Particle horizon in a power-law universe ( $R (t) = A t^{p}, D_{p} (t) = c t / (1 - p), 0 < p < 1$ )

The radial null integral from the Big Bang is

χ_{p} = c \int_{0}^{t} d t^{'} / R (t^{'}) = c t^{1 - p} / [A (1 - p)]

. Multiplication by the scale factor gives the proper radial horizon distance. It is

3 c t

for a flat matter-only model and

2 c t

for a flat radiation-only model. A finite initial time different from zero or an earlier expansion era changes the result.

Table of contents
- Angular particle-horizon size in a power-law universe Particle horizon in a power-law universe
  - Particle-horizon radius versus patch diameter Angular particle-horizon size in a power-law universe

Angular particle-horizon size in a power-law universe ( $θ_{p} = r^{1 - p} / (1 - r^{1 - p}), r = t_{e} / t_{0}$ )

 0  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.

Particle-horizon radius versus patch diameter

 0  0

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.