scale factor

Imagine the whole universe printed on a rubber sheet that you can stretch evenly in every direction. To describe how stretched it is at any moment, you only need a single number: a multiplier that says how much bigger the sheet is now compared to some reference time. That number is the scale factor, written a(t). It is the universe's master dial, turning the abstract idea of expansion into one quantity you can track through time.

By convention the scale factor is set to a = 1 today. At an earlier time when the universe was half as spread out, a = 0.5; when distances were one-tenth of today's, a = 0.1; at the very beginning of the Big Bang, a approached 0. Crucially, every distance between unbound galaxies scales in lockstep with a(t): if a doubles, all such distances double. This makes the scale factor the precise mathematical heart of expansion. It connects directly to redshift, since the wavelength of light stretches by exactly the same factor that space stretched while the light travelled — the relation is 1 + z = 1/a, so a galaxy seen at redshift z = 3 emitted its light when the universe was a = 1/4 of its present size.

The scale factor matters because the entire history and future of the cosmos can be told as the story of how a(t) changes — slow at first, decelerating under gravity, then accelerating under dark energy. The Friedmann equations are precisely the rules that govern how a(t) evolves. A caution worth keeping: the scale factor describes relative sizes, not an absolute size of the universe; a = 0.5 does not mean the universe was a particular number of metres across, only that all cosmic distances were half their present values.