Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 45 / 3 / a / Solution

In an Einstein-de Sitter universe, a slightly overdense spherical region initially expands nearly with the Hubble flow, with its early physical radius proportional to $t^{2/3}$. Its extra self-gravity gradually slows its expansion relative to the background. The radius reaches a maximum at spherical-collapse turnaround, where the coherent radial velocity is zero; it then contracts. The ideal pressureless top-hat would collapse to a point, but real collisionless matter undergoes crossing and relaxation into a bound system with random motions and virial equilibrium.

The bound-shell solution illustrates the stages:At $\theta =\pi$, $R_m=2A$ and $t_m=\pi B$; the formal cold collapse occurs at $\theta =2\pi$, $t_c=2t_m$. For a uniform isolated sphere, let $U=-3G{M}^2/(5R)$. At turnaround $K_m=0$, whereas the final virial theorem gives $2K_v+U_v=0$. Conservation of energy gives $U_m=K_v+U_v=U_v/2$, henceThe usual spherical-collapse prescription associates virialization with roughly the formal collapse epoch, not with a literal instantaneous halt of the cold shell. The sketch distinguishes the formal point collapse from the schematic settling at the virial radius:

The plateau onset is illustrative. A real cluster's relaxation, ongoing accretion and nonuniform density profile replace the cold solution, and surface pressure or energy exchange can change the half-radius estimate. The radius here is physical, not comoving.