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.

For an Einstein-de Sitter universe normalized by $a(t_0)=1$, $a=(t/t_0{)}^{2/3}$. Its current comoving particle horizon is $3c{t}_0$. A photon emitted inward at that location at $t_0$ covers $3c{t}_0[(t/t_0{)}^{1/3}-1]$ in comoving distance by time $t$. Equating these distances gives arrival at $8t_0$, a travel time of $7t_0$.

Neglecting curvature and dark energy, integration of the Friedmann equation for radiation in cosmology and pressureless matter gives the displayed comoving particle horizon. At matter-radiation equality, $a_{\mathrm{eq}}={\Omega }_{r,0}/{\Omega }_{m,0}$, this is $2(\sqrt{2}-1)$ times the radiation-only extrapolation $c{a}_{\mathrm{eq}}/(H_0\sqrt{{\Omega }_{r,0}})$. Its associated inverse length differs by an order-one factor from the matter-radiation equality scale $a_{\mathrm{eq}}{H}_{\mathrm{eq}}/c$.