Evolution equations dynamical system of the Lema\^itre--Tolman--Bondi metric containing coupled dark energy

TL;DR

This paper models inhomogeneous spherically symmetric universes with coupled dark energy using a dynamical system approach, analyzing their evolution, critical points, and conditions for a bounce, providing insights into complex cosmic dynamics.

Contribution

It introduces a 7-dimensional autonomous dynamical system for LTB models with coupled dark energy, analyzing inhomogeneities and bounce conditions in detail.

Findings

Identified critical points and their stability in the dynamical system.

Analyzed evolution of energy density and curvature profiles.

Numerically found initial conditions leading to a spherical bounce.

Abstract

We consider inhomogeneous spherically symmetric models based on the Lema\^{i}tre-Tolman-Bondi (LTB) metric, assuming as its source an interactive mixture of ordinary baryonic matter, cold dark matter and dark energy with a coupling term proportional to the addition of energy densities of both dark fluids. We reduce Einstein's field equations to a first order 7-dimensional autonomous dynamical system of evolution equations and algebraic constraints. We study in detail the evolution of the energy density and spatial curvature profiles along the phase space by means of two subspace projections: a three-dimensional projection associated with the solutions of the Friedman-Lema\^\i tre-Robertson-Walker metric (invariant subspace) and a four-dimensional projection describing the evolution of the inhomogeneous fluctuations. We also classify and study the critical points of the system in comparison with previous work on similar sources, as well as solving numerically the equations for initial energy density and curvature profiles that lead to a spherical bounce whose collapsing time we estimate appropriately.