Observational constraint on the interacting dark energy models including the Sandage-Loeb test

Two types of interacting dark energy models are investigated using the type Ia supernova (SNIa), observational $H(z)$ data (OHD), cosmic microwave background (CMB) shift parameter and the secular Sandage-Loeb (SL) test. We find that the inclusion of SL test can obviously provide more stringent constraint on the parameters in both models. For the constant coupling model, the interaction term including the SL test is estimated at $\delta=-0.01 \pm 0.01 (1\sigma) \pm 0.02 (2\sigma)$, which has been improved to be only a half of original scale on corresponding errors. Comparing with the combination of SNIa and OHD, we find that the inclusion of SL test directly reduces the best-fit of interaction from 0.39 to 0.10, which indicates that the higher-redshift observation including the SL test is necessary to track the evolution of interaction. For the varying coupling model, we reconstruct the interaction $\delta (z)$, and find that the interaction is also negative similar as the constant coupling model. However, for high redshift, the interaction generally vanishes at infinity. The constraint result also shows that the $\Lambda$CDM model still behaves a good fit to the observational data, and the coincidence problem is still quite severe. However, the phantom-like dark energy with $w_X<-1$ is slightly favored over the $\Lambda$CDM model.