= 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 as
$$
\boxed{\dot{\mathbf r}=H\mathbf r+\mathbf v_{\rm pec},\qquad\mathbf v_{\rm pec}=a\dot{\mathbf x}.}
$$
Thus <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.
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