the scale factor
Imagine a rubber sheet with a grid drawn on it, galaxies pinned at the grid crossings. As the sheet stretches, the galaxies stay at their grid coordinates while every physical distance between them grows. The single number that says how much the sheet has stretched at a given time is the scale factor. It is the mathematical heart of an expanding universe: one function of time that keeps score of how big space is.
The scale factor a(t) is a dimensionless function, conventionally set to a = 1 today, such that the physical distance between two objects at rest in the cosmic flow is d(t) = a(t) times their fixed comoving separation. When a doubles, all such distances double. Its fractional rate of change is exactly the Hubble parameter, H(t) = a_dot / a, where a_dot is the time derivative of a; evaluated today this is the Hubble constant H_0. Wavelengths of light stretch in the same proportion, giving 1 + z = a_now / a_then, and densities of ordinary matter dilute as 1/a^3 while radiation dilutes faster, as 1/a^4, because its wavelengths stretch too.
The scale factor is where cosmology hides its dynamics: how a(t) grows over time is dictated by the Friedmann equations, and the mix of matter, radiation, and dark energy determines whether that growth decelerates or, as we now observe, accelerates. A subtlety worth stressing: comoving coordinates do not expand, only physical distances do, and gravitationally bound systems like galaxies and rulers do not stretch with a because their internal forces hold them fixed.
At the epoch of the cosmic microwave background the scale factor was about a = 1/1100, so the universe was roughly 1100 times smaller in every linear dimension and its temperature was about 1100 times higher than the 2.725 K we see today, close to 3000 K, hot enough that atoms had not yet formed.
One number, a(t), sets sizes, redshifts, temperatures, and densities across cosmic time.
Only the ratio of scale factors is physical; the choice a = 1 today is a convention. And a bound object such as a galaxy or an atom does not grow with a, because internal forces, not the cosmic expansion, set its size.