Does the universe have a limit or is it infinite? People have been asking this question ever since they began to think about the universe. Amazingly, this question can be answered - sort of.
The key concept here is that of an cosmic horizon (sometimes also called the cosmological horizon). We already encountered a similar concept - the event horizon - when we discussed black holes (see Lesson 7). But the cosmic horizon is somewhat different: it appears because expanding universe has a finite age, and so the light (moving with finite speed - the speed of light) can only travel a finite distance during the lifetime of the universe. Let's say, the universe is 13 billion years old. Then the horizon will be at about 13 billion light years from us: galaxies that are farther away than the horizon distance are not visible to us - the light from them simply did not have enough time to reach us!
In reality, the horizon distance is not exactly 13 billion light years even if the universe is exactly 13 billion years old, because the universe expands. In the past, all distances were smaller, so the light could have passed them in a shorter time (the light got a kind of "free ride" on the expanding space). For example, today the horizon is actually at a distance of about 40 billion light years, not 13. However, this is a minor point because we are not going to use the specific value of the horizon distance in this class. All you need to know is that the horizon exists and is at a very large distance from us (tens of billions of light years).
The resolution of Olber's paradox comes from a combination of the concepts of cosmic expansion, the horizon, and look-back time. The sky is not infinitely bright because the starlight from distant galaxies is redshifted so much that it is cool and invisible. In fact, we can see radiation coming from a time before stars and galaxies even existed.
In addition, we can only see stars and galaxies within the cosmic horizon, and not all of the stars and galaxies in the universe. Each of these two reasons alone is sufficient to resolve the Olber's paradox, so in the Big Bang theory the paradox is solved with a big safe margin!
Remember: the expansion of the universe means that the space itself stretches with time. What does this expansion do to the horizon distance? Well, that depends on the speed with which the space expands. In some sense, it is similar to running down the escalator that is actually moving up. If you run as fast as you can, and the escalator is moving up slower than you run down, you will go down. But if the escalator is moving up faster than you can run down, you will end up going up! The same is true for the light of a distant galaxy as it moves through the expanding universe: because the speed of light is fixed (thanks to uncle Albert!). If a part of the universe is not expanding away from us too fast, the light will reach us, sooner or later. But if a part of the universe is expanding fast enough, the light will never reach us-- just as you will never reach the bottom of the escalator if the escalator moves too fast. This kind of expansion of the universe is sometimes called "superluminal" - "faster than light", but keep in mind, no violation of Einstein's Relativity Theory takes place - uncle Albert only forbids energy or information to be send faster than the speed of light, but says nothing about the rate with which space itself can stretch.
Now you can make a guess of what happens when the expansion of the universe decelerates - the universe expands slower and slower with time. In this case the horizon distance grows with time, as light from progressively more distant galaxies can go down the cosmic escalator (because the escalator is slowing down). That means that galaxies that were once (billions of years ago) beyond the horizon of the Earth are within the horizon today. In a decelerating universe, the number of galaxies within the horizon increases with cosmic time.
But if the expansion of the universe accelerates (and we will learn later that we have good reasons to think it indeed does at the present time), then sooner or later the cosmic escalator speed will exceed the speed of light, and the horizon will start moving inward. Suppose you have a friend (or a foe) who is running down the escalator a few steps behind you, and the escalator is moving up faster and faster. ou may still reach the bottom whereas your friend (or foe - whichever you prefer) will still have a couple of steps to make when the escalator starts moving up so fast as to carry that guy up even if he runs at fast as he can. There is a fascinating conclusion: if the expansion of the universe accelerates and the horizon is moving towards us, then galaxies that were visible at one moment (i.e. they were inside the horizon) will become invisible billions of years later (the horizon will move in front of them and they end up outside the cosmic horizon)! In an accelerating universe, the number of galaxies within the horizon decreases with cosmic time.