As Alyssa and Hope explained in the previous walkthrough on Academia, you will need to go out and find five of the seven Graviton Cores hidden in the world to advance farther in the story. Upon finding these Cores, you must return them to Alyssa.
The first of the Graviton Cores is located in New Bodhum, where you began the game. Once you’ve warped there, head to the Tidal Shallows area, which is located up the beach, over the stone ramp, and to the right. This is the section that rose from the water upon your first victory against Gogmagog. From here, search for the Chocobo. The Graviton Core is located around the bend from the Chocobo, next to a metal hatch at the water’s edge.
The next Graviton Core is located in Bresha Ruins. You’ll have to climb on a Chocobo to get this one. Once you’ve mounted the Chocobo, ride it to the Echoes of the Past area and examine the back end of the wall to the right of the area. You’ll notice that there seems to be something behind the wall. Jump over the short section of wall with the Chocobo to land on a grassy knoll. In the northern corner of the area, use Moogle Hunt to find the Core.
The third Core is located in Oerba. Again, you’ll have to climb on a Chocobo to get this one. This Core is found in the center of the map, on top of a building. You can find the building by finding Chocolina; the rooftop in question is situated just behind her. Jump onto the roof and fly to the gap in the wall to find the Graviton Core.
The fourth Core is in Academia 400AF. When you return here, you will begin at the Gate in the upper section of the New Town area. Go to the moving ramp near Chocolina and follow it to the bottom of the area. There will be a few spots that you’ll have to jump across along the way, but nothing too taxing. The fourth Core will await you there.
The fifth and final core is located in Yaschas Massif 100AF. You’ve been to Yaschas Massif before, but never in 100AF, so you’ll need to unlock the area before you can get the Core. In order to do this, go back to New Town in Academia. Make sure that you have at least one Wild Artefact in your inventory when you do. You’ll have to head to the south, following a blue moving walkway near Chocolina. After the blue walkway ends, ride the red walkway to the right to go down a level. Now turn south to reach the Gate. Use a Wild Artefact to get to Yaschas Massif circa 100AF. Once you arrive, go to the Pass of Paddra area in the northwest. Watch out for some fairly difficult enemies as you make your way through the area. Keep heading northwest to find the Core in a canyon, near some truncated stumps protruding from the ground.