  Consider compact objects --such as neutron star or black hole binaries-- in
\emph{full, non-linear} general relativity. In the case with zero cosmological
constant $\Lambda$, the gravitational radiation emitted by such systems is
described by the well established, 50+ year old framework due to Bondi, Sachs,
Penrose and others. However, so far we do not have a satisfactory extension of
this framework to include a \emph{positive} cosmological constant --or, more
generally, the dark energy responsible for the accelerated expansion of the
universe. In particular, we do not yet have an adequate gauge invariant
characterization of gravitational waves in this context. As the next step in
extending the Bondi et al framework to the $\Lambda >0$ case, in this paper we
address the following questions: How do we impose the `no incoming radiation'
condition for such isolated systems in a gauge invariant manner? What is the
relevant past boundary where these conditions should be imposed, i.e., what is
the \emph{physically relevant} analog of past null infinity
$\mathcal{I}^{-}_{0}$ used in the $\Lambda=0$ case? What is the symmetry group
at this boundary? How is it related to the Bondi-Metzner-Sachs (BMS) group?
What are the associated conserved charges? What happens in the $\Lambda \to 0$
limit? Do we systematically recover the Bondi-Sachs-Penrose structure at
$\mathcal{I}^{-}_{0}$ of the $\Lambda=0$ theory, or do some differences persist
even in the limit? We will find that while there are many close similarities,
there are also some subtle but important differences from the asymptotically
flat case. Interestingly, to analyze these issues one has to combine conceptual
structures and mathematical techniques introduced by Bondi et al with those
associated with \emph{quasi-local horizons}.
