Constraints on interacting dark energy revisited: implications for the Hubble tension

We revisit a class of coupled dark energy models where dark energy interacts with dark matter via phenomenological energy exchange terms. We include the perturbation of the Hubble rate in the interaction term, which was absent in previous works. We also consider more recent data sets such as cosmic microwave background (CMB) anisotropies from \text{Planck} 2018, type I-a supernovae (SNIa) measurements from Pantheon+ and data from baryon acoustic oscillations (BAO), and redshift space distortions (RSD). One of the models presents a strong incompatibility when different cosmological datasets are used. We analyze the influence of the SH0ES Cepheid host distances on the results and, although for one model the discrepancy of $H_0$ is reduced to $3.2\sigma$ when compared to the value obtained by the \textit{Planck} collaboration and $3.6\sigma$ when compared to the SH0ES team, joint analysis is incompatible. Including BAO with RSD shows incompatibility with SH0ES for all models considered here. We perform a model comparison and although there is a preference for interacting dark energy over $\Lambda$CDM for all the models for joint analysis CMB+BAO+RSD+SNIa, most of the 2D contours do not overlap. We conclude that the models of interactions in the dark sector considered in this paper are not flexible enough to fit all the cosmological data including values of $H_0$ from SH0ES in a statistically acceptable way. Therefore, the addition of one extra degree of freedom (i.e. the coupling between dark matter and dark matter) does not help enough to alleviate the already existing tension in the vanilla $\Lambda$CDM, suggesting that the models would need to be modified to include further flexibility of predictions to help elucidate this issue.