Dynamical properties of the conformally coupled flat FRW model

TL;DR

This paper analyzes the dynamics of a flat FRW cosmological model with a conformally coupled massive scalar field, revealing instability of stationary solutions and asymptotic behaviors resembling radiation and matter dominance.

Contribution

It provides a Lyapunov-like function for the model's equations and characterizes the asymptotic regimes, including stability analysis and long-term behavior.

Findings

All stationary solutions are unstable.

Initial conditions lead to an expanding universe with scalar field oscillations.

Identifies asymptotic solutions for radiation and matter dominated phases.

Abstract

In this paper we study the dynamical behaviour of a simple cosmological model defined by a spatially flat Robertson-Walker geometry, conformally coupled with a massive scalar field. We determine a Lyapunov-like function for the non-linear evolution equations. From this function we prove that all the stationary solutions are unstable. We also show that all initial conditions, different from the stationary points, originate an expanding universe in the asymptotic regime, with a scale parameter $a(t)$ that goes to infinity and the scalar field $\phi (t)$ that goes to zero in an oscillatory way . We also find two asymptotic solutions, valid for sufficiently large values of time. These solutions correspond to a radiation dominated phase and to a matter dominated phase, respectively.