Reconstruction of the interaction term between dark matter and dark
  energy using SNe Ia
  We apply a parametric reconstruction method to a homogeneous, isotropic and
spatially flat Friedmann-Robertson-Walker (FRW) cosmological model filled of a
fluid of dark energy (DE) with constant equation of state (EOS) parameter
interacting with dark matter (DM). The reconstruction method is based on
expansions of the general interaction term and the relevant cosmological
variables in terms of Chebyshev polynomials which form a complete set
orthonormal functions. This interaction term describes an exchange of energy
flow between the DE and DM within dark sector. To show how the method works we
do the reconstruction of the interaction function expanding it in terms of only
the first six Chebyshev polynomials and obtain the best estimation for the
coefficients of the expansion assuming three models: (a) a DE equation of the
state parameter $w =-1$ (an interacting cosmological $\Lambda$), (b) a DE
equation of the state parameter $w =$ constant with a dark matter density
parameter fixed, (c) a DE equation of the state parameter $w =$ constant with a
free constant dark matter density parameter to be estimated, and using the
Union2 SNe Ia data set from "The Supernova Cosmology Project" (SCP) composed by
557 type Ia supernovae. In both cases, the preliminary reconstruction shows
that in the best scenario there exist the possibility of a crossing of the
noninteracting line Q=0 in the recent past within the $1\sigma$ and $2\sigma$
errors from positive values at early times to negative values at late times.
This means that, in this reconstruction, there is an energy transfer from DE to
DM at early times and an energy transfer from DM to DE at late times. We
conclude that this fact is an indication of the possible existence of a
crossing behavior in a general interaction coupling between dark components.
