 Solution

Put $H=2/(3t)$ and let $f_B={\rho }_B/\bar{\rho }$, $f_D=1-f_B$ be constant background matter fractions. Define the total matter perturbation $S=f_B{\Delta }_B+f_D{\Delta }_D$ and the relative mode $D={\Delta }_D-{\Delta }_B$. Adding with the background weights and subtracting the two given equations respectively givesThe total and relative density modes of two pressureless matter fluids are consequentlyThese are the general consequences of the supplied equations; a single ratio additionally needs initial-mode assumptions.

For the usual dark-matter-dominated growing-mode approximation, take $f_B\ll 1$, neglect the decaying modes, and set the relative-velocity mode $E=0$. Then ${\Delta }_D\propto t^{2/3}\propto R$. With ${\Delta }_B(t_r)=0$ at recombination, $C={\Delta }_D(t_r)$, so the baryon catch-up approximation after recombination isAt redshifts $z,z_0\gg 1$, replacing $1+z$ by $z$ gives the printed ${\Delta }_B\simeq {\Delta }_D(1-z/z_0)$. The printed relation is a high-redshift approximation with a specified relative mode, not an exact identity for arbitrary solutions of the two-fluid equations.

For example, retaining a finite baryon fraction with a growing total mode and constant relative mode gives, with $x=R/R(t_r)$,Alternatively, if initially both ${\Delta }_B=0$ and ${\dot{\Delta }}_B=0$ while the dark-matter growing mode is already present, then in the dark-matter-dominated limit the relative transient is not zero andThis shows why the missing initial-velocity convention matters. TODO: the source does not supply the initial relative velocity or a baryon-fraction approximation; those inputs are needed to select the quoted special case uniquely. The general solution and conditional specializations above remain valid.

The physical conclusion is robust: after decoupling, baryonic gas can fall into pre-existing dark-matter potential wells and its fractional density perturbation catches up. The absolute constant offset need not vanish; it becomes small relative to the growing common mode. The pressureless approximation applies on scales above the baryonic Jeans scale and while perturbations remain linear; subsequent collapse requires nonlinear treatment.^[ref]