> If the expansion rate is dropping, surely it is headed towards zero? Expansion rate could be dropping but converging to a non-zero value. That seems unlikely

> If the expansion rate is dropping, surely it is headed towards zero? Expansion rate could be dropping but converging to a non-zero value. That seems unlikely to me, but it's an answer that would fit that description just fine. The language is a bit imprecise, though, which I expect is the problem. The (to me) obvious technical interpretation of "expansion rate is headed towards zero" is that d size(t)/dt -> 0 as t -> infinity, but the (again, to me) obvious non-technical interpretation is "expansion will completely stop at some point". So "*not* headed towards zero" means "derivative isn't going to zero", or "expansion never quite stops", respectively. The derivative of ln(t) does go to zero, but it has unbounded growth, so it fails the first test but passes the second. The universe experiencing logarithmic expansion seems reasonable enough.