Observational constraints and differential diagnosis for cosmic evolutionary models

In this paper, we have proposed a plotting method based on the " natural plotting rule " (NPR) which can be used to distinguish different cosmological scenarios more efficiently and obtain more useful information. By using the NPR, we have avoided the blindness to use different diagnostics when discovering that some scenarios can be hardly differentiated from each other, and develop a logical line to adopt different diagnostics. As a concrete instance, we take this method based on the NPR to distinguish several Cardassian scenarios from the base cosmology scenario, and one from the other. We place constraints on three Cardassian cosmological scenarios and their flat versions by utilizing the Type Ia supernovae (SNe Ia), baryonic acoustic oscillations (BAO), cosmic microwave background (CMB) radiation, observational Hubble parameter (OHD) data-sets as well as the single data point from the newest event GW150914, and discover that our results are more stringent than previous results for constraining the cosmological parameters of the Cardassian scenarios. We find that the flat original Cardassian (FOC) and original Cardassian (OC) scenarios can only be distinguished in the plane of $\{\Omega_m,S_3^{(1)}\}$ at the present epoch, however, if applying the NPR to plot hierarchically for these Cardassian scenarios in the plane of $\{S_3^{(1)},S_4^{(1)}\}$, we can obtain more detailed information and distinguish the two scenarios better than before. More importantly, from the planes of $\{S_4,S_4^{(2)}\}$, $\{S_5^{(1)},S_5^{(2)}\}$, $\{S_3^{(2)},S_4^{(2)}\}$, $\{\Omega_m,S_3^{(1)}\}$,$\{z,\omega\}$ $\{\epsilon(z),S_3^{(1)}\}$ and $\{z,Om\}$, we dsicover that the flat modified polytropic Cardassian (FMPC) scenario can be directly removed from the possible candidates of dark energy phenomenon, since its evolutional behavior deviates from the base cosmology scenario too much.