Radiative falloff in Einstein-Straus spacetime

The Einstein-Straus spacetime describes a nonrotating black hole immersed in a matter-dominated cosmology. It is constructed by scooping out a spherical ball of the dust and replacing it with a vacuum region containing a black hole of the same mass. The metric is smooth at the boundary, which is comoving with the rest of the universe. We study the evolution of a massless scalar field in the Einstein-Straus spacetime, with special emphasis on its late-time behavior. This is done by numerically integrating the scalar wave equation in a double-null coordinate system that covers both portions (vacuum and dust) of the spacetime. We show that the field's evolution is governed mostly by the strong concentration of curvature near the black hole, and the discontinuity in the dust's mass density at the boundary; these give rise to a rather complex behavior at late times. Contrary to what it would do in an asymptotically flat spacetime, the field does not decay in time according to an inverse power law.