Expanding Coordinate Grids

R(t) is the scale factor (which we write as a(t), with a=1 now). l(t) is the proper distance (which we write as r) and r is the comoving (current) distance (which we write as ro). These two galaxies are anchored to the grid -- they have no peculiar velocities.

As the universe expands, light crosses the expanding grid (always moving locally at speed c). The redshift is simply R(tobs) / R(tem), the ratio of the scale factors between observing and emission.


Figure from OU text "Galaxies and Cosmology"