 Solution

For pressureless matter with curvature and a possible constant cosmological constant, write the background Friedmann equation asDifferentiating on the expanding branch givesConsequentlyThis proves that the Hubble parameter as a solution of the dust growth equation is valid, including constant-$\Lambda$ and curvature contributions to the background. It does not extend without change to a radiation background or arbitrary dynamical dark energy.

In an Einstein-de Sitter universe, $a\propto t^{2/3}$, $H=2/(3t)$ and $4\pi G\bar{\rho }=2/(3t^2)$. The linear matter perturbation growth equation becomesSubstitute $D=t^p$. The indicial equation is $p(p-1)+4p/3-2/3=0$, with roots $2/3$ and $-1$. ThereforeThe Hubble-parameter solution is the decaying mode in the Einstein-de Sitter universe.