In figuring out the rough timeline for my sci-fi, I've run into an interesting dilemma. As it stands right now, There are about 90 years in between one of my civilizations discovering space travel and them building an empire spanning several dozen planets and two or three hundred star systems in their neighborhood of the galaxy. Now, with the level of technology that they start with, I'm not too worried about their ability to reach and establish outposts and industry on these planets. The problem comes when I look at the list of three or four city-planets in the new empire.
So, to the main issue in this question. Assuming that establishing the colony is easy enough and no diseases or natural disasters wreck the population, how long would it take for a colony of sentients with human-like lifespans and reproductive cycles to turn an Earth-like planet into an ecumenopolis?
Notes and rules:
The empire in question starts with a planet with a population of about 20 trillion.
The empire in question starts its expansion with conquest and federation, adding 4 other developed planets and 1 ecumenopolis with 120 trillion inhabitants in the ten years after gaining space-faring status. The inhabitants of these other planets are of different species. They can still be used for colonization purposes, though they would not be the preferred type.
Cloning technology is available but seen as immoral by the empire.
Don't depopulate the starting planets by more than 10% of their populations.
Food is plentiful.
Robotics and large scale industry makes building the cities relatively easy.
Minimum population density for me to consider it an ecumenopolis is 2000 people per square mile.
Let me know if any other details are needed.