Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 9B / Solution

For small $\theta$,Put $u=(6t/B{)}^{1/3}$. Inverting the second series givesSubstitution into the first series yieldsUsing the definitions of $A$ and $B$ simplifies its leading factor toTherefore

The homogeneous sphere containing the same mass has radius $\bar{R}=R_0{\Omega }_{m,0}^{1/3}a(t)$. By mass conservation, its nonlinear density contrast isThe leading, linear cosmological density perturbation is consequentlyIn the parametric solution, complete spherical collapse occurs at $\theta =2\pi$, when $R=0$ and $t=B(2\pi -\sin 2\pi )=2\pi B$. Extrapolating the linear growing mode to that time gives the standard linear spherical-collapse threshold