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.
تحميل البحث