Upon checking a maw for the first time in Batallia Downs, Sauromugue Champaign, or Rolanberry Fields, you will trigger the first mission for Wings of the Goddess. You will obtain a Pure White Feather key item, which is your "pass" through the maws.
Abyssea Cavernous Maws do not require you to enter/exit like Maws that give you access to the past. You can not enter Abyssea zones Maw without first doing the quest outlined above and you have to be at least level 30 in order to enter Abyssea Cavernous Maws. There are links below for the full walk thru and a list of all Maws. Skyfiredancer (talk) 02:22, October 24, 2015 (UTC)