Analysis of energy conditions in linear and square-root f ( T, T ) gravity models

We investigate the viability of modified teleparallel Formula: see text gravity through a systematic analysis of energy conditions in a spatially flat FLRW universe. Two representative models are considered: a linear form Formula: see text = Formula: see text + Formula: see text and a nonlinear square-root form Formula: see text = Formula: see text + Formula: see text. Using current observational values of the Hubble and deceleration parameters, we derive explicit constraints on the null (NEC), weak (WEC), dominant (DEC), and strong (SEC) energy conditions. The results show that NEC, WEC, and DEC are satisfied within well-defined regions of the parameter space, while SEC is consistently violated, providing a natural explanation for late-time cosmic acceleration. The corresponding equation of state parameter remains in the range –1 Formula: see text –Formula: see text, indicating a quintessence-like behavior compatible with observational constraints. A key result of this work is that, in the square-root model, the energy conditions become independent of the torsion coupling parameter Formula: see text, revealing a non-trivial decoupling between geometric corrections and matter contributions. These findings suggest that Formula: see text gravity offers a consistent framework for explaining accelerated cosmic expansion, while providing clear constraints on model parameters from fundamental physical conditions.