Linear matter perturbation growth equation ($\ddot{\delta }+2H\dot{\delta }-4\pi G{\bar{\rho }}_m\delta =0$)

For pressureless sub-Hubble perturbations, combine the linearized cosmological continuity equation, cosmological Euler equation and gravitational Poisson equation with matter as its source. The expansion contributes a $2H$ friction term and matter self-gravity contributes the negative coefficient of $\delta$. The background expansion can include radiation or dark energy; treating their perturbations as additional sources requires a more complete system.

Use a monotone linear growth factor $b(t)$ as time, $\mathbf{u}=\dot{\mathbf{x}}/\dot{b}$, and $\psi =a\phi /(Cb)$ with $C=3H_0^2{\Omega }_{m0}/2$ and $a_0=1$. The cosmological Poisson equation becomes ${\nabla }^2\psi =\delta /b$; the continuity equation becomes ${\partial }_b\delta +\nabla \cdot [(1+\delta )\mathbf{u}]=0$. The material derivative is $D_b={\partial }_b+\mathbf{u}\cdot \nabla$. Substitution into the pressureless cosmological Euler equation, followed by the growth equation and $\dot{b}=Hbf$, proves the displayed motion equation, with $f=d\ln b/d\ln a$. The density derivative in the conservative continuity equation is Eulerian, not material.

For a pure scalar growing density contrast with time-independent boundary conditions, ${\nabla }^2\psi ={\delta }_{\ast }$ fixes the scaled peculiar gravitational potential up to an irrelevant constant. The linearized continuity equation fixes the longitudinal scaled velocity to $-\nabla \psi$. The remaining homogeneous velocity is ${\mathbf{u}}_h=\mathbf{C}(\mathbf{x})/(a^2\dot{b})$; a divergence-free such field does not change the growing density at linear order. Excluding this decaying velocity component is necessary for time-independent $\mathbf{u}$. In an Einstein-de Sitter universe, $b=a$ and ${\mathbf{u}}_h\propto a^{-3/2}$, furnishing a counterexample if the pure growing-velocity hypothesis is omitted.

For dust, curvature and a constant cosmological constant, write $H^2=A{a}^{-3}+B{a}^{-2}+C$. Then $\dot{H}=-3A{a}^{-3}/2-B{a}^{-2}$ and $\ddot{H}=H(9A{a}^{-3}/2+2B{a}^{-2})$, which imply $\ddot{H}+2H\dot{H}-3A{a}^{-3}H/2=0$. Thus $H$ is a solution of the pressureless linear matter perturbation growth equation. In an Einstein-de Sitter universe, $H\propto t^{-1}$ is the decaying mode and the independent growing mode is $t^{2/3}$. Additional dynamical components need not preserve this identity.