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.
