Global dynamics and asymptotics for monomial scalar field potentials and perfect fluids

TL;DR

This paper analyzes the global dynamics of scalar fields with monomial potentials coupled with perfect fluids in flat FLRW cosmology, using dynamical systems techniques to understand asymptotic behaviors and attractor solutions.

Contribution

It introduces a new three-dimensional dynamical systems reformulation for scalar fields and fluids, providing a comprehensive global description and asymptotic analysis of solutions.

Findings

Global solution space characterized using dynamical systems

Attractor corresponds to an unstable center manifold of a de Sitter fixed point

Approximate solutions improved with Padé approximants and compared to slow-roll

Abstract

We consider a minimally coupled scalar field with a monomial potential and a perfect fluid in flat FLRW cosmology. We apply local and global dynamical systems techniques to a new three-dimensional dynamical systems reformulation of the field equations on a compact state space. This leads to a visual global description of the solution space and asymptotic behavior. At late times we employ averaging techniques to prove statements about how the relationship between the equation of state of the fluid and the monomial exponent of the scalar field affects asymptotic source dominance and asymptotic manifest self-similarity breaking. We also situate the `attractor' solution in the three-dimensional state space and show that it corresponds to the one-dimensional unstable center manifold of a de Sitter fixed point, located on an unphysical boundary associated with the dynamics at early times. By deriving a center manifold expansion we obtain approximate expressions for the attractor solution. We subsequently improve the accuracy and range of the approximation by means of Pad\'e approximants and compare with the slow-roll approximation.