# What is the scale factor $a(t)$?

Scale factors are used to refer to the velocity of expansion of the universe, curvature of the universe, etc. Scale factors can be defined in the form $$D(t)=a(t)D(t_o)$$, meaning that scale factors can be used as a ratio of distance between any two galaxies. But at the same time, I don't understand that scale factors can also be represented by curvature radius. The change in the scale factor means that the distance between the two galaxies changes, which means you can consider the scale factor as a distance or radius of the universe.

To sum up, is the scale factor the distance between ants on the balloon? Could you illustrate the difference between distance and scale factor?

The Euclidean plane in polar coordinates has the metric $$dr^2 + r^2 dθ^2$$. The unit sphere in polar coordinates has the metric $$dr^2 + \sin^2 r\,dθ^2$$, where $$r$$ is the latitude (zero at a pole) and $$θ$$ the longitude. A cone in polar coordinates has the metric $$dr^2 + k^2r^2 dθ^2$$ for some $$0.
These metrics are all of the form $$dr^2 + f(r)^2 dθ^2$$, and describe surfaces with a certain kind of rotational symmetry. You could think of them as "surfaces you can make on a potter's wheel".