Global phase space analysis for a class of single scalar field bouncing solutions in general relativity

Abstract

We carry out a compact phase space analysis of a non-canonical scalar field theory whose Lagrangian is of the form $F(X)-V(\phi)$ within general relativity. In particular, we focus on a kinetic term of the form $F(X)=\beta X^m$ with power law potential $V_0 \phi^n$ and exponential potential $V_0 e^{-\lambda\phi/M_{Pl}}$ of the scalar field. The main aim of this work is to investigate the genericity of nonsingular bounce in these models and to investigate the cosmic future of the bouncing cosmologies when they are generic. A global dynamical system formulation that is particularly suitable for investigating nonsingular bouncing cosmologies is used to carry out the analysis. We show that when $F(X)=\beta X^m$ ($\beta<0$), nonsingular bounce is generic for a power law potential $V(\phi) = V_0 \phi^n$ only within the parameter range $\left\lbrace \frac{1}{2}<m<1,\,n<\frac{2m}{m-1}\right\rbrace$ and for an exponential potential $V(\phi) = V_0 e^{-\lambda\phi/M_{Pl}}$ only within the parameter range $\left\lbrace\frac{1}{2}<m\leq1\right\rbrace$. Except in these cases, nonsingular bounce in these models is not generic due to the non-existence of global past or future attractors. Our analysis serves to show the importance of a global phase space analysis to address important questions about nonsingular bouncing solutions, an idea that may and must be adopted for such solutions even in other theories.