I would like to know why the night sky isn't bright all the time. The stars in the universe give off heat and this heat has to go somewhere, right? Usually light accompanies heat, and this heat that doesn't escape the universe must be heating it up and thus the night sky should be bright as day. (I am assuming that 10+ billion years is good enough to heat up the universe.)
You've stumbled across a famous problem referred to as Olbers' Paradox, the solution of which you will probably find quite surprising.
To respond to the question the way you phrased it, I guess I would simply say "no, 10+ billion years is not enough time to heat up the universe." More specifically, the fact that the universe is only around 13 billion years old means that an individual star can only heat up a region of space within a distance of about 13 billion light-years from it. Anything farther, and the light just wouldn't have had enough time to get there yet. Therefore, when we look at the night sky, we only see light coming from stars within about 13 billion light-years of us, and the total amount of light produced by all these stars is not enough to make the night sky particularly bright.
A more detailed description of Olbers' paradox allows you to conclude that if the universe (a) were big enough so that every line of sight ended in a star, (b) were infinitely old, (c) were static and not expanding and (d) if several other simple assumptions were satisfied, then the entire night sky would be roughly as bright as the surface of our sun!
Therefore, the simple observation that the night sky is dark allows us to say something very profound about our universe: it cannot be infinitely big, infinitely old and static all at the same time! Many people used to believe that the universe was this way, and it turns out that all they would have had to do to convince themselves otherwise was think carefully about a simple question that every child asks: "Why is the sky dark at night?" This is definitely an argument for the old adage that there's no such thing as a stupid question!
As it turns out, we still don't know whether the universe is infinitely big, but we do have a good grasp on the age of the universe and we do know that it is expanding, and of the two, it is the finite age of the universe which seems to provide the main explanation for Olbers' paradox.
I read an interesting article in The New York Times recently which said that the first person known to have explained Olbers' paradox was the poet Edgar Allan Poe! In his largely forgotten prose poem "Eureka" (published in 1848) Poe wrote the following (quoted on this page, which also contains other examples of Poe's prescience):
Were the succession of stars endless, then the background of the sky would present us an uniform luminosity, like that displayed by the Galaxy - since there could be absolutely no point, in all that background, at which would not exist a star. The only mode, therefore, in which, under such a state of affairs, we could comprehend the voids which our telescopes find in innumerable directions, would be by supposing the distance of the invisible background so immense that no ray from it has yet been able to reach us at all.
Sounds better than when I try to explain it, doesn't it?
This page was last updated June 27, 2015.