Reconstruction, Thermodynamics and Stability of $\Lambda$CDM Model in $f(T,\mathcal{T})$ Gravity

TL;DR

This paper reconstructs the $ ext{Lambda}$CDM model within $f(T, ext{T})$ gravity, analyzing its thermodynamics, stability, and how it describes cosmological evolution, revealing conditions for energy-momentum conservation and thermodynamic laws.

Contribution

It introduces a novel reconstruction of $ ext{Lambda}$CDM in $f(T, ext{T})$ gravity, exploring stability and thermodynamics, and identifies conditions for conservation and physical viability.

Findings

The $ ext{Lambda}$CDM action includes linear, constant, and non-linear terms in $f(T, ext{T})$.

Energy-momentum conservation requires $F_g$ to be linear in $ ext{T}$.

The model is stable under perturbations except for the geometric part in de Sitter space.

Abstract

We reconstruct the $\Lambda$CDM model for $f(T,\mathcal{T})$ Theory, where $T$ is the torsion scalar and $\mathcal{T}$ the trace of the energy-momentum tensor. The result shows that the action of $\Lambda$CDM is a combination of a linear term, a constant ($-2\Lambda$) and a non-linear term given by the product $\sqrt{-T}F_g\left[(T^{1/3}/16\pi G)\left(16\pi G\mathcal{T}+T+8\Lambda\right)\right]$, with $F_g$ being a generic function. We show that to maintain conservation of energy-momentum tensor should impose that $F_g[y]$ must be linear on the trace $\mathcal{T}$. This reconstruction decays in the $f(T)$ Theory for $F_g\equiv Q$, with $Q$ a constant. Our reconstruction describes the cosmological eras to the present time. The model present stability within the geometric and matter perturbations for the choice $F_g=y$, where $y=(T^{1/3}/16\pi G)\left(16\pi G\mathcal{T}+T+8\Lambda\right)$, except for geometric part to de Sitter model. We impose the first and second laws of thermodynamics to the $\Lambda$CDM and find the condition where they are satisfied, that is, $T_A,G_{eff}>0$, however where this is not possible for cases where we choose, leading to a breakdown of positive entropy and Misner-Sharp energy.