 Particle horizon in a power-law universe

The radial null integral from the Big Bang is ${\chi }_p=c{\int }_0^{t}d{t}^{\prime }/R(t^{\prime })=c{t}^{1-p}/[A(1-p)]$. Multiplication by the scale factor gives the proper radial horizon distance. It is $3ct$ for a flat matter-only model and $2ct$ for a flat radiation-only model. A finite initial time different from zero or an earlier expansion era changes the result.