This high enough density is the density at which the schwarzchild radius of the object is less than the actual radius of the object.
Mathematically,
[tex]r_s=\frac{2Gm}{c^2}=r[/tex]
If you assume the object to be a uniform sphere, we can write its mass as the product of density and volume:
[tex]m=\rho V=\frac{4/3}\rho\pi r^3[/tex]
And substituting into the first equation,
[tex]r=\frac{2G(\frac{4/3}\rho\pi r^3){c^2} [/tex]
[tex]\rho=\frac{3}{8}\frac{c^2}{G\pi r^2}[/tex]