Smoothed matter density variance (${\sigma }^2(M,z)=\frac{1}{2{\pi }^2}{\int }_0^{\infty }{k}^{2}P(k,z)|W(kR){|}^{2}dk$)

The smoothed matter density variance measures the root-mean-square linear density contrast after applying a window corresponding to a Lagrangian mass scale. For a homogeneous isotropic field it is ${\sigma }^2=(1/(2{\pi }^2))\int k^{2}P(k)|W(kR){|}^{2}dk$. The variance convention, growth epoch and smoothing mass-density relation must agree with the collapse barrier used in a halo abundance calculation.