Averaging Theory and Dynamical Systems in Cosmology: A Qualitative Study of Oscillatory Scalar-Field Models

TL;DR

This paper applies averaging theory and dynamical systems techniques to analyze oscillatory scalar-field cosmological models across various geometries, providing insights into their long-term behavior and attractor structures.

Contribution

It introduces a novel application of averaging methods to cosmological models, establishing a near-identity conjugacy between oscillatory and averaged flows under mild assumptions.

Findings

Effective systems preserve asymptotic behavior.

Late-time attractors depend on geometry.

No autonomous drift when leading averaged vector field vanishes.

Abstract

We study cosmological models using dynamical systems and averaging methods, encompassing flat and open FLRW geometries as well as the LRS Bianchi types I, III, and V. Under mild regularity and frequency-scaling assumptions, we obtain a near-identity conjugacy between the oscillatory flow and an averaged slow flow, with $\| \mathbf{x}(t)-\bar {\mathbf{x}}(t)\| =\mathcal{O}(H(t))$. The effective systems preserve the original asymptotics and yield geometry-dependent late-time attractor classifications. A corollary addresses the case in which the leading averaged vector field vanishes, so the system exhibits no autonomous drift at order $H^0$.