Coupled dark energy with perturbed Hubble expansion rate

The coupling between dark sectors provides a possible approach to mitigate the coincidence problem of cosmological standard model. In this paper, dark energy is treated as a fluid with a constant equation of state, whose coupling with dark matter is proportional the Hubble parameter and energy density of dark energy, that is, $\bar{Q}=3\xi_x\bar{H}\bar{\rho}_x$. Particularly, we consider the Hubble expansion rate to be perturbed in the perturbation evolutions of dark sectors. Using jointing data sets which include cosmic microwave background radiation, baryon acoustic oscillation, type Ia supernovae, and redshift-space distortions, we perform a full Monte Carlo Markov Chain likelihood analysis for the coupled model. The results show that the mean value with errors of interaction rate is: $\xi_x=0.00305_{-0.00305-0.00305-0.00305}^{+0.000645+0.00511+0.00854}$ for $Q^{\mu}_A\parallel u^{\mu}_c$; $\xi_x=0.00317_{-0.00317-0.00317-0.00317}^{+0.000628+0.00547+0.00929}$ for $Q^{\mu}_A\parallel u^{\mu}_x$, which means that the recently cosmic observations favored small interaction rate which is up to the order of $10^{-3}$. Moreover, in contrast to the coupled model with unperturbed expansion rate, we find perturbed Hubble expansion rate could bring about negligible impact on the model parameter space.