Herodotus has gone on ahead to the Temple of Apollo. The temple is in the Sanctuary of Delphi. Find him outside the doors to the temple and speak with him.
After a brief cut scene you will find yourself within the temple. Make your way into the depths until you reach the main chamber, at which point you will enter dialogue with one of the cultists. After the brief discussion, approach other cultists with search icons above their head to learn more about the cult. It does not matter which dialogue options you choose. When you are done talking, head east beneath the waterfall to uncover more information from the letters that you can steal from a small box on the floor. Next you should head north west, where one cultist is interrogating another. Decide what should be done with him and then make your way back to the centre of the chamber.
When you reach the centre of the chamber, interact with the artefact. A cut scene will initiate in which you learn that the apparent leader of the cult of Kosmos is your sibling. When the cut scene ends you will have the opportunity to speak with Herodotus, who will suggest the next course of action. When the dialogue with Herodotus ends, the quest will complete.