Divergence-free deceleration and energy conditions in non-minimal $f(R,T)$ gravity

We investigate the divergence-free parametric form of the deceleration parameter within the simplest non-minimal matter-geometry coupling in $f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the energy-momentum tensor. Specifically, we consider the linear model $f(R,T) = R + 2\lambda T$, where $\lambda$ governs the interaction between matter and geometry. Using this parametric form, we derive the Hubble parameter as a function of redshift $z$ and incorporate it into the modified Friedmann equations. Constraining the model with OHD and Pantheon data, we obtain precise estimates for $H_0$, the present deceleration parameter $q_0$, and its evolutionary component $q_1$, confirming a smooth transition between cosmic deceleration and acceleration. Further, we analyze the evolution of the energy density $\rho$ and total EoS parameter $\omega$ for different $\lambda$ values, highlighting deviations from $\Lambda$CDM and the role of $\lambda$ in shaping cosmic dynamics. In addition, we examine energy conditions, finding that the NEC and DEC are satisfied throughout evolution, while the SEC is violated at late times, supporting the observed acceleration. Our findings demonstrate that this divergence-free parameterization within $f(R,T)$ gravity offers a viable framework for explaining late-time cosmic acceleration while maintaining key observational and theoretical constraints.