Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 66 / 4 / ii / Solution

Well before matter-radiation equality, radiation domination makes the integrand approximately constant:At equality, ${\Omega }_{r,0}{a}_{\mathrm{eq}}^{-4}={\Omega }_{m,0}{a}_{\mathrm{eq}}^{-3}$, so $a_{\mathrm{eq}}\simeq 1/3000$ and ${\Omega }_{r,0}\simeq 10^{-4}$. Extrapolating the radiation-only result gives $r_p(a_{\mathrm{eq}})\simeq 133\mathrm{Mpc}$. This overestimates the horizon because matter is no longer negligible at equality and increases the expansion rate.

Retain both matter and radiation, still neglecting the insignificant early cosmological constant. The particle horizon through radiation-matter equality isThereforeThis is a comoving distance; the proper horizon at equality is approximately $110/3000\mathrm{Mpc}\simeq 37$ kiloparsecs.

For today's horizon, extrapolating matter domination givesThis is an order-of-magnitude estimate: dark energy is significant late, and radiation matters early. Numerical integration of the complete expression with the supplied density parameters and ${\Omega }_{r,0}\simeq 10^{-4}$ givesBoth neglected positive-density terms increase $H$ at fixed $a$ and reduce the horizon relative to the matter-only extrapolation.