There are a lot of ill-informed rivenless Eidolon [Rubico] builds using Heavy Caliber instead of mods like Vigilante Armaments (+multishot is really good.) or (primed) Cryo Rounds (also good) - both are better and have no downsides. Rivens aren't needed, but if you do get one replace whichever mod based on stats IF it is more DPS. We aren't using Galvanized mods since there aren't any other mobs to kill to proc their bonuses, and without them Galvanized mods are usually straight weaker.
[Vile Acceleration] is for Burst DPS, which matters a lot more than sustained DPS. Can be swapped with [Vigilante Armaments] for better sustained but lower burst. It also offsets [Critical Delay], which gives us more crit chance and allows us to CONSISTENTLY crit over [Point Strike], which is extremely good.
Exilus is personal preference.