A drop of ink spreads in a glass of water: its edges move faster than its center, and the drop's shape changes. Likewise, 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 uniform sphere with a sharp boundary, which has a single expansion rate. But real clumps are smooth: density gradually falls from center to edge, and each shell feels its own gravitational pull.
The authors introduce two numbers for each point: the local contrast, how much denser the material is here compared to the cosmic average, and the internal contrast, how much denser than average is everything inside that point. It turns out that the velocity of expansion perpendicular to the radius depends only on the internal contrast, while the velocity along the radius also depends on the difference between the local and internal contrasts. This difference directly indicates how steeply the density changes from point to point: if it is large, the shells stretch more strongly.
For quick estimates, a simple three-term approximation is proposed that errs by no more than 0.7% over a wide range of cosmologies. This means that instead of lengthy computer calculations, one can instantly obtain velocities for a given density profile. Previously, for each new profile one had to solve the equations of motion anew; now it suffices to plug numbers into a ready-made formula.
Practical benefit: in cosmological simulations and analytical models of structure formation, spherical profiles are often needed. The new result allows rapid computation of velocities without re-solving the dynamics, saving time and simplifying interpretation of data on peculiar velocities of galaxies.
The ink drop in the glass comes to mind again: now its spreading can be described by a single function, knowing only the distribution of ink at each moment. Likewise, a dark matter clump, its expansion is fully determined by the density profile and one universal function, the same for all smooth spherical shapes.
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.