 Logarithmic linear growth rate

For scale-independent linear growth $\delta (\mathbf{k},a)=D(a){\delta }_{\mathrm{ini}}(\mathbf{k})$, the logarithmic linear growth rate measures fractional amplification per logarithmic change of the scale factor. In an Einstein-de Sitter universe, $D\propto a$ and $f=1$. The cosmological Poisson equation for conserved pressureless matter gives $\psi \propto D/a$ at fixed comoving coordinate, and therefore ${\psi }^{\prime }=\mathcal{H}(f-1)\psi$. This relates suppression of matter growth to the Integrated Sachs-Wolfe effect.