More on McVittie's Legacy: A Schwarzschild - de Sitter black and white   hole embedded in an asymptotically $\Lambda$CDM cosmology
  Recently Kaloper, Kleban and Martin reexamined the McVittie solution and argued, contrary to a very widely held belief, that the solution contains a black hole in an expanding universe. Here we corroborate their main conclusion but go on to examine, in some detail, a specific solution that asymptotes to the $\Lambda$CDM cosmology. We show that part of the boundary of the solution contains the inner bifurcation two - sphere of the Schwarzschild - de Sitter spacetime and so both the black and white hole horizons together form a partial boundary of this McVittie solution. We go on to show that the null and weak energy conditions are satisfied and that the dominant energy condition is satisfied almost everywhere in the solution. The solution is understood here by way of a systematic construction of a conformal diagram based on detailed numerical integrations of the null geodesic equations. We find that the McVittie solution admits a degenerate limit in which the bifurcation two - sphere disappears. For solutions with zero cosmological constant, we find no evidence for the development of a weak null singularity. Rather, we find that in this case there is either a black hole to the future of an initial singularity or a white hole to its past. 
Kayll Lake and Majd Abdelqader
gr-qc astro-ph.CO hep-th