cosmology - How large is the universe?

We know that the age of the universe (or, at least the time since the Big Bang) is roughly 13.75 billion years. I have heard that the size of the universe is much larger than what we can see, in other words, much larger than the observable universe. Is this true? Why would this not conflict with what is given by Special Relativity (namely, that faster-than-light travel is prohibited)?

If it is true, then what is the estimated size of the universe, and how can we know?

Answer

It is indeed more useful to cite the age of the universe, because this defines the region in space which is observable, a 13.75 billion lightyear sphere (approximately).

Clearly, however, the entire universe could be more than 13.75 billion years across in diameter; that number is merely a radius. For example, let's suppose a naive view of the expansion of the universe which doesn't include inflation or dark energy. At the moment of the big bang, photons rush off in every direction at the speed of light (again, naive cosmology - ignore the fact that the universe is opaque for 300,000 years).

These photons are all moving out from one point at the speed of light. We imagine, then, an expanding sphere whose surface is defined by the furthest point which light has so far reached. This sphere is expanding in volume very quickly indeed - the radius is expanding at the speed of light.

So by now, 13.7 billion years later, the radius is 13.7 billion light years. The diameter of the sphere is twice that, 27.4 billion light years. The volume is volume of a sphere with radius r=13.7 GLy, which is $4\pi r^3 / 3 =57.4$ billion cubic lightyears.

This is shows that the universe can easily be much larger than 13.75 billion light years across. Also, note that, if the earth is formed on the expanding sphere, no one from earth will ever be able to "catch up" and see the other side of the sphere, since that side is still expanding. This is what people mean when they say that the universe is larger than we can see.

Now, this answer is wrong. Do not go quoting these numbers. It doesn't take into account inflation or the expansion of the universe. No one knows enough about either of those two effects to give a really good precise number for the size of the universe, but you can be certain that they only result in a bigger universe. One lower-bound for the radius of the universe is 39 billion light years, based on some analysis of the cosmic microwave background.