Cosmological effects on $f(\bar{R},\bar{T})$ gravity through a non-standard theory

This study aims to investigate the impact of dark energy in cosmological scenarios by exploiting $f(\bar{R},\bar{T})$ gravity within the framework of a {\it non-standard} theory, called {\it {\bf K-}essence} theory, where $\bar{R}$ represents the Ricci scalar and $\bar{T}$ denotes the trace of the energy-momentum tensor associated with the {\bf K-}essence geometry. The Dirac-Born-Infeld (DBI) non-standard Lagrangian has been employed to generate the emergent gravity metric $({\bar{G}}_{\mu \nu })$ associated with the {\bf K-}essence. This metric is distinct from the usual gravitational metric $(g_{\mu \nu })$. It has been shown that under a flat FLRW background gravitational metric, the modified field equations and the Friedmann equations of the $f(\bar{R},\bar{T})$ gravity are distinct from the usual ones. In order to get the equation of state (EOS) parameter $\omega$, we have solved the Friedmann equations by taking into account the function $f(\bar{R},\bar{T})\equiv f(\bar{R})+\lambda \bar{T}$, where $\lambda$ represents a parameter within the model. We have found a relationship between $\omega$ and time for different kinds of $f(\bar{R})$ by treating the kinetic energy of the {\bf K-}essence scalar field (${\dot{\phi }}^2$) as the dark energy density which fluctuates with time. Surprisingly, this result meets the condition of the restriction on ${\dot{\phi }}^2$. By presenting graphical representations of the EOS parameter with time, we show that our model is consistent with the data of $SNIa$+$BAO$+$H(z)$ within a certain temporal interval.