= 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:
$$
R=A(1-\cos\theta),\qquad t=B(\theta-\sin\theta),\qquad A^3=GMB^2.
$$
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 $U=-3GM^2/(5R)$. At turnaround $K_m=0$, whereas the final <virial theorem> gives $2K_v+U_v=0$. Conservation of energy gives $U_m=K_v+U_v=U_v/2$, hence
$$
\boxed{R_v=R_m/2.}
$$
The 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:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-45-collapse.png]
{title=Spherical proto-cluster radius: expansion, turnaround, formal collapse and schematic virialization}
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.
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