The interacting dark energy model could propose a effective way to avoid the coincidence problem. In this paper, dark energy is taken as a fluid with a constant equation-of-state parameter ${w}_{x}$. In a general gauge, we could obtain two sets of different perturbation equations when the momentum transfer potential is vanished in the rest frame of dark matter or dark energy. There are many kinds of interacting forms from the phenomenological considerations; here, we choose $Q=3H{$\xi${}}_{x}{$\rho${}}_{x}$, which owns the stable perturbations in most cases. Then, according to the Markov chain Monte Carlo method, we constrain the model by currently available cosmic observations, which include cosmic microwave background radiation, baryon acoustic oscillation, type Ia supernovae, and $f{$\sigma${}}_{8}(z)$ data points from redshift-space distortion. Jointing the geometry tests with the large scale structure information, the results show a tighter constraint on the interacting model than the case without $f{$\sigma${}}_{8}(z)$ data. We find the interaction rate in $3$\sigma${}$ regions: ${$\xi${}}_{x}={0.00372}_{$-${}0.00372$-${}0.00372$-${}0.00372}^{+0.000768+0.00655+0.0102}$. It means that the recently cosmic observations favor a small interaction rate between the dark sectors, snf at the same time, the measurement of redshift-space distortion could rule out a large interaction rate in the $1$\sigma${}$ region.

We study a novel interacting dark energy − dark matter scenario where the anisotropic stress of the large scale inhomogeneities is considered. The dark energy has a constant equation of state and the interaction model produces stable perturbations. The resulting picture is constrained using different astronomical data aiming to measure the impact of the anisotropic stress on the cosmological parameters. Our analyses show that a non-zero interaction in the dark sector is allowed while a non-interaction scenario is recovered within 68% CL. The anisotropic stress is also constrained to be small, and its zero value is permitted within 68% CL. The dark energy equation of state, w_x, is also found to be close to ‘−1’ boundary. However, from the ratio of the CMB TT spectra, we see that the model has a mild deviation from the ΛCDM cosmology while such deviation is almost forbidden from the CMB TT spectra alone. Although the deviation is not much significant, but from the present data, we cannot exclude such deviation. Overall, at the background level, the model is close to the ΛCDM cosmology while at the level of perturbations, a non-zero but a very small interaction in the dark sector is permitted. Perhaps, a more accurate conclusion can be made with the next generation of surveys. We also found that the region w_x < −1, is found to be effective to release the tension on H₀. Finally, from the Bayesian analysis, we find that ΛCDM remains in still preferred over the interacting scenarios.

The coupling between dark sectors provides a possible approach to mitigate the coincidence problem of the 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, $\overline{Q}=3{\xi }_x\overline{H}{\overline{\rho }}_x$. In particular, we consider the Hubble expansion rate to be perturbed in the perturbation evolutions of dark sectors. Using joint data sets which include cosmic microwave background radiation, baryon acoustic oscillation, type Ia supernovae, and redshift-space distortions, we perform a full Markov chain Monte Carlo likelihood analysis for the coupled model. The results show that the mean value with errors of the interaction rate is ${\xi }_x=0.00305_{-0.00305-0.00305-0.00305}^{+0.000645+0.00511+0.00854}$ for $Q_A^{\mu }\parallel u_c^{\mu }$ and ${\xi }_x=0.00317_{-0.00317-0.00317-0.00317}^{+0.000628+0.00547+0.00929}$ for $Q_A^{\mu }\parallel u_x^{\mu }$, which means that the recent cosmic observations favor a small interaction rate which is up to the order of ${10}^{-3}$. Moreover, in contrast to the coupled model with an unperturbed expansion rate, we find the perturbed Hubble expansion rate can bring about a negligible impact on the model parameter space.