Large Scale Structure

Window Function

In order to study the statistical properties of the density fluctuation field at a certain scale R, we use the window function. As an example, to compute the variance of the density contrast at scale R, we convolve the window function in the Fourier space with the power spectrum.

Top Hat

This function returns the top hat window function in the real space.

Equation 1. 

WTH(r,R) = 3 4πR3 1  R 0 > R. (8)

The mass enclosed within the volume selected by this window function is MTH(R) = 4π 3 ρ¯R3,where ρ¯(z) is the mean density of the universe at redshift z.

The top-hat window function in the Fourier space is given by

Equation 2. 

Wth(k,R) = 3 (kR)3(sinkR (kR)coskR) (9) = 3 (kR)j1(kR), (10)

where jν(kR) is the spherical Bessel function.

The first derivative with respect to R

Equation 3. 

dWTH(k,R) dR = 9 k3R4(sinkR(kR)coskR)+ 3 kR2 sinkR.

Gaussian

This function returns the gaussian window function in the real space,

Equation 4. 

WG(r,R) = (2πR2)32 exp r2 2R2 .

The mass enclosed within the volume selected by this window function is MG(R) = (2π)32ρ¯(z)R3, where ρ¯(z) is the mean density of the universe at redshift z.

This function returns the gaussian window function in the Fourier space,

Equation 5. 

WG(k,R) = exp k2R2 2 .

This function returns the derivative with respect to R of the gaussian window function in the real space,

Equation 6. 

dWG(k,R) dR = k2Rexp k2R2 2 .

Transfer Function

NcTransferFuncBBKS

NcTransferFuncEH

NcTransferFuncCAMB

NcTransferFuncPert