Novel approach towards the large-scale stable Interacting Dark-Energy
  models and their Astronomical Bounds
  Stability analysis of interacting dark energy models generally divides its
parameters space into two regions: (i) $w_x \geq -1$ and $\xi \geq 0$ and (ii)
$w_x \leq -1$ and $\xi \leq 0$, where $w_x$ is the dark energy equation of
state and $\xi$ is the coupling strength of the interaction. Due to this
separation, crucial information about the cosmology and phenomenology of these
models may be lost. In a recent study it has been shown that one can unify the
two regions with a coupling function which depends on the dark energy equation
of state. In this work we introduce a new coupling function which also unifies
the two regions of the parameter space and generalises the previous proposal.
We analyse this scenario considering the equation of state of DE to be either
constant or dynamical. We study the cosmology of such models and constrain both
scenarios with the use of latest astronomical data from both background
evolution as well as large scale structures. Our analysis shows that a non-zero
value of the coupling parameter $\xi$ as well as the dark energy equation of
state other than `$-1$' are allowed. However, within $1\sigma$ confidence
level, $\xi = 0$, and the dark energy equation of state equal to `$-1$' are
compatible with the current data. In other words, the observational data allow
a very small but nonzero deviation from the $\Lambda$-cosmology, however,
within $1\sigma$ confidence-region the interacting models can mimick the
$\Lambda$-cosmology. In fact we observe that the models both at background and
perturbative levels are very hard to distinguish form each other and from
$\Lambda$-cosmology as well. Finally, we offer a rigorous analysis on the
current tension on $H_0$ allowing different regions of the dark energy equation
of state which shows that interacting dark energy models reasonably solve the
current tension on $H_0$.
