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We are now able to obtain the full system of equations the govern the background expansion in this model:  \begin{align}  & E^2(z) = g(z)[\Omega_m(z) + \Omega_\Lambda(z)],\\ \Omega_\Lambda(z)], \label{syshubble}\\  & (\Omega_m + \Omega_\Lambda) dg + g d \Omega_\Lambda = 0, \label{sysomol}  \\ & \Omega_\Lambda(z) = \Omega_\Lambda^0 + \nu [E^2(z) - 1], \label{syslambda}  \\ & \Omega_m(z) = \Omega_m^0 (1+z)^{3(1+w_n)},  \end{align}  The first equation is the equivalent of the Friedmann equation in the $\Lambda CDM$ model, the second is the diffrerential form of the Bianchi equation \ref{bianchiconstraint}, the third is just e rewrite of \ref{secondlambdalaw} using the density parameter and the last equation is a rewrite of the standard equation for $\rho_m$ generalized to include relativistic ($w_m = \frac{1}{3}$) and nonrelativistic matter ($w_m = 0$).