Past exam of the short options paper of the physics Prelims of the University of Oxford / 2022 / S02 / 3 / c / Solution

Normalize the scale factor to $a(t_0)=1$ and set $a_d=10^{-3}$. In the same matter-only model $a=(t/t_0{)}^{2/3}$, soIts present proper size after expansion with the scale factor isThe present comoving distance to decoupling isThe angular diameter distance is $D_A=a_d{\chi }_d$, so the angular size of a transverse length equal to one particle horizon radius isIf “the horizon” denotes the diameter of the whole causally connected ball, its angular diameter is twice this, approximately $3.74^{\circ }$. The degree-scale conclusion is the same. Neglecting early radiation is an explicit approximation inherited from the question.