ASYMPTOTIC BEHAVIOR OF COMPLEX SCALAR FIELDS IN A FRIEDMAN-LEMAITRE UNIVERSE

TL;DR

This paper analyzes the asymptotic behavior of complex scalar fields governed by Einstein-Klein-Gordon equations in a Friedman-Lemaître universe, identifying conditions for inflationary stages through phase space analysis.

Contribution

It provides a detailed phase portrait and asymptotic analysis of coupled Einstein-Klein-Gordon equations, including singular points and inflationary solutions.

Findings

Identification of singular points in phase space

Asymptotic behavior of solutions at infinity

Emergence of inflationary stages in solutions

Abstract

We study the coupled Einstein-Klein-Gordon equations for a complex scalar field with and without a quartic self-interaction in a curvatureless Friedman-Lema\^{\i}\-tre Universe. The equations can be written as a set of four coupled first order non-linear differential equations, for which we establish the phase portrait for the time evolution of the scalar field. To that purpose we find the singular points of the differential equations lying in the finite region and at infinity of the phase space and study the corresponding asymptotic behavior of the solutions. This knowledge is of relevance, since it provides the initial conditions which are needed to solve numerically the differential equations. For some singular points lying at infinity we recover the expected emergence of an inflationary stage.