= Open radiation universe particle horizon
{title2=$d_{h0}=\frac{\operatorname{arsinh}\sqrt{(1-\Omega_0)/\Omega_0}}{H_0\sqrt{1-\Omega_0}}$}
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.
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