Past exam of the relativistic Astrophysics and Cosmology course of the University of Cambridge / 2012 / Paper 1 / 2 / a

Density parameters at the same epoch share a common critical-density denominator. Thus compare the densities directly: ${\rho }_m={\rho }_{m0}(1+z{)}^3$ and ${\rho }_{\gamma }={\rho }_{\gamma 0}(1+z{)}^4$. Equality givesThis is the photon-only matter-radiation equality appropriate to the specified radiation density, not the equality including all relativistic species.