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

Abstract

We study the stability of the cosmological scalar field models by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. In this approach, we describe the time evolution of the scalar field cosmologies in geometric terms, by performing a “second geometrization” and considering them as paths of a semispray. By introducing a nonlinear connection and a Berwald-type connection associated with 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 for Jacobi stability in scalar field cosmologies. We consider the Jacobi stability properties of the scalar fields with exponential and Higgs type potential. 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. We also investigate the stability of the phantom quintessence and tachyonic scalar field models, by lifting the first-order system to the tangent bundle. It turns out that in the presence of a power law potential both of these models are Jacobi unstable during the entire cosmological evolution.