 Photon-only matter-radiation equality

Nonrelativistic matter density grows as $(1+z{)}^3$, whereas freely redshifting photon energy density grows as $(1+z{)}^4$. Their equality gives the displayed redshift. Radiation inferred solely from a cosmic microwave background temperature excludes the relic neutrino energy density, so the resulting redshift differs from total matter-radiation equality. Comparing density parameters at one epoch is equivalent to comparing the densities because their common critical-density denominator cancels.