Einstein-de Sitter universe

An Einstein-de Sitter universe is spatially flat and contains only pressureless matter. In conformal time, $a\propto {\tau }^2$ and $\mathcal{H}=a^{\prime }/a=2/\tau$.

Spherical secondary infall follows concentric collisionless shells around a primordial overdensity. Each shell initially expands with the Hubble flow, turns around, collapses, and subsequently crosses other shells.

The turnaround radius of a shell is its maximum physical radius before collapse. At a given cosmic time, the currently turning shell defines a natural similarity length.

If the initial fractional excess mass is a power law in enclosed mass, spherical secondary infall admits a similarity reduction in which radii scale by the current turnaround radius and masses by the mass of the currently turning shell.