Global dynamics and evolution for the Szekeres system with nonzero cosmological constant term

TL;DR

This paper analyzes the global dynamics of the Szekeres system with a nonzero cosmological constant, revealing conditions for attractors and their relation to the de Sitter universe through Hamiltonian reduction and dynamical systems techniques.

Contribution

It introduces a Hamiltonian framework for the Szekeres system with cosmological constant and reduces it to a 2D system, providing new insights into its global behavior and physical evolution.

Findings

Existence of attractors only for positive cosmological constant and specific I0 values

Trajectories tend to infinity otherwise

Attractor in the finite regime corresponds to de Sitter universe

Abstract

The Szekeres system with cosmological constant term describes the evolution of the kinematic quantities for Einstein field equations in $\mathbb{R}^4$. In this study, we investigate the behavior of trajectories in the presence of cosmological constant. It has been shown that the Szekeres system is a Hamiltonian dynamical system. It admits at least two conservation laws, $h$ and $I_{0}$ which indicate the integrability of the Hamiltonian system. We solve the Hamilton-Jacobi equation, and we reduce the Szekeres system from $\mathbb{R}^4$ to an equivalent system defined in $\mathbb{R}^2$. Global dynamics are studied where we find that there exists an attractor in the finite regime only for positive valued cosmological constant and $I_0<2.08$. Otherwise, trajectories reach infinity. For $I_ {0}>0$ the origin of trajectories in $\mathbb{R}^2$ is also at infinity. Finally, we investigate the evolution of physical properties by using dimensionless variables different from that of Hubble-normalization conducing to a dynamical system in $\mathbb{R}^5$. We see that the attractor at the finite regime in $\mathbb{R}^5$ is related with the de Sitter universe for a positive cosmological constant.