hmf.density_field.filters.Filter.sigma¶
- Filter.sigma(r, order=0, rk=None)[source]¶
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)
\[\sigma^2_n(R) = \frac{1}{2\pi^2} \int_0^\infty dk\ k^{2(1+n)} P(k) W^2(kR)\]