Resolving FLRW cosmology through effective equations of state in $f(T)$ gravity

This article explores the cosmological scenario of the universe in the context of the $f(T)$ power law model, where $T$ represent the torsion scalar. To obtain the deterministic solution of the field equations we parametrized the effective Equation-of-State with two parameters $m$ and $k$ as suggested by A. Mukherjee in a flat FLRW environment. We impose constraints on the free parameters of the derived solution by utilizing MCMC analysis assuming the $CC, Pantheon+SH0ES$, and $CC+Pantheon+SH0ES$ as data samples. We explore the dynamics of cosmological parameters. The evolutionary profile of the deceleration parameter exhibits the transition %from the decelerated to the accelerated phase. The effective Equation-of-State parameter indicates the model remains in the quintessence era and gradually becomes the Einsteins-de-Sitter model. In addition to this, we also explore the jerk, snap, and lerk parameters. Furthermore, the $Om(z)$ diagnostic shows that the model has a consistent positive slope across the entire evolution, but resembles the standard $\Lambda$CDM model in the near future. At last, we conclude that the power law function of the modified $f(T)$ gravity model in the framework of the FLRW universe aligns more closely with the $\Lambda$CDM model for given observational data.