Top-hat filter

A top-hat filter smooths a field by averaging uniformly inside a chosen ball and assigning zero weight outside. For radius $R$, its normalized three-dimensional real-space weight is $W_R=3/(4\pi R^3)$ inside the ball. Applying it to the density contrast gives the overdensity on that scale and its variance. For uniform background density $\bar{\rho }$, the associated mass is $M=(4\pi /3)\bar{\rho }{R}^3$. A real-space top-hat differs from a sharp-k smoothing filter.