Religion and babies: are believers to blame for overpopulation?

In the context of rising global population (see my more detailed comments here), what effect does religious affiliation have on fertility?

Hans Rosling is one of the key figures behind the fascinating work at GapMinder, which aims to bring global statistics to life for ordinary people. He is something of a regular at TED and his talks are always entertaining and interesting. I have posted a few before.

Warning: the method he uses is applicable only at the broadest (national) level and so flattens some interesting exceptions, but other more detailed studies do support his basic claim here.