A closer look at interacting dark energy with statefinder hierarchy and
  growth rate of structure
  We investigate the interacting dark energy models by using the diagnostics of
statefinder hierarchy and growth rate of structure. We wish to explore the
deviations from $\Lambda$CDM and to differentiate possible degeneracies in the
interacting dark energy models with the geometrical and structure growth
diagnostics. We consider two interacting forms for the models, i.e., $Q_1=\beta
H\rho_c$ and $Q_2=\beta H\rho_{de}$, with $\beta$ being the dimensionless
coupling parameter. Our focus is the I$\Lambda$CDM model that is a
one-parameter extension to $\Lambda$CDM by considering a direct coupling
between the vacuum energy ($\Lambda$) and cold dark matter (CDM), with the only
additional parameter $\beta$. But we begin with a more general case by
considering the I$w$CDM model in which dark energy has a constant $w$
(equation-of-state parameter). For calculating the growth rate of structure, we
employ the "parametrized post-Friedmann" theoretical framework for interacting
dark energy to numerically obtain the $\epsilon(z)$ values for the models. We
show that in both geometrical and structural diagnostics the impact of $w$ is
much stronger than that of $\beta$ in the I$w$CDM model. We thus wish to have a
closer look at the I$\Lambda$CDM model by combining the geometrical and
structural diagnostics. We find that the evolutionary trajectories in the
$S^{(1)}_3$--$\epsilon$ plane exhibit distinctive features and the departures
from $\Lambda$CDM could be well evaluated, theoretically, indicating that the
composite null diagnostic $\{S^{(1)}_3, \epsilon\}$ is a promising tool for
investigating the interacting dark energy models.
