There are few studies because most research on chickens is done by the commercial sector, and they are not interested in hens past 2 years old.
Some wild bird species are still fertile in their 20s, 30s, 40s and beyond.
I have 7 year olds who had offspring this year. No decline in their egg shell quality. They lay fewer eggs than they used to, but almost all their eggs are fertile.
My roos are younger - none over 3 currently. They don't thrive as well as hens when older, for social, flock dynamic reasons rather than biological ones (younger males make ex-doms' lives a misery).