Angular particle-horizon size in a power-law universe (${\theta }_p=r^{1-p}/(1-r^{1-p}),r=t_e/t_0$)

Divide the particle horizon in a power-law universe at emission by the angular diameter distance in a flat power-law universe. The result is the small angular extent of one radial horizon length. Matter-only and radiation-only histories have powers $1/3$ and $1/2$, respectively. These are idealized single-era histories; a real recombination horizon and distance depend on different portions of the expansion history.

The proper radial particle horizon is the distance travelled by a null signal since the initial boundary, measured on the current time slice. A spherical patch with that radius has twice that physical diameter. Dividing one or the other by an angular diameter distance therefore differs by a factor of two. A question using horizon size should state which linear extent is intended.