= Solution
For a pressureless spherical mass shell with fixed enclosed mass and no shell crossing, the <shell theorem> gives $\ddot R=-GM/R^2$. A <cosmological constant> is absent in the <Einstein-de Sitter universe>. Parametric differentiation gives
$$
\dot R=\frac AB\frac{\sin\theta}{1-\cos\theta},\qquad
\ddot R=-\frac A{B^2(1-\cos\theta)^2}.
$$
Since $R^2=A^2(1-\cos\theta)^2$, the equation of motion holds provided
$$
\boxed{A^3=GMB^2.}
$$
The expanding branch runs from $\theta=0$ to the <turnaround of spherical collapse> $\theta=\pi$; the recollapsing branch runs from $\pi$ to $2\pi$. Hence
$$
\boxed{A=\frac{R_{\max}}2,\quad B=\frac{t_{\max}}\pi,\quad t_{\rm coll}=2t_{\max}.}
$$
The common bang-time origin selects the growing overdensity, without an extra arbitrary shift in the time parameter.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-55-spherical-collapse.png]
{title=Spherical-collapse radius versus time, showing turnaround and the distinct virialized endpoint}
{height=480}
In the <Einstein-de Sitter universe>, $H=2/(3t)$ and $\bar\rho=1/(6\pi Gt^2)$. An unperturbed homogeneous sphere containing the same mass thus has $R_{\rm bg}^3=(9/2)GMt^2$. Comparing its volume with the perturbed sphere gives
$$
\frac\rho{\bar\rho}=\frac{R_{\rm bg}^3}{R^3}
=\frac92\frac{(\theta-\sin\theta)^2}{(1-\cos\theta)^3}.
$$
At maximum expansion,
$$
\boxed{\frac{\rho_{\rm ta}}{\bar\rho(t_{\max})}=\frac{9\pi^2}{16}=\left(\frac{3\pi}{4}\right)^2\simeq5.55.}
$$
For the non-dissipative, homologous <spherical-collapse model>, the potential energy of a uniform sphere is $W=-3GM^2/(5R)$. At turnaround the bulk kinetic energy is zero, so $E=W_{\rm ta}$. At <virial equilibrium>, $2K+W=0$ gives $E=W_{\rm vir}/2$. Conservation of energy implies $W_{\rm vir}=2W_{\rm ta}$, and therefore $R_{\rm vir}=R_{\max}/2$. This comparison assumes the same potential-energy structure coefficient; it is the usual top-hat model rather than an exact claim about every halo profile.
Halving the radius multiplies the density by eight. During the interval to $t_{\rm coll}=2t_{\max}$, the background density falls by four. Thus
$$
\boxed{\Delta_{\rm vir}\equiv\frac{\rho_{\rm vir}}{\bar\rho(t_{\rm coll})}=8\times4\times\frac{9\pi^2}{16}=18\pi^2\simeq178.}
$$
The formal pressureless trajectory reaches zero radius; the physical virialized object instead has a finite radius. Its density is evaluated at the same collapse epoch, not at the instantaneous singular solution's density.
For a physical estimated virial radius and enclosed mass, $\rho_h=3M/(4\pi r_{\rm vir}^3)$. Equating this to $\Delta_{\rm vir}\bar\rho_{m,0}(1+z_f)^3$ gives a <halo density estimate of formation redshift>:
$$
\boxed{1+z_f\simeq\left[\frac{3M}{4\pi r_{\rm vir}^3\Delta_{\rm vir}\bar\rho_{m,0}}\right]^{1/3}.}
$$
In the pure <Einstein-de Sitter universe>, $\bar\rho_{m,0}$ equals the present <critical density>. The inference assumes the measured density retains the collapse-epoch normalization. Later accretion, mergers and a changing virial-radius convention can change it, so it is a model-dependent formation estimate rather than a unique historical date.
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