Cosmological Analysis of $f(R,Σ, T)$ Gravity via $O_m(z)$ Reconstruction: Implications into Dark Energy Dynamics

Abstract: This study investigates the cosmological implications of $f(R,\Sigma,T)$ gravity by reconstructing the Hubble parameter from a logarithmic parameterization of the $O_m(z)$ diagnostic. Our approach offers a model-independent way to probe the nature of dark energy and differentiate it from a cosmological constant. We derive the field equations for $f(R,\Sigma,T) = R + \Sigma + 2\pi\eta T$ within the homogeneous, isotropic, and spatially flat Friedmann-Robertson-Walker (FRW) metric. A comprehensive analysis of key physical parameters, including the equation of state (EoS) parameter $\omega(z)$, the $(\omega-\omega')$-plane, the squared sound speed $\vartheta^2$, and various energy conditions (Null, Dominant, Strong), is presented. Our findings reveal a dynamic EoS parameter that consistently remains within the quintessence regime ($-1 < \omega < -1/3$), approaching $\omega = -1$ in the far future, thereby avoiding phantom behavior and maintaining the weak energy condition. The model successfully reproduces the cosmic transition from deceleration to acceleration, as indicated by the deceleration parameter $q(z)$ crossing zero. While the model aligns well with observational data for cosmic expansion, the analysis of $\vartheta^2$ indicates classical instability, a point requiring further theoretical refinement. Overall, this work demonstrates the viability of $f(R,\Sigma,T)$ gravity as a framework capable of describing the universe's accelerated expansion, consistent with current cosmological observations, while offering a dynamic alternative to the $\Lambda$CDM model.