 Solution

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.