# 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 FLRW metric with its scale factor is a way of describing the large-scale shape of spacetime. Its physical significance is in the shape it describes, nothing more or less.

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".

The universe on a large scale has that kind of shape, or the Lorentzian version of it. You often see pictures like this: which, while pretty heavy on the artistic license, do correctly convey that spacetime has a shape that can be described by a metric of this form.

The distance between the stars/galaxies on this surface as a function of time can be understood in terms of the scale factor, but it's not caused by the scale factor. The scale factor is just a function that shows up in the quasi-polar coordinate system shown by the yellow lines. You can put different coordinates on the spacetime and describe the motion in a different way.