Particle horizon through radiation-matter equality ($r_p(a)=\frac{2c}{H_0{\Omega }_{m,0}}\left(\sqrt{{\Omega }_{r,0}+{\Omega }_{m,0}a}-\sqrt{{\Omega }_{r,0}}\right)$)

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 $a_{\mathrm{eq}}{H}_{\mathrm{eq}}/c$.