Constraints on interacting dynamical dark energy and a new test for
  $\Lambda$CDM
  We consider a generic description of interacting dynamical Dark Energy,
characterized by an equation of state with a time dependent coefficient $w(t)$,
and which may interact with both radiation and matter. Without referring to any
particular cosmological model, we find a differential equation which must be
satisfied by $w(t)$ and involving the function $Q(t)$ which describes the
interaction between Dark Energy and the other cosmological fluids. The relation
we find represents a constraint for various models of interacting dynamical
Dark Energy. In addition, an observable is proposed, depending on kinematic
variables and on density parameters, which may serve as a new test for
$\Lambda$CDM.
