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., $Q_1=beta Hrho_c$ and $Q_2=beta Hrho_{de}$, 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.