Dark $f(\mathcal{R},\varphi,\chi)$ universe with Noether symmetry

Using the Noether symmetry approach, we investigate $f(\mathcal{R},\varphi,\chi)$ theories of gravity, where $\mathcal{R}$ is the scalar curvature, $\varphi$ is the scalar field, and $\chi$ is the kinetic term of $\varphi$. Based on the Lagrangian for $f(\mathcal{R},\varphi,\chi)$ gravity, we obtain the determining equations. We consider $f(\mathcal{R},\varphi,\chi)$ models of a flat Friedmann–Robertson–Walker universe. Using the obtained solutions, we find conserved quantities. In the framework of this scenario, the continuity equation is extremely important for analyzing the energy density and pressure. Using the first integral of motion, we present a graphical analysis of the energy density, pressure component, and parameter of the equation of state. The negativity of the pressure observed in the considered cases in fact suggests that this theory can describe a Noether universe with dark matter.