Matter-radiation equality

Matter-radiation equality is the epoch at which the background matter and radiation densities are equal. Its horizon wavenumber $k_{\mathrm{eq}}$ sets the turnover scale of the cold-dark-matter transfer function.

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.

The matter-radiation equality scale is the comoving horizon wavenumber at equality. Modes at larger wavenumber entered during radiation domination and had their matter growth suppressed.