Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 67 / 1 / i

A comoving coordinate labels a position relative to the homogeneous expansion. For two nearby comoving positions separated by $\Delta \chi$, their proper distance in cosmology at time $t$ is $a(t)\Delta \chi$. Thus a fixed comoving separation expands physically with the scale factor, while a fixed proper separation corresponds to a shrinking comoving separation. Normalize $a_0=1$, so $a(z)=1/(1+z)$.

At a common cosmological redshift, a transverse proper size $l$ subtends $\theta \simeq l/D_A$, where $D_A$ is the angular diameter distance. At $z=3$, a $5$ comoving Mpc separation has proper size $5/(1+3)=1.25$ Mpc. HenceThe proper-size object has the larger angular size, by a factor of four. This comparison does not require a numerical angular diameter distance, since it cancels.