A drop of ink spreads in a glass of water: its edges move faster than its center. Similarly, a clump of dark matter collapsing under its own gravity expands unevenly, with different shells flying apart at different speeds. Previously, astrophysicists could accurately describe only the simplest case: a homogeneous sphere with a sharp boundary. But real clumps are smooth: density gradually falls from the center to the edge, and each shell feels its own gravitational pull.
For quick estimates, a simple formula has been proposed that errs by no more than 0.7%. This means that instead of lengthy computer calculations, you can instantly obtain velocities for a given density profile. Previously, for each new profile you had to solve the equations of motion anew; now you just plug numbers into a ready-made formula.
For a shell in a void wall, the local contrast can be positive while the internal contrast is negative: the shell is denser than average, but inside it there is still a mass deficit. This difference drives the expansion anisotropy.