$a$. We might expect that this exotic background fluid smoothly interpolates between a dust dominated phase $\rho\sim a^{-3}$ and a dark energy dominated phase $\rho\sim a^{-3(1+w_X)}$ where $w_X$ is a constant and should be taken as any possible value in the range $(-1.46,-0.78)$. It can be expected that the energy density of the NGCG should be elegantly expressed as $$\rho_{\rm Ch}=  \left[A a^{-3(1+w_X)(1+\alpha)} + {B a^{-3 (1 +
\alpha)}}\right]^{1 \over 1 + \alpha}~.\label{chrho}$$ The derivation of the Eq. (\[chrho\]) should be the consequence of substituting the equation of state Eq. (\[eqstate\]) into the energy conservation equation of the NGCG for an homogeneous and isotropic spacetime, this requires the function $\tilde{A}(a)$ to be of the form $$\tilde{A}(a)=-w_XAa^{-3(1+w_X)(1+\alpha)}~,\label{A}$$ where $A$ is a positive constant, and the other positive constant $B$ appears in Eq. (\[chrho\]) an integration constant. One can see explicitly that this model recovers the GCG model as the equation-of-state parameter $w_X$ taken to be $-1$, and an ordinary $X$CDM model can be reproduced by taking the parameter $\alpha$ to be zero. The parameter $\alpha$ is called [*interaction parameter*]{} of the model as will be shown below.

The NGCG scenario involves an interacting XCDM picture. For showing this, we first decompose 