= Particle horizon through radiation-matter equality
{title2=$r_p(a)=\frac{2c}{H_0\Omega_{m,0}}\left(\sqrt{\Omega_{r,0}+\Omega_{m,0}a}-\sqrt{\Omega_{r,0}}\right)$}
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>, $a_{\rm eq}=\Omega_{r,0}/\Omega_{m,0}$, this is $2(\sqrt2-1)$ times the radiation-only extrapolation $ca_{\rm eq}/(H_0\sqrt{\Omega_{r,0}})$. Its associated inverse length differs by an order-one factor from the <matter-radiation equality scale> $a_{\rm eq}H_{\rm eq}/c$.
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