= Formation redshift from halo mean density
{title2=$1+z_f=[3M/(4\pi R^3\Delta_f\rho_0)]^{1/3}$}
In a spherical-overdensity halo convention, $3M/(4\pi R^3)=\Delta_f\rho_0(1+z_f)^3$, so $1+z_f$ follows from the displayed cube root. At fixed measured $M,R,\rho_0$, changing the adopted overdensity changes $1+z_f\propto\Delta_f^{-1/3}$. It does not change the hydrostatic <temperature> estimate from the same measured <mass> and radius. A <redshift> inferred this way is a model formation epoch, not a direct age measurement.
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