  This paper is motivated by the non-linear stability problem for the expanding
region of Kerr de Sitter cosmologies in the context of Einstein's equations
with positive cosmological constant. We show that under dynamically realistic
assumptions the conformal Weyl curvature of the spacetime decays towards future
null infinity. More precisely we establish decay estimates for Weyl fields
which are (i) uniform (with respect to a global time function) (ii) optimal
(with respect to the rate) and (iii) consistent with a global existence proof
(in terms of regularity). The proof relies on a geometric positivity property
of compatible currents which is a manifestation of the global redshift effect
capturing the expansion of the spacetime.
