hmf.density_field.filters.Filter.sigma

Filter.sigma(r, order=0, rk=None)

Calculate the nth-moment of the smoothed density field, \(\sigma_n(r)\).

Note

This is not \(\sigma_n^2(r)\)!

Parameters

• r (float or array_like) – The radii of the spheres at which to calculate the nth moment.

• order (int, optional) – The order of the moment. Default 0 corresponds to common mass variance.

Returns

sigma – The square root of the moment at r.

Return type

array_like

Notes

The general definition for the nth-moment of the smoothed density field is (see Bardeen et al. 1986, Eq 4.6c)

\[\sigma^2_n(R) = \frac{1}{2\pi^2} \int_0^\infty dk\ k^{2(1+n)} P(k) W^2(kR)\]