Can $f(T)$ gravity theories mimic $\Lambda$CDM cosmic history

TL;DR

This paper investigates whether $f(T)$ gravity theories can replicate the entire cosmic history of the $ ext{Lambda}$CDM model, including radiation, matter, and dark energy dominated eras, by reconstructing the function $f(T)$ accordingly.

Contribution

The study reconstructs the $f(T)$ function that can reproduce the $ ext{Lambda}$CDM cosmic history and analyzes the stability of these models during different cosmic eras.

Findings

Reconstructed $f(T)$ functions mimic $ ext{Lambda}$CDM expansion history.

Identified critical lines corresponding to radiation and matter dominance.

Derived stability conditions for the reconstructed models during each era.

Abstract

Recently the teleparallel Lagrangian density described by the torsion scalar T has been extended to a function of T. The $f(T)$ modified teleparallel gravity has been proposed as the natural gravitational alternative for dark energy to explain the late time acceleration of the universe. In order to reconstruct the function $f(T)$ by demanding a background $\Lambda$CDM cosmology we assume that, (i) the background cosmic history provided by the flat $\Lambda$CDM (the radiation ere with $\omega_{eff}=1/3$, matter and de Sitter eras with $\omega_{eff}=0$ and $\omega_{eff}=-1$, respectively) (ii) the radiation dominate in the radiation era with $\Omega_{0r}=1$ and the matter dominate during the matter phases when $\Omega_{0m}=1$. We find the cosmological dynamical system which can obey the $\Lambda$CDM cosmic history. In each era, we find a critical lines that, the radiation dominated and the matter dominated are one points of them in the radiation and matter phases, respectively. Also, we drive the cosmologically viability condition for these models. We investigate the stability condition with respect to the homogeneous scalar perturbations in each era and we obtain the stability conditions for the fixed points in each eras. Finally, we reconstruct the function $f(T)$ which mimics cosmic expansion history.