Spherical collapse model

Summary

The spherical collapse model follows a uniform spherical overdensity in an expanding universe as it slows down, turns around and collapses. The sphere behaves like a small closed universe, so its radius has an exact cycloid solution. It gives the critical linear overdensity $\delta_c \approx 1.686$ for collapse and the virial overdensity $18\pi^2 \approx 178$ of a halo. These two numbers are the input of the Press–Schechter formalism and of many halo definitions in structure formation.