# Dynamics of a spherically symmetric inhomogeneous coupled dark energy model with coupling term proportional to non relatvistic matter

## TL;DR

This paper investigates the dynamics of a spherically symmetric inhomogeneous coupled dark energy model with a coupling term proportional to non-relativistic matter, analyzing its evolution, stability, and possible cosmological scenarios.

## Contribution

It introduces a novel quasi-local scalar variables approach to study coupled dark energy models with a specific coupling term proportional to cold dark matter density.

## Key findings

- Presence of a past attractor in a non-physical region for energy flow from matter to dark energy.
- Physical past attractor when energy flows from dark energy to matter.
- Numerical solutions show various scenarios including expansion, dark energy consumption, and structure formation.

## Abstract

Quasi--local scalar variables approach is applied to a spherically symmetric inhomogeneous Lema\^\i tre--Tolman--Bondi metric containing a mixture of non-relativistic cold dark matter and coupled dark energy with constant equation of state. The quasi--local coupling term considered is proportional to the quasi--local cold dark matter energy density and a quasi--local Hubble factor-like scalar via a coupling constant $\alpha$. The autonomous numerical system obtained from the evolution equations is classified for different choices of the free parameters: the adiabatic constant of the dark energy $w$ and $\alpha$. The presence of a past attractor in a non-physical region of the energy densities phase-space of the system makes the coupling term non physical when the energy flows from the matter to the dark energy in order to avoid negative values of the dark energy density in the past. On the other hand, if the energy flux goes from dark energy to dark matter, the past attractor lays in a physical region. The system is also numerically solved for some interesting initial profiles leading to different configurations: an ever expanding mixture, a scenario where the dark energy is completely consumed by the non-relativistic matter by means of the coupling term, a scenario where the dark energy disappears in the inner layers while the outer layers expand as a mixture of both sources, and, finally, a structure formation toy model scenario, where the inner shells containing the mixture collapse while the outer shells expand.