How the expansion of the universe determines the causal structure of McVittie spacetimes

We present a theorem which allows one to recognize and classify the asymptotic behavior and causal structure of McVittie metrics for different choices of scale factor, establishing whether a black hole or a pair black-white hole appears in the appropriate limit. Incidentally, the theorem also solves an apparent contradiction present in the literature over the causal structure analysis of the McVittie solution. Although the classification we present is not fully complete, we argue that this result covers most if not all physically relevant scenarios.