= Density modes of a collapsing pressureless sphere
{title2=$\delta_g\sim R^{-3/2},\quad\delta_d\sim R$}
For <pressureless homologous spherical collapse>, let $\phi$ be its <cycloid> parameter. The <density contrast> equation becomes $\delta_{\phi\phi}-\tan(\phi/2)\delta_\phi-3\delta/(1+\cos\phi)=0$. One solution is $\delta_g=\sin\phi/(1+\cos\phi)^2$. <Reduction of order> gives an independent solution that decays at collapse,
$$
\delta_d=\delta_g\int_\phi^\pi\frac{(1+\cos s)^3}{\sin^2s}\,ds
=\delta_g\left[4\cot(\phi/2)+\sin\phi-3(\pi-\phi)\right].
$$
For $\epsilon=\pi-\phi$, the integral is $\epsilon^5/40+O(\epsilon^7)$, while $\delta_g\sim4\epsilon^{-3}$. Therefore $\delta_d\sim\epsilon^2/10$. Since $R\sim R_0\epsilon^2/4$, the displayed powers follow. The growing branch is unique up to scale and addition of the decaying branch; many mixed solutions grow, so it is not literally the only function with growth. Finite perturbations eventually leave the linear regime.
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