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Subtracting the baryon equation from the cold-dark-matter equation cancels their common gravitational source and gives the baryon--cold-dark-matter relative density modeBecause the background fractions ${\bar{\rho }}_c/{\bar{\rho }}_m$ and ${\bar{\rho }}_b/{\bar{\rho }}_m$ are constant after recombination, their weighted sum givesDuring matter domination, $a\propto t^{2/3}$. The first equation gives $a^2\dot{D}=\text{constant}$ and henceThe cosmic-time matter density modes giveSolving the definitions for the individual contrasts,soThus $\boxed{f=1}$. Both species feel the same gravitational potential after baryon pressure becomes negligible; the growing total mode overtakes the constant relative mode, so baryons catch up with cold dark matter.

During matter domination,The perturbation equation becomesSubstitution of the power law ${\delta }_k\propto t^{\alpha }$ giveswhose roots are $\alpha =2/3$ and $\alpha =-1$. Hence the cosmic-time matter density modes give

During matter domination, $a\propto t^{2/3}$ and $4\pi G{\bar{\rho }}_m=2/(3t^2)$. Trying ${\delta }_m\propto t^p$ giveswhose roots are $p=2/3$ and $p=-1$. Thus the cosmic-time matter density modes are

During asymptotic cosmological constant energy domination, $H$ is constant and ${\bar{\rho }}_m$ is negligible, soThe matter density modes during cosmological-constant domination are thereforeThe growing matter-era solution freezes to a constant.

Cold dark matter has $c_s=0$. During matter domination,The density equation becomesTrying the power law $\delta =t^p$ giveswhose roots are $p=2/3$ and $p=-1$. Thus the cosmic-time matter density modes areand the leading growing mode is $\delta \propto t^{2/3}\propto a(t)$.