Jacobi Stability Analysis of Scalar Field Models with Minimal Coupling to Gravity in a Cosmological Background

Abstract

We perform the study of the stability of the cosmological scalar field models, by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. In the KCC approach we describe the time evolution of the scalar field cosmologies in geometric terms, by performing a "second geometrization", by considering them as paths of a semispray. By introducing a non-linear connection and a Berwald type connection associated to the Friedmann and Klein-Gordon equations, five geometrical invariants can be constructed, with the second invariant giving the Jacobi stability of the cosmological model. We obtain all the relevant geometric quantities, and we formulate the condition of the Jacobi stability for scalar field cosmologies in the second order formalism. As an application of the developed methods we consider the Jacobi stability properties of the scalar fields with exponential and Higgs type potential. We find that the Universe dominated by a scalar field exponential potential is in Jacobi unstable state, while the cosmological evolution in the presence of Higgs fields has alternating stable and unstable phases. By using the standard first order formulation of the cosmological models as dynamical systems we have investigated the stability of the phantom quintessence and tachyonic scalar fields, by lifting the first order system to the tangent bundle. It turns out that in the presence of a power law potential both these models are Jacobi unstable during the entire cosmological evolution.