Angular diameter distance in a flat power-law universe ($d_A(t_e,t_0)=c{t}_e^p(t_0^{1-p}-t_e^{1-p})/(1-p)$)

For $R=A{t}^p$, $0<p<1$, integrate the radial photon path from emission to reception and multiply by the scale factor at emission. This gives the displayed angular diameter distance. The arbitrary normalization $A$ cancels, and the distance is not obtained by multiplying light speed by elapsed cosmic time.