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 (A) and cold dark matter (CDM), with the only additional parameter beta. But we begin with a more general case by considering the IwCDM 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 IwCDM 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-3((1))-epsilon plane exhibit distinctive features and the departures from Lambda CDM could be well evaluated, theoretically, indicating that the composite null diagnostic {S-3((1))-epsilon} is a promising tool for investigating the interacting dark energy models. We also compare our results with the observed uncertainties on diagnostic parameters. We find that current observations still do not have sufficient precisions to completely distinguish I Lambda CDM models from the Lambda CDM model. Anyway, our work points out what precisions of measurements should be achieved to distinguish the I Lambda CDM models from the Lambda CDM model.