 Solution

The outer factor is explicitly $a_{\mathrm{bg}}/a_m=\frac{1}{4}(6\pi t/t_m{)}^{2/3}$, the normalized scale factor of an Einstein-de Sitter universe, with the same enclosed mass and initial bang time as the region. The correction describes a fractional reduction of the region's radius relative to this background:For fixed mass, density varies as inverse radius cubed, so ${\rho }_{\mathrm{region}}/{\rho }_{\mathrm{bg}}=(1-ϵ{)}^{-3}=1+3ϵ+O({ϵ}^2)$. The growing linear density contrast is thereforeThe full spherical collapse reverses its expansion at $t_m$ and reaches zero radius at $t=2t_m$, corresponding to $\theta =2\pi$. In the ideal cold, pressureless model this makes the nonlinear density contrast infinite. Extrapolating the linear growing mode to that same time instead gives the finite linear spherical-collapse thresholdThis is an extrapolation of a linear perturbation, not an accurate finite-radius approximation at the moment of collapse. Virial equilibrium requires relaxation of the cold coherent motion and replaces the ideal singular endpoint.