At what scale is the universe expanding?
The cosmic scale factor also know as the Robertson-Walker scale factor.
I have to admit, I thought this was a bit of a trick question as the whole universe is expanding it must be expanding at a universal scale. However, that is too simplistic.
The universe does not just expand as we see it in 3 dimensions. Space time itself is expanding so that not only are objects moving away from each other due to their own motion, but the space itself between the objects is in fact "stretching".
Consider the very beginning of the universe according to the Big Bang theory. The sheer mass and relative small volume of the original point of expansion would have had such a high amount of gravity that any expansion would have been impossible. Essentially there was so much mass in such a relatively small volume, that it should have turned into a black hole. So how did it start expanding?
The answer to that is that space time expanded, and rapidly.
Imagine 2 baseballs beside each other on a field. Gravity is keeping them together, and no force applied directly to the balls can separate them. Now an earthquake opens a chasm between them, moving them apart. The was no force on the actual baseballs directly to separate them but now they are apart. That is how the big bang had to have started. There is extra evidence, as scientists have mathematically determined that for the first 5 billion years of the universe, the expansion rate was slowing. Essentially that means that gravity was trying to pull everything back together, to form a black hole, but the stretching of space was at a greater rate. After the first 5 billion years the distance between objects caused gravity to lessen and the rate of expansion increased.
So they gave the scale factor of universal expansion a new name to reflect how the universe was expanding in more than just 3 dimensions. They called the scale the cosmic scale or the Robertson-Walker scale.