Le gain de culture
 
If a culture becomes a significant minority within a country, they can gain the status as 
accepted culture. The criteria for "significant minority" is that it represents 20% or more of the 
base tax value in 
all your 
core provinces (not just the ones you own) and that you own at least one core of that culture. As of v.1.3, a culture loses it's accepted status if it drops below 5%. These percentages can be changed in 
defines.txt.