How quickly will it expand? One premise is that it will expand at the speed of light, because that's the fastest speed at which information can travel. There are also tantalizing experiments on quantum disentanglement that show some effect at rates faster than the speed of light, even much faster, perhaps theoretically instantaneously. Interestingly enough, though, this is not the transmission of information, but the transmission of profound quantum randomness, which doesn't accomplish our purpose of communicating intelligence. You need to transmit information, not randomness. So far nobody has actually shown true transmission of information at faster than the speed of light, at least not in a way that has convinced mainstream scientific opinion.
If, in fact, that is a fundamental barrier, and if things that are far away really are far away, which is to say there are no shortcuts through wormholes through the universe, then the spread of our intelligence will be slow, governed by the speed of light. This process will be initiated within 200 years. If you do the math, we will be at near saturation of the available matter and energy in and around our solar system, based on current understandings of the limitations of computation, within that time period. However, it's my conjecture that by going through these other dimensions that Alan and Paul talked about, there may be shortcuts. It may be very hard to do, but we're talking about supremely intelligent technologies and beings. If there are ways to get to parts of the universe through shortcuts such as wormholes, they'll find, deploy, and master them, and get to other parts of the universe faster. Then perhaps we can reach the whole universe, say 10^80 protons, photons, and other particles that Seth Lloyd estimates represents on the order of 10^90 bits, without being limited by the apparent speed of light.
If the speed of light is not a limit, and I do have to emphasize that this particular point is a conjecture at this time, then within 300 years, we would saturate the whole universe with our intelligence, and the whole universe would become supremely intelligent and be able to manipulate everything according to its will. We're currently multiplying computational capacity by a factor of at least 10^3 every decade. This is conservative as this rate of exponential growth is itself growing exponentially. Thus it is conservative to project that within 30 decades (300 years), we would multiply current computational capacities by a factor of 10^90, and thus exceed Seth Lloyd's estimate of 10^90 bits in the Universe. We can speculate about identity will this be multiple people or beings, or one being, or will we all be merged? but nonetheless, we'll be very intelligent and we'll be able to decide whether we want to continue expanding. Information is very sacred, which is why death is a tragedy. Whenever a person dies, you lose all that information in a person. The tragedy of losing historical artifacts is that we're losing information. We could realize that losing information is bad, and decide not to do that any more. Intelligence will have a profound effect on the cosmological destiny of the universe at that point.
I'll end with a comment about the SETI project. Regardless of this ultimate resolution of this issue of the speed of light and it is my speculation (and that of others as well) that there are ways to circumvent it if there are ways, they'll be found, because intelligence is intelligent enough to master any mechanism that is discovered. Regardless of that, I think the SETI project will fail it's actually a very important failure, because sometimes a negative finding is just as profound as a positive finding for the following reason: we've looked at a lot of the sky with at least some level of power, and we don't see anybody out there. The SETI assumption is that even though it's very unlikely that there is another intelligent civilization like we have here on Earth, there are billions of trillions of planets. So even if the probability is one in a million, or one in a billion, there are still going to be millions, or billions, of life-bearing and ultimately intelligence-bearing planets out there.
If that's true, they're going to be distributed fairly evenly across cosmological time, so some will be ahead of us, and some will be behind us. Those that are ahead of us are not going to be ahead of us by only a few years. They're going to be ahead of us by billions of years. But because of the exponential nature of evolution, once we get a civilization that gets to our point, or even to the point of Babbage, who was messing around with mechanical linkages in a crude 19th century technology, it's only a matter of a few centuries before they get to a full realization of nanotechnology, if not femto and pico-engineering, and totally infuse their area of the cosmos with their intelligence. It only takes a few hundred years!
So if there are millions of civilizations that are millions or billions of years ahead of us, there would have to be millions that have passed this threshold and are doing what I've just said, and have really infused their area of the cosmos. Yet we don't see them, nor do we have the slightest indication of their existence, a challenge known as the Fermi paradox. Someone could say that this "silence of the cosmos" is because the speed of light is a limit, therefore we don't see them, because even though they're fantastically intelligent, they're outside of our light sphere. Of course, if that's true, SETI won't find them, because they're outside of our light sphere. But let's say they're inside our light sphere, or that light isn't a limitation, for the reasons I've mentioned, then perhaps they decided, in their great wisdom, to remain invisible to us. You can imagine that there's one civilization out there that made that decision, but are we to believe that this is the case for every one of the millions, or billions, of civilizations that SETI says should be out there?
That's unlikely, but even if it's true, SETI still won't find them, because if a civilization like that has made that decision, it is so intelligent they'll be able to carry that out, and remain hidden from us. Maybe they're waiting for us to evolve to that point and then they'll reveal themselves to us. Still, if you analyze this more carefully, it's very unlikely in fact that they're out there.
You might ask, isn't it incredibly unlikely that this planet, which is in a very random place in the universe and one of trillions of planets and solar systems, is ahead of the rest of the universe in the evolution of intelligence? Of course the whole existence of our universe, with the laws of physics so sublimely precise to allow this type of evolution to occur is also very unlikely, but by the anthropic principles, we're here, and by an analogous anthropic principle we are here in the lead. After all, if this were not the case, we wouldn't be having this conversation. So by a similar anthropic principle we're able to appreciate this argument. I'll end on that note.