NumCosmoWindowTophat

A top-hat window function.

This object implements the NcWindow class for a top-hat window function.

The top-hat window function in the real space is defined as, \begin{equation} W_{TH}(r, R) = \frac{3}{4\pi R^3}, \,\,\,\, \mathrm{for} \,\,\, r \leq R \,\, , \end{equation} and 0 otherwise. The mass enclosed within the volume selected by this window function is $$M_{TH}(R)= \frac{4\pi}{3}\bar{\rho} R^3 \, ,$$ where $\bar{\rho}(z)$ is the mean density of the universe at redshift $z$.

When the function nc_window_eval_fourier() is applied, it returns the top-hat window function in the Fourier space, \begin{equation} W_{th}(k, R) = \frac{3}{(kR)^3} \left[ \sin (kR) - (kR)\cos (kR)\right] = \frac{3}{(kR)} j_1(kR), \end{equation} where $j_\nu(kR)$ is the spherical Bessel function. The function nc_window_deriv_fourier() returns the derivative with respect to R in Fourier space, \begin{equation} \frac{\mathrm{d} W_{TH}(k, R)}{\mathrm{d} R} = -\frac{9}{k^3 R^4} \left[ \sin (kR) - (kR)\cos (kR) \right] + \frac{3}{k R^2} \sin (kR) \, . \end{equation} The derivative with respect to R in real space is performed by nc_window_eval_realspace().