I don’t think anyone (at least not anyone in this thread) thinks we’re alone in the universe. RickJay’s comment was simply that the distances are so vast that for practical matters, we might as well be alone. Even if there are a thousand other planets with intelligent life just in our galaxy, the difficulties in making contact with them are just too great.
As an exercise to the reader, I propose the following:
A civilization sends out colony ships which take 100 years to reach the nearest star. The civilization can send out colony ships every ten years, to a maximum of ten colonies before all (relatively) nearby stars are claimed.
Each colony takes 100 years to become firmly established, before it starts sending out colony ships, to a maximum of ten before all nearby unclaimed stars are taken. How long does this Fibonacci-like progression take to span a empire covering a sizable fraction of a galaxy, i.e. 10 billion star systems?
Assuming that eventually the “frontier” becomes an essentially flat wave front, then it’s advancing through the galaxy at the rate of one solar system every 200 years. So 200 years to travel an average of five light-years gives us ~ four million years to span the galaxy, down to half that if the starting point was near the center.
Bryan Ekers is talking about a civilisation that increases its population efforts tenfold every 200 years, however, this won’t mean it could span the galaxy any faster - they would quite quickly reach a point where their capacity to expand their efforts far outstrips the number of available local stars to send their colony ships to (because some of them would also be in the local vicinity of their peer colonies), so yes, the wavefront can’t move any faster than the fastest part of it and it can’t grow exponentially for long.
In any event, if it takes a billion years minimum for life to form and evolve into a technologically advanced state allowing interstellar travel, the additional time required to spread through a significant portion of a galaxy is (at a few million years) a mere rounding error by comparison. I take this to mean that if we did pick up signals from an advanced civilization, it’s probably too late to keep them from coming here and assimilating the heck outta us when they get around to it. We’d have to hope that they destroyed themselves before they mastered interplanetary travel (and we’re just picking up signals from their early-stage post-industrial forays into radio transmission) and that we can avoid destroying ourselves before we can establish our first interstellar colony.
The race is on!
Aw, I was hoping we’d pick up a video signal of Lrrrrpip, who is pair-bonded to Reekee (leader of a musical collective, and native of a different planet). In the video signal, we see Lrrrrpip disguising her(?)self as a K!bn performer in order to infiltrate the collective.