By carbon 14 dating
To determine year of death, the researchers used radiocarbon levels in soft tissues.Unlike tooth enamel, soft tissues are constantly being made and remade during life.
Radiocarbon levels in teeth formed before then contained less radiocarbon than expected, so when applied to teeth formed during that period, the method was less precise.
Thus, their radiocarbon levels mirror those in the changing environment.
The researchers found that certain soft tissues — notably blood, nails and hair — had radiocarbon levels identical to the contemporary atmosphere.
If either assumption is wrong carbon-14 dating doesn't work.
The first assumption is fairly reasonable, though some scientists have recently called into question whether or not nuclear decay rates were accelerated in the past (this might change the rate of carbon-14 decay).