= Solution
In the growing-mode <spherical-collapse model>, use $\theta-\sin\theta=\theta^3/6-\theta^5/120+\cdots$ and $1-\cos\theta=\theta^2/2-\theta^4/24+\cdots$. Their ratio in <2/c> gives $\delta=3\theta^2/20+O(\theta^4)$. Meanwhile $t\propto\theta-\sin\theta$, and the linear growing <density contrast> scales as $t^{2/3}$. Matching at small $\theta$ therefore fixes
$$
\boxed{\delta_L=\frac35\left[\frac34(\theta-\sin\theta)\right]^{2/3}.}
$$
This is a linearly extrapolated growing-mode amplitude at any epoch, including epochs where the actual <density contrast> is nonlinear. It carries the initial-condition qualification in <2/c>.
Numerical comparison is
$$
\begin{array}{c|rrr}
\theta&\delta&\delta_L&1-\delta_L/\delta\\\hline
1/4&0.00942684&0.00935549&0.00757\\
\pi/2&0.466138&0.340810&0.2689\\
2\pi/3&1.01186&0.568083&0.4386\\
\pi&4.55165&1.06241&0.7666
\end{array}
$$
The last column measures the fractional underestimate of the actual <density contrast>. Linear theory is accurate at $\delta\ll1$, but substantially underpredicts it by the time $\delta$ approaches unity. At $\theta=\pi$, the shell reaches <spherical-collapse turnaround>; $1+\delta=9\pi^2/16\simeq5.55$.
At $\theta=2\pi$, $r=0$ and the ideal pressureless <density> diverges at finite time $t_c=2\pi B$. Yet the extrapolated linear value remains finite:
$$
\boxed{\delta_c=\frac35\left(\frac{3\pi}{2}\right)^{2/3}=1.68647.}
$$
This is the <linear spherical-collapse threshold>, not the true <density contrast> at collapse. Actual nonspherical collisionless systems undergo <shell crossing> and <virialization> rather than becoming uniform point singularities. The ideal <virial theorem> and <conservation of energy> estimate a final radius $r_m/2$ if the structural form is preserved.
\Image[/past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2021-spherical-collapse.png]
{title=Parametric spherical collapse and the failure of the extrapolated linear density contrast}
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