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Think you need around 60 settlements (military alliance). Could be wrong though.
Every time you take a new settlement, you acquire points for your imperium rating. Once you build up enough points, your imperium rating will level up and you'll get a message from the advisor saying: 'your empire grows strong etc'.
In the first game, Chaos invasion will have a chance to trigger once you hit level 4 imperium, and the chance increases as you're imperium rating increase. Archaon triggers when you reach level 7 imperium.
They've done some rebalancing for Mortal Empires, but you can expect something similar.
The bigger your empire grows, the higher the chance of Chaos invasion. Ironically, for my high elf campaign, I am at turn 164 and still no chaos invasion because I hardly expanded.
Marisher was right, around 35 and 55 for both events to trigger.
Alliances do not count into this, but vassals do.
Edit: What Mr. Ape describes is the (completely correct) technical explanation.
In these games, Imperium is only intended by settlement count, though (yours and your vassals) by a rate of 1 per settlement, so no difference in capital/minor settlements.