Large-scale (in-) stability Analysis of an Exactly Solved Coupled
  Dark-Energy Model
  Assuming a non-gravitational interaction amongst the dark fluids of our
universe namely, the dark matter and dark energy, we study a specific
interaction model in the background of a spatially flat
Friedmann-Lema\^itre-Robertson-Walker geometry. The interaction model, as we
found, solves the background evolution in an analytic way when the dark energy
takes a constant barotropic equation of state, $w_x$. In particular, we analyze
two separate interaction scenarios, namely, when the dark energy is a fluid
other than the vacuum energy (i.e., $w_x \neq -1$) and when it is vacuum energy
itself (i.e., $w_x = -1$). We found that the interacting model with $w_x \neq
-1$ produces stable perturbation at large scales for $w_x < -1$ with the
coupling strength $\xi<0$. Both the scenarios have been constrained with the
latest astronomical data having distinct origin. The analyses show that a very
small interaction with coupling strength is allowed and within 68.3\%
confidence-region, $\xi =0$ is recovered. The analyses further show that a
large coupling strength significantly affects the large scale dynamics of the
universe while according to the observational data the interaction models are
very well consistent with the $\Lambda$-cosmology. Furthermore, we observe that
for the vacuum interaction scenario, the tension on $H_0$ is not released while
for the interacting dark energy scenario with $w_x < -1$, the tension on $H_0$
seems to be released partially because of the high error bars in $H_0$.
Finally, we close the work with the Bayesian evidence which shows that the
$\Lambda$CDM cosmology is favored over the two interacting scenarios.
