[gaukes]

Well, they wouldn't be looking for Earth specifically, they'd be looking for more resources which are everywhere. An intelligent and energy-hungry enough species would want to harness the power of every sun it can reach.

That growth would be exponential because acquiring resources provides more energy which allows them to acquire more resources even faster.

If an intelligent species doubled the number of stars they control every 2 million years, they'd have every star in the milky way in around 50 million years.

That's still like 0.2% the lifetime of the galaxy.