Cosmic evolution in the background of non-minimal coupling in $f(R,T,R_{\mu \nu }{T}^{\mu \nu })$ gravity

In this manuscript, we are interested to address the issue of cosmic expansion in the background of matter-geometry coupling. For this purpose we consider $f(R,T,Q)$ modified theory (where $R$ is the Ricci Scalar, $T$ is the trace of energy-momentum tensor (EMT) $T_{uv}$ and $Q=R_{uv}{T}^{uv}$ is interaction of EMT $T_{\mu \nu }$ and Ricci Tensor $R_{uv}$). We formulate modified field equations in the background of flat Friedmann-Lemaître-Robertson-Walker (FLRW) model which is defined as $d{s}^2=d{t}^2-a\left(t{)}^2\right(d{x}^2+d{y}^2+d{z}^2)$, where $a\left(t\right)$ represents the scale factor. In this formalism energy density is found using covariant divergence of modified field equations. $\rho$ involves a contribution from non-minimal matter geometry coupling which helps to study different cosmic eras based on equation of state (EOS). Furthermore, we apply the energy bounds to constrain the model parameters establishing a pathway to discuss the cosmic evolution for best suitable parameters in accordance with recent observations.