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In a matter-dominated Einstein-de Sitter universe, the growing linear density contrast is proportional to the scale factor. When $|\delta |\sim 1$, 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 isThis 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 $18{\pi }^2\simeq 178$ 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.^[ref]

Let $t_0$ be the initial epoch, let $r(t)$ be the physical radius of the outer matter shell, and put $r(t_0)=r_0$. The background Einstein-de Sitter universe has critical density ${\rho }_b=3H_i^2/(8\pi G)$ at that epoch. Its mass inside $r_0$ is $M_b=M-\delta M$, where $M$ is the actual conserved mass. Hence the background Friedmann equation impliesFor spherical collapse of pressureless matter, with no shell crossing, the shell theorem gives $\ddot{r}=-GM/r^2$. Its conservation of energy first integral isConsequentlyFor $\delta M>0$, the initially expanding branch reaches cosmological turnaround at $r_{\mathrm{ta}}=A/B=M{r}_0/\delta M$. Since $dt=dr/\dot{r}$ on that branch, the elapsed cosmic time isThis is spherical turnaround from an unperturbed Hubble velocity. The integral measures the comoving matter observers' time interval in the pressureless model. The integration variable is the physical radius, not the radius divided by the background scale factor. If an absolute cosmic time is wanted, add $t_0$ to the displayed interval.

Set $u=Br/A$, retaining $r_0$ in $B$. Replacing only the lower integration limit by zero givesSubstitution of the constants yieldsTo quantify the approximation, write $ϵ=\delta M/M\ll 1$. The true lower limit is $u_0=B{r}_0/A=ϵ$, and an elementary integration by substitution, $u={\sin }^2\chi$, givesSince $\arcsin \sqrt{ϵ}-\sqrt{ϵ(1-ϵ)}=\frac{2}{3}{ϵ}^{3/2}+O({ϵ}^{5/2})$, the approximate time is too large by a relative fraction $\frac{4}{3\pi }{ϵ}^{3/2}+O({ϵ}^{5/2})$. This explains why changing the lower limit introduces little error for a small excess mass. The cosmological turnaround time diverges as $\delta M\to 0^{+}$, consistently with the absence of finite cosmological turnaround in an exactly critical Einstein-de Sitter universe.

In an Einstein-de Sitter universe, give an initially spherical, pressureless region total mass $M$ and excess mass $0<\delta M\ll M$, while holding its initial expansion velocity at the unperturbed Hubble flow value at physical radius $r_0$. Before shell crossing, the outer shell has ${\dot{r}}_0^2=2G(M-\delta M)/r_0$ and conserved specific orbital energy $-G\delta M/r_0$. The shell theorem and conservation of energy give ${\dot{r}}^2=2GM/r-2G\delta M/r_0$, so its cosmological turnaround radius is $M{r}_0/\delta M$. With $A=2GM$, $B=2G\delta M/r_0$ and $ϵ=\delta M/M$, the elapsed cosmic time is exactlyReplacing the square bracket by $\pi /2$ gives the displayed approximation, with relative overestimate $4{ϵ}^{3/2}/(3\pi )+O({ϵ}^{5/2})$. Holding the initial expansion velocity fixed is an essential condition: a pure growing-mode linear cosmological density perturbation has an initial peculiar velocity as well as an excess mass, changing the binding energy at the same order.