Halo density estimate of formation redshift ($1+z_f\simeq [{\rho }_h/({\Delta }_{\mathrm{vir}}{\bar{\rho }}_{m,0}){]}^{1/3}$)

If a halo retains the mean density acquired at collapse, $3M/(4\pi R^3)={\Delta }_{\mathrm{vir}}{\bar{\rho }}_{m,0}(1+z_f{)}^3$ estimates its formation redshift. In the non-dissipative spherical-collapse model during matter domination, ${\Delta }_{\mathrm{vir}}=18{\pi }^2$. Subsequent mass growth, profile evolution and the reference-density convention limit this inference.