Reconstructing the dark matter and dark energy interaction scenarios
  from observations
  We consider a class of interacting dark energy models in a flat and nonflat
FLRW universe where the interaction is characterized by the modified evolution
of the pressureless dark matter as $a^{-3+\delta (a)}$, $a$ being the FLRW
scale factor and $\delta (a)$ quantifies the interaction rate. By assuming the
most natural and nonsingular parametrization for $\delta (a)$ as $\delta \left(
a\right) =\sum_{i} \delta_i (1-a)^i $, where $\delta _{i}$'s ($i=0,1,2,3,..$)
are constants, we reconstruct the expansion history of the universe for three
particular choices of the DE sector using different cosmological datasets. Our
analyses show that the non-interacting scenario is consistent with the
observations while the interaction is not strictly ruled out. We reconstruct in
the following way. We start with the first two terms of $\delta (a)$ above and
constrain $\delta_0$, $\delta_1$. Then we consider up to the second order terms
in $\delta (a)$ but fix $\delta_0$, $\delta_1$ to their constrained values and
constrain $\delta_2$; similarly we constrain $\delta_3$, and finally we
constrain $(\delta_0, \delta_1, \delta_1, \delta_3)$ by keeping all of them to
be free as a generalized case. Our reconstruction technique shows that the
constraints on $\delta_2$ (fixing $\delta_0$ and $\delta_1$) and $\delta_3$
(fixing $\delta_0$, $\delta_1$ and $\delta_2$) are almost zero for any
interaction model and thus the effective scenario is well described by the
linear parametrization $\delta (a)\simeq \delta_{0}+\delta _{1}(1-a)$.
Additionally, a strong negative correlation between $\delta_0$, $\delta_1$ is
observed independently of the dark energy fluid and the curvature of our
universe.
