Comoving perturbations of a homologously collapsing cloud ($\mathbf{k}(t)=\mathbf{q}/R(t)$)

A Fourier mode advected with a homologous spherical flow has phase $\mathbf{q}\cdot \mathbf{r}/R(t)$. Its material derivative in the background vanishes because $\dot{\mathbf{k}}=-(\dot{R}/R)\mathbf{k}$. Thus contraction shortens its physical wavelength while preserving its phase on each fluid element. Linearizing the continuity equation and pressureless momentum equation gives $\dot{\delta }=-i\mathbf{k}\cdot {\mathbf{u}}_1$ and ${\dot{\mathbf{u}}}_1+(\dot{R}/R){\mathbf{u}}_1=-i\mathbf{k}{\Phi }_1$. Combining these with the Poisson equation for Newtonian gravity yields $\ddot{\delta }+2(\dot{R}/R)\dot{\delta }-4\pi G\rho \delta =0$ for nonzero compressive wavenumber. It is a local bulk calculation, without the moving cloud-boundary conditions.

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.