  Results on the isolation of the radiative degrees of freedom of the
gravitational field with a positive cosmological constant in full General
Relativity are put forward. Methods employed in a recent geometric
characterisation of gravitational radiation are used and, inspired by
Ashtekar's work on asymptotically flat space-times, a space of connections is
defined. Ground differences emerge due to the space-like character of the
conformal boundary, and one has to put into play a fundamental result by
Friedrich concerning the initial value problem for space-times with a positive
cosmological constant. Based on this, half of the radiative degrees of freedom
are identified; remarkably, they utterly determine the gravitational radiation
content for space-times with algebraically special rescaled Weyl tensor at
infinity. Directions for defining the phase space in the general case are
proposed.
