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An Einstein-de Sitter universe has $a\propto t^{2/3}$, $H=2/(3t)$, and homogeneous density $\rho =1/(6\pi G{t}^2)$. HenceBefore the perturbation appreciably changes the motion, every shell follows the Hubble flow,

By Newton's shell theorem, only matter currently inside a shell contributes to its radial acceleration. Labelling matter by initial radius givesBefore shell crossing, the enclosed mass of a given shell is $M_i$. Its initial specific energy isAt the turnaround radius, $E_i=-G{M}_i/r_{\ast }$, soThe radial Kepler solution from the Big Bang to apocenter givesAfter turnaround a shell falls inward. Shell crossing makes its enclosed mass time dependent; shells then oscillate through the center, form caustics near successive apocenters, and phase mix into a halo.

For self-similar secondary infall, writeSubstitution in the shell equation givesUsing $G{t}_{\ast }^2/r_{\ast }^3={\pi }^2/(8M_i)$,

Now imposeThe case $ϵ=0$ is a scale-independent fractional overdensity, for which all shells turn around together in the ideal model. The case $ϵ=1$ has constant excess mass $\delta M_i=M_0$, corresponding to a central point-mass seed outside its core.

Since $r_{\ast }/r_i=(M_i/M_0{)}^{ϵ}$, while $r_i^3=(9/2)G{t}_i^2{M}_i$, the turnaround formulas giveThe shell turning at time $t$ therefore hasFor a shell with $\tau =t/t_{\ast }(M_i)$,Consequently its autonomous similarity equation isThe exponent $2/(3ϵ)$ follows directly from the preceding mass ratio; the printed $2ϵ/3$ in the final displayed equation is inconsistent with the supplied initial condition and preceding requested result. The corrected nonlinear equation is integrated numerically after turnaround.

Use the Fourier convention $f(\mathbf{x})={\int }_{\mathbf{k}}f(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{x}}$ with ${\int }_{\mathbf{k}}=\int d^{3}k/(2\pi {)}^3$. Neglect anisotropic stress and assume the pressureless velocity is irrotational, soFourier transforming $\nabla \cdot (\delta \mathbf{v})$ in the nonlinear continuity equation giveswhere the alpha mode-coupling kernel and its symmetrization areTaking the divergence of the Euler equation giveswith the symmetric beta mode-coupling kernel

In the Einstein-de Sitter universe, $a^{\prime }=\mathcal{H}a$ and ${\mathcal{H}}^{\prime }=-{\mathcal{H}}^2/2$. At linear order the ansatz gives ${\tilde{\theta }}^{(1)}={\tilde{\delta }}^{(1)}$. At second order, the continuity and Euler equations becomewhere each integral includes the momentum-conserving measure above. Eliminating ${\tilde{\theta }}^{(2)}$ yieldswith the standard perturbation theory density kernel

For an Einstein-de Sitter universe normalized by $a(t_0)=1$, $a=(t/t_0{)}^{2/3}$. Its current comoving particle horizon is $3c{t}_0$. A photon emitted inward at that location at $t_0$ covers $3c{t}_0[(t/t_0{)}^{1/3}-1]$ in comoving distance by time $t$. Equating these distances gives arrival at $8t_0$, a travel time of $7t_0$.

For a pure scalar growing density contrast with time-independent boundary conditions, ${\nabla }^2\psi ={\delta }_{\ast }$ fixes the scaled peculiar gravitational potential up to an irrelevant constant. The linearized continuity equation fixes the longitudinal scaled velocity to $-\nabla \psi$. The remaining homogeneous velocity is ${\mathbf{u}}_h=\mathbf{C}(\mathbf{x})/(a^2\dot{b})$; a divergence-free such field does not change the growing density at linear order. Excluding this decaying velocity component is necessary for time-independent $\mathbf{u}$. In an Einstein-de Sitter universe, $b=a$ and ${\mathbf{u}}_h\propto a^{-3/2}$, furnishing a counterexample if the pure growing-velocity hypothesis is omitted.

A pressureless bound spherical shell first expands, reaches $\dot{R}=0$ at its maximum radius, and recollapses. In the cycloidal spherical-collapse model, turnaround occurs at $\theta =\pi$, with $R_{\max }=2A$ and $t_{\max }=\pi B$. Its mean density is $9{\pi }^2/16$ times the background density in an Einstein-de Sitter universe.