= Solution
The outer factor is explicitly $a_{\rm bg}/a_m=\tfrac14(6\pi t/t_m)^{2/3}$, 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:
$$
\frac{a_{\rm lin}}{a_{\rm bg}}=1-\epsilon(t),\qquad
\epsilon(t)=\frac1{20}\left(6\pi\frac{t}{t_m}\right)^{2/3}.
$$
For fixed mass, <density> varies as inverse radius cubed, so $\rho_{\rm region}/\rho_{\rm bg}=(1-\epsilon)^{-3}=1+3\epsilon+O(\epsilon^2)$. The growing linear <density contrast> is therefore
$$
\delta_{\rm lin}(t)=\frac3{20}\left(6\pi\frac{t}{t_m}\right)^{2/3},\qquad
\boxed{\delta_{\rm lin}(t_m)=\frac3{20}(6\pi)^{2/3}\simeq1.062.}
$$
The 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 threshold>
$$
\boxed{\delta_{\rm lin}(2t_m)=\delta_c=\frac3{20}(12\pi)^{2/3}\simeq1.686,\qquad
\delta_{\rm nonlin}(2t_m)\longrightarrow+\infty.}
$$
This 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.
Back to article page
