= Solution
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. Hence
$$
\boxed{\theta_{5\,\mathrm{proper\ Mpc}}=4\,\theta_{5\,\mathrm{comoving\ Mpc}}.}
$$
\b[The 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.
Back to article page
