Density modes of a collapsing pressureless sphere (${\delta }_g\sim R^{-3/2},{\delta }_d\sim R$)

For pressureless homologous spherical collapse, let $\varphi$ be its cycloid parameter. The density contrast equation becomes ${\delta }_{\varphi \varphi }-\tan (\varphi /2){\delta }_{\varphi }-3\delta /(1+\cos \varphi )=0$. One solution is ${\delta }_g=\sin \varphi /(1+\cos \varphi {)}^2$. Reduction of order gives an independent solution that decays at collapse,For $ϵ=\pi -\varphi$, the integral is ${ϵ}^5/40+O({ϵ}^7)$, while ${\delta }_g\sim 4{ϵ}^{-3}$. Therefore ${\delta }_d\sim {ϵ}^2/10$. Since $R\sim R_0{ϵ}^2/4$, 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.