 Solution

A comoving coordinate labels a location in the expanding background and is constant for an object moving with the Hubble flow. For example, a radial comoving geodesic coordinate $\chi$ gives proper distance in cosmology $d(t)=a(t)\chi$ along the constant-cosmic-time slice. In a flat background one can write physical coordinates $\mathbf{r}=a(t)\mathbf{x}$ and split the physical velocity asThus proper distance in cosmology is measured using the spatial metric at a specified epoch, whereas a comoving distance factors out the overall expansion. A proper distance on a simultaneous spatial slice is not automatically the luminosity distance or a distance directly inferred from light received today. In a curved spatial geometry $a\chi$ is the radial proper distance, while the sphere's circumference involves the corresponding curved radial function.