SharpKEllipsoid

class hmf.density_field.filters.SharpKEllipsoid(k, power, **model_parameters)

Fourier-space top-hat window function with ellipsoidal correction.

See Schneider, Smith, Reed 2013.

Refer to Filter for more details.

Methods

__init__(k, power, **model_parameters)

a3(r)

Short-axis scale with ellipsoidal correction.

a3a1(e, p)

Compute the short:long axis ratio of an ellipsoid.

a3a2(e, p)

Compute the short:medium axis ratio of an ellipsoid given its ellipticity/prolateness.

dlnr_dlnm(r)

Compute the derivative of log radius with log mass.

For the usual \(m\propto r^3\) mass assignment, this is just 1/3.

Parameters:

r (array_like) – Radii.

dlnss_dlnm(r)

Compute the logarithmic slope of mass variance with mass.

This is an important quantity, and is used directly to calculate \(\frac{dn}{dm}\).

Parameters:

r (array_like) – Radii.

dlnss_dlnr(r)

Compute the derivative of the mass variance with radius.

Parameters:

r (array_like) – Radii

Returns:

dlnss_dlnr – The derivative of the the mass variance with radius.

Return type:

array_like

Notes

Given a prescription for how radius grows with mass (typically with a log-slope of 1/3, and set in dlnr_dlnm()), this specifies the quantity \(\frac{d \ln \sigma^2}{d\ln m}\).

The general formula is

\[\frac{d\ln \sigma^2}{d\ln R} = \frac{1}{\pi^2\sigma^2} \int_0^\infty W(kR) \frac{dW(kR)}{d\ln(kR)} P(k)k^2 dk\]

dw_dlnkr(kr)

Compute the derivative of the (fourier-transformed) filter with \(\ln(kr)\).

Parameters:

kr (array_like) – Scale(s) at which the derivative is evaluated.

Notes

In terms of \(\frac{dw^2}{dm}\), which is a commonly used quantity, this has the relationship

\[w\frac{dw}{d\ln r} = \frac{2}{r}\frac{dw^2}{dm}\frac{dm}{dr}.\]

em(xm)

Compute the average ellipticity of a patch as a function of peak tensor.

gamma(r)

Bardeen et al. 1986 equation 6.17.

classmethod get_models() → dict[str, type]

Get a dictionary of all implemented models for this component.

k_space(kr)

Fourier-transform of the real-space filter.

Parameters:

kr (array_like) – The scales at which to return the filter function

Returns:

w – The filter in fourier space, len(kr)

Return type:

array_like

mass_to_radius(m, rho_mean)

Return radius of a region of space given its mass.

Parameters:

• m (array_like) – Masses

• rho_mean (float) – Mean density of the Universe.

Returns:

r – The corresponding radii to m

Return type:

array_like

Notes

The units of m don’t matter as long as they are consistent with rho_mean.

nu(r, delta_c=1.68647)

Peak height, \(\frac{\delta_c^2}{\sigma^2(r)}\).

Parameters:

• r (array_like) – Radii

• delta_c (float, optional) – Critical overdensity for collapse.

pm(xm)

Compute the average prolateness of a patch as a function of peak tensor.

r_a3(rmin, rmax)

Spline interpolant for radius-to-axis ratio relationship.

Created from computed a3 values over the radius range.

radius_to_mass(r, rho_mean)

Return mass of a region of space given its radius.

Parameters:

• r (array_like) – Radii

• rho_mean (float) – Mean density of the Universe.

Returns:

m – The corresponding masses to r

Return type:

float or array of floats

Notes

The units of r don’t matter as long as they are consistent with rho_mean.

real_space(R, r)

Filter definition in real space.

Parameters:

• R (float) – The smoothing scale

• r (array_like) – The radial co-ordinate

sigma(r, order=0)

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)

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

xi(pm, em)

Ellipsoid axis ratio correction factor.

Computed from ellipticity and prolateness parameters.

xm(g, v)

Peak of the distribution of x.

Here, x is the sum of the eigenvalues of the inertia tensor (?) of an ellipsoidal peak, divided by the second spectral moment.

Equation A6. in Schneider et al. 2013