Open radiation universe particle horizon ($d_{h0}=\frac{\mathrm{arsinh}\sqrt{(1-{\Omega }_0)/{\Omega }_0}}{H_0\sqrt{1-{\Omega }_0}}$)

For a radiation-filled expanding universe with $0<{\Omega }_0<1$ and no cosmological constant, set $x=a/a_0$. The particle horizon radial proper distance is $H_0^{-1}{\int }_0^1[{\Omega }_0+(1-{\Omega }_0)x^2{]}^{-1/2}dx$. An inverse-hyperbolic-sine antiderivative gives the formula. The spatially flat limit is $d_{h0}=H_0^{-1}$; radial proper distance is distinct from the area radius of the corresponding horizon sphere.