I just got my copy of the Eberron Rising sourcebook and am keen to run a game. I love the idea of a noir detective style story, but it seems to me that it will revolve around a lot of role-playing and investigation, but relatively few fights.
How should I award experience in a game like this?
As a DM in 5E, you have massive freedom to craft your campaign and reward players as you see fit. Even when working from a source book such as Eberron, you can adjust encounters as you see fit to add to the story or present additional challenge.
XP was never designed as only a combat reward, but rather, a reward for your PCs overcoming an obstacle to their adventure goal. This can be outright defeat of an enemy in combat, roleplay encounters, puzzle-solving, political machinations; anything your players come up with to address a problem. Your role in PC progression is to determine the difficulty of these problems, and to reward the PCs with the appropriate degree of "growth" for having overcome them; this growth is represented as Experience Points.
In short, the simple answer is: "Award experience however you like!"
