findings

Identified conditions for the existence of separatrices in inflaton models.

Derived asymptotic inflationary solutions and approximations.

Established growth criteria for potentials lacking separatrices.

Abstract

We consider separatrix solutions of the differential equations for inflaton models with a single scalar field in a zero-curvature Friedmann-Lema\^{\i}tre-Robertson-Walker universe. The existence and properties of separatrices are investigated in the framework of the Hamilton-Jacobi formalism, where the main quantity is the Hubble parameter considered as a function of the inflaton field. A wide class of inflaton models that have separatrix solutions (and include many of the most physically relevant potentials) is introduced, and the properties of the corresponding separatrices are investigated, in particular, asymptotic inflationary stages, leading approximations to the separatrices, and full asymptotic expansions thereof. We also prove an optimal growth criterion for potentials that do not have separatrices.