  We construct dynamical many-black-hole spacetimes with well-controlled
asymptotic behavior as solutions of the Einstein vacuum equation with positive
cosmological constant. We accomplish this by gluing Schwarzschild-de Sitter or
Kerr-de Sitter black hole metrics into neighborhoods of points on the future
conformal boundary of de Sitter space, under certain balance conditions on the
black hole parameters. We give a self-contained treatment of solving the
Einstein equation directly for the metric, given the scattering data we
encounter at the future conformal boundary. The main step in the construction
is the solution of a linear divergence equation for trace-free symmetric
2-tensors; this is closely related to Friedrich's analysis of scattering
problems for the Einstein equation on asymptotically simple spacetimes.
