Abstract
This paper rigorously examines the potential of the [Formula: see text] theory as a promising framework for understanding the dark sector of the universe, particularly in relation to cosmic acceleration. The [Formula: see text] theory extends gravitational dynamics by incorporating both the torsion scalar [Formula: see text] and the trace of the energy-momentum tensor [Formula: see text]. Further, we explore the functional form [Formula: see text], where [Formula: see text] is a free parameter that modulates the matter’s influence on spacetime evolution. To evaluate this model, we employ an effective EoS parameter dependent on redshift [Formula: see text], to solve the field equations and analyze the evolution of the Hubble parameter [Formula: see text]. Using a joint dataset ([Formula: see text]) and the Markov Chain Monte Carlo (MCMC) method with Bayesian analysis, we obtain the best-fit parameter values: [Formula: see text], [Formula: see text] and [Formula: see text], which align well with current observational data. Our findings indicate a deceleration parameter of [Formula: see text], supporting a present-day accelerated expansion phase, with a transition redshift [Formula: see text] marking the universe’s shift from deceleration to acceleration. Moreover, we confirm a positive cosmic fluid energy density, reinforcing stability, and find an EoS parameter value of [Formula: see text], consistent with quintessence-driven acceleration. These results underscore the viability of [Formula: see text] as a robust framework for addressing the accelerating universe and dark energy dynamics, paving the way for future investigations into its cosmological implications.