That's a description of a cyclical closed Universe; it made sense when Weinberg wrote "The First Three Minutes" because at that time we had no definitive data on whether or not the Universe was open, closed, or flat.
Since that time, however, the data have pretty much shut the door on the closed Universe. The W-MAP data in particular indicate the matter density of the Universe is exquisitely close, if not exactly equal, to the critical value, which corresponds with a flat Universe that never collapses, especially when the accelerating expansion is considered.
I know, and I don't like it. A universe that starts, produces us, and then fades away with a long goodbye into the big sleep. That's all there is? What's it all about, Alfie? It bothers me that the universe is the cosmic equivalent of a one-night stand.