As far as I know, the classic paper applying the Hawking singularity theorem to our universe is this one: Hawking and Ellis, "The Cosmic Black-Body Radiation and the Existence of Singularities in Our Universe," Astrophysical Journal, vol. 152, p. 25, 1968, http://articles.adsabs.harvard.edu/full/1968ApJ...152...25H The version of the theorem they use is one that says that if the SEC holds, there are no CTCs, and a trapped surface exists, then an incomplete geodesic exists. The paper assumes Λ=0. For Λ≠0, the SEC is violated. Does anyone know of a paper that connects the dots between observation and a singularity theorem for Λ≠0 in the same way that this paper did for Λ=0? E.g., this review article http://www.livingreviews.org/lrr-1998-11 doesn't seem to mention the issue.