Comoving particle horizon ($r_p(t)=c{\int }_0^t\frac{d{t}^{\prime }}{a(t^{\prime })}$)

The comoving particle horizon is the particle horizon expressed in comoving coordinates, assuming a finite initial cosmic-time boundary at zero. In a spatially flat FLRW metric, it is the distance a photon could have travelled since that boundary. The corresponding proper distance in cosmology at time $t$ is $a(t)r_p(t)$. It differs from the future-directed cosmological event horizon and from the instantaneous comoving Hubble radius.