Past exam of the general relativity and cosmology course of the University of Oxford / 2013 / 3 / b / Solution

At matter-radiation equality, ${\rho }_{M,0}{a}^{-3}={\rho }_{R,0}{a}^{-4}$, so $a_{\mathrm{eq}}={\Omega }_{R,0}/{\Omega }_{M,0}$. Evaluating the integrated square-root relation givesThis assumes both present densities are positive and uses the same Big Bang origin for conformal time.