Mode interactions in scalar field cosmology

TL;DR

This paper analyzes the dynamics of scalar field cosmologies near a critical transition point, revealing universal geometric structures that underpin inflationary behavior and connect model perturbations to observable inflation parameters.

Contribution

It introduces a geometric unfolding framework that classifies inflationary models and derives slow-roll parameters directly from this structure, independent of specific potentials.

Findings

Universal relations for inflation parameters derived from geometric variables

Classification of perturbations via unfolding parameters

Structural explanation for near scale-invariance of primordial perturbations

Abstract

We study the dynamics of spatially homogeneous Friedmann--Robertson--Walker universes filled with a massive scalar field in a neighbourhood of the massless transition $s=1$. At this point the Einstein--scalar system exhibits a codimension--two Hopf--steady--state organising centre whose versal unfolding describes all small deformations of the quadratic model. After reduction to the centre manifold, the dynamics is governed by two slow geometric modes $(r,z)$: the Hopf amplitude $r$, measuring the kinetic departure from de Sitter, and the slowly drifting Hubble mode $z$. We show that the standard slow--roll parameters follow directly from these unfolding variables, $\epsilon\sim\tfrac32 r^{2}$ and $\eta\sim z$, so that the spectral tilt, tensor--to--scalar ratio, and scalar amplitude arise as universal functions of $(r,z)$, independently of the choice of potential. The two unfolding parameters $(\mu_{1},\mu_{2})$ classify all perturbations of the quadratic model and can be interpreted physically as controlling the tilt and curvature deformations of generic polynomial inflationary potentials. Thus the near scale--invariance of primordial perturbations emerges as a structural property of the unfolding of the organising centre, providing a potential--independent mechanism for an early phase of accelerated expansion. We discuss the implications of this geometric framework for the interpretation and classification of inflationary models.