= Solution
In a matter-dominated <Einstein-de Sitter universe>, the growing linear <density contrast> is proportional to the scale factor. When $|\delta|\sim1$, linear evolution no longer describes collapse. An overdense region initially expands more slowly than the background, reaches <cosmological turnaround>, and then contracts under self-gravity.
The <spherical top-hat collapse model> supplies a useful idealization: pressureless shells obey $\ddot r=-GM/r^2$. At formal collapse, the linearly extrapolated growing mode is
$$
\boxed{\delta_c=\frac35\left(\frac{3\pi}{2}\right)^{2/3}\simeq1.686.}
$$
This is a linear extrapolation, not the true finite nonlinear <density> at that instant. In the ideal energy-conserving spherical model, <virial equilibrium> gives a virial <radius> about half the turnaround <radius>; its mean <density> at the collapse time is $18\pi^2\simeq178$ times the cosmic matter <density>.
Real perturbations are nonspherical: tides and anisotropic collapse produce sheets, filaments and haloes. Collisionless <dark matter> develops <shell crossing> and phase mixing rather than radiatively losing its orbital energy. Baryonic <gas> can shock, cool and collect within the resulting potential wells. Hierarchical merging then assembles larger systems where small-scale fluctuations have sufficient initial power. <Press-Schechter formalism> connects a collapse threshold with the statistics of the smoothed initial field, but halo abundances and internal structure need more than the single spherical model. These ideas apply after perturbations become nonlinear; one must not extend pressureless subhorizon equations through all radiation-era or horizon-scale conditions.<Cosmology by David Tong>[Section3.3: <gravitational collapse> and structure formation]{ref}
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